Chapter 7

PORTFOLIO THEORY

The Benefits of Diversification

Outline

• Diversification and Portfolio Risk

• Portfolio Return and Risk

• Measurement of Co movements in Security Returns

• Calculation of Portfolio Risk

• Efficient Frontier

• Optimal Portfolio

• Riskless Lending and Borrowing

• The Single Index Model

Gains and Losses

Some investments can be very successful.

£10,000 invested in September 2001 in Lastminute.com would have been worth £134,143 in August 2003

$10,000 invested in August 1998 in Cephalon would have been worth $107,096 in September 2003 (and $180,000 in 2007)

And Losses

Losses in value can be even more spectacular

$10,000 invested in September 2000 in Palm Inc. would have been reduced to $91 by April 2003

A holding in July 2000 of £15 million in Exeter Equity Growth Fund would have been worth £72,463 in August 2003 (the share price fell from 103.50 to 0.50)

Diversification and Portfolio RiskBefore we look at the formula for portfolio risk, let us understand somewhat intuitively how diversification influences risk. Suppose you have Rs.100,000 to invest and you want to invest it equally in two stocks, A and B. The return on these stocks depends on the state of the economy. Your assessment suggests that the probability distributions of the returns on stocks A and B are as shown in Exhibit 7.1. For the sake of simplicity, all the five states of the economy are assumed to be equiprobable. The last column of Exhibit 7.2 shows the return on a portfolio consisting of stocks A and B in equal proportions. Graphically, the returns are shown in Exhibit 7.2.  

 Probability Distribution of Returns

 State of the Probability Return on Return on Return on Economy Stock A Stock B Portfolio  1 0.20 15% -5% 5% 2 0.20 -5% 15 5% 3 0.20 5 25 15% 4 0.20 35 5 20% 5 0.20 25 35 30%

  

Returns on Individual Stocks and the Portfolio

Expected Return and Standard Deviation

Expected Return

Stock A : 0.2(15%) + 0.2(-5%) + 0.2(5%) +0.2(35%) + 0.2(25%) = 15%Stock B : 0.2(-5%) + 0.2(15%) + 0.2(25%) + 0.2(5%) + 0.2(35%) = 15%Portfolio ofA and B : 0.2(5%) + 0.2(5%) + 0.2(15%) + 0.2(20%) + 0.2(30%) = 15%

Standard Deviation

Stock A : σ2A = 0.2(15-15)2 + 0.2(-5-15)2 + 0.2(5-15)2 + 0.2(35-15)2 + 0.20

(25-15)2 = 200 σA = (200)1/2 = 14.14% Stock B : σ2

B = 0.2(-5-15)2 + 0.2(15-15)2 + 0.2(25-15)2 + 0.2(5-15)2 + 0.2(35-15)2

= 200 σB = (200)1/2 = 14.14%

Portfolio : σ2(A+B) = 0.2(5-15)2 + 0.2(5-15)2 + 0.2(15-15)2 + 0.2(20-15)2

+ 0.2(30-15)2 = 90 σA+B = (90)1/2 = 9.49%

Portfolio Expected Return n

E(RP) = wi E(Ri) i=1

where E(RP) = expected portfolio return

wi = weight assigned to security i

E(Ri) = expected return on security i n = number of securities in the portfolio

Example A portfolio consists of four securities with expected returns of 12%, 15%, 18%, and 20% respectively. The proportions of portfolio value invested in these securities are 0.2, 0.3, 0.3, and 0.20 respectively.

The expected return on the portfolio is:

E(RP) = 0.2(12%) + 0.3(15%) + 0.3(18%) + 0.2(20%)

= 16.3%

Relationship Between Diversification and Risk

Risk

Unique

Risk

Market Risk

No. of Securities 1 5 10 20

Market Risk Versus Unique Risk

• Basic insight of modern portfolio theory:

Total risk = Unique risk + Market risk

•The unique risk of a security represents that portion of it total

risk which stems from firm-specific factors.

•The Market risk of a stock represents that portion of its risk

which is attribute to economy wide factors

Equations for Portfolio Risk

In symbols

E(Rp) = wi E(Ri) i=1

 But

p2 wi

2i 2

Thanks to the inequality shown in Eq. (7.1), investors can

achieve the benefit of risk reduction through diversification. Before

we discuss how this can be accomplished let us first understand how

comovements in security returns are measured

 

Portfolio Risk

The risk of a portfolio is measured by the variance (or standard

deviation) of its return. Although the expected return on a portfolio

is the weighted average of the expected returns on the individual

securities in the portfolio, portfolio risk is not the weighted average

of the risks of the individual securities in the portfolio (except when

the returns from the securities are uncorrelated).

