Subsections
14 "Portfolio" Problem,
Optimal Investment of Resources
1 Introduction
The previous examples illustrated competition and inspection
processes in economical, social and ecological
problems. Here the optimal investment of available resources is considered. Investment problems depend on the nature of
resources to be invested. An important part of any investment
problem is a proper definition of utility functions that
determine the profittorisk relation.
Here we consider an illustrative example
how to invest some fixed capital in
Certificates of Deposit (CD) and Stocks.
The portfolio problem is to maximize the average
utility of a wealth. That is obtained by optimal distribution of
available capital between different objects with uncertain
parameters [Mockus et al., 1997]. Denote by the part of the capital
invested into an object . The returned wealth
is . Here
and
is an interest rate. Denote by the
reliability of investment. Here is the insolvency probability.
is the utility the wealth . Denote by the expected
utility function. depends on the capital distribution
. If is continuous,
the expected utility function
Here is probability density of wealth . If the wealth is discrete
, the expected utility function
Here is the number of discrete values of wealth . is the probability that the wealth will be returned, if the capital
distribution is . We search for such capital distribution
which provides the greatest expected utility of the returned
wealth:
One may define probabilities of discrete values of
wealth
by exact expressions. For example,
Here
.
From expression (14.5)
Here is the number of different values of
wealth .
One determines
approximately by the Monte Carlo approach:
Here
where
Here is the number of Monte Carlo samples.
is a random number uniformly distributed on the unit
interval. In this case
Investing in CD, the interests are defined by contracts.
Only the reliabilities
of banks are uncertain.
Investing in stocks, in addition to reliabilities
of companies, their future stock rates are uncertain, too. The predicted stock rates are defined by a coefficient that shows the relation between the present and the predicted stock rates. The prediction "horizon" is supposed to be the same as the maturity time of CD.
To simplify the model suppose that one predicts
different values of relative stock rates
with corresponding estimated probabilities
.
In this case,
one may define probabilities of discrete values of
wealth
by exact expressions.
The expressions for CD remains the same. Therefore,
we shall consider only stocks assuming that and .
Then
Here
.
The reliabilities , the stock rate predictions and their estimated probabilities are defined by
experts, possibly, with the help of time series models such as ARMA (see Section 15).
For example, maximal values of multistep prediction
are considered as "optimistic" estimates. The minimal values as "pessimistic" ones. The average values of multistep prediction are regarded as "realistic" estimates.
Here is a simplest illustration were and . In this case from (14.5) (14.10)
the probabilities of wealth returns
are
Here
.
The optimal portfolio depends on the utility function .
Consider, for example, the optimal portfolio for three different
utility functions.
The first utility function is linear
This function is for "rich" persons. Rich persons want to maximize the average wealth. They are not emotional about
accidental losses or gains. In the linear case
(14.11), the optimal portfolio is to invest all the
capital in an object with the highest product .
The
second utility function is for "prudent" persons which averse
risk
Here is a risk threshold.
denotes the
maximal return of invested capital (see expression
(14.4)). If
then, in the
riskaverse case (14.12), an optimal decision is
. Here one divides the capital equally
between all the objects^{}.
The third utility function is for "risky" persons. Risky persons are
ready to risk for the great win .
Here one invests all the capital in the object with highest
wealth return. Therefore, , if
.
