University of Amsterdam
MSc Mathematics
Master Thesis
Dynamic Risk Budgeting in
Investment Strategies: The Impact
of Using Expected Shortfall
Instead of Value at Risk
Supervisors:
Bert Kramer
Peter Spreij
Author:
Wout Aarts
5870151
January 10, 2016
Acknowledgements
Writing this Masters thesis in Mathematics has been a challenging endeavor
and one which could not have been finished without the incredible support I
received from numerous people throughout the process. First of all, I would
like to express my gratitude to Ortec Finance for giving me the opportunity to write my thesis for the company. More specifically, I would like to
give a special thanks to Bert Kramer from Ortec Finance for providing me
with direction in my research and ensuring that it would be useful for one
of Ortec Finance’s clients. Martin van der Schans’ guidance in teaching
me programming in Python and his in-depth explanations of how risk management theory and mathematics intersect, has been invaluable throughout
this research. From the UvA I would like to extend my gratitude to Peter
Spreij for the assistance in understanding complex mathematical theory, a
challenging yet crucial part for graduating from a Mathematics master’s.
Further, I wish to acknowledge the support provided by Aqsa Hussain, particularly in the final stages of my research, and for reading my thesis as a
second viewer before submission. Finally, I am very grateful to all the colleagues at Ortec Finance; for all the help and patience, explanations about
basic theory about risk management and the informal chats.
1
Dynamic Risk Budgeting in Investment Strategies: The Impact
of Using Expected Shortfall Instead of Value at Risk
Wout Aarts
Abstract
In this thesis we formalize an investment strategy that uses dynamic risk
budgeting for insurance companies. The dynamic component of the investment strategy includes the property that the amount of risk taken by an
insurance company depends on its financial position. This investment strategy is applied to examine the effect of using Expected Shortfall (ES) instead
of Value at Risk (VaR) for the risk budgeting component in the investment
strategy. This effect is measured by observing return-risk characteristics of
1000 economic scenarios over a five year time horizon. Further, a detailed
proof of the theorem that ES is a coherent risk measure is provided. We
compare ES-optimal portfolios with VaR-optimal portfolios by looking at
the return-risk ratio, which is the realized return per unit of risk. The first
observation is that the available risk budget is smaller when ES is used
instead of VaR to determine shocks. This results in more weight being allocated to asset classes which are more risk averse, affecting both the realized
return and the risk that the insurer takes. We find that, when using the
formalized risk budgeting investment strategy, the return-risk characteristics of ES-optimal portfolios are preferable as ES also takes tail risk into
account.
Keywords: Expected Shortfall, Value at Risk, Coherent Risk Measure,
Solvency II, Investment Strategy, Risk Budgeting
2
Contents
1 Introduction
4
2 Background
2.1 Asset Liability Management . . . . . . . . . . . . . . . . . . .
2.2 Solvency II Assumptions . . . . . . . . . . . . . . . . . . . . .
6
6
7
3 Properties of the Gaussian Distribution
3.1 Linear Correlation . . . . . . . . . . . .
3.2 Value at Risk . . . . . . . . . . . . . . .
3.3 Aggregation of VaR . . . . . . . . . . .
3.4 Quantile functions . . . . . . . . . . . .
3.5 Log-normal Return Rate . . . . . . . . .
4 Dynamic Risk Budgeting Investment
4.1 General Case . . . . . . . . . . . . .
4.2 Stylized Case . . . . . . . . . . . . .
4.3 Scenario Analysis . . . . . . . . . . .
5 Risk Measures
5.1 Coherent Risk Measures . . . . .
5.2 Value at Risk . . . . . . . . . . .
5.3 Expected Shortfall . . . . . . . .
5.4 Coherence of Expected Shortfall
5.5 Stylized Case . . . . . . . . . . .
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Strategy
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6 Conclusion
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6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6.2 Concluding Statements . . . . . . . . . . . . . . . . . . . . . . 39
6.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
7 Appendix
42
7.1 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . 42
7.2 Python Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . 48
7.3 Miscellaneous Theorems and Proofs . . . . . . . . . . . . . . 55
3
1
Introduction
In the current financial climate, insurance companies are finding it increasingly challenging to be financially sound. This is due to low interest rates,
declining business and the introduction of the Solvency II Directive - a new
prudential regulatory framework which will come into force on 1 January
2016 [13]. One of the objectives of Solvency II is to prevent insurance companies from becoming insolvent, which happens when the available capital
is insufficient to cover the risks of the liabilities. The liabilities are future
payments for the insured.
In this new regime, insurers will have to establish technical provisions
to cover expected future claims from policyholders. In addition to this,
insurers must have sufficient available resources to cover a Solvency Capital
Requirement (SCR) which is the economic capital needed as a risk budget
for the investments and liabilities. The SCR is established using the risk
measure Value at Risk [6].
Value at Risk with confidence level α (VaRα ) is defined as follows: for
a given risk class (asset or liability), time horizon, and confidence level α,
the VaRα is a threshold loss value, such that the probability that the loss
on the risk class over the given time horizon does not exceed this value is α.
The SCR in Solvency II is based on a Value-at-Risk measure calibrated to
a 99.5% confidence level over a 1-year time horizon. The SCR supposedly
covers all risks that an insurer faces (e.g. insurance, market, credit and
operational risk) and will take full account of any risk mitigation techniques
applied by the insurer (e.g. reinsurance and securitisation) [6].
When VaRα is used as risk measure for a certain asset class, an assumption can be made about the shock in return rate. In the context of this
research, shock is referred to as the maximum assumed loss. In Solvency
II, the shock is used as the risk charge - defined as the amount of money
invested in a certain asset class multiplied by the risk charge gives the SCR.
Several of the assumptions the quantitative requirements in Solvency
II are based on are often contested. These assumptions have a significant
impact on the actual aggregated risk [8]. Therefore making it important for
insurance companies to review their current policies and look for alternative
ways for estimating risk. For instance, the use of risk measure VaRα is
criticized for not being a coherent risk measure. One implication of working
with a non-coherent risk measure is that the benefits of diversification on
the overall risk of the portfolio are not reflected properly. This is because
VaRα does not hold information outside of the confidence level, while some
risk classes might have heavy-tailed distributions.
Further, the correlation in the tail (outside of the confidence interval) is
much higher than within the confidence level. This means that in a worst
case scenario, there is a higher probability of losses of different asset classes
exceeding the confidence level altogether.
4
An alternative risk measure is Expected Shortfall with confidence level α
(ESα ), which is the expected loss of a risk class in the (1 − α)-fraction worst
cases. Thus, the risk measure ESα represents the expected loss conditional
on the loss exceeding VaRα . Since ESα is in fact a coherent risk measure,
many risk managers prefer ESα over VaRα [10]. In this thesis, a detailed
proof of the coherence of ESα is outlined. This proof is based on some
fundamental results from probability theory.
Several insurance companies prefer a dynamic investment strategy where
the weight allocation per asset class changes over time. This could mean
the amount of risk an insurer takes depends on its financial position. In this
thesis, the insurers financial position is represented by a value similar to the
Solvency ratio, which will be defined later on. An alternative method of
influencing the dynamic strategy is by determining the risk appetite. The
risk appetite will be described as an insurers expected solvency ratio after
a maximum shock.
The aim of this research is twofold. Firstly, an optimal dynamic investment strategy using risk budgets is formalized. Therefore, we have to
check which part of the strategy should be dynamic, what external factors
influence the strategy and how the investor’s risk appetite can be taken into
account. This will be summarized into an optimization problem.
Secondly, the impact of changing the risk measure from VaRα to ESα
will be investigated. In this comparison, the focus will be on changing the
risk measure on the assets side of the insurers balance sheet. We will use
the investment strategy defined in the first part to determine the impact of
using ESα instead of VaRα , by assessing the return-risk characteristics of
both. Subsequently, differences between the VaRα -optimal portfolios and
ESα -optimal portfolios will be highlighted by analyzing the tail risks of the
realized return and the corresponding measured risk based on ESα .
Although similar research comparing VaRα with ESα has been conducted, this research also involves the additional element of a dynamic risk
budgeting strategy. Dynamic risk budgeting strategies are becoming increasingly popular for insurers to implement into their investment policies.
Thus, in this thesis, we will evaluate the potential for a Dutch insurance
company to implement this into their policy.
5
2
Background
This section provides a context for the new Solvency II regulation as well as
a general background on ALM. We discuss the Economic Scenario Generator
and its role in measuring risk, and subsequently evaluate several mathematical assumptions Solvency II is based on.
2.1
Asset Liability Management
Solvency II is a new supervisory framework that will come into force starting
2016 for all insurance companies in Europe. This regulation was initiated
by the European Insurance and Occupational Pension Authority (EIOPA)
and has the primary objective to reduce the risks for insurers to become
insolvent [3]. Insolvency is a state of being unable to cover liabilities which
are expected future cash flows, given the insurer’s current economic capital.
The economic capital that the insurer uses to pay their customers is invested
in asset classes like stock, real-estate, sovereign bonds and credits. The
liabilities, as well as the economic capital, are prospective, thus estimated
values. For this reason an accurate estimation of what these values will be
and how they will evolve in the future is necessary [3].
To obtain insight into the future development of assets and liabilities,
often an Economic Scenario Generator (ESG) is used. An ESG uses Monte
Carlo simulation and several stochastic models to generate future scenarios
[16]. Most insurance companies do not possess an ESG, therefore outsource
their Asset Liability Management (ALM) studies to companies like Ortec
Finance, who are specialized in this kind of research.
A typical ALM study performed by Ortec Finance is as follows: an
insurance company delivers data concerning their liabilities, expectations with respect to future new business -, and their current investment strategy.
Then, Ortec Finance will model this in a way, such that the ALM model is
able to provide results pertaining to the solvency and portfolio value over
the next 5 to 60 years (5 year is more relevant for insurers, whereas 60
years is more relevant for pension funds). The ALM model is able to show
important information concerning a great amount of possible scenarios (in
general, for insurers the number of scenarios is 1000 or 2000).
Using such a great number of scenarios results in relevant statistical
values, including the 0.5th percentile which the Solvency II Directive uses
for risk measurement as well. The 0.5th percentile of the loss distribution is
a risk measure which is used for solvency II and will hereafter be referred
to as VaR. The risk measure VaR is probably the most widely used risk
measure in risk management [10].
6
2.2
Solvency II Assumptions
The Solvency II Directive is a new supervisory framework initiated by EIOPA,
which puts demands on the required economic capital, risk management and
reporting standards of insurance companies [3]. The Solvency II Directive
consists of three pillars. The first pillar describes the quantitative requirements every insurer should meet, covering both assets as liabilities. The
second pillar describes the standards of risk management and governance
and the third pillar concerns the transparency for supervisors and public
reporting standards [5]. In this thesis, the term Solvency II will only refer
to the first pillar.
In general, an insurance company has to deal with a great amount of
uncertainties. When assessing liabilities, a proper estimation is important in
order to gain insight into how much technical provision is needed for covering
payments to its clients. On the asset side, interest rates, credit spreads,
equity movements, real estate prices and exchange rates are considered great
risk factors. This thesis will focus on the risk factors on the asset side.
As mentioned in the introduction, the buffer that an insurer needs to hold
according to Solvency II is called the SCR, and is based on the VaR with
confidence level 99, 5%, or VaR0,995 . The SCR is established by assuming
the VaR0,995 -value to be the maximum shock of an asset class, and using
this as the risk charge.
For example, suppose the maximum shock of the asset class ’stock’ in
Solvency II is calculated to be 40%. In the worst case, this assumes that
this asset class can drop 40% in value within one year. If an insurer has
allocated 1.5 million euro in stock, we have: SCRstock = 1.5 · 0.40 = 0.60
million euro. In this case, the risk charge on investments in stock is 40%.
For all asset classes, EIOPA has established the correlation matrix P
for the corresponding return rates. This correlation is used to determine
SCRmarket which is the required capital to cover the market risk. In Solvency
II, the aggregation of risk factors is calculated as if the risk factors are
normally distributed. This means that if for asset class i, the VaR0.995 value si and an insurer invests xi euro in this asset class, we have
sX X
sX X
SCRmarket =
ρij si xi sj xj =
ρij SCRi SCRj ,
j
i
j
i
where ρij is the correlation between asset class i and j which represent asset
classes like stock, real-estate, credit spread and sovereign bonds.
When SCRmarket is determined, it can be used to calculate SCRtotal
which is the total capital charge to cover the insurer’s risk including liabilities. SCRtotal is calculated by the same aggregation formula as SCRmarket .
For calculating SCRtotal , instead of summing over the asset classes, the sum
is over all risk classes (i.e. market, life, non-life, operational, default). For
7
the risk aggregation, EIOPA has established correlation matrix Q = (qkl )k,l .
This results in
sX X
SCRtotal =
qkl SCRk SCRl ,
l
k
where k, l represent the risk classes.
In the Solvency II Directive the term SCRintangibles is also included, presenting the capital requirement for intangible asset risk. This aspect will
not be analyzed in this thesis. The determination of the SCR of all risk
factors can be found in the supplement to the Solvency II Directive of 2015
[14].
As aforementioned, the risk aggregation formula is based on multivariate
normal distributions. In reality, the random variables an insurance company
copes with are heavy-tailed and have a much higher dependency in the tail,
as opposed to the multivariate normal distribution. If one asset class has a
high tail dependency with another asset class, and it drops in value more
than the assumed maximum shock, then the correlation in this last 0.5th
percentile is increased enormously. As a result, in a scenario where asset
values drop outside of the 99, 5% confidence level, the highly correlated tail
events have a much higher probability of occurring at the same time, than
is assumed in Solvency II.
8
3
Properties of the Gaussian Distribution
Since Solvency II uses a formula based on a multivariate Gaussian distribution for aggregating risk and taking dependency relations into account,
in this chapter we summarize the relevant properties of the Gaussian distribution. Firstly, dependency relations and linear correlation are defined.
Secondly, upon defining risk measure VaR, we see how risk based on VaR
can be aggregated when a Gaussian distribution is assumed and how this
leads to the aggregation formula used to calculate SCRtotal . Thirdly, we see
how VaR is a quantile when a Gaussian distribution is assumed. Lastly,
a return formula based on log-normal return rates is outlined, which will
be revisited in chapter 4 where the optimal risk budgeting strategy will be
defined. In this section, the primary focus is on asset risk.
3.1
Linear Correlation
Using the linear correlation coefficient is a very rudimentary, but also simple
way of describing risk dependencies in a single number. Let the covariance
of random variable X1 and X2 be given by:
Cov(X1 , X2 ) = E (X1 − E[X1 ])(X2 − E[X2 ]) .
For the variance, we write:
V(X1 ) = E (X1 − E[X1 ])2 = Cov(X1 , X1 ).
The linear correlation coefficient ρ is defined as follows [11]:
Cov(X1 , X2 )
ρ(X1 , X2 ) = p
,
V(X1 ) × V(X2 )
where X1 , X2 are non-degenerate random variables.
Let X and Y be multivariate random variables with finite variance. In
case of multiple dimensions the so-called correlation matrix needs to be
applied. This is a symmetric and positive semi-definite matrix given by


