Optimal; Optimal portfolio selection; Portfolio; Selection; Cash flow; Capital at risk; Risk;

Dhaene, Jan

Goovaerts, Marc

Vanduffel, Steven

Vyncke, David

English

Research Papers in Economics

103
Tijdschrift voor Economie en Management
Vol. L, 1, 2005
Optimal Portfolio Selection for Cash-Flows
with Bounded Capital at Risk
by D. VYNCKE, M. GOOVAERTS, J. DHAENE and S. VANDUFFEL
Marc Goovaerts
KULeuven, 
AFI Actuarial Science, 
Leuven and University 
of Amsterdam, 
The Netherlands
David Vyncke
KULeuven, 
AFI Actuarial Science, 
Leuven
Steven Van Duffel
KULeuven, 
AFI Actuarial Science, 
Leuven
Jan Dhaene
KULeuven, 
AFI Actuarial Science, 
Leuven and University 
of Amsterdam, 
The Netherlands
ABSTRACT
We consider a continuous-time Markowitz type portfolio problem that consists 
of minimizing the discounted cost of a given cash-ﬂ  ow under the constraint of 
a restricted Capital at Risk. In a Black-Scholes setting, upper and lower bounds 
are obtained by means of simple analytical expressions that avoid the classical 
simulation approach for this type of problems. The problem is easily extended 
to cope with more general discount processes.
Keywords: Black-Scholes model, Capital at Risk, portfolio optimization, Value 
at Risk.
*  The authors would like to acknowledge the ﬁ  nancial support of the Onderzoeksfonds 
K.U. Leuven (GOA/02: Actuariële, ﬁ  nanciële en statistische aspecten van afhankelijk-
heden in verzekerings- en ﬁ  nanciële portefeuilles).104
I. INTRODUCTION
The portfolio selection algorithm as introduced in Markowitz (1959) 
uses a mean-variance analysis to ﬁ  nd optimal portfolios. In this meth-
od, a portfolio is called optimal if it yields the largest return among all 
portfolios with the same variance or, vice versa, if it has the smallest 
variance among all portfolios with the same return. For long term in-
vestments, however, the use of the variance as a risk measure leads to a 
smaller proportion of risky assets in a portfolio than one would expect. 
Based on the empirical observation that stock indices are growing 
faster than riskless rates in the long run, the proportion of risky assets 
should increase with the duration of the investment period. But since 
the variance increases with time, the proportion of risky assets will 
decrease. Therefore, Emmer et al (2001) propose to use the Capital 
at Risk (CaR) as an alternative risk measure and derive a closed-form 
formula to calculate the optimal CaR-constrained portfolio.
The Capital at Risk of a portfolio is commonly deﬁ  ned as the differ-
ence between the mean of the proﬁ  t-loss distribution and a small quan-
tile of this distribution (the so-called Value at Risk), but Emmer et al 
(2001) use a different deﬁ  nition which limits the possibility of excess 
losses over the riskless investment. We will also follow this approach 
and show how optimal CaR-constrained portfolios can be obtained for 
cash-ﬂ  ows in a Black-Scholes setting.
In a Black-Scholes market the stock prices   {  S  i  (t)}  t  0   for i = 1, …, d, 
evolve from the following equations:
d S  i  (t) =   S  i  (t)   (   m  i dt +  ∑ 
j=1
   
d
    s  ij d W  j  (t)  )  ,    S  i  (0) =   s  i  ,  i = 1, …, d
where W(t) is a standard d-dimensional Brownian motion, m = (m1, …, 
md)' is the vector of stock-appreciation rates, and s =   (  s  ij  )  1i, jd   is the 
matrix of stock-volatilities.
Denoting the fraction of the wealth X(p, t) that is invested in as-
set i by pi(t), the wealth process of an initial unit amount follows the 
dynamic
  dX(p, t) = X(p, t) (p'mdt + p'sdW(t)),  X(p, 0) = 1.  (1)
As in Emmer et al (2001), we assume that the fractions in the dif-
ferent stocks remain constant on [0, T], i.e. p(t) = p = (p1, …, pd)'. So, 105
since the stock prices evolve randomly, one has to follow a dynamic 
trading strategy to keep the fractions of wealth invested in the different 
stocks constant. Solving the SDE (1) yields
  X(p, t) = exp {Y(p, t)} (2)
where Y(p, t) equals 
  Y(p, t) = p'mt – p's2p    t  __  2     + p'sW(t). (3)
In the following section we extend some results obtained by Em-
mer et al (2001) to an actuarial context where consecutive payments 
have to be made. But implementing the mean-CaR criterion in a multi-
period model inevitably turns the portfolio selection problem into a 
very complex problem for which no analytical solutions exist. Con-
sequently, this type of portfolio selection problem is generally tackled 
by means of simulation of the corresponding stochastic processes. In 
section III we construct close approximations to the model in order to 
avoid this excessive time-consuming approach. Section IV concludes 
with a numerical illustration of the approximations.
II. EXTENSION  TO  CASH-FLOWS
In an insurance setting, at certain points in time an amount is with-
drawn from or added to the money invested. We consider a cash-ﬂ  ow 
ct denoting the total payments for each year t (e.g. pensions in a pen-
sion fund). Throughout the paper we will assume that ct  0 (t = 1, 
…, T).
The present value of the cash-ﬂ  ow equals
  V =  ∑ 
t=1
    
