A Stochastic Discount Factor Valuation
Approach to VAR Models
Specification and Estimation of the Deflator technique
by
Jessica Oosthuizeni
B.Sc. Financial Mathematics, University of Johannesburg
November 2008
A thesis submitted in partial fulfillment of the requirements for the degree of
Master of Science in Quantitative Finance and Actuarial Sciences
Faculty of Economics and Business Administration
Tilburg University
Through completing an internship project in cooperation with
ORTEC Centre for Financial Research (OCFR)
Supervisors
Prof. Hans Schumacherii
Dr. Hens Steehouweriii
i
Anr.: s.683101; [email protected]
Tilburg University; [email protected]
iii
ORTEC Centre for Financial Research; [email protected]
ii
Table of Contents
LIST OF FIGURES ............................................................................................................................... iii
LIST OF TABLES................................................................................................................................. iv
LIST OF APPENDICES ......................................................................................................................... v
Acknowledgements ......................................................................................................................... vi
Abstract ............................................................................................................................................ 1
1.
Introduction .............................................................................................................................. 2
2.
Background and Relevance ....................................................................................................... 5
2.1
2.1.1.
Classical ALM ................................................................................................................ 7
2.1.2.
Value-based ALM .......................................................................................................... 8
2.2
3.
Asset Liability Management and Risk Management ............................................................... 5
Vector AutoRegressive Models ............................................................................................... 9
2.2.1.
Univariate Autoregressive Models ............................................................................ 9
2.2.2.
Multivariate Autoregressive Models ....................................................................... 10
Methodology of the Stochastic Discount Factor ...................................................................... 11
3.1
The Fundamentals of Asset Pricing ....................................................................................... 11
3.1.1.
The Replicating Principle ......................................................................................... 12
3.1.2.
Risk and the Risk Premium....................................................................................... 13
3.1.3.
No-arbitrage conditions............................................................................................ 17
3.1.4.
Multi-period Asset Pricing ........................................................................................ 18
3.1.5.
Bond Pricing ............................................................................................................... 19
3.2
Three ways of Pricing ............................................................................................................ 20
3.3
Risk Neutral Valuation .......................................................................................................... 21
3.4
Stochastic Discount Factor Valuation ................................................................................... 24
3.4.1.
The Radon-Nikodym Process and Girsanov’s Theorem ....................................... 24
3.4.2.
The Pricing Kernel Process ....................................................................................... 25
3.4.3.
Discrete Pricing Kernel ............................................................................................. 26
3.5
Estimation of a suitable Stochastic Discount Factor ............................................................. 27
3.5.1.
A SDF approach to asset pricing using panel data ............................................... 27
3.5.2.
A neural network approximation to the Pricing Kernel ...................................... 30
3.5.3.
Asset pricing with observable stochastic discount factors ................................. 32
i
3.5.4.
3.6
4.
VAR model as a SDF Model ................................................................................................... 36
Model Specification and Estimation ........................................................................................ 36
4.1
Data ....................................................................................................................................... 37
4.2
General Setup and Specification of the Model ..................................................................... 38
4.2.1.
State dynamics and the VAR model ......................................................................... 38
4.2.2.
Stochastic Discount Factor Model ........................................................................... 39
4.2.3.
Modelling the Financial Products and the Valuation thereof ............................. 41
4.3
Estimation of Parameters ..................................................................................................... 46
4.3.1.
Initial parameterization ........................................................................................... 46
4.3.2.
Optimization Process ................................................................................................ 48
4.3.3.
Out-of-Sample Test .................................................................................................... 52
4.4
5.
Pricing Kernel and affine term structure of interest rates ................................. 34
Application and Results......................................................................................................... 54
4.4.1.
Pension Products........................................................................................................ 55
4.4.2.
Valuation of the Pension Products .......................................................................... 56
4.4.3.
Practical Application using Pricing Models ........................................................... 56
4.4.4.
Results.......................................................................................................................... 58
Conclusion............................................................................................................................... 60
References ...................................................................................................................................... 64
Appendices ..................................................................................................................................... 67
Notes and Remarks ......................................................................................................................... 72
ii
LIST OF FIGURES
Figure 1: Current business flows ............................................................................................................. 3
Figure 2: Aspiring business flows ............................................................................................................ 4
Figure 3: General ALM problem formulation (Source: Steehouwer, 2005) ............................................ 6
Figure 4: Scenario Graphs - The future evolution of the time paths for each state variable ............... 40
Figure 5: The Residual Sum of Squares for each Optimization Method ............................................... 51
Figure 6: Results from the valuation of a Basic Pension Product ......................................................... 59
Figure 7: Present Values for each basic financial asset using Initial Parameters ................................. 70
Figure 8: Present Values for each basic financial asset using the Calibrated Parameters ................... 70
Figure 9: Residual Sum of Squares for basic financial assets using Initial Parameters ......................... 71
Figure 10: Residual Sum of Squares for basic financial assets using Calibrated Parameters ............... 71
iii
LIST OF TABLES
Table 1: Statistical Summary (Source: ORTEC Finance BV) ................................................................... 39
Table 2: The Residual Sum of Squares for the Optimization Methods ................................................. 49
Table 3: Results from the Out-of-Sample test for system A: Initial Parameters .................................. 53
Table 4: Results from the Out-of-Sample test for system B: All Zeros ................................................. 54
iv
LIST OF APPENDICES
Appendix A: Proof of the Excess Return in equation (12)
Appendix B: Deriving the Radon-Nikodym process
Appendix C: Deriving the Pricing Kernel Process
Appendix D: A Comparison between the Present Values using Initial Parameters
Appendix E: A Comparison between the Present Values using Calibrated Parameters
Appendix F: The Residual Sum of Squares using Initial Parameters
Appendix G: The Residual Sum of Squares using Calibrated Parameters
v
Acknowledgements
I would like to make use of this opportunity to extend my gratitude towards all those involved in the
successful completion of this paper. It has been a long and difficult road and I can finally end of a
rewarding year with a gratifying result. However, the thesis project would not have been possible
without the help of my two supervisors; Prof. Hans Schumacher and Dr. Hens Steehouwer. Prof.
Schumacher, thank you for your patience and willingness to teach me with your wisdom. It has been
a humbling experience to be in the presence of such an intelligent teacher. You have been a great
mentor to me. Hens, I thank you for allowing me the opportunity to grow beyond my capabilities
and realize my own potential. Your trust and faith in my abilities have inspired me to take on greater
challenges in life. I valued your support and guidance throughout the project greatly. I am thankful
to Dr. Henk Hoek who greatly assisted me in the modelling process. I appreciate your readiness to
always lend a helping hand. You took interest in my project and really supported me in my efforts to
achieve the objectives set out. Though, all of this would not have been possible if it was not for a
special person, Prof. Lorraine Greyling, who made it her goal to allow me and three fellow students
the opportunity to study abroad. Thank you for taking on the challenge to make a success of the
dual scholarship program between the University of Johannesburg and Tilburg University. Not only
did you deal with all the arrangements but you also provided the financial means. Thank you for
your continual faith in my success as a student. Last but not least, to my family and friends, thank
you for always believing in me. I am grateful to the special people in my life, my family and Chris
Blignaut, who never denied me the opportunity to follow my dream even though it meant parting
from them. To my Tilburg family; Tarryn Roos, Cristina Arango, Evi Mesaikou and Vicky Silva Cortes
you have made my time in Holland a truly unforgettable experience. Thank you for your positive
influences in my life and I love you all dearly. I also would like to thank Liezel Marais, for her
constant support, understanding and motivation.
vi
Abstract
The stochastic discount factor (SDF) plays a major part in the pricing of assets by providing an
appropriate discounting for the future expected cash flows to the time of valuation. Vector
AutoRegressive (VAR) models are often used for asset liability management (ALM) by generating
scenarios to determine the risk and return profile of pension funds and insurance companies. By
combining these two models the future scenarios from the VAR model can be used for valuation
purposes. An advantage of the VAR/SDF approach is that there is an automatic consistency between
on the one hand the scenarios from the VAR model that are used to calculate classical (ALM) risk and
return numbers and, on the other hand, the valuation results that are obtained by applying the SDF
on the same problem. This method provides a way to value assets (or liabilities) under the “real
world probabilities”. However, comparing this technique to another class of pricing assets, the riskneutral valuation or arbitrage free models, then this technique is somewhat more tedious. Various
methods to estimate an appropriate SDF are explored. The most appropriate method is chosen to
construct a SDF corresponding to a realistic VAR model. The parameters are calibrated using basic
financial assets to form a closed form approximation for the SDF. Estimated prices of complex
pension products are computed using the closed form approximation. As a final result, the estimated
prices are compared to those of the risk neutral approach.
KEYWORDS: Stochastic discount factor, Time series VAR models, Pricing kernel, Deflator, Asset
Liability Management (ALM), Value-based ALM, Scenario analysis, Asset pricing
JEL CODES: C32, C51, C52, C61, G12
1
1. Introduction
The stochastic discount factor (SDF) provides a general framework for pricing assets as described by
Harrison and Kreps (1979), Hansen and Richard (1987), Hansen and Jagannathan (1991) and
Cochrane (2001). The various asset pricing models differ primarily due to their choice of discount
factor and by rightfully specifying the stochastic discount factor one manages to include most of the
theories presently in use in the financial world (Smith & Wickens, 2002). This is a very admirable
property. Many economic models can be used in an attempt to price assets and by simply choosing
the appropriate SDF, a fitting Pricing Equation results. The question now arises that given a specific
model what SDF should be used and more importantly how to estimate the SDF in this particular
model setting.
In practice the vector autoregressive (VAR) model is frequently applied for the purpose of managing
risk in an asset liability management (ALM) framework where the future scenarios are generated by
forward iteration of the historical dynamics (Steehouwer, 2005). The popularity of the VAR model to
model macroeconomic time series is based on its simple linear structure ((Campbell, Lo, &
MacKinlay, 1997) and (Juselius, 2003)). The VAR setting plays an integral part for this paper as it
forms the model support on which the Pricing Equation will be based. From the Pricing Equation the
focus turns to the stochastic discount factor which is primarily used to price the expected discounted
value of a future cash flow.
The aim and focus of each of these two concepts differ significantly as the SDF model is used for
valuation, most commonly in a risk neutral world, and the VAR model in the ALM study is concerned
with scenario simulations to analyse risk exposure and asset holdings. Apart from this being an
apparent difference between the two concepts, another important difference plays a role. While
the SDF is comonly set in a continuous framework, the risk management’s VAR model is set in
discrete time. Already some contrasts arise between the two methods but since both of these
models exhibit positive characteristics, a combination of the two may give rise to a consistent
valuation of embedded options from the scenarios generated in the VAR model. Campbell and
Shiller ((1987) & (1988)) and Campbell, Lo and MacKinlay (1997) applied the VAR model while doing
much econometric work on asset pricing to test market efficiency. This combination will shortly be
explored further. Hence, the focus for this study will be based on how to estimate an appropriate
SDF for a VAR model so that it can be used for consistent discounted valuation of contingent claims
or embedded options in future scenarios. Valuation takes place specifically at the current time, thus
giving rise to value based ALM ((Kortleve & Ponds, 2006) and (Hoevenaars & Ponds, 2007)). These
2
apparent differences will no doubt give rise to some interesting challenges during the estimation
and calibration process of a suitable SDF.
In an attempt to motivate the layout of the study and the starting point of the approach to be
followed, a graphical illustration is provided. Assume that the general objectives of any company
can currently be split in two major activity flows; first flow being that of “risk management” and the
second being “valuation”. The former flow focuses on scenario analysis by forward iteration using
Monte Carlo simulations to generate future scenarios in the VAR model setting. This is known as
traditional ALM. The latter flow is based on the asset pricing framework which is used primarily to
price the expected discounted value of a future cash-flow, i.e. valuation. In other words the current
price is obtained by discounting backwards by the appropriate term structure to the current time. In
a risk neural world the money-market account is a fitting tool for discounting. An example of such a
valuation takes place in the 1-factor Hull-White Black-Scholes (1HW-BS) model setting which is a well
known risk- neutral model. (For a full overview of the 1HW-BS model see Hull (2005)). In the current
situation typically these two flows are detached. The reason for the apparent separation is not only
because of the differences mentioned in the previous paragraph, but also more importantly the two
different concepts are based on different model structures. Therefore, they use different scenario
sets causing the evaluation of the two flows to be split. This can be seen from the graphical
representation in Figure 1. These kinds of activities are typically associated with insurance products
where the future expectations and the valuation of contingent claims can be assessed separately for
PRESENT TIME
PRESENT TIME
VALUATION
Determine asset prices by discounting back
using the money market account as numéraire
SCENARIO SET
OF FUTURE
CASHFLOWS
VAR MODEL
ALM & RISK MANAGEMENT
Use scenario analysis for the purpose of ALM by
applying Monte Carlo simulations
SCENARIO SET
OF FUTURE
CASHFLOWS
1HW-BS MODEL
balance sheet purposes.
Figure 1: Current business flows
This separation of the two techniques gives rise to consistency problems and therefore the focus
now turns to efforts to combine these two flows. By estimating the most suitable SDF for the VAR
model setting, a more consistent evaluation of embedded options can possibly be obtained.
3
PRESENT TIME
VAR MODEL
VALUATION
Determine asset prices by discounting back
using the stochastic discount factor
SAME SCENARIO SET OF
FUTURE CASHFLOWS
ALM & RISK MANAGEMENT
Use scenario analysis for the purpose of ALM by
applying Monte Carlo simulations
Figure 2: Aspiring business flows
The VAR model is regularly used for simulating scenarios of financial and economic variables that are
important in determining the risk and return profile of pension funds and insurance companies. By
combining this model with the stochastic discount factor (SDF), also known as the pricing kernel or
deflator, these scenarios can also be used for valuation purposes. By using the same scenario set,
the VAR/SDF approach gives rise to an advantageous property.
There will be an automatic
consistency between on the one hand the scenarios from the VAR model that are used to calculate
classical (ALM) risk and return numbers and, on the other hand, the valuation results that are
obtained by applying the pricing kernel on the same problem. This is known as Value-based ALM.
Figure 2 provides a depiction of this and provides a broad outline to explain the setting, motivation
and objectives of the research.
The motivation behind the development of the stochastic discount factor model can be drawn from
the introduction of new accounting standards in the pension industry. This requires that assets and
liabilities are valued at a fair and market consistent value (Hoevenaars R. P., 2008). These marketconsistent1 valuation methods together with an increasing demand for transparency in pension
schemes have led to a once-in-lifetime change to pension schemes. The irrefutable pension reform
taking place challenges all those involved with the implementation of a new way of thinking (Nijman
& Koijen, 2006). Both valuation and risk management will be influenced. The central initiative
behind the market-consistent approach to valuation relies on the fact that many pension liabilities
share common characteristics with traded financial assets. Therefore these market prices can be
used to derive sensible prices for pension liabilities (Hibbert, Morrison, & Turnbull, 2006). To obtain
this practically the VAR model can be extended to include the SDF so that a more consistent model
structure is achieved. This method has the potential to value financial assets at fair market prices.
Therefore the objectives are to construct a realistic SDF for the VAR model in a practical way and
specify the parameter estimations. An empirical application of the pricing kernel to basic financial
products might give some indication of how well this pricing method performs in a stochastic
1
Fair value, market consistent and market value are used synonymously.
4
framework. Under the law of one price the risk neutral valuation method and the SDF valuation
method must give rise to the same price in an identical model framework (Schumacher, 2007). A
comparison between the two models might reveal how well the stochastic discount factor method
performs. It is uncertain apriori how sensitive the prices of assets are to using a different model
structure.
In order to obtain some insights regarding these sensitivities an analysis can be
performed by using the SDF to value a realistic complex pension product. The popular 1-factor HullWhite Black-Scholes risk neutral model can be constructed for comparative purposes ((Hull,
2005)&(Jarrow & Yildirim, 2003)).
In short this is a feasibility study to inspect whether the stochastic discount factor (SDF) method in
fact possesses the ability to price assets and liabilities fairly.
The implied changes to risk
management and valuation by the introduction of fair value principles can be examined by
extending the VAR model to model the dynamics of the SDF. Up to now the SDF approach, also
referred to as the pricing kernel method, is still fairly out of favour due to the laborious complexity
relating to the modelling process. During the construction and estimation of the proposed model
some issues and problems will be encountered. They will be dealt with in the most practical way
possible. The layout of the paper is ordered as follows. Section 2 starts with a background
explanation of important topics which are relevant to the research conducted. This provides the
reader with additional information on risk management and value-based ALM (Asset Liability
Management) necessary to understand later chapters, but do not form the main part of the paper.
Section 3 provides an in-depth discussion on the methodology of the stochastic discount factor as an
asset pricing model. In addition the risk neutral valuation method is discussed as an alternative
asset pricing model so that it can be implemented as a comparison to the estimated prices obtained
by implementing the SDF. Various methods to estimate a suitable stochastic discount factor is also
considered in Section 3. To conclude the chapter the most appealing method from the “inventory
list” is chosen. Specification and estimation of the chosen stochastic discount factor model takes
place in Section 4. Model parameters are calibrated on basic financial assets to create a closed-form
approximation to the SDF. As a final result an attempt is made to value a complex pension product
containing an embedded option. Section 5 concludes with a summary of the most important results.
2. Background and Relevance
2.1 Asset Liability Management and Risk Management
Asset and Liability Management (ALM) is a helpful tool in the approach to risk management. This
entails estimating the future expected returns in different economic states by scenario modelling
5
and the risks involved in each respective state of the world. According to Steehouwer (2005) a
typical ALM problem on a general level can be categorized into three parts, firstly the objectives and
constraints of the stakeholders, secondly the policy instruments and thirdly the risk and return
factors which are generally characterized by uncertainties. Figure 1 provides a graphical illustration
of the general ALM problem.
Stakeholders Objectives and Constraints
Risk and Return Factors
Policy Instruments
Figure 3: General ALM problem formulation (Source: Steehouwer, 2005)
Firstly, there are various stakeholders each with their own objectives and constraints. These
objectives and constraints typically clash.