ρ(X1 , Y1 ) · · · ρ(X1 , Yn )


..
..
..
ρ(X, Y ) = 
.
.
.
.
ρ(Xn , Y1 ) · · · ρ(Xn , Yn )
This linear correlation coefficient is also known as Pearson’s linear correlation. It measures only linear stochastic dependency of two random variables and it holds that −1 ≤ ρ(X, Y ) ≤ 1.
9
3.2
Value at Risk
Consider the probability space (Ω, F, P) on which the following (random)
variables are defined:
Let At be P
the total asset value
Pn at time t for a portfolio of n asset classes.
n
w
A
=
Define At =
i=1 Xt,i , so Xt,i = wt,i At where Xt,i is the
i=1 t,i t
amount ofPmoney invested in asset class i at time t and for the weights wt,i
we have ni=1 wt,i = 1. Let Xt = (Xt,1 , ..., Xt,n )> represent the portfolio
at time t. In some cases it is convenient to look at the loss distribution
L = −X. This depends on the context and will be stated when necessary.
Lastly, for asset class i we have return rt,i with mean E[rt,i ] = µi .
In this thesis, the investment strategy is defined by the way wt = (wt,1 , ..., wt,n )
is rebalanced in every step t → t + δt. The rebalancing happens annually,
thus the step t → t + 1 represents a one year interval.
The measure VaR is based on the maximum possible loss of an investment, which is derived from statistics of historical values. The maximum
possible loss can be defined as inf{l ∈ R : FL (l) = 1} [10].
However, in most models, the support of FL is unbounded which leads
to the maximum loss being infinity. That is why a quantile of the loss
distribution is taken, and we get the VaR. We extend “maximum loss” to
“maximum loss which is not exceeded with a given high probability”, the
so-called confidence level, and get the following definition [10]:
Definition 3.1. Let α ∈ (0, 1). The VaR of a portfolio at the confidence
level α is given by the smallest number l such that the probability that the
loss L exceeds l is no larger than (1 − α). Formally,
VaRα (L) = inf{l ∈ R : P(L > l) ≤ 1 − α} = inf{l ∈ R : FL (l) ≥ α}.
In Solvency II, the confidence interval of α = 0.995 is used. Sometimes,
VaR is defined on the basis of a return X instead of loss. In that case we
have VaRα (X) = inf{x ∈ R : P(X < x) ≥ 1 − α} which holds an equivalent
definition. In this thesis, both definitions will be used depending on the
context. In some cases it is more convenient to have the worst case in the
left tail and in some cases in the right tail. In actuarial science it is common
to use the VaR definition based on expected loss L. Further, throughout
this thesis, when no α is given, one can assume α = 0.995.
The drawback of VaR0.995 is that it does not contain information about
the values outside the confidence interval. For this reason, VaR is not a
coherent risk measure. This will be discussed in more detail in the chapter
5.
3.3
Aggregation of VaR
The matrix of linear correlations as described in section 3.1, can be used to
characterize the level of interdependence of marginal losses. For an investor
10
it is important, that when the non-diagonal elements are low, the level of
diversification that can be realized is high, with incremental exposure to a
risk component [8].
Let Y1 , ..., Yn random variables withPstandard deviation σi for i = 1, ..., n
and correlation matrix P. When Y = ni=1 Yi we have
σY2 = (σ1 , ...σn )P(σ1 , ...σn )>
where σY2 denotes the variance of Y [8]. Now assume a portfolio
X is a
P
multivariate normally distributed random variable. For A = i Xi we have
v
uX
n
u n X
VaR(A) = t
ρi,j VaR(Xi )VaR(Xj ),
i=1 j=1
Note that the random variable Xi in this definition has already taken the
weight in asset class i into account. Thus, VaR(Xi ) is the amount of money
which an investor can possibly loose after a maximum assumed shock si (on
the given confidence interval). Since the shock si defines the risk charge, we
have VaR(Xi ) = asi wi = SCRi where a is the initial total asset value of the
portfolio. Thus, when we write w s for the component-wise multiplication
of the vectors w and s in matrix form we have
q
VaR(A) = SCRmarket = a (w s)> P(w s),
where P is the correlation matrix of the asset classes.
3.4
Quantile functions
For the next propositions, theory regarding quantile functions will be highlighted. The cumulative distribution function of the standard normal distribution is given by
Z x
1 2
1
e− 2 t dt.
Φ(x) = √
2π −∞
In words, this describes the probability that a standard normally distributed random variable X will be found to have a value less than or equal
to x. Notation: when we have µ 6= 0 or σ 6= 1 we denote the cdf by Φ(x, µ, σ).
Definition 3.2. (generalized inverse and quantile function)
1. Given some increasing function T : R → R, the generalized inverse of
T is defined by T ← (y) := inf{x ∈ R : T (x) ≥ y}, where the infimum
of an empty set is defined to be infinity.
11
2. Given some distribution function F , the generalized inverse F ← is
called the quantile function of F . For α ∈ (0, 1) the α-quantile of F is
given by
qα (F ) := F ← (α) = inf{x ∈ R : F (x) ≥ α}
If F is increasing we have qα (F ) = F −1 (α), which holds for F = Φ.
Proposition 1. Assume the loss distribution FL is normal with mean µ
and variance σ 2 . For α ∈ (0, 1) we can write
VaRα (L) = µ + σΦ−1 (α),
where Φ−1 (α) is the α-quantile of Φ.
Proof. By definition of generalized inverse and right continuity of FL we
know that a point x0 ∈ R is the α-quantile of FL if and only if the following
two conditions are satisfied: FL (x0 ) ≥ α; FL (x) < α for all x < x0 [10].
Since FL is strictly increasing, we have to show that FL (VaRα ) = α. We
have
L − µ
FL (VaRα ) = P(L ≤ VaRα ) = P
≤ Φ−1 (α) = Φ(Φ−1 (α)) = α.
σ
This implies that the quantile function qa (FL ) has the property the random variable L will exceed µ + σΦ−1 (α) with probability 1 − α, and will lie
outside of the interval µ ± σΦ−1 (α) with probability 2(1 − α).
Proposition 2. Suppose we have random variables Y1 , ..., Yn with Yi ∼
N (0,P
σi2 ), covariance matrix Σ, weight vector w = (w1 , ..., wn )> and define
Y = i wi Yi . We have
√
VaRα (Y ) = Φ−1 (α; 0; w> Σw)
Proof. By proposition 1 we have VaR(Yi ) = Φ−1 (α)σi .
v
uX
n
u n X
t
VaR(Y ) =
ρi,j wi wj VaR(Yi )VaR(Yj ),
i=1 j=1
v
uX
n
u n X
t
=
ρi,j wi wj σi σj Φ−1 (α)Φ−1 (α),
i=1 j=1
v
uX
n
u n X
−1
t
= Φ (α)
ρi,j wi wj σi σj ,
√
i=1 j=1
= Φ−1 (α) w> Σw,
√
= Φ−1 (α; 0, w> Σw).
12
This
√ proves in the case of µ = 0, VaR(Y ) is a is a percentile of the mix
N (0, w> Σw).
This shows that when µ = 0, the VaR of a multivariate normally distributed random variable is a quantile of the normal√distribution with the
mean equal to zero and standard deviation equal to w> Σw.
3.5
Log-normal Return Rate
In this subsection, we define a formula for the geometric return of a portfolio,
based on “Arithmetic and Geometric Mean Rates of Return in Discrete
Time” by Arie ten Cate, 2009 [4].
Let At be the total value of a portfolio at time t. Let rt be the return
on the portfolio using the current investment mix w (which in this subsection will not variate over time). Assume the sequence of return rates rt is
independently distributed. Given that w is constant, by definition we have
At = (1 + rt )At−1 .
We assume P(rt < −1) = 0 and E[rt ] = µ for all t. At time t = T , by
independence of rs and As−1 for all s we have:
E[AT ] = E[(1 + rT )AT −1 ] = E[(1 + rT )]E[AT −1 ] = (1 + µ)E[AT −1 ].
Repeating this down to t = 0 we get
E[AT ] = (1 + µ)T A0 ,
and
h A i1/T
T
E
− 1 = µ,
A0
The arithmetic mean is
µ̂ar
T
T
1X
1 X At
=
rt =
−1 ,
T
T
At−1
t=1
t=1
which implies
E[µ̂ar ] = µ.
Let 1 + µ̂g be the geometric mean of the 1 + ri , we get
1 + µ̂g =
T
Y
1 + rt
t=1
1/T
=
A 1/T
T
A0
.
One can show µ̂g ≤ µ̂ar . Now assume that all rt are IID and assume that
1 + rt is log-normal, we get
log(1 + rt ) ∼ N (µ0 , σ 2 ).
13
It follows that P(rt ≤ −1) = 0, since the logarithm is undefined at zero. We
get
log(AT /A0 ) = log
T
Y
t=1
1+rt =
T
X
log(1+rt ) ∼
t=1
T
X
N (µ0 , σ 2 ) ∼ N (T µ0 , T σ 2 ),
t=1
which leads to
log(1+µ̂g ) = log(AT /A0 )1/T = (1/T ) log(AT /A0 ) ∼ (1/T )N (T µ0 , T σ 2 ) ∼ N (µ0 ,
It follows from the formulas for the expectation and variance of a log
normal variable that
1
0
2
E[1 + rt ] = eµ + 2 σ ,
1
0
2
E[1 + µ̂g ] = eµ + 2T σ ,
0
2
2
V(1 + rt ) = (eσ − 1)e2µ +σ .
We have
1
0
2
E[1 + µ̂g ] = eµ + 2T σ ,
0
1
= eµ − 2 σ
2 + 1 σ2 + 1
2
2T
= (1 + µ)e
σ2
1
− 12 σ 2 + 2T
,
σ2
.
Thus, for the geometric return, we define
1
E[µ̂g ] = (1 + µ)e− 2 σ
2+ 1
2T
σ2
− 1.
In the next section this formula will be used.
PExpected return µ will be
the arithmetic mean of total asset value µ = i wi µi with corresponding
marginal weights wi and marginal expected returns µi . The standard de2 = (w σ)> P(w σ), where σ is the vector of
viation will be given by σw
marginal standard deviations and w σ component-wise multiplication of
w and σ .
14
1 2
σ )
T
4
Dynamic Risk Budgeting Investment Strategy
In this section we check how to allocate weights in a portfolio given a certain risk budget, when the objective is to maximize the total return of the
portfolio. This allocation depends on the insurers financial state and risk
appetite. One of the properties in this investment strategy is when the insurer has a higher Solvency ratio, they should be able to take more risk. We
will use the SCR formulas as defined in the Solvency II Directive.
Firstly, the general case will be introduced. Secondly, a four dimensional
case with realistic numbers will be formalized, which will be used to analyze
the dynamic risk budgeting strategy. Throughout this thesis, this four dimensional case will be used to examine results about the effect of different
risk measures using the ESG of Ortec Finance.
4.1
General Case
Assume we have n assets classes. The objective is to find optimal allocation
of w when the following parameters are known:
• Initial total assets a
• Initial equity e
• Expected returns µ = (µ1 , ..., µn )> of the asset classes 1, ..., n
• Standard deviations σ = (σ1 , ..., σn )> of the asset classes 1, ..., n
• Shocks s = (s1 , ..., sn )> as a percentage of the initial allocation (in
Solvency II the shock of asset class i is determined by VaR(ri ) and
this also defines the risk charge for asset class i)
• Correlation matrix P, which is an n × n matrix n asset class return
rates
• Risk appetite parameter γ (which will be defined later on)
We want to find the allocation w that maximizes the expected return
R of the investments. To formalize an optimization problem, we require a
formula that represents the expected return. Using the arithmetic return
wµ̇, which is based solely on the marginal expected returns will directly
result in all weight being in the asset class with the highest expected return
when formalizing this into an optimization problem.
Therefore, the return formula needs to have the property that a higher
volatility results in a lower expected return. In section 3, we have seen
E[µ̂g ] has the volatility in a negative component, hence this representation
of expected return suffices this property. Thus, we write
15
1
2
1
2
R(w, µ, σ) = (1 + (w · µ))e− 2 σw + 2T σw − 1,
2 = (w σ)P(w σ)> and the inner product µ · w represents the
where σw
arithmetic return rate of the portfolio.
By Solvency II regulation, the assumed maximum shock based on weights
w, risk charges s and correlation matrix P can be calculated by
q
sw = (w s)> P(w s).
The objective is to define a dynamic investment strategy. This means
that the amount of risk the investor will take, depends on the solvency ratio
and the risk appetite. Therefore, a risk appetite parameter γ is introduced.
The parameter γ represents the solvency ratio after enduring a maximum
shock. This means the solvency ratio of the insurer drops to γ when a shock
sw is realized.
The risk budget RB induced by Solvency II is given by
RBsolvency = SCRtotal .
The insurers available risk budget RB is defined as
RBavailable := e − γSCRtotal .
By solvency II regulation we have
SCRtotal = asw .
In our optimal investment strategy we want the risk budget to be used
to its maximum. Thus, we set RBsolvency = RBavailable . Combining these
equations results in
sw =
e
.
(1 + γ)a
which will lead to one of the constraints in the optimization problem. Define
smax = max{si : i ∈ 1, ..., n}. Additional to last constraint we need to bound
sw with smax , thus we get
n
o
e
sw = min smax ,
.
(100% + γ)
Without this addition, the optimization problem might not have any feasible
solution, because of the following proposition:
Proposition 3. Let smax = max{si : i ∈ 1, ..., n} and smin = min{si :
i ∈ 1, ..., n}. If shock sw is calculated by the Solvency II risk aggregation
formula, it must hold that smin ≤ sw ≤ smax .
16
Proof.
v
uX
n
u n X
sw = t
wi si ρi,j wj sj
j=1 i=1
≤
sX X
j
= smax
smax wi ρij smax wj
i
sX
wj
j
≤ smax
X
wi ρij
i
sX
wj
j
X
wi
i
= smax
The proof for smin ≤ sw is analogous to sw ≤ smax , replacing ≤ with ≥ and
max with min.
This shows that for all possible weight allocations w, the shock will
always be bounded by smin and smax .
e
In the case that (1+γ)a
≥ smax (resp. ≤ smin ), the optimization can
return the weight vector where all weight is allocated to the asset class
that has highest (resp. lowest) shock (and this will be the case when the
asset class where the highest shock is assumed also hasPthe highest expected
return). Additionally, we have the usual constraints i wi = 1 and wi ≥ 0
for all i. The last constraint is the actual Solvency Capital Requirement,
thus
asw ≤ SCRtotal .
e
For deriving the constraint sw = (1+γ)a
we used that asw = SCRtotal . Since
the factor SCRtotal cancels in this derivation, this constraint is used such that
after determining the optimal weights, the resulting capital charge does not
exceed SCRtotal .
Combining all above constraints and estimation of return, we arrive at
the following optimization problem:
17
Optimization Problem. Given total assets a, equity e, expected return
vector µ, standard deviation vector σ, shock vector s, risk appetite parameter
γ and correlation matrix P, the following optimization problem gives an
optimal allocation w based on the log-normally distributed geometric return
rates.
maximize R(w, µ, σ),
w
X
subject to
wi = 1,
i
wi ≥ 0,
n
smin ≤ sw = min smax ,
e o
,
(1 + γ)
asw ≤ SCRtotal .
Here we have formulated an optimization problem in terms of the given
values a, e, µ, σ, s, P, γ. This optimization will be used to analyze the return
risk characteristics of a Dutch insurance company.
4.2
Stylized Case
In this subsection, we will analyze a stylized case which involves four asset
classes. This subsection will provide insight on what the effect of the initial
values a, e, SCRtotal is on the allocation w when the optimization in section
4.1 is used. Therefore, the data of a Dutch insurance company will be used
to illustrate the effect of the dynamic risk budgeting strategy.
Since the allocation w is optimized once per time interval, the time
element t can be disregarded in this subsection. In the second part we will
analyze this optimization using 1000 scenarios over a five year time frame.
These scenarios are generated by Ortec Finance’s scenario generator.
Suppose we have four asset classes: stock, real-estate, spread and government bonds. We denote the shock by s which is based on VaR0.995 , the
expected arithmetic return by µ and the standard deviation by σ. We assume the following values which are based on data from Ortec Finance’s
ALM model:
18