These examples
are abstract. An average person behaves "risky," if
only a small part of his resources is involved. The same person
behaves prudently, if all his wealth is at stake. There is
a point between areas of risky and prudent behavior. At this point an average person
behaves like the "rich" one. Here is an example
Here is a boundary point between risky and prudent areas.
In this section the optimal insurance is regarded as a special case
of optimal investment.
Then the expected utility at the end of the next year
where
is a probability to own next year the wealth ,
is an utility function of the wealth .
Supose that
and
Here is the value of the object ,
the product denotes insurance charge of the object .
is insurance policy of the object .
, if the object survives,
, if not.
is a survival probability of the object .
For example:
,
,
where
.
Utility functions are different for different persons
and organizations. An individual utility function is defined by
a lottery
. Here is the
probability to win the best event .
is the
probability to get the worst one . Denote by the "ticket
price" of this lottery. There are two possible decisions:
 keep the ticket money ,
 bay a ticket and risk losing
this money while hoping to win a greater wealth with
probability .
Denote by a "hesitation" probability, when one cannot
decide which decision to prefer. One defines the "hesitation"
probability by this condition
Here the symbol denotes the "hesitation." If utilities and , the utility of the "ticket" is equal to the hesitation
probability [Fishburn, 1964].
Suppose, for example, that event is to keep all the investment capital, , in a safe; no risk, no
profit. Assume that the event means doubling the
capital, . The event means losing all the
capital, .
Denote by the hesitation probability. Then
. If and then the
utility of the capital . Here one obtained
capital utilities at three points: , and .
To define a
reasonable approximation of the utility function , we
need at least two additional points. For example, points and . One defines the corresponding utilities by the
hesitation probabilities and . These are
obtained by two hesitation lotteries
and
Here one obtains utility values
.
The remaining capital utility values are defined by the linear interpolation
In consulting offices, the "psychological tests" defining capital utilities are not always convenient. Then one of the four "typical"
utility functions can be selected. The typical utility functions
represent the risky, the average, the rich and the prudent
persons. The selection depends on observable personal traits.
The same capital utility function (14.14) could be
used for all customers. Then one defines customer differences by different
border points, namely:
(see Figure 14.2).
The software [Petrikis and Juodgalvyte, 1997] is on the website (see Section
4) and is run by Internet
 open the line 'GMJ1' on the 'Software systems' page (see
Figure 3.1),
 start the applet,
 open the
'Methods' menu, choose the method and its parameters (see Figure
14.1),
 open the 'Tasks' menu, chose the task
'Portfolio',
 select the character : prudent, rich, risky,
average (see Figure 14.2),
 select the method:
Monte Carlo, Deterministic (see Figure 14.2),
 fix
the predicted stock rates by updating or keeping the default
values determined using ARMA model (see Figure 14.2),
 click at the 'Operation' label at the top of page,
 click at the 'Run' button at the bottom of page,
 read
results of the optimal investment, where 'Iteration' means the
number of iterations to reach the optimal investment, 'F(x)'
shows the expected utility with the minus sign, the following
lines show the relative investments^{} (see Figure 14.3),
 open the
'convergence' window (see Figure 14.4) which show how
the best utility depends on the iteration number,
 open the
'projection' window (see Figure 14.5) showing how
the utility depends on the first variable,
 open the
'spectrum' window showing how a frequency of utilities depends
on the iteration number^{}.
Figure 14.1:
The control window.

