Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 1.

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 1


Try these problems

Ch 6Problem 6Problem 12Problems 14-16 (see p 157)

Ch 7Problem 1Problem 5

Portfolio Diversificationand the

Capital Asset Pricing Model

Prof. Ian GiddyNew York University

New York University/ING Barings


Equity Risk and Return: Summary

Investors diversify, because you get a better return for a given risk.

There is a fully-diversified “market portfolio” that we should all choose

The risk of an individual asset can be measured by how much risk it adds to the “market portfolio.”


Capital Allocation Possibilities:Treasuries or an Equity Fund?

rf=7%

E(rP)

=17%

P=27%

10%

P

Expected Return

Risk

7%

THE EQUITY FUND


We Can Buy Some T-bills and Some of the Risky Fund...

C.A.L.

SLOPE=0.37

E(R)

SD

17%

14%

18.9% 27%

ONE PORTFOLIO:

30% Bills, 70% Fund

E(R)=.3X7+.7X17=14%

SD=.7X27=18.9%

rf=7%


...Or Buy Two Risky Assets

A

E(r)

B


Diversification

Asset F Asset G Portfolio of Assets F and

GReturn

Time

Return

Time

Return

Time

kkk


Portfolio Return...

To compute the return of a portfolio: use the weighted average of the returns of all assets in the portfolio, with the weight given each asset calculated as

(value of asset)/(value of portfolio).

The portfolio return E(Rp) is:

E(Rp) = (w1k1)+(w2k2)+ ... (wnkn) = wj kj

where wj = weight of asset j, kj = return on asset j


...and Risk (Standard Deviation)

Portfolio return is the weighted average of all assets’ returns,

But portfolio standard deviation is normally less than the weighted average of all assets’ standard deviations!

The reason: asset returns are imperfectly correlated.


Measuring Portfolio Risk

The variance of a 2-asset portfolio is:

where wA and wB are the weights of A and B in the portfolio.

P2

A2

A2

B2

B2

A B A B A B = w + w + 2 w w


Return and Risk, Generalized

Portfolio return:

where wi are the weights of each asset in the portfolio. (Expected return is simply the weighted sum of the individual asset returns.)

Portfolio variance:

When i = j, the term wiwjFiFjDij becomes wi2Fi

2.

E(R ) = w E(R )pi=1

n

i i

P2

i=1

n

j=1

n

i j i j ij = w w


Covariance and Correlation

cov,

[,

( )][,

( )]K I

iPi r

K iE r

KrI i

E rI

K IK I

K I,

cov,

The correlation coefficient scales the covariance to a value between -1 and +1:


Risk and Return of Stocks, Bonds and a Diversified Portfolio

Rate of Return

State Prob. Equity Bond Portfolio

Recession 1/3 -7% +17% +5%

Normal 1/3 +12% +7% +9.5%Boom 1/3 +28% -3% +12.5%

Expected Return 11% 7.0% 9.0%Variance 204.7% 66.7% 9.5%Standard Deviation 14.3% 8.2% 3.1%


The Correlation Between Stock and Bond Returns Covariance

= 0.3333(-7-11)(17-7) + 0.3333(12-11)(7-7) +0.3333(28-11)(-3-7)

= -116.67

Correlation

= -116.66 / 14.3(8.2) = -0.99

p R E R R E Rss

n

s e e s b b1

, ,( ) ( )

cov ,e b

e b


Portfolio Return and Standard DeviationGiven:

WS = 0.5 RS = 12% S = 25%

WB = 0.5 RB = 9% B = 12%

and S,B = 0.2

Rp = 0.5(12)+0.5(9) = 10.5%

P = [(0.5)2(25) 2+(0.5) 2(12) 2+2(0.5)(0.5)(25)(12)(0.2)]1/2

= (156.25+36+30)1/2

= (222.25) 1/2

= 14.91%


Attainable Set of Risk/Return Combinations

A

E(r)

B

AB 1AB 0

AB 1


The Minimum-Variance Frontier of Risky Assets

Efficient frontier

Individual assets

Global minimum-variance portfolio

E(r)


The Efficient Frontier of Risky Assets with the Optimal CAL

Efficient frontier

CAL(P)E(r)

rf


The Capital Asset Pricing Model (CAPM)

CAPM Says:The total risk of a financial

asset is made up of two components.

A. Diversifiable (unsystematic) risk

B. Nondiversifiable (systematic) risk

The only relevant risk is nondiversifiable risk.

CAL(P)E(r)

rf


Types of Risk

Portfolio

Risk

kp

Number of Securities (Assets) in Portfolio1 5 10 15 20 25

}}{

TOTAL RISK

NONDIVERSIFIABLE RISK

DIVERSIFIABLE RISK


The Model: CAPM

The CAPM (Capital Asset Pricing Model) links together nondiversifiable risk and return for all assets:

A. Beta Coefficient (b) is a relative measure of nondiversifiable risk; an index of the change of an asset's return in response to a change in the market return

B. Market Return (km) is the return on the market portfolio of all traded securities


Security Market Line

Nondiversifiable Risk, 0 .50 1.0 1.5 2.0 . . .

