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Computational Finance 1/23

Panos Parpas

Single Period Markowitz Model

381 Computational Finance

Imperial College

London


Topics Covered

Notation and Terminology

Random variables, mean, (co)variance, correlationRandom variables, mean, (co)variance, correlation

Asset return, portfolio return and riskAsset return, portfolio return and risk

Portfolio Optimisation

Optimal asset allocation, risk management Optimal asset allocation, risk management

Short sale, diversificationShort sale, diversification

Mean-Variance model, efficient frontier


Terminology

Random Variable y is a random variable, takes finite number of values, yj for j=1,2,…,m.

a probability (associated with each event) represents the relative chance of an occurrence of yj.such that

Expected Value (Mean value or mean) average value obtained by regarding probabilities as frequencies

Variance measure of possible deviation from the mean

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m

jj ,,2,1for 0 and 1

1

22

22

2

)(

)(2)(

var

yyE

yyyEyE

yyE(y)

m

jjj ypyEy

1

)(

relative frequency

finite number ofpossibilities

Expected value of squared variable

how much y tends to vary from its mean


Terminology

Covariance of two random variables

Correlation between two random variables

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2121

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))((cov

yyyyE

yyyyE),y(y

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correlated negatively0

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correlated positively 0

)var()var(

),cov(

12

12

12

12

21

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yy

yy),ycor(y

21 and yy

21 and yy

sign defines direction of

the relationship


Asset Returns

Asset: investment instrument that can be bought and sold

uncertain asset prices – the return is random uncertainty described in probabilistic terms

Asset return: you purchase an asset today and sell it next year

Total return on this investment is defined as

Rate of return:

Rate of return acts much like an interest rate

0

1

investedamount

recievedamount return total

x

xR

0

01

x

xxr

invested amount

invested amount-recieved amount return ofrate

01 )1(1 xrxrR


Returns

Returns Returns Time Period r1 r2

1 6 82 4 23 7 114 3 -15 8 126 2 -27 11 138 -1 -3

MEAN 5 5VARIANCE 3.77964 6.36209


Portfolio Return

Consider n risky assets to form a portfolio and invest among n assetsSelect an amount invested in the ith asset

The amounts invested can be expressed as fractions of the total investment

is fraction or weight of asset i in portfolio

Let the total return of asset i be . The amount of money gained at the end of the period on asset i is

The total return of the portfolio The rate of return of portfolio

0x

n

iii xxnix

1000 such that ,1for

where21 ,00 iii wn,,ixwx

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00 xwRxR iiii

10

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xwRR

1

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Portfolio Return Both the total return and the rate of return of the portfolio of assets are equal to the weighted sum of the corresponding individual asset returns, with the weight of an asset being its relative weight in the portfolio.

Expected return of the portfolio is the weighted sum of the individual expected rates of return

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Example

Consider a portfolio of a risk-free asset and two risky stocks and three equally likely states

Find the expected return of the portfolio formed by 3 assets

Scenario tree with 3 states

States

(s) T-Bill

Return% Stock A Return%

Stock B Return %

(i=1) (i=2) (i=3) s1 “boom” 5 16 3 s2 “normal” 5 10 9 s3 “recession” 5 1 15

root represents today

future uncertainty is discretised

by 3 events (states)

3

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Example: Expected Returns

The expected return of risk free asset is For stock A and B, expected return of the portfolio

The expected return of the portfolio

For equally weighted portfolio

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Risk

Risk: a chance that investment’s actual return will be different than

expected –includes losing some or all of original investment

risk averse and risk-seeking (loving)

Systematic risk: influences a large number of assets, such as political

events – impossible to protect yourself against this risk

Unsystematic risk: (specific risk) affects very small number of assets.

For example, news that affects a specific stock such as a sudden strike

Diversification is the only way to protect yourself

Others: credit (default) risk, foreign exchange risk, interest rate risk,

political risk, market risk


Diversification

Risk management technique

Mixes a wide variety of investments within a portfolio

Objective is to minimize the impact that any one security will

have on overall performance of the portfolio

For the best diversification, portfolio should be spread among many different assets; cash, stocks, bonds

securities should vary in risk and

securities should vary by industry to minimize unsystematic risk to different

companies


Portfolio Risk

For n securities, portfolio risk – variance of the portfolio return

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Expected value of squared variable- how much rate of return of portfolio tends to vary from its mean


Example: Variance of Portfolio

The variance and covariance are calculated as

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Example: Variance of Portfolio

The portfolio risk is

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For equally weighted portfolio27

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Short Sale

It is possible to sell an asset that you do not own through the short selling (shorting) the asset. In order to implement short selling

borrow the asset from the owner such as brokerage firm

sell the borrowed asset to someone else, receiving an amount x0

at a later date purchase the asset for x1, and return the asset to lender

The short selling is profitable if the asset price declines

Risky since the potential for loss is unlimited –if asset value increases

Although prohibited within certain financial institutions, not universally

forbidden

1001 then , If xxprofitxx

0101 then , If xxlossxx


Example:Short-sale

Assume that company X has a poor outlook next month. The stock is now trading at £65, but you see it trading much lower than this price in the future.

You decide to take risk and trade on this stock. Two things can happen; stock price can go up or down

Predicted Stock price £40

Predicted Stock price £85

Action taken Price(£) Cost(£) Price(£) Cost(£) Borrowed 100

shares of X & sell 65 6500 65 6500

Bought back 100 shares of X

40 -4000 85 -8500

Profit/Loss 2500 -2000

Make Money

Lose Money

RISKY: no guarantee that the price of a short stock will drop


Optimal Portfolio

Portfolio theory • effects of investor decisions on security prices • relationship that should exist between the returns and risk.

• possible to have different portfolios varying levels of risk &return• decide how much risk you can handle and allocate (or diversify) portfolio according to this decision

Harry Markowitz: 1990 Nobel Prize winner in Economic Sciences• published in 1952 Journal of Finance titled “Portfolio Selection”• formalized an integrated theory of diversification, portfolio risks, efficient & inefficient portfolios

Maximize (expected return of portfolio)

(weighted sum of individual expected rates of return)

Minimize (portfolio risk)

(expected value of squared variable- how much rate of return of portfolio

tends to vary from its mean)


The Markowitz Model

Construct portfolio under uncertainty

Suppose that there are n assets with random rates of return

with mean values

The portfolio consists of n assets

are the portfolio weights

such that

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niwi ,,2,1for

11

n

iiw

nrrr ,,, 21


Maximum Expected Wealth Problem

niw

w

rw

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iii

w

,,2,1 0

1

to subject

max

1

1

•No risk –maximize expected portfolio return –risk-neutral approach


Minimum Variance Portfolio

niw

w

ww

i

n

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n

jijji

n

iw

,,2,1 0

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to subject

min

1

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•Take risk into account


Mean-Variance Optimisation Problem

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n

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to subject

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• trade-off between expected rate of return &variance of rate of return in a portfolio• naturally leads to diversification by taking risk attitudes in to account


Efficient Frontier

-10

-5

0

5

10

15

20

25

30

0 1 2 3 4 5 6 7 8 9

Active Risk %/AnnumAbs

olut

e R

etur

n %

/Ann

um• Varying the desired level of return and repeatedly solving QP problems identifiesthe minimum variance portfolio for each value of : efficient portfoliosr

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Business

risk risk

risk unsystematic risk

portfolio risk varianceof

portfolio risk fornsecurities

levels of risk return

rate risk

variance of portfolio

weighted portfolio