© Markus Rudolf Page 1 Intertemporal Surplus Management BFS meeting Internet-Page: Intertemporal...

© Markus Rudolf Page 1

Intertemporal Surplus Management

BFS meeting

Internet-Page: http://www.whu.edu/banking


1. Basics setting2. One period surplus management model3. Intertemporal surplus management model4. Risk preferences, funding ratio, and currency

beta5. Results

William T. Ziemba, University of British ColumbiaMarkus Rudolf, WHU Koblenz



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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

Basic settingAssumptions



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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

Basic settingReferences



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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

Basic settingReferences continuous time finance



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The variance of the asset portfolio

The variance of the liabilities

The covariance between the asset portfolio and the liabilities:

The vector of portfolio fractions of the risky portfolio:

The vector of expected asset returns:

The covariance matrix of the assets:

The vector of covariances between the assets and the liabilities:

The unity vector:

2A

2L

AL

n ,,1

nA EE ,,1

nnn

n

V

1

111

nLLALV ,,1

ne 1,,1

One period surplus management modelNotation



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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):

The expected surplus return:

The surplus variance:

Lt

tAt

tt

ttt

t

tt

t

tttt

t

ttSt R

FR

LA

LLL

A

AA

A

LALA

A

SSR ~1~

~~~~~~ 11111

LAS EF

ERE1~

ALLAS FFRVar 1

21~ 22

2

One period surplus management modelBasic setting



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The expected surplus return in terms of assets returns:

The variance of the surplus return in terms of asset returns:

The Lagrangian:

The optimum condition in a one period setting:

LASS EF

rreERE1~

ALLSS VFF

VRVar 1

21~ 22

2

SLAALL EE

FrreV

FFVL

112

1 22

reVVVF AAL 11

2

1




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Interpretation of the result:

The optimum portfolio consists out of two portfolios:

1. This is the tangency portfolio in the CAPM framework

2. 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.

V reA 1

V VAL 1




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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

The state variable follows a geometric Brownian motion:

This setting is equivalent to Merton (1973).

~~

( , ) , ~

~~

( , ) , ~

RdA

AE Y t dt Y t dz

RdL

LE Y t dt Y t dz

A A A A

L L L L

dY

YE dt dzY Y Y

~~

dz z dtA A~ ~

dz z dtL L~ ~

dz z dtY Y~ ~

Intertemporal surplus management modelObjective function



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Substituting these definitions into the definition for the surplus return yields:

dtztYF

ztYdttYEF

tYE

L

Ld

FA

Ad

A

SdR

LLAALA

S

~,1~,,

1,

~1~~~

dtAEF

ESdE LA

1~

22 ~1~1~ dtAz

FzdtAE

FEESdE LLAALA

dtAzF

zE LLAA

2

2~1~ weil dt2 = dt3/2 =0

dtAzzEF

zEF

zE LALALLAA

222

222 ~~2

1~1~

Intertemporal surplus management modelTransformations



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Because and and

Furthermore, analogously:

0~~ LA zEzE 1~~ 22 LA zEzE

dtAFF

SdE ALLA

22

222 1

21~

ALLLAALALALA ERERERREzzE ~~~~~~ follows:

dtYAF

dtzzEYF

AdtzzEAYYdSdE

LYAY

YLYLYAYA

1

~~~~~~

Intertemporal surplus management modelTransformations



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The J-function is the expected lifetime utility in the decision period t ...

by applying Itô's lemma. It follows the fundamental partial differential equation of intertemporal portfolio optimization:

J S Y t E U S Y d E U S Y d U S Y d

E U S Y d J S dS Y dY t dt

U S Y t dt E J S dS Y dY t dt

U dt E J J dS dYJ J dt J dS J

tt

T

tt

t dt

t dt

T

tt

t dt

t

t S Y t SS

, , max , , max , , , ,

max , ,~,

~,

, , max~,

~,

max~ ~ ~

1

2

1

22 YY SYdY J dSdY o dt2 2~ ~ ~

01

2

1

22 2 2

U dt E J dS dYJ J dt J dS J dY J dSdY o dtt S Y t SS YY SYmax~ ~ ~ ~ ~ ~

Intertemporal surplus management modelJ - function



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Substituting in the definitions of expected returns, variances and covariances:

Substituting in the following definitions ...

provides:

01 1

2

12

1

1

2

1

22

2 2

2 2 2

max

J EFE A J E J J

F FA

J Y JF

A Y U

S A L Y Y t SS A L AL

YY Y SY AY LY

E re r V V VA A A AL AL AY AY 2

01 1

2

12

1

1

2

1

22 2

2 2

max

J re rFE A J E J J V

F FV A

J Y J VF

A Y U

S A L Y Y t SS L AL

YY Y SY AY LY

Intertemporal surplus management modelOptimum condition



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Differentiating this expression with respect to the vector of portfolio fractions yields ...