SML

}Market RiskPremium: 4%

} Asset Z’s Risk Premium: 6%

1716151413121110987654321

Rz =

Rm =

RF =

Required

Return, R(%)

RF m z


The Equation for the CAPM

Rj = RF + j (Rm - RF)where:

Rj = Required return on asset j;

RF = Risk-free rate of return

j = Beta Coefficient for asset j;

Rm = Market return

The term [j(Rm - RF)] is called the risk premium and (Rm-RF) is called the market risk premium


Silicon Graphics

Consider an investment in Silicon Graphics. It has a Beta of 2.0 (riskier than the average stock). If the T-bill rate is 5% and the S&P return is 10%, what is the required return for Silicon Graphics stock?

kj = .05 + [2.0 x (.10-.05)] = .05 + [2.0 x (.05)]

= .05 + .10 = .15 or 15%


Graphic Depiction of CAPM

REQUIRED

RETURN

15% Rj

with bj = 2.0

10% Rm

5% RF

0 .50 1.0 1.5 2.0 . . .

SecurityMarketLine

}Market RiskPremium: 5% }Stock’s

RiskPremium: 10%

Beta (Nondiversifiable Risk)

SML = Rj= .05 + j(.10-.05)

Given: RF = 5%; Rm = 10%


Interpreting Beta

Market Beta = 1.0 = average level of riskA Beta of .5 is half as risky as averageA Beta of 2.0 is twice as risky as averageA negative Beta asset moves in opposite

direction to market


Finding Beta: Example

You have found the following data relative to the stock of the Telmex Corp. and current conditions:

Required/expected return = 20%

Market portfolio return = 11%

Risk premium for market portfolio = 6%

What is the Beta of Telmex stock?


Determine the Risk-Free Rate

Algebraic Solution Graphic Solution

Rm - RF = .06 .11 - RF = .06

RF = .05

Rm =11%

Rf = 5%} 6%} 5%

1.0Beta

SML

Ri = Rf + i(RM - Rf)


Plug into SML Formula

.20 = .05 + [Beta x (.11 - .05)]

.15 = Beta x (.06)

.15 = Telmex Beta 2.5 .06

Portfolio TheoryAssignment

Prof. Ian GiddyNew York University

New York University/ING Barings


Try these problems

Ch 6Problem 6Problem 12Problems 14-16 (see p 157)

Ch 7Problem 1Problem 5


BKM Chapter 6, Problem 6

Gold

Stocks

Optimal CALE(r)

P

A. If G,S<+1, gold is still an attractive asset to hold as part of a portfolio.

E(r)

Optimal CAL

Gold

Stocks

P

B. If G,S=+1, a portfolio of stocks and bills only dominates a portfolio with gold in all instances


BKM Chapter 6, Problem 12

RA-RfSince B’s error is small, diversification effect is less than for A, which has large unsystematic risk. RM-Rf

RB-Rf

RM-Rf

Stock A has a large error term so would be very risky if all funds were in this one basket.

A

B


BKM, Chapter 6, Problem 14

Most diversification achieved with 1st 20 stocks

By choosing low-correlated assets in the portfolio, risk may not be affected significantly. But would these be the best-return stocks?

Portfolio

Risk

kp

Number of Securities (Assets) in Portfolio1 5 10 15 20



The risk/number of stocks relationship is nonlinear, so risk increases as number of stock is further reduced

Portfolio

Risk

kp

Number of Securities (Assets) in Portfolio1 5 10 15 20



Hennessy’s portfolioE(r)

Limiting Hennessy’s holdings may have little impact on the risk of the total portfolio



E(RP) = Rf + [E(RM) - Rf]

20 = 5 + (15-5)

=15/10 = 1.5



S&PReturn

LibertyTravel

Nynex

Weakmarket

5% 2% 3.5%

Strongmarket

20% 32% 14%



A ) The beta is the change in the stock return per change in the market return. Therefore:

Aggressive = (2-32)/(5-20) = 2.00

Defensive = (3.5-14)/(5-20)

= .70B ) The expected return is an average of the

two possible outcomes:E(RAgg.) = .5(2+32) = 17%

E(RDef.) = .5(3.5+14) = 8.75%



C / The SML is determined by the market expected return of .5(20+5) = 12.5%, with a beta of 1, and the bill return of 8%.

Therefore, the equation for the security market line is:

E(R) = 8 + (12.5 - 8)



SMLE(r)

M

D

A

8%

12.5%

.7 1.0 2.0

17%



D / Stock SML E(R) Analyst E(R) AlphaAgg. 17% 17% 0Def. 11.15% 8.75% -2.4%

SMLE(r)

M

D

A

8%

12.5%

.7 1.0 2.0

D



E / The hurdle rate is determined by the project beta .7. The correct discount rate is 11.15%, the fair return on the defensive stock.


Equity Risk and Return: Summary

Investors diversify, because you get a better return for a given risk.

There is a fully-diversified “market portfolio” that we should all choose

The risk of an individual asset can be measured by how much risk it adds to the “market portfolio”

The CAPM tells us how the required return relates to the relevant risk.



www.giddy.org



www.giddy.org

Ian Giddy

NYU Stern School of Business

Tel 212-998-0704; Fax 212-995-4220

[email protected]

http://www.giddy.org

Copyright ©1997 Ian H. Giddy Portfolio Diversification and the CAPM 1.

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