CHAPTER 6 PORTFOLIO RISK AND RETURN: PART II

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FORMULAS FOR PORTFOLIO RISK AND RETURN

1

2

1, 1

1

2

2 2 2

1 , 1,

2

Cov( , )

1

Given: Cov( , ) and Cov( , )

Then:

N

p i ii

N

p i ji j

N

ii

ij i j i

N N

p i i i j ij i ji i j i j

p p

E R w E R

w w i j

w

i j i i

w w w

EXHIBIT 6-1 PORTFOLIO RISK AND RETURN

Portfolio Weight in

Asset 1 Weight in

Asset 2 Portfolio Return Portfolio Standard Deviation

X 25.0% 75.0% 6.25% 9.01% Y 50.0 50.0 7.50 11.18 Z 75.0 25.0 8.75 15.21 Return 10.0% 5.0% Standard deviation 20.0% 10.0% Correlation between

Assets 1 and 2 0.0

%01.9)20)(.10)(.0)(75(.)10)(.20)(.0)(25(.10.75.20.25. 2222 X

PORTFOLIO OF RISK-FREE AND RISKY ASSETS

Combine risk-free

asset and risky asset

Capital allocation line (CAL)

Superimpose utility

curves on the CAL

Optimal Risky

Portfolio

EXHIBIT 6-2 RISK-FREE ASSET AND PORTFOLIO OF RISKY ASSETS

DOES A UNIQUE OPTIMAL RISKY PORTFOLIO EXIST?

Identical Expectations

Different Expectations

Single Optimal Portfolio

Different Optimal

Portfolios

CAPITAL MARKET LINE (CML)

Capital Allocation Line

(CAL)

Market Portfolio Is the Risky Portfolio

Capital Market Line (CML)

EXHIBIT 6-3 CAPITAL MARKET LINE

M

Rf

Exp

ecte

d Po

rtfo

lio R

etur

n E

(Rp)

Standard Deviation of Portfolio p

CML

Individual Securities

Points above the CML are not achievable Efficient

frontier E(Rm)

CML: RISK AND RETURN

1 1

1

1

1p f m

p m

E R w R w E R

w

m fp f p

m

E R RE R R

By substitution, E(Rp) can be expressed in terms of σp, and this yields the equation for the CML:

EXAMPLE 6-1 RISK AND RETURN ON THE CML

Mr. Miles is a first time investor and wants to build a portfolio using only U.S. T-bills and an index fund that closely tracks the S&P 500 Index. The T-bills have a return of 5%. The S&P 500 has a standard deviation of 20% and an expected return of 15%.1. Draw the CML and mark the points where the investment in the market is 0%, 25%, 75%, and 100%.2. Mr. Miles is also interested in determining the exact risk and return at each point.

EXAMPLE 6-2 RISK AND RETURN OF A LEVERAGED PORTFOLIO WITH EQUAL LENDING AND BORROWING RATES

Mr. Miles decides to set aside a small part of his wealth for investment in a portfolio that has greater risk than his previous investments because he anticipates that the overall market will generate attractive returns in the future. He assumes that he can borrow money at 5% and achieve the same return on the S&P 500 as before: an expected return of 15% with a standard deviation of 20%. Calculate his expected risk and return if he borrows 25%, 50%, and 100% of his initial investment amount.

SYSTEMATIC AND NONSYSTEMATIC RISK

Nonsystematic Risk

Systematic Risk

Total Risk

Can be eliminated by diversification

RETURN-GENERATING MODELS

Different Factors

Return-Generating Model

Estimate of Expected Return

GENERAL FORMULA FOR RETURN-GENERATING MODELS

11 2

k k

i f ij j i m f ij jj j

E R R E F E R R E F

Factor weights or factor loadings

Risk factors

All models contain return on the market portfolio as a key factor

THE MARKET MODEL

i f i m f

i f i m f i

i i i m i

E R R E R R

R R R R e

R R e

Single-index model

The difference between expected returns and realized

returns is attributable to an error term, ei.

The market model: the intercept, αi, and slope coefficient, βi, can be estimated by

using historical security and market returns. Note αi = Rf(1 – βi).

CALCULATION AND INTERPRETATION OF BETA

, ,2 2

Cov ,

0.026250 0.70 0.25 0.15 0.70 0.25 1.170.02250 0.02250 0.15

i m i m i m ii mi

m m m

i

R R

Market’s Return

Asset’s Beta

Asset’s Return

EXHIBIT 6-6 BETA ESTIMATION USING A PLOT OF SECURITY AND MARKET RETURNS

Market Return

Slope = β𝑖 [Beta]

Ri

Rm

CAPITAL ASSET PRICING MODEL (CAPM)

3% 1.5 9% 3% 12.0%

3% 1.0 9% 3% 9.0%

3% 0.5 9% 3% 6.0%

3% 0.0 9% 3% 3.0%

i f i m f

i

i

i

i

E R R E R R

E R

E R

E R

E R

Beta is the primary determinant of expected return

ASSUMPTIONS OF THE CAPMInvestors are risk-averse, utility-maximizing, rational individuals.

