5
Capital Lecture Asset Pricing Model
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Key Concepts and Skills
Understand variance as a measure of total risks Understand the modern portfolio theory Understand Capital asset pricing model (CAPM) Know how to calculate expected returns based on CAPM
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Lecture outline
Returns of individual assets and risks of individual assets. Returns of portfolios and risks of portfolios Traditional Portfolio Theory Modern Portfolio Theory Betas of individual assets Betas of portfolios Security Market Line Capital Asset Pricing Model
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Returns
The simplest measure of return is the holding period return.
Holding period = return
Ending _ Beginning + Income value value
Beginning value
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Holding Period Return
Example : Buy 100 shares at $25 per share Dividend of $0.10 per share Sell the shares at $30 per share
Time
Holding period return =
$30 - $25 + $0.10 $25
= 20.4%
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Calculating Returns -- Historical
You bought a stock for $35 and you received dividends of $1.25. The stock is now selling for $40.
What is your dollar return? What is your percentage return?
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Calculating Returns -- Prospective
Econ. Prob. 0.10 0.20 0.40 Alta -22.0% -2.0 20.0 Repo 28.0% 14.7 0.0 Am F. 10.0% -10.0 7.0 MP -13.0% 1.0 15.0
Bust
Below avg. Avg.
Above avg.
Boom
0.20
0.10
35.0
50.0
-10.0
-20.0
45.0
30.0
29.0
43.0
1.00
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Expected Rate of Return
^ R = expected rate of return. R = RiPi.
i=1 ^
n
^ RAlta = 0.10(-22%) + 0.20(-2%) + 0.40(20%) + 0.20(35%) + 0.10(50%) = 17.4%.
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Risk
A common measure of risk is variance. The standard deviation is the square root of the variance.
n
Ri Variance
i 1
R
2
n
2000 2001 2002 2003 2004 2005 Average
Stock A 1% 9% 8% 1% 9% 2% 5%
Stock B 6% 4% 6% 7% 3% 4% 5%
VALATILITY
10% 9% 8% 7% 6% 5% 4% 3% 2% 1% 0% 2000 2001 2002 2003 2004 2005
STOCK A
STOCK B
2000 2001 2002 2003 2004 2005 Average Return Variance
Stock A Stock B 1% 6% 9% 4% 8% 6% 1% 7% 9% 3% 2% 4% 5% 5% 0.00164 0.00024
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Prospective version
Variance =
Alta
(Ri R)2 Pi
i=1
n
^
= [(-22 - 17.4)20.10 + (-2 - 17.4)20.20 + (20 - 17.4)20.40 + (35 - 17.4)20.20 + (50 - 17.4)20.10]1/2 = 20.0%.
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Portfolios
A portfolio is a collection of assets The portfolio's return is the weighted average of the returns for each asset in the portfolio
n
RP
w1 R1 w2 R2 ... wn Rn
i 1
wi Ri
Compute the portfolio's risk using the same formulas as for an individual asset
Stock A Stock B P (5:5) P (3:7) P (7:3) 1% 6% 2000 3.5% 4.5% 2.5% 9% 4% 2001 6.5% 5.5% 7.5% 8% 6% 2002 7.0% 6.6% 7.4% 1% 7% 2003 4.0% 5.2% 2.8% 9% 3% 2004 6.0% 4.8% 7.2% 2% 4% 2005 3.0% 3.4% 2.6% Average 5% 5% 5.0% 5.0% 5.0% Variance 0.00164 0.00024 0.00029 0.00011 0.00067
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Average Returns
Investment Large stocks Average Return 12.4%
Small Stocks
Long-term Corporate Bonds Long-term Government Bonds U.S. Treasury Bills Inflation
17.5%
6.2% 5.8% 3.8% 3.1%
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Risk Premiums
The risk premium is the return over and above the risk-free rate Treasury bills are considered to be riskfree Historical Risk Premiums
Large stocks: 12.4 3.8 = 8.6% Small stocks: 17.5 3.8 = 13.7% Long-term corporate bonds: 6.2 3.8 = 2.4% Long-term government bonds: 5.8 3.8 = 2.0%
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Portfolio Selection
Traditional Portfolio Theory: investors assess the risks and returns of individual securities to construct their portfolios.
Modern Portfolio Theory: investors select portfolios based on the portfolios overall riskreturn characteristics
Traditional
Modern
2000 2001 2002 2003 2004 2005 Average
Variance
Stock A Stock B P (5:5) P (3:7) P (7:3) 1% 6% 3.5% 4.5% 2.5% 9% 4% 6.5% 5.5% 7.5% 8% 6% 7.0% 6.6% 7.4% 1% 7% 4.0% 5.2% 2.8% 9% 3% 6.0% 4.8% 7.2% 2% 4% 3.0% 3.4% 2.6% 5% 5% 5.0% 5.0% 5.0%
0.00164 0.00024 0.000290 0.000114 0.000674
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Diversification
An investor can reduce certain risk simply by investing in many anti-correlated securities.
Don't put all your eggs in one basket."
VALATILITY
10% 9%
STOCK A
STOCK B
8%
7% 6%
5%
4% 3% 2% 1% 0% 2000 2001 2002 2003 2004 2005
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Efficient Frontier
An investor should select a portfolio that lies on the efficient frontier.
