Intertemporal Surplus Management
William T. Ziemba, University of British Columbia
Markus Rudolf, WHU Koblenz
Internet-Page: http://www.whu.edu/banking
Intertemporal Surplus Management
1. Basics setting
2. One period surplus management model
3. Intertemporal surplus management model
4. Risk preferences, funding ratio, and currency
beta
5. Results
© Markus Rudolf
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Intertemporal Surplus Management
Basic setting
Assumptions
Pension Funds and life insurance companies have the legal order to invest
in order to guarantee payments; the liabilities of such a company a
determined by the present value of the payments
The growth of the value of the assets under management has to be
orientated at the growth of the liabilities
Liabilities and assets are characterized by stochastic growth rates
Surplus Management: Investing assets such that the ratio between assets
and liabilities always remains greater than one; i.e. such that the value of the
assets exceeds the value of the liabilities in each moment of time
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Intertemporal Surplus Management
Basic setting
References
Andrew D. Roy (1952), Econometrica, the safety first principle, i.e.
minimizing the probability for failing to reach a prespecified (deterministic)
threshold return by utilizing the Tschbyscheff inequality
Martin L. Leibowitz and Roy D. Henriksson (1988), Financial Analysts
Journal, shortfall risk criterion: revival of Roy's safety first approach under
stronger assumptions (normally distributed asset returns) and application on
surplus management
William F. Sharpe and Lawrence G. Tint (1990), The Journal of Portfolio
Management, optimization of asset portfolios respecting stochastic liability
returns, a closed form solution for the optimum portfolio selection
Robert C. Merton (1993) in Clotfelter and Rothschild, "The Economics of
Higher Education", optimization of a University's asset portfolios where the
liabilities are given by the costs of its activities; the activity costs are
modeled as state variables
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Intertemporal Surplus Management
Basic setting
References continuous time finance
Robert C. Merton (1969 and 1973), The Review of Economics and
Statistics and Econometrica, Introduction of the theory of stochastic
processes and stochastic programming into finance, developement of the
intertemporal capital asset pricing model, the base for the valuation of
contingent claims such as derivatives
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Intertemporal Surplus Management
One period surplus management model
Notation
The variance of the asset portfolio
2
A
The variance of the liabilities
 L2
The covariance between the asset portfolio and the liabilities:
 AL
The vector of portfolio fractions of the risky portfolio:
   1,, n 
 A   E1,, E n 
  11   1n 


V  
 


 n1   nn 
The vector of expected asset returns:
The covariance matrix of the assets:
The vector of covariances between the assets and the liabilities: V    ,, 
AL
1L
nL
e   1
, ,1   n
The unity vector:
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Intertemporal Surplus Management
One period surplus management model
Basic setting
Goal of the model: Identify an asset portfolio (consisting out of n risky and one
riskless asset) which reveals a minimum variance of the surplus return
The surplus return definition according to Sharpe and Tint (1990):






~
~
~
~
~
St  1  St
At  1  Lt  1  At  Lt 
At  1  At
Lt  1  Lt
1 ~
~
~
RSt 


 Lt
 R At  RLt
At
At
At
At Lt
Ft
The expected surplus return:
 
1
~
E RS E A  E L
F
The surplus variance:
 
1 2
1
~
2
Var RS  A

 L  2   AL
F
F2
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Intertemporal Surplus Management
One period surplus management model
Basic setting
The expected surplus return in terms of assets returns:
1
~
E RS  E S    A  re   r  E L
F
 
The variance of the surplus return in terms of asset returns:
1 2
1
~
Var RS   S2   V 
 L  2   V AL
F
F2
 
The Lagrangian:
L   V 
1
F2
 L2  2
1
1


  V AL     A  re   r  E L  E S 
F
F


The optimum condition in a one period setting:

1 1

V V AL  V 1 A  re 
F
2
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Intertemporal Surplus Management
One period surplus management model
Basic setting
Interpretation of the result:
The optimum portfolio consists out of two portfolios:
1
1. V   A  re This is the tangency portfolio in the CAPM framework
1
2. V VAL
This is minimum surplus return variance portfolio
The concentration on one of these portfolios is dependent on the Lagrange
multiplier This is the grade of appreciation of an additional percent of expected
surplus return in terms of additional surplus variance (a risk aversion factor: how
much additional risk is an investor willing to take for an additional percent of
return).
Usual portfolio theory results in the context of surplus management.
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Intertemporal Surplus Management
Intertemporal surplus management model
Objective function
Goal of the model: Maximize the lifetime expected utility of a surplus management
policy. It is assumed that the input parameters of the model fluctuate randomly in
time depending on a state variable Y.
The asset and the liabilitiy returns follow Itô-processes: standard Wiener
processes
~
~ ~
dz
dA
~
A z A dt
RA 
 E A (Y , t )dt   A Y , t   dz~A
A
~
dL
~
dz~L  ~
z L dt
RL 
 E L (Y , t )dt   L Y , t   dz~L
L
The state variable follows a geometric Brownian motion:
~
dY
 EY dt   Y dz~Y
Y
dz~Y  ~
zY dt
This setting is equivalent to Merton (1973).
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Intertemporal Surplus Management
Intertemporal surplus management model
Transformations
Substituting these definitions into the definition for the surplus return yields:
~
~
~
dS dA 1 dL
~
RS 

 
A
A F L
1
1




  E A Y , t   E L Y , t dt    A Y , t z~A   L Y , t z~L  dt
F
F




1
~ 

E dS   E A  E L   A  dt
F


 
 
2
 
1
1

~2



E dS  E   E A  E L   A  dt    A z~A   L z~L  A dt  
F
F



 
 
2

1
  2
~
~
 E   A z A   L zL    A  dt
weil dt2 = dt3/2 =0
F
 

 2
   A
E

© Markus Rudolf
z~A2  
1
F2
 
 L2  E z~L2 

1
 2   A   L  E z~A  z~L   A 2  dt
F

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Intertemporal surplus management model
Transformations
   
~
~
~2
~2
Because E z A   E zL   0 and E z A  E zL  1 and
~
~
~
~
 A   L  E z~A  z~L   E R A  RL  E R A  E A RL  E L   AL





follows:
 
 2

1 2
1
~
E dS 2    A

 L  2  AL   A 2  dt
F


F2
Furthermore, analogously:


A
~ ~
E dS  dY  AY A Y E z~A  z~Y dt  Y L Y E z~L  z~Y dt
F
1


   AY   LY   A  Y  dt
F


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Intertemporal Surplus Management
Intertemporal surplus management model
J - function
The J-function is the expected lifetime utility in the decision period t ...
T
T

 t  dt




J  S ,Y , t   max Et   U  S ,Y ,  d   max Et   U  S ,Y ,  d   U  S ,Y ,  d 


t

 t

t  dt
 t  dt

~
~
 max Et   U  S ,Y ,  d  J S  dS ,Y  dY , t  dt 

 t

~
~
 U  S ,Y , t  dt  max Et J S  dS ,Y  dY , t  dt





1
1 2 ~2
~
~
~ ~
~


 U dt  max Et  J  J S dS  dYJY  J t dt  J SS dS 2  JYY
dY  J SY dSdY  o dt  
2
2



by applying Itô's lemma. It follows the fundamental partial differential equation
of intertemporal portfolio optimization:
1
1 2 ~2
~
~
~ ~
~


0  U dt  max Et  J S dS  dYJY  J t dt  J SS dS 2  JYY
dY  J SY dSdY  o dt  
2
2



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Intertemporal Surplus Management
Intertemporal surplus management model
Optimum condition
Substituting in the definitions of expected returns, variances and covariances:
1
1
1
1

0  max  JS  EA  EL   A  JY EY  Jt  JSS   2A  2  2L  2  AL   A2


F 
2
F
  
F
1 2 2 2
1
 JYY
 Y  Y  JSY   AY   LY   A  Y  U 


2
F

Substituting in the following definitions ...
E A     A  re  r
 2A   V
 AL   VAL
 AY   VAY
provides:
1
1
1
1



