OPTIMISATION OF THE SECURITIES PORTFOLIO AS A
PART OF THE RISK MANAGEMENT PROCESS
Srečko Devjak*
Received: 8. 03. 2004
Accepted: 9. 04. 2004
Review
UDC: 336.763
Securities of Slovene companies are listed at the Ljubljana Stock Exchange.
Market capitalisation at the Ljubljana Stock Exchange has been growing since
1996 due to new listings of equities. On the basis of financial data time series for
listed equities, the financial investor can calculate a risk for each individual
security with a selected risk measure and can determine an optimal portfolio,
subject to selected constraints. In this paper, we shall consequently determine an
optimal portfolio of equities for the financial investor, investing his assets only in
selected equities listed at the Ljubljana Stock Exchange. Selecting an appropriate
risk measure is especially important for a commercial bank in a risk management
process. Commercial banks can use internal models in the risk management
process and for the purpose of capital charges as well. An optimal portfolio will be
calculated, using a non-linear mathematical model.
1. INTRODUCTION
The goal of this paper is to determine an optimal portfolio of equities for
the financial investor, investing his assets only in selected equities, which are
listed at the Ljubljana Stock Exchange. We are going to use standard deviation
as a risk measure in the mathematical model.
In this paper, we will understand a commercial bank as a financial investor.
Commercial banks have two possibilities in order to calculate capital charges
for the market risk they are being exposed to. The first approach is the
standardised approach, which has to be used by banks in case they do not have
*
Srečko Devjak, Bank of Slovenia, Banking supervision department, Slovenska 35,
1505 Ljubljana, Slovenia, Phone: + 386 1 47 19 522, Fax: + 386 1 47 19 727,
Email: [email protected]
81
Management, Vol. 9, 2004, 1, pp. 81-91
S. Devjak: Optimisation of the securities portfolio as a part of the risk management process
an internal model. If a bank uses an internal model for risk management
purposes, it can use several risk measures in order to measure risk. Each risk
measure has its strengths and its weaknesses. Consequently, the volume of risk
calculated using a specific risk measure will vary among risk measures. Since
the process of risk management in a commercial bank is based on a calculated
value of risk measure, it is very important for a commercial bank to fully
understand the interpretation of a selected risk measure. If the volume of risk
varies using different risk measures, the decisions upon changes in the positions
in a portfolio will be different.
Harry Markowitz (1952) introduced standard deviation as a risk measure.
His work is the foundation of the portfolio theory. This is why we are going to
calculate an optimal portfolio using standard deviation as a risk measure.
1. THEORETICAL BACKGROUND
By purchasing a certain security, the financial investor takes the risk that
the actual rate of return on his investment will differ from the expected rate of
return. The higher the risk of an individual security, the higher the required rate
of return. Suppose the risk of an individual security is measured by the variance
of its rate of return. If we mark the expected rate of return of a security by
E ~
r , and its actual rate of return by ~r , then the variance of the rate of return
on this security VAR~
r  equals (Mramor, 1991, p. 45):

2
VAR~
r  E ~
r -E ~
r

When the financial investor invests all his assets in one risky security, then
a higher rate of return variance for this security also represents a higher level of
risk for his entire assets. Assets with several different risky securities will,
therefore, be less risky for an investor than assets containing only one risky
security.
ri and the rate of
Let us now mark the rate of return on a risky security by ~
r . When the financial
return on the financial investor’s entire assets by ~
investor invests all his assets in n risky securities, then the expected rate of
return on his assets equals the weighted arithmetic mean of the expected rates of
return on all risky securities in the portfolio. If ai is a share of all the assets
invested in a risky security, the following applies:
82
Management, Vol. 9, 2004, 1, pp. 81-91
S. Devjak: Optimisation of the securities portfolio as a part of the risk management process
E ~
r   E a1  ~
r1  a2  ~
r2    an  ~
rn   a1  E ~
r1   a2  E ~
r2     an  E ~
rn 
in
E ~
r    ai  E ~
ri 
i 1
under the condition that:
in
a
i 1
i
1
If the risk of financial assets is measured by the variance of its rate of
return, the following applies (Mramor, 1991, p. 46):

