Portfolio Theory and Capital Asset Pricing Model

Prof. Ashok Thampy

IIMB

Markowitz Portfolio Theory

• Combining stocks into portfolios can reduce standard deviation, below the level obtained from a simple weighted average calculation.

• Correlation coefficients make this possible.

• The various weighted combinations of stocks that create this standard deviations constitute the set of efficient portfoliosefficient portfolios.

Markowitz Portfolio Theory

Coca Cola

Reebok

Standard Deviation

Expected Return (%)

35% in Reebok

Expected Returns and Standard Deviations vary given different weighted combinations of the stocks

Efficient Frontier

Standard Deviation

Expected Return (%)

•Each half egg shell represents the possible weighted combinations for two stocks.

•The composite of all stock sets constitutes the efficient frontier

Efficient Frontier

Standard Deviation

Expected Return (%)

•Lending or Borrowing at the risk free rate (rf) allows us to exist outside the

efficient frontier.

rf

Lending

BorrowingT

S

Efficient Frontier

A

B

Return

Risk (measured as )

Efficient Frontier

A

B

Return

Risk

AB

Efficient Frontier

A

BN

Return

Risk

AB

Efficient Frontier

A

BN

Return

Risk

AB

Goal is to move up and left.

WHY?

ABN

Efficient Frontier

Return

Risk

Low Risk

High Return

High Risk

High Return

Low Risk

Low Return

High Risk

Low Return

Efficient Frontier

Return

Risk

Low Risk

High Return

High Risk

High Return

Low Risk

Low Return

High Risk

Low Return

Portfolio Risk

)rx()r(x Return PortfolioExpected 2211

)σσρxx(2σxσxVariance Portfolio 21122122

22

21

21

Portfolio Risk

n

1iii )r(x Return Portfolio Expected

n

ji 1,ij)σ( jixxVariance Portfolio

Portfolio RiskThe shaded boxes contain variance terms; the remainder contain covariance terms.

1

2

3

4

5

6

N

1 2 3 4 5 6 N

STOCK

STOCKTo calculate portfolio variance add up the boxes

Limits of Diversification

covariance average x 1/N) - (1 variance average x (1/N) variance Portfolio

)covariance .(average (N variance average Variance Portfolio 2

221

).1

NNx

NN

As the number of stocks in the portfolio becomes very large, the portfolio variance tends towards the average covariance.

Portfolio Diversification

Suppose you make a portfolio constructed by taking equalProportions of n assets; that is xi = 1/n for each i. then The corresponding portfolio return and variance is :

n

1ii)(r

n1

Return Portfolio Expected

n

σ)σ(

1 2

1,ij2

n

jinVariance Portfolio

Question : Find the minimum variance portfolio

abba

abab

abba

abba

ababaaaa

abaabaaa

abbabbaa

xx

Solving

xxxx

xxxx

xxxx

p

p

p

22

0)21(2)1(22

)1(2)1(

2

22

2

22

2

22

2

22222

22222

and

:get we this

Return

Risk

Risk Free

Return, = rrf

Efficient Portfolio

Market Return = rm

The one-fund theorem: There is a single fund F of risky assets such that any efficient portfolio can be constructed as a combination of the fund F and the risk free asset.

F

Capital Market Line

Return

Risk

.

rf

Risk Free

Return =

Efficient Portfolio

Market Return = rp

pσ

Slope = (rp-rf)/ pσ The portfolio that maximizes theSlope gives the efficient portfolio.

The capital market line is mathematically expressed asFollows:

asset. efficient arbitrary anof returnof rate theof deviation standard the and

value expected the are and and return,of rate market theof deviation standard and

values expected the are and where

r

r

rrrr

MM

M

fMf

Capital Asset Pricing Model

2)(

M

iMifMifi

i

rrrr

r

where

:satisfies i asset anyof return,of rateexpected the efficient, is M portfolio market theIf :CAPM The

Portfolio Beta

Beta of a portfolio is the weighted average beta of individualAssets in the portfolio.

Security Market Line

Return

.

rf

Risk Free

Return =

Efficient Portfolio

Market Return = rm

BETA1.0

Security Market LineReturn

BETA

rf

1.0

SML

SML Equation = rf + B ( rm - rf )

Systematic and Unsystematic Risk

risk icunsystemat risk systematic

Variance

VarrVarrVar

Cov

E

rrrr

iMiii

Mi

i

ifMifi

)()()(

0),(

0)(

)(

22

Testing the CAPM

Avg Risk Premium 1931-65

Portfolio Beta1.0

SML

30

20

10

0

Investors

Market Portfolio

Beta vs. Average Risk Premium

Testing the CAPM

Avg Risk Premium 1966-91

Portfolio Beta1.0

SML

30

20

10

0

Investors

Market Portfolio

Beta vs. Average Risk PremiumIn the period 1966-91, returnhas not been proportionate to betaas predicted by the CAPM-SML.