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Risk, Return, and CAPM

Finance 221Summer 2006

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Review

• Trade-off between risk and return

• What is the difference between systematic and unsytematic risk?

• Why are investors compensated only for systematic risk?

• What is a risk premium?

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Average Return and St. Dev. for Individual Securities, 1994-2003

0

5

10

15

20

25

0 10 20 30 40 50 60

Average Return (%)

Standard Deviation (%)

Wal-MartAnheuser-Busch

Archer Daniels MidlandAmerican Airlines

Standard Deviation captures total risk 4

Review: Diversification

• Research suggests that investors are not as diversified as they should be

• Can also diversify across asset class: a portfolio composed of 80% stocks and 20% bonds gets 97% of S&P 500 return with 85% of the risk.

• International diversification; Correlation between US and Brazil Markets – 0.09 in late 1970s, 0.68 in early 1990s. Home bias (Japan – 98%).

• Campbell, Lettau, Malkiel and Xu (2001): market volatility relatively constant over 1962-1997, firm-specific volatility doubled. Correlations declined from 0.28 to 0.08 in the sample period.

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Today: CAPM (Capital Asset Pricing Model)

Nobel Prize in Economics, 1990, Awarded to Harry Markowitz, Merton Miller, and William Sharpe for “their pioneering work in the theory of financial economics”. ⇒ CAPM

A stock’s beta (β), a measure of how much a stock’s returns vary in response to changes in the overall market, is an important determinant of the stock’s expected return: a forecast of the return that an asset will earn over some period of time.

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Expected Returns

• Expected returns reflect our estimate of how an investment will perform in the future. Think about the expected returns of stocks vs. bonds.

• Expected returns are not observable. This means we need to develop methods to estimate expected returns.

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Expected Returns

Decisions must be based on expected returns

Methods used to estimate expected return

Historical approach

Probabilistic approach

Risk-based approach8

Historical Approach for Estimating Expected Returns

Assumption: the distribution of expected returns will be similar to historical distribution of returns.

Using 1900-2003 annual returns, the average risk premium for U.S. stocks relative to Treasury bills is 7.6%. Treasury bills currently offer a 2% yield to

maturity

Expected return on U.S. stocks: 7.6% + 2% = 9.6%

Can historical approach be used to estimate the expected return of an individual stock?

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Historical Approach for Estimating Expected ReturnsAssume General Motors long-run average return is

17.0%. Treasury bills average return over same period was 4.1%

GM historical risk premium: 17.0% - 4.1% = 12.9%

GM expected return = Current T-bill rate + GM historical risk premium = 2% + 12.9% = 14.9%

Limitations of historical approach for

individual stocks

May reflect GM’s past more than its future

Many stocks do not have a long history to forecast expected return

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Probabilistic Approach for Estimating Expected ReturnsIdentify all possible outcomes of returns and assign

a probability to each possible outcome:

GM Expected Return = 0.20(-30%) + 0.70(15%) +0.10(55%) = 10%

For example, assign probabilities for possible states of economy: boom, expansion, recession and project

the returns of GM stock for the three states

55%10%Boom

15%70%Expansion

-30%20%Recession

GM ReturnProbabilityOutcome

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Risk-Based Approach for Estimating Expected Returns1. Measure the risk of the asset 2. Use the risk measure to estimate the expected

return

How can we capture the systematic risk component of a stock’s volatility?

1. Measure the risk of the asset

• Systematic risks simultaneously affect many different assets

• Investors can diversify away the unsystematic risk• Market rewards only the systematic risk: only

systematic risk should be related to the expected return

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• Collect data on a stock’s returns and returns on a market index

• Plot the points on a scatter plot graph– Y–axis measures stock’s return– X-axis measures market’s return

• Plot a line (using linear regression) through the points

Risk-Based Approach for Estimating Expected Returns

Slope of line equals beta, the sensitivity of a stock’s returns relative to changes in overall market returns

Beta is a measure of systematic risk for a particular security.

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-30%

-20%

-10%

0%

10%

20%

30%

-30% -20% -10% 0% 10% 20% 30%

Slope = Beta = 1.44

Scatter Plot for Returns on Sharper Image and S&P 500

S&P 500 weekly returns

Shar

per

Imag

e w

eek l

y re

turn

s

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-15%

-10%

-5%

0%

5%

10%

15%

-15% -10% -5% 0% 5% 10% 15%

Beta = 0.11

Scatter Plot for Returns on ConAgra and S&P 500

S&P 500 weekly returns

ConA

gra

wee

kly

retu

rns

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Beta

Beta is a standardized measure of the systematic risk of an individual asset

Sharper Image – β = 1.44 ⇒ On average, if the market’s return moves by 1%, the return for Sharper Image moves in the same direction by 1.44% ⇒ Very sensitive to overall market movements.

ConAgra – b = 0.11 ⇒ On average, if the market’s return moves by 1%, the return for ConAgra moves in the same direction by 0.11% ⇒ Less sensitive to market movements.

Which stock has the highest systematic risk? 16

Risk-Based Approach for Estimating Expected Returns

Beta measures systematic risk and links the risk and expected return of an asset.