Consider the following examples to illustrate this
confliction; a Defined Contribution (DC) pension scheme where the only stakeholder is the individual
who would like to maximise his retirement payout while being constrained by the amount of risk
exposure he is willing to undergo in order to avoid falling below a minimum pension level; and a
Defined Benefit (DB) pension scheme where the sponsors strive for elevated levels of equity
exposure in order to lower contributions in the long run. However, in the absence of adequate
surpluses both the beneficiaries and the regulating authorities in this case will be far less
enthusiastic over such an equity concentrated asset allocation. This is due to the high level of risk
involved in this case. Another example is that of a commercial bank or life insurance company trying
to obtain the optimal duration policy. The stakeholders are the shareholders attempting to gain high
and stable returns, while the regulating authorities act on behalf of the clients and the wellbeing of
the financial system by safeguarding them from extensive exposure to solvency risk. From this it is
obvious that these contrasting objectives and constraints have considerable influences and play an
integral role in the ALM problem (Steehouwer, 2005).
Next the role of the policy instruments, such as the funding policy, strategic asset allocation and the
indexation policy are considered. The decision maker(s) (for example the board members of a
pension fund) have to make an assessment of what the optimal funding strategy is, how much
indexation can be granted and what the course of action should be for investing. These are just
some of the policy instruments among many others that need to be carefully considered in order to
reach the objectives and constraints of the stakeholders. The outcomes generated from ALM can
6
help the decision makers to form an opinion about the attractiveness around the implementation of
certain policy instruments (Steehouwer, 2005).
Last but not least is the risk and return factors. This is the reason for the existence of the ALM
problem, because without these uncertainties there would be no reason for ALM studies and
without any risk the concept of risk premiums would not even exist. The uncertainties of the real
world make ALM such an interesting topic.
Scenario analysis and ALM are interconnected. Typically scenario analysis is applied in ALM by
implementing Monte Carlo simulations or stochastic simulations to model the uncertainties of the
world, i.e. the economic risk and return factors. As a rule of thumb a large number of economic
scenarios are normally modelled, where an economic scenario is essentially the potential future
development of all the important macroeconomic variables. For that reason scenario analysis
provides valuable information about these uncertainties by supplying probability distributions for
the essential variables. With the help of a specific economic model, like for instance the VAR Model,
together with the applicable policy instruments these scenarios produce stochastic simulations of
important economic data like returns on asset classes, inflations and interest rates with respect to
the objectives and constraints of the stakeholders. Therefore, ALM and scenario analysis enable
decision makers to examine and weigh up the risk and return effects of various policies against each
other so that the most efficient and effective strategic policy can be reached ((Hoevenaars & Ponds,
2007), (Steehouwer, 2005)) 2.
2.1.1. Classical ALM
In the pension industry, ALM is being used to investigate and assess the performance of pension
deals and possible alternatives. Hoevenaars and Ponds (2007) and Ponds and Van Riel (2007)
portray ALM as the cornerstone of a pension fund policy. To corroborate this idea the purpose of
ALM is to find an optimal pension contract with the most favourable policies and the best risk
sharing options. In essence a pension contract is a warranty of what guarantees (benefits) will be
provided whilst defining the risk and reward allocation. It explicitly stipulates the necessary assetmix and funding policy needed for adequate financing and lays down the risk allocation rules to
determine how risks can be best shared over the various stakeholders by identifying who will be
responsible for bearing risks. The contribution of ALM now becomes invaluable as the outcomes
resulting from scenario analysis provide some clarification about the practicality and sustainability of
2
A more efficient policy results in a higher return at the same level of risk, and/or the same level of return with
fewer risk. For a policy to be effective the policy makes optimal use of the risk budget, i.e. the risk does not
exceed the risk budget just to obtain a higher return and is also not smaller than the allocated level of risk
appetite which will result in a lower expected return (Steehouwer, 2005).
7
the pension contract(s). Hence, the attractiveness of a pension contract can be explored by carrying
out a sensitivity analysis to survey the various expected values and relevant risk measures for the
key variables. The consequences of a particular policy variant in terms of asset mix, contribution
policy and indexation rules can be evaluated and the necessary steps can be implemented to take
care of specific limitations like the funding ratio. The fundamental nature of ALM is to arrive at
optimal pension deals.
2.1.2. Value-based ALM
Value-based ALM builds forth on the success of classical ALM. By adding additional information in
the form of present values of future cash flows, the value-based ALM include a new dimension to
assessing pension plans. To obtain the present values, the future outcomes from the scenario
analysis performed for classical ALM are discounted back to the present with a suitable risk adjusted
discount rate (Kortleve & Ponds, 2006). The discounting can be performed by either making use of
risk neutral valuation or a pricing kernel specification (or deflators). Since the same outcomes are
used for both techniques, the value-based ALM approach in actuality just works on top of classical
ALM and can therefore only lead to valuable insights (Hoevenaars & Ponds, 2007). The present
values are also known as economic value. With the new dimension, the decision makers can
evaluate which policy variants are fair in terms of economic value. Although questions may be
raised over what “fair” really implies. For instance when buying a pension product is the price
reflecting the fair benefits that will be received. Is the price determined fairly such that it represents
value for money? This brings allow the concept of financial fairness. The proposed SDF/VAR
approach (that will be discussed later on) provides a promising way to value pension products fairly
by making use of the concept of value-based ALM.
Essentially the focus of value-based ALM is on the economic value and the extra insight it brings
about. To illustrate this insightfulness consider how equities are “valued” in classical ALM opposed
to value-based ALM. In classical ALM equities are construed as attractive asset holdings as it
typically does better than the riskless asset. On average these risky assets earn higher returns
resulting in subsequent cheaper funding, higher benefits and the potential lowering of contribution
rates. However, due to the risky nature of equities its performance can drastically deteriorate in bad
economic times. This can lead to even more strained economic hardship for a pension fund that
greatly depends on equities. Resulting issues like increasing contribution rates and lowering benefits
come at a very high price in times like these. The contribution of value-based ALM now becomes
apparent as it reveals that holding elevated levels of equity potentially exposes the stakeholders to
undisclosed levels of risk. In other words, stakeholders often lose more by investing in risky equities.
8
Hence this brings the sustainability of the pension fund into question. Effectively high expected
returns and low average contribution rates may have low economic value for certain stakeholders.
The advantages of value-based ALM follow almost intuitively. The first benefit is the integration
with the financial market and the value it presently attaches to future cash flows. A pension deal
with high benefits and low contribution rates overall will have a high present value, while a
contribution rate that frequently increases in poor times will have a lower present value. Therefore
the economic value of all policy decisions can be evaluated by looking at the present value of
contributions and benefits and the surpluses or losses it will bring about.
Also, the added
transparency and straightforward interpretation of present values are in line with the fair value
principles of the new pension-fund accounting. Therefore there are less scepticism regarding the
financial fairness of pension schemes and relating products. A second important advantage is that of
intergenerational transfers. Value-based ALM gives a clearer picture of who will gain and who will
loose from changing the current pension deal. In other words it facilitates the detection of possible
value transfers between various stakeholders resulting from policy changes. Hence, value-based
ALM can be helpful in constructing better pension deals which can only improve the attractiveness
and sustainability of pension funds ((Kortleve & Ponds, 2006) and (Hoevenaars & Ponds, 2007)).
2.2 Vector AutoRegressive Models
2.2.1. Univariate Autoregressive Models
The Autoregressive (AR) process based on Gaussian errors has frequently been a popular choice as a
description of macroeconomic time series ((Campbell, Lo, & MacKinlay, 1997) and (Juselius, 2003)),
owing its attractiveness to its straightforward linear structure. A simplified interpretation of the
autoregressive models is given by Steehouwer (2005) where he portrays the models as having basic
counterparts in the world of (deterministic) differential equations which improves the likelihood for
an easy evaluation of the dynamics of the AR process. In a stochastic framework, where these types
of models are typically viewed as complex, it is exactly these particular features of the AR models
that greatly assist the estimation procedure.
Furthermore, the AR process has its name to thank to the unambiguous set-up of the model. It is
called autoregressive because it uses current and past known values of the variable to describe the
future evolution of the particular variable (forward iteration)3, i.e. for a AR process of order 1 the
value of the variable at time is described by its previous value at time − 1. The AR(1) process is
typically represented by the following formulation:
3
Future values can also be used to back out the past values of the variable in the autoregressive models.
(backward iteration)
9
= + Β
+ (1)
The constant term and the autoregressive coefficient Β are crucial parameters of the AR(1)
process necessary to model the dynamics of this process where the dynamics involve important
characteristics obtained from historical data. The white noise is modelled by the error term .
Certain assumptions regarding the error term are assumed, i.e. the error process are assumed to be
independent identically-distributed (i.i.d.) sampled from a normal distribution with a zero expected
mean and a constant variance of ; ~0, . Furthermore all intertemporal correlations are
equal to zero.
= 0
= = 0 for ≠ 0
(2)
The AR process, as stated in equation (1), is also known as a linear first-order difference equation.
Difference equations express a variable at time in terms of the previous value of that variable,
thereby transmitting identifiable information from the previous value to the subsequent value of the
variable at time . This also holds true for relating the coming value of the variable, the lead, to its
preceding value. The order of the difference equation depends on the number of lead or lagged
variables being used, for example equation (1) uses only the first lag and therefore is a first-order
difference equation.
2.2.2. Multivariate Autoregressive Models
Identical to many other economic models, there is a multivariate version of the AR model which
enables the simultaneous modelling of several state variables. This is unanimously known as the
vector autoregressive (VAR) model. Just as the univariate case, the multivariate approach has been
utilized time and again. There are many reasons for this. Its flexibility, ease of estimation and overall good fit of macroeconomic data are just some of the reasons. Another important motive why the
VAR model receives continual attention is the opportunity to combine long-run and short-run
information in the data by taking advantage of the cointegration property (Juselius, 2003). The
multivariate VAR model of order 1 with state variables follows from equation (1) such that
10
,
",
,
= , = ⋮ ,Β = ! ⋮
⋮
",
,
⋯
⋱
⋯
",
⋮ %
",
,
,
s=t
Σ
= ⋮ , = 0, &' = ( for *
s≠t
0
,
(3)
The properties of the -variate stochastic time series process - are determined by the above
parameters where represents the constant vector of size × 1, Β is a × matrix and is a
× 1 vector of the independent identically-distributed (i.i.d) multivariate error process sampled
from the Choleski decomposition of the × covariance matrix Σ of the error terms.
The
multivariate error terms for the VAR model is obtained by sampling from the independent standard
normal distribution to form a vector / of random variables such that /~0, 0 and then computing
the vector 1 that is a linear combination of / such that 1 = 2 ' / + 3 with 2 ' 2 = Σ and 1~3, Σ,
where 2 is the upper diagonal matrix of the Choleski decomposition, 0 is the identity matrix, 3 is
equal to 0 and the covariance matrix is
( )
 E ε 12,t
E (ε 1,t ε 2 ,t ) L E (ε 1,t ε k ,t )


E
E ε 22,t
M E (ε 2 ,t ε k ,t )
(
)
ε
ε
2 , t 1, t
Σ=


M
L
O
M


2
E ε k ,t 
 E (ε k ,t ε 1,t ) E (ε k ,t ε 2 ,t ) L
( )
( )
.
3. Methodology of the Stochastic Discount Factor
3.1 The Fundamentals of Asset Pricing
The first fundamental theorem of asset pricing as explained by Harrison and Kreps(1979), Hansen
and Richard(1987), Hansen and Jagannathan(1991) and Cochrane(2001) states that if the
assumption of no-arbitrage holds, there exists a non-negative stochastic discount factor (SDF) such
that the current price of an asset equals the expected discounted value of the asset’s stochastic
future cash-flows.
4 = 56 76
1 = 5686 11
(4)
(5)
4 is the price of an asset at time . The future payoff and future gross return at time + 1 is
represented by 76 and 86 , respectively4. The SDF is used to discount the expected future cash
flows (payoffs and returns) to current time where the expectation is taken with respect to
information available at time . 56 embodies the one-period SDF from time period to + 1 and
is guaranteed to be strictly positive where 0 < 56 ≤ 1 as a necessary consequence of the
absence of arbitrage opportunities ((Harrison & Kreps, 1979), (Kreps, 1981) and (Ross, 1978)). Keep
in mind that the Pricing Equation holds for a vector containing the future payoffs as well as for a
vector of returns. The resulting price will then also be a vector.
The second fundamental theorem of asset pricing defines that if and only if the market is complete
with respect to the basic assets, then there exists a unique SDF. This implies that the SDF is in fact
distinctive and lies in the payoff space, where the payoff space is the set containing all the assets
with resulting cash flows that are marketed and available in the market. In general this unique SDF
is referred to as the pricing kernel ((Charlier, Melenberg, & Schumacher, 2007) and (Harrison &
Kreps, 1979)). Accordingly Rosenberg and Engle (2002, p. 2) define the first fundamental theorem of
asset pricing in terms of the pricing kernel as follows: “the asset pricing kernel summarizes investor
preferences for payoffs over different states of the world. In the absence of arbitrage, all asset prices
can be expressed as the expected value of the product of the pricing kernel and the asset payoff.”
To clarify the relationship between the SDF and the pricing kernel an explicit formulation is specified
by Charlier, Melenberg and Schumacher (2007). In line with the current notation the SDF will be
denoted by 56. With the aim to facilitate the specification of both the SDF and the pricing kernel
;<
in one equation, the unique pricing kernel will be identified as 56
. Then the SDF is simply the sum
of the unique pricing kernel and some arbitrary white noise = which is assumed to be an element of
the set of all payoffs orthogonal to the payoff space ℳ.
;<
56 = 56
+ =; = ∈ ℳ @
(6)
3.1.1. The Replicating Principle
It is understandable why the first fundamental theorem of asset pricing is considered to be such an
essential theorem. It forms the foundation for the valuation of all future expected cash flows. From
4
Future gross returns are obtained by dividing future cash-flows with current price 4 ;
4 = 56 76 ⇒
4 56 76 76
DE
=
= B56 C
4
4
4
⇒ 1 = 56 86 ; where gross returns are 86 =
12
76
.
4
this Pricing Equation the payoff of any asset can be priced to obtain a fair and arbitrage free
valuation of the payoffs. Not only is the SDF a fitting valuation method by allowing the pricing of any
asset, but there is another convenient attribute of the SDF pricing method. Given a complete
market the SDF possesses the ability to find the portfolio that mimics the future stochastic payoffs
on the asset. This is known as the replicating portfolio or the hedge portfolio of the underlying basic
assets. Essentially the replicating portfolio prices any asset or liability by mimicking (hedging) each
of the future stochastic payoffs (Charlier, Melenberg, & Schumacher, 2007). Then the value of the
replicated asset or liability is equal to the value of the replicating portfolio. For instance finding the
hedge for primitive assets such as stocks and bonds is effortless since the hedge is just an
investment in the asset itself. However, finding the replicating portfolio for options and complex
assets and obligations is not so trivial. This is where the SDF really attains its power by greatly
assisting the valuation process. Hence the replicating principle is simply another pricing method that
is more convenient in a particular context. This concept was first introduced by the groundbreaking
work of Black and Scholes (1973) and Merton (Theory of Rational Option Pricing, 1973) on the theory
of option pricing. The advantages of using a replicating portfolio are that an arbitrary discount rate
is not required and the term structure of interest rates is automatically taken into account. In a
complete market with a unique pricing kernel the replicating portfolio is unique and as a result the
hedge is perfect. In incomplete markets a perfect hedge cannot be found due to the existence of
unsystematic risks. Unsystematic risks refer to the likelihood that an asset’s cash flows depend on
stochastic variables that are uncorrelated with the market. As an example of an incomplete market
with unsystematic risks consider pension funds and insurance products that are exposed to longevity
risk and mortality risk. In general the unsystematic risks of financial assets can be diversified away.
But for pension liabilities this is not the case. To cover the remaining risk the pension plan sponsor
can make contribution payments. The question now arises of how to evaluate the complex assets
and obligations. Naturally a good place to start is to set up a measure of risk as a valuation tool. By
introducing a price of risk, the hedge portfolio can replicate the future stochastic payoffs with
minimum risk.
3.1.2. Risk and the Risk Premium
Intuitively the concept of risk plays an important role in the finance world. Risk can be divided into
two categories, namely that of essential and non-essential risk (Charlier, Melenberg, & Schumacher,
2007). Essential risk fulfils an important part in financial trading because it accommodates the
possibility of higher returns. It is also known as unsystematic risk because it relates to only a small
part of the market, like a company, and it can be hedged away with a well diversified portfolio of
financial assets. While essential risk can be perceived as ‘acceptable’ risk, in general investors want
13
to evade non-essential risk. Non-essential risk, also known as systematic risk, is typically economy
wide. This means that it is a common risk relating to the market portfolio which affects all investors
and cannot be diversified. Systematic risk is unpredictable and uncertain and therefore appropriate
compensation cannot be premeditated. To motivate shareholders and investors to take on the
additional risk requires a reasonable compensation.
The risk premium provides the needed
incentive. In order to provide the appropriate payment the value of the risk premium must be
determined. The risk premium can be explained by undiversified risk; for that reason if general
market risk is present then the risk premium reflects this. To deduce a measure of risk the SDF plays
a very important role. The Pricing Equation from equation (4) can be rewritten as:
4 = 56 76 + 2LM 56, 76
(7)
By setting the 56 = 1⁄8 where 8 stands for the gross one period short term interest
N
N
return5, then equation (7) can now be converted into:
4 =
76
8
N
+ 2LM 56, 76 (8)
The first term on the right hand side in equation (8) represents the standard discounted present
value. Logically this can be perceived as the asset’s price in a risk neutral world. The second term
consequently quantifies the risk. Hence it symbolizes the risk adjustment. From the covariance
term, it follows that a negative correlation between an asset’s payoff and the SDF will lead to a low
price. Conversely, a positive correlation will result in a high price (Smith & Wickens, 2002).
