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Efficient Diversification

CHAPTER 6

1- Diversification and Portfolio Risk 2- Asset Allocation with two risky assets 3- The optimal risky Pf with a risk free asset 4- A single factor valuation model

Diversification and Portfolio Risk

• Market risk

– Systematic or Nondiversifiable

• Firm-specific risk

– Diversifiable or nonsystematic

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Figure 6.1 Portfolio Risk as a Function of the Number of Stocks

Figure 6.2 Portfolio Risk as a Function of Number of Securities

2- Asset Allocation with two risky assets

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Two Asset Portfolio Return – Stock and Bond

ReturnStock

htStock Weig

Return Bond

WeightBond

Return Portfolio

r

wr

w

r

S

S

B

B

p

rwrwr SSBBp

Covariance

r1,2 = Correlation coefficient of

returns

Cov(r1r2) = r1,2s1s2

s1 = Standard deviation of

returns for Security 1

s2 = Standard deviation of

returns for Security 2

Correlation Coefficients: Possible Values

If r = 1.0, the securities would be

perfectly positively correlated

If r = - 1.0, the securities would be

perfectly negatively correlated

Range of values for r 1,2

-1.0 < r < 1.0

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Two Asset Portfolio St Dev – Stock and Bond

Deviation Standard Portfolio

Variance Portfolio

2

2

,

22222 2

s

s

rsssss

p

p

SBBSSBSSBBp wwww

Numerical Example: Bond and Stock

Returns

Bond = 6% Stock = 10%

Standard Deviation

Bond = 12% Stock = 25%

Weights

Bond = .5 Stock = .5

Correlation Coefficient

(Bonds and Stock) = 0

Return and Risk for Example

Return = 8%

.5(6) + .5 (10)

Standard Deviation = 13.87%

[(.5)2 (12)2 + (.5)2 (25)2 + …

2 (.5) (.5) (12) (25) (0)] ½

[192.25] ½ = 13.87

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Return and Risk for Example: What are the characteristics of the Pf with min variance?

• Consider a Pf P including i and j where weights are equal to wi and wj respectively wi + wj =1 & wj =1- wi

var(p) = wi ² var (i) + (1-wi)²var(j) + 2wi (1-wi) cor (i,j).si.sj

To Minimize the variance: var(P)/wi = 0

var(p)/ wi = 2wi.var(i) - 2.var(j) + 2wi.var(j) + 2.cor(i,j).si.sj – 4wi. cor (i,j). si.sj

=> 2wi.[var(i) + var(j) - 2.cor (i,j).si.sj]

+ 2. cor (i,j).si.sj - 2var(i) = 0

var(j) - cor (i,j).si.sj

• wi = _____________________________________

var (i) + var(j) - 2.cor (i,j).si.sj

Return and Risk for Example: What are the characteristics of the Pf with min variance?

var(j) - cor (i,j). si.sj

• wi = _____________________________________

var (i) + var(j) - 2.cor (i,j).si.sj

In our case, cor =0

var(j)

• wi = ______________________

var (i) + var(j)

wi=.122/(.122 +.252) = 0.1873 for stocks& 0.8127 for bonds

E(R) =0.8127(6) + .1873 (10) = 6.75%

Std-dev = [0.81272 (12) 2 + .18732 (25) 2] ½ = 10.81%

Figure 6.3 Investment Opportunity Set for Stock and Bonds

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Figure 6.4 Investment Opportunity Set for Stock and Bonds with Various Correlations

Assignment Exercise (EC) 2

a- Do you believe the mean return & variance of the

stock in PANEL B to be more than, less than the values

computed in PANEL A? Why?

b- Calculate the new values of E(R) & Var of PANEL B

c- Calculate Covar between stock and bond funds of PANEL B?

Panel A Probability Stock fds R% Bond fds R% Recession 0.3 -11 16 Normal 0.4 13 6 Boom 0.3 27 -4

Panel B Probability Stock fds R% Bond fds R%

Recession 0.3 -16 16 Normal 0.4 13 6 Boom 0.3 30 -4

3- The optimal risky Pf with a risk free asset

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Extending to Include Riskless Asset

• The optimal combination becomes linear

• A single combination of risky and riskless assets will dominate

Figure 6.5 Opportunity Set Using Stock and Bonds and Two Capital Allocation Lines

Dominant CAL with a Risk-Free Investment (F)

CAL(B) dominates CAL(A) – it has a larger slope Slope = (E(R) - Rf) / s

[ E(RB) - Rf) / sB ] > [E(RA) - Rf) / sA]