T
    c  t  e  
–Y(p, t)  . (4)
with expected value given by
  V0 = E[V] =  ∑ 
t=1
    
T
    c  t  e  
–m(p)t +   s 
2  (p)t   (5)
where m(p) = p'm and s2(p) = p's2p. Obviously, we want to select a 
portfolio that minimizes V0. Note however that there are restrictions 106
on the amounts of risky stocks that can be bought, due to the require-
ments of control authorities. In addition, also due to regulators, the 
probability of “ruin” has to be restricted. Let e denote the maximum 
probability of “ruin allowed’’. Denoting the 1 – e quantile of the dis-
counted cash-ﬂ  ow by Qe(p), i.e.
 Pr    [  ∑ 
t=1
    
T
    c  t  e  
–Y(p, t)      Qe(p) ]   = 1 – e (6)
we deﬁ  ne the Capital at Risk as
 CaR  =  Qe(p) –  ∑ 
t=1
    
T
    c  t  e  
–rt   (7)
where r is a constant reference interest rate, e.g. the riskless interest 
rate.
So, if we assume that the provision for the future payment obliga-
tions is the 1 – e quantile, then the CAR is the required provision in 
excess of the required provision in case of riskless investments.
By taking into account the extra cost of the CaR, we come to the fol-
lowing optimization problem:
  min     
p
       V  0   + u(CaR),  subject  to    p  j    0,  ∑ 
j=1
   
d
    p  j   = 1  ,  CaR   C
where the increasing function u(·) denotes the supplementary cost 
of the Capital at Risk and where C denotes the maximum CaR al-
lowed.
Since the quantile Qe(p) is very hard (or even impossible) to obtain 
due to the dependency structure between the random variables Y(p, t), 
t = 1, …, T, in (4), it seems impossible to solve this optimization prob-
lem without using Monte Carlo simulation. In the next section, we will 
show how this excessive time-consuming approach can be avoided by 
using easily computable approximations to Qe(p).107
III. AVOIDING SIMULATION
Instead of calculating the exact quantile of the distribution, we will 
look for bounds, in the sense of “more favourable/less dangerous’’ 
and “less favourable/more dangerous’’, with a simpler structure. This 
technique is common practice in the actuarial literature. When lower 
and upper bounds are close to each other, together they can provide 
reliable information about the original and more complex variable. 
The notion “less favourable’’ or “more dangerous’’ will be deﬁ  ned by 
means of the convex order.
Deﬁ  nition 1  A random variable V is smaller than a random variable 
W in convex order if
  E [u(V)]  E [u(W)], (8)
for all convex functions u: ˙ " ˙ : x 7 u(x), provided the expectations 
exist. This is denoted as
 V  cx W. (9)
Since convex functions are functions that take on their largest val-
ues in the tails, the variable W is more likely to take on extreme values 
than the variable V, and thus W is more dangerous.
In Vyncke et al (2001) and Kaas et al (2001) upper and lower 
bounds for present value functions are constructed. These bounds 
in convex order turn out to be rather close to the exact present value 
distribution.
Proposition 1  Consider a sum of random variables
V = X1 + X2 + … + Xn,
and deﬁ  ne the related stochastic quantities
  Vu =   F    X  1    –1   (U) +   F    X  2    –1   (U) + … +   F    X  n    –1   (U) (10)
  Vl = E[X1|Z] + E[X2|Z] + … + E[Xn|Z], (11)108
where U is a random variable, uniformly distributed on [0,1], and 
where Z can be any random variable for which the expectations exist. 
The following relations then hold:
Vl cx V cx Vu.
Proof: see  Vyncke  et al (2001) and Dhaene et al (2002).
It is clear that the lower bound Vl will perform at best if Z and V are 
very similar, so we choose
  Z =  ∑ 
t=1
    