As a matter of convenience it might be more suitable to work with asset returns instead of asset
prices. Alternatively, the risk term can then also be expressed in terms of the asset’s gross return
(Charlier, Melenberg, & Schumacher, 2007). Note that gross returns can either be described in
nominal or real terms and since the discount factor is stochastic, the corresponding discount factor
must also be defined in nominal or real terms (Smith & Wickens, 2002). Rewriting equation (5) leads
to the following:
Assume a risk free asset is available that pays €1 in future states of the world. Then the gross return on the
risk free asset at time is equal to the future cash flow divided by its price, but since it is the risk free asset the
payoff is 1 in all future states.
5
86& =
N
76& 76& 1
1
=
=
=
;0 ≤ Q
4
56& 76& 56& ∙ 1 56& Hence it can be deduced that the gross return on the risk free asset is time independent and is therefore
N
simply 8 known at time .
14
1 = 56 86 + 2LM 56, 86 (9)
The expected return on any asset can then be defined as:
86 =
1 − 2LM 56, 86 56 (10)
The conditional covariance between MS6 and RS6 can be estimated as the covariance of the error
terms in the joint model of MS6 and RS6 and possibly other variables. Smith and Wickens explain
that equation (10) holds whether the asset is risky or not. In the case of a risk free asset the return is
known with certainty. Assuming that the risk free asset pays €1 in the next period ( + 1, then the
return 86 is simply equal to the risk free return 8 known at time . The Pricing Equation in terms
N
of gross returns in equation (5) can then be written as:
1 = U56 8 V
N
(11)
By rewriting equation (11) it naturally follows that the expected SDF is equal to 1 over the risk free
return, so 56 = 1⁄8 (Also see footnote 5). Now considering a risky asset, the expected
N
return has to be higher than the riskless return to compensate for the exposure to additional risk.
Therefore, the excess return can be computed by subtracting the risk free return from the expected
return on the risky asset. The expected risky return 86 is not the risk free return and is unknown.
The excess return follows algebraically by substituting equation (11) into (10), (See appendix A for
the algebraic proof):
U86 − 8 V = −8 2LM W56 , U86 − 8 VX
N
N
N
(12)
Equivalently, because the covariance term on the left can be split and the covariance of the known
risk free return is zero, then equation (12) can also be written as
U86 − 8 V = −8 2LM 56 , 86 .
N
N
(13)
Consider an alternative representation where the risk free return is expressed as 8 = 1 + Y ,
N
N
where Y is the risk free rate. Similarly, assume that the expected return on the risky asset can be
N
15
expressed as 86 = 1 + Y66. A simple substitution into equation (12) yields an equivalent
result:
UY6 − Y V = −U1 + Y V2LM W56 , UY6 − Y VX
N
N
N
(14)
Both equations (12) and (14) represent the excess return over the risk free return and logically
contain a covariance term representing the risk. The extra return over the risk free return identifies
the compensation that investors require for ventures in risky assets. Therefore the expected excess
return computes the price associated to the risk, namely the risk premium.
In general if market risk is present the risk premium can be explained by the undiversified risk. So
the risk premium measures the exposure of all assets to the risk of the general market portfolio. The
risk premium can be represented as the product of the market price of risk and the quantity of risk.
The quantity of risk term is thought of as a regression coefficient which captures the exposure to
risk. To express the exposure to risk in terms of correlation coefficients, introduce the standard
deviation of the SDF and the risky return. By reordering equation (13) (or (12)), then the risk
premium can be expressed as the product of the quantity and market price of risk.
U86 − 8 V =
N
2LM 56 , 86 −[\ 56 [\ 86
∙
56
[\ 56[\ 86 = " ]
" =
] =
2LM 56 , 86
[\ 56 [\ 86
−[\ 56[\ 86
56 (15)
Where the risk exposure of the asset to the undiversified risk is measured by " and the price of risk
is represented by ] . In the literature this method is well-known as the beta pricing for discount
factors (Charlier, Melenberg, & Schumacher, 2007).
As mentioned above, a negative correlation between the asset return and the discount factor lead to
high returns. This is because the market price of risk ] < 0. Naturally risk-averse investors require
a positive compensation. As a result the risk-premium must be non-negative and consequently
6
Please note this is not the same as the log returns discussed later in the paper, but merely another
representation of the normal returns. Take care not to confuse the two terms. However, effort has been
made to denote this properly as far as possible.
16
Cov 56, 86 ≤ 0. Similar to the correlation relationship between asset payoffs and the SDF,
the covariance term of the asset’s return and the SDF influence the expected excess asset returns.
The source of the risk essentially comes from the covariance term. Assume the risk premium is
positive. Then if the stock return on the asset is in a low state (because of high ) the SDF will take
high values, and vice versa (Smith & Wickens, 2002). However, it is also possible that there is a
positive correlation between the SDF and returns. Then the compensation is not positive any longer
because of the negative market price of risk. This offers a discount to investors with investments in
the risky assets. Therefore a positive correlation offers a discount while a negative correlation acts
as a risk premium. In general investors like to have assets with a negative correlation to the market
risk because it offers a hedge against the market risk. Consequently these negatively related assets
with the stock market are valued more. Recollect that the implicit discount rate for payoffs is low
when the stock returns are in low states. In “bad” states of the world where the stock price is low,
the payoffs will thus have a relatively high value. De Jong (2004) explains that derivatives with a
negative correlation that primarily pay in “bad” states, like put options, will be more valuable.
Therefore these derivatives will be relatively expensive compared to their (undeflated) expected
payoff. Conversely, when derivatives have a positive correlation and pay in “good” states of the
world, like call options, this will be valued less.
3.1.3. No-arbitrage conditions
Very often it might be more convenient to express the Pricing Equation in terms of logged returns7.
However, the corresponding discount factor also has to be defined in logarithmic terms. Therefore
`6 = ln 56 when Y6 = ln 86, such that the SDF and the gross returns are jointly log-
normally distributed. This assumption is commonly used, owing its popularity to its analytical
convenience and because it is a reasonably good approximation.
Recall that the lognormal
Hence, ln / = b + ⁄2.
Relying on the theoretical
distribution where ln / ~b, has an expected value such that / = c-db + ⁄2
(Campbell, Lo, & MacKinlay, 1997).
underpinnings of the lognormal distribution, equation (5) then becomes
1 = 56 86 = c-df gln56 86 h + ½jkY gln5686hl.
Where
b = gln56 86 h and = jkY gln5686 h
Normal gross returns are defined as 86 = 76 ⁄4 = 46 − 4 ⁄4 , and the logarithm of gross returns
are expressed as:
46
D = ln1 + 86 = ln86 Y6 = ln 46 − ln 4 = ln C
4
7
17
Now taking the logarithms lead to
ln 56 86 = gln5686h + ½jkY gln56 86 h
0 = `6 + Y6 + ½jkY `6 + ½jkY Y6 + 2LM `6, Y6 .
(16)
Accordingly when the expected return is equal to the risk free rate so that Y6 = Y , then the
N
expectation of the return is simply the risk free rate and the variance of the return is zero. In
addition the covariance term which represents the risk falls away. Therefore equation (16) becomes
`6 + ½jkY `6 = −Y
N
(17)
To obtain the no-arbitrage condition subtract equation (17) from (16), and remember UY V = Y .
UY6 − Y V + ½jkY Y6 = −2LM `6 , Y6
N
N
N
(18)
Equation (18) contains an extra term on the left compared to the excess return equation (12). This
extra term is known as the Jensen’s effect, which is in actual fact half the conditional variance of
returns. This is a result from taking expectations of a non-linear function. Again, just as in equation
(14), the right hand side symbolize the risk premium (Smith & Wickens, 2002). As mentioned before,
the different asset pricing models differ mainly because of their choice of SDF. The no-arbitrage
condition can now be implemented to derive the parameters of a fitting SDF.
3.1.4. Multi-period Asset Pricing
The general principles of the SDF methodology have been discussed for a single-period setting.
These principles can also be extended to a multi-period setting. Recall that the first fundamental
theorem of asset pricing defines the current price at time as the expected discounted future
stochastic cash flows given that the absence of arbitrage holds. In the multi-period setting the
Pricing Equation becomes the sum of the one-period valuations.
p
4 = B n 5,o 7o E
oq6
(19)
p
1 = B n 5,o 8o E
oq6
18
(20)
Here, 4 is the real price of an asset at time . 76& and 86& once again represent the future payoff
and future gross return at time + Q with Q ≥ 0. The price is determined by taking the expectation
at time and discounting with the non-negative multi-period 5,6& . The multi-period SDF from
time period to + Q is by nature the cumulative one period SDF’s (Campbell, Lo, & MacKinlay,
1997) and can be expressed recursively as follows:
6&
5,6& = s 5o
,o
oq6
Alternatively the SDF can be expressed in log-terms where `,6& = log U5,6& V:
(21)
6&
`,6& = n `o
,o
oq6
(22)
Equations (21) and (22) can be seen as the term structure of the stochastic discount factors.
Because the discount factor is stochastic it depends on the statistical characteristics of the future
expected returns. In general equation (21) can be used for the valuation of normal returns, while
equation (22) is appropriate when lognormal returns are used. A fitting Pricing Equation for the
lognormal returns will then be of this form:
p
1 = B n c-dU`,o + Yo VE.
oq6
(23)
3.1.5. Bond Pricing
Now that the multi-period has been discussed it is convenient to look at the pricing of a primitive
asset. Consider an t-period zero coupon bond. The Pricing Equation can be extended recursively
such that
4
u
4
u
= W56& 46&
u
&
is the price of the t-period zero coupon bond. 46&
t − Q-period bond in the future where ≤ Q.
X
u
&
(24)
represents the price of a remaining
56& is necessarily the multi-period SDF as
expressed in equation (21). This proves to be a very handy result and the Pricing Equation certainly
demonstrates a great deal of versatility in its application.
19
3.2 Three ways of Pricing
It is now very clear that the Pricing Equation together with the SDF plays an integral role in finance.
The importance of this approach becomes apparent as a helpful tool in the valuation to different
asset classes. Recollect that in the introduction two separate business flows where discussed,
namely that of ALM/Risk management flow and the Valuation flow. The focus mainly falls on the
latter flow as the Pricing Equation will be further discussed as the method to pricing assets. There
are three variations of the Pricing Equation namely state vector pricing, also better known as Arrow
Debreu pricing, stochastic discount factor pricing and risk neutral pricing (Charlier, Melenberg, &
Schumacher, 2007). An important consequence of the no-arbitrage assumption is that it implies that
the law of one price must hold. Remarkably all three methods are equivalent and will give the same
price for a given future payoff discounted in the same model. This is abiding with the law of one
price. However, the difference between the different variations of the Pricing Equation is rooted in
the way their definitions are constructed. A short description of each method’s definition together
with advantages and disadvantages follows.
The first method, the Arrow Debreu pricing method, was developed from the pioneering work by
Arrow (1964) and Debreu (1959). This pricing method can best be described as a time-state
preference model where equilibrium exists. Famous for its contribution to the theoretical advances
in the economics of investment under uncertainty, the model introduces primitive securities, each
paying €1 in one specific state of nature and nothing otherwise (Campbell, Lo, & MacKinlay, 1997).
Hence, the connotation of state vector pricing. The Arrow Debreu model used equilibrium in one
model to construct equilibrium in another model (Friesen, 1979). As a result this pricing method
greatly facilitated a better understanding of economic equilibrium in an uncertain environment.
Although the state vector pricing model is relatively easy in its approach, the limitation to this model
is that it is best used in a single period setting.
From the state vector pricing the stochastic discount factor pricing method can be derived. This is
done by dividing the state price densities with the real world probabilities to obtain a deflator pricing
system that takes into account real world probabilities. The advantage of this method as a valuation
tool is that there is an automatic consistency between the real world scenarios and the present
value of the future expected cash flow in these scenarios. Also, the real risk is taken into account
necessitating an appropriate compensation for the risk exposure.
The disadvantage of this
technique is that it’s very sensitive to the specific time and state of the world especially due to the
risk exposure, and therefore the computation can be cumbersome.
20
The third method is the risk neutral approach. This approach is constructed by adjusting the real
world probabilities in the deflator technique to risk neutral probabilities. By doing so, one assumes
that cash flows are risk neutral and will have the same probability of paying regardless of the time
and state of the world. Analytical simplicity is a key advantage point of the last method and is often
applied in the literature as well as in practice. However, the reality is that expected future cash
flows are unknown and with uncertainty comes risk. Therefore these payoffs can never be entirely
risk free. The future cash flows carry an inherent factor of risk because the cash flows are stochastic.
So even though the risk neutral method is the most common technique due to computational ease,
there is room for inconsistent estimations.
An explanation of the two most commonly used methods in stochastic multi-period settings, the
risk-neutral valuation method and the stochastic discount factor valuation method, will be discussed
in detail. The former method is most commonly used in the finance world due to its simplicity whilst
the focus of this paper is on the latter method, particularly since the stochastic discount factor
(deflator) method is more complex due to its adjustment to risk. An analysis follows.
3.3 Risk Neutral Valuation
Risk-neutral valuation (RNV)8 forms the cornerstone of asset pricing. This method is based on the
assumption that there are no-arbitrage opportunities. The concept of a numéraire plays an integral
part in Risk-Neutral Valuation. A numéraire is an asset with a strictly positive payoff and a strictly
positive price. Hence it satisfies the assumption of no-arbitrage. Now given a numéraire exists the
stochastic discount factor can equally be defined in terms of an equivalent probability measure ℚ.
This is known as the so-called risk-neutral probability measure if the related numéraire is the risk-
free asset. Importantly the equivalent probability measure ℚ has the same structure as the original
(real) probability measure ℙ, i.e. if the probability is positive under the original measure ℙ it is also
positive under the new probability measure ℚ. Therefore the price of an asset equals the expected
cash-flows, under the risk-neutral measure, discounted at the term-structure of interest rates.
7
ℚ 76&
= C
D
6&
76&
7 = ℚ
D
C
6&
(25)
(26)
7 is current value of the realization of future cash flows. The expectation of the stochastic future
cash flows 76& at future time + Q, Q ≥ 0, is taken at current time under the equivalent
8
Equivalent terms are market-consistent, economic valuation, and fair value.
21
probability measure ℚ. is the numéraire given that a basic asset is available such that all payoffs
and the price associated to the basic asset is positive in all states of the world. ((Charlier, Melenberg,
& Schumacher, 2007) and (Schumacher, 2007)). Closer inspection of equation (25) reveals that it is a
martingale because the expectation of the next period cash-flows is equal to the current value of the
cash-flows. Thus the RNV method can also be defined in terms of the equivalent martingale
measure. Hence, 7 ⁄ is the relative price with respect to the numéraire where only time
discounting is taken into account under the equivalent martingale measure.
When the numéraire is the risk-free asset, the resulting probability distribution is referred to as the
risk neutral probability distribution such that all expected returns are equal to the return on the riskfree asset. Assuming the expected value of the risk-free asset is equal to the current value
multiplied with the 1-period risk-free interest rate such that
6 = U1 + Y V
N
(27)
then amending equation (25) proves that under the risk-neutral probability the expected returns are
equal to the risk-free return.
1
7
=
ℚ 7 U1 + Y N VQ 6&
N Q
U1 + Y V ℚ 76&
= C
D
7
N Q
U1 + Y V = 86& N &
ℚ
U8 V = 86& ℚ
(28)
Where 8 = U1 + Y V and 86 is the gross return on the risk-free asset and the cash flows for
N
Q = 1, respectively.9
N
As proof the gross return of a risk-free asset is defined as the future payoff divided by the price. Assuming
the risk-free asset (satisfying all constraints of a numéraire) is a bond, x, such that the price, 4y , is normalized
to 1, then the future payoff, x6 , is simply the risk-free interest rate multiplied by 1:
9
8N =
N
x6 U1 + Y V1
=
= U1 + YN V
1
4y
Therefore the gross return of the risk-free asset is equal to the one period risk-free interest rate. The gross
return is also known as the risk-free rate.
22
For all asset classes the risk neutral expected returns are equal to the risk free rate. In other words
the asset classes are inflated under the risk neutral probabilities. Therefore deflating the cash flows
with the risk free rate is fitting. Effectively there is no need for a risk premium. This is because the
risk adjustment takes place inherently by using the new adjusted risk neutral probabilities z ℚ . No
further risk adjustment is needed since the aversion to risk is already included in the risk adjusted
probabilities. To see this, assume a derivative of the underlying basic asset exists which is a financial
instrument. Then the price of the derivative is calculated by the RNV method as the expected payoff
of the derivative in a risk neutral world, discounted at the risk-free interest rate. In the risk neutral
world, the expected return on the stock is set equal to the risk free rate (de Jong, 2004). By setting
7 = 4 in equation (26), because the current value of the cash flow is simply just the price of the
expected future cash flow, and using the result obtained from (28), then the following equation
follows:
76
4 = ℚ
B N E
8
(29)
where 4 is the price of the expected value of the derivative 76 at time discounted at the risk-free
rate 8 . Equivalently, by using 8 = U1 + Y V the risk neutral method can be expressed in terms of
N
N
N
continuous discounting10 such that
{|
76X.
4 = ℚ
Wc
}
(30)
The RNV is a very convenient and powerful method for the valuation of a variety of asset classes like
contingent claims and derivatives. However, it is natural to imagine that the risk adjustment is
unpredictable and depends on the states of the world. For instance in a bad economic scenario, the
average risk averse investor values risk adjustment more than in a good state and expects prices to
reflect this. Because certain payoffs are inflated more due to the risk adjustment, it therefore
follows that certain states of the world subsequently require more discounting than others (Hansen
L. P., 2001). Although the RNV method implicitly inflates and deflates for the risk adjustment, it may
be more informative to model the risk adjustment explicitly under real world probabilities. This
brings about the stochastic discount factor (SDF) valuation.
10
Since log1 + - ≈ - for - close the zero, then discounting in discrete terms by
approximately equivalent to c
}
{|
.
23
U1 + YN V
are
3.4 Stochastic Discount Factor Valuation
To take into account the real world probabilities ℙ, the risk adjustment has to be modelled explicitly.