where the following constants are defined:

a e V re b e V V c e V VA AY AL 1 1 1

J

AJV re

YJ

AJV V

FV V

aJ

AJbYJ

AJcF

S

SSA

SY

SSAY AL

S

SSM

SY

SSY L

1 1 11

1

Intertemporal surplus management modelOptimum portfolio weights



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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:

2. The hedge portfolio for the state variable Y which corresponds to Merton's (1973) intertemporal CAPM:

3. A hedge portfolio for the fluctuations of the liabilities:

MA

A

V re

e V re

1

1

Y AY

AY

V V

e V V

1

1

L AL

AL

V V

e V V

1

1

Intertemporal surplus management modelFour fund separation theorem



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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.

Intertemporal surplus management modelInterpretation



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Because this portfolio reveals the maximum correlation with the state variable:

which is identical with the hedge portfolio. Furthermore:

21

..max~

,~

~

AAY

n

VtsVY

Yd

R

R

Cov

AYAYAAY VVVVL

VVL 12

2

102

AYAYAY

A

AYAYAY VVVVVV 1

2222

Intertemporal surplus management modelWhy is V-1VAY a hedge portfolio



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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.

Risk preferences, funding ratio, and currency betaFour fund theorem



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T

tt d

SEtYSJ

StYSU

max,,,,

,YASSS

Assumption of HARA-utility function (U HARA <=> J HARA)

, where

Note that this implies the class of log-utility for approaching 0:

Under this assumption we have:

SSSS

tYSU lnln0

lim0

lim,,0

lim

.

1

1

21

dY

dAJ

dY

dA

dA

dSJ

dY

YAdSJ

SJSJ

SSSSSY

SSS

Risk preferences, funding ratio, and currency betaDifferent utility functions



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The holdings of the market portfolio are:

which simplifies in the log utility case to:

The holdings of the state variable hedge portfolio are:

The holdings of the liability hedge portfolio is independent of the class of

utility function.

FA

S

JA

J

SS

S 11

1

1

1

Y

A

SS

SY

R

R

YdY

AdA

JA

JY~

~

/

/

FA

S

JA

J

SS

S 11

1

Risk preferences, funding ratio, and currency betaPortfolio holdings



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The portfolio holdings:

1. The market portfolio by the amount of

2. The liability hedge portfolio by the amount of

3. The hedge portfolio by the amount of:

4. The cash equivalent portfolio

Fc

1

Fa

A

Sa

JA

Ja

SS

S 11

Y

A

SS

SY

R

Rb

YdY

AdAb

AdY

dAY

b

S

A

dYYAS

dY

bJA

JYb ~

~1

2

Risk preferences, funding ratio, and currency betaPortfolio holdings



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Assuming the regression model without constant provides:

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.