Measurement Of Comovements

In Security Returns

• To develop the equation for calculating portfolio risk we

need information on weighted individual security risks and

weighted comovements between the returns of securities

included in the portfolio.

• Comovements between the returns of securities are

measured by covariance (an absolute measure) and

coefficient of correlation (a relative measure).

Covariance

COV (Ri , Rj) = p1 [Ri1 – E(Ri)] [ Rj1 – E(Rj)]

+ p2 [Ri2 – E(Rj)] [Rj2 – E(Rj)]

+

+ pn [Rin – E(Ri)] [Rjn – E(Rj)]

•

• • •

IllustrationThe returns on assets 1 and 2 under five possible states of nature are given below State of nature Probability Return on asset 1 Return on asset 2 1 0.10 -10% 5%

2 0.30 15 12 3 0.30 18 19

4 0.20 22 15 5 0.10 27 12 The expected return on asset 1 is : E(R1) = 0.10 (-10%) + 0.30 (15%) + 0.30 (18%) + 0.20 (22%) + 0.10 (27%) = 16% The expected return on asset 2 is : E(R2) = 0.10 (5%) + 0.30 (12%) + 0.30 (19%) + 0.20 (15%) + 0.10 (12%) = 14% The covariance between the returns on assets 1 and 2 is calculated below :

State of nature

(1)

Probability

(2)

Return on asset 1

(3)

Deviation of the return on

asset 1 from its mean

(4)

Return on asset 2

(5)

Deviation of the return on asset 2 from

its mean (6)

Product of the deviations

times probability

(2) x (4) x (6)

1 0.10 -10% -26% 5% -9% 23.4 2 0.30 15% -1% 12% -2% 0.6 3 0.30 18% 2% 19% 5% 3.0 4 0.20 22% 6% 15% 1% 1.2 5 0.10 27% 11% 12% -2% -2.2 Sum = 26.0

Thus the covariance between the returns on the two assets is 26.0.

Coefficient Of Correlation

Cov (Ri , Rj)

Cor (Ri , Rj) or ij = i j

ij

i j

ij = ij . i . j

where ij = correlation coefficient between the returns on

securities i and j

ij = covariance between the returns on securities

i and j i , j = standard deviation of the returns on securities

i and j

=

Graphical Portrayal of Various Types of Correlation Relationships

Portfolio Risk : 2 – Security Case

p = [w12 1

2 + w22 2

2 + 2w1w2 12 1 2]½

Example : w1 = 0.6 , w2 = 0.4,

1 = 10%, 2 = 16%

12 = 0.5

p = [0.62 x 102 + 0.42 x 162 +2 x 0.6 x 0.4 x 0.5 x 10 x 16]½

= 10.7%

Portfolio Risk : n – Security Case

p = [ wi wj ij i j ] ½

Example : w1 = 0.5 , w2 = 0.3, and w3 = 0.2

1 = 10%, 2 = 15%, 3 = 20%

12 = 0.3, 13 = 0.5, 23 = 0.6

p = [w12 1

2 + w22 2

2 + w32 3

2 + 2 w1 w2 12 1 2

+ 2w2 w3 13 1 3 + 2w2 w3 232 3] ½

= [0.52 x 102 + 0.32 x 152 + 0.22 x 202

+ 2 x 0.5 x 0.3 x 0.3 x 10 x 15

+ 2 x 0.5 x 0.2 x 05 x 10 x 20

+ 2 x 0.3 x 0.2 x 0.6 x 15 x 20] ½

= 10.79%

Risk Of An N - Asset Portfolio

2p = wi wj ij i j

n x n MATRIX 1 2 3 … n 1 w1

2σ12 w1w2ρ12σ1σ2 w1w3ρ13σ1σ3 … w1wnρ1nσ1σn

2 w2w1ρ21σ2σ1 w2

2σ22 w2w3ρ23σ2σ3 … w2wnρ2nσ2σn

3 w3w1ρ31σ3σ1 w3w2ρ32σ3σ2 w3

2σ32 …

: : :

n wnw1ρn1σnσ1 wn2σn

2

Dominance Of Covariance

As the number of securities included in a portfolio increases, the

importance of the risk of each individual security decreases whereas

the significance of the covariance relationship increases.

