We consider a portfolio built according to the Capital Market Line of the Capital-Asset-Pricing Model. The universe of asset classes include marketable shares and bonds only. We investigate losses that emerge when the rate of return of the portfolio is lower than that required to fulfil a defined obligation. We will classify these losses and calculate upper limits for them.portfolio CAPM loss

U. Spreitzer

V. Reznik

English

Research Papers in Economics

CAP_Loss_g.doc p.1 22.01.2004
An investigation of a portfolio-loss under the CAPM
V. Reznik and U. Spreitzer
Abstract:
We consider a portfolio built according to the Capital Market Line of the Capital-Asset-Pricing
Model. The universe of asset classes include marketable shares and bonds only. We investigate
losses that emerge when the rate of return of the portfolio is lower than that required to fulfil a
defined obligation. We will classify these losses and calculate upper limits for them. 
1  Introduction
In Germany, pension liabilities that are financed internally by so called “book reserves”,
support fund or pension funds are almost entirely underwritten by the
Pensionssicherungsverein auf Gegenseitigkeit (PSV aG). This institution levies an insurance
premium proportional to the value (measured by the accrued actuarial liability using the
Entry Age Normal Valuation Method with fixed assumptions) of the legally vested benefit
obligation. The funding method used by the PSV aG can be described as a Pay-As-You-Go
method until retirement commencement. For retirees, however, the full present value (on
very conservative assumptions) is provided to a consortium of insurers, that account for
experience gains and losses in an own profit and loss statement towards the PSV aG
(Terminal Funding). The central question to ask therefore is: what percentage of the
obligation is used as a premium (over the last 30 years, the percentage has been fairly
volatile), and is this premium a good measure of the risk not to fulfil the obligation?
Several reports 
1 still suggest a premium which is proportional to the value of the legally
vested obligation but that this premium should be modified in accordance with the different
pension vehicle (pension fund, support fund, book reserve, etc.) in operation. 
In this publication we give a definition of this risk or loss, based on the so called lower partial
moments as published by Fishburn
2.CAP_Loss_g.doc p.2 22.01.2004
We consider a portfolio consisting of bonds and shares. Portfolios containing derivatives and
other instruments such as futures, options, etc. are beyond the scope of this paper. We
describe the shares as assets with a variable rate of return. We assume that the capital market
can be described with the Capital Asset Pricing Model (CAPM)
3, and that the investment of
the portfolio corresponds to the Capital Market Line of the CAPM. Our model is a one
periodic model. We give a measure of this loss as a function of expected rate of return,
variance, etc. of the assets.
2  Selection of losses
We have selected three different losses: the average loss LA , the maximum loss LM and the
value-at-risk-loss LE. We assume that the rate of return R and the probability density p(R) of
the portfolio are random variables with real values. From this we conclude that the following
integrals which define the losses are well defined. 
2.1  The average loss LA
We define LA 
1)  ∫
∞ −
− =
Rnec
nec A dR R p R R I L ) ( ) (
Rnec is the rate of return, which is necessary to fulfil obligations. I is the amount of the
investment. The probability to obtain a loss larger than 0 is:
2)  ∫
∞ −
nec R
dR R p ) (
with
3)  1 ) ( = ∫
∞
∞ −
dR R p .
We use 1) instead of:CAP_Loss_g.doc p.3 22.01.2004
4)  ∫
+∞
∞ −
− = dR R p R R I L nec A ) ( ) (
because  the yield above Rnec,is not of immediate interest. Rather, we focus on the value of
the loss.
2.2  The maximum loss LM
We define LM 
5)  ∫
∞ −
− =
min
) ( ) (
R
nec M dR R p R R I L .
We call Rmin the minimum rate of return. This means that for a rate of return above Rmin we
assume that the loss is not relevant.
2.3  The value-at-risk loss LV
LV is the loss which is caused by the very low rates of return from the portfolio. We define it
similar to the Value-at-Risk
4. Therefore we consider the 1-γ-Quantil, which is defined as
follows: 
6)  γ = ≥ − ) ( VaR R R P nec .
VaR is the rate of return, for which only γ of all R are below. LV is therefore
7)  ∫
−
∞ −
− =
VaR Rnec
nec V R d R p R R L ) ( ) ( ) (
Concerning value-at-risk there is also the following definition
4:
8)  σ γ z VaR =
with z γ is the γ -Quantil of standard normal distribution.
3  Estimation of loss of the portfolio
We consider a portfolio whose investment is a combination of the bonds and the so called
tangency  portfolio which consists of assets only. The rate of return of this (pension fund)
portfolio RPF is:CAP_Loss_g.doc p.4 22.01.2004
9)  TP PF R R R ) 1 ( 0 α α − + =
the expected rate of return is EPF (or µPF)
10)  TP PF R µ α α µ ) 1 ( 0 − + = ;
and the standard deviation σPF
11)  TP PF σ α σ ) 1 ( − = ;
with RTP is the rate of return, µTP the expected rate of return, σTP the standard deviation of the
Tangency portfolio and R0 the rate of return of the bonds. Also we assume 0≤α≤1. This
investment on this tangent is also called an investment on the capital market line
3. Using 1)
we obtain for the loss LA:
12)  ∫
∞ −
− + − + − − − =
nec R
TP TP TP nec A R R d R R p R R R I L ) ) 1 ( ( ) ) 1 ( ( ) ) 1 ( ( 0 0 0 α α α α α α .
We define
13)  ) ) 1 ( ( ) ( 0 TP TP PF R R p R p α α − + ≡ .
In the following we will use only pPF(RTP) and we call this p(RTP). From this we conclude:
14)  ∫
−
−
∞ +
− − − − =
α
α
α α α
1
0
0
) ( ) ) 1 ( ( ) 1 (
R R
TP TP TP nec A
nec
dR R p R R R I L .
LM and LV we can calculate in a analogue manner.
4  Loss of the portfolio, when the rate of return of the portfolio is normal
distributed
We assume, that RTP is “normally” distributed: 
15) 
TP
TP
TP TP
TP
R
R p
σ π
σ
µ
2
2
) (
exp
) (
2
2





