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Unformatted text preview: The Capital Asset Pricing Model
(Chapter 12 of the textbook) 1 Mean Variance Optimization with Mean Variance Optimization with Riskless Borrowing and Lending
return
CM L
100% stocks Optimal Risky Portfolio rf
100% bonds σ All investors have the same CML because they all have the same optimal risky portfolio given the riskfree rate. 2 The Separation Property The Separation Property
return
CM L
100% stocks Optimal Risky Porfolio rf
100% bonds σ The separation property implies that portfolio choice can be separated into two tasks: (1) determine the optimal risky portfolio, and (2) selecting a point on the CML. 3 return Optimal Risky Portfolio with a Risk Optimal Risky Portfolio with a Risk Free Asset CM 0 L
ML 1 C
100% stocks Second Optimal Risky Portfolio r r 1 f 0 f First Optimal Risky Portfolio 100% bonds σ
By the way, the optimal risky portfolio depends on the riskfree rate as well as the risky assets.
4 Consider a portfolio of N stocks: Each stock has variance σ2. The covariance between any pair of stocks is β. An investor holds equal amounts in each stock. What is the portfolio variance? What happens as N tends to infinity? (i.e. a well diversified portfolio). Moral: When a security is held in a welldiversified portfolio, the covariance of its returns with the rest of the portfolio is a more appropriate measure of the risk of the security that its variance.
5 Risk and Covariance Risk and Covariance Definition of Risk When Investors Definition of Risk When Investors Hold the Market Portfolio Researchers have shown that the best measure of the risk of a security in a large portfolio is the beta (β )of the security. Beta measures the responsiveness of a security to movements in the market portfolio. βi = Cov( Ri , RM ) σ ( RM )
2
6 Estimating β with regression Estimating with regression
Security Returns ine L ic t r is cte a ar Ch Slope Slope = βi Return on Return market % market Ri = α i + β i Rm + e i
7 Stock Research in Motion Nortel Networks Bank of Nova Scotia Bombardier Investors Group. Maple Leaf Foods Roger Communications Canadian Utilities TransCanada Power Estimates of β for Selected Estimates of Stocks
Beta 3.04 3.61 0.28 1.48 0.36 0.25 1.17 0.08 0.08 8 The Formula for Beta The Formula for Beta
βi =
Cov( Ri , RM ) σ ( RM )
2 Clearly, your estimate of beta will depend upon your choice of a proxy for the market portfolio. 9 11.9 Relationship between Risk and 11.9 Relationship between Risk and Expected Return (CAPM) Expected Return on the Market:
R M = RF + Market Risk Premium
• Expected return on an individual security: R i = RF + β i × ( R M − RF )
Market Risk Premium This applies to individual securities held within welldiversified portfolios. 10 Expected Return on an Individual Expected Return on an Individual Security This formula is called the Capital Asset Pricing Model (CAPM)
Expected return on a security R i = RF + β i × ( R M − RF )
= RiskBeta of the + × free rate security Market risk premium • Assume β i = 0, then the expected return is RF. • Assume β i = 1, then Ri = R M
11 Relationship Between Risk & Relationship Between Risk & Expected Return
Expected return R i = RF + β i × ( R M − RF ) RM RF
1.0 β R i = RF + β i × ( R M − RF )
12 13.5% 3%
β i = 1. 5 RF = 3%
1.5 β Expected return Relationship Between Risk & Relationship Between Risk & Expected Return R M = 10%
13 R i = 3% + 1.5 × (10% − 3%) = 13.5% Arbitrage Pricing Theory
Arbitrage arises if an investor can construct a zero investment portfolio with a sure profit. Since no investment is required, an investor can create large positions to secure large levels of profit. In efficient markets, profitable arbitrage opportunities will quickly disappear.
14 12.1 Factor Models: Announcements, 12.1 Factor Models: Announcements, Surprises, and Expected Returns The return on any security consists of two parts. 1) the expected or normal return: the return that shareholders in the market predict or expect 2) the unexpected or risky return: the portion that comes from information that will be revealed . Examples of relevant information: Statistics Canada figures (e.g., GNP) 15 A way to write the return on a stock in the coming month is:
R = R +U where R is the expected part of the return U is the unexpected part of the return 12.1 Factor Models: 12.1 Factor Models: Announcements, Surprises, and Expected Returns 16 12.1 Factor Models: Announcements, 12.1 Factor Models: Announcements, Surprises, and Expected Returns Any announcement can be broken down into two parts, the anticipated or expected part and the surprise or innovation: Announcement = Expected part + Surprise. The expected part of any announcement is part of the information the market uses to form the expectation, R of the return on the stock. 17 A systematic risk is any risk that affects a large number of assets, each to a greater or lesser degree. An unsystematic risk is a risk that specifically affects a single asset or small group of assets. Unsystematic risk can be diversified away. Examples of systematic risk include uncertainty about general economic conditions, such as GNP, interest rates, or inflation. On the other hand, announcements specific to a company, such as a gold mining company striking gold, are examples of unsystematic risk.
18 12.2 Risk: Systematic and 12.2 Risk: Systematic and Unsystematic We can break down the risk, U, of holding a stock into two components: systematic risk and unsystematic risk: σ Total risk; U 12.2 Risk: Systematic and 12.2 Risk: Systematic and Unsystematic
R = R +U becomes ε
Nonsystematic Risk; ε Systematic Risk; m R = R+m+ε where m is the systematic risk ε is the unsystematic risk
n
19 Systematic risk is referred to as market risk. m influences all assets in the market to some extent. ε is specific to the company and unrelated to the specific risk of most Corr ( ε ε ) = 0 other companies.
i, j 12.2 Risk: Systematic and 12.2 Risk: Systematic and Unsystematic 20 12.3 Systematic Risk and Betas 12.3 Systematic Risk and Betas The beta coefficient, β , tells us the response of the stock’s return to a systematic risk. In the CAPM, β measured the responsiveness of a security’s return to a specific risk factor, the return on the market portfolio. βi = Cov( Ri , RM ) σ ( RM )
2
21 • We shall now consider many types of systematic risk. 12.3 Systematic Risk and Betas 12.3 Systematic Risk and Betas For example, suppose we have identified three systematic risks on which we want to focus: 1. Inflation 2. GDP growth 3. The dollarpound spot exchange rate, S($, £) Our model is: + ε R = R+m
R = R + β I FI + βGDP FGDP + βS FS + ε βI is the inflation beta βGDP is the GDP beta βS is the spot exchange rate beta ε is the unsystematic risk 22 Systematic Risk and Betas: Systematic Risk and Betas: Example
R = R + βI FI + βGDP FGDP + βS FS + ε Suppose we have made the following estimates:
1. 2. 3. β I = 2.30 β GDP = 1.50 β S = 0.50. ε = 1% Finally, the firm was able to attract a “superstar” CEO and this unanticipated development contributes 1% to the return. = R − 2.30 × FI + 1.50 × FGDP + 0.50 × FS + 1% R
23 Systematic Risk and Betas: Example Systematic Risk and Betas: Example We must decide what surprises took place in the systematic factors. If it was the case that the inflation rate was expected 3%, but in fact was 8% during the time period, then FI = Surprise in the inflation = actual – expected = 8% 3% = 5%
24 Systematic Risk and Betas: Systematic Risk and Betas: Example
R = R − 2.30 × 5% + 1.50 × FGDP + 0.50 × FS + 1% If it was the case that the rate of GDP growth was expected to be 4%, but in fact was 1%, then FGDP = Surprise in the rate of GDP growth = actual – expected = 1% 4% = 3% R = R − 2.30 × 5% + 1.50 