0  max  J S     A  re  r  E L   A  J Y EY  J t  J SS   V  2  2L  2  VAL   A 2



F
2
F
  
F
1
1

 J YY  Y2  Y 2  J SY   VAY   LY   A  Y  U 


2
F

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Intertemporal Surplus Management
Intertemporal surplus management model
Optimum portfolio weights
Differentiating this expression with respect to the vector of portfolio fractions  yields ...
J
YJ
1
   S V 1   A  re  SY V 1VAY  V 1VAL
AJ SS
AJ SS
F
 a
JS
YJ
1
 M  b SY  Y  c  L
AJ SS
AJ SS
F
where the following constants are defined:
a  e V 1   A  re
© Markus Rudolf
b  e V 1VAY
c  e V 1VAL
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Intertemporal Surplus Management
Intertemporal surplus management model
Four fund separation theorem
Maximizing lifetime expected utility of a surplus optimizer can be realized by
investing in three portfolios:
1. The market portfolio which is the tangency portfolio in a classical CAPM
framework:
M 
V 1  A  re
eV 1  A  re
2. The hedge portfolio for the state variable Y which corresponds to Merton's
(1973) intertemporal CAPM:
Y 
V 1V AY
eV 1V AY
3. A hedge portfolio for the fluctuations of the liabilities:
© Markus Rudolf
Page 15
L 
V 1V AL
eV 1V AL
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Intertemporal Surplus Management
Intertemporal surplus management model
Interpretation
Classical result of Merton (1973): Both, the market portfolio and the hedge
portfolio for the state variable, are hold in accordance to the risk aversion towards
fluctuations in the surplus and the state variable.
New result of the intertemporal surplus management model: The weight of the
hedge portfolio of the liability returns if c/F which implies that all investors choose a
hedging opportunity for the liabilities independent of their preferences. The only
factor which influences this holding is the funding ratio of a pension fund.
The reason: All pension funds are influenced by wage fluctuations in the same
way, whereas market fluctuaions only affect such investors with high exposures in
the market.
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Intertemporal surplus management model
Why is V-1VAY a hedge portfolio
Because this portfolio reveals the maximum correlation with the state variable:
~ 
 R

  1  dY~ 
   V AY  max s.t . V   2
Cov     ,
A

  ~  Y 


  Rn 



L
1 1
2
L   V AY    V   A

 V AY  2V  0 
V V AY

2
which is identical with the hedge portfolio. Furthermore:
2V  V AY 2 V   V AY 2 
© Markus Rudolf
 AY
  AY V 1V AY   AY 
2
A
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Risk preferences, funding ratio, and currency beta
Four fund theorem
All investors hold four portfolios:
• The market portfolio
• The liabilities hedge portfolio
• One portfolio for each state variable
• The cash equivalent
The composition out of these portfolios depends on preferences, which are hardly
interpretable.
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Risk preferences, funding ratio, and currency beta
Different utility functions
Assumption of HARA-utility function (U  HARA <=> J  HARA)
T S  


U S,Y , t  
J S,Y , t   max E t 
d 




, where SS   SAY , 
t 

Note that this implies the class of log-utility for  approaching 0:
S
S


lim U S,Y , t  lim
 lim S   ln S ln S
 0
 0   0
Under this assumption we have:
J S  S  1 J SS    1  S   2
dS AY  1
dS dA
dA
J SY 
 J SS 

 J SS 
.
dY
dA dY
dY
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Intertemporal Surplus Management
Risk preferences, funding ratio, and currency beta
Portfolio holdings
The holdings of the market portfolio are:

JS
S
1


A  J SS
A  1 1  
1

 1  
 F
which simplifies in the log utility case to:

JS
S
1

 1
A  J SS
A  1
F
The holdings of the state variable hedge portfolio are:
~
Y  J SY
RA
dA / A


~
A  J SS
dY / Y
RY
The holdings of the liability hedge portfolio is independent of the class of
utility function.
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Intertemporal Surplus Management
Risk preferences, funding ratio, and currency beta
Portfolio holdings
The portfolio holdings:
1. The market portfolio by the amount of
a
2. The liability hedge portfolio by the amount of
c
JS
S
1

 a   a  1  
A  J SS
A
 F
1
F
3. The hedge portfolio by the amount of:
1


Y  d
dY 
Y
Y  J SY
 S AY 

b
 b 
 b 
A
A  J SS
S2
dA
~
R
dA
A
dY  b 
 b  ~ A
A
dY Y
RY

4. The cash equivalent portfolio
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Intertemporal Surplus Management
Risk preferences, funding ratio, and currency beta
Hedge ratio and portfolio holdings
Assuming the regression model without constant provides:
~
RA
~
~ ~
~
~ ~
R A   R A , RY  RY   R A , RY  ~
RY
~
~
E R A  RY
 AY
~ ~
  R A , RY  