 −
−
= .
Calculations with other densities of probability may be done in other publications.
From this we conclude:CAP_Loss_g.doc p.5 22.01.2004
16) 
∫
∫
∫
−
−
∞ −
−
−
∞ −
−
−
∞ −





 −
− 





− + −
−
− −
=





 −
− 





−
−
− −
=





 −
− − − −
−
=
α
α
α
α
α
α
σ
µ
µ µ
α
α
σ π
α
σ
µ
α
α
σ π
α
σ
µ
α α
σ π
α
1
2
2
1
0
2
1
2
2
0
2
1
2
2
0
0
0
0
2
) (
exp
) 1 ( 2
) 1 (
2
) (
exp
) 1 ( 2
) 1 (
2
) (
exp ) ) 1 ( (
2
) 1 (
R R
TP
TP
TP TP
TP TP TP
nec
TP
R R
TP
TP
TP TP
TP
nec
TP
R R
TP
TP
TP TP
TP nec
TP
A
nec
nec
nec
dR
R
R
R R
I
dR
R
R
R R
I
dR
R
R R R I L
[]
[] ∫
∫
∫
∫
−
−
∞ −
−
−
∞ −
−
−
∞ −
−
−
∞ −
− 




 −
− −
−
− 




 −
− 





−
−
− −
=
= 




 −
− −
−
+ 




 −
− 





−
−
− −
=
α
α
α
α
α
α
α
α
µ
σ
µ
µ
σ π
α
σ
µ
µ
α
α
σ π
α
σ
µ
µ
σ π
α
σ
µ
µ
α
α
σ π
α
1
2
2 2
1
2
2
0
2
1
2
2 2
1
2
2
0
2
0
0
0
0
) (
2
) (
exp
2
) 1 (
2
) (
exp
) 1 ( 2
) 1 (
2
) (
exp
2
) 1 (
2
) (
exp
) 1 ( 2
) 1 (
R R
TP TP
TP
TP TP
TP TP
TP
R R
TP
TP
TP TP
TP
nec
TP
R R
TP
TP
TP TP
TP TP
TP
R R
TP
TP
TP TP
TP
nec
TP
nec
nec
nec
nec
R d
R
R I
dR
R R R
I
dR
R
R I
dR
R R R
I
using
17)  π 2
2
exp
2
= 





− ∫
∞
∞ −
dx
x
we obtain
18)  π
σ
µ
σ
µ
σ
µ
σ
2 ) (
2
) (
exp
2
) (
exp
1
2
2
2
2
=
−





 −
− = 




 −
− ∫ ∫
∞
∞ −
∞
∞ − TP
TP TP
TP
TP TP
TP
TP
TP TP
TP
R
d
R
dR
R
and CAP_Loss_g.doc p.6 22.01.2004
19) 






−
− − −
− = 





−
−
− −
≤





 −
− 





−
−
− −
∫
−
−
∞ −
) 1 (
) 1 (
) 1 ( 2
) 1 ( 2
) 1 (
2
) (
exp
) 1 ( 2
) 1 (
0 2 0
2
1
2
2
0
2
0
α
µ α α
α π µ
α
α
π
α
σ
µ
µ
α
α
σ π
α
α
α
TP nec
TP
nec
R R
TP
TP
TP TP
TP
nec
TP
R R
I
R R
I
dR
R R R
I
nec
.
Using
20)  [] [] [ ]
2
exp
2
exp
exp
2 2
2 a a
dx x x
a a −
=
−
= − −
∞ − ∞ −∫
we conclude
21) 
[]
[]
