2
VAR~
r  E ~
r -E ~
r


2
VAR~
r   E a1  ~
r1  a2  ~
r2    an  ~
r   Ea1  ~
r1  a2  ~
r2    an  ~
rn 

2
2
 i  n
 i  n

 in
 
 
~
~
~
VARr   E   ai  ri   E   ai  ri    E   ai  ri  E ai  ri  

 i 1
 
 
 i 1
 i 1




2
VAR~
r    ai2  E ~
ri  E ~
ri   2 ai a j  E ~
ri  E ~
ri   ~
rj  E ~
rj 
n
i 1
n
i j
VAR~
r    ai2  VAR~
ri   2 ai a j  COV ~
ri ,~
rj 
n
n
i 1
i j
The variance of the rate of return on an individual security was marked by
VAR~
ri  , and the covariance of the rate of return on an individual pair of
 
ri ,~
rj . The covariance of the rate of return on
securities was marked by COV ~
two securities shows the changes of the rate of return on one security, in relation
to the rate of return of the other security. The more negative the covariance of
the two securities, the higher the possibility is of reducing the variance of the
rate of return of the entire assets by purchasing these two securities. The
standardised value of the covariance is a correlation coefficient.
Equity is a risky security bringing the financial investor a return in the
form of a dividend and capital gain. Three different rates of return on equity can
83
Management, Vol. 9, 2004, 1, pp. 81-91
S. Devjak: Optimisation of the securities portfolio as a part of the risk management process
be distinguished. This is a dividend rate of return, calculated as a relation
between the paid dividend in a certain time period and the uniform price of this
equity in the beginning of the analysed time period.
The uniform price of equity listed at the Ljubljana Stock Exchange is
calculated as a weighted arithmetic mean of its prices within an individual
trading day, where the relations between the quantity of an individual deal and
the entire daily traded quantity with this equity serve as weights. The
calculation of the uniform price does not take into account block deals and
applications.
The capital rate of return of equity is defined as a relation between the
changes of the uniform price within a certain time period and the uniform price
for this equity in the beginning of the analysed time period. The total rate of
return on equity is the sum of its dividend rate of return and capital rate of
return.
3. DESIGNING A MATHEMATICAL MODEL
The present financial analysis will include equities that are listed at the
Ljubljana Stock Exchange and for which the longest time series of data are
available. These are equities of the following companies: BTC, Droga Portorož,
Gradbeno Podjetje Grosuplje, Kolinska, Lek, Luka Koper, Mercator, Nika, SKB
and Terme Čatež. BTC and Mercator are trading companies, Droga Portorož
and Kolinska are food companies, Gradbeno Podjetje Grosuplje is a
construction company, Lek is a pharmaceutical company, Luka Koper is a
transport company, Nika and SKB are financial brokerage companies, and
Terme Čatež is a tourist company. The companies included in this financial
analysis, therefore, belong to seven different sections of the industry. The
number of these sections is quite small. Therefore, the financial investor has
highly limited chances for a diversification of his financial assets.
3.1. Variables in the model
All variables in the model can be divided into two groups. The first group
consists of the exogenous variables, while the endogenous or decision variables
are in the second group. In the process of a portfolio optimisation, the type and
the fraction of security in the portfolio have to be determined. For this reason,
the mathematical model will include the following decision-related variables:
84
Management, Vol. 9, 2004, 1, pp. 81-91
S. Devjak: Optimisation of the securities portfolio as a part of the risk management process
BTC
DRPG
GPG
KOLR
LEKA
LKPG
MELR
NIKA
SKBB
TC RG
… fraction of assets invested in BTC equities,
… fraction of assets invested in Droga Portorož equities,
…fraction of assets invested in Gradbeno Podjetje Grosuplje equities,
… fraction of assets invested in Kolinska equities,
… fraction of assets invested in Lek equities,
… fraction of assets invested in Luka Koper equities,
… fraction of assets invested in Mercator equities,
… fraction of assets invested in NIKA equities,
