We base our analysis on an investor, usually a retiree, endowed with a certain amount of wealth W, who considers both his own consumption needs (fixed periodic withdrawals) and the requirement of his heirs (defined bequest). For this purpose he pursues the following in-vestment strategy. The part F is invested in a set of investment funds with the target to achieve an accumulated wealth at the end of a certain time horizon of at least the original amount of wealth W (or the fraction ), measured in real terms. As certain investment risks are implied, we allow for the probability of falling short of the target and implement it into our model as a risk control parameter. The remaining part MM of the original wealth is invested in money market funds in order to avoid additional investment risks and deliver fixed periodic withdrawals until the end of the respective time horizon. The optimal investment strategy is the investment fund allocation that satisfies the probability of shortfall and mini-mizes F, while maximizing the fixed periodic withdrawals. We outline this investment prob-lem in a mathematical model and illustrate the solution for a reasonable choice of empirical parameters.

Albrecht, Peter

Weber, Carsten

English

Research Papers in Economics

SONDERFORSCHUNGSBEREICH504
Rationalit¨ atskonzepte,
Entscheidungsverhalten und
¨ okonomische Modellierung
Universit¨ at Mannheim
L 13,15
68131 Mannheim
No. 03-09
Combined Accumulation- and Decumulation-Plans
with Risk-Controlled Capital Protection
Peter Albrecht
and Carsten Weber
May 2003
Financial support from the Deutsche Forschungsgemeinschaft, SFB 504, at the University of
Mannheim, is gratefully acknowledged.
Sonderforschungsbereich 504, email: risk@bwl.uni-mannheim.de
Sonderforschungsbereich 504, email: carsten.weber@bwl.uni-mannheim.deCombined Accumulation- and Decumulation-Plans 
with Risk-Controlled Capital Protection 
 
Peter Albrecht and Carsten Weber 
 
University of Mannheim,  
Chair for Risk Theory, Portfolio Management and Insurance 
D - 68131 Mannheim (Schloss), Germany 
Telephone: 49 621 181 1680 
Facsimile: 49 621 181 1681 
E-mail: risk@bwl.uni-mannheim.de 
 
Abstract 
We base our analysis on an investor, usually a retiree, endowed with a certain amount of 
wealth W, who considers both his own consumption needs (fixed periodic withdrawals) and 
the requirement of his heirs (defined bequest). For this purpose he pursues the following in-
vestment strategy. The part F is invested in a set of investment funds with the target to 
achieve an accumulated wealth at the end of a certain time horizon of at least the original 
amount of wealth W (or the fraction  ( )W h 1- ), measured in real terms. As certain investment 
risks are implied, we allow for the probability of falling short of the target and implement it 
into our model as a risk control parameter. The remaining part MM of the original wealth is 
invested in money market funds in order to avoid additional investment risks and deliver fixed 
periodic withdrawals until the end of the respective time horizon. The optimal investment 
strategy is the investment fund allocation that satisfies the probability of shortfall and mini-
mizes F, while maximizing the fixed periodic withdrawals. We outline this investment prob-
lem in a mathematical model and illustrate the solution for a reasonable choice of empirical 
parameters.  
 
Keywords: Financial Engineering, Value-at-Risk, Capital Protected Accumulation Plans  2
1.  Investment problem 
An investor of a certain age, usually a retiree of about 60 years, possesses a certain amount of 
wealth W, e.g. 100.000€, which he invests according to the following requirements: 
•  A minimal part F is invested in a set of investment funds, or asset categories, with the 
target to achieve an accumulated wealth at the end of a certain time horizon, e.g. 20 
years of at least the original amount of wealth W or the fraction  ( )W h 1- , measured in 
real terms (capital protection in real terms for a defined bequest). 
•  The remaining part MM of the original wealth is invested in money market funds, out of 
which an annual annuity due is withdrawn until the end of the respective time horizon 
(annuization for individual consumption needs). 
Figure 1: Illustration of the investment problem 
 
It is evident, that the amount of F determines the amount of MM and therefore also the annu-
ity due. In order to maximise the annuity due, the investor has to choose an investment strat-
egy, minimizing the amount of F, while meeting the above-mentioned investment require-
ments. 
original amount of wealth W 
part of wealth F 
fund investment 
 
capital protection 
in real terms 
part of wealth MM 
money market investment 
 
annual annuity due 
in real terms   3
In the following, we develop a general solution for this respective investment problem. For 
clarity reasons, we present our model in detail in the appendix against a theoretical back-
ground. 
2.  Methodology 
2.1  Condition of risk-controlled capital protection in real terms 
In case of fund investment under risk, the reach of the respective investment target is not only 
determined by the average investment return, but also by the volatility of the fund. Therefore, 
it is necessary to specify a condition, that incorporates the capital protection for a defined be-
quest under risk. Capital protection for fund investment under risk can not be guaranteed with 
full certainty, but only to a distinct degree of certainty, being represented by a probability. 
Thus, we propose the following criterion of risk control based on the shortfall probability. 
This condition of risk-controlled capital protection in real terms is orally defined as: 
 
At the end of a previously fixed time horizon, the desired fraction of the original amount of 
wealth ( )W h 1- may fall short merely in a maximum of a out of 100 investment outcomes. 
 
The parameter a is a confidence coefficient, that has to be individually defined by the inves-
tor, e.g.  % 10 %, 5 %, 1 = a . This means, that the desired fraction of wealth is failed in no more 
than 1%, 5% or 10% of all possible investment scenarios. In this way, the shortfall probability 
of the desired fraction of capital protection can be controlled. The determination of the Value-
at-Risk of the distribution of wealth at the end of the time horizon constitutes the focus of our 
methodology. For a mathematical formalization, the reader is referred to the appendix. 
Table 2 summarizes the general procedure of our formalization in order to generate the mini-
mal amount of F and the corresponding investment fund allocation as well as fundamental   4
factors that influence the investment problem. 
Figure 2: Formalization of the procedure  
 
For further concretion of the general procedure, we limit our analysis to the case of three dif-
ferent investment funds or asset categories. 
 
2.2 Case of three investment funds 
For the simultaneous development of three investment funds, e.g. a representative stock, bond 
or property fund, we assume a trivariate geometric Brownian motion. Since the distribution of 
F at the end of the time horizon can not be determined in an analytical way, the Value-at-Risk 
confidence 
coefficient 
time horizon 
risk-controlled  
fund investment 
calculation of the 
Value-at-Risk 
condition of risk-
controlled capital 
protection 
stochastic process of 
fund investment 
average investment 
return, volatility and 
correlation of funds 
investment  
fund allocation 
optimal risk-
controlled fund in-
vestment 
F 
risk control 
optimization 
selection   5
is not analytically definable either and therefore has to be generated in a Monte Carlo-
Simulation. 
In consequence, the determination of the minimal amount of F and the corresponding optimal 
investment fund allocation can not be achieved analytically either. Like in Albrecht/Maurer 
(2002), we use the standard approach of restraining the possible investment fund allocations 
to a representative number and vary the investment weights of each fund by steps of 5%, 
which results in 231 investment fund allocations. 
 
stock funds  0%  0%  … 0%  5%  … 5%  10%  … 10%  … 95%  95%  100% 
bond funds  0%  5%  … 100%  0%  … 95%  0%  … 90%  … 0%  5%  0% 
property funds  100%  95%  … 0%  95%  … 0%  90%  … 0%  … 5%  0%  0% 
Table 1: Representative investment fund allocations 
 
3.  Results 
The following results refer to the case of three investment funds and are based on the parame-
ters for continuous investment returns in real terms. 
 
average investment returns of stock fund  8%, 5% resp. 
average investment return of bond fund  4% 
average investment return of property fund  3.3% 
   
volatility of stock fund  25% 
volatility of bond fund  6% 
volatility of property fund  2% 
   
correlation between stock and bond funds  0.2 
correlation between stock and funds  -0.1 
correlation between bond and property funds  0.6 
   
sales charge of stock fund  5%   6
sales charge of bond fund  3% 
sales charge of property fund  5% 
Table 2: Specification of parameters 
We use the empirical results for the German market of Maurer/Schlag (2002) and Sebastian 
(2003) for our simulations, however, we projected the average returns to be slightly lower and 
the volatilities to be slightly higher in our prospective model calculations. With regard to the 
stock fund we take two alternative scenarios into account. On the one hand, we consider an 
average investment return of 8% in real terms, which often represents the standard estimate of 
average stock returns for very long time horizons as used in Pye (2000). On the other hand, 
the conservative projection of 5% in real terms serves to obtain information on the sensitivity 
of the results. 
Tables 3 and 4 contain the minimum amounts of F, the corresponding optimal investment 
fund allocations and the annual annuity dues based on a continuous real money market return 
of 1.5% according to the respective time horizons as well as the confidence coefficients. From 
now on we deal with the transformation of the confidence coefficients a into degrees of cer-
tainty  ( ) a - 1 . The results refer to average real stock returns of 8% and 5% and assume full 
capital protection of  ( ) 1 h 1 = - . As outlined in the appendix, the desired fraction of capital 
protection does not affect the determination of the optimal investment fund allocation, but 
merely the amount of F and the annual annuity due. 
 
 
 
 
 
   7
  time horizon in years 
  5  10  15  20  25 
  ( 5% 0% 95% )  ( 5% 0% 95% )  ( 10% 5% 85% )  ( 10% 5% 85% )  ( 15% 15% 70% )
95%  94.851,07  81.533,17  69.232,59  58.189,23  48.499,59 
  1.060,91  1.973,80  2.273,47  2.401,72  2.451,92 
  ( 5% 0% 95% )  ( 10% 5% 85% ) ( 15% 20% 65% ) ( 20% 30% 50% ) ( 25% 40% 35% )
90%  93.189,78  79.201,37  66.248,61  54.455,79  43.912,82 
d
e
g
r
e
e
 
o
f
 
c
e
r
t
a
i
n
t
y
 
 
1
.
4
0
3
,
2
1
 
2
.
2
2
3
,
0
4
 
2
.
4
9
3
,
9
6
 
2
.
6
1
6
,
1
8
 
2
.
6
7
0
,
2
9
 
Table 3: Results for an average real stock return of 8% 
 
  time horizon in years 
  5  10  15  20  25 
  ( 5% 0% 95% )  ( 5% 0% 95% )  ( 5% 0% 95% )  ( 5% 5% 90% )  ( 5% 5% 90% ) 
95%  95.552,87  82.774,21  71.127,18  60.889,03  51.978,41 
  916,31  1.841,16  2.133,47  2.246,63  2.286,29 
  ( 5% 0% 95% )  ( 5% 5% 90% )  ( 5% 5% 90% )  ( 5% 10% 85% )  ( 10% 20% 70% ) 
90%  93.888,81  80.701,59  68.909,11  58.607,26  49.446,32 
d
e
g
r
e
e
 
o
f
 
c
e
r
t
a
i
n
t
y
 
 
1
.
2
5
9
,
1
8
 
2
.
0
6
2
,
6
9
 
2
.
2
9
7
,
3
7
 
2
.
3
7
7
,
7
0
 
2
.
4
0
6
,
8
4
 
Table 4: results for an average real stock return of 5% 
 
All in all, the following plausible dependencies can be observed: 
•  The longer is the time horizon, the larger is the share invested in stock and bond funds. 
•  The longer is the time horizon, the smaller is the amount of F, that has to be invested in 
the risky investment funds and the larger is the amount of MM disposable for the annu-
ity due.   8
•  The higher is the degree of certainty, the lower is the share of stock and bond funds and 
the larger is the share of property funds as the least risky type of investment fund. 
•  Using lower average real stock returns leads to consistently larger amounts of F to be 
invested in risky investment funds and in general to a lower share of stock funds for the 
optimal investment fund allocation. 
 