The combination of B & F dominates A & F The CAL equation for any portfolio on the combination

of B & F is thus: E(Rp) = Rf + [E(RB) - Rf) / sB] sp

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What are the characteristics of the Pf that maximizes the slope

(E(Ri)-Rf) var(j) - (E(Rj)-Rf) cor(i,j).si.sj

• wi = __________________________________

(E(Ri)-Rf) var(j) + (E(Rj)-Rf) var(i) -

(E(Ri)-Rf + E(Rj)-Rf) cor(i,j).si.sj

If i is the weight of stocks, and j is for bonds, and Rf=5%

i = 67.01% and j=32.99%

E(R)=8.68% and s = 17.97%

S= (8.68-5)/17.97 = 0.20

Figure 6.6 Optimal Capital Allocation Line for Bonds, Stocks and T-Bills

CAL(O) has the best risk/return or the largest slope. Regardless of risk preferences combinations of O & F

dominate

What about a Pf P including 45% T Bills and 55% Pf O?

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Figure 6.7 The Complete Portfolio

Figure 6.9 Portfolios Constructed from Three Stocks A, B and C

efficient frontier

• Construct a risk/return plot of all possible portfolios

– Those portfolios that are not dominated constitute the efficient frontier

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Fig 6.10 The Efficient Frontier of Risky Assets & Individual Assets

The Northwestern-most Pfs in terms of E(R) & Stdev

from the universe of securities

(i.e. Pfs that maximize E(R) at each level of Pf Risk)

Extending Concepts to All Securities

• The optimal combinations result in lowest level of risk for a given return

• The optimal trade-off is described as the efficient frontier

• These portfolios are dominant

• The optimal combination becomes linear.

• A single combination of risky and riskless assets will dominate.

Extending to Include Riskless Asset

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Efficient Frontier (cont’d)

• When a risk-free investment is available, the shape of the efficient frontier changes

– The expected return and variance of a risk-free rate/stock return combination are simply a weighted average of the two expected returns and variance

• The risk-free rate has a variance of zero

Efficient Frontier (cont’d)

Standard Deviation

Expected Return

Rf A

B

C

Efficient Frontier (cont’d)

• The efficient frontier with a risk-free rate:

– Extends from the risk-free rate to point B

• The line is tangent to the risky securities efficient frontier

– Follows the curve from point B to point C

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In class exercise A pension fd Mgr considers three MFs (Stock Fd, Bond Fd & T-Bills=5.5%)

E(R) std-dev Stock fund (S) 15% 35% Cor (S,B)=0.15

Bond fund (B) 9% 23%

a- Tabulate and draw the Invtt opportunity of S & B

b- Draw a tangent from the Rf, what about Optimal Pf?

c- What is the reward-to-variability ratio of the best feasible CAL (E(R) – Rf)/stdev ?

d- Suppose the Pf must yield 12% and be efficient, that is on the CAL, what is its std-dev? What is the % invested in T-bill and each of the two risky assets?

4- Single Factor Model

ri = E(Ri) + ßiF + e

Ri = the excess return on a security (ri-rf)

ßi = index of a securities’ particular return to the factor

F= some macro factor; in this case F is unanticipated movement; F is commonly related to security returns

Assumption: a broad market index like the S&P500 is the common factor

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Single Index Model

Risk Premium Market Risk Premium

or Index Risk Premium

i = the stock’s expected return if the

market’s excess return is zero

ßi(rm - rf) = the component of return due to

movements in the market index

(rm - rf) = 0

ei = firm specific component, not due to market

movements

a

( ) ( ) e r r r r i f m i i f i - - b a

Let: Ri = (ri - rf)

Rm = (rm - rf)

Risk premium

format

Ri = ai + ßi(Rm) + ei

Risk Premium Format

Figure 6.11 Scatter Diagram for Dell

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Figure 6.12 Various Scatter Diagrams

Components of Risk

• Market or systematic risk: risk related to the macro economic factor or market index

• Unsystematic or firm specific risk: risk not related to the macro factor or market index

• Total risk = Systematic + Unsystematic

Measuring Components of Risk

si2 = bi

2 sm2 + s2(ei)

where;

si2 = total variance

bi2 sm

2 = systematic variance

s2(ei) = unsystematic variance

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Total Risk = Systematic Risk + Unsystematic Risk

Systematic Risk/Total Risk = r2

ßi2 s

m2 / s2 = r2

bi2 sm

2 /( bi2 sm

2 + s2(ei)) = r2

Examining Percentage of Variance

Advantages of the Single Index Model

• Reduces the number of inputs for diversification

• Easier for security analysts to specialize