These risk adjustments are present because some states of the world are discounted more than
others. The resulting prices then reflect this. Hence, the SDF varies with states of the world that are
realized at date and determines the associated risk adjustments (Hansen L. P., 2001). The SDF
valuation method takes the form as represented in equations (4) and (5) under the real world
probabilities ℙ such that
4 = ℙ
56 76 .
(31)
The same approach as Ang and Piazzesi (2003) is followed to develop a term structure model for
56 to explicitly model the risk adjustment under real world probabilities. Assume the absence of
arbitrage. This guarantees the existence of an equivalent martingale measure ℚ (or risk-neutral
measure). Then the price of any asset at time that is not paying any dividends in period + 1
satisfies equation (30), where the expectation is taken under ℚ (Harrison & Kreps, 1979). The risk
neutral measure ℚ is equivalent to the original measure ℙ such that they correspond on which
events are possible. The no-arbitrage assumption implies that the law of one price holds which
further implicates that the stochastic discount pricing method and the risk-neutral pricing method
give the same price. However, a change of measure is necessary to accommodate the real world
probabilities. Taking expectations under different measures is an important notion in finance. To
assist this process of changing measures the Radon-Nikodym derivative is introduced (Schumacher,
2007).
3.4.1. The Radon-Nikodym Process and Girsanov’s Theorem
To assume the existence of the Radon-Nikodym derivative is equivalent to the absence of arbitrage
opportunities and allows the pricing of any asset in the economy (Ang & Piazzesi, 2003). The RadonNikodym derivative  of ℚ with respect to ℙ converts the risk-neutral measure to the real-world
measure such that ℚ 7 = ℙ 7. Hence, the Radon-Nikodym derivative  can be expressed as
€ℚ⁄€ℙ. The change of measure can also be defined in terms of processes such that the RadonNikodym process f l with 0 ≤ . Then
ℙ
ℚ
76 = C
6
7 D
 6
(32)
It is important to point out that the Radon-Nikodym process must be positive as implied by the
equivalence to absence of arbitrage. Furthermore, the expectation of the process f l must be
24
equal to 1 such that it is a ℙ-martingale ℙ
6 =  . Assume the Radon-Nikodym derivative is then
given by (Proof is given in Appendix B)
1
6 =  c-d C− ]' ] −] D
2
(33)
] represents the time-varying market price of risk associated with which is the source of
uncertainty. Now substituting the risk neutral equation (30) into (32) give the following result:
{|
{
ℚ
76X = ℙ
BCc |
Wc
}
}
6
D 76E.

(34)
From this it can be seen that the Radon-Nikodym process together with the risk free rate rightfully
captures the dynamics of the SDF process 56 (Compare equation (34) with (31)). Then the SDF
can be represented as
56 = c-dU−Y V
N
6

(35)
Substitution of equation (33) into (35) lead to
1
N
56 = c-d C−Y − ]' ]S − ]' 6D
2
(36)
An important point of the SDF valuation method is that in this process, the discount factor 56
includes both the time discounting and the change from the original measure to the equivalent risk
neutral measure (Schumacher, 2007). Alternatively the SDF can also be derived from the theory of
partial differential equations. This is known as the pricing kernel process and the result is analogous
to the result obtained from the Radon-Nikodym process.
3.4.2. The Pricing Kernel Process
Once again assume that no-arbitrage opportunities exist.
To determine the dynamics of the
discount factor, Schumacher describes a continuous process 5 by 5 = 1 and
€5 = −5 Y € + ]' €‚ .
(37)
;<
The process 5 is called the pricing kernel. (Think of 5 as equivalent to 56
, because in general
one aims to model the unique discount factor without the white noise). In order to satisfy the
absence of arbitrage assumption, then it is necessary to assume that the market price of risk ] for
25
stock returns satisfies the generalized form of the Black-Scholes equation such that it is equal to the
Sharpe ratio11 (Schumacher, 2007). By using the Euler approximation scheme the partial differential
equation of the pricing kernel process then becomes
1
56 = 5 c-d ƒ−Y − ]' ] − ]' ∆‚ …
2
(38)
The process 5 represents the stochastic discount factor which will be used as a deflator for the
payoffs in the real world. Y is the short term rate at time and ] is the time-varying market price
of risk for stock returns. ‚ is a standard Wiener Process, where ‚ = 0 and the increment ∆‚ is
assumed to be normally distributed with mean 0 and variance 1. This can be symbolized by . De
Jong (2004) suggests a deflator with a similar structure for the Black-Scholes model where stock
prices follow a log-normal distribution. The deflator is typically stochastic because it varies with the
random variables driving the stock returns. For the one-period setting = 0 and ∆ = 1 the deflator
takes the form
1
N
5 = c-d ƒ−Y − ]' ] − ]' …
2
(39)
where ~0,1 is the random variable that defines the stock returns, and ] is the Sharpe ratio.
The short term rate Y is assumed to be the risk free rate Y for the next period.
N
3.4.3. Discrete Pricing Kernel
Additionally by using the second order Taylor approximation, the (continuously compounded)
deflator can then (if necessary) be approximated as:
5=B
1
1+
N
Y
E†
1
1
+ ]' ‡
+ ]' ]
=B
1
N
8
1
E†
‡
'
1 + ] + ]' ]
From this discrete representation it is clear that the one-period deflator is the product of the risk
free rate and a stochastic term. The shocks on the stock returns, as well as the market price of risk,
influence the stochastic term. This supports the intuition that the discount factor must include some
stochastic part, over and above the risk free discounting, to adjust for the additional risk in the real
world. To summarize, the SDF valuation method uses real world probabilities, but discounts with a
risk corrected discount factor.
Sharpe ratio of a stock is the market price of risk, defined as the risk premium divided by the standard
deviation of the relative return. The risk premium is the relative expected return minus the risk free rate, 8N .
11
]=
b − 8N
26
3.5 Estimation of a suitable Stochastic Discount Factor
In the literature there is a vast amount of academic papers available on the topic of estimation of
the SDF. This specific part also forms a crucial part of this paper. The goal is to specify and estimate
a suitable pricing kernel for a consistent valuation of the contingent claims obtained from the
scenario analysis in the VAR model. The VAR model is typically used for risk management purposes
while valuation is used in pricing of assets. A problem of consistency arises in the current setting
where prices are valued on the fair value principle where the risk-free interest rate is used as a
deflator for risk neutral payoffs, while in the VAR model real world scenarios are estimated. In order
to bridge this problem and value real world scenarios in a more appropriate and consistent manner,
various techniques for estimations of the stochastic discount factor is evaluated and organized in a
list. This list can be viewed as a sort of inventory summarizing various methods of estimation
applied in the academic literature. From the list an applicable method will be chosen to apply
practically in order to calibrate the parameters for a consistent valuation of the VAR model’s real
world scenarios. The purpose of this study is to derive a SDF which can be applied to the VAR model
to obtain consistent valuations (prices) of the real world scenario generated cash-flows.
3.5.1. A SDF approach to asset pricing using panel data
The approach to the SDF estimation proposed by Araujo, Fernandes and Issler ((2005) & (2006))
applies the Pricing Equation in a panel-data framework to develop an innovative consistent
estimator of the SDF mimicking portfolio12. If for all assets at all time periods the Pricing Equation is
valid, then this offers a basis to construct an estimator of the SDF mimicking portfolio in a panel-data
framework where there are ample number of assets and of time periods. This approach depends
heavily on the actuality that the logarithm is the common feature 1`of every asset return in the
economy as described by Hansen and Singleton (1983). The determinants of the logarithm of asset
returns are obtained from an exact Taylor expansion of the Pricing Equation. This common feature
of asset returns acts as a sort of identification strategy to recover the logarithm of the mimicking
portfolio. By imposing reasonable standard restrictions to curb the behaviour of asset returns, a
consistent estimator of the SDF is obtained which is a straightforward function of the arithmetic and
geometric averages of the asset returns. This results in an advantageous non-parametric function
making it possible to analyse intertemporal asset pricing without specifying precise preferences or
using consumption data. For an overview of similar approaches the authors suggest the following
papers by Hansen and Jagannathan (1991) and Campbell (1993).
12
The terms SDF, SDF mimicking portfolio and mimicking portfolio are used analogously.
27
To better understand the approach an explanation of the mimicking quality of the mimicking
portfolio follows. In particular, the authors label the SDF mimicking portfolio as a “SDF generator”.
Economic theory states that under mild conditions the SDF, 56 from equation (5), exists and is
unique in complete markets. This is universally known as the pricing kernel (Charlier, Melenberg, &
Schumacher, 2007). However, although markets are typically incomplete, a unique discount factor
∗
can still exist in an incomplete market setting by defining the SDF mimicking portfolio, 56
where
∗
56 = 56
+ M6 with UM6 8‰,6 V = 0, therefore the connotation of a “SDF generator”.13
Unsurprisingly certain assumptions are imposed on the SDF mimicking portfolio to obtain this “SDF
generator”.
∗
The first assumption is in line with finance theory and states that the Pricing Equation must hold
which is equivalent to the “law of one price”.
∗
The second assumption requires that the SDF as well as the SDF mimicking portfolio must be
∗
strictly positive 56; 56
> 0 as a necessary condition to use the logarithms of the SDF.
The “no-arbitrage” assumption is naturally a stronger assumption than the “law of one price”.
∗
Thirdly, there exists a risk-free rate 86 which can be measured with the sigma-algebra
N
relating to the conditional set used in working out the conditional moments ∙. This is a
necessary assumption to evaluate the equity-premium puzzle. (For an overview on the equity
premium puzzle see the pioneering work of Hansen and Singleton (1983), and Mehra and
Prescott (1985)).
∗
The fourth and last assumption imposes weak stationarity to manage the extent to which the
data is time-series and cross-sectional dependent. By presuming that the natural logarithm of
the product of the SDF mimicking portfolio and the vector of asset returns f‹t5∗ Œ l is
covariance stationary with finite first and second moments, time series dependence is limited.
To control the degree of cross-sectional dependence let ‰, = lnU5 8‰, V − lnU5 8‰, VŽ
‘
and assume lim‘→“ ‘” ∑‘
‰q ∑—q–U‰, —, V– = 0 where N is the total number of assets with
asset ˜, ™ = ℕ. The limit guarantees that convergence of the probability of cross-sectional
means.
An important aspect of this assumption is that it allows for conditional
heteroskedasticity in all asset returns which is evident in most empirical work on asset returns.
The main result is given by Proposition 1:
“If the vector process f‹t5∗ Œ l satisfies assumptions 1 to 4, the realization of the SDF mimicking
portfolio at time t, denoted by 5∗ , can be consistently estimated as , › → ∞ using:
13
The SDF generator is comparable to equation 6.
28
∗ =
5
8žŸ
1 p žŸ ž ∑
› —qU8— 8— V
£
ž ‘
where 8žŸ = ∏‘
‰q 8‰, and 8 = ‘ ∑‰q 8‰, are respectively the geometric average of the reciprocal
¢
of all asset returns and the arithmetic average of all asset returns.”
(Araujo, Fernandes, & Issler (2006, p. 8) & (2005, p.11))
∗ are worth mentioning. This estimator of the
of asset returns. Some striking features of 5
In essence the estimator of the SDF is the equally-weighted reciprocal of the cross-sectional average
realizations of the mimicking portfolio is fully non-parametric and only implements asset returns in
∗ is “preference-free” despite the no
intertemporal substitution puzzles. In other words the 5
its approach, making it suitable to directly analyse various preference specifications or to investigate
arbitrage implication following from the mild restriction of the Pricing Equation. This independence
from any parametric assumptions about preferences implies that there is no misspecification risk
when choosing an improper functional form for the estimation of the SDF. Also, the SDF mimicking
portfolio does not depend on consumption data therefore avoiding excess smoothness. In general
the problem with excess smoothness, which is typically found in consumption-based estimates, is
∗ is allowed to be
Interestingly, asset returns used in the estimation of 5
that it causes troubling economic puzzles. As a consequence this approach shows little signs of the
equity-premium puzzle.
heteroskedastic, opening up the approach to high-frequency data.
Also an advantage of this
estimator is that it filters out the common component of asset returns, avoiding accumulating
measurement error.
A second important proposition is one that follows from the close relationship between the SDF and
the risk-free rate. Due to the close relationship a consistent estimation of the risk-free rate can be
based on the consistent estimator of the SDF. The Proposition 2 follows;
∗ as in Proposition 1 above offers a consistent estimator of the risk-free rate, 8 , as follows”
“Using 5
N
8 =
1
 Ž
5
(Araujo, Fernandes, & Issler, 2005, p. 15)
29
N
This approach certainly has some impressive attributes making it a promising method to derive a
consistent SDF for panel data. Remarkably, the combination of economic theory (Pricing Equation)
with basic econometric tools (panel-data framework) leads to a consistent estimator of 5 which is
“preference-free” which bring about reasonable empirical results and which shows little
resemblance to the equity-premium puzzle. (A complete explanation with technical details of the
assumptions and proofs of the propositions are included in the papers by Araujo, et.al. ((2006)
&(2005)).
3.5.2. A neural network approximation to the Pricing Kernel
A new approach to no arbitrage and arbitrage pricing is put forth by Bansal and Viswanathan (1993)
where a low-dimensional nonnegative nonlinear pricing kernel is estimated by means of a neural
network approximation. As a starting point the authors evaluate the influential idea proposed by
Ross (1976) and Merton (1973) that expected returns of a portfolio can be represented by just a few
significant state variables denoted as factors, thus implying that the only risks being priced are those
relating to the factors influencing the portfolio. As a consequence of this idea, the unconditional
arbitrage-pricing theory (APT) of Ross (1976) assumes that the expected returns (payoffs) are linear
in the factors and idiosyncratic risk. However APT implies the existence of a pricing kernel that is
possibly a nonlinear function of only the factors or state variables. The linear construction of payoffs
together with the no-arbitrage restriction, are strict assumptions necessary to obtain security prices
with a linear association with the factors.
The authors condemn the use of such stringent
assumptions as ineffectual since it unnecessarily limits pricing to only primitive securities with linear
structures while derivative securities with non-linear structures, such as levered equity and holding
period returns, cannot be priced. The APT model is not the only model with restrictions of linear
payoff structures and low dimensionality. Various other models with a linear combination of factor
payoffs and low dimensionality are also assessed. The renowned CAPM and ICAPM models, as well
as the minimum variance pricing kernel by Hansen and Jagannathan (1991), are criticized by the
authors for failing to abide by the no-arbitrage restriction. Since most payoff structures are
nonlinear, the above mentioned methods / models with linear payoff structures are denounced as
inadequate pricing tools and cannot accommodate the nonlinearity together with low
dimensionality.
The existence of derivative securities with nonlinear payoffs implies that a low-dimensional linear
pricing kernel need not exist. By only imposing the two most crucial assumptions, no-arbitrage and
low dimensionality, Bansal and Viswanathan (1993) uncover a fitting pricing kernel for nonlinear
payoff structures. They take a different approach by utilizing the notion of equilibrium pricing
30
models to find a nonnegative (as a necessary consequence of no-arbitrage), nonlinear pricing kernel
satisfying low-dimensionality, hence reaching an asset-pricing theory with testable implications. In
their own words, “the main contribution of this paper is to identify a pricing kernel that prices all
securities (including derivative securities), depends only on a few economy-wide factors, and satisfies
the restriction of no arbitrage. Hence, this pricing kernel embodies in it all the restrictions of interest
and provides a simple way to test the theory. Further, this pricing kernel is capable of pricing
dynamic trading strategies.” (Bansal & Viswanathan, 1993, p. 1233)
A description of the assumptions and the ensuing requirements provide a background for the model
setting followed by the authors. The assumptions are discussed briefly.
∗
To obtain a nonlinear pricing kernel that is a function of a small number of factors, the existence
of an individual whose marginal rate of substitution is a function only of a few factors is a vital
condition. This personifies the ideas of Ross (1976) and outlines the first assumption.
∗
Secondly, it is assumed that the factors provide adequate information for predicting future
factor realizations.
∗
In addition, to be able to simplify the economic interpretation of the nonlinear APT and to
attain testable restrictions, another assumption is assumed. The third assumption makes it
possible to write the pricing kernel in terms of observables by defining the existence of a
continuous, one-to-one and onto mapping from the set of factors to a subset of all the
securities that are free of non-factor risk. This assumption of invertibility has a less restrictive
implementation than the assumption of linearity in payoffs.
The following restrictions on a pricing kernel are implied by the nonlinear APT theory and explained
as follows;
∗
The first restriction involves that the pricing kernel satisfies the orthogonality conditions such
that the pricing kernel can price any one period ahead payoff.
∗
The second restriction follows from the no-arbitrage assumption so that the pricing kernel is
necessarily nonnegative thereby implying that positive payoffs have positive prices.
∗
Low-dimensionality of the pricing kernel is the third restriction which put into practice the idea
of Ross (1976) and Merton (1973) that only a few factors containing the risks are priced.
Now that the assumptions and restrictions are clear, the estimation of the pricing kernel follows
naturally. The semi-nonparametric estimation (SNP) procedure developed by Gallant and Tauchen
(1989) is implemented because the exact form of the pricing kernel is unknown. Conveniently this
31
approach embeds a large class of parametric asset-pricing models including the APT, CAPM and
ICAPM which act as nested models. Additionally, the SNP approach also tolerates the restrictions of
no arbitrage. The Generalized Method of Moments (GMM) (Hansen L. P., 1982) is applied in the
estimation process to avoid making distributional assumptions and pre-specifying the number of
factors. Since the exact functional form is unidentified, neural networks are used to approximate
the unknown function and the GMM is used to estimate the neural net. The estimated neural net is
a powerful and consistent estimator of the unknown pricing kernel which is the main objective.
3.5.3. Asset pricing with observable stochastic discount factors
In the approach by Smith and Wickens (2002) they suggest an observable stochastic discount factor
approach to pricing assets and put forth a valid argument why the VAR model might be an
inappropriate SDF model. This approach may offer a possible tactic for the Frequency Domain
Factor modelling because, while the SDF can be used explicitly or implicitly such as in the CAPM14
and CCAPM15 with observable factors, in this approach an attempt is made to concentrate on the
explicit use of the SDF model with observable macroeconomic factors. A descriptive account of the
proposed approach by the authors now follows.