… fraction of assets invested in SKB equities,
… fraction of assets invested in Terme Čatež equities.
The names of decision variables selected in this model are identical to the
security indexes from the Ljubljana Stock Exchange listings. Our goal is to find
an optimal portfolio. A portfolio can be considered efficient, when no other
portfolio exists with the same rate of return and a lower rate of return variance,
or when no other portfolio exists with a higher rate of return at the same rate of
return variance.
In our portfolio optimisation, we will assume that at a certain required total
rate of return on financial assets, the financial investor minimises the variability
of the total rate of return on these assets. For this reason, the mathematical
model will include the following exogenous variables: expected total rate of
return on an individual equity listed at the Ljubljana Stock Exchange, total rate
of return variance for each equity, and total rate of return covariance for each
pair of the analysed equities. On the basis of a minimum required total rate of
return for a portfolio of equities, we will determine such a combination of the
shares of equities in the portfolio where the rate of return variance of the
portfolio reaches its lowest value.
In Slovenia, financial investors are not liable to pay tax on capital gain
from selling a security since this capital gain was realised after the expiry of the
three-year period from the day of the purchase of the security. For this reason,
this financial analysis will assume that the maturity of the financial investment
is not shorter than three years. This assumption, in turn, allows the present
analysis to ignore tax on paid dividends and the transaction costs.
In the calculation of the expected total rate of return on equities, the
analysis will take into account the adaptive expectations of financial investors.
The total expected rate of return on each equity in the financial analysis for the
next period will be calculated as a weighted arithmetic mean of the realised total
rates of return in the previous periods. The expected total rate of return
calculated in this way is based on the data on the realised annual rates of return
85
Management, Vol. 9, 2004, 1, pp. 81-91
S. Devjak: Optimisation of the securities portfolio as a part of the risk management process
in the period from the beginning of 1997 to the end of October 2001. The rates
of return from January 2001 to October 2001 have been treated as an
approximation of the annual rates of return on an individual equity for 2001.
The values of the exogenous variables will be calculated by the SPSS
programme on the basis of a data series on the annual rates of return for each
equity included in the financial analysis.
3.2. Optimal portfolio of selected equities
Table 1: Covariance matrix of total rates of return
a
ce
NIK
GPG
SKB
TCR
RPG
OL
EK
KP
TC
EL
BT
65
48
68
1
7
0
6
4
8
3
DR
93
21
11
7
8
5
9
6
8
2
GP
29
19
27
0
5
0
6
9
9
3
KO
47
87
03
6
9
6
6
0
2
5
LE
64
08
85
4
6
9
0
9
9
8
LK
83
67
78
8
8
9
2
9
3
2
ME
85
28
94
3
2
3
5
8
2
0
NI
82
93
84
5
3
9
7
4
3
5
SK
93
90
02
8
1
9
7
8
7
8
TC
84
02
64
8
1
7
3
5
8
4
a.
Th
Source: Author’s calculation.
Table 2: Correlation matrix of total rates of return
io
R
O
E
K
E
I
K
C
T
P
P
L
K
P
L
K
B
R
C
G
B
0
0
6
0
3
9
2
8
7
0
D
0
0
1
4
8
1
8
1
3
0
G
6
1
0
7
4
5
0
4
7
8
K
0
4
7
0
0
2
2
8
1
2
L
3
8
4
0
0
1
4
7
9
8
L
9
1
5
2
1
0
7
4
5
7
M
2
8
0
2
4
7
0
1
2
0
N
8
1
4
8
7
4
1
0
2
4
S
7
3
7
1
9
5
2
2
0
5
T
0
0
8
2
8
7
0
4
5
0
Source: Author’s calculation.
86
Management, Vol. 9, 2004, 1, pp. 81-91
S. Devjak: Optimisation of the securities portfolio as a part of the risk management process
Figures in the correlation matrix are standardised values of figures in the
covariance matrix. In our case, the financial investor can invest his savings in
10 different equities listed at the Ljubljana Stock Exchange. In a non-linear
mathematical model, the objective function is the variance of the total rate of
return of the equities portfolio, determined by the following equation:
sBTC  DRPG  ~
sDRPG  GPG  ~
sGPG  KOLR  ~
sKOLR
 BTC  ~