Figure 14.2:
The input fields.

Figure 14.3:
The output field.

Figure 14.4:
The convergence window.

Figure 14.5:
The projection window.

The example of predicted stock rate of Vilnius Bankas is
presented in the file 'onestep?VB.txt'.
Nine approximation points are used:
. Four utility functions are available. They
differ at the approximation points.
They represent different persons.
 prudent person
 :
,
 risky person
 :
,
 rich person
 :
,
 average person

.
Users select one of them.
Two investment possibilities are considered: Certificates of
Deposit (CD) and shares. Considering CD, the two parameters
should be fixed: the interest and the reliability (bank's
survival probability). These parameters are defined in the file
. Considering shares, only one parameter is
used, the predicted stock rate at the time of CD maturity. That is correct, if
the investor is confident^{} about the predicted stock rates at the maturity time. The default values of
expected changes of stock rates are zero. The predicted stock
rates are given in the files like this one:
.
Input file = VB.txt
Pradzia = 26.000000 Prognoze = 30.297932 </BODY></HTML>
Here 'VB' denotes the Vilnius Bank stock rates. 'Pradzia' denotes
the rates now. 'Prognoze' denotes the predicted rates. The
difference between predicted and present stock rates,
divided by the present stock rate and expressed in percents, is
considered as a "stock interest." That is an important simplification. However, it helps to consider
both shares and CD in the same way. That is convenient for
calculations.
Predicted rates in files are
obtained by the ARMA model using the Vilnius Bank data. The predicted
values can be changed manually by clicking and writing on the
corresponding fields of the 'Tasks' page.
The stock rates of the following major Lithuanian jointstock
companies, including the banks, are considered.
 Bank "Vilniaus Bankas"
 : data file VB.txt,
 Bank
"Snoras"
 : data file Snoras.txt,
 Bank "Hermis"
 : data file
Hermis.txt,
 Savings bank "LTB"
 : data file LTB.txt,
 Agricultural bank "LZUB"
 : data file LZUB.txt,
 Bank "Siauliu
Bankas"
 : data file Siauliu.txt,
 Economic bank "Ukio Bankas"
 data file Ukio.txt,
 Dairy "Rokiskio Suris"
 : data file
Rokiskio.txt,
 Brewery "Kalnapilis"
 data file Kalnapilis.txt,
 Dairy "Birzu Pienas"
 : data file BirzuAB.txt.
The first record of the data files^{} shows closing rates on January 01, 1998. The
last record defines closing rates on November 14, 1998. Two hundred twenty nine sessions of Lithuanian stock exchange are recorded in the data. Using this
data ARMA model makes ninty day predictions what corresponds to
three month CD.
Optimization results are in Tables 14.1, 14.2, and 14.3. Tables 14.1 and 14.2 both are for a prudent person. Table 14.1 is obtained by the Monte Carlo method. Table 14.2 shows results of the deterministic method. Table 14.3 is obtained by the deterministic method for a risky person.
The "Optimal investment" column is just a copy of the output field (see Figure 14.3). The output field represents optimal values of internal variables. The internal variables are proportional to the optimal parts of the capital.
Actual optimal parts are obtained by normalization ^{} and are shown
in the "Optimal part" column.
Table 14.1:
Monte Carlo method. Prudent person. Optimal portfolio
obtained after 892 iterations. The
utility is 0.773.
Company 
Optimal investment 
Optimal part 
Vilniaus
Bankas, CD 
0.157 
0.0251 
Snoras, CD 
0.354 
0.0566 
Hermis,
CD 
0.402 
0.0643 
Litimpex, CD 
0.724 
0,1159 
Ukio Bankas,
CD 
0.334 
0.0535 
Siauliu Bankas, CD 
0.842 
0.1348 
Medicinos Bankas, CD 
0.096 
0.0153 
State bonds 
0.318 
0.0509 
Vilniaus Bankas, shares 
0.626 
0.1002 
Snoras,
shares 
0.677 
0.1083 
Hermis, shares 
0.293 
0.0469 
Ukio Bankas,
shares 
0.466 
0.0746 
Siauliu Bankas, shares 
0.461 
0.0738 
Rokiskio Suris, shares 
0.108 
0.0169 
Birzu Pienas,
shares 
0.061 
0.0097 
Kalnapilis, shares 
0.146 
0.02337 

Table 14.2:
Exact o method. Prudent person. Optimal portfolio
obtained after 892 iterations. The
utility is 0.776.
Company 
Optimal investment 
Optimal part 
Vilniaus
Bankas, CD 
0.699 
0.0755 
Snoras, CD 
0.453 
0.0489 
Hermis,
CD 
0.836 
0.904 
Litimpex, CD 
0.795 
0,0859 
Ukio Bankas,
CD 
0.366 
0.0395 
Siauliu Bankas, CD 
0.783 
0.0846 
Medicinos Bankas, CD 
0.926 
0.1001 
State bonds 
0.589 
0.0636 
Vilniaus Bankas, shares 
0.937 
0.1001 
Snoras,
shares 
0.361 
0.0390 
Hermis, shares 
0.343 
0.0370 
Ukio Bankas,
shares 
0.002 
0.0002 
Siauliu Bankas, shares 
0.944 
0.1020 
Rokiskio Suris, shares 
0.760 
0.0822 
Birzu Pienas,
shares 
0.07 
0.00075 
Kalnapilis, shares 
0.052 
0.0056 