  
0.51
s0
s1  0.23

 
s=
s2  = 0.12 ,
0
s3
  

µ0
0.076
µ1  0.052
 

µ=
µ2  = 0.036 ,
µ3
0.022
  

σ0
0.22
σ1   0.12 
 

σ=
σ2  = 0.095 .
σ3
0.011
In these vectors, stock is indexed by 0, real-estate by 1, spread by 2 and
government bonds by 3. Note that the risk charge of government bonds is
zero. By letting s3 = 0 ensures there is no risk charge on the government
bonds.
Further, let total assets a = 40000, equity e = 6000, SCRtotal = 2900
(for these values, we have: one unit is one million euro) and γ = 1.3. These
are realistic values representing the current state of an insurers portfolio
that will lead to the allocation w using the the optimization in section 4.1.
Computing this in python gives the following weight vector:
w = (7.9%, 17.8%, 11.1%, 63.2%)>
The script that is used to get these numbers can be found in appendix,
section 7.2.
The allocation is determined by two factors, namely the ratio β =
2
e
, which equals 0.071 with the current values of a, e, γ and the
(1+γ)a
other factor is the constraint
asw ≤ SCR
total .
2
2
1
e
= ae 1+γ
We have β = (1+γ)a
. In this ratio, ae determines the
insurers financial position. With the initial values we used we have ae =
6000
1
40000 = 0.15. The factor 1+γ determines the risk appetite. With current
1
values this factor is 2.3
.
The following graphs show the cumulative weights of the four asset
classes. On the left we see the cumulative weight graph that variates in
e
1
1
a and the other factor is regarded as the constant 1+γ = 2.3 . On the right
we see the cumulative weight graph that variates in risk appetite and the
first factor is the constant ae = 0.15:
19
In the left graph, we can see that when ae = 0 all weight will allocated
to government bonds, the risk free asset class (this is not a realistic number,
since this would mean the equity equals zero). When ae increases, more
weight will be on the risky asset classes.
In the right graph, we see that when γ increases, more weight is on
the risk free asset class. When γ increases, the insurer wants to have a
higher solvency ratio after a maximum shock, thus the investment strategy
becomes more risk averse. In the appendix, a sensitivity analysis on the
initial parameters can be found.
In the optimization we observe a big allocation in real estate (green). In
practice, it is common for insurance companies to cap the allocation to a
certain percentage due to real estate being a relatively illiquid asset class.
An illiquid asset class means shares in this asset class cannot easily be sold
or exchanged for money without a substantial loss in value. A realistic cap
on the weight allocation is 15%. In the next graph we illustrate what the
influence on the risk budgeting strategy is, when real estate is capped at
15%. Again, on the left we see the cumulative weights when ae variable, and
on the right we see the cumulative weights when the risk appetite is variable.
We observe that the w2 increases significantly, especially in the cases
where more risk budget is available: when ae high and when γ is low. This
effect is due to the optimization using all available risk budget. Since the
allocation on real estate is capped at 15%, the allocation to credit spread
will increase. This increase will be more than only the difference between
the initial allocation without the cap, and the 15%, since otherwise the total
risk budget is not fully used.
20
This is because credit spread is a less riskier asset class than real estate.
In this respect, credit spread not only compensates for the weight which
is not allocated to real estate. Additionally, it compensates for the extra
available risk budget, in redistributing the weight which was also initially
allocated to sovereign bonds.
4.3
Scenario Analysis
In this subsection, an algorithm using the optimization formalized in section
4.1 is provided, modelling an insurers portfolio. For the scenario analysis we
use the following data for 1000 scenarios over a five year time frame: Initial
total assets a0,ultimo , initial equity e0,ultimo and one is able to calculate initial
SCR0,total using the initial values. Note that in this algorithm, for total
assets at , SCRt,total and equity et we distinguish primo year (start of year)
and ultimo year (end of year).
In year t, for every scenario we have return rates rt,i for all asset classes
i, cash flow rates Ct which is the net income and costs, SCRt,i for i ∈{
default, life, health, non-life } and the primo- and ultimo technical provision
Vt,primo , Vt,ultimo which is the expected capital that is required for covering
all liabilities. Note that SCRt,market depends on wt−1 . The data for these
scenarios are obtained from the Ortec Finance ALM model. In the scenario
analysis, we use the Solvency II correlation matrix Q to calculate SCRt,total .
This means, if we let
bt = SCRt,market , SCRt,def ault , SCRt,lif e , SCRt,health , SCRt,non−lif e
>
,
and we have:

1
0.25

Q=
0.25
0.25
0.25

0.25 0.25 0.25 0.25
1
0.25 0.25 0.50

0.25
1
0.25
0 
,
0.25 0.25
1
0 
0.50
0
0
1
this leads to
SCRt,total
q
= b>
t Qbt ,
which agrees with the Solvency II Directive [14].
Assume we know at−1,ultimo , et−1,ultimo and SCRt−1,total . For time t =
1, 2, 3, 4, 5, we do the following steps in every scenario:
1. at,primo = at−1,ultimo + Ct ,
et,primo = et−1,ultimo + Ct .
2. Determine SCRt,total based on SCRt,i , Q, at,primo , ae,primo , where
SCRt,market is based on wt−1 , using the the assset classes correlation
matrix P.
21
3. Use the optimization in section 4.1 to find optimal wt , using et,primo ,
at,primo and SCRt,total .
P
4. Let a0t,ultimo = at,primo i wt,i (1 + rt,i ).
5. Investment return at time t is given by: IRt := a0t,ultimo − at,primo .
6. et,ultimo = et,primo + 0.75(IRt + Vt,primo − Vt,ultimo ) − 0.25Ct .
7. at,ultimo = et,ultimo + Vt,ultimo .
This algorithm globally agrees with the ALM model employed by Ortec
Finance. Essentially this means: firstly the values ’total assets’ and ’equity’
evolve from ultimo year to primo next year by adding the cash flow, secondly
the weights are rebalanced and thirdly these values evolve from primo year
to ultimo year by adding the investment return.
In step 6, the factor of 0.75 is incorporated to assess the tax which is
charged over the net profit. The term 0.25Ct is subtracted, because otherwise the tax over C would be double since it is already taken into account
in step 1.
We are able to check financial stability by looking at the return-risk-ratio
in year t. Denote Rt for the return in year t and define
θt =
Rt
.
swt
Note: The ratio θt is the ratio of two percentages, namely the realized return
and the assumed maximum shock which is dependent on w. The return in
a
year t is calculated by at,ultimo
− 1 and the maximum shock swt is calculated
t,primo
p
>
by swt = (wt s) P(wt s). For each year t we look at the median of
θt of the 1000 scenarios. It is more useful to analyze the median instead
of the mean p
as there are scenarios which have wt = (0, 0, 0, 1) for some t.
This means (wt s)> P(wt s) = 0 and we would have θt = ∞, which
increases the mean of θt over 1000 scenarios enormously and therefore does
not provide representative information.
t
θt
Median of θt :
1
2
3
0.59 1.11 1.05
4
0.90
5
0.82
In this table we see that in year 2 and 3 the insurer has a return-risk ratio
bigger than one, which means the realized return in these years is bigger
than the assumed maximum shock sw . The shock sw agrees with the risk
charge measured by Solvency II. In year 1, 4 and 5, we see sw is bigger that
the realized return. In next chapter we will see what the impact of measuring
risk with ES instead of VaR is, by comparing the return-risk ratios.
22
To get insight on how these values of θt are formed, we also check the
medians of the returns and the calculated shocks.
Median of return in year t:
t
1
2
3
4
5
Rt 0.0308 0.0258 0.0317 0.0307 0.0340
Median of swt :
t
1
2
swt 0.0520 0.0279
3
0.0339
4
0.0411
5
0.0487
Note that for the calculation of θt , in every scenario and for all t the
ratio θt is established and afterwards the median of θt over all scenarios is
observed. This is why dividing the Rt median with the swt median does not
lead to the same values as θt .
Since the θt values are higher than the fraction of the median return and
median swt , one might deduce returns higher (resp. lower) than the median
occur at the same moments as the swt higher (resp. lower) than the median.
The high returns seem to affect the return-risk ratio more than swt , which
results in the θt being higher when the median is observed after calculating
the ratios, than when it is the other way around. This implies that higher
returns are realized when a bigger risk is taken. In this table the return
medians are more constant whereas the medians of swt fluctuate more.
23
5
Risk Measures
In a previous section we have seen what role the risk measure VaR has
in ALM and Solvency II. In this section we investigate what properties
risk measures intuitively should have and how this can be translated into
mathematics. Firstly, the definition of coherent risk measure is explained.
Secondly, we see VaR is not a coherent risk measure, for it does not suffice
subadditivity, a property of a coherent risk measure. Thirdly, we define ES
formally and prove that ES is a coherent risk measure. Lastly, we investigate
the effect of using ES instead of VaR on the dynamic investment strategy
defined in section 4.
5.1
Coherent Risk Measures
Let L0 (Ω, F, P) be the set of all randoms variables on (Ω, F) which are almost surely finite. Financial risks are represented by a set M ⊂ L0 (Ω, F, P)
[10]. The random variables in the set M can be interpreted as portfolio
losses over time horizon ∆. Throughout this thesis, the time horizon is one
year.
Risk measures are real-valued functions R : M → R, where R(L) represents the amount of capital that should be added to a position with loss
given by L ∈ M, so that the position becomes acceptable to an external
or internal risk controller. Positions with R(L) ≤ 0 are acceptable without
injection of capital; if R(L) < 0, capital might even be withdrawn [10].
The notion of coherent risk measure is an important concept in this
context as it is where the economic rationale is translated to mathematics.
The definition of coherent risk measure is introduced by Artzner et al., 1999
[2]. Note that the original definition is based on random variables in general,
whereas in this thesis it is based on the loss distribution, which leads to a
sign change when compared to the definition in Artzner et al., 1999.
Definition 5.1. Given the random variables L1 , L2 ∈ M, a risk measure
R whose domain includes M is called a coherent risk measure if it satisfies
the following conditions:
• Monotonicity: If L1 ≤ L2 almost surely, then R(L1 ) ≤ R(L2 )
• Translation Invariance: For any constant c ∈ R, R(L1 +c) = R(L1 )+c
• Positive Homogeneity: For any λ > 0, R(λL1 ) = λR(L1 )
• Subadditivity: R(L1 + L2 ) ≤ R(L1 ) + R(L2 )
Intuitively, we want our risk measure to satisfy monotonicity because if
a portfolio leads to a higher loss, it requires more risk capital. Secondly, we
want our risk measure to be translation invariant, for if a constant quantity
24
of value c is added to (or subtracted from) a portfolio loss L, the risk must
increase (resp. decrease) by the same amount [2]. Consider a position with
loss L and R(L) > 0. Adding the amount of capital R(L) to the position
leads to the adjusted loss L̃ = L − R(L), with R(L̃) = R(L − R(L)) =
R(L) − R(L) = 0, so that the position L̃ is acceptable without further
injection of capital [10]. In addition, the risk measure should be positive
homogeneous. This ensures that risk scales linearly with position size [12].
The last property for a risk measure to be coherent is subadditivity. This
property captures the idea that a merger of risk should not create additional
risk [2]. Suppose investing in asset class 1 leads to loss L1 and investing in
asset class 2 leads to loss L2 whilst also assuming that a risk manager wants
to make sure the total risk is less than M . When using subadditive risk
measure R(·) he can choose bounds M1 , M2 such that M1 + M2 ≤ M ,
and impose the constraint that R(Li ) ≤ Mi . The risk measure R(·) being
subadditive ensures that R(L) ≤ M1 + M2 ≤ M . If an investor uses a
non-subadditive risk measure it would seem more convenient to open two
accounts and invest in both asset classes separately.
Note that a positive homogeneous, subadditive risk measure is convex.
Definition 5.2. A risk measure R is convex, if for all L1 , L2 ∈ M and
λ ∈ [0, 1] we have
R(λL1 + (1 − λ)L2 ) ≤ λR(L1 ) + (1 − λ)R(L2 ).
Convexity agrees with the notion that we need diversification to spread
the risk. As explained by Föllmer, 2010 [7], consider the case where one
investment strategy leads to loss L1 , and another investment strategy leads
to loss L2 . If the investor diversifies, spending the fraction λ using the first
strategy and 1 − λ using the second strategy, the investor obtains λL1 + (1 −
λ)L2 . We want this diversification to decrease the risk of investing in the
portfolios using these fractions separately.
5.2
Value at Risk
Having defined some relevant properties of risk measures in general, we are
now able to analyze which of these properties VaR carries. In this subsection
we see that in general VaR satisfies the conditions monotonocity, translation
invariance and positive homogeneity. Moreover, we see an example of VaR
that illustrates VaR is not a coherent risk measure as it does not satisfy the
characteristic of subadditivity. Recall:
VaRα (L) = inf{l ∈ R : P(L > l) ≤ 1 − α} = inf{l ∈ R : FL (l) ≥ α}
Proposition 4. The risk measure VaR satisfies monotonocity, translation
invariance and positive homogeneity.
25
Proof. Let L1 , L2 ∈ M as defined above. Firstly, for monotonocity, let
L1 ≤ L2 . We have VaR(L1 ) = inf{l ∈ R : P(L1 > l) ≤ 1 − α} ≤ inf{l ∈ R :
P(L2 > l) ≤ 1 − α} = VaR(L2 ). For translation invariance let c ∈ R. We
have VaR(L1 + c) = inf{l ∈ R : P(L1 + c > l) ≤ 1 − α} = inf{l ∈ R : P(L1 >
l) ≤ 1 − α} + c = VaR(L1 ) + c.
Lastly, for positive homogeneity let λ > 0. We have
VaR(λL1 ) = inf{l ∈ R : P(λL1 > l) ≤ 1 − α},
= inf{l ∈ R : P(L1 > l/λ) ≤ 1 − α},
= inf{l ∈ R : P(L1 > l0 ) ≤ 1 − α},
(where l/λ = l0 )
= inf{λl0 ∈ R : P(L1 > l0 ) ≤ 1 − α},
(since l = λl0 )
= λinf{l0 ∈ R : P(L1 > l0 ) ≤ 1 − α},
= λVaR(L1 ).
The following example by McNeil, Frey, Embrechts [10], shows that VaRα
is not a coherent risk measure in a case where α = 0.95, since the subadditivity property does not hold for VaRα .
Example 5.1. Consider portfolio of 100 defaultable corporate bonds. The
current price of each bond is 100. If there is no default, a bond pays an
amount of 105 in one year; otherwise there is no repayment. Hence Li , the
loss of bond i, is equal to 100 when the bond defaults and to −5 otherwise.
The distribution of the loss of Li , i ∈ {0, . . . , 100} are IID and one unit of
bond i has the following distribution:
Li =
−5 with probability 0.98
100 with probability 0.02
Denote Yi = 1 if bond i defaults and Yi = 0 in case it does not default.
We can rewrite the distribution of Li to the following loss distribution
Li = 100Yi − 5(1 − Yi ) = 105Yi − 5.
Now consider two portfolios with a current value equal to 10 000. In
portfolio A the weight is concentrated on bond 1, thus A consists of 100
units of bond 1. This implies
LA = 100L1 ,
so VaR0.95 (LA ) = 100VaR(L1 ). Now P(L1 ≤ −5) = 0.98 ≥ 0.95 and for
l < −5 we have P (L1 ≤ l) = 0 < 0.95. Hence, VaR0.95 (L1 ) = −5 and
therefore VaR0.95 (LA ) = −500.
26
This means that the capital charge for this portfolio is -500. Thus, if
the risk controller has a negative equity of 500, this is still an acceptable
strategy at a 95% confidence level.
In portfolio B the weight is completely diversified and there is one unit
on each bond.
100
100
X
X
LB =
Li = 105
Yi − 500,
i=1
i=1
and hence:
100
100
X
X
VaR0.95 (LB ) = VaR0.95 105
Yi − 500 = 105q0.95
Yi − 500.
i=1
i=1
P100
P100
We know
Yi ≤ 5 ≈
i ∼ Bin(100, 0.02) and we have P
i=1 YP
i=1
P100 0.984 < 0.95 and P( 100
i=1 Yi ≤ 4) ≈ 0.949 < 0.95. So q0.95
i=1 Yi = 5
hence
VaR0.95 (LB ) = 525 − 500 = 25.
This means the investor needs an additional risk capital of 25 to satisfy a
regulator working with VaR0.95 as a risk measure. Thus, we get
VaR0.95 (LA ) = −500,
VaR0.95 (LB ) = 25.
In this example we see the non-subadditivity of VaR:
VaR0.95
100
X
Li > 100VaR0.95 (L1 ) =
i=1
100
X
VaR0.95 (Li ),
i=1
where the last equality is true since the Li are IID implying this risk measure
is not coherent.
Intuitively, portfolio B is less risky since it is diversified and therefore
should have a lower risk. Here we see that for portfolio B, an investor would
need an extra risk capital of 25. Meanwhile, for portfolio A, a risk capital
of 500 could be withdrawn and would still be acceptable. This indicates a
drawback of non-coherent risk measures.
5.3
Expected Shortfall
Having recognized the consequence of VaR being non-subadditive, we look
at the coherent risk measure ES. In this section, firstly, we provide a formal
definition for ES and subsequently investigate several properties of ES(L)
when L is normally distributed. Secondly, we prove that ES is a coherent
risk measure. Lastly, we calculate shocks of the asset classes in our stylized
case using ES. This way we can compare the impact of using ES instead of
VaR on the dynamic risk budgeting strategy.
27
The following definitions and theorems can be found in McNeil, Frey,
Embrechts [10].
Definition 5.3. For a loss L with E(|L|) < ∞ and distribution function FL
the Expected Shortfall at confidence level α ∈ (0, 1) is defined as
Z 1
1
ESα (L) =
qu (FL )du,
1−α α
where qu (FL ) = FL← (u) is the quantile function of FL . ES is thus related to
VaR by
Z 1
1
VaRu (L)du.
ESα (L) =
1−α α
Instead of fixing a particular confidence level α, VaR is averaged over all
levels u ≥ α and thus some properties of the tail of the loss distribution are
regarded. Obviously, we have VaRα ≤ ESα .
For continuous loss distributions an even more intuitive expression can
be derived which shows that ES can be interpreted as the expected loss that
is incurred in the event that VaR is exceeded [10].
Proposition 5. For an integrable loss L with continuous distribution function FL and any α ∈ (0, 1) we have
ESα =
E(L; L ≥ qα (L))
= E(L|L ≥ VaRα ).
1−α
Here we have used the notation E(X; A) := E(X 1A ) for a generic integrable random variable X and a generic set A ∈ F.
Proof. Denote by U a random variable with uniform distribution on the
interval [0, 1]. By a property of the uniform distribution we know that the