References 
ALBRECHT, P. and MAURER, R. (2002), Self-Annuitization, Consumption Shortfall in Re-
tirement and Asset Allocation: The Annuity Benchmark. In Journal of Pension Eco-
nomics and Finance 1, pp.269 – 288. 
MAURER, R. and SCHLAG, C. (2001), Investmentfonds-Ansparpläne: Erwartetes Versor-
gungsniveau und Shortfallrisiken. In Der Langfristige Kredit 12/2001, pp.440 – 445. 
PYE, G.B. (2000), Sustainable Investment Withdrawals. In Journal of Portfolio Management, 
Summer 2000, pp.73 – 83. 
SEBASTIAN, S. (2003), Inflationsrisiken von Aktien-, Renten- und Immobilieninvestments, 
Bad Soden/Taunus.   9
Appendix: Fundamental Methodology 
In the general case of N investment funds or asset categories, the development of the value of 
each fund during n years is determined by 
    N k t U n V
n
t
k k ..., , 1 , ) ( exp : ) ( 1
1
= ￿
ł
￿
￿
Ł
￿
= + ￿
=
.  (1) 
Assuming a multivariate geometric Brownian motion, the vectors of continuous real returns 
( ) ( ) ( ) ( ) t U , , t U , , t U N k 1 L L  with  n , , 1 t L =  are i.i.d. as 
    ( ) ) ( N ~ U , , U , , U N k 1 ￿
m
,
L L .  (2) 
The vector of real log-returns has a multivariate normal distribution. 
Given  ( ) N k x x x , , , , x 1 L L =  with  1 x 0 k £ £  and ￿ =1 xk  represents the vector of shares 
invested in each of the N investment funds or asset categories and  % a 100 k  with  n , , 1 k L =  
the respective sales charges, we obtain the following wealth in real terms per invested unit 
after a time horizon of n years 
    ￿
= +
+
= +
N
k k
k
k n a
n V
x V
1 1
) ( 1
) ( 1
x
.  (3) 
The condition of risk-controlled capital protection in real terms is defined as 
    [ ] ( ) ( ) a = - £ + W h 1 ) ( V 1 F P n
x
  (4) 
with  ( ) 1 h 1 0 £ - <  being the desired degree of capital protection. (For example, ( ) 9 . 0 h 1 = -  
demonstrates a capital protection in real terms of 90%.) 
Given  ( )
x
a Q  represents the a-quantile of the random number  ( )
x
n V 1+ , we obtain  
   
) ( Q
W ) h 1 (
) ( F F
x
x
a
-
= = .  (5) 
Then it is necessary to determine the investment fund allocation x*, that yields 
    min! ) ( F ﬁ
x
.     10 
Since equation (5) applies,  ) ( F
x
 is at its minimum, when  ) ( Q
x
a  is at its maximum. Thus, 
regardless of the desired degree of capital protection in real terms, from a formal perspective, 
we merely have to find the investment fund allocation x*, that yields 
    max! ) ( Q ﬁ a
x
.  (6) 
According to equation (5),  ) ( Q
x
a  determines the minimum amount of F for a given confi-
dence coefficient a. Capital protection is only feasible, if the amount of F can be financed by 
the original wealth W, implying  W F £ . Because of equation (5), the necessary condition to 
be fulfilled, is 
    ( ) h 1 *) ( Q - ‡ a
x
.  (7) 
Therefore, our analysis can be conducted independent from the amount of original wealth W 
and based on one unit of wealth instead. For this purpose, it is sufficient to analyze the quan-
tile  ) ( Q
x
a  and fix the desired degree of capital protection in real terms.  
The average development of the value of the optimal fund investment  ) ( F
x
*
 ultimately is 
    [ ] ￿
=
+
+
=
N
1 k
) v m ( n
k
k
2
k 2
1
k e
a 1
* x
F *) ( F E
x
,  (8) 
while  ( ) k m m = ,  ( ) kj v = S  and  kk
2
k v v = . 
Finally, we describe the amount of MM in real terms invested in money market funds, that 
yields the annual annuity due after a time horizon of n years 
  W
Q
)] h 1 ( Q [
Q
W ) h 1 (
W F W
a
a
a
- -
=
-
- = - .  (9) 
Given i represents the deterministic annual continuous real money market rate, the annual real 
annuity due is determined by  
 
1 q
1 q
q W
Q
) h 1 ( Q
R
n
1 n
-
- - -
=
-
a
a   (10)   11 
with 
i e q = : . Evidently, the annual real annuity due is positive, if  ( ) h 1 Q - > a . SONDERFORSCHUNGSBereich504 WORKING PAPER SERIES
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Inflationsrisiken von Aktien-, Renten- und Immobilieninvestments,

Investmentfonds-Ansparpläne: Erwartetes Versorgungsniveau und Shortfallrisiken.

Self-Annuitization, Consumption Shortfall in Retirement and Asset Allocation: The Annuity Benchmark.

Sustainable Investment Withdrawals.

Combined Accumulation- and Decumulation-Plans with Risk-Controlled Capital Protection

http://www.sfb504.uni-mannheim.de/publications/dp03-09.pdf

Combined Accumulation- and Decumulation-Plans with Risk-Controlled Capital Protection

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