risk, return, and capm

Berlin, 04.01.2006 Fußzeile 1

Professor Dr. Rainer StachuletzCorporate Finance

Berlin School of Economics and Law

Risk, Return, and

CAPM


Expected Returns

Decisions must be based on expected returns

Methods used to estimate expected return

Historical approach

Probabilistic approach

Risk-based approach


Historical Approach for Estimating Expected Returns

Assume 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?


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 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


Probabilistic Approach for Estimating Expected Returns

Identify 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


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


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.


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


Scatter Plot for Returns ConAgra and S&P 500

-15%

-10%

-5%

0%

5%

10%

15%

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

Beta = 0.11

S&P 500 weekly returns

ConA

gra

weekl

y r

etu

rns


Rf

Capital

Market Line

Risk

Average

Return


rM

M

Slope CML:

2M

fM rr

Individual Stock A:

M,Acov

CAPM


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



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.


Risk and Expected ReturnsSecurity Market Line

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

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

ß = 1.5•

•

14

Estimating the Risk Free Rate

UK INTEREST RATES

425/32-423/3243/4 – 411/16Treasury Bills

One

Year

Six

months

Three

months

One

month

7 days

notice

Over-

Night

Feb 17

425/32-423/3243/4 – 411/16Treasury Bills

One

Year

Six

months

Three

months

One

month

7 days

notice

Over-

Night

Feb 17

Two prices are quoted, one for selling one for buying. Take the middle value

Two prices are quoted, one for selling one for buying. Take the middle value

Extract of UK interest rate data from the Financial Times (17 February, 2005)

15

The Steps Towards the Estimation of Beta Using Ordinary Least Squares Regression

16


Return

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 0.5 1 1.5 2 2.5 3 3.5Beta

M arket return = 8.25%

BA expected return = 11.825%

BA actual return = 13.5%

Security Market Line

BP expected return = 7.52%

BP actual return = 2.56%

Return

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 0.5 1 1.5 2 2.5 3 3.5Beta

M arket return = 8.25%

BA expected return = 11.825%

BA actual return = 13.5%


BP expected return = 7.52%

BP actual return = 2.56%

17

Arbitrage Drivers and the Linearity of the Security Market Line


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?


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) = w1 E (R1)+ w2 E (R2)+…+wn E (Rn)


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



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



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


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.


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

risk, return, and capm

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