random variable FL← (U ) has distribution function FL . We have to show that
R1
that E(L; L ≥ qα (L)) = α FL← (u)du. Now
E(L; L ≥ qα (L)) = E(FL← (U ); FL← (U ) ≥ FL← (α)) = E(FL← (U ); U ≥ α);
in the last equality we used the fact that FL← is strictly increasing since FL
R1
is continuous. Thus we get E(FL← (U ); U ≥ α) = α FL← (u)du. The second
representation follows since for a continuous loss distribution
This proposition can be used to calculate ES for a Gaussian loss distribution.
Proposition 6. If L ∼ N (µ, σ 2 ) and α ∈ (0, 1) then
ESα = µ + σ
φ(Φ−1 (α))
,
1−α
where φ is the density of the standard normal distribution.
28
Proof. We have
L − µ
L − µ L − µ
ESα = µ + σE
≥ qα
σ
σ
σ
!
Hence, it suffices to compute ES for the standard normal random variable
L̃ := L−µ
σ . Here we get:
ESα =
5.4
1
1−α
Z
∞
lφ(l)dl =
Φ−1 (α)
1
φ(Φ−1 (α))
[−φ(l)]∞
Φ−1 (α) =
1−α
1−α
Coherence of Expected Shortfall
Now, in order to prove that ES is a coherent risk measure, we need to analyze
results where the following theory is used: Fatou’s Lemma, the Strong Law
of Large Numbers and the Glivenko-Cantelli theorem. These results can be
found in the appendix.
The complexity of proving that ES is a coherent risk measure lies with the
subadditivity property. To prove subadditivity, it is convenient to represent
ES as a limit of discrete loss distribution functions using proposition 8.
These loss distribution functions are essentially, all the VaRλ -values with
λ ∈ (α, 1). Thus, the sum of these loss distribution quantiles multiplied by
an averaging scaling factor is exactly the expected loss, given that the loss
is outside the confidence interval [0, α].
In proposition 7, the convergence of the discrete case to the continuous
case is proved, which is a result obtained by W.R. van Zwet, 1980 [15]. To
prove this theorem, which is based on order statistics, we look at a theorem
on convergence and accompanying definitions which form the base of Van
Zwet’s theorem.
Having seen Van Zwet’s result and the representation of ES as a limit of
discrete loss functions, we are able to prove the subadditivity of ES. Firstly,
it is proven for the discrete case which is represented using order statistics,
such that Van Zwets theorem can be applied. Secondly, Van Zwet’s theorem
concerning convergence proves subadditivity in the continuous case.
We start by giving some definitions that support lemma 1, which is a
basic result in measure theoretic probability regarding convergence which is
used by Van Zwet for his theorem about order statistics [15].
Let U1 , U2 , ... be IID random variables according to the uniform distribution on the interval (0, 1). For N = 1, 2, ..., let U1:N < U2:N < ... < UN :N
denote the ordered U1 , ..., UN . We have Lebesgue measurable functions
JN : (0, 1) → R, N = 1, 2, ..., and a Borel measurable function g : (0, 1) → R.
Further, we define gN : (0, 1) → R, N = 1, 2, ..., by gN (t) = g(U[N t]+1:N ),
where [N t] denotes the integer part of N t. Additionally, in the case that
29
[N t] + 1 > N we let gN (t) = g(UN ). In the proof, t is be a number between
0 and 1.
Lemma 1. With probability 1, gN converges
to g in Lebesgue measure, i.e.
P limN →∞ {t : |gN (t) − g(t)| ≥ δ} = 0 = 1 for every δ > 0.
Proof. Choose > 0. By Lusin’s theorem [9] (also appendix section 7.3),
for a measurable function g̃ : [0, 1] → R there exists a compact E ⊂ [0, 1]
such that g̃ restricted to E is continuous and λ(E) > 1 − . Now let B =
(0, 1) \ E which is Borel. We have B ⊂ (0, 1), λ(B) ≤ and let g = g̃ on
(0, 1) ∩ B c . Define g̃N (t) = g̃N (U[N t]+1:N ) and BN = {t : U[N t]+1:N ∈ B},
so that gN = g̃N on (0, 1) ∩ B c . Since λ(BN ) = λ{t : U[N t]+1:N ∈ B} =
1 PN
i=1 1{Ui ∈B} = PN (B), where PN denotes the empirical distribution of
N
U1 , ..., UN , it follows from the strong law that lim supN λ(BN ) ≤ with
probability 1. In view of the Glivenko-Cantelli theorem (see appendix) and
the continuity of g̃, this implies that with probability 1 we have for every
δ > 0: lim supN λ{t : |gN (t) − g(t)| ≥ δ} ≤ λ(B) + lim supN λ(BN ) +
lim supN λ{t : |g̃N (t) − g̃(t)| ≥ δ} ≤ 2. Since > 0 is arbitrary, we have
P limN →∞ {t : |gN (t) − g(t)| ≥ δ} = 0 = 1 for every δ > 0, thus the lemma
is proved.
We use this lemma to prove Van Zwet’s theorem about order statistics.
For this theorem we first need to define the normed space Lp . For 1 ≤ p ≤ ∞,
Lp is the Lebesgue space of measurable functions f : (0, 1) → R with finite
R1
norm ||f ||p = { 0 |f |p dλ}1/p for 1 ≤ p < ∞ and ||f ||∞ = ess sup|f | for
p = ∞.
The purpose of this note is to show that under integrability assumptions
on JN and g,
Z
1
JN (gN − g)dλ =
MN :=
N
X
0
Z
i/N
Z
JN dλ −
g(Ui:N )
(i−1)/N
i=1
1
JN gdλ
0
converges to zero for N → ∞ with probability 1, which is Van Zwet’s first
result. In addition to this, if JN converges in an appropriate sense to a
function J which shares the integrability properties of JN , Van Zwets second
result states that
Z
Z
1
1
JN gN dλ −
M̃N :=
0
Jgdλ
0
also converges to zero with probability 1.
Proposition 7 (Van Zwet, 1980 [15]). Let 0 = t0 < t1 < ... < tk = 1
−1
and > 0. For j = 1, ..., k, let 1 ≤ pj ≤ ∞, p−1
= 1 and define
j + qj
intervals Aj = (tj−1 , tj ) and Bj = (tj−1− , tj+ ) ∩ (0, 1). Suppose that, for
j = 1, ..., k, JN 1Aj ∈ Lpj for N = 1, 2, ..., g 1Bj ∈ Lqj and either
1. 1 < pj ≤ ∞ and supN ||JN 1Aj ||pj < ∞, or
30
2. pj = 1 and {JN 1Aj : N = 1, 2, ...} is uniformly integrable.
Then limN →∞ MN = 0 with probability 1. If, moreover, there exist
R t a function J with JN 1Aj ∈ Lpj for j = 1, ..., k, such that limN →∞ 0 JN dλ =
Rt
0 Jdλ for every t ∈ (0, 1), then also limN →∞ M̃N = 0 with probability 1.
In this proof, some of the arguments do not go into specific detail. The
extended proof can be found in Van Zwet, 1980 [15].
Proof. Consider an index j with 1 < pj ≤ ∞, so qj < ∞. Choose δ ∈ (0, ]
and define Cj = (tj−1 − δ, tj + δ) ∩ (0, 1). The Glivenko-Cantelli theorem
and the strong law of large numbers ensure that with probability one [17]
Z
Z
1 X
qj
qj
lim sup
|gN | dλ ≤ lim sup
|g(Ui )| 1Cj ∪Ui =
|g|qj dλ < ∞.
N
N
N
Aj
Cj
i
R1
R1
Since δ ∈ (0, ] is arbitrary, this implies that 0 |gN |qj 1Aj dλ → 0 |g|qj 1Aj dλ
with probability 1 by Fatou’s lemma.
By
R 1Vitali’s convergence theorem (see appendix, section 7.3) this implies
that 0 |gN − g|qj 1Aj dλ → 0, and Hölder’s inequality yields ||JN ||pj ||gN −
R1
g||qj 1Aj → 0 with probability 1. Thus, we can conclude that 0 JN (gN −
g)1Aj dλ → 0 with probability 1.
For an index j with pj = 1 and qj = ∞, the Glivenko-Cantelli theorem ensures that lim supN ||gN 1Aj ||∞ ≤ ||g 1Bj ||∞ < ∞ with probability 1.
Because of the uniform integrability of JN 1Aj and Lemma 1, we have with
probability 1,
Z
lim sup |MN | ≤ δ lim sup ||JN 1Aj ||1 + 2||g 1Bj ||∞ lim sup
|JN |
N
N
N
{|gn −g|1Bj >δ}
= δ lim sup ||JN 1Aj ||1
N
R1
for every δ > 0. Since supN ||JN 1Aj ||pj < ∞, we find that 0 JN (gN −
g)1Aj dλ → 0 with probability 1.
This proves the first statement of the theorem. The second
statement
R1
is apparent, as the assumptions of the theorem imply that 0 JN g 1Aj dλ →
R1
0 Jg 1Aj dλ for j = 1, ..., k.
This convergence result is the basis of the representation of ESα as the
limit of a sum defined by the order statistics of the loss distribution functions.
Intuitively, one might say these loss distribution functions are the functions
VaRλ with λ ∈ (α, 1). The partition 0 = t0 < t1 < ... < tk = 1 in Van
Zwet’s theorem is used for splitting the range of integration (0, 1) into (0, α]
and (α, 1). The next result from Acerbi and Tasche, 2002, gives the desired
representation of ESα .
31
Proposition 8 (Acerbi and Tasche, 2002 [1]). Let α ∈ (0, 1) be fixed. For
a sequence (Li )i∈N of IID random variables, E[Li ] < ∞ for all i we have
P[n(1−α)]
lim
n→∞
Li,n
= ESα
[n(1 − α)]
i=1
a.s.,
where L1,n ≥ ... ≥ Ln,n are the order statistics of L1 , ..., Ln and where
[n(1 − α)] denotes the largest integer not exceeding n(1 − α).
Proof. The “with probability 1” part of this lemma is essentially a special
case of proposition 7 with 0 = t0 < (1 − α) = t1 < t2 = 1, J(t) = 1[α,1) (t),
JN (t) = 1 [N α]−1 (t), g(t) = F −1 (t), and p1 = p2 = ∞. This proves the
N
,1
’almost sure convergence part’. Concerning the L1 -convergence note that
n
[n(1−α)]
X
X
Li:n ≤
|Li |.
i=1
i=1
Pn
−1
|Li | converges to E[L1 ] in L1.
P[n(1−α)]
This implies uniform integrability for
Li:n .
i=1 |Li | and for i=1
Together with the already proven almost sure convergence, this implies assertion.
By the strong law of large numbers n
n−1
i=1
P
n
In this lemma we see that ESα can be regarded as the limiting average
of the [n(1 − α)] upper order statistics from a sample of size n from the loss
distribution. Thus, ESα can be estimated when there is a large sample size
and [n(1 − α)] is large. With this representation we can prove that ESα is
a coherent risk measure, where the proof of ES being subadditive can be
found in McNeil, Frey, Embrechts [10].
Theorem 1. Expected shortfall is a coherent measure of risk.
Proof. We have to prove ES satisfies monotonicity, translation invariance,
positive homogeneity and subadditivity. The first three properties follow
directly from VaR having these properties, since the operation of integration
is linear and increasing. Thus, we only have to prove subadditivity.
Consider a generic sequence of random variables L1 , ..., Ln with associated order statistics L1,n ≥ ... ≥ Ln,n and note that for arbitrary m satisfying 1 ≤ m ≤ n we have
m
X
Li,n = sup{Li1 + ... + Lim : 1 ≤ i1 < ... < im ≤ m}.
i=1
Now consider two random variables L and L̃ with joint distribution function F and a sequence of iid bivariate random vectors (L1 , L̃1 ), ..., (Ln , L̃n )
with the same distribution function F . Writing (L + L̃)i := Li + L̃i and
32
(L + L̃)i,n for an order statistic of (L + L̃)1 , ..., (L + L̃)n we observe that we
must have
m
X
(L + L̃)i,n = sup{(L + L̃)i1 + ... + (L + L̃)im : 1 ≤ i1 < ... < im },
i=1
≤ sup{Li1 + ... + Lim : 1 ≤ i1 < ... < im }
=
m
X
i=1
+ sup{L̃i1 + ... + L̃im : 1 ≤ i1 < ... < im },
m
X
Li,n +
L̃i,n .
i=1
By setting m = [n(1 − α)] and letting n → ∞ we infer from Proposition
8 that ESα (L + L̃) ≤ ESα (L) + ESα (L̃). Now that we have seen ES is
subadditive, the proof of ES being a coherent risk measure is complete.
Since we have seen ES is a coherent risk measure as opposed to VaR, we
are able to examine the effect of using ES instead of VaR in Example 5.1
regarding defaultable corporate bonds.
Example 5.2. Recall, we have a portfolio with a current value of 10 000.
The loss distribtution of the 100 IID random variables is given by
−5 with probability 0.98
Li =
100 with probability 0.02
In portfolio A all the weight is concentrated on bond 1, thus A consists
of 100 units of bond 1. In portfolio B the weight is completely diversified
and there is one unit on each bond.
We have
LA = 100L1 ,
so
ES0.95 (LA ) = 100ES(L1 ).
Now P(L1 ≤ −5) = 0.98 ≥ 0.95 and for l < −5 we have P (L1 ≤ l) =
0 < 0.95. Recal VaR0.95 (L1 ) = −5, hence
Z 1
1
ES0.95 (L1 ) =
VaRu (L1 )du,
1 − 0.995 0.995
−5(0.98 − 0.95) + 100(1 − 0.98)
=
,
0.05
= 37
and therefore ES0.95 (LA ) = 3700.
In portfolio B the weight is completely diversified and there is one unit
on each bond.
33
100
X
LB =
Li = 105
100
X
i=1
Yi − 500,
i=1
therefore we get
ES0.95 (LB ) = ES0.95 105
100
X
Yi − 500
i=1
= 105 · ES0.95
100
X
Yi − 500
i=1
1
= 105 ·
1 − 0.95
Z
1
VaRu (B)du − 500, (B ∼ Bin(100, 0.02))
0.95
Here we need the following numbers,
P(B ≤ 4) ≈ 0.9491,
P(B ≤ 5) ≈ 0.9845,
P(B ≤ 6) ≈ 0.9959,
P(B ≤ 7) ≈ 0.9991,
P(B ≤ 8) ≈ 0.9998,
..
.
which implies
VaRu (B) = 5 for u ∈ [0.9500, 0.9845)
VaRu (B) = 6 for u ∈ [0.9845, 0.9959)
VaRu (B) = 7 for u ∈ [0.9959, 0.9991)
VaRu (B) = 8 for u ∈ [0.9991, 0.9998)
..
.
Thus
Z 1
VaRu (B)du = 5 × (0.9845 − 0.9500) + 6 × (0.9959 − 0.9845)
0.95
+ 7 × (0.9991 − 0.9959) + 8 × (0.9998 − 0.9991) + · · · ,
≈ 0.28,
which we use to continue the calculation
105
× 0.28 − 500,
ES0.95 (LB ) ≈
0.05
≈ 88.
34
This implies we have
VaR(LA ) = −500,
VaR(LB ) = 25,
ES(LA ) = 3700,
ES(LB ) ≈ 88,
As we can see in this example, ES captures our intuitive notion that risks
lurk in the tail of a distribution and is readily interpreted as an estimate of
the average loss that we could experience with a given probability [12]. In
portfolio A, the risk controller is unable to see the possible risk when VaR
is used. The expected loss is -500 which means it is expected that the risk
controller earns 5 on every unit he puts in. Meanwhile the expected loss,
conditional on the loss being outside of the confidence level, is very high,
better illustrating the tail risk.
In portfolio B we see the values of ES and VaR do not differ much, since
when using VaR, the effect of the loss being outside of the confidence level
is already partially observable. When ES is used, the expected loss only
increases by (approximately) 63 in comparison to when VaR is used.
5.5
Stylized Case
In this section we analyze the return-risk characteristics when ES is used
as a risk measure instead of VaR. To compare VaR with ES, the same data
is used as in the scenario analysis of section 4. Firstly, the behaviour of
the weights depending on ae and the risk appetite is illustrated by looking
at the cumulative weight graphs, when shocks are established with ES instead of VaR. Secondly, we look at the return-risk ratio when risk (in the
denominator) is measured using ES. Here, we are able to compare ES-based
optimization with VaR-based optimization, by looking at the return-risk
characteristics when the risk budgeting investment strategy formulated in
chapter 4 is used.
Using ES as risk measure we find the following shock vector, based on
the data of the same Dutch insurance company as in chapter 4,
  