Quantitative Expression for Dominance of Covariance

1 2 3 … n 1

2

3:n

w12σ1

2 w1w2ρ12σ1σ2 W1w3ρ13σ1σ3

w1wnρ1nσ1σn

w2w1ρ21σ2σ1 w22σ2

2 W2w3ρ23σ2σ3

3

w2wnρ2nσ2σn

:

w3w1ρ31σ3σ1 w3w2ρ32σ3σ2 W32σ3

2 ………

wnw1ρn1σnσ1

wn2σn

2

:

Var(Rp) = Σ w12 Var(Ri) + Σ Σ wi wj Cov (Ri, Rj)

i=1 i=1 j=1 i¹ j

 If a naïve diversification strategy is followed wi = 1/n. Under such a strategy  n n n Var (Rp) =1/n Σ 1/n Var (Ri) + Σ Σ 1/n2 Cov (Ri, Rj)

i=1 i=1 j=1 j¹i

 The average variance term and the average covariance term may be expressed as follows:  n

Var = 1/n Σ Var (Ri)

i=1  1 n n

Cov = Σ Σ Cov (Ri, Rj)n(n-1) i=1 j=1

i¹jHence

1 n-1Var (Rp) = Var + Cov

n n As n increases, the first term tends to become zero and the second term looms large. Put differently, the importance of the variance term diminishes whereas the importance of the covariance term increases.

Portfolio Proportion of AwA

Proportion of BwB

Expected returnE (Rp)

Standard deviationp

1 (A) 1.00 0.00 12.00% 20.00%

2 0.90 0.10 12.80% 17.64%

3 0.759 0.241 13.93% 16.27%

4 0.50 0.50 16.00% 20.49%

5 0.25 0.75 18.00% 29.41%

6 (B) 0.00 1.00 20.00% 40.00%

Efficient Frontier For A Two Security-Case

Security A Security B

Expected return 12% 20%

Standard deviation 20% 40% Coefficient of correlation -0.2

Portfolio Options And

The Efficient Frontier

••12%

20%

20% 40%Risk, p

Expected return , E(Rp)

1 (A)

6 (B)

23

Feasible Frontier Under Various Degrees Of

Coefficient of Correlation

•

12%

20%

A (WA = 1)

Standard deviation, p

Expected return , E (Rp)

= – 1 .0

= 1 .0 = 0

B (WB = 1)

Minimum Variance Portfolio

Most investors (and portfolio managers) invest in two broad categories of financial assets viz., bonds and stocks. So, an important practical issue is: what is the proportion of bonds (and, by derivation, stocks) that minimises portfolio variance? To answer this question, let us look at the risk of a portfolio consisting of two assets, viz., bonds and stocks: 

Var (Rp) = w2bσ2

b + w2sσ2

s + 2wbwS ρbsσBσS

 where Var (Rp) is the variance of the portfolio consisting of bonds and stocks, wb is the proportion invested in bonds, wS is the proportion invested in stocks (wS =1-wb), σB is the standard deviation of returns from bonds, σS is the standard deviation of returns from stocks, and ρbs is the coefficient of correlation between the returns from bonds and stocks.

Minimum Variance Portfolio -2

The value of wb that minimises portfolio variance is:

To illustrate the above formula, let us consider the following data: E(rB) = 8%, E(rs) = 15%, σB = 0%, σS = 20%.Given the above data the expected return and standard deviation of a portfolio consisting of bonds and stocks is:

E(rp) = wb . 8 + ws . 15 

σp = [ w2b . 100 + w2

s . 400 + 2wb.ws.pbs . 200 ]1/2

 where E(rp) is the expected portfolio return, σp is the portfolio standard deviation, wb and ws are the proportions invested in bonds and stocks, ρBS is the coefficient of correlation between the returns on bonds and stocks. 

wB (min) =σ2

s - σbσS ρbS

σ2b+ σ2

s - 2σbσS ρbS

Minimum Variance Portfolio -3

The minimum portfolio variance for various correlations is shown below

Correlation

ρ = - 1.0 ρ = 0 ρ = 0.5

Minimum Variance Portfolio

wb (min) 0.6667 0.8000 1.000E (rp) 10.33% 9.400 8.000σp 0 12.00% 10.00%

Efficient Frontier For The

n-Security Case

Standard deviation, p

Expected return , E (Rp)

AO

N

M

X

F

•

•

•

•

•

B

D

Z•

•

•

Approaches to Determining the Efficient Frontier

• Graphical Analysis

• Calculus Analysis

• Quadratic Programming Analysis

Quadratic Programming Analysis

Technically, the quadratic programming approach manipulates the portfolio weights to determine efficient portfolios. The procedure followed is as follows. A desired expected return, say 9 percent, is specified. Then all portfolios (combinations of securities) that produce 9 percent expected returns are considered and the portfolio that has the smallest variance (standard deviation) of return is chosen as the efficient portfolio. This is continued for other levels of portfolio return, 10 percent, 11 percent, 12 percent, and so on, until all the possible expected returns are considered. Alternatively, the problem can be solved by specifying various levels of portfolio variance (standard deviation) and choosing the portfolios that offer the highest expected return for various levels of portfolio variance (standard deviation).