The process of pricing assets can conveniently be done by applying the general framework offered
by the SDF model. To help specify the SDF model the use of single and multiple factors, and of latent
and observed factors are often applied which forms a support on which the SDF model can be based.
For this reason the SDF model is a factor pricing model. However, single factor models are often
unsuitable in most situations in particular for term-structures while latent factors involve to some
extent arbitrary specifications of generating processes leading to some problems with economic
interpretation. Smith and Wickens (2002) propose to use multiple factors that are observable
stochastic discount factors in their new pricing approach and employ a multivariate GARCH-in-mean
(MGM) process to model the joint distribution of excess returns and the factors. The advantage of
using the SDF model as a factor pricing model is that this model allows for the factors to be linear
functions of the conditional covariances between the factors and the excess return on the risky
asset. Other factor pricing models, like the well-known asset pricing theory (APT) (Ross, 1976), do
not possess this quality. It is in fact this conditional covariation between the factors and returns
that give rise to the risk premium, and inspires the use of the multivariate context where having
conditional covariances in the conditional mean process is vital. Accordingly, the authors describe
how the affine factor models, CIR (Cox, Ingersoll and Ross) and Vasicek models, are noticeable
exceptions because they meet this condition, but at a price.
14
15
Capital Asset Pricing model
Consumption-based inter-temporal capital asset pricing model
32
Although the VAR models are commonly used in practice (Campbell, Lo, & MacKinlay, 1997), the
authors have a different opinion of the general single equation and VAR models. While the paper
discusses various traditional approaches of the different implementations of the SDF model for
FOREX, equity and bonds together with the new proposed approach, it also mentions reasons why
the VAR model is not a valid way to estimate asset pricing models or to test market efficiency. In
general the VAR model is not a SDF model and no apparent attempt is made to restrict the model to
satisfy the condition of no-arbitrage. Accordingly they deem the VAR model inadequate to model
the risk premium. One of the main reasons for the inadequacy is just that, the VAR model’s inability
to provide a suitable risk premium for the risk-averse investor.
To model the risk premium
conditional covariances are needed and the VAR model does not include this therefore being
inefficient in its approach.
However, under certain assumptions it is possible that the VAR model can be a SDF model. There
are two main exceptions, the first being under the assumption of risk neutrality where there is no
need for a risk premium. The second exception is if one of the variables of the VAR model is excess
returns, then implicitly the risk premium is the lag structure on the right-hand side. The lagged
variables act as a fitting proxy for the risk premium and can be interpreted as the factors in a linear
factor model. If the risk premium can be shown to be a linear function of the variables in the VAR
model, then it can also be exempt and the VAR model can be a SDF model. In fact the Vasicek and
CIR affine factors models of the term structure also satisfy the latter condition; therefore the authors
suggest these affine factor models as alternative methods to the VAR model (Smith & Wickens,
2002).
This new way of formulating and estimating the SDF model by using observable factors in a
multivariate GARCH-in-mean process offers a broad formulation making it suitable for comparing
and testing different types of SDF models. Attractively this approach recognizes the sources of risk
making it possible to assess its importance and providing the opportunity of tilting portfolios to
provide a hedge against these risks. However, there are also some problems relating to the
proposed approach. In the SDF model risk is determined from the conditional covariation between
the excess returns and the factors. Conditional heteroskedasticity in the factors is important to
assure the risk premium is time-varying and an important shortcoming of using observable
macroeconomic factors is that they do not exhibit much heteroskedasticity, especially at lower
frequency data. The lack of heteroskedasticity in macroeconomic variables together with the
tendency that statistical models tend to be highly parameterized makes it difficult to obtain
statistically well-determined models (Smith & Wickens, 2002).
33
3.5.4. Pricing Kernel and affine term structure of interest rates
The final approach uses the pricing kernel to define an affine term structure of interest rates.
Cochrane and Piazzesi (2008) make use of the yield curves to determine the value of assets with
future expected cash flows. Assume the term structure of interest rates is time varying and can
therefore be used to price financial assets such as a Bond at fair value. According to Hoevenaars
(2008) interest rates are one of the most important risk factors in a fair value world. Cochrane and
Piazzesi attempt to decompose the yield curve into expected interest rate and risk premium
components so that the risk factors become apparent. To do so the term structure of bond yield
curves is derived by means of the pricing kernel such that they are affine in the state variables of the
vector autoregressive (VAR) model.
An extension of the VAR model to include the stochastic discount factor leads to the derivation of
term structures of interest rates that are affine in the state variables. Assume the return dynamics
are modelled in a first-order VAR model such that
= + Β
+ (40)
where and are both × 1 vertices containing the error terms and state variables,
respectively (see Section 2.2 for details). The vector of error terms is assumed to be independent
identically-distributed with mean 0 and variance covariance matrix ¤ ( ~¥0, Σ). Cochrane and
Piazzesi (2002) introduce a specification for the discount factor that necessarily corresponds to the
first-order VAR model in equation (40). Subsequently arbitrage opportunities are ruled out and the
stochastic discount factor takes the form such that
1
56 = c-d C−Y − ]' Σ]S − ]' 6D.
2
(41)
Y is the one-period yield and ]S the market price of risk. Given this representation of the pricing
kernel, modelling is indistinguishable from the specification of the prices of risk. Assume the market
price of risk ] is an affine process of the state variables as specified by Ang and Piazzesi (2003) such
that
] = ] + ] .
(42)
] symbolizes a -vector and ] is a × matrix which depend on the state variables modelled by
. Furthermore, assume that the one-period short rate is an affine function of all the state
variables such that
34
Y = −¦ − ¦' (43)
By substitution the pricing kernel is rewritten as
1
56 = c-d C−¦ − ¦' − ]' Σ]S − ]' 6D.
2
(44)
The model structure implies that if the prices of risk are affine in state variables then the term
structure of interest rates are also affine in the state variables. To illustrate this consider the case
presented by Ang and Piazzesi (2003) and Cochrane and Piazzesi (2008) where an t −period Zero
Coupon Bond is priced with the Pricing Equation as specified in equation (24). In general bond prices
can be expressed as exponential affine functions of the state variables such that
4
u
4
u
= c-d§u + xu' (45)
represents the price of the t −period Zero Coupon Bond. The coefficients §u and xu can be
computed recursively by the following difference equations:
1
§u6 = −¦ + §u + xu' − Σ] + xu' Σxu
2
'
= −¦' + xu' Β − Σ] xu6
(46)
with § = −¦ and x = −¦ . The dynamics of the VAR model are taken into account in both
equations by and x. (See (Ang & Piazzesi, 2003) and (Cochrane & Piazzesi, 2002) for details on the
derivation of the difference equations). To obtain the affine term structure of interest rates the
continuously compounded yield /u can be determined from the price of the t −period Zero Coupon
Bond.
/u = −
log 4
t
u
=−
§u
xu '
− C D = §¨u + x©u' t
t
(47)
Therefore the yields are modelled as affine functions of the state variables from the VAR model.
The affine term structure of interest rates concludes the inventory list to find a suitable estimation
for a SDF. The latter method clearly has the most potential to obtain the goals set out in the
introduction since it is set in the same model structure. A discussion of the VAR model as a SDF
model follows.
35
3.6 VAR model as a SDF Model
Assume the joint dynamics of annual inflation rate, nominal annual interest rates and stock returns
on a market index are modelled in a first-order VAR model. Recall from section 3.4 that the pricing
kernel is a very powerful tool for valuation for all assets for which the joint distribution of the pricing
kernel and future payoffs of the assets has to be modelled. Furthermore the modelling of nominal
and real interest rates relies on specifications of the pricing kernel (Nijman & Koijen, 2006). By
extending the VAR model a model to specify the term structure from the pricing kernel can be
obtained. In this way an economic environment is constructed that incorporates the time-varying
risk factors (Hoevenaars R. P., 2008). This implies that the term structures used in the Pricing
Equation in (4) and (5) to value assets will be consistent with the relevant risk factors affecting the
state variables. It is clear from equation (44) that the market price of risk relates the shocks in the
underlying state variables to the discount factor making it stochastic and time dependent. Nijman
and Koijen (2006) support the explicit specification of the pricing kernel since it is more preferable to
ensure consistency of the discount factor with real state probabilities.
4. Model Specification and Estimation
The approach to specifying the model for the SDF valuation in a VAR model setting can best be
described by a 2-step estimation process. The method as proposed by Hoevenaars (2008) firstly
estimates the parameters of the VAR model by maximum likelihood. However, this is not necessary
since the purpose of the study is to focus on the valuation process. Thus parameters of the VAR
process have been predetermined by using Yule-Walker estimations and are fixed.
Next the
parameters of the SDF model are estimated conditional on the VAR parameters. This step is of
interest and requires careful consideration.
Three primitive financial assets are used in the
modelling process in order to estimate comparable prices. The first asset is a Zero Coupon Bond, the
second is an Inflation Linked Zero Coupon Bond and last is a basic equity holding. Comparing the
estimated prices from the VAR/SDF approach to observed prices gives an indication of the model fit
and determines if it is suitable. By valuating each individual financial asset with the risk neutral
pricing model acts as a stand-in for the observed prices. The residual sum of squares provide a
measure to determine how much different the estimated prices are from the observed prices. An
optimization procedure leads to a possible calibration of the SDF parameters. To evaluate the
performance of the estimated parameters an out of sample test is conducted. Finally, the model
specification with estimated parameters is applied to a basic pension product. A nominal (no
indexation) and a real (full indexation) pension product are priced using the calibrated parameters.
36
The estimated price is compared to the observed price just as before. However, if the model
specification and estimation procedure is proficient, then the difference between the observed and
estimated price should be insignificantly small. As a final result, a complex pension product with
embedded options is valued using VAR/SDF approach with the calibrated parameters.
Before the specification and estimation process begins, certain assumptions must be specified. The
first assumption lies at the heart of asset pricing. That is the absence of arbitrage opportunities.
This implies that the law of one price holds. Next, to create a self-consistent model, assume that the
expected stock return implied in the VAR is the same as the one implemented in the Pricing Equation
before the date of maturity = 0*, *›h. All cash streams are modelled implicitly. This includes
in (5). Furthermore, for each of the three assets it is assumed that there are no payouts or inflows
dividend payments in the case of equity returns. The evaluation procedure now follows.
4.1 Data
The historical data set contains three economic variables, sampled annually from 1970-200716. All
the series concern end-of-year data and are measured in level. The first data series is Price Inflation.
The data for the Netherlands Price Inflation is obtained from Bloomberg17. It is assumed that the
Price and Wage Inflation are set equal. The second data set, the 10-year Nominal Interest Rate for
the Netherlands is also obtained from Bloomberg18. A flat yield curve is assumed for the purpose of
the study. As a result of the flat yield curve the nominal interest rate is the same over all maturities.
Therefore the long term nominal interest rate proxies as the short term nominal interest rate at time
. Lastly, the third data set is the return on equities. The stock returns for Europe is obtained from
an internet database19, where the “MSCI Europe Gross Index – local” is used to sample the data for
1970-2001 and the “MSCI Europe Gross Index – Euro” for 2002-2007. The data set is used as the
state variables in the VAR time-series model. The historical data has already been implemented to
estimate the parameters in the VAR model by using Yule-Walker estimation techniques. Therefore it
is not necessary to redo this process. Note that the long term mean values of the VAR model have
been set at forward looking expected values rather than on averages of the historical data.
Conversely the volatility and correlation dynamics are taken directly from historical data. The future
evolution of all the series can now be computed in the VAR model to obtain the scenario set.
16
Source: ORTEC Finance BV
Bloomberg: OENLC005
18
Bloomberg: OENLR006
19
http://www.mscibarra.com
17
37
4.2 General Setup and Specification of the Model
4.2.1. State dynamics and the VAR model
The VAR model plays an integral part in the analysis because it forms the setting on which the SDF
model is estimated. The popular time series model is used to generate the future evolution of the
state variables. This forms the future scenario set generated from the VAR model that will later be
used in the valuation process. The parameters and the data statistics of this model are specified as
estimated by the Yule-Walker method20 (Steehouwer, 2005). The parameters for the VAR model
are:
,ª
ª = !,ª % ,
«,ª
0.775696600
Β = ¬ 0.037386933
−1.161459400
0.0025124508
= ¬0.0055833823±,
0.0382836500
0.083652566 −0.0171560680
0.831542400
0.0050897898 ±
1.709851100 −0.0431063410
,
s=t
Σ
= ¬, ± , = 0, &' = ( for *
s≠t
0
«,
0.0002167606500
Σ = ¬ 7.1798562e − 005
−0.0007258713500
7.1798562e − 005
0.0001025311500
−0.0005069793300
−0.00072587135
−0.00050697933 ±
0.03827066100
where ª represent the vector of initial values for the VAR model. The last value of each series in
the data set is used as the initial values. is the constant term and Β is the autoregressive
coefficient matrix of the VAR model. The error term is assumed to be independent identicallydistributed (I.I.D.) and sampled from a normal distribution with mean zero and variance Σ. The
vector of multivariate error term is sampled from the Choleski decomposition of the covariance
matrix Σ of error terms.
The unconditional historical data statistics and the unconditional model statistics for the future
evolution of the VAR model is given in Table 1. The statistical data portrays the average and
standard deviation for the individual data series and are in line with what can be expected for a
realistic model. The long term average annual level for Price Inflation, Nominal Interest Rate and
MSCI Stock Returns are 2%, 4% and 8%, respectively. While Price Inflation and Nominal Interest
Rate are not so volatile, the Stock Returns are much more volatile with a standard deviation of just
below 20%. This is a reasonable number for the instability relating to stock returns. The correlation
20
Note that the Yule-Walker estimations were conducted by ORTEC Finance BV. These parameters values are
fixed and only used as input in the analysis.
38
is measured as the covariance relationship between the different state variables divided by the
product of the standard deviation of each corresponding state variable. The correlation of a variable
with itself is called the auto-correlation. This is naturally the auto-covariance divided by the variance
of the respective state variable and is simply equal to 1. Another measure of similarity between two
state variables is the cross-correlation where one of the variables is lagged one period. The 1st order
auto-correlation is the diagonal entries in the cross-correlation matrix. Steehouwer (2005) provides
an extensive description on how to model these data statistics.
Mean
Price Inflation
Nominal Interest rate
MSCI Stock Returns
Price Inflation
Nominal Interest rate
MSCI Stock Returns
St.dev
Correlation
HISTORICAL DATA STATISTICS
3.64%
2.74%
1
6.91%
1.99%
0.616
1
13.02% 19.78%
-0.162
-0.068
1
UNCONDITIONAL MODEL STATISTICS
2.00%
2.74%
1
4.00%
1.99%
0.616
1
8.00%
19.78%
-0.162
-0.068
1
Cross-Correlation
0.833
0.556
-0.047
0.547
0.859
0.076
-0.254
-0.015
-0.028
0.833
0.556
-0.047
0.547
0.859
0.076
-0.254
-0.015
-0.028
Table 1: Statistical Summary (Source: ORTEC Finance BV)
Now that the statistical aspects and model dynamics have been discussed the focus turns from the
past to the future. Monte Carlo methods are used to model the future evolution of the time paths
for each state variable. Graphical representations of the future scenarios for each state variable
show the development of the VAR model over a 20 year horizon with 1000 scenarios. In each graph
one scenario of the evolution of the time path is highlighted (with stars) to indicate the flow of an
individual time series. Characteristically, a time series is modelled from its last position therefore
creating a path over time. The straight white line in each graph represents the mean value of the
relevant state variable and converges to the long run averages as listed in Table 1. These scenario
graphs depict the evolution of the VAR model taking into account the model dynamics. Therefore
the future scenario set is consistent with historical data. After inspecting the results it is safe to say
that the VAR model is a realistic model. The scenario graphs can be found in Figure 4.
4.2.2. Stochastic Discount Factor Model
The modelling of the discount factor is the essence of this study. The dynamics of the stochastic
discount factor depend on the state variables in the VAR model, therefore it is stochastic (Campbell,
Lo, & MacKinlay, 1997). It is useful to think of the SDF as an additional state variable being modelled
at different future points in time and for each specific scenario, therefore relating it to the errors at
that point in time. The appropriate SDF for the scenarios generated in the VAR model is specified in
39
Price Inflation
0.15
0.1
0.05
0
-0.05
-0.1
0
2
4
6
8
10
12
14
16
18
20
14
16
18
20
14
16
18
20
Horizon
Nominal Interest Rate
0.12
0.1
0.08
0.06
0.04
0.02
0
-0.02
-0.04
0
2
4
6
8
10
12
Horizon
MSCI Europe Stock Return
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
0
2
4
6
8
10
12
Horizon
Figure 4: Scenario Graphs - The future evolution of the time paths for each state variable
section 3.4 and 3.6. ((Nijman & Koijen, 2006), (Hoevenaars R. P., 2008) & (Cochrane & Piazzesi,
2008)). There are four parameters to estimate namely ¦ , ¦' , ] and ]' . The delta parameters
model the short rate effect and contain a constant value as well as a time dependent value.
Consider the case where the time-dependent market price of risk ] in equation (44) is set to zero
for all . Then the deflator contains no risk adjustment. As a result only time discounting is taken
into account and thus the short rate dynamics is comparable to the risk neutral discounting such
that
Y ≡ −¦ − ¦' .
N
40
However the stochastic payoffs are modelled with real world probabilities and are not risk neutral.
Therefore discounting only for time is insufficient and will not correctly deflate the real world
payoffs. One can imagine that the inclusion of the shocks substantially improves the fit of the
approximate discount factor. Therefore a non-zero market price of risk can be expected. The
market price of risk ] essentially relates the exposure to risk to the valuation of payoffs. A non-zero
] affects the long run mean of the risk premium while a non-zero matrix ] affects the timevariation (Ang & Piazzesi, 2003). In short the delta parameters model the time discounting effect
associated with the short rate while the lambda parameters capture the risk adjustment related to
the error terms.