T
    c  t  e  
–p'mt   Y(p, t) (12)
which can be seen to be a ﬁ  rst order approximation of V.
For the (1 – ε)-quantiles of Vu and Vl we ﬁ  nd 
 Q   e   *  (p) =
  ∑ 
t=1
    
T
    c  t   exp   { –m(p)t + r(p, t) s(p)   √
_
  t   F 
–1  (1 – e) +   ( 1 –   
  r 
2  (p, t)
  ______  2    )     s 
2  (p)t  } 
(13)
with m(p) = p'm, s2(p) = p's2p and where the parameters r(p, t) are 
given by
r(p, t) = Corr(Y(p, t), Z) =   
  ∑
i=1  
t
    b  i  
  _______ 
  √
______
  t   ∑
i=1  
T
    b  i   2    
  ,  with    b  i   =  ∑ 
t=i
    
T
    c  t  e  
–m(p)t  
in case of the lower bound Vl, and r(p, t)  1 in case of the upper 
bound Vu. Note that r(p, t) depends on p only through m(p). Since 
0  r(p, t)  1 and s(p)  0, the quantile   Q  e   *  (p) is an increasing func-
tion of s(p). This implies that the adjusted Capital at Risk 
    CaR 
*   =   Q  e   *  (p) –  ∑ 
t=1
    
T
    c  t  e  
–rt   (14)
is also increasing in s(p) for both approximations. Note, however, 
that the adjusted CaR isn’t necessarily increasing with the planning 109
horizon T. Figure 1 shows the adjusted CaR (e = 0.05) for a pure stock 
portfolio, i.e. a portfolio consisting of one asset with a strictly positive 
volatility, for a cash-ﬂ  ow ct=100 (t=1, …, T) for different values of 
T. In case the return m equals 0.1, the adjusted CaR increases with T 
(for both upper and lower bound), but if m = 0.18 then the CaR* ﬁ  rst 
increases and then decreases with time. As in Emmer et al (2001), at 
some point in time the CaR* even becomes negative which means that 
the pure stock strategy should be preferred over the risk-free strategy 
if the planning horizon is beyond that point in time. 
From (5) it can be seen that also V0 is increasing in s(p). So logically 
assuming that the cost function u(·) is an increasing function, we see 
that the adjusted objective function V0 + u(CaR*) is increasing in s(p). 
This implies that the adjusted optimization problem
  min     
p
       V  0   + u(  CaR 
*  ),  subject  to    p  j    0,  ∑ 
j=1
   
d
    p  j   = 1  ,   Car 
*    C
years
C
a
R
51 0 1 5 2 0 2 5
0
5
0
0
1
0
0
0
FIGURE 1
Capital at Risk of the pure stock portfolio as a funstion of the planning horizon. 
Both upper (grey) and lower bound (black) are depicted for m = 0.10 (solid line) 
and m = 0.18 (dashed line). The volatility equals 0.20 and the risk free interest rate 
equals 0.05.110
can be solved by minimizing s2(p) for each m(p) = m, and choosing 
the solution which minimizes the adjusted objective function. So, 
solving this optimization problem boils down to successively solving 
a quadratic program. Because of the speciﬁ  c nature of the optimization 
problem, the solution will also be part of the mean-variance efﬁ  cient 
frontier.
IV. NUMERICAL  ILLUSTRATION
In this section we illustrate the method by considering a portfolio con-
sisting of 5 risky stocks and 1 riskless bond. The stock-appreciation 
rates m and stock-volatilities s are given by
stock 1 2345
m 0.1346 0.1659 0.1895 0.2014 0.095
s 0.1585 0.2293 0.3368 0.4299 0.0686
and their correlation matrix equals
 
    (   
  
1
 
  
  0.7217     
0.2571
 
 
  
  -0.0719     
0.408
       
 
0.7217
 
  
  1     
0.1436
 
 
  
  -0.083     
0.1419
        
 
0.2571
 
  
  0.1436     
1
   
 
  
