Risk, Return, and CAPM

Professor XXXXXCourse Name / Number

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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 approach

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Historical Approach for Estimating Expected ReturnsAssume that 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 Returns

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

S&P 500 weekly returns

Sharp

er

Imag

e w

eekl

y r

etu

rns

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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

weekl

y r

etu

rns

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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 Returns

Expected 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

Portfolio E(R) $ Invested Weights

IBM 10% $2,500 0.125

GE 12% $5,000 0.25

Sears 8% $2,500 0.125

Pfizer 14% $10,000 0.5

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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Portfolio Risk

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

constituent securities.

Portfolio Beta $ Invested Weights

IBM 1.00 $2,500 0.125

GE 1.33 $5,000 0.25

Sears 0.67 $2,500 0.125

Pfizer 1.67 $10,000 0.5

ß P = (0.125)(1.00) + (0.25)(1.33) + (0.125)(0.67) + (0.50)(1.67) = 1.38But portfolio volatility is not the same as the

weighted average of all portfolio security volatilities

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Security Market Line

Portfolio E(R) Beta

Risk-free asset Rf 0

Market portfolio E(Rm) 1

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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Security Market Line and CAPMThe 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)

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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

i

E(RP)

RF

SML

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

•A - Undervalued

•

•

•RM

=1.0

•B

•A

• B - Overvalued

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Efficient Markets

Efficient 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.

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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.

Decisions should be made based on expected returns.

Expected returns can be estimated using historical, probabilistic, or risk approaches.

Portfolio expected return/beta equals weighted average of the expected returns/beta of the

assets in the portfolio.

CAPM predicts that the expected return on a stock depends on the stock’s beta, the risk-

free rate and the market premium.

Risk, Return, and CAPM