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Risk, Return, and CAPM. Professor Dr. Rainer Stachuletz Corporate Finance Berlin School of Economics and Law. Expected Returns. Methods used to estimate expected return. Decisions must be based on expected returns. Historical approach. Probabilistic approach. Risk-based approach. - PowerPoint PPT Presentation
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Berlin, 04.01.2006 Fußzeile 1
Professor Dr. Rainer StachuletzCorporate Finance
Berlin School of Economics and Law
Risk, Return, and
CAPM
Berlin, 04.01.2006 Fußzeile 2
Expected Returns
Decisions must be based on expected returns
Methods used to estimate expected return
Historical approach
Probabilistic approach
Risk-based approach
Berlin, 04.01.2006 Fußzeile 3
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?
Berlin, 04.01.2006 Fußzeile 4
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
Berlin, 04.01.2006 Fußzeile 5
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
Berlin, 04.01.2006 Fußzeile 6
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
Berlin, 04.01.2006 Fußzeile 7
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.
Berlin, 04.01.2006 Fußzeile 8
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
Berlin, 04.01.2006 Fußzeile 9
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
Berlin, 04.01.2006 Fußzeile 10
Rf
Capital
Market Line
Risk
Average
Return
Risk-Based Approach for Estimating Expected Returns
rM
M
Slope CML:
2M
fM rr
Individual Stock A:
M,Acov
CAPM
Berlin, 04.01.2006 Fußzeile 11
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
Berlin, 04.01.2006 Fußzeile 12
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.
Berlin, 04.01.2006 Fußzeile 13
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
The Security Market Line
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%
Security Market Line
BP expected return = 7.52%
BP actual return = 2.56%
17
Arbitrage Drivers and the Linearity of the Security Market Line
Berlin, 04.01.2006 Fußzeile 18
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?
Berlin, 04.01.2006 Fußzeile 19
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)
Berlin, 04.01.2006 Fußzeile 20
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
Berlin, 04.01.2006 Fußzeile 21
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
Berlin, 04.01.2006 Fußzeile 22
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
Berlin, 04.01.2006 Fußzeile 23
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.
Berlin, 04.01.2006 Fußzeile 24
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.
Berlin, 04.01.2006 Fußzeile 25
• 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
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