2. Use the risk measure to estimate the expected return:

• Plot beta against expected return for two assets:- A risk-free asset that pays 4% with certainty,

with zero systematic risk and- An “average stock”, with beta equal to 1, with

an expected return of 10%.• Draw a straight line connecting the two points.• Investors holding a stock with beta of 0.5 or 1.5,

for example, can find the expected return on the line.

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Risk and Expected ReturnsExpected returns

•

•10%

1

Risk-free asset

• • • •0.2 0.4 0.6 0.8 21.2 1.4 1.6 1.8

• • • • •

Beta

•4%

•18%

•14%

“average”stock

What is the expected return for stock with beta = 1.5?

ß = 1.5•

•

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Portfolio Expected Returns

The portfolio expected return equals the weighted average of the portfolio assets’ expected returns

E(Rp) = w1E(R1)+ w2E(R2)+…+wnE(Rn)

• w1, w2 , … , wn : portfolio weights• E(R1), E(R2), …, E(RN): expected returns of

securities

Expected return of a portfolio with N securities

How does the expected return of a portfolio relate to the expected returns of the securities in the

portfolio?

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Portfolio Expected Returns

$10,000

$2,500

$5,000

$2,500

$ Invested

0.1258%Sears

0.514%Pfizer

0.2512%GE

0.12510%IBM

WeightsE(R)Portfolio

E(Rp) = (0.125)(10%) + (0.25)(12%) + (0.125)(8%) + (0.5)(14%) = 12.25%

E(Rp) = w1E(R1)+ w2E(R2)+…+wnE(Rn)

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Short Selling

• Portfolio weights must sum up to one• Not all weights have to be positive. A negative

weight ⇒ Short Selling

Rocket.com and BricksNMortar Inc. both sell for $10 per share. You expect returns on Rocket.com to be 25%, but returns on BricksNMortar Inc. to be only 5%. You have $1,000 to invest.

Borrow 50 shares of BricksNMortar --- sell immediately. Buy 150 shares of Rocket.com.

%35%)25()5.1(%)5()5.0()( =×+×−=pRE

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Short Selling

$1,350Net Cash Earned($1,350-$1,000)/$1,000 = 0.35 = 35%Rate of Return =

(price = $10.50)-$525Return Borrowed Shares

(price = $12.50)$1875Sell Rocket SharesEnd of Year

(150 shares)$1500Rocket Shares(50 shares)$500Borrowed

$1000Initial InvestmentBeginning of Year

*This is called leveraging 22

Portfolio Risk

Portfolio risk is the weighted average of systematic risk (beta) of the portfolio constituent securities.

$10,000$2,500

$5,000

$2,500

$ Invested

0.1250.67Sears0.51.67Pfizer

0.251.33GE

0.1251.00IBM

WeightsBetaPortfolio

ß P = (0.125)(1.00) + (0.25)(1.33) + (0.125)(0.67) + (0.50)(1.67) = 1.38

But portfolio volatility is not the same as the weighted average of all portfolio security volatilities

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Security Market Line

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0

Beta

E(Rm)Market portfolio

RfRisk-free asset

E(R)Portfolio

Portfolio composed of the following two assets:

• An asset that pays a risk-free return Rf, , and • A market portfolio that contains some of every

risky asset in the market.

Security market line: the line connecting the risk-free asset and the market portfolio

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The Security Market Line

βi

E(RP)

RF

SML

Slope = E(Rm) - RF = Market Risk Premium•

•

•RM

β =1.0

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Security Market Line and CAPM

The two-asset portfolio lies on security market line

Given two points (risk-free asset and market portfolio asset) on the security market line, the equation of the

line:

E(Ri) = Rf + ß [E(Rm) – Rf]

• Return for bearing no market risk

• Portfolio’s exposure to market risk

• Reward for bearing market risk

The equation represents the risk and return relationship predicted by the Capital Asset Pricing

Model (CAPM) 26

The Security Market Line

• In equilibrium, all assets lie on this line.• If individual stock or portfolio lies above the

line:• Expected return is too high.• Investors bid up price until expected return

falls.• If individual stock or portfolio lies below SML:

• Expected return is too low.• Investors sell stock driving down price until

expected return rises.

Plots relationship between expected return and betas

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The Security Market Line

βi

E(RP)

RF

SML

Slope = E(Rm) - RF = Market Risk Premium

•A - Undervalued

•

•

•RM

β =1.0

•B

•A

• B - Overvalued

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CAPM: Examples

1. The risk-free rate is currently 5%, and the expected risk premium on the market portfolio is 7%. What is the expected return on a stock with a beta of 1.2?

2. You believe that a particular stock has an expected return of 15%. The stock’s beta is 1.2, the risk-free rate is 3%, and the expected market risk premium is 6%. Based on this, is the stock overvalued or undervalued?

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Efficient MarketsEfficient market hypothesis (EMH): in an efficient

market, prices rapidly incorporate all relevant information

Financial markets much larger, more competitive, more transparent, more homogeneous than product

markets

Much harder to create value through financial activities

Changes in asset price respond only to new information. This implies that asset prices move

almost randomly.30

Efficient Markets

CAPM gives analyst a model to measure the systematic risk of any asset.

If asset prices unpredictable, then what is the use of CAPM?

On average, assets with high systematic risk should earn higher returns than assets with low systematic

risk.

CAPM offers a way to compare risk and return on investments alternatives.