~
~
~
~
VARr   VAR  LEKA  sLEKA  LKPG  sLKPG  MELR  sMELR 
  NIKA  ~
sNIKA  SKBB  ~
sSKBB  TCRG  ~
sTCRG






In the model, the expected total rate of return of individual security will be
marked by ~
si , whereas i signifies an individual security. The objective function
has consequently been determined by using the following general equation:
VAR~
r    ai2  VAR~
ri   2 ai a j  COV ~
ri ,~
rj 
n
n
i 1
i j
In determining an optimal portfolio, our mathematical model will include
the following constraints:
1. Restriction of the required total rate of return on the portfolio
For the financial investors who are unwilling to take a risk; the higher the
risk, the less useful the investment.
For this reason, the model will assume that the minimum required total
rate of return on the portfolio equals the average nominal rate of return on a
bank deposit in Slovenia with a maturity of one year. In this case, it seems
reasonable to take into account the average annual nominal passive interest
rates for all commercial banks in a banking system, starting from the year
for which the annual rates of return on securities are available.
In the calculation of the average annual nominal interest rates for
deposits with a maturity of one year, we are going to use the monthly data
from the Bank of Slovenia Bulletin. This interest rate is calculated as a
weighted arithmetic mean on the basis of the stated interest rates of
commercial banks in the banking system, while the states of selected
passive items in the balance sheet of an individual commercial bank are
used as weights.
87
Management, Vol. 9, 2004, 1, pp. 81-91
S. Devjak: Optimisation of the securities portfolio as a part of the risk management process
The calculation of the average annual nominal interest rate for banking
deposits was made for the pe riod from January 1997 to October 2001. The
average annual nominal interest rate of a deposit in Slovenia was 13.65%.
Therefore, the following applies:
BTC  E ~
sBTC   DRPG  E ~
sDRPG   GPG  E ~
sGPG   KOLR  E ~
sKOLR  
~
~
~
  LKPG  E s   MELR  E s   NIKA  E ~s
 LEKA  E s
LEKA
LKPG
MELR
NIKA