s0
0.57
s1  0.26
 

sES = 
s2  = 0.14
s3
0
We take the same initial values as in the VaR-based scenario analysis. Thus,
let total assets a = 40000 and equity e = 6000 and let the SCRtotal = 2900.
35
The optimization gives the following weight vector:
w = (7.2%, 15.4%, 9.1%, 68.3%)> .
Now we compare the optimized allocation when using ES instead of VaR.
Again, on the left we see the cumulative weights where ae is variable, and on
the right we see the cumulative weights when the risk appetite is variable.
6000
Recall that the initial values are ae = 40000
= 0.15, γ = 1.3.
Which can be compared to the cumulative weight graphs of the optimization
based on VaR:
Here we see the behaviour of w is quite similar. The ES-based optimization
is more risk-averse, since there is less weight in stock and there is more
weight in sovereign bonds and spread. Now we have established the impact
on the strategy, we can compare the return-risk characteristics.
For the comparison of ES with VaR, we use the same scenarios as in
chapter 4. Denote RtES (resp. RtVaR ) as the return in year t with ES-based
(resp. VaR-based) optimization.
The return-risk ratio is defined as follows
θtES =
RtES
.
sES
wt
Looking at the median of the return-risk ratio θtES , return RtES and shocks
sES
wt we find that over all scenarios:
36
Median θtES of all scenarios:
t
1
2
3
4
ES
θt
0.551 1.086 1.047 0.881
5
0.836
Median RtES of all scenarios:
t
1
2
3
ES
Rt
0.0287 0.0254 0.0312
4
0.0305
5
0.0336
Median sES
wt of all scenarios:
t
1
2
3
ES
swt 0.0520 0.0270 0.0331
4
0.0405
5
0.0468
We can compare this to the medians of the ratio θtVaR , return RtVaR and
shock sES
wt over all scenarios of θ when optimization is executed with VaR
and risk is measured with ES. Thus, we denote
θtVaR =
RtVaR
.
sES
wt
This leads to the following medians.
Median θtVaR of all scenarios:
t
1
2
3
4
θtVaR 0.521 0.974 0.923 0.787
Median RtVaR of all scenarios:
t
1
2
3
RtVaR 0.0308 0.0258 0.0317
Median sES
wt of all scenarios:
t
1
2
3
sES
0.0592
0.0317
0.0386
wt
5
0.719
4
0.0307
4
0.0467
5
0.0340
5
0.0555
Firstly, we observe that the median of θtES is higher than the median
of θtVaR for all t. Secondly, the returns are slightly higher when the optimization is VaR-based. Thirdly, the measured risk is also higher when the
optimization is VaR-based. Based on these observations, one might deduce
the return-risk ratio is favourable when the optimization is ES-based, due
to this strategy being more risk averse. The VaR-based optimization yields
slightly higher returns, however, this does not compensate for the additional
tail risk taken when considering the return-risk ratio.
To elaborate on this last observation, the actual tail risk is examined.
Therefore we check the realized returns of all years in all scenarios focusing
on the 0.5th and the 5th percentile. For the realized returns in these tail
regions, we also observe the corresponding risk based on ES which is used
to calculate the risk-return ratio θ. In Appendix 7.2, the subsection labelled
“Ratio Check”, one is able to see how these results are obtained.
Observing only the lowest 0.5% realized returns, with their corresponding
37
risks and ratios, we take the averages for both strategies resulting in the
following table:
0.5% tail
VaR-based optimization ES-based optimization
Realized return
-0.090
-0.051
Risk (based on ES) 0.160
0.128
θ
-0.572
-0.430
Comparing these results, we see that in this tail region, the average
realized return of VaR-optimal portfolios is -9.0%, whereas the ES-optimal
portfolios have an average of -5.1%. This means almost half of the loss is
realized when using ES-based optimization.
The measured risk for the VaR-based optimization is 16.0%, which is
higher than the measured risk of 12.8%. From this observation, one might
deduce that when the optimization is ES-based, more weight is allocated to
asset classes that are less risky in the tail region as well.
To analyze the sensitivity of the quantile, we check how these values differ
from the 5% lowest values of realized returns with corresponding ES-based
risk and ratios. This leads to the following averages:
5% tail
VaR-based optimization ES-based optimization
Realized return
-0.022
-0.014
Risk (based on ES) 0.094
0.071
θ
-0.217
-0.192
The averages of the 5% lowest realized returns and the corresponding
measured risks for ES-optimal portfolios are much closer to the VaR-optimal
portfolios, illustrating that the risk lurks in the 0.5% tail.
In conclusion, we see that when ES is used instead of VaR in the risk
budgeting strategy to determine shocks, the available risk budget becomes
smaller. This leads to a more risk averse weight allocation, resulting in
ES-optimal portfolios being better prepared for risk that lurks in the tail.
This also illustrates that a risk controller using VaR, should be aware of the
tail risk since VaR does not contain information outside of the confidence
interval.
38
6
Conclusion
In this section, the main findings of this thesis are summarized, followed by
concluding remarks. The limitations of this research will also be outlined,
as well as areas for further research in the field.
6.1
Summary
This thesis has introduced an investment strategy for insurance companies
that optimizes the return-risk ratio, namely maximizing the expected return
without exceeding SCRtotal . In the first part of this thesis, an investment
strategy is proposed where the amount of risk that the insurer takes is proportional to their current financial position. This investment strategy is
based on an optimization problem that uses assumptions about statistical
values of asset class returns, i.e. expected return, standard deviation, correlation and shocks. The outcome of this optimization is an optimal weight
allocation.
In the second part of this thesis, firstly, the concept of a coherent risk
measure is investigated. Secondly, an example of how the benefits of diversification are not taken into account when risk is measured using VaR,
illustrates that VaR is not a coherent risk measure. Thirdly, ES is introduced and a detailed proof of the coherence of this risk measure is given.
This proof requires the Law of Large Numbers in context of order statistics.
An estimator of ES is provided, which is used to show the convergence of
a discrete representation of ES. Further, it is shown that in the extended
example which illustrates the incoherence of VaR, using ES does in fact lead
to a proper valuation of risk that follows the economic rationale.
Lastly, the impact of using ES instead of VaR to measure the shocks
is investigated, by examining the return-risk characteristics. The returnrisk ratio, where risk is measured using ES, of the ES-optimal portfolio is
compared to the VaR-optimal portfolio. The results are based on the risk
budgeting investment strategy as proposed in the first part of the thesis.
The return-risk characteristics are explored by reviewing the tail risk in
both strategies. Therefore, we look at the lowest 0.5% and 5% realized
returns and the corresponding measured risk based on ES and return-risk
ratios.
6.2
Concluding Statements
This research concludes that VaR-optimal portfolios face significantly more
tail risk than ES-optimal portfolios. When ES is used instead of VaR to
determine the shocks, the available risk budget is smaller. This means more
weight will be allocated to asset classes which are more risk averse, affecting
both the realized return and the risk that the insurer takes.
39
The first part concludes that the optimal weight allocation can be summarized in an optimization problem. The objective function to be maximized, is the expected log-normal return rate that depends on weight, expected return and standard deviation. The constraints in the optimization
problem ensure the available risk budget is fully utilized, without exceeding
the total SCR.
The second part concludes that ES yields a more accurate estimation of
risk than VaR does. A comparison of two portfolios - one which is diversified
and the other which is concentrated on one asset class - illustrates how
VaR might significantly underestimate the risk when compared to ES. In
the scenario analysis, we find that an ES-optimal portfolio will have more
weight allocated to less riskier asset classes than a VaR-optimal portfolio,
resulting in the decrease of both the realized return and the measured risk.
Moreover, this scenario analysis illustrates the limitation of not using a
coherent risk measure since it does not contain information outside of the
confidence interval.
Therefore, the overall conclusion of this research is that when using the
risk budgeting investment strategy formulated in the first part of this thesis, ES-optimal portfolios are more risk averse than VaR-optimal portfolios,
indicating that VaR-optimal portfolios encounter significantly more tail risk.
6.3
Limitations
This research has been confronted with three noteworthy limitations. Firstly,
the stylized case only uses four asset classes, which defines the total assets.
In an actual insurers portfolio, besides “stock”, “real-estate”, “sovereign
bonds” and “credits” there are typically more asset classes. For example,
the Dutch insurance company that is used for the stylized case also has asset
classes like: “mortgages”, “liquid resources”, “currency”, “swap”.
In future research, the stylized case can be extended using a greater
selection of asset classes. In this thesis, this is simplified because the focus
is on the impact of risk budgets. Therefore, in order to analyze the effect of
different shocks, the behaviour of all asset classes is assumed to be similar.
If one has the objective to research how optimal risk budgets should be
established per asset class, taking the differences between those asset classes
into account, the stylized case should be extended with modeling different
aspects of the asset classes.
Secondly, the expected geometric return that is used in the formulation of
the optimization problem, is calculated under the assumption that the return
rates are log-normally distributed. However, in reality insurers deal with
heavy tailed distributions. Furthermore, the correlation matrix of Solvency
II is used, since the data set used for this research was too small to deduce
the actual correlation values.
The third limitation is that the data used is based on 1000 scenarios
40
with rates of five years, taken from Ortec Finance’s scenario generator.
This means that for all return rates the 99.5%-quantile contains information
about the 5 worst cases per year. The 99.5%-quantile is used for calculating
the values for VaR and ES. One could argue that five values per year is
not enough to calculate the shock. For future research, it might therefore
be crucial to use a bigger scenario set, where scenarios are spanning over a
longer time period, resulting in more accurate values of VaR and ES.
Aside from the given limitations, this research illustrates the importance
of understanding the differences which exist between ES and VaR in a risk
budgeting investment strategy, with ES being a more favorable method as
it provides a better estimation of the tail risk.
41
7
Appendix
7.1
Sensitivity Analysis
In this section, we execute a sensitivity analysis on the parameters we use
in our optimization, formulated in chapter 4. The parameters we analyze
are expected return and standard deviation. In the optimization we use the
following values

