PORTFOLIO THEORYReturns

Risk

PORTFOLIO Individual security is described by two

dimensions Return; as measured by Expected Value, Risk; as measured by Standard Deviation

A combination of two or more securities is called a portfolio

Portfolio return and risk are also described by Mean and Standard Deviation.

STOCK RETURNS & RISK

Since expected returns for Security 1 and 2 are same with Security 1 having lower risk, every rationale investor would invest all his money in Security 1 only.

Scenario Probability RETURNS % SHARE 1 SHARE 2

Scenario A 0.5 7 13Scenario B 0.5 11 5

Expected Return 9% 9%Variance 4 16Std. Dev. 2% 4%

TRADITIONAL VIEW What is the rationale for Security 2 to exist? Should one buy Security 2 at all?TRADITIONAL VIEW

Never put all eggs in one basket. Everything can not go bad simultaneously.

PORTFOLIO EFFECT Harry Markowitz provided the theoretical

background to the traditionalists by naive assumptions of investors’ attitude towards risk and return, and quantification of a) risk by standard deviation and b) return by expected value.

PORTFOLIO EFFECT Let a Portfolio of Security 1 and 2 be formed

How the risk has been reduced to zero. The losses of one security are offset by gains of

the other and vice-versa, keeping returns unaffected in all scenarios.

Scenario Prob Retn X1

Retn X2

Proportion Exp. Retn2 1

Scenario A 0.5 7 13 14 13 9Scenario B 0.5 11 5 22 5 9Expected Return=9% and Risk i.e Variance=0 (NIL)

CO-VARIANCE Risk of the portfolio is dependent upon

how securities forming the portfolio are related to each other rather than their individual risk.

Portfolio risk in terms of co-variance is defined as

The relationship of Sec 1 and Sec 2 is given by the co-efficient of correlation ρas

∑=

−−=N

1n2211n21 )xx)(xx(p)x,x(Cov

21

2121

)x,x(Cov)x,x(σσ

=ρ

CORRELATION Co-efficient of correlation ρ lies between

± 1 and describes the nature of relationship of returns of two securities.

ρ = + 1; Perfect positive correlation: The returns of two securities move by the same amount and in the same direction

ρ = -1; Perfect negative correlation:Thereturns of two securities move by the same amount but in the opposite direction

ρ = 0; Returns of the two securities have no correlation

PORTFOLIO RETURN & RISK The portfolio return rp and portfolio risk σp

of 2 securities is given by

Covariance of 1 and 2 is=0.5*(7-9)(13-9)+0.5*(11-9)(5-9)= -8

Coefficient of correlation ρ = -8/4*2=-1 And with f1 = 1/3 and f2=2/3 we get

rp = 9% and σp= 0

212122

22

21

21p

2211p

ff2ff

x*fx*fr

σρσ+σ+σ=σ

+=

PORTFOLIO EFFECT Combining two securities leads to reduction in

risk if they are less than perfectly correlated. No portfolio effect is seen if coefficient of

correlation is +1. Smaller the coefficient of correlation greater is

the portfolio effect. Combining negatively correlated securities

produce extremely good portfolios. With careful selection of proportions it is

possible to reduce the risk to zero. Necessary conditions for no risk portfolio:

ρ = -1 and f1/f2=σ2/ σ1

FEASIBLE PORTFOLIOS

Sec 2Return

ρ =-1

-1< ρ <+1ρ =-1 ρ =+1

Sec 1

Risk

Depending upon the correlation coefficient, a combination of risk and return along the lines shown can be achieved by varying proportions of investment in 1 & 2.

By combining 2 securities it is possible to have portfolio with risk lower than that of either Sec 1 or Sec 2.

EFFICIENT FRONTIER Combining more than 2 securities will

convert the feasible set of portfolios from line to a “Space”.

Rationale behaviour of investors (seeking lower risk for the same return and higher return for the same risk) will eliminate inefficient portfolios. Few portfolios will dominate others and these will form an EFFICIENT FRONTIER.

Efficient Frontier can be arrived at by using Quadratic Programming.

DIVERSIFICATION By adding securities in a portfolio the risk

can be reduced: To what level the risk can be reduced. How many securities are adequate. How to identify securities to be included or

excluded in/from a portfolio. Markowitz diversification will suggest

Reduction in risk as number of securities in the portfolio increase.

Selection of securities is based on coefficient of correlation, those with negative/low correlation must be included in the portfolio.

POWER OF DIVERSIFICATION The variance of the portfolio of n securities is

If we assume that all variances are equal to σ and all coefficients of correlation equal to ρ then all covariances are equal to ρσ2. The variance of the portfolio of n equally weighted portfolio is:

22 2p

22

p

2ji21

; n eargl with ,n

1nn

....Cov(2,1)Cov(1,2) and .... ,n/1...ff

ρσ≅σρσ−

+σ

=σ

ρσ===σ==σ=σ===

∑ ∑∑

∑∑

= ≠==

= =

+σ=σ=

=σ

n

1j

n

ji,1iji

n

1i

2i

2ip

n

1j

n

1ijip

)j,i(Cov*f*f*f j,i gSegregatin

)j,i(Cov*f*f

POWER OF DIVERSIFICATION When n is large the importance of individual

variance declines and portfolio variance is more dependent upon covariances.

When there is perfect correlation (ρ=1) then portfolio variance σp=σ; implies no reduction in risk irrespective of the portfolio size.

When there is no correlation (ρ = 0) the variance of the portfolio σp=σ/√n; implies that risk reduces as number of securities are increased.

The decrease in portfolio risk is smaller and smaller with each addition of security. The impact of increasing from 1 to 2 securities is much larger than that of from 100 to 101st.

SIMPLE DIVERSIFICATION An empirical study was done by constructing

different portfolios of 40 securities with equal investment out of 470 stocks listed at NYSE.

OBSERVATIONS: There was reduction in risk till 15/20 securities

were added. Very nominal reduction in risk was possible

thereafter. Risk could not be reduced to zero irrespective of

the number of securities in the portfolio. The performance of portfolios with careful

selection of securities and those with random selection was almost same.

SIMPLE DIVERSIFICATION Diversification across industries is no better than

Simple Diversification. There is no perceptible reduction in risk beyond

8/10 securities.Total Risk

Diversifiable Risk

Non-Diversifiable Risk10 15 20

Nos. of Securities

PORTFOLIO VARIANCE The risk that is diversified away is the

Unsystematic Risk of the stocks. Non Diversifiable risk is SYSTEMATIC Risk; The

risk that is common to all stocks, external to the firm, and uncontrollable.

Therefore while devising portfolio the Systematic risk of the share as measured by its Beta assumes greater importance.

Portfolio Variance is given by

∑∑==

+σβ=σn

1i

2i

2i

2m

2n

1iii

2p ef*)f(

SUPERFLUOUS DIVERSIFICATION Maximum reduction in risk can be obtained

with 10-15 randomly selected stocks. Further addition and research is superfluous and disadvantageous. Unnecessary and avoidable cost of research Desire to add securities results in holding lack-

lustre securities resulting in sub-optimisation Frequent changes and large number mean

higher transaction costs eroding gains. Difficult to monitor large number of securities.

Larger the number greater is inertia and confusion.