~
E RY2
 Y2






 


~
Y  J SY
RA

b
 b  ~  b   R A , RY   b  AY
A  J SS
RY
2
Y
The risk tolerance against the state variable equals the negative hedge ratio of
the portfolio against the state variable. If the state variable is an exchange rate,
the risk tolerance equals the negative currency hedge ratio.
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Intertemporal Surplus Management
Risk preferences, funding ratio, and currency beta
Hedge ratio and portfolio holdings
Assumption 1: The state variable is the exchange rate risk
Assumption 2: There are k foreign currency exposures. Each foreign currency is
represented by a state variable.
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Intertemporal Surplus Management
Risk preferences, funding ratio, and currency beta
Hedge ratio and portfolio holdings
The the following modifications
have to be implemented:
(a)
The state variables
with returns:
~
~
RY1 ,, RYk
(b)
The risk tolerances
for state variables:
Y1J SY1
Yk J SYk
~ ~
~ ~

   R A , RY1 ,,
   R A , RYk
AJ SS
AJ SS
(c)
The covariances
between the state
variables and the
assets:
V AY1 ,,V AYk
(d)
The hedge portfolios:
Y1 
(e)
The b-coefficients:
b1  e V 1V AY1 ,, bk  e V 1V AYk
© Markus Rudolf

V 1V AY1
e V
1
V AY1
Page 24

,,Yk 


V 1V AYk
e V 1V AYk
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Risk preferences, funding ratio, and currency beta
Hedge ratio and portfolio holdings
Substituting these expressions into the equation for the portfolio allocation
provides the optimum asset holdings of an internationally diversified pension
fund ...


