Ch 6

Efficient Diversification

Diversification and Portfolio Risk

Total risk:Market risk

Systematic or Nondiversifiable Firm-specific risk

Diversifiable or nonsystematic or unique

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

Exercise 421. Risk that can be eliminated through diversification is called ______

risk. A) unique B) firm-specific C) diversifiable D) all of the above

2. The risk that can be diversified away is ___________. A) beta B) firm specific risk C) market risk D) systematic risk

Two Asset Portfolio Return – Stock and Bond

ReturnStock

htStock Weig

Return Bond

WeightBond

Return Portfolio

rwrwr

S

S

B

B

p

rwrwr SSBBp

Covariance

r1,2 = Correlation coefficient of returns

r1,2 = Correlation coefficient of returns

Cov(r1r2) = r1,2s1s2Cov(r1r2) = r1,2s1s2

s1 = Standard deviation of returns for Security 1s2 = Standard deviation of returns for Security 2

s1 = Standard deviation of returns for Security 1s2 = 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

Two Asset Portfolio St Dev – Stock and Bond

Deviation Standard Portfolio

Variance Portfolio

2

2

,

22222 2

p

p

SBBSSBSSBBp wwww

rp = Weighted average of the n securitiesrp = Weighted average of the n securities

sp2 = (Consider all pair-wise

covariance measures)sp

2 = (Consider all pair-wise covariance measures)

In General, For an n-Security Portfolio:

Numerical Example: Bond and Stock

ReturnsBond = 6% Stock = 10%

Standard Deviation Bond = 12% Stock = 25%

WeightsBond = .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

Figure 6.3 Investment Opportunity Set for Stock and Bonds

Minimum variance portfolio

Ws = σB

2 - Cov(rS, rB) / (σs2 + σB

2 -2Cov(rS, rB))

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

Extending to Include Riskless Asset

The optimal combination becomes linearA 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(O) dominates other lines -- it has the best risk/return or the largest slope

Slope = (E(R) - Rf) / s[ E(RP) - Rf) / s P ] > [E(RA) - Rf) / sA]

Regardless of risk preferences combinations of O & F dominate

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

Figure 6.7 The Complete Portfolio

Figure 6.8 The Complete Portfolio – Solution to the Asset Allocation Problem

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

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

Figure 6.10 The Efficient Frontier of Risky Assets and Individual Assets

Exercise 22

1. Adding additional risky assets will generally move the efficient frontier _____ and to the _______. A) up, right B) up, left C) down, right D) down, left

2. Rational risk-averse investors will always prefer portfolios ______________. A) located on the efficient frontier to those located on the capital market line B) located on the capital market line to those located on the efficient frontier C) at or near the minimum variance point on the efficient frontier D) Rational risk-averse investors prefer the risk-free asset to all other asset choices.

Exercise331. The standard deviation of return on investment A is .10 while the standard deviation of

return on investment B is .05. If the covariance of returns on A and B is .0030, the correlation coefficient between the returns on A and B is __________. A) .12 B) .36 C) .60 D) .77

2. Consider two perfectly negatively correlated risky securities, A and B. Security A has an expected rate of return of 16% and a standard deviation of return of 20%. B has an expected rate of return 10% and a standard deviation of return of 30%. The weight of security B in the global minimum variance is __________. A) 10% B) 20% C) 40% D) 60%

Exercise321. Which of the following correlations coefficients will produce the least

diversification benefit? A) -0.6 B) -1.5 C) 0.0 D) 0.8

2. The expected return of portfolio is 8.9% and the risk free rate is 3.5%. If the portfolio standard deviation is 12.0%, what is the reward to variability ratio of the portfolio? A) 0.0 B) 0.45 C) 0.74 D) 1.35

Single Factor Modelri = E(Ri) + ßiF + e

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

Single Index Model

Risk Prem Market Risk Prem or Index Risk Prem

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

( ) ( ) errrr ifmiifi+-+=- ba

Let: Ri = (ri - rf)

Rm = (rm - rf)Risk premiumformat

Ri = ai + ßi(Rm) + ei

Risk Premium Format

Figure 6.11 Scatter Diagram for Dell

Figure 6.12 Various Scatter Diagrams

Components of RiskMarket or systematic risk: risk related to the

macro economic factor or market indexUnsystematic or firm specific risk: risk not related

to the macro factor or market indexTotal 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

Total Risk = Systematic Risk + Unsystematic RiskSystematic Risk/Total Risk = r2

ßi2 s

m2 / s2 = r2

bi2 sm

2 / (bi2 sm

2 + s2(ei)) = r2

Examining Percentage of Variance