Y

AYAYYAA

R

RRRRRRR ~

~~,~~~,~~

22~

~~~,~

Y

AY

Y

YAYA

RE

RRERR

2

,~

~

Y

AYYA

Y

A

SS

SY bRRbR

Rb

JA

JYb

Risk preferences, funding ratio, and currency betaHedge ratio and portfolio holdings



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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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kYY RR ~,,~1

k

kYA

SS

SYkYA

SS

SYRR

AJ

JYRR

AJ

JY ~,~,,~,~1

11

kAYAY VV ,,1

k

kk

AY

AYY

AY

AYY

VVe

VV

VVe

VV

1

1

1

1

,,

1

11

kAYkAY VVebVVeb 111 ,,

1

The the following modifications have to be implemented:

(a) The state variableswith returns:

(b) The risk tolerances for state variables:

(c) The covariancesbetween the statevariables and the assets:

(d) The hedge portfolios:

(e) The b-coefficients:




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Substituting these expressions into the equation for the portfolio allocation

provides the optimum asset holdings of an internationally diversified pension

fund ...

where

Problem: The portfolio beta depends on the asset allocation vector . No

analytical solution is possible.

Solution: Approximating the portfolio weights by a numerical procedure.

L

k

iYYAiM F

cRRbF

aii

1,

11

1

n

lYAlYA ili

RRRR1

,,




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Case studyTable 1:

Descriptive statistics

The stock data is based on MSC indices and the bond data on JP Morgan indices (Switzerland on SalomonBrothers date). The wage and salary growth rate is from Datastream. Monthly data between January 1987 andJuly 2000 (163 observations) is used. All coefficients are in USD. The average returns and volatilities are inpercent per annum.

Meanreturn

Volatility BetaGBP

Beta JPY BetaEUR

BetaCAD

BetaCHF

Stocks USA 13.47 14.74 0.18 0.05 0.35 -0.59 0.35

UK 9.97 17.96 -0.47 -0.36 -0.29 -0.48 -0.14

Japan 3.42 25.99 -0.61 -1.11 -0.45 -0.44 -0.43

EMU countries 10.48 15.8 -0.32 -0.27 -0.26 -0.45 -0.11

Canada 5.52 18.07 0.05 -0.02 0.27 -1.44 0.34

Switzerland 11.56 18.17 -0.14 -0.32 -0.26 0.13 -0.32

Bonds USA 5.04 4.5 -0.03 0 -0.06 -0.04 -0.06

UK 6.86 12.51 -0.92 -0.44 -0.8 -0.39 -0.59

Japan 3.77 14.46 -0.53 -1.04 -0.75 0.09 -0.72

EMU countries 7.78 10.57 -0.69 -0.42 -0.93 -0.06 -0.74

Canada 5.16 8.44 -0.09 0.03 -0.02 -1.14 0.02

Switzerland 3.56 12.09 -0.67 -0.53 -1.03 0.19 -0.99

Exchange GBP 0.11 11.13 1 0.37 0.86 0.42 0.64

rates in JPY -2.75 12.54 0.48 1 0.65 0.04 0.61

USD EUR* 1.13 10.08 0.7 0.42 1 0.1 0.79

CAD 0.79 4.73 0.08 0 0.02 1 -0.02

CHF 0.32 11.57 0.69 0.52 1.04 -0.13 1

Wages and salaries 5.71 4.0 0 0.01 0 -0.01 0

*: ECU before January 1999



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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 ofinterest of 2% per annum is assumed.

Marketportfolio

Liabilityhedge

portfolio

Hedgeportfolio

GBP

Hedgeportfolio

JPY

Hedgeportfolio

EUR

Hedgeportfolio

CAD

Hedgeportfolio

CHF

Stocks USA 83.9 -30.4 3.9 -4.5 -7.5 60.9 -5.1

UK -14.8 60.6 -31.0 -1.1 -8.6 -85.3 1.9

Japan -6.7 2.7 6.1 8.9 -2.4 -4.0 -0.7

EMU -19.2 -68.3 35.3 12.8 23.5 138.3 -3.8

Canada -39.2 5.1 14.6 13.6 5.5 -2.6 -0.9

Switzerl. 21.6 1.5 -24.8 -14.6 -14.7 -100.7 1.0

Bonds USA 14.8 126.0 -126.4 -28.6 -41.2 -627.3 -35.9

UK -9.7 -56.8 189.7 7.9 -3.4 97.5 -8.5

Japan 0.6 39.1 -31.2 133.9 -3.6 -30.1 6.4

EMU 138.5 9.6 -16.4 -38.0 97.3 -170.1 34.0

Canada 7.9 5.0 -11.8 -32.6 -7.9 679.6 0.9

Switzerl. -77.7 5.9 91.9 42.4 62.9 143.8 110.7



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Case study

Table 3Weightings 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. Theweightings of the eight funds are in percent.

Funding ratio 0.9 1 1.1 1.2 1.3 1.5

Market portfolio -11.5% 0.0% 6.3% 12.3% 18.2% 24.6%Liability hedge portfolio 14.8% 13.3% 12.1% 11.1% 10.2% 8.9%

Hedge portfolio GBP -0.6% -0.5% -0.5% -0.5% -0.4% -0.4%

Hedge portfolio JPY 1.2% 1.1% 1.0% 0.9% 0.9% 0.8%

Hedge portfolio EUR 0.3% 0.2% 0.1% 0.0% 0.0% -0.1%

Hedge portfolio CAD -0.1% -0.1% -0.1% -0.1% -0.1% -0.1%

Hedge portfolio CHF 1.1% 1.0% 0.9% 0.8% 0.8% 0.7%

Riskless assets 94.9% 85.1% 80.2% 75.3% 70.4% 65.6%

Portfolio beta against GBP -0.01 -0.01 -0.01 -0.01 -0.01 -0.01

Portfolio beta against JPY 0.02 0.02 0.02 0.02 0.02 0.02

Portfolio beta against EUR 0.01 0.00 0.00 0.00 0.00 0.00

Portfolio beta against CAD -0.02 -0.02 -0.01 -0.01 -0.01 -0.01

Portfolio beta against CHF 0.02 0.01 0.01 0.01 0.01 0.01



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,

2

22

2

222

dttYdYdY

dttF

tYtAtdYtdS

dttFtF

tAtdStdS

dttF

tAdSE

Y

YLLYAY

LAALLA

LA

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