  
0.076
µ0
µ1  0.052

 
µ=
µ2  = 0.036 ,
0.022
µ3
  

σ0
0.22
σ1   0.12 
 

σ=
σ2  = 0.095 .
σ3
0.011
µ = (µ0 , µ1 , µ2 , µ3 )> = (0.076, 0.052, 0.036, 0.022)> ,
σ = (σ0 , σ1 , σ2 , σ3 ) = (0.22, 0.12, 0.095, 0.011)> .
In the first subsection, we analyze the sensitivity of the expected return. Firstly, we check what happens when we add 0.01 to the marginal
expected returns, and then we investigate the effect of subtracting 0.01 of
the marginal expected returns. Secondly, we do a similar check for the
standard deviations, only then with adding and subtracting 0.05.
Return rates parameter
Optimization with µ and σ:
Optimization with µ + (0.01, 0, 0, 0)> and σ:
42
Optimization with µ + (0, 0.01, 0, 0)> and σ:
Optimization with µ + (0, 0, 0.01, 0)> and σ:
Optimization with µ + (0, 0, 0, 0.01)> and σ:
43
Optimization with µ − (0.01, 0, 0, 0)> and σ:
Optimization with µ − (0, 0.01, 0, 0)> and σ:
Optimization with µ − (0, 0, 0.01, 0)> and σ:
Optimization with µ − (0, 0, 0, 0.01)> and σ:
44
Standard deviation parameter
Optimization with µ and σ:
Optimization with µ and σ + (0.05, 0, 0, 0)> :
Optimization with µ and σ + (0, 0.05, 0, 0)> :
45
Optimization with µ and σ + (0, 0, 0.05, 0)> :
Optimization with µ and σ + (0, 0, 0, 0.05)> :
Optimization with µ and σ − (0.05, 0, 0, 0)> :
46
Optimization with µ and σ − (0, 0.05, 0, 0)> :
Optimization with µ and σ − (0, 0, 0.05, 0)> :
Optimization with µ and σ − (0, 0, 0, 0.05)> :
Firstly, we observe that the expected return and standard deviation of
stock are the least sensitive to parameters. This might be because stock has
the highest values for both parameters, and this will in all cases be the asset
class that gets a high weight allocation in the cases where ae is high or γ is
low.
The second observation is that when real-estate has a higher (resp. lower)
expected return or lower (resp. higher) standard deviation, the additional
(resp. eliminated) allocated weight is in credit spread, and the other way
around. For instance, the case where the optimization is executed with
µ−(0, 0, 0.01, 0)> = (0.076, 0.052, 0.026, 0.022)> the expected return of real-
47
estate and credit spread are very close. Since the standard deviation of
credit spread is not much lower, this results in no weight allocated to credit
spread. One might conclude, the expected return is not high enough in this
case considering the risk induced by the extra volatility.
7.2
Python Scripts
Portfolio Algorithm
In this section the algorithm that is used for the scenario analysis is shown.
Firstly the initual values are given. This are based on Ortec Finance’s ALM
model. Secondly, the optimization from section 4.1 is given. Thirdly, the required data is loaded. Lastly, a data frame is generated that holds the initial
values for all scenarios, and the algorithm of section 4.3 is programmed.
# -*- coding: utf-8 -*"""
Created on Tue Jun 21 12:46:10 2015
@author: Wout
"""
from __future__ import division
from __future__ import print_function
from pandas import *
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import os
import general_export_reader as ger
import plot_tools as pt
import Formules
import pdb
import scipy as sp
if True: #values gained from ALS/EXCEL
cor_mrkt = np.array([[1, 0
, 0
, 0],
[0, 1
, 0.75, 0],
[0, 0.75, 1
, 0],
[0, 0
, 0
, 1]])
48
cor_life= np.array([[1,
0.25,0.25,0.25,0.25],
[0.25, 1,0.25,0.25,0.50],
[0.25,0.25,1,0.25,0],
[0.25,0.25,0.25,1,0],
[0.25,0.50,0,0,1]])
exp_return = np.array([0.075642,0.052234,0.035625, 0.021967])
stdev = np.array([0.221907, 0.116928, 0.055463, 0.010718])
schokken = np.array([0.5101188, 0.228609, 0.119293, 0])
cvar = np.array([0.567559, 0.2635, 0.13838, 0])
schokken_risk = np.array([0.5101188, 0.228609, 0.119293])
cor_mrkt_risk = np.array([[1, 0
, 0
],
[0, 1
, 0.75],
[0, 0.75, 1
]])
#################
OPTIMISATION ALGORITHM #######################
def get_weights(rho, s, mu, sigma, b, scrt, ta):
N = mu.shape[0]
con=list()
con.append( {’type’: ’eq’, ’fun’ : lambda x : Formules.VaR(x, s, rho)-b} )
con.append( {’type’: ’eq’, ’fun’ : lambda x :np.sum(x)-1} )
con.append( {’type’: ’ineq’, ’fun’ : lambda x :scrt
- np.sqrt(np.dot(x*s, rho.dot(s*x)))*ta} )
bounds = [(0,None) for k in range(N)]
x0=np.ones(N)/N
sol=sp.optimize.minimize(lambda x:
-1*Formules.log_georeturn(Formules.r_a(x,mu),
Formules.log_variance(Formules.r_a(x,mu),
Formules.Variance(rho, sigma, x))),
x0, method =’SLSQP’ ,
constraints = con,
bounds=bounds )
return sol[’x’]
####################################################################
def beta(eigenvermogen,assets,gamma):
return np.divide(eigenvermogen,((1+gamma)*assets))
49
if __name__ == ’__main__’:
if True:
sc_set =dict()
if True:
varnames= [’pukkasstroom’,’life’,’nonlife’,’health’,
’operational’, ’market’, ’default’, ’vvpprimo’, ’vvpultimo’,
’aandelen’, ’spread’, ’swap’, ’staats’, ’oghuis’, ’ogwinkel’,
’ogkantoor’]
for varname in varnames:
sc_set[varname]=pd.read_excel(’alsdata2.xlsx’, varname,
index_col=None, na_values=[’NA’]).T
df=pd.read_excel(’alsdata2.xlsx’, varname,
index_col=None, na_values=[’NA’]).T
pass
sc_set = pd.concat(sc_set.values(), axis=1,
keys=sc_set.keys(),names=[’Name’, ’Scenario Nr.’])
simulation_start = sc_set.index[0]#.date()
if True:
sc_nrs = sc_set.columns.get_level_values(’Scenario Nr.’).unique()
scenarios_mix = pd.DataFrame(index=sc_set.index, columns = sc_nrs)
scenarios_mix.iloc[0] = np.ones(len(sc_nrs))/len(sc_nrs)
weights_index=range(2013,2019)
weights_varnames=[’w1’, ’w2’, ’w3’, ’w4’, ’eigenvermogen’,
’belegdvermogen’, ’belegresultaat’, ’beleg’, ’risicovrij’,
’assets’, ’scr_tot’, ’riskratio_asset’,’riskratio_asset_cvar’,
’riskratio_total’, ’asset_shock’,’asset_shock_cvar’,
’betacheck’, ’asset_return’]
n_var=len(weights_varnames)
n_scenarios=1000
scenario_nrs=range(0,n_scenarios)
weights_columns=pd.MultiIndex.from_product([weights_varnames, scenario_nrs],
names=[’Name’, ’Sc. nr.’])
weights = pd.DataFrame(data = 0,index=weights_index, columns = weights_columns)
weights.loc[2013,’eigenvermogen’] = EV_init
weights.loc[2013,’belegdvermogen’] = BV_init
weights.loc[2013,’beleg’] = beleg_init
weights.loc[2013,’risicovrij’] = risicovrij_init
weights.loc[2013,’assets’] = assets_init
weights.loc[2013,’scr_tot’] = 2900
weights.loc[2013,’w1’] = w_init[0]
50
weights.loc[2013,’w2’] = w_init[1]
weights.loc[2013,’w3’] = w_init[2]
weights.loc[2013,’w4’] = w_init[3]
for t in scenarios_mix.index:
if t>2013:
for sc_nr in sc_nrs:
a_t_primo = weights.loc[t-1,(’assets’,sc_nr)]
+ sc_set.pukkasstroom.loc[t,sc_nr]
e_t_primo = weights.loc[t-1,(’eigenvermogen’,sc_nr)]
+ sc_set.pukkasstroom.loc[t,sc_nr]
z=np.array([weights.loc[t-1,(’w1’,sc_nr)] * a_t_primo ,
weights.loc[t-1,(’w2’,sc_nr)] * a_t_primo,
weights.loc[t-1,(’w3’,sc_nr)] * a_t_primo,
weights.loc[t-1,(’w4’,sc_nr)] * a_t_primo])
scr_market=np.sqrt(np.dot(z*schokken,
cor_mrkt.dot(schokken*z)))
x=np.array([scr_market,sc_set.default.loc[t,sc_nr],
sc_set.life.loc[t,sc_nr],
sc_set.health.loc[t,sc_nr],
sc_set.nonlife.loc[t,sc_nr]]) #x is vector met alle SCR
scr_totaal = np.sqrt(np.dot(x, cor_life.dot(x)))
weights.loc[t,(’scr_tot’,sc_nr)] = scr_totaal
bet= beta(e_t_primo,a_t_primo,1.3)
weights.loc[t,(’betacheck’,sc_nr)] = bet
if bet < np.min(schokken):
bet = 0.00001
elif bet > np.max(schokken):
bet = np.max(schokken)
else:
bet=bet
ta = np.max([0,a_t_primo])
wt =
get_weights(cor_mrkt, schokken, exp_return,
stdev, bet, scr_totaal, ta)
if t == 2016 and sc_nr==500:
pass
weights.loc[t,(’w1’,sc_nr)]=wt[0]
51
weights.loc[t,(’w2’,sc_nr)]=wt[1]
weights.loc[t,(’w3’,sc_nr)]=wt[2]
weights.loc[t,(’w4’,sc_nr)]=wt[3]
X_a= weights.loc[t,(’w1’,sc_nr)] * a_t_primo*
(1+sc_set.aandelen.loc[t,sc_nr])