Why Is the Feasible Region Broken-egg Shaped-1

To see why the feasible region has a broken-egg shape (or umbrella shape), let us look at two securities shown in Exhibit 7.10.

Exhibit 7.10  Two Securities Portfolio

  

Expected return, E(r)

Standard deviation, σ

A

B

Why Is the Feasible Region Broken-egg Shaped-2

The expected return of a portfolio comprising of A and B (wa + wb = 1) is:

E(RP) = wa E(Ra) + wb E(Rb)

The standard deviation of the portfolio return is:

σ P= [wa2 σ a

2 + wb2 σ b

2 + 2 wa wbρab σ a σ b ]½

Note that while the expected portfolio return is not affected by Pab, the coefficient

of correlation between the returns on A and B, the standard deviation of

portfolio return is affected by Pab.

Why Is the Feasible Region Broken-egg Shaped-3

Now consider two cases:Case 1:  The returns of securities A and B are perfectly positively correlated.Case 2:  The returns of securities A and B are less than perfectly positively correlated.

In case 1, ρab, = 1. So σ p = [wa

2 σ a2 + wb

2 σ b2 + 2 wa wb σ a σ b ]½

= [waσ a + wb σ b]

Graphically, this means that the portfolio risk-return profile plots as the straight line joining A and B in Exhibit 7.10.

In case 2, ρab, < 1. So

σ p = [wa2 σ a

2 + wb2 σ b

2 + 2 wawb ρab σ a σ b ]½

Since ρab < 1, σ p will be less than [wa σ a + wb σb]

profile plots as the broken curved line joining A and B in Exhibit 7.10.Since ρab is typically less than 1, most of the portfolio risk-return profiles are like the curved line. This implies that the feasible region would have an umbrella like shape.

Risk-Return Indifference Curves

Expected return, E(Rp)

Standard

deviation σ p,

IQ IP

Utility Indifferences Curves

Expected return, E(Rp)

Standard

deviation σ p,

Ip4

Ip3

Ip2 Ip1

R

A

B

S

Optimal Portfolio

•

•

•

•

•

•P*

Q*

A

O

N

M

X

Standard deviation, p

Expected return , E (Rp)

IQ3

IQ2 IQ1

IP3 IP2IP1

•

•

•

•

•

•

A

O

N

M

X

Standard deviation, p

Expected return , E (Rp)

•

•

••

•

•

• •

Rf

u

D

EV

G

B

CF Y

I

II

•

S

Riskless Lending And

Borrowing Opportunity

Thus, with the opportunity of lending and borrowing, the efficient frontier changes. It is no longer AFX. Rather, it becomes Rf SG as it domniates AFX.

Separation Theorem

• Since Rf SG dominates AFX, every investor would do well to choose some combination of Rf and S. A conservative investor may choose a point like U, whereas an aggressive investor may choose a point like V.

• Thus, the task of portfolio selection can be separated into two steps:

1. Identification of S, the optimal portfolio of risky securities.

2. Choice of a combination of Rf and S, depending on one’s risk attitude.

This is the import of the celebrated separation theorem

Single Index Model

Information - Intensity of the Markowitz model n Securities

n Variance terms & n(n -1) /2Covariance terms

Sharpe’s ModelRit = ai + bi RMt + eit

E(Ri) = ai + bi E(RM)var (Ri) = bi

2 [var (RM)] + var (ei)cov (Ri ,Rj) = bi bj var (RM)

Markowitz Model Single Index Modeln (n + 3)/2 3n + 2

E (Ri) & var (Ri) for each bi , bj var (ei) for security n( n - 1)/2 each security & covariance terms E (RM) & var (RM)

Summing up• Portfolio theory, originally proposed by Markowitz, is the

first formal attempt to quantify the risk of a portfolio and develop a methodology for determining the optimal

portfolio.

• The expected return on a portfolio of n securities is : E(Rp) = wi E(Ri)

• The variance and standard deviation of the return on an n-security portfolio are:

p2 = wi wj ij i j

p = wi wj ij i j ½

• A portfolio is inefficient if (and only if) there is an alternative with (i) the same E(Rp) and a lower p or (ii) the same p and

a higher E(Rp), or (iii) a higher E(Rp) and a lower p

• Given the efficient frontier and the risk-return

indifference curves, the optimal portfolio is found at the

tangency between the efficient frontier and a utility

indifference curve.

• If we introduce the opportunity for lending and borrowing at the risk-free rate, the efficient frontier changes dramatically. It is simply the straight line from the risk-free rate which is tangential to the broken-egg shaped feasible region

representing all possible combinations of risky assets.

• The Markowitz model is highly information-intensive.

• The single index model, proposed by Sharpe, is a very helpful simplification of the Markowitz model.