4.2.3. Modelling the Financial Products and the Valuation thereof
The VAR model and SDF model are both influential models and in order to implement them it is now
necessary to specify the financial products. Three basic financial assets are chosen for the valuation
process. First asset is a Zero Coupon Bond, followed by an Inflation Linked Zero Coupon Bond and
lastly a Stock is modelled. The prices of the three assets is estimated using the VAR/SDF approach
where the real world payoffs are discounted using the pricing kernel method. These VAR/SDF
valuations are known as the estimated prices. Theoretically the risk neutral valuation method and
the stochastic discount factor method in a particular model structure should price an asset such that
is gives the same price. This is a consequence of the law of one price. Therefore in order to have a
measure of comparison the risk neutral valuation technique is also applied to price all three financial
assets. This acts as a stand-in for observable prices in the market. A discussion of the modelling
process now follows.
a) ESTIMATED PRICES
Zero Coupon Bond
One of the most fundamental assets for modelling prices in financial markets is bonds. This financial
asset owes its popularity to the fact that it is assumed to be a risk-free asset and favourable
assumptions can be made which simplifies the modelling process. In general bonds pay out a
reoccurring coupon payment until the time of expiration. This can be annually, monthly or any other
time period which is specified in the financial contract. The contract expires at the time of maturity
when the face value is paid out. The multi-period Pricing Equation in (19) is used to model the
expected cash flows of a bond with coupon payments.
41
4©y
p
= B n 5,o 2o + 5,p xp E
oq6
(48)
4©y is the estimated price of a Bond at current time and xp is the face value of the bond paying at
time of maturity. The coupon payments are represented by 2o where the sum is taken over all the
coupon payments. One can think of the coupon payments as a series of Zero Coupon Bonds paying
the required amount to cover the coupon payment. Each coupon payment is redeemed at the
corresponding date of expiration. Then equation (48) can be modelled as the sum of the expected
payoffs. Zero Coupons Bonds are very easy to model due to the simplicity of its cash flows. For a
Zero Coupon Bond there are no coupon payments. The only cash flow is the face value paying at
time of maturity. Therefore in the valuation process it is assumed that the coupon payments are
zero. As a matter of convenience it is assumed that valuation only takes place at = 0 such that
there are no intermediate valuations of Zero Coupon Bonds at some time in the future.
Inflation Linked Zero Coupon Bond
Another important financial asset is one that provides compensation to investors for inflation. The
Inflation Linked Bond has a similar structure as a coupon paying Bond. In addition the inflation cover
is modelled such that the expected cash flows for the Inflation Linked Bond are adjusted to inflation.
p
o
p
4©µ¶y = B n 5,o ¬2o B s 1 + ˜t·‰ E± + 5,p ¬0¸xp B s 1 + ˜t·‰ E±E
oq6
‰q6
‰q6
(49)
4©µ¶y is the estimated price of the Inflation Linked Bond at current time = 0 such that there are no
intermediate valuations. The inflation compensation is represented by 1 + ˜t·‰ and is time
dependent. 0¸xp is the face value of the Inflation Linked Bond paying at expiration time › and 2o is
the coupon payments. Once again as a matter of simplicity it is assumed that the coupon payments
are zero.
Equity Returns
In practice stock returns are volatile and carry with it a large degree of unpredictability. However,
on the basis of empirical data certain statements in a statistical sense can be made concerning the
distribution of stock returns. The stock returns are modelled to incorporate the riskiness associated
with it into the discount factor. In principle it can be assumed that for a basic equity holding one can
expect to receive the MSCI Europe stock return as calculated by the VAR model. Consider an equity
42
holding that is worth €1 and receives stochastic returns at future points in time. The price formation
calculates the expectation of these future stochastic returns at time as
4©¹º
p
p
= B5,p s 8o E = B5,p s 1 + Yo E
oq6
oq6
(50)
where the estimated price is 4©¹º . The time dependent return on the stock is 8o which is modelled
as a state variable in the VAR model as Yo . To comply with equation (5) the estimated price if
modelled correctly should be equal to 1.
Conveniently the prices in equation (48), (49) and (50) can be estimated by approximating the
expectation through Monte Carlo simulation of the state vector influencing the payoffs as well as the
&
stochastic discount factor simultaneously. Let Q represent the scenario such that 5,p
denotes the
cumulative stochastic discount factor from time to › in replication Q Q = 1, … , [. Assuming there
are no coupon payments then the prices are approximated by taking the empirical mean such that
4žµ¶y
¹
4žy
¹
1
&
= nU5,p
xp& V
[
&q
p
1
&
= n B5,p
¬0¸xp& B s 1 + ˜t·‰ & E±E
[
&q
4ž¹º
‰q6
¹
p
1
&
= n B5,p
s 1 + Yo & E .
[
&q
oq6
(51)
Consequently the payoffs are scenario and time dependent where the superscript Q indicates the
scenario and the subscript denotes the time. As [ tends to infinity, the empirical mean for each
asset will converge to the true price of the asset on the basis of the weak law of large numbers
(Nijman & Koijen, 2006).
b) OBSERVED PRICES
The “observed” price for each of the three assets is calculated based on the principles of the risk
neutral approach. This means that the probabilities are altered under the change of measure so that
the risk adjustment is modelled implicitly. As a result there is no risk premium because the risk is
not priced explicitly. Additionally under the risk neutral measure the future payoffs are discounted
43
with respect to the current term structure. Hence the risk free rate is appropriate for discounting.
Assume a flat term structure.
Zero Coupon Bond
The “observed” price of the Zero Coupon Bond is reproduced in the pricing model. Typical future
cash flows relating to a Coupon Bond include the coupon rate and the face value of the Bond. Both
these types of cash flows are predetermined such that they are in fact deterministic. Thus pricing
under the risk neutral measure is in fact equivalent to the pricing model. The risk free rate is known
at time so that the coupon payments as well as the face value of the Bond are discounted to
current time . The “observed” price is given by
4y
p
= n c o{| 2o + c p{| xp .
oq6
}
}
(52)
There are no coupon payments 2o for a Zero Coupon Bond and xp is the face value of the Bond. It is
once again assumed that valuation only takes place at time = 0.
Inflation Linked Zero Coupon Bond
Following in order the Inflation Linked Bond is also valued in the pricing model. Assume the
expected inflation is equal to the known inflation at current time such that the inflation
compensation is neither time dependent nor stochastic. Thus the compensation for inflation is
deterministic. The “observed” price for the Inflation Linked Bond is calculated by
4µ¶y
p
= n c o{| g2o 1 + ˜t·S o h + c p{| g0¸xp 1 + ˜t·S o h
oq6
}
}
(53)
Again the coupon payments 2o are assumed to be zero and valuation takes place at time = 0 with
no intermediate valuations in the future. The inflation compensation is represented by ˜t·S and is
known at time . 0¸xp is the face value of the Inflation Linked Bond.
Equity Returns
The “observed” price of a basic stock is simply the value of the stock at current time . This is
because there is no need to openly identify the risk premium under the risk neutral measure. The
risk is included in the change of measure and is therefore not priced explicitly. Hence the expected
return is equal to the current value of the stock normalized to 1.
44
p{|
4¹º = ℚ
8 T X
Wc
}
p{| 1
1 = ℚ
+ Y T X
Wc
}
(54)
It is important to realize an important difference in modelling the “observed” prices compared to the
estimated prices. The estimated prices are modelled using stochastic payoffs and a stochastic
discount factor that depend on the state of the world at a certain time and scenario. A proxy to the
“observed” prices are constructed using risk altered payoffs and a constant discount rate, namely
the risk free rate. The latter prices are naturally not dependent on the time or scenario since it
simply represents the observed prices today for a particular asset.
c) RESIDUAL SUM OF SQUARES
To determine how good a measure the estimated prices are, it is necessary to compare it with the
observed prices.
The residual sum of squares (RSS) is a powerful technique that provides a
convenient way to evaluate the discrepancy between the observed and estimated prices. For every
future point in time the estimated price is subtracted from the observed price. The difference is
then squared to ensure that negative values don’t influence the overall value by creating a cancelling
effect when the differences are added up over time. Any inconsistencies will thus give rise to a big
residual sum of squares which will indicate that the relevant estimated price is inefficient. The
residual sum of squares are modelled as
½
8[[ = n n U4p‰ − 4©p‰ V ; ˜ = x, 0¸x, [8 .
‰ pq6
(55)
4p‰ and 4©p‰ represent the observed and estimated price of a financial asset ˜, respectively. The
subscript › is an index indicating the horizon and not the time. ¾ is the overall horizon value. This
means that both the observed price as well as the estimated price is a matrix made up of row
vectors for each financial asset ˜ where ˜ represents a Zero Coupon Bond, Inflation Linked Zero
Coupon Bond or a Stock. For each financial asset the observed and estimated price is a row vector
consisting of the corresponding observed/estimated price at yearly horizon points › where
›=g + 1, ¾h. To clarify this concept equation (55) is extended in the following way.
Let \ ≔ 4p‰ − 4©p‰ where
45
y
g46
µ¶y
4p‰ = !g46
¹º
g46
y
46
⋯
µ¶y
46 ⋯
¹º
46
⋯
y
4½
4½y h
µ¶y
4½
4½µ¶y h%
¹º
4½
4½¹º h
y
g4©6
µ¶y
= !g4©6
¹º
g4©6
y
4©6
⋯
µ¶y
4©6 ⋯
¹º
4©6
⋯
y
4©½
4©½y h
µ¶y
4©½
4©½µ¶y h%.
¹º
4©½
4©½¹º h
and
4©p‰
Then the residual sum of squares can be expressed as
½
8[[ = n n \‰p
; ˜ = x, 0¸x, [8 .
‰ pq6
Subtraction between the observed and estimated price in ‚ takes place in an element-wise manner
between the two row vectors relating to the financial asset ˜. To conclude the residual sum of
squares provide valuable insights into the suitability of the of the VAR/SDF approach to modelling
the estimated prices.
4.3 Estimation of Parameters
4.3.1. Initial parameterization
In order to start the evaluation process the parameters for the SDF must be specified. For a first
initial evaluation the parameters are restricted to only a few variables by making some assumptions
about the short rate dynamics and the risk adjustment.
Short Rate Dynamics
Assume that the one-period short rate Y is affine in the state variables such that
Y = −¦ − ¦' ¦ is a constant term while ¦' is a 1 × 3 row vector and incorporates time dependence into the
model. The short rate takes into account the time discounting effect relating to the state variables.
The starting value for the delta parameters is based on the presupposition that the time dependent
short term nominal interest rate is comparable to the risk free rate. Therefore the parameters are
set equal to
¦ = 0 and ¦' = g0 1 0h
46
such that the nominal interest rate is the only factor in the short rate dynamics of the discount
factor. Even though the nominal interest rate is compared to the risk free rate, the discounting with
the nominal interest rate from the VAR model is changing over time and therefore making it possible
to incorporate time stochasticity in the valuation problem.
Risk Adjustment
When modelling the cash flows of financial products under real world probabilities there is an
expected amount of risk. In order to evaluate the adjustment to risk, the market price of risk is
determined. Assume the market price of risk ] is an affine process of the state variables as
specified by Ang and Piazzesi (2003) such that
] = ] + ] .
] symbolizes a 3 × 1 vector and ] is a 3 × 3 matrix which depend on the 3 state variables
modelled by .
By following the same trend as for the short rate dynamics then certain
assumptions are drawn over from the risk neutral valuation technique. The very popular risk neutral
Black Scholes model makes an important conjecture that the market price of risk for stock returns is
constant and does not depend on the state of the economy. Since the exact relationship between
the state variables with the risk terms is unknown then this is as good a starting point as any.
Suppose then the market price of risk is time independent such that the risk adjusting lambda
parameters are
0
]  = ¬ 0± ,
]
0 0 0
] = ¬0 0 0±.
0 0 0
The vector valued ] represents the constant time independent part of the risk adjustment. The
time dependence relating to the state variables in the VAR model is modelled by the matrix ] . The
Sharpe ratio for Stock returns such as in the Black Scholes model is used to emulate the behaviour of
the time dependent market price of risk for stock returns. This explains its position as the third
element of the column vector ] .
Preliminary Results
The initial parameters are now used to value the three basic assets and then compared to the
observed prices. The present value estimations for each financial asset valued at time = 0 are
illustrated in the graphs in appendix D. Every asset’s present values are calculated both in terms of
the estimated prices as well as the observed prices. The observed prices provide a much needed
comparison tool. From the present value graphs it becomes clear how poorly the initial parameters
47
are performing. The estimated prices for the Bond assets do not deviate too far from the observed
prices, especially in the case of the Zero Coupon Bond. However, it is clear from the present value of
Equity Returns in the last graph (appendix D) that the initial parameters used to estimate the prices
are inapt. This can be expected because the lambda parameters capturing the risk adjustment have
been set equal to zero in the previous paragraph. Therefore the market price of risk has been
inadequately specified. In all three graphs it is apparent that the expected future cash flows under
real world probabilities are insufficiently deflated. The apparent differences between the observed
and estimated prices give rise to arbitrage opportunities. Consider the present values of the equity
returns. To buy a stock that will pay €1 in 20 years has an estimated price of €2 at time = 0. A
rational investor will try to take advantage of the arbitrage opportunities by selling a stock holding
worth €1 for €2. Therefore it is not reasonable to have an estimated price different from the
observed price. Furthermore a graphical portrayal of the residual sum of squares for each financial
asset in appendix F shows the inconsistencies between the observed and estimated prices for the
relevant asset. From these graphs it is clear that the initial parameters perform poorly in its attempt
to appropriately discount the expected future stochastic payoffs. The residual sum of squares are
increasing over time and showing no signs of coming to an end. The realization is of course that the
delta and lambda parameter values are based on very strict assumptions which do not allow for
much freedom in the modelling process. These parameter values are only used as starting values.
An indispensable step that follows next is the optimization procedure where more fitting parameter
values are estimated.
4.3.2. Optimization Process
In the optimization of the parameters it is important to identify a goal to work towards. Therefore it
is necessary to specify an objective function which will fulfil this task. It is only fitting to imagine that
with the current problem at hand one would like to minimize the inconsistencies between the
observed and estimated prices so that both methods give rise to the same price. Since the
estimated prices are modelled using the deflator technique the attention is focussed on the
stochastic discount factor with unknown parameters. The objective function is defined as
min
ÀÁ, À¢ ,ÂÁ ,¢
8[[
(56)
so that the residual sum of squares is minimized by optimizing ¦ , ¦ , ] and ]. In order to
strengthen the validity of the estimated SDF parameters, a horizon of 65 years is chosen for the
calibration process. The reason for this is that when dealing with pension products long term
obligations can typically be expected. The three basic financial assets are used in the optimization
48
and calibration procedure. Once again the initial parameters are implemented as a starting value.
For interest sake another approach as a starting value is also tried. This is done to determine which
parameters play an integral role and which don’t. This might lead to insightful conclusions of
whether the parameter can be set equal to zero for the foreseeable future. The comparable second
approach starts by setting all parameters equal to zero. The results for the optimization process are
summarized in Table 2. A cross (x) indicates which parameters are estimated in the table. In the
case where a parameter is not marked with a cross then the value simply stays equal to the initial
start value. In system A the delta and lambda parameters are set equal to the initial values as
specified in section 4.3.1. and in system B they are all set equal to zero. The average error between
the estimated and observed price as well as the residual sum of square for each system and every
optimization method is listed in Table 2.
¦
rf
x
¦
-
]
-
]
-
OPTIMIZATION METHOD
1
2
3
x
4
x
5
x
6
7
8
9
x
x
x
x
x
x
x
x
10
11
12
13
14
15
x
x
x
16
17
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
A: INITIAL PARAMETERS
B: ALL ZEROS
Ave error
RSS
Ave error
RSS
5.3084
1051.0631
1447.9905
286702.128
0.0525
10.3974
8.5829
1699.4081
0.0282
5.5787
0.0339
0.0007
0.1366
0.001
6.7184
0.1988
0.0005
0.1015
0.0722
14.3021
0.0003
0.0154
3.0539
0.0004
0.0554
0.0709
0.0004
0.0836
0.0003
0.0614
0.0003
0.0588
0.0003
0.0546
0.0004
0.0815
0.0002
0.0491
0.0002
0.0002
0.0494
0.0003
0.0491
0.0542
0.0002
0.0406
0.005
0.9981
0.0002
0.0416
0.0002
0.0415
0.0002
0.0436
0.0002
0.048
0.0002
0.0346
0.0002
0.039
0.0002
0.0359
0.0002
0.0349
0.0002
0.031
0.0002
0.0331
Table 2: The Residual Sum of Squares for the Optimization Methods
For each of the two different starting systems the optimization gives similar results. Hence, the
optimization process does not depend primarily on the starting values, although good starting values
can greatly assist the process. However, it is visible that the initial parameters in system A on the
whole tend to have lower residual sum of squares. This is trivial because the initial parameters in
system A intrinsically include short rate dynamics and a constant market price of risk, namely the
49
Sharpe ratio for stock returns, as a starting value. This implies that the short rate and market price
of risk is present in the stochastic discount factor even in cases where the optimization method is
not optimizing with respect to ¦ and ]. Although system A overall gives better results, some
remarkable points of system B are pointed out.
The first remark is that the residual sum of squares (RSS) in method 5B and 6B are very high
compared to the corresponding RSS in 5A and 6A. Recall that system A models the short rate
dynamics in such a way that it is time dependent and equivalent to the risk free rate. In system B
the short rate dynamics are set equal to zero. Hence modelling without the short rate dynamics give
higher RSS which indicates it is an unsuitable method. It is clear that the parameter specification for
the short rate dynamics play an integral part in the valuation process. Method 12B displays the
same behaviour. To further support the importance of the short rate dynamics compare (5B with 8B
and 10B), (6B with 9B and 11B) and (12B with 15B and 16B). In all three cases the inclusion of the
short rate dynamics in the optimization process greatly reduces the RSS. To conclude modelling with
only the risk adjustment parameters is not a good idea. Conversely, the short term rate performs
fairly well on its own compared to only optimizing the risk adjustment parameters. Next point to
consider is the case where the market price of risk is zero in 4B and 7B (3B is excluded because ¦ is
optimized which is simply a scalar and performs poorly). The short rate does not perform too badly
on its own. Although it does not give rise to the best RSS, it manages to do better on its own
compared to the risk adjustment parameters and can be used alone to discount future payoffs if
necessary. However, it is not recommended that either the delta parameters or the lambda
parameters are used on their own. This becomes apparent in both systems A and B as more
freedom is added to the model parameters the better it performs. The best result is obtained in
number 17A and 17B. This is the case where all parameters are set free. Both method 16 and 15 for
systems A and B are not far behind.
