presentation msc thesis

Optimal portfolio construction using stocks from BSE and estimation market risk using VaR methodology

M. Sc. thesis

Marko Ćulić

[email protected]

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 andestimation 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 ConstructionMarket Risk in Financial Markets

Estimation Market Risk using Value at RiskEstimation 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

IntroductionOptimal Portfolio Construction

Market Risk in Financial MarketsEstimation Market Risk using Value at Risk

Estimation Market Risk of Optimal Portfolio using VaR

Risk and returnIntroduction

Optimal Portfolio Construction



• 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 returnIntroduction




• 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

jiji

ijji

N

i

ii

N

ji

ijji wwwww1,1

22

1,

2 σσσσ

ijj

N

jiji

iji

N

i

ii www ρσσσ ∑∑≠

==

+=1,1

22

WVCVWT ⋅⋅=Π

2σ

W

VCV

DiversificationIntroduction




• 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

jiji

ijji

N

i

ii

N

ji

ijji wwwww1,1

22

1,

2 σσσσ

• Suppose1)

2)

3)

0=ijσ

Nwi /1=

0 1

22→=

∞→Π

∞→i

NN NLimLim σσ

( ) [ ]ji

ji

ijNN

RRCovN

NNLimLim ,

1 2

22

→

−=

≠∞→

Π∞→

σσ

Nii ,...,2,1 ,22== σσ

• Suppose1)

2)

3)

0≠ijσ

Nii ,...,2,1 ,22== σσ

Nwi /1=

OptimizationIntroduction




• 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

ijjiww1,

2 min2

1 min

2

1σσ

ExampleIntroduction




• 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 portfolioIntroduction




• Modification of the MPT using primary and secondary stock selection

• Primary selection criteria: (A or B) and CA: 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





• Results of the selection process (only part)

Company name SymbolFree 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

RecommendationsIntroduction








Potential for diversificationCorrelation coefficients in the period 08.01.2007 – 30.04.2008 (332 data)

Optimization processIntroduction




Portfolio P1

• No short position

• Equal weighted portfolio

• Small risk

• Quite good sector breakdown





Portfolio P2

• Minimization the variance of the portfolio


• The smallest risk

• Expected portfolio return decreased

• Number of shares in portfolio decreased

• Sector breakdown changed





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





Portfolio P4

• Maximization the expected return of the portfolio


• Only one stock, with the highest expected return

• Performances of theportfolio P4 is described by characteristics of VinoŽupa





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





Portfolio P6

• Maximization the Sharpe ratio


• The highest Sharpe ratio

• Very high expected return

• Small number of stocks

• Bad sector breakdown





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





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

BELEX15BELEXlinePortfolio P720 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

BELEX15BELEXlinePortfolio P7

Political influence to the financial market





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


-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


Political influence to the financial market

Market RiskIntroduction



• Market risk is a consequence of the changes in security price infinancial 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 VaRIntroduction


Estimation Market Risk using Value at Risk


• 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%

ExampleIntroduction




• 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 factorsIntroduction




• 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 VaRIntroduction




• 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 VaRIntroduction




• 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 clusteringIntroduction




• 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

ExampleIntroduction




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 P7Introduction



• Estimation market risk of optimal portfolio (P7) using followingmethods:– 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 VaRIntroduction



-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 VaRIntroduction



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%Returns of portfolio

EWMA VaRIntroduction



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%Returns of portfolio

Results of the modelsIntroduction



Movements of VaR 99% models, in comparison to the real movements of the optimal portfolio return

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 modelsIntroduction



Movements of VaR 95% models, in comparison to the real movements of the optimal portfolio return


Number of

exceeding

Average

VaR




-8,00%

-4,00%

0,00%

4,00%

8,00%

12,00%



Modified conditionsIntroduction



• 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)




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




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%






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%



ConclusionIntroduction



• 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

ConclusionIntroduction



• 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

Thanks!

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