(1) Number of Stocks in Portfolio 1 2 4 6 8 10 20 30 40 50 100 200 300 400 500 1,000
(2) Average Standard Deviation of Annual Portfolio Returns 49.24 37.36 29.69 26.64 24.98 23.93 21.68 20.87 20.46 20.20 19.69 19.42 19.34 19.29 19.27 19.21
(3) Ratio of Portfolio Standard Deviation to Standard Deviation of a Single Stock 1.00 .76 .60 .54 .51 .49 .44 .42 .42 .41 .40 .39 .39 .39 .39 .39
Average annual Standard deviation (%)
49.2
Diversifiable risk 23.9 19.2 Nondiversifiable risk
1
10
20
30
40
1,000
Number of stocks in portfolio
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Systematic Risk
Variance and standard deviation measure the volatility of returns, and the total risk. Total risk =systematic risk + unsystematic risk Market Risk, Non-diversifiable Risk, or Systematic Risk: factors Risk that affect a large number of assets. Asset-specific Risk, Diversifiable Risk, or Unsystematic Risk : Risk factors that affect a limited number of assets
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Linkage to Capital Asset Pricing Model
Consequently, the total risk for a diversified portfolio is essentially equal to the systematic risk.
The expected return on a risky asset depends only on that asset's systematic risk since unsystematic risk can be diversified away.
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Measuring Systematic Risk, beta
We use the beta coefficient to measure systematic risk.
The beta () is a measure of how the returns of the individual security, i, are expected to vary and covary with that of the market. A beta of 1 implies the asset has the same systematic risk as the overall market A beta < 1 implies the asset has less systematic risk than the overall market A beta > 1 implies the asset has more systematic risk than the overall market
What does beta tell us?
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Total versus Systematic Risk
Consider the following information:
Security C Security K
Standard Deviation 20% 30%
Beta 1.25 0.95
Which security has more total risk? Which security has more systematic risk? Which security should have the higher expected return?
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Portfolio Beta -- Example
Suppose you have $15,000 to invest and you have purchased securities in the following amounts. What are your portfolio weights in each security?
$2000 of DCLK $3000 of KO $4000 of INTC $6000 of KEI
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Consider the previous example and four securities with the following betas
Security DCLK KO INTC KEI Beta 4.03 0.84 1.05 0.59
What is the portfolio beta?
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Relationship: Expected Returns and Betas
Rf=8%, RA=20%, A =1.6 What is the risk free asset beta, and what is the market portfolio beta?
30% 25%
E(RA)
Expected Return
20% 15% 10% Rf 5% 0% 0 0. 5 1 1. 5 Beta 2
A
2. 5
3
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Reward-to-Risk Ratio (Sharpe Ratio):
The reward-to-risk ratio is the slope of the line illustrated in the previous example
Slope = (E(RA) Rf) / ( A 0) Reward-to-risk ratio for previous example = (0.20 0.08) / (1.6 0) = .075
The security market line (SML) is the representation of market equilibrium
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Market Equilibrium
In equilibrium, all assets and portfolios must have the same reward-to-risk ratio and they all must equal the reward-to-risk ratio for the market
E ( RA ) R f
A
E ( RM R f )
M
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Capital Asset Pricing Model
The capital asset pricing model (CAPM) defines the relationship between risk and return E(RA) = Rf + A(E(RM) Rf) E(RM) Rf = market risk premium
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CAPM -- Example
If the risk-free rate is 6.15% and the market risk premium is 9.5%, what is the expected return for each?
Security DCLK KO INTC KEI Beta 4.03 0.84 1.05 0.59
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CAPM -- Example
Consider an asset with a beta of 1.2, a risk-free rate of 5% and a market return of 13%.
What is the reward-to-risk ratio in equilibrium? What is the expected return on the asset?
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Use the historical stock returns to calculate the beta for PQU.
Year 1 2 3 4 5 6 7 8 9 10 Market 25.7% 8.0% -11.0% 15.0% 32.5% 13.7% 40.0% 10.0% -10.8% -13.1% PQU 40.0% -15.0% -15.0% 35.0% 10.0% 30.0% 42.0% -10.0% -25.0% 25.0%
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Calculating Beta for PQU
50% 40% 30% 20% 10% 0% -10% -20% -30% -30% -20% -10% 0% 10%
PQU Return
rPQU = 0.8308 rM + 0.0256 R = 0.3546 20% 30% 40% 50%
2
Market Return
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What is beta for PQU?
The regression line, and hence beta, can be found using a calculator with a regression function or a spreadsheet program. In this example, b = 0.83.
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Many analysts use the S&P 500 to find the market return. Analysts typically use four or five years' of monthly returns to establish the regression line. Some analysts use 52 weeks of weekly returns.
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Finding Betas on the Web
Go to Thomson ONE--Business School Edition using the information on the card that comes with your book. Enter the ticker symbol for a Stock Quote, such as IBM or Dell, then click GO.
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Other Web Sites for Beta
Go to http://finance.yahoo.com Enter the ticker symbol for a Stock Quote, such as IBM or Dell, then click GO. When the quote comes up, select Key Statistics from panel on left.
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Expected Return versus Market Risk: Which investment is best?
Security Alta Market Am. Foam Repo Men
Expected Return (%) 17.4 15.0 13.8 1.7
Risk, b 1.29 1.00 0.68 -0.86
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Use the CAPM to calculate required return.
Assume rRF = 8%; rM = rM = 15%. RPM = (rM - rRF) = 15% - 8% = 7%.
rAlta = 8.0% + (7%)(1.29) = 17%. rM = 8.0% + (7%)(1.00) = 15.0%. rAm. F. = 8.0% + (7%)(0.68) = 12.8%. rRepo = 8.0% + (7%)(-0.86) = 2.0%.
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Expected versus Required Returns
Alta Market Am. F. Repo
Exp. r 17.4 15.0 13.8 1.7
Req. r 17.0 15.0 12.8 2.0
Undervalued Fairly valued Undervalued Overvalued
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ri (%)
rM = 15 rRF = 8 Repo
. . .
T-bills 1
Alta
.
Market
Am. Foam
.
0
-1
2
Risk, bi