The concluding remarks are that the initial parameters in system A result in better RSS for the
majority of the optimization methods. The only strikingly different case is 8A where is seems that
setting the entries of the ¦ vector equal to zero results in a better residual sum of squares. When
no parameters are optimized as in method 1 the resulting RSS and average error are very large. This
clearly indicates the need for an optimization process and that the starting values for the parameters
are inappropriate. Another special case is method 2 where the future expected cash flows under
real world probabilities are discounted only with the risk free rate know at time . The results
indicate that discounting only with the risk free rate is not sufficient with large inconsistencies
between the observed and estimated prices.
50
A visual representation of the results in Table 2 provides a better overview of which optimization
method produces the smallest residual sum of squares (RSS). The histogram in Figure 5 indicates a
downward trend as more freedom is allowed in the optimization of the parameters. Methods 17,
15, 16, 12 and 13 (based on blue blocks with solid outline) in decreasing order seem to offer the best
optimized parameters for the stochastic discount factor used in the valuation of financial assets.
In general the first 6 methods face the problem of being too restrictive in the parameter
optimization process and as a result is under-fitting. On the other end of the histogram method 17 is
entirely unrestricted. Although this method provides the lowest RSS, the predicament arises of
over-fitting the parameter values. This occurs when all the parameters are set free to be optimized
and a closed form approximation is obtained for the parameter values. It is natural to assume that
somewhere in the middle is an appropriate specification of the parameters. The problem with overfitting a model is that parameters are being optimized such that a specific objective function relating
to a specific state of the economy is minimized. This means that certain factors are taken into
account which is not characteristic of the overall relationship between the stochastic discount factor
and the VAR model parameters. To deal with the problem of over-fitting an out-of-sample test is
conducted.
RSS for each Optimization Method
Residual Sum of Squares
Initial Parameters
All Zeros
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
1
2
3
4
5
6
7
8
9
10
11
Optimization Method
Figure 5: The Residual Sum of Squares for each Optimization Method
51
12
13
14
15
16
17
4.3.3. Out-of-Sample Test
In the previous section attention was given to optimize the parameters of the stochastic discount
factor to minimize the RSS. This procedure gave rise to a set of calibrated prices. However, because
the calibration process was conducted in a closed form approximation, it is normal to be concerned
about the calibrated parameters over-fitting the empirical data in the model. To investigate just
how well the calibrated parameters really perform the prices are again constructed. This time the
calibrated parameters are fixed beforehand and used to determine the stochastic discount factor. In
an effort to manage the problem of over-fitting an out-of-sample test is conducted. The historical
values of 2006 are used as initial input for the state variables in the VAR model. This implies that the
risk free rate as well as the inflation rate known at time changes. Naturally this affects the
observed prices more than it does the estimated prices. This is because in the construction of a
suitable proxy for observed prices a fixed inflation rate is used to compensate for inflation while the
risk adjusted stochastic payoffs are discounted by the fixed risk free rate. On the other hand, the
estimated prices are computed in the VAR model using new starting values. Therefore the out-ofsample test will lead to different prices for both the observed and estimated values. If the calibrated
parameters capture economic wide effects instead of over compensating for specific effects relating
to the closed-form approximation, then the out-of-sample tests results should show similar
outcomes than the original tests.
The results for the out-of-sample test in Table 3 and Table 4 clearly show that the calibrated
parameters performed much worse in the valuation process resulting in much higher RSS’s. This can
be expected since the parameters for the pricing kernel were calibrated in a closed-form
approximation. This means that the same empirical data that was used to estimate the parameters
in the first place, was again valued using the calibrated parameters. Obviously when the initial
inputs change the calibrated parameters fall short of sufficiently minimizing the discrepancies
between observable and estimated prices. Even though the RSS’s are high, it is important to
mention some significant insights gained from conducting the out-of-sample test. By sorting the
residual sum of squares for each optimization method in an ascending order, it can be observed
which optimization process performs well in both the original test and the out-of-sample test.
52
A: INITIAL PARAMETERS
#
17
15
16
12
13
14
10
11
9
6
8
7
5
4
3
2
1
¦
x
x
¦
x
x
x
x
x
x
x
x
x
x
x
x
x
x
rf
Original test ordered
RSS
] ] Ave error
0.031
x
x
0.0002
x
x
0.0002
0.0346
x
x
0.0002
0.0359
x
x
0.0002
0.0406
x
0.0002
0.0416
x
0.0002
0.0436
x
0.0002
0.0491
0.0494
x
0.0002
x
0.0003
0.0546
x
0.0003
0.0554
x
0.0003
0.0614
0.0004
0.0709
x
0.0005
0.1015
0.0007
0.1366
0.0282
5.5787
0.0525
10.3974
5.3084
1051.063
#
17
15
12
16
13
11
6
9
8
10
5
14
7
4
3
2
1
¦
x
x
x
x
x
x
x
x
rf
Out of sample test ordered
¦ ] ] Ave error
x
x
x 0.001064
x
x
0.0011
x
x
0.0011
x
x
x
0.0015
x
x
0.0015
x
x
0.0016
x
0.0016
x
0.0016
x
0.0017
x
x
0.002
x
0.0022
x
x
0.0025
x
0.0052
x
0.0064
0.0257
0.0501
5.1208
RSS
0.2106
0.2131
0.2234
0.2875
0.2884
0.3089
0.3096
0.3165
0.3356
0.4011
0.4407
0.4984
1.0384
1.2631
5.0983
9.9277
1013.914
Table 3: Results from the Out-of-Sample test for system A: Initial Parameters
From Table 3 it can be seen that the 5 “best” optimization methods with the lowest RSS’s for the
original test lead to the 5 “best” optimization methods for the out-of-sample test. Optimization
method 17 with the most freedom in its parameters performs best in both the original and the outof-sample test. This supports the use of a stochastic discount factor model with both the short rate
dynamics and a market price of risk that consist of constant and time-dependent parameters.
Furthermore the 5 “best” optimization methods accommodate for the risk adjustment in the
valuation process. The most striking piece of evidence discovered is the fact that every single one of
the top 5 optimization methods contains the constant ] of the time-dependent market price of risk.
In some sense this demonstrates that the market price of risk can in fact be time-independent as
assumed in the Black-Scholes model (Black & Scholes, 1973). The fact that optimization method 14
(where ] is not optimized) significantly drops in its ability to minimize the residual sum of squares
also illustrates the importance of the long run mean of the risk premium.
The next “best”
optimization methods all contain the time-varying ] of the market price of risk indicating the
importance of the market price of risk. System B in Table 4 where the parameters are initially set to
zero provide similar insights as system A where initial parameters are used.
53
B: ALL ZEROS
#
17
16
15
13
14
10
11
8
9
7
4
12
6
3
5
2
1
¦
x
x
x
x
x
x
x
x
rf
¦
x
x
x
x
x
x
x
x
Original test ordered
Ave error
RSS
] ]
x
0.0331
x
0.0002
x
x
0.0002
0.0349
x
x
0.0002
0.039
0.0415
x
0.0002
x
0.0002
0.048
x
0.0002
0.0491
0.0542
x
0.0003
x
0.0003
0.0588
x
0.0004
0.0815
0.0004
0.0836
0.001
0.1988
x
x
0.005
0.9981
x
0.0154
3.0539
0.0339
6.7184
x
0.0722
14.3021
8.5829
1699.408
1447.991 286702.1
#
17
15
8
16
13
11
9
10
14
12
7
4
6
3
5
2
1
¦
x
x
x
x
x
x
x
x
rf
Out of sample test ordered
RSS
¦ ] ] Ave error
x
x
x
0.0011
0.2084
x
x
0.0011
0.2219
x
0.0013
0.2554
x
x
x
0.0013
0.2596
x
x
0.0014
0.2756
x
x
0.0015
0.2883
x
0.0018
0.3567
x
x
0.002
0.4014
x
x
0.0023
0.4515
x
x
0.0063
1.2421
x
0.0072
1.429
x
0.0094
1.8537
x
0.0215
4.2542
0.0331
6.5465
x
0.0676
13.3872
8.5834
1699.509
1447.407 286586.7
Table 4: Results from the Out-of-Sample test for system B: All Zeros
With the completion of the Out-of-sample tests, the SDF/VAR approach as discussed in Section 4.2 is
modelled by using the calibrated parameters from the optimization method 17 is system A. This is
done in order to determine whether or not the calibrated parameters indeed improve the fit of the
SDF to the data. If so, then there should be a significant improvement from the results obtained in
appendices D and F. This is in fact the case. The results can be found in appendix E and appendix G.
The present values of the each of the three financial assets obtained by implementing the calibrated
parameters in appendix E undeniably provide a better valuation technique.
The differences
between the two valuation methods for each financial asset are considerably decreased.
A
considerable improvement in the present values can be seen in the valuation of the equity returns.
Appendix G provides a graphical representation of the residual sum of squares (RSS) for both the
bonds and the equity returns. Instead of having a systematic increase in the discrepancies between
the two valuation methods, namely risk neutral and pricing kernel method, the remaining variations
can be cause by white noise.
4.4 Application and Results
Finally, now that the calibration process is completed the estimated stochastic discount factor
method can be applied to value a realistic pension product. In general pension funds, like many
other institutions, are confronted with the reality that inflation seriously affects their ability to offer
54
sufficient pension rights. The potential deterioration of the financial position of many pension funds
has lead to the introduction of new international accounting standards. This necessitates the
valuation of financial assets and liabilities at market value (Nijman & Koijen, 2006). Since pension
liabilities are generally to some extent indexed according to inflation, this implies that an investment
strategy that is somewhat related to the liabilities ought to be implemented. This presents both
valuation and risk management with a new set of challenges. In addition solvency requirements
together with a growing demand for more transparency concerning the implicit arrangements in a
pension deal have made pension reforms unavoidable (Hoevenaars R. P., 2008). It is clear that
financial institutions, like pension funds, are faced with the challenge to accommodate the recent
industry developments. This implicates that valuation takes place at market value, while risk
management extends from basic ALM analysis to value-based ALM. Conducting such a value-based
ALM analysis possesses the ability to reveal value transfers between stakeholders. The main issue
now becomes how to determine the fair value of inflation-sensitive pension rights. Before the
valuation process commences three pension products are discussed.
4.4.1. Pension Products
Consider the case where no inflation compensation is granted to inflation-sensitive pension rights.
This implies that the pension privileges of the holder of such a pension product won’t be protected
against inflation risk. Naturally one can expect that the value of this nominal pension product will
not be valued as highly. It is somewhat obvious that it would be preferable to adjust the pension
rights to inflation in order to provide protection against inflation risk. This can be done by providing
pension products with full (price-) indexation. The value of the real pension product with full
indexation can be expected to be higher. Besides these two extreme cases that offer either no- or
full- indexation, recently pension products have also been designed that offer indexation to inflation
only in certain scenarios (Nijman & Koijen, 2006). One such a pension product provides inflation
indexation conditional on the funding ratio of the pension scheme. The funding ratio of a pension
scheme expresses the pension plan’s assets to its liabilities as a fraction. For instance the pension
product can be defined to provide no indexation if the funding ratio is below 100% and full
indexation in scenarios where the funding ratio is above 130%. The interval where the funding ratio
falls between 100% and 130% can be defined such that it contains the option to receive a proportion
of the inflation indexation. This means the pension product with conditional indexation depending
on the scenario also contains an embedded option. These three pension products must now be
valued.
55
4.4.2. Valuation of the Pension Products
The valuation of pension liabilities is truly imperative. Not only does the valuation of pension
liabilities facilitate the process of determining a fair premium, but can also improve risk
management. By identifying how the value of assets and liabilities will fluctuate due to market
shocks, adequate measures can be put in place to protect indexation rights from inflation risks. In
addition market valuation of pension rights increases the transparency of pension schemes and
makes it possible for supervisory authorities to determine the solvency position. Assuming that
absence of arbitrage holds then a pension liability can be valued by drawing from the market value
of financial assets with a similar risk and duration. The market value of nominal pension rights is
rather simple to construct by means of market data for nominal bonds. The market risks to which
the nominal pension obligations are exposed are comparable to the risks in a corresponding
portfolio of nominal bonds. Assuming that longevity risk can be diversified and the maturity
structures are matched, then the market price of nominal pension rights can be constructed (Nijman
& Koijen, 2006). The valuation of inflation-sensitive pension products is more complex. Real
pension products can be valued by using a portfolio of bonds indexed to price inflation. However,
the market for inflation indexed financial instruments are still not well developed causing a
reasonable amount of scepticism regarding the liquidity of the market prices. So unless a sufficiently
liquid market for inflation indexed bonds is available the valuation process will still be somewhat
tricky. Nevertheless, according to Nijman and Koijen (2006) there are a number of traded inflation
indexed bonds for a limited amount of maturities available in certain markets like the UK, US and
Sweden. From the complexity involved in the valuation of the real pension product, one can imagine
that the valuation of the conditional indexation pension product brings about more difficulties.
There aren’t any traded financial products that mimic the payoffs of the conditional indexed pension
product with embedded options. Clearly the value of this pension obligation then cannot be derived
directly from market prices. In attempt to value the conditionally indexed pension obligations the
pricing kernel and the risk neutral approach as pricing models will be implemented.
4.4.3. Practical Application using Pricing Models
In the construction of a modelling framework for the valuation of complex pension products a
horizon of 65 years is chosen. This is motivated by the fact that pension products are in general a
long term obligation.
Moreover the optimization process calibrated the parameters for the
stochastic discount factor pricing method on 65 years and 500 scenarios. A scenario set of 500 time
paths is again applied in the valuation process. It is assumed for simplicity that expected inflation is
equal to current realized inflation. This indicates that the real interest rate equals the nominal
interest rate minus the current realized inflation. Normally the realized and expected inflation
56
would be modelled separately. Furthermore, the real and nominal interest rates are assumed to be
flat and the wage inflation is assumed to be equal to price inflation. For the sake of comparison the
nominal and real pension products are valued at market value. The nominal and real market values
can be seen as the actuarial reserves that use actuarial principles to construct an analytic valuation
method. In addition two pricing models, namely the Hull-White Black-Scholes model and the Pricing
kernel model, are also implemented. The former pricing model is based on risk neutral assumptions
while the latter model is the pricing kernel valuation approach as specified in Section 4.2.2. As a
matter of practically assume that no new entrants join the pension scheme such that the accrued
rights/liabilities do not change.
The nominal and real market values
The market value of a nominal pension product can be derived from market data by taking the value
of a bond with matching cash flows (Nijman & Koijen, 2006). All future cash flows of the nominal
bond based on the current nominal accrued pension rights are discounted with the current nominal
term structure. The future cash flows do not include (expected) inflation. This is also known as the
nominal actuarial reserve. In a similar way, the market data for inflation-indexed bonds can be used
to derive the value of a real pension product. Once again the cash flows of the inflation-indexed
bonds are matched to the current real accrued pension rights. The real value of accrued obligations
is the value of pension rights when full indexation would always be granted. However, there are two
ways to calculate the real actuarial reserve. First, the future expected cash flows that include
expected inflation can be discounted with the nominal term structure. Alternatively, the future
expected cash flows without expected inflation can be discounted with the real term structure. The
second method is favoured. As determined by actuarial principles, the expectations of the future
cash flows can be calculated by using mortality probabilities. Therefore it is not necessary to
simulate the value of actuarial reserves because an analytical expression is available. Unfortunately,
this application does not apply to the condition indexation pension products. This is because there
are no assets with similar characteristics to match the cash flows (Nijman & Koijen, 2006). Pricing
models where simulations are used can be implemented to value such products, such as the risk
neutral Hull-White Black-Scholes model and the Stochastic Discount Factor (SDF) model.
The Hull-White Black-Scholes Model
The first pricing model considered to price the condition inflation-sensitive pension product is the
popular risk neutral 1-factor Hull-White interest rate model (1HW-BS) (Hull, 2005). This model has
been extended with the Black Scholes model with a stochastic interest rate for the stock prices and a
consistent inflation model based on Jarrow and Yildirim (2003). In case of both nominal and real
57
interest rates two Hull-White models are assumed; one for the nominal rate and one for the real
rate. Also, if more than one equity variable is considered, there will be a Black Scholes model for
each equity category. The 1HW-BS model can be calibrated on either market prices or on “real
world scenarios”. The market prices of derivative instruments, like swaptions, are often used for
calibration. In this case all information available at the time that market prices are taken is
absorbed. Alternatively, “real world scenarios” can be generated in a first order VAR model. The
VAR model has been estimated on historical data of price inflation, long term interest rate and
equity returns for the MSCI Europe. In calibrating the 1HW-BS model the same initial (flat) nominal
and real term structures is used as in the SDF pricing model. The calibration of model parameters
has been done on the second method using “real world scenarios” such that the historical
information will result in the volatilities and correlations of the risk neutral scenario set. Once the
parameters have been calibrated the model can be used to generate risk neutral scenarios. The risk
neutral scenario set is calculated such that the average scenario returns on equity are calculated
according to the initial nominal term structure. The average inflation scenarios are obtained by
taking the difference between the initial nominal and real term structures. By using a risk neutral
scenario set the market prices can be calculated by discounting the relevant cash flows by the one
year nominal interest rate. This is equivalent to discounting with the initial nominal term structure.
This scenario set from the 1HW-SB model is model with risk adjusted probabilities. The stochastic
discount factor (SDF) model uses real world probabilities to model the scenario set of state variables.