k
1
1

  a1   M   bi  R A , RYi Yi  c  L
F
F

i 1
where
 R A , RYi  
  l  R Al , RYi 
n
l 1
Problem: The portfolio beta depends on the asset allocation vector . No
analytical solution is possible.
Solution: Approximating the portfolio weights by a numerical procedure.
© Markus Rudolf
Page 25
BFS meeting
Intertemporal Surplus Management
Case study
Table 1:
Descriptive statistics
The stock data is based on MSC indices and the bond data on JP Morgan indices (Switzerland on Salomon
Brothers date). The wage and salary growth rate is from Datastream. Monthly data between January 1987 and
July 2000 (163 observations) is used. All coefficients are in USD. The average returns and volatilities are in
percent per annum.
Mean Volatility
return
Stocks
Bonds
Exchange
rates in
USD
USA
UK
Japan
EMU countries
Canada
Switzerland
USA
UK
Japan
EMU countries
Canada
Switzerland
GBP
JPY
EUR*
CAD
CHF
Wages and salaries
13.47
9.97
3.42
10.48
5.52
11.56
5.04
6.86
3.77
7.78
5.16
3.56
0.11
-2.75
1.13
0.79
0.32
5.71
14.74
17.96
25.99
15.8
18.07
18.17
4.5
12.51
14.46
10.57
8.44
12.09
11.13
12.54
10.08
4.73
11.57
4.0
Beta
Beta JPY
GBP
0.18
-0.47
-0.61
-0.32
0.05
-0.14
-0.03
-0.92
-0.53
-0.69
-0.09
-0.67
1
0.48
0.7
0.08
0.69
0
0.05
-0.36
-1.11
-0.27
-0.02
-0.32
0
-0.44
-1.04
-0.42
0.03
-0.53
0.37
1
0.42
0
0.52
0.01
Beta
EUR
0.35
-0.29
-0.45
-0.26
0.27
-0.26
-0.06
-0.8
-0.75
-0.93
-0.02
-1.03
0.86
0.65
1
0.02
1.04
0
Beta
CAD
-0.59
-0.48
-0.44
-0.45
-1.44
0.13
-0.04
-0.39
0.09
-0.06
-1.14
0.19
0.42
0.04
0.1
1
-0.13
-0.01
Beta
CHF
0.35
-0.14
-0.43
-0.11
0.34
-0.32
-0.06
-0.59
-0.72
-0.74
0.02
-0.99
0.64
0.61
0.79
-0.02
1
0
*: ECU before January 1999
© Markus Rudolf
Page 26
BFS meeting
Intertemporal Surplus Management
Case study
Table 2:
Optimum portfolios of an internationally diversified pension fund
The portfolio holdings are based on equation (9). All portfolio fractions are percentages. A riskless rate of
interest of 2% per annum is assumed.
Stocks
Bonds
USA
UK
Japan
EMU
Canada
Switzerl.
USA
UK
Japan
EMU
Canada
Switzerl.
© Markus Rudolf
Market
portfolio
Liability
hedge
portfolio
83.9
-14.8
-6.7
-19.2
-39.2
21.6
14.8
-9.7
0.6
138.5
7.9
-77.7
-30.4
60.6
2.7
-68.3
5.1
1.5
126.0
-56.8
39.1
9.6
5.0
5.9
Hedge
portfolio
GBP
3.9
-31.0
6.1
35.3
14.6
-24.8
-126.4
189.7
-31.2
-16.4
-11.8
91.9
Hedge
portfolio
JPY
Hedge
portfolio
EUR
-4.5
-1.1
8.9
12.8
13.6
-14.6
-28.6
7.9
133.9
-38.0
-32.6
42.4
-7.5
-8.6
-2.4
23.5
5.5
-14.7
-41.2
-3.4
-3.6
97.3
-7.9
62.9
Page 27
Hedge
portfolio
CAD
60.9
-85.3
-4.0
138.3
-2.6
-100.7
-627.3
97.5
-30.1
-170.1
679.6
143.8
Hedge
portfolio
CHF
-5.1
1.9
-0.7
-3.8
-0.9
1.0
-35.9
-8.5
6.4
34.0
0.9
110.7
BFS meeting
Intertemporal Surplus Management
Case study
Table 3
Weightings of the funds due to different funding ratios
The weightings of the portfolios according to equation (16), where =0, i.e. log utility, is assumed. The
weightings of the eight funds are in percent.
Funding ratio
Market portfolio
Liability hedge portfolio
Hedge portfolio GBP
Hedge portfolio JPY
Hedge portfolio EUR
Hedge portfolio CAD
Hedge portfolio CHF
Riskless assets
Portfolio beta against GBP
Portfolio beta against JPY
Portfolio beta against EUR
Portfolio beta against CAD
Portfolio beta against CHF
© Markus Rudolf
0.9
-11.5%
14.8%
-0.6%
1.2%
0.3%
-0.1%
1.1%
94.9%
-0.01
0.02
0.01
-0.02
0.02
1
0.0%
13.3%
-0.5%
1.1%
0.2%
-0.1%
1.0%
85.1%
-0.01
0.02
0.00
-0.02
0.01
1.1
1.2
1.3
1.5
6.3%
12.1%
-0.5%
1.0%
0.1%
-0.1%
0.9%
80.2%
-0.01
0.02
0.00
-0.01
0.01
12.3%
11.1%
-0.5%
0.9%
0.0%
-0.1%
0.8%
75.3%
-0.01
0.02
0.00
-0.01
0.01
18.2%
10.2%
-0.4%
0.9%
0.0%
-0.1%
0.8%
70.4%
-0.01
0.02
0.00
-0.01
0.01
24.6%
8.9%
-0.4%
0.8%
-0.1%
-0.1%
0.7%
65.6%
-0.01
0.02
0.00
-0.01
0.01
Page 28
BFS meeting
Intertemporal Surplus Management
 

E dS   At     A  L   dt
F t  


2
  
dS t   dS t   A 2 t     A2  2 L  2  AL A L
F t 
F t 

   

dS t   dY t   At   Y t     AY  LY L Y   dt
F t 



  dt


dY  dY  Y 2 t    Y2  dt ,
© Markus Rudolf
Page 29
BFS meeting