Markets are frictionless, including no transaction costs or taxes.

Investors plan for the same single holding period.

Investors have homogeneous expectations or beliefs.

All investments are infinitely divisible.

Investors are price takers.

EXHIBIT 6-7 THE SECURITY MARKET LINE (SML)

1.0

M E(Rm)

Rf

Expe

cted

Ret

urn

Beta

βi = βm

Slope = Rm – Rf

E(Ri)

SML

The SML is a graphical representation of the CAPM.

PORTFOLIO BETA

1

(0.40 1.50) (0.60 1.20) 1.32N

p i ii

w

Portfolio beta is the weighted sum of the betas of the component securities:

3% 1.32 9% 3% 10.92%

p f p m f

p

E R R E R R

E R

The portfolio’s expected return given by the CAPM is:

APPLICATIONS OF THE CAPM

CAPM Applications

Estimates of Expected

Return

Performance Appraisal

Security Selection

PERFORMANCE EVALUATION: SHARPE RATIO AND TREYNOR RATIO

• Focus on total risk

Sharpe Ratio

• Focus on systematic risk

Treynor Ratio

Sharpe ratio p f

p

R R

Treynor ratio p f

p

R R

PERFORMANCE EVALUATION: M-SQUARED (M2)

•Identical rankings•Expressed in percentage terms

Sharpe Ratio

2 mp f m f

p

M R R R R

PERFORMANCE EVALUATION: JENSEN’S ALPHA

Estimate portfolio beta

Determine risk-adjusted return

Subtract risk-adjusted return

from actual return

p p f p m fR R R R

EXHIBIT 6-8 MEASURES OF PORTFOLIO PERFORMANCE EVALUATION

Manager Ri σi βi E(Ri) Sharpe Ratio

Treynor Ratio

M2 αi

X 10.0% 20.0% 1.10 9.6% 0.35 0.064 0.65% 0.40% Y 11.0 10.0 0.70 7.2 0.80 0.114 9.20 3.80 Z 12.0 25.0 0.60 6.6 0.36 0.150 0.84 5.40 M 9.0 19.0 1.00 9.0 0.32 0.060 0.00 0.00 Rf 3.0 0.0 0.00 3.0 – – – 0.00

EXHIBIT 6-11 THE SECURITY CHARACTERISTIC LINE (SCL)

Ri – Rf

Exce

ss S

ecur

ity Re

turn

Excess Market Return

[Beta]

Rm – Rf

Excess Returns

Jensen’s Alpha

EXHIBIT 6-12 SECURITY SELECTION USING SML

A (𝑅= 11%,β = 0.5)

C (𝑅= 20%,β = 1.2)

B (𝑅= 12%,β = 1.0)

Ri

Beta 𝛃𝒎 = 1.0

15%

Rf = 5%

Retu

rn o

n In

vest

men

t

SML

Undervalued

Overvalued

DECOMPOSITION OF TOTAL RISK FOR A SINGLE-INDEX MODEL

2 2 2 2 2 2 2i

Total variance = Systematic variance + Nonsystematic variance

2Cov ,

i fi m fi

i m ei m i i m ei

R R R R e

R e

Zero

EXHIBIT 6-13 DIVERSIFICATION WITH NUMBER OF STOCKS

5 1

Varia

nce

Number of Stocks

Systematic Variance

Total Variance

Non-Systematic Variance

Variance of Market Portfolio

10 20 30

WHAT SHOULD THE RELATIVE WEIGHT OF SECURITIES IN THE PORTFOLIO BE?

2Information ratio i

ei

Greater Non-Systematic Risk →Lower Weight

Higher Alpha →Higher Weight

LIMITATIONS OF THE CAPM

• Single-factor model• Single-period modelTheoretical

• Market portfolio• Proxy for a market portfolio• Estimation of beta• Poor predictor of returns• Homogeneity in investor expectations

Practical

EXTENSIONS TO THE CAPM: ARBITRAGE PRICING THEORY (APT)

1 ,1 ,p F p K p KE R R

Risk Premium for Factor 1

Sensitivity of the Portfolio to Factor 1

FOUR-FACTOR MODEL

, , , ,it i i MKT t i SMB t i HML t i UMD tE R MKT SMB HML UMD

Size Anomaly

Momentum Anomaly

Value AnomalySystematic

Risk

SUMMARY

• Portfolio risk and return• Optimal risky portfolio and the capital market line

(CML)• Return-generating models and the market model• Systematic and non-systematic risk• Capital asset pricing model (CAPM) and the

security market line (SML)• Performance measures• Arbitrage pricing theory (APT) and factor models