The Stochastic Discount Factor Method
The second pricing method used to price the inflation-indexed pension product is the pricing kernel
method. The model uses real world probabilities to generate the scenario set in the VAR model for
the inflation rate, the long term nominal interest rate and the stock returns. The structure of the
SDF is defined in equation (44).
The parameters of the SDF have been calibrated using an
optimization technique to form a closed-form approximation. Out-of-sample tests revealed that the
SDF that allow the most freedom in setting the parameters perform the best in terms of minimizing
the residual sum of squares. Therefore, for the purpose of obtaining “comparable” results, the
calibrated parameters obtained from the 5 “best” optimization methods have been chosen to
perform the evaluation for the pricing kernel pricing approach.
4.4.4. Results
Finally the different pricing methods can be applied to realistically value the pension product. The
three different pension products as discussed above are considered.
The first two products
comprise of a nominal pension product with no indexation and a real pension product with
58
unconditional indexation. The third product involves the pricing of a complex pension product with
an embedded option. Consider the case where a pension fund offers the possibility of a conditional
indexation based on the financial position of the fund. If the funding ratio drops below 100% then
no indexation is given, while a funding ratio above 130% enjoys full indexation rights. The pension
fund proposes an option such that indexation takes place proportionally when the funding ratio falls
in the interval 100% to 130%.
Figure 6 provides a graphical representation of the prices determined by each of the different pricing
methods. The nominal and real market values, or equivalently the actuarial reserves, are symbolized
by “Nom MV” and “Real MV”, while “1HW-BS” denotes the risk neutral 1-factor Hull-White BlackScholes pricing model where a risk neutral scenario set is generated. The 5 “best” optimization
methods with the corresponding calibrated parameters are also used to determine the value of each
of the three pension products1. This is symbolized by “SDF” where the number indicates which
optimization method with corresponding calibrated parameters where used. Depending on how
well the calibrated parameters performed in minimizing the residual sum of squares, then this
corresponds to the value in the 5 numbered columns. Naturally 1 indicates the best method and 5
the worst out of the 5 variations of calibrated parameters considered.
Price in € millions
Valuation of a Basic Pension Product with
Embedded Options
500
450
400
350
300
250
200
150
100
50
0
1
2
3
4
5
1HWBS
SDF 17
SDF 15
SDF 16
SDF 12
SDF 13
Real
MV
Unconditional Indexation
423.12
441.38
439.36
439.52
441.72
441.61
446.27
Conditional Indexation
407.93
401.29
404.27
401.11
408.62
397.42
303.14
295.81
295.26
295.56
297.18
294.02
Nom
MV
No Indexation
323.06
Figure 6: Results from the valuation of a Basic Pension Product
1
Only calibrated parameters from system A is applied where the initial parameters as set in Section 4.3.1 has
been used as starting values in the calibration process
59
It is clear from the results in Figure 6 that the valuations of the three pension products are very
similar. Each of the estimated prices (“1HW-BS” and “SDF’s”) for the nominal pension product
where no inflation indexation is granted is very close to the nominal actuarial reserve as represented
by “Nom MV”. The same can be said for the real pension product with full indexation, where the
estimated prices are comparable to the real actuarial reserve, “Real MV”. In both these instances
though the valuations obtained from the actuarial principles are slightly higher than estimated
prices.
Another noticeable difference includes the price of the real pension product with
unconditional indexation. The “1HW-BS” pricing method results in a lower price for the real pension
product compared to the other valuations techniques. This might be explained by the fact that the
two pricing models, the 1HW-BS model and the SDF method, have very different model structures
and by definition will give different valuations. It is best to compare the pricing methods individually
with the corresponding actuarial reserve which act as proxy for the market values. However, it is
debatable whether the prices of the nominal and real market values are in fact liquid. Especially in
the case of real market values where a market for inflation indexed bonds is incomplete, resulting in
problems with illiquidity. This implies that the market values can be used for comparative reasons
but with caution. However, another reason for the apparent differences can be explained by the
fact that a finite number of scenarios where used. Only 500 scenarios where used which is in fact a
small number considering that Monte Carlo estimation techniques require a large amount of
simulations to converge to the true values. This together with the different model structures can be
responsible for white noise in the modelling process.
Call to mind that the conditionally indexed pension product in fact possesses an embedded option.
Inflation is indexed conditionally on the state of the funding ratio in every scenario. If the funding
ratio is between 100% and 130%, then the option to receive proportional inflation indexation can be
“valued”. In Figure 6 the “value” of the option can be obtained by simply subtracting the price of the
conditional indexation pension product with the price of the nominal pension product. In this way
the option of receiving conditional indexation as opposed to no indexation is given a “value”. This
interesting result concludes the application analysis.
5. Conclusion
In an attempt to comply with the recent industry developments an innovative method to asset
pricing is examined. This approach is known as the stochastic discount factor (SDF) method or
equivalently the pricing kernel method. The introduction of new accounting standards requires that
assets and liabilities are valued at market value. This involves challenges both for valuation and risk
60
management. While valuation methods need to put a fair value on assets and liabilities, the
implications for risk management involve the extension from the basic ALM analysis to value-based
ALM. With the use of the SDF approach a way of valuating assets and liabilities in a marketconsistent manner becomes attainable. By using a scenario set generated by a first-order VAR
model, the future cash flows of financial assets relating to the state variables can be constructed.
These future expected cash flows can be discounted by using the SDF.
management by giving rise to a value-based ALM approach.
This facilitates risk
There is an advantageous to
implementing a pricing kernel where the dynamics are obtained by extending the VAR model. The
proposed VAR/SDF approach provides an automatic consistency between on the one hand the
scenarios from the VAR model used for calculating risk and return numbers, and on the other hand,
the valuation results that are obtained by applying the SDF on the same scenario set. The automatic
consistency provides the VAR/SDF approach with the potential to value products at more consistent
values. Consequently this may offer a possible solution to some of the issues relating to financial
fairness. Furthermore, in addition to giving rise to fair prices, the value-based ALM also sheds some
light on generational accounting. This necessarily implicates risk management.
The advantages relating to the value-based ALM analysis motivates why it is important to find a
suitable stochastic discount factor for discounting. Various estimations methods to find the most
appropriate SDF was discussed. The approach followed by Cochrane and Piazzesi ((2002) & (2008)),
Nijman and Koijen (2006) and Hoevenaars (2008) where the VAR model is extended to derive the
dynamics of the SDF was chosen. Once the VAR/SDF method was specified, the aspects relating to
the implementation of the modelling process was dealt with. The parameters of the SDF depend on
the state variables of the VAR model such that a closed-form approximation was implemented to
calibrate the parameters. Three basic financial assets were used in the calibration over a horizon of
65 years with 500 scenarios simulations. The objective function used to estimate the parameters
was set to minimize the residual sum of squares between estimated prices and observed prices.
Observed prices where modelled using risk neutral principles, while the estimated prices were
obtained from the proposed VAR/SDF approach. In the estimation of parameters concerns were
raised over the problem of over-fitting the model. An out-of-sample analysis was done to shed some
light on this problem. Results obtained from the out-of-sample tests indicated that the pricing
kernel with the most freedom in setting parameter values performed best in terms of minimizing the
discrepancies between the observed and estimated prices. Therefore both the short rate dynamics
and the risk adjustment fulfil an important role in determining the deflator. By using the calibrated
parameters with the most freedom (as determined in optimization method 17 system A), the
VAR/SDF approach once more attempted to value the three basic financial assets. Results indicate
61
that the VAR/SDF approach possesses the capability to provide a possible way to value financial
assets at market consistent value.
As a final result, a complex pension product was valued using the proposed VAR/SDF method.
Comparisons with the actuarial reserves and the values obtained from the 1-factor Hull-White BlackScholes (1HW-BS) model provide a general sense of how well the pricing method is performing. The
resulting estimated prices correspond with the comparing prices.
However, certain issues
necessarily indicate that differences between the prices of the different valuation techniques can
naturally be expected. Firstly, the VAR/SDF approach as a valuation method and the 1HW-BS pricing
method are based on different model structures. Secondly, the values calculated by actuarial
principles provoke some scepticism. This is due to the lack of complete markets with financial
derivatives available to mimic the cash flows of an inflation-sensitive product, such as pension
liabilities. These long-standing actuarial reserves valuing pension products at market values are in
fact the cause of concern for financial fairness. If complete markets are available, then the actuarial
reserves at market value provide sufficiently important information.
However, in incomplete
markets for inflation-sensitive products this method is not a favourable one giving rise to illiquidity
problems in prices. It brings along with it doubt over how fair a price really is. In fact the goals
behind the new accounting standards are to increase consistency and transparency in order to
obtain a fair market value. So even though an evaluation of the estimated prices (obtained for the
VAR/SDF approach) with the prices from 1HW-BS model and actuarial reserves present valuable
comparisons, a word of caution is expressed. The illiquidity of market prices for inflation-sensitive
products makes reasonable assessments difficult. A third reason for different prices is of an
operational nature. Implementation of the Monte Carlo method making use of VAR scenarios are in
general a slow method. A certain amount of modelling error can be expected because a relative
small scenario set was considered. By including variance reduction techniques the modelling error
can possibly be improved.
Some problems experienced during the quantitative analysis are mentioned. One such a problem
was caused by the slow convergence rate of the Monte Carlo method. In general it is preferred to
use a much larger scenario set than the one used. As mentioned before variance reduction
techniques can be applied to assist in obtaining a more accurate valuation method. Another
problem encountered included the valuation at different points in time, in particular a 10 year rollover bond. In general it should hold that at any time in the future it should be possible to obtain a
price for such a financial asset by taking the expectation over all future cash flows from that point in
time onwards. However, valuation only took place at current time, = 0. Future research can focus
62
on finding appropriate ways to use the VAR/SDF valuation method to value at any time in the
future. Further difficulties were experienced in setting up an appropriate optimization process to
calibrate the parameters of the SDF. The calibration process is very time-consuming and sensitive to
changes in the horizon while only 3 assets were used over a 65 year horizon with a small scenario
set. This implies that the inclusion of more assets and liabilities over changing horizons will
necessitate an extensive and prolonged analysis. For future research, an analysis can be performed
to determine the exact relationship of the SDF parameters to the VAR state variables. Possibly if
some of the parameters are set equal to 0, then the speed of the calibration process can be
improved. By identifying economic wide characteristics present in the data can facilitate the process
to determine which SDF parameters are vital and which are insignificant. Lastly, it is recommended
to perform such a valuation analysis using the proposed VAR/SDF approach without assuming a flat
curve for nominal and real interest rates. Since a realistic world uses changing term structures, the
inclusion of non-flat term structure can have significant influences on the performance of the model.
In conclusion the proposed VAR/SDF approach performed surprisingly well in providing a fair price to
financial assets. Even though the intermediate steps are very sensitive to changes, the final price
valuations are somewhat less sensitive to using a different model structure. The added advantage of
automatic consistency between the VAR scenario set and the dynamics of the SDF make the
approach even more attractive. The main contribution of the VAR/SDF approach is that it provides a
consistent method to value inflation-sensitive products with embedded options. These prices are
potentially fair and provide greater transparency in the implicitly valuations of pension schemes.
63
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66
Appendices
A. Proof of the Excess Return in equation (12)
86 =
1 − 2LM 56, 86 56 (10)
1 = 56 86 = U56 8 V
N
(11)
The excess return follows algebraically by substituting equation (11) into (10) and substituting
56 = 1⁄8 :
N
86 =
U568 V − 2LM 56, 86 56 N
By expanding equation (11)
U56 8 V = 56 U8 V + 2LM U56, 8 V
N
Then
N
N
56 U8 V + 2LM U56, 8 V − 2LM 56 , 86 86 =
56 N
N
Term by term division on the right hand side result in:
86 = U8 V +
N
2LM U56, 8 V − 2LM 56, 86 56
N
Now rewriting the above equation by gathering the expectation terms on the left, substituting for
56 with 1⁄8 and rearranging the covariance terms leads to the following:
N
U86 − 8 V = −8 Ã2LM 56 , 86 − 2LM U56 , 8 VÄ
N
N
U86 − 8 V = −8 2LM W56 , U86 − 8 VX
N
N
N
N
∎
B. Deriving the Radon-Nikodym process
Use Girsanov’s theorem to introduce a continuous process  by  = 1 and € = − ] €‚ . ]
represents the market price of risk and is a process adapted to the standard Brownian motion ‚ .
The partial differential equation contains no drift term and assures that the Radon-Nikodym process
remains positive. Now, setting  = log  and using the famous Itô rule:
ۮ =
Then
1 Ç Æ
ÇÆ
€ +
€g, h
2 Ç- Ç-
€ log  =
€ log  =
1
1
€ −
€g, h

2 (57)
1
1
− ] €‚ −
− ] €

2 1
€ log  = − ]' ] € − ] €‚ .
2
Now taking the integral on both sides leads to
6&
È
6&
€ log & = È
1
ƒ− ]'& ]& €Q − ]& €‚& …
2
Because the process ] is time-dependent it cannot be taken outside of the integral.
To
approximate this equation, use the Euler approximation scheme where the time-step is discretized
such that + Q = + ∆. The time step is assumed to be infinitely small and converging to 0.
6∆
È
6∆
1 6∆ '
€ log & = − È
]& ]& €Q − È
]& €‚&
2 Then the exact approximation is
1
log 6∆ = log  + C− ]' ] D ∆ + −] ∆‚ + L∆
2
where L∆ measures the error and converges to 0 as the time step converges to 0. Assume the
measuring error is 0 and the time step is 1. Taking the exponential of the equation gives
where ∆‚ is denoted by .
1
6 =  c-d C− ]' ] −] D
2
∎
C. Deriving the Pricing Kernel process
Assume the continuous process 5 by 5 = 1 and
€5 = −5 Y € + ]' €‚ Again using the famous Itô rule in (57) and the partial differential equation in (37), the dynamics of
the pricing kernel can then be computed. Set Æ = log 5 , such that
ۮ =
€ log 5 =
6&
È
1
1
€5 −
€g5, 5h
5
25
1
1
−5 ]' €
U−5 Y € + ]' €‚ V −
5
25 6&
€ log 5 = È
1
ƒC−Y − ]' ] D € − ]' €‚ …
2
Assume the time step is discretized and infinitely small such that + Q = + ∆. Assume the
approximation error is zero. The Euler Approximation scheme is used to approximate an exact
expression that links 56∆ with 5
6∆
log 56∆ = log 5 È
1
ƒC−Y& − ]'& ]& D €Q − ]'& €‚& …
2
Then taking the exponential leads to
1
56∆ = 5 c-d ƒC−Y − ]' ] D ∆ − ]' ∆‚ …
2
where ∆‚ is denoted by and the time step is 1.
1
56 = 5 c-d ƒ−Y − ]' ] − ]' …
2
∎
D. A Comparison between the Present Values using
E. A Comparison between the Present Values using
Initial Parameters
Calibrated Parameters
Present Value of a Zero Coupon Bond
Present Value of a Zero Coupon Bond
1
1
Observed Prices
Estimated Prices
0.9
0.9
0.8
0.8
Price in €
Price in €
Observed Prices
Estimated Prices
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0
2
4
6
8
10
12
14
16
18
0.4
20
0
2
4
6
8
16
18
20
Observed Prices
Estimated Prices
0.95
0.9
0.9
0.85
0.85
Price in €
Price in €
14
1
Observed Prices
Estimated Prices
0.95
0.8
0.8
0.75
0.75
0.7
0.7
0.65
0.65
2
4
6
8
10
12
14
16
0
18
2
4
6
8
10
12
14
16
14
16
18
Maturity in years
Maturity in years
Present Value of Equity Returns
Present Value of Equity Returns
2
12
Present Value of a Inflation Linked Zero Coupon Bond
Present Value of a Inflation Linked Zero Coupon Bond
1
0
10
Maturity in years
Maturity in years
1.06
Observed Prices
Estimated Prices
1.04
1.8
Price in €
Price in €
1.02
1.6
1.4
1
0.98
0.96
1.2
0.94
0.92
Observed Prices
Estimated Prices
1
0
2
4
6
8
10
12
14
16
18
20
Maturity in years
Figure 7: Present Values for each basic financial asset using Initial Parameters
0.9
0
2
4
6
8
10
12
18
20
Maturity in years
Figure 8: Present Values for each basic financial asset using the Calibrated
Parameters
F. The Residual Sum of Squares using Initial Parameters
G. The Residual Sum of Squares using Calibrated
Parameters
Residual Sum of Squares for Bonds
-3
x 10
1.5
Zero Coupon Bond
Inflation Linked Zero Coupon Bond
Difference measured in terms of Prices
Difference measured in terms of Prices
3.5
3
2.5
2
1.5
1
0.5
0
0
2
4
6
8
10
12
14
16
18
x 10
Residual Sum of Squares for Bonds
-3
Zero Coupon Bond
Inflation Linked Zero Coupon Bond
1
0.5
0
20
0
2
4
6
8
Maturity in years
Residual Sum of Squares for Equity Returns
x 10
1
Difference measured in terms of Prices
Difference measured in terms of Prices
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
18
20
4
6
8
10
12
14
16
18
16
18
20
16
18
20
1
0
20
0
2
4
6
8
10
12
14
Maturity in years
Comparing the Residual Sum of Squares for all 3 assets
1.5
Zero Coupon Bond
Inflation Linked Zero Coupon Bond
Equity Returns
Difference measured in terms of Prices
Difference measured in terms of Prices
16
2
Maturity in years
1
14
Equity Returns
0.9
2
12
Residual Sum of Squares for Equity Returns
-4
Equity Returns
0
0
10
Maturity in years
0.8
0.6
0.4
0.2
x 10
Comparing the Residual Sum of Squares for all 3 assets
-3
Zero Coupon Bond
Inflation Linked Zero Coupon Bond
Equity Returns
1
0.5
0
0
2
4
6
8
10
12
14
16
18
Maturity in years
Figure 9: Residual Sum of Squares for basic financial assets using Initial
Parameters
20
0
0
2
4
6
8
10
12
14
Maturity in years
Figure 10: Residual Sum of Squares for basic financial assets using Calibrated
Parameters
Notes and Remarks
72