  0.0255     
-0.0829
     
 
-0.0719
 
  
  -0.083     
0.0255
  
 
  
  1     
-0.1154
       
  
0.408
 
  
  0.1419     
-0.0829
 
 
  
  -0.1154     
1
    ) 
The riskless bond yields a 0.05 return and we assume that the cost 
function is given by 
u(x) =   {   0.2x      0.05x       x  0      x  0 
This can be interpreted as follows: the insurance company has to pay 
a dividend of 20% to its shareholders if the CaR is positive and earns 
5% (the risk-free interest rate) if the CaR is negative.
First, we consider a cash-ﬂ  ow ct = 100 (t=1, …, 20). For e = 0.05, 
the proportions (in %) based on the upper bound Vu are very close to 
those based on Vl, as can be seen from the following table. Note that p0 
indicates the proportion that is invested in the riskless bond.111
appr.
e
Vu
0.05
Vl
0.05
Vu
0.01
Vl
0.01
p0 0.00 0.00 0.00 0.00
p1 0.00 0.00 0.00 0.00
p2 48.28 48.01 37.28 45.59
p3 27.16 27.30 20.44 24.35
p4 24.57 24.70 18.39 21.97
p5 0.00 0.00 23.90 8.10
m 18.10 18.11 16.03 17.37
s 18.23 18.25 13.90 16.67
V0 595.13 595.10 621.03 602.11
CaR* -139.10 -235.56 52.46 -0.12
cost 588.17 583.33 631.52 602.10
In Figure 2 the optimal portfolios based on Vu and Vl are indicated by 
a circle and a rectangle respectively. For e = 0.01, the proportions ap-
pear to be less efﬁ  cient for this kind of cash-ﬂ  ow (see also Figure 3).
Next, we consider an increasing cash-ﬂ  ow ct = 5t (t=1, …, 20). 
Apart from rounding errors, we see that the proportions for the Vu ap-
proximation equal those of the Vl approximation in case of e = 0.05. 
For e = 0.01, the largest difference in proportion is approximately 8% 
(see also Figures 4 and 5).
appr.
e
Vu
0.05
Vl
0.05
Vu
0.01
Vl
0.01
p0 0.00 0.00 0.00 0.00
p1 0.00 0.00 0.00 0.00
p2 48.01 48.01 45.73 48.28
p3 27.30 27.30 24.42 27.16
p4 24.70 24.70 22.04 24.57
π5 0.00 0.00 7.82 0.00
m 18.11 18.11 17.39 18.10
s 18.25 18.25 16.72 18.23
V0 183.89 183.89 187.21 183.91
CaR* -155.02 -184.89 -0.19 -32.75
cost 176.14 174.67 187.20 182.27112
Finally, we consider a decreasing cash-ﬂ  ow ct = 105-5t (t=1, …, 20). 
For e = 0.05 (see Figure 6) as well as for e = 0.01 (see Figure 7), the 
method appears to perform quite well. 
appr.
e
Vu
0.05
Vl
0.05
Vu
0.01
Vl
0.01
p0 0.00 0.00 0.00 0.00
p1 0.00 0.00 0.00 0.00
p2 48.52 48.28 34.96 40.04
p3 25.73 27.16 19.35 21.74
p4 23.24 24.57 17.40 19.59
p5 2.52 0.00 28.30 18.64
m 17.84 18.10 15.66 16.48
s 17.67 18.23 13.15 14.81
V0 442.18 440.98 459.03 451.36
CaR* 0.61 -51.86 91.55 43.59
cost 442.30 438.38 477.34 460.08
REFERENCES
Dhaene J., Denuit M., Goovaerts M.J., Kaas R. & Vyncke D., 2002, The Concept of 
Comonotonicity in Actuarial Science and Finance: Theory, Insurance: Mathematics and 
Economics, to appear.
Emmer S., Klüppelberg C., Korn R., 2001. Optimal Portfolios with Bounded Capital at 
Risk, Mathematical Finance 11, 4, 365-384.
Kaas R., Dhaene J., Goovaerts M.J., 2000. Upper and Lower Bounds for Sums of Random 
Variables, Insurance: Mathematics and Economics 27, 151-168.
Markowitz H., 1959. Portfolio Selection: Efﬁ  cient Diversiﬁ  cation of Investments (John 
Wiley & Sons, New York).
Vyncke D., Goovaerts M. & Dhaene J., 2001. Convex Upper and Lower Bounds for Present 
Value Functions, Applied Stochastic Models for Business and Industry, 17, 149-164.113
return m
v
o
l
a
t
i
l
i
t
y
 