 SKBB  E ~
sSKBB   TCRG  E ~
sTCRG  
~
~
sGPG 
 BTC  sBTC  DRPG  sDRPG  GPG  ~



~
~
~
 E   KOLR  sKOLR  LEKA  sLEKA  LKPG  sLKPG 
  13,65
  MELR  ~
~
~
~
sMELR  NIKA  sNIKA  SKBB  sSKBB  TCRG  sTCRG 

By simplifying the equation, we get the following result:
16,36 BTC  17 ,702 DRPG  8,49GPG  15,51KOLR 
 20,972 LEKA  15,038LKPG  36,342MELR  39,45 NIKA 
 6,376SKBB  12,238TCRG  13,65
2. Restriction of the sum of shares of risky securities in a portfolio
For determining the shares of risky securities in a portfolio, the following
applies:
BTC  DRPG  GPG  KOLR  LEKA  LKPG  MELR  NIKA 
 SKBB  TCRG  1
3. Restriction of negative values
In our case, all decision variables can only assume non-negative values,
which can be indicated in the following way:
BTC, DRPG, GPG, KOLR, LEKA, LKPG, MELR, NIKA, SKBB, TCRG  0
An optimal portfolio includes 6,5% of Gradbeno Podjetje Grosuplje
equities, 41,9% of Kolinska equities, 8,0% Lek equities, 11,9% Luka Koper
equities, 6,6% of Mercator equities, 21,4% of SKB bank equities and 3,6% of
Terme Čatež equities.
By using the LINGO computer programme, an optimal solution for our
minimisation programme was determined after 81 iterations. In an optimal
88
Management, Vol. 9, 2004, 1, pp. 81-91
S. Devjak: Optimisation of the securities portfolio as a part of the risk management process
solution, the variance of the total rate of return on the portfolio of equities is
very small:
VAR~
r   8,939  10- 6  percentage points 
2
The standard deviation from the minimum required total rate of return on
the portfolio will be calculated by using the following equation:
2
σ ~
r   VAR~
r   8,939667  106  percentage points 
 0,003  percentage points
Let us now take a look at the structure of an optimal portfolio by individual
sections of the industry. It consists of seven different equities belonging to all
seven branches of the industry. For the financial investor who purchases the
securities included in this financial analysis, this means the maximum possible
diversification of a portfolio of equities by the branches of industry, as well as
the maximum reduction of its non-systematic risk.
Table 3: Slack and surplus variables and Lagrange multipliers
Row
1
2
3
4
5
6
7
8
9
10
11
12
13
Slack or Surplus
0.8939667E-05
0.0000000
0.0000000
0.0000000
0.0000000
0.6476873E-01
0.4187352
0.8038623E-01
0.1192469
0.6650877E-01
0.0000000
0.2140374
0.3631682E-01
Dual Price
1.000000
-0.1868325E-04
0.1926822E-03
0.0000000
0.0000000
0.0000000
0.0000000
0.0000000
0.0000000
0.0000000
0.0000000
0.0000000
0.0000000
Source: Author’s calculation.
The SLACK OR SURPLUS column shows the values of the slack or
surplus variables by rows for each condition in the mathematical model after its
transformation in the linear equation, where the first row is the variance of the
total rate of return of the portfolio of securities. The condition for the minimum
89
Management, Vol. 9, 2004, 1, pp. 81-91
S. Devjak: Optimisation of the securities portfolio as a part of the risk management process
required total rate of return of the portfolio is in the second row of the model,
where the value of the surplus variable equals 0. This means that the expected
total rate of return of the portfolio is 13.65%.
3.3. Sensitivity analysis
In a minimisation non-linear programme, the value of the DUAL PRICE
variable for a certain row equals the negative value of the Lagrange multiplier
for that particular row.
This means that if we increase the value of the right side of an individual
condition for the relatively small value bi , the optimal value of our target
function increases for approximately  bi i , i.e. it is reduced for
approximately bi i .
In our case, the condition for the required total rate of return on the
portfolio of securities is particularly interesting. When the right side of this
condition is increased by 0.5 percentage points (i.e. to 14.15%), the value of the
objective function is reduced for approximately bi i , that is to
approximately:
Δbi λi
 0,5 percentage points    1,868  10 5  percentage points  
 9 ,342  10 6  percentage points 
2
Thus, the variance of the total rate of return of the portfolio is increased for
approximately 9,342  10 6  percentage points 2 , i.e. to approximately:
8,939  10- 6  percentage points   9 ,342  10 6  percentage points  
2
2
 1,828  10-5  percentage points 
2
In this case, the standard deviation of the total rate of return of the portfolio
of securities would be increased to approximately:
2
σ ~
r   VAR~
r   1,828  105  percentage points
 0,004 percentage points
90
Management, Vol. 9, 2004, 1, pp. 81-91
S. Devjak: Optimisation of the securities portfolio as a part of the risk management process
4. CONCLUSION
At a required minimum total rate of return, the variability of the total rate
of return on an optimal portfolio of equities is very small. For this reason, this
particular form of saving is competitive for Slovenian financial investors with a
bank deposit of the same maturity.
On the basis of the calculated total rate of return variance for the portfolio
of equities, we may conclude that at the minimum required total rate of return
used in this financial analysis, the risk of this portfolio is negligible.
REFERENCES
1. Košmelj, B.; Rovan, J.: Statistično sklepanje. Ljubljana: Ekonomska
Fakulteta, 1997.
2. Markowitz, H. M.: Portfolio selection. Malden: Blackwell Publishers, 1991.
3. Markowitz, H. M.: Portfolio selection. Journal of Finance, 1952 (7). 77-91.
4. Mishkin F. S.; Eakins, S. G.: Financial markets and institutions. Third
edition. Addison Wesley Longman, 2000.
5. Mramor, D.: Finančna politika podjetja. Teoretični prikaz. Ljubljana:
Gospodarski Vestnik, 1991.
6. Prohaska, Z.: Finančni trgi. Ljubljana: Ekonomska Fakulteta, 1999.
7. Official publications of the Bank of Slovenia.
OPTIMIZACIJA PORTFOLIA VRIJEDNOSNICA KAO DIO PROCESA
UPRAVLJANJA RIZIKOM
Sažetak
Vrijednosnice slovenskih poduzeća uvrštene su u kotaciju Burze u Ljubljani. Tržišna
kapitalizacija Burze u Ljubljani u stalnom je porastu od 1996. godine zbog uvrštenja
novih vlasničkih vrijednosnica. Na temelju vremenskih serija financijskih podataka za
uvrštene vrijednosnice, investitori mogu izračunati rizik za pojedinu vrijednosnicu,
koristeći odabrani indikator rizika, ali se, na jednak način, te uz odgovarajuća
ograničenja, može odrediti i optimalni portfolio. U ovom se radu pokušava odrediti
optimalni portfolio vlasničkih vrijednosnica za investitora koji svoj kapital ulaže
isključivo u vrijednosnice uvrštene u kotaciju Burze u Ljubljani. Izbor odgovarajuće
mjere rizika posebno je značajan za komercijalne banka u procesu upravljanja rizikom.
Naime, komercijalne banke mogu koristiti interne modele u procesu upravljana rizikom
i u svrhu određivanja cijene kapitala. Za izračun optimalnog portfolia koristi se
nelinearni matematički model.
91
92