−
− − −
−
−
=
=
−





 −
−
− −
=
= − 




 −
− −
−
∫
∫
−
−
∞ −
−
−
∞ −
2
0
0
2 ) 1 (
) 1 (
2
1
exp
2
) 1 (
)
2
(
2
) (
exp
2
2 ) 1 (
) (
2
) (
exp
2
) 1 (
0
2
) 1 (
2
2 2
1
2
2 2
TP
TP nec TP
R R
TP
TP TP
TP
TP TP
TP
TP TP TP
R R
TP TP
TP
TP TP
TP TP
TP
R R
I
R
d
R R
I
R d
R
R I
nec
nec
σ α
µ α α
π
σ α
σ
µ
σ
µ
σ
µ
π
σ α
µ
σ
µ
µ
σ π
α
α
α
α
α
Using 19) and 21) we can give an upper limit for LA:
22) 
















−
− − −
−
−
+






−
− − −
− ≤
2
2 ) 1 (
) 1 (
2
1
exp
2
) 1 (
) 1 (
) 1 (
) 1 (
0
2
0 2
TP
TP nec TP
TP nec
A
R R
I
R R
I L
σ α
µ α α
π
σ α
α
µ α α
α
This can be written as:
23) 
















−
−
+ − ≤
2
2
) ; (
2
1
exp
2
) 1 (
) ; ( ) 1 (
2
2
TP
TP TP
TP A
f
I f I L
σ
µ α
π
σ α
µ α α
with 
24) 
) 1 (
) 1 (
) ; (
0
α
µ α α
µ α
−
− − −
=
TP nec
TP
R R
f
or asCAP_Loss_g.doc p.7 22.01.2004
25) 
=
 



 











−
−
−
−
−
+
 


 


−
−
−
−
−
−
− ≤
2
2 2 ) 1 ( 2
1
exp
2
) 1 (
) 1 (
) 1 (
) 1 ( ) 1 (
) 1 (
0
2
0 2
TP
TP
TP
nec TP
TP nec
A
R R
I
R R
I L
σ
µ
σ α
α
π
σ α
α
µ α
α
α
α
α
=












 



 











−
−
−
−
−
+
− − − −
=
2
2 2 ) 1 ( 2
1
exp
2
) 1 (
) 1 ( ) )( 1 (
0
2
2
0
TP
TP
TP
nec TP
TP nec
R R
R R
I
σ
µ
σ α
α
π
σ α
µ α α α
We define the function gA (µTP, σTP):
26) 
 



 











−
−
−
− − =
2
2 2 ) 1 ( 2
1
exp
2
) (
0
TP
TP
TP
nec TP
TP A
R R
g
σ
µ
σ α
α
π
σ
µ α .
and therefore follows:
27)  ( ) ) ( ) 1 ( ) )( 1 (
2
0 α α α α A nec A g R R I L − − − − ≤ .
LM and LV we can measure similarly.
We want to know how the investment on the Capital Market Line changes from bonds to
assets and vice versa when the investor demands that the losses should not exceed a value
which he defined a priori. We define L, which is an upper limit for LA, LM or LV, as
proportional to the investment I, with a rate λ:
28)  I L LA λ = ≤
with
29)  ( ) ) ( ) 1 ( ) )( 1 (
2
0 α α α α λ A nec g R R − − − − = .
We have limited α as follows
30)  1 0 ≤ ≤α .
When α = 1; i.e. that agent invests only in bonds we obtain:CAP_Loss_g.doc p.8 22.01.2004
31)  0 = A L .
When α = 0, i.e. investor invests only in the tangency portfolio and we obtain:
32) 
 



 


















 −
− + − = − = = ≤
2
2 2
1
exp
2
) (
TP
TP nec TP
TP nec A nec A
R
R I I g R I L L
σ
µ
π
σ
µ λ .
Using 22) or 27) we can decide whether a loss which is given a priori can be reached with the
characteristics of the portfolio as α, µTP and σTP. Assuming, that the tangency portfolio (also
µTP and σTP) is fixed (e.g. DAX 30) we can decide whether changing α can reduce the loss.
5  Conclusion
Using a one-period model we have defined the loss which an  investor  can suffer when he
wants to fulfil obligations by using the returns on his portfolio. We measured these losses as
a function of the composition of the portfolio, of the expected rate of return of the assets, of
the rate of return of the bonds and of the variance of the asset.
The model prepared in this paper has limitations. In expanding this model the following
points should be taken into consideration:
•  More than one periodic model.
•  A different composition of the portfolio e.g. including futures, options, etc.
•  Different distributions of probabilities in rates of return.
                                                
1Gerke W., Heubeck K., Gutachten zur künftigen Funktionsfähigkeit der Insolvenzsicherung durch den
Pensions-Sicherungs-Verein VVaG, BetrAV 5, 433-491 (2002).
2 Fishburn, P.C., Mean risk analysis with risk associated with below-target returns, American Economic Review
67, 116-26 (1977).
3 Sharpe, W.F., Capital Asset Prices: A Theory of Market Equilibrium under conditions of risk, The Journal of
Finance 19, 425-42 (1964).
4 Klüppelberg, C., Korn R., Optimale Portfolios mit beschränktem Value-at-Risk, Solutions 3, 23 - 32 (1999).

An investigation of a portfolio-loss under the CAPM

http://128.118.178.162/eps/fin/papers/0402/0402013.pdf

An investigation of a portfolio-loss under the CAPM

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics