Presentation MSc thesis

Optimal portfolio construction using stocks from BSE
and estimation market risk using VaR methodology
M. Sc. thesis
Marko Ćulić
culic_marko@yahoo.co.uk
Belgrade, 2009

Agenda
• Introduction
• Optimal portfolio construction
• Market risk in financial market
• Estimation market risk using VaR
• Estimation market risk of optimal portfolio
• Conclusion

The goal
• Focus on portfolio construction using Markowitz optimization and
estimation market risk of optimal portolio using Value at Risk
• Application of MPT in transition markets is difficult
– Low liquidity
– Short time series data
– Non stable VCV matrix
• Some possible solutions
– Primary selection based on liquidity criteria
– Secondary selection based on recommendations by researchers
– Optimization process
– Reoptimization
Introduction
Optimal Portfolio Construction
Market Risk in Financial Markets
Estimation Market Risk using Value at Risk
Estimation Market Risk of Optimal Portfolio using VaR

The notation
• Formula for holding period return,
• Formula for variance of sample,
• The variance of a random variable,
• The covariance between two random variables,
• The correlation coefficient between two random variables,
Introduction

The notation
• Variance – covariance matrix for 3 assets
• Correlation matrix for 3 assets
Introduction

Risk and return
• Investment – the current commitment of funds to assets that will be
held over the future time period. Investors expect that future
payments will compensate the investors for
– The time the funds are obtained,
– The expected rate of inflation
– The uncertainty of the future payments
• Every investment involves some degree of uncertainty
– Future selling price is unknown, future dividends are unknown, future
cash flows are unknown
– Might sell asset due to emergency
– Reinvestment rate might change
– Increase in inflation changes the purchasing power of money
Introduction

Risk and return
Introduction
• We used expected return and standard deviation as return and risk
measures in investment decision making process
• Expected return of the portfolio
where
is portfolio weight of the i-th asset in the portfolio
is the expected return of the i-th asset in the portfolio
• Negative weights mean short position
[ ] [ ] 1,
11
=⋅= ∑∑ ==
Π
N
i
i
N
i
ii wREwRE
iw
[ ]iRE

Risk and return
Introduction
• First mathematical definition of the risk in portfolio analysis was
introduced by Markowitz in “Portfolio selection”, 1952
• The variance of the portfolio
• Using matrix form
where
is vector of weights
is Variance – covariance matrix, dimension NxN
∑∑∑
≠
===
Π +==
N
ji
ji
ijji
N
i
ii
N
ji
ijji wwwww
1,1
22
1,
2
σσσσ
ijj
N
ji
ji
iji
N
i
ii www ρσσσ ∑∑
≠
==
+=
1,1
22
WVCVW T
⋅⋅=Π
2
σ
W
VCV

Diversification
Introduction
• The variance of the portfolio consists of two parts
• Portfolio with the large number of assets reduces risk only to second
part of the previous formula. Two cases
∑∑∑
≠
===
Π +==
N
ji
ji
ijji
N
i
ii
N
ji
ijji wwwww
1,1
22
1,
2
σσσσ
• Suppose
1)
2)
3)
0=ijσ
Nwi /1=
0
1 22
→=
∞→
Π
∞→
i
NN N
LimLim σσ
( ) [ ]ji
ji
ij
NN
RRCov
N
NN
LimLim ,
1 2
2
2
→




 −
=
≠
∞→
Π
∞→
σσ
Nii ,...,2,1,22
== σσ
• Suppose
1)
2)
3)
0≠ijσ
Nii ,...,2,1,22
== σσ
Nwi /1=

Optimization
Introduction
• Suppose that we have N assets and that we know
– Expected rate of return for each asset
– Standard deviations for each asset
– Coefficient correlation between each two assets
• The problem is to find portfolio that minimize the variance
• Additional constraints in optimization problem
– We can target expected portfolio return
– Sum of the weights equal to 1
– No short selling
– Include only stocks with BUY recommendation
∑=
Π =
N
ji
ijjiww
1,
2
min
2
1
min
2
1
σσ

Example
Introduction
• Suppose that can invest in the following investments
– Asset 1 with expected return of 0,1 and standard deviation of 0,3.
– Asset 2 with expected return of 0,2 and standard deviation of 0,5.
Correlation between Asset 1 and Asset 2 is 0,2.
– Asset 3 is the risk free asset with expected return of 0,05
• Find the optimal mean-variance portfolio for the given expected
portfolio return of 0,2
• Solution
Return of the portfolio
Variance of the portfolio
L function
Percentage to invest in each asset

Optimal portfolio
Introduction
• Modification of the MPT using primary and secondary stock selection
• Primary selection criteria: (A or B) and C
A: Stock liquidity – average monthly volume minimum 2% of free float
B: Number of trading days in last 125 days minimum 50%
C: Free float market capitalization minimum EUR 5m
• Secondary selection criteria: > HOLD
– Filtrate stocks with potential for growth
– Descriptive recommendation SELL, HOLD, BUY, STRONG BUY
OptimizationPrimary selection
• Liquidity
• Number of trading days
• Free Float MCap
Secondary selection
• HOLD
• BUY
• STRONG BUY
42 stocks 30 stocks 16 stocks

Optimal portfolio
Introduction
• Results of the selection process (only part)
Company name Symbol
Free float
MCap in EUR
Average
free float
liquidity
Trading days
in last 125
days
Primary
selection
Secondary
selection
> 5.000.000 >2,00% >50,00% AND(A;
OR(B;C))
>SELL
(A) (B) (C)
1 Agrobacka AGBP 662.943 4,38% 13,60% - -
2 Agrobanka AGBN 119.158.797 2,90% 100,00% OK SELL
3 Agrocoop AGRC 1.645.359 1,34% 60,80% - -
4 AIK banka AIKB 452.911.404 3,08% 100,00% OK HOLD
5 Alfaplam ALFA 16.993.168 1,52% 81,60% OK SELL
6 Bambi-Banat BMBI 28.351.657 1,23% 84,00% OK HOLD
7 Banini BNNI 8.880.441 17,90% 48,80% OK SELL
8 Cacanska banka CCNB 26.236.142 0,46% 53,60% OK HOLD
9 Credy banka CYBN 8.075.317 0,86% 42,40% - -
10 Dijamant DJMN 20.512.145 0,32% 39,20% - -
11 Energoprojekt ENHL 177.665.483 0,93% 100,00% OK BUY
12 Fidelinka FIDL 9.140.838 1,09% 80,80% OK SELL

Recommendations
Introduction

Optimal portfolio
Introduction
Potential for diversification
Correlation coefficients in the period 08.01.2007 – 30.04.2008 (332 data)

Optimization process
Introduction
Portfolio P1
• No short position
• Equal weighted portfolio
• Small risk
• Quite good sector
breakdown

Introduction
Portfolio P2
• Minimization the variance
of the portfolio
• The smallest risk
• Expected portfolio return
decreased
• Number of shares in
portfolio decreased
• Sector breakdown changed

Introduction
Portfolio P3
• Minimization the variance
of the portfolio
• Shares from Prime market
up to 70%
• Portfolio P3 shifts to the
right in risk-return diagram
• Slightly higher risk in
comparison to P2
• Number of shares in
portfolio decreased to 10
• Sector breakdown
dramatically changed

Introduction
Portfolio P4
• Maximization the expected
return of the portfolio
• Only one stock, with the
highest expected return
• Performances of the
portfolio P4 is described
by characteristics of Vino
Župa

Introduction
Portfolio P5
• Maximization the expected
return of the portfolio
• Shares from Prime market
have 70%
• Only two stocks in
portfolio, one from the
Prime market and the
second one is Vino Župa

Introduction
Portfolio P6
• Maximization the Sharpe
ratio
• The highest Sharpe ratio
• Very high expected return
• Small number of stocks
• Bad sector breakdown

Introduction
Portfolio P7
• Maximization the Sharpe
ratio
• Additional constraints
• Sharpe ratio droped
• Small risk
• Small adjusted beta
• All stocks in portfolio
• Very good sector
breakdown
( )
( )
( ) %10
%8
%6
≤
≤
≤
BUYSTRONGw
BUYw
HOLDw
i
i
i

Introduction
Movements of indices and portfolio P7 before construction The main performance in the period
Jan 8th 2007 – Apr 30th 2008 (332 data)
600
1.000
1.400
1.800
2.200
2.600
Jan-07 Mar-07 May-07 Jul-07 Sep-07 Nov-07 Jan-08 Mar-08
BELEX15
BELEXline
Portfolio P7
20 per. Mov. Avg. (Portfolio P7)
20 per. Mov. Avg. (BELEXline)
20 per. Mov. Avg. (BELEX15)
-8,00%
-4,00%
0,00%
4,00%
8,00%
12,00%
Jan-07 Mar-07 May-07 Jul-07 Sep-07 Nov-07 Jan-08 Mar-08 May-08
BELEX15
BELEXline
Portfolio P7
Political influence to the
financial market

Introduction
Movements of indices and portfolio P7 in the period from the
construction up to September 19 2008
The main performance in the period
May 5th 2008 – Sept 18th 2008 (100 data)
700
850
1.000
1.150
1.300
May-08 Jun-08 Jul-08 Aug-08 Sep-08 Oct-08
BELEX15
BELEXline
Portfolio P7
-12,75%
-22,82%
-27,28%
-8,00%
-4,00%
0,00%
4,00%
8,00%
12,00%
May-08 Jun-08 Jul-08 Aug-08 Sep-08 Oct-08
BELEX15
BELEXline
Portfolio P7
Political influence to the
financial market

Market Risk
Introduction
• Market risk is a consequence of the changes in security price in
financial market
• Market risk represents a risk of unpredictable and negative changes
in the market prices of securities
• Example – movements of GM stock in the period 1971 – 2009

Definition of VaR
Introduction
• Value at Risk is a measure that summarizes the expected maximum
loss of portfolio over a target horizon within a given confidence
interval
• Important details of this definition
– VaR is an estimate, not strictly defined value
– VaR assumes that trading positions in portfolio are fixed for the given
time horizon
– VaR is not estimated for the worst – case loss
• We need two parameters for estimation VaR
– The holding period, e.g. 1 day or 10 days
– The confidence level, e.g. 95% or 99%

Example
Introduction
• Suppose that we are interested in estimating daily VaR for a single
position with market value of EUR 1mill and volatility ( ) of 0,1%
on a daily basis. Also, confidence interval is 99% ( )
σα ⋅⋅=01,0;1 ValueMarketVaR
%1,032,2000.000.1 ⋅⋅=
000.232=
σ
32,2=α

VaR factors
Introduction
• Level of alpha depends of risk averse level of each institution
• Higher alpha implies higher VaR, which require higher capital and
higher rating of institution
• Basel II recommend 99% confidence level
• The second factor is time horizon
• It depends of liquidity and value of position
• For trading portfolio standard is 1 day horizon
Alpha Confidence level
3,432 99,97%
2,326 99%
1,881 97%
1,645 95%

Types of VaR
Introduction
• Historical simulation VaR
– Non parametric approach
– Empirical distribution obtained from the observed data
• Parametric VaR
– We assume that returns corresponds to some of the theoretical distribution
– Normal distribution is good approximation for stock returns
– Student t-distribution fit better the data from the fat tailed distribution
• Monte Carlo VaR
– Assumes normal distribution for the data
– Numerous scenarios for the future movements of market variables
randomly created
– Takes the highest loss with the specific probability value

Parametric VaR
Introduction
• Suppose that we are interested in estimating VaR at 95% confidence
level for a holding period of 1 day. We estimate and over this
horizon to be 0,004 and 0,01, respectively. Also, if the value of
portfolio is EUR 1mill, than
• If we are interested in a confidence level of 99%
• If we are interested in longer time horizon, than VaR over 10 days
σµ

Volatility clustering
Introduction
• In order to estimate volatility we can use
– Unconditional volatility
• Standard deviation p.a.
• Depends of the data set
• React slowly to the market shocks
• Efekat duha
– Conditional volatility
• Exponentially Weighted Moving Average (EWMA)
• Introduced by RiskMetrics Group
• The latest observations carry the highest weight
• React fast to the market shocks
• RiskMetrics volatility of asset i at time t:
• Lambda is dickey factor
• Cutoff point, n

Example
Introduction
Movements of the exchange rate of dinar against euro Daily log returns of the exchange rate
Unconditional and EWMA volatility of the exchange rate

VaR of portfolio P7
Introduction
• Estimation market risk of optimal portfolio (P7) using following
methods:
– Standard historical VaR
– Analytical unconditional VaR
– Analytical EWMA VaR
• Holding period – 1 day
• Confidence level: 95%, 99%
• The validity of the model was tested on 100-day sample, since there
was no time series that was long enough
• Basel Committee recommends the sample of 250 data
• Optimal portfolio was observed in the period from May 5th 2008 to
September 19th 2008 (100 trading days)