s
0.05 0.10 0.15 0.20
0
.
0
0
.
1
0
.
2
0
.
3
0
.
4
return m
9
5
%
 
C
a
R
0.05 0.10 0.15 0.20
0
1
0
0
0
2
0
0
0
3
0
0
0
4
0
0
0
return m
c
o
s
t
0.05 0.10 0.15 0.20
1
0
0
0
1
5
0
0
2
0
0
0
2
5
0
0
FIGURE 2
Optimal portfolios for ct =100 (t=1, …, 20) with e = 0.05.
return m
v
o
l
a
t
i
l
i
t
y
 
s
0.05 0.10 0.15 0.20
0
.
0
0
.
1
0
.
2
0
.
3
0
.
4
return m
9
9
%
 
C
a
R
0.05 0.10 0.15 0.20
0
2
0
0
0
4
0
0
0
6
0
0
0
8
0
0
0
return m
c
o
s
t
0.05 0.10 0.15 0.20
5
0
0
1
5
0
0
2
5
0
0
3
5
0
0
FIGURE 3
Optimal portfolios for ct =100 (t=1, …, 20) with e = 0.01.
return m
v
o
l
a
t
i
l
i
t
y
 
s
0.05 0.10 0.15 0.20
0
.
0
0
.
1
0
.
2
0
.
3
0
.
4
return m
9
5
%
 
C
a
R
0.05 0.10 0.15 0.20
0
5
0
0
1
0
0
0
1
5
0
0
2
0
0
0
2
5
0
0
return m
c
o
s
t
0.05 0.10 0.15 0.20
2
0
0
4
0
0
6
0
0
8
0
0
1
0
0
0
FIGURE 4
Optimal portfolios for ct = 5t (t=1, …, 20) with e = 0.05.114
return m
v
o
l
a
t
i
l
i
t
y
 
s
0.05 0.10 0.15 0.20
0
.
0
0
.
1
0
.
2
0
.
3
0
.
4
return m
9
9
%
 
C
a
R
0.05 0.10 0.15 0.20
0
2
0
0
0
4
0
0
0
6
0
0
0
return m
c
o
s
t
0.05 0.10 0.15 0.20
5
0
0
1
0
0
0
1
5
0
0
2
0
0
0
FIGURE 5
Optimal portfolios for ct = 5t (t=1, …, 20) with e = 0.01.
return m
v
o
l
a
t
i
l
i
t
y
 
s
0.05 0.10 0.15 0.20
0
.
0
0
.
1
0
.
2
0
.
3
0
.
4
return m
9
5
%
 
C
a
R
0.05 0.10 0.15 0.20
0
5
0
0
1
0
0
0
1
5
0
0
return m
c
o
s
t
0.05 0.10 0.15 0.20
6
0
0
8
0
0
1
0
0
0
1
2
0
0
FIGURE 6
Optimal portfolios for ct = 105 – 5t (t=1, …, 20) with e = 0.05.
return m
v
o
l
a
t
i
l
i
t
y
 
s
0.05 0.10 0.15 0.20
0
.
0
0
.
1
0
.
2
0
.
3
0
.
4
return m
9
9
%
 
C
a
R
0.05 0.10 0.15 0.20
0
1
0
0
0
2
0
0
0
3
0
0
0
4
0
0
0
return m
c
o
s
t
0.05 0.10 0.15 0.20
6
0
0
8
0
0
1
2
0
0
1
6
0
0
FIGURE 7
Optimal portfolios for ct = 105 – 5t (t=1, …, 20) with e = 0.01.

Convex Upper and Lower Bounds for Present Value Functions, Applied Stochastic Models for Business and Industry, 17, 149-164.113 return m v o l a t i l i t y s 0.05 0.10 0.15 0.20

Optimal Portfolios with Bounded Capital at Risk,

Portfolio Selection: Efﬁ cient Diversiﬁ cation of Investments

Upper and Lower Bounds for Sums of Random Variables,

Optimal portfolio selection for cash-flows with bounded capital at risk.

https://lirias.kuleuven.be/bitstream/123456789/122846/1/TEM_2005-1_10_Vyncke.pdf

Optimal portfolio selection for cash-flows with bounded capital at risk.

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