X_woning = a_t_primo* (1+sc_set.oghuis.loc[t,sc_nr])
* og_vast[0]
X_kantoor = a_t_primo* (1+sc_set.ogkantoor.loc[t,sc_nr])
* og_vast[1]
X_winkel = a_t_primo* (1+sc_set.ogwinkel.loc[t,sc_nr])
* og_vast[2]
X_v = weights.loc[t,(’w2’,sc_nr)]
* (X_woning + X_kantoor + X_winkel)
X_sp = weights.loc[t,(’w3’,sc_nr)] * a_t_primo*
(1+sc_set.spread.loc[t,sc_nr]+sc_set.swap.loc[t,sc_nr])
X_st = weights.loc[t,(’w4’,sc_nr)]
* a_t_primo* (1+sc_set.staats.loc[t,sc_nr])
a_t_tijdelijk=np.sum([X_a,X_v,X_sp, X_st ])
weights.loc[t,(’belegresultaat’,sc_nr)] = a_t_tijdelijk - a_t_primo
weights.loc[t,(’eigenvermogen’,sc_nr)] = np.max([ 0 , e_t_primo +
0.75*(weights.loc[t,(’belegresultaat’,sc_nr)]
+ sc_set.vvpprimo.loc[t,sc_nr]
- sc_set.vvpultimo.loc[t,sc_nr])
- 0.25*sc_set.pukkasstroom.loc[t,sc_nr]])
weights.loc[t,(’assets’,sc_nr)] =
weights.loc[t,(’eigenvermogen’,sc_nr)]
+ sc_set.vvpultimo.loc[t,sc_nr]
Ratio Check
The following script is used in section 5.5 where df1 represents the data
frame that contains the realized return of every scenario every year. The
variable df2 represents the data frame that contains the calculated shock
based on expected shortfall. The variable q is the examined quantile. The
objective of this script is to investigate tail risk, by looking at the lowest q%
realized returns or highest q% measured risk.
52
def ratio_check(df1,df2,q):
quant = np.percentile(df1,q)
return = []
shock = []
ratio = []
for i in df1.index: #jaren
for j in df1.columns: #scenarios
if df1.loc[i,j]<quant:
return.append(df1.loc[i,j])
shock.append(df2.loc[i,j])
theta = np.divide(df1.loc[i,j],df2.loc[i,j])
ratio.append(theta)
y = pd.DataFrame([return,shock,ratio], index = [’returns’,’shocks’, ’theta’] )
return y
The output of this script, when the optimization is based on VaR and
with q = 0.5% resp. q = 5% is
ratio_check(weights.asset_return, weights.asset_shock_cvar, 0.5)
Out[2]:
0
1
2
3
4
5
6
returns -0.128774 -0.180772 -0.092739 -0.083175 -0.128945 -0.081483 -0.095603
shocks
0.170550 0.170550 0.170550 0.170550 0.170551 0.170550 0.170550
theta
-0.755052 -1.059931 -0.543762 -0.487684 -0.756054 -0.477762 -0.560553
7
8
9
returns -0.090343 -0.067693 -0.059687
shocks
0.170550 0.170550 0.170550
theta
-0.529713 -0.396908 -0.349968
...
...
...
...
20
21
22
-0.122263 -0.060558 -0.069306
0.170550 0.170550 0.133192
-0.716875 -0.355075 -0.520344
23
24
25
26
27
28
29
returns -0.073353 -0.088542 -0.079825 -0.073868 -0.059064 -0.133160 -0.054113
shocks
0.170550 0.127915 0.089021 0.112180 0.170550 0.170550 0.170550
theta
-0.430095 -0.692191 -0.896702 -0.658478 -0.346315 -0.780764 -0.317287
[3 rows x 30 columns]
ratio_check(weights.asset_return, weights.asset_shock_cvar, 5)
Out[3]:
0
1
2
3
4
5
6
returns -0.013313 -0.007719 -0.005017 -0.013158 -0.009117 -0.007849 -0.006732
shocks
0.059167 0.059167 0.059167 0.059167 0.059167 0.059167 0.059167
theta
-0.225013 -0.130466 -0.084796 -0.222385 -0.154095 -0.132655 -0.113780
53
7
8
9
returns -0.021118 -0.007227 -0.003684
shocks
0.059167 0.059167 0.059167
theta
-0.356915 -0.122148 -0.062257
...
...
...
...
290
291
292
-0.005810 -0.024021 -0.014620
0.170550 0.170550 0.170550
-0.034067 -0.140847 -0.085722
293
294
295
296
297
298
299
returns -0.041691 -0.054113 -0.016520 -0.005939 -0.008520 -0.022604 -0.004540
shocks
0.098235 0.170550 0.079636 0.071301 0.054480 0.170550 0.059990
theta
-0.424406 -0.317287 -0.207438 -0.083296 -0.156395 -0.132537 -0.075676
[3 rows x 300 columns]
Here we see the 30 resp. 300 lowest realized returns (which is the 0.5% resp.
5% percentile of 1000 scenarios × 6 years) when based on VaR with the
corresponding shock and return-risk ratio.
The output of this script, when the optimization is based on ES and
with q = 0.5% resp. q = 5% is
ratio_check(weights.asset_return, weights.asset_shock_cvar, 0.5)
Out[2]:
0
1
2
3
4
5
6
returns -0.027608 -0.027713 -0.028129 -0.094455 -0.066574 -0.056968 -0.029034
shocks
0.050177 0.042604 0.152204 0.154389 0.159123 0.149188 0.109645
theta
-0.550209 -0.650474 -0.184812 -0.611795 -0.418381 -0.381856 -0.264804
7
8
9
returns -0.046488 -0.029642 -0.090790
shocks
0.163243 0.151918 0.149774
theta
-0.284776 -0.195118 -0.606180
...
...
...
...
20
21
22
-0.092640 -0.031341 -0.038260
0.153986 0.095638 0.074941
-0.601613 -0.327708 -0.510539
23
24
25
26
27
28
29
returns -0.065291 -0.034631 -0.066862 -0.026814 -0.060457 -0.058749 -0.067784
shocks
0.098932 0.086571 0.076792 0.117449 0.091748 0.151155 0.157126
theta
-0.659959 -0.400034 -0.870681 -0.228303 -0.658948 -0.388668 -0.431396
[3 rows x 30 columns]
ratio_check(weights.asset_return, weights.asset_shock_cvar, 5)
Out[3]:
0
1
2
3
4
5
6
returns -0.010411 -0.005202 -0.002998 -0.009006 -0.005932 -0.004143 -0.005090
shocks
0.051965 0.051965 0.051965 0.051965 0.051965 0.051965 0.051965
54
theta
-0.200350 -0.100100 -0.057685 -0.173310 -0.114156 -0.079735 -0.097946
7
8
9
returns -0.017110 -0.005704 -0.002606
shocks
0.051965 0.051965 0.051965
theta
-0.329262 -0.109766 -0.050154
...
...
...
...
219
220
221
-0.001002 -0.019926 -0.006471
0.069990 0.066226 0.086018
-0.014323 -0.300873 -0.075232
222
223
224
225
226
227
228
returns -0.017740 -0.067784 -0.008546 -0.014037 -0.020184 -0.004887 -0.002045
shocks
0.061322 0.157126 0.062812 0.065378 0.087461 0.046079 0.052285
theta
-0.289299 -0.431396 -0.136051 -0.214700 -0.230771 -0.106068 -0.039105
[3 rows x 229 columns]
Here we see the 30 resp. 300 lowest realized returns (which is the 0.5%
resp. 5% percentile of 1000 scenarios × 6 years) when based on ES with the
corresponding shock and return-risk ratio.
7.3
Miscellaneous Theorems and Proofs
Strong Law of Large Numbers
The strong law of large numbers states that the sample average converges
almost surely to the expected value, thus X → µ almost surely, when n →
∞. That is
P lim X n = µ = 1.
n→∞
Lusin’s Theorem
For an interval [a, b], let f : [a, b] → R be a measurable function. Then, for
every > 0, there exists a compact E ⊂ [a, b] such that f restricted to E is
continuous and
λ(E) > b − a − ε.
Note that E inherits the subspace topology from [a, b]; continuity of f restricted to E is defined using this topology.
Glivenko-Cantelli
Assume that X1 , X2 , ... are are independent and identically-distributed random variables in R with common cumulative distribution function F (x).
The empirical distribution function for X1 , . . . , Xn is defined by
n
Fn (x) =
1X
I(−∞,x] (Xi ),
n
i=1
55
where IC is the indicator function of the set C. For every (fixed) x, Fn (x) is
a sequence of random variables which converge to F (x) almost surely by the
strong law of large numbers, that is, Fn converges to F pointwise. Glivenko
and Cantelli strengthened this result by proving uniform convergence of Fn
to F .
Thus the theorem states
kFn − F k∞ = sup |Fn (x) − F (x)|−→0 a.s.
x∈R
Vitali’s Convergence Theorem
Let (X, F, µ) be a positive measure space. If
1. µ(X) < ∞
2. {fn } is uniformly integrable
3. fn (x) → f (x) almost everywhere as n → ∞ and
4. |f (x)| < ∞ almost everywhere
then the following hold:
1. f ∈ L1 (µ)
R
2. limn→∞ X |fn − f |dµ = 0 (L1 -convergence)
Hölders Inequality
Let (S, Σ, µ) be a measure space and let p, q ∈ [1, ∞] with 1/p + 1/q = 1.
Then, for all measurable real- or complex-valued functions f and g on S,
kf gk1 ≤ kf kp kgkq .
Fatou’s Lemma
Let f1 , f2 , f3 , ... be a sequence of non-negative measurable functions on a
measure space (S, Σ, µ). Define the function f : S → [0, ∞] by
s ∈ S.
f (s) = lim inf fn (s),
n→∞
Then f is measurable and
Z
Z
f dµ ≤ lim inf
S
n→∞
fn dµ .
S
Note: The functions are allowed to attain the value +∞ and the integrals
may also be infinite.
56
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