Historical VaR
Introduction
-8,00%
-4,00%
0,00%
4,00%
8,00%
12,00%
Jan-08 Feb-08 Mar-08 Apr-08 May-08 Jun-08 Jul-08 Aug-08 Sep-08
Historical VAR 95%
Historical VAR 99%
Returns of portfolio
Movements of historical simulation VaR, in comparison to
the real movements of the optimal portfolio return
Results of Historical 1-day VaR (100 data)
Number of
exceeding
Average
VaR
Date of
exceeding
Historical
VaR 99%
1 3,45% Sept 16th
Historical
VaR 95% 5 2,26%
May 13th
May 15th
May 16th
Sept 16th
Sept 18th

Analytical VaR
Introduction
Movements of analytical VaR, in comparison to the real
movements of the optimal portfolio return
Results of Analytical 1-day VaR (100 data)
Number of
exceeding
Average
VaR
Date of
exceeding
Historical
VaR 99%
1 3,56% Sept 16th
Historical
VaR 95% 3 2,53%
May 15th
Sept 16th
Sept 18th
-8,00%
-4,00%
0,00%
4,00%
8,00%
12,00%
Analytical VAR 95%
Analytical VAR 99%

EWMA VaR
Introduction
Movements of analytical EWMA VaR, in comparison to the
real movements of the optimal portfolio return
Results of Analytical EWMA 1-day VaR (100 data)
Number of
exceeding
Average
VaR
Date of
exceeding
Historical
VaR 99%
1 3,65% Sept 16th
Historical
VaR 95% 4 2,58%
Aug 27th
Sept 15th
Sept 16th
Sept 18th
-8,00%
-4,00%
0,00%
4,00%
8,00%
12,00%
EWMA VAR 95%
EWMA VAR 99%

Results of the models
Introduction
Movements of VaR 99% models, in comparison to the real
Results of 1-day VaR 99% models (100 data)
Number of
exceeding
Average
VaR
Historical VaR 1 3,45%
Analytical VaR 1 3,56%
Analytical EWMA VaR 1 3,65%
-8,00%
-4,00%
0,00%
4,00%
8,00%
12,00%
Historical VAR 99%
Analytical VAR 99%
EWMA VAR 99%
Portfolio returns

Results of the models
Introduction
Movements of VaR 95% models, in comparison to the real
Number of
exceeding
Average
VaR
-8,00%
-4,00%
0,00%
4,00%
8,00%
12,00%
Historical VAR 95%
Analytical VAR 95%
EWMA VAR 95%
Portfolio returns

Modified conditions
Introduction
• Extremely high positive portfolio return of 11,22% on May 12th as a
consequence of parliamentary elections
• Political factor influenced on the movements in the financial market
and increased volatility
• Neutralize short-term political effect
• We decided to change the portfolio return on May 12th with dummy
return of 0,16% (average daily return from Jan 9th 2007 to May 11th
2008)

Modified conditions
Introduction
10,00%
20,00%
30,00%
40,00%
50,00%
60,00%
EWMA volatility p.a. real data
EWMA volatility p.a. without extreme return
EWMA volatility p.a. of the optimal portfolio
17,50%
20,00%
22,50%
25,00%
27,50%
30,00%
Unconditional volatility p.a. real data
Unconditional volatility p.a. without extreme return
Unconditional volatility p.a. of the optimal portfolio

Modified conditions
Introduction
Movements of VaR 99% models, in comparison to the movements
of the optimal portfolio after introduction dummy variable
Number of
exceeding
Average
VaR
-8,00%
-4,00%
0,00%
4,00%
8,00%
12,00%
Historical VAR 99%
Analytical VAR 99%
EWMA VAR 99%
Portfolio returns

Modified conditions
Introduction
Movements of VaR 95% models, in comparison to the movements
of the optimal portfolio after introduction dummy variable
Number of
exceeding
Average
VaR
-8,00%
-4,00%
0,00%
4,00%
8,00%
12,00%
Historical VAR 95%
Analytical VAR 95%
EWMA VAR 95%
Portfolio returns

Conclusion
Introduction
• MPT offers a solution to the rational investors how to construct their
own portfolio and as its aim has the portfolio optimization in
accordance with the specified investment goals
• While constructing a portfolio, individual securities and their
properties are not as important as their mutual interaction
• Individual security risk can be diversified, but the systemic risk can
not be removed by diversification

Conclusion
Introduction
• Market risk is the most important risk in stock portfolio
management
• VaR is a unique, statistical measure of possible losses in the
portfolio value and it represents a loss measure that can occur due to
the «normal» market movements
• VaR offers a consistent and integrated approach to the market risk
management. Higher management became more aware of the
relation between the taken risks and realized profits

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