Tables 14.1 and 14.2
show that the optimal distribution of capital obtained by
the approximate method differ from that defined by the exact one.
This is important disadvantage of
the approximate Monte Carlo method.
Note, that the best values of utility functions are close for both methods. They are
0.773, for the Monte Carlo method, and 0.776, for the deterministic one. This means that the optimal values of utility functions are not very sensitive
to moderate changes in the capital distribution. The reason is
that here investment alternatives are not too different.
Table 14.3:
Exact method. Risky person. Optimal investment
obtained after 159 iterations. The utility is 0.291.
Company 
Optimal investment 
Optimal part 
Vilniaus
Bankas, CD 
0.907 
0.107 
Snoras, CD 
0.083 
0.0098 
Hermis, CD 
0.857 
0.101 
Litimpex, CD 
0.438 
0,0517 
Ukio Bankas,
CD 
0.024 
0.00283 
Siauliu Bankas, CD 
0.737 
0.0870 
Medicinos Bankas, CD 
0.739 
0.872 
State bonds 
0.656 
0.0774 
Vilniaus Bankas, shares 
0.762 
0.0899 
Snoras,
shares 
0.179 
0.0211 
Hermis, shares 
0.894 
0.1055 
Ukio Bankas,
shares 
0.451 
0.0532 
Siauliu Bankas, shares 
0.098 
0.011 
Rokiskio Suris, shares 
0.946 
0.112 
Birzu Pienas,
shares 
0.019 
0.00224 
Kalnapilis, shares 
0.076 
0.0089 

Comparing Table 14.2 representing the prudent person and Table 14.3 representing the risky one, we see differences. The best value of utility function is equal to 0.776, for the "prudent." It is equal to 0.291, for the "risky."
This difference is explained by assumptions made to normalize utility
functions. The normalization assumes that the maximal utility is a unit and the
minimal one is zero. Therefore, the best value of utility function of the
risky person happens to be less then that of the prudent one.
However, lower utilities does not mean that
risky persons are getting less satisfaction. One needs more
sophisticated normalization techniques
to compare correctly the satisfaction
scales of different persons by their utility functions.
The first step is to drop the "confidence" assumption that one predicts the stock rates exactly. To do this,
the variance of the predicted stock rates should be included into the updated Portfolio Model.
The important future task is the sensibility analysis.
For example, by considering three scenarios: pessimistic, optimistic and realistic. In the pessimistic
scenario lower probabilities for CD's and lower interest rates for stocks are considered.
In the optimistic case one sets higher values for these parameters. The realistic case is in the middle. Important task is to represent visually the "stability" of securities ^{}.
If the optimal values of some securities are not too different in all three scenarios,
they are considered as stable.
Here we consider an illustrative example
how to invest some fixed capital
to minimize the damages of the road accidents.
The problem is split in two stages.
In the first stage, the resources are distributed between groups of accidents defined by their causes.
In the second stage, the geographical distribution of funds inside a single group is regarded.
The four causes of road accidents are considered:
 intoxication,
 speeding,
 ignoring safety belts,
 bad roads "black spots."
Denote by a part of the funds spent to reduce accidents related to the cause .
In this section, we shall assume for simplicity that the total amount of funds is unit.
Here index denotes intoxication, means speeding, denotes safety belts, and denotes
black spots.
Denote by
the expected number of accidents in relation to the part of funds .
Suppose that
where
Here is the actual number of accidents
when the part of funds was .
Denote the estimate of expected losses from all four groups of accidents
when the distribution of funds between causes are defined by the vector
Here
is the estimate of expected damages
of an accident of the group .
If data is available, then is just average value.
If not, then is some expert estimate.
The distribution between black spots will be regarded.
Denote the distribution vextor by
,
where is a number of black spots. is the part of funds invested to improve the black spot .
Denote by
the expected number of accidents in relation to the part of funds .
where
Here is the actual number of accidents
at the black spot when the part of funds was .
The formal definition is
Here
is the estimate of expected damages
of an accident at the black spot .
If data is available, then is just average value.
If not, then is some expert estimate.
The direct minimization of the dimensional utility
function (14.27) is difficult if is large. In the case (14.27) the dynamic programming is an appropriate optimization technique.
The dynamic programming is a
conventional technique to optimize sequential decisions
[Bellman, 1957].
Applying the dynamic
programming to specific problems, one
develops specific algorithms, as usual.
1 Bellman's Equations
One starts from the last black spot defining by a part of funds left for the spot .
Then
Here the relation shows how the maximal utility that can be obtained if the part of funds designated for the set of black spots starting from is distributed in the best possible way. The relation
shows how the optimal part of funds for
the black spot depends on the part of funds designated for the set of black spots starting from .
Here .
Reurrent equations (14.32) and (14.33)
define the optimal parts of funds as a function of . The actual optimal values of these funds are obtained by substitutions assuming that the total amount of funds is unit.
The minimization of the road accidents is just a good ilustration.
Similar procedures can be used in many other problems of optimal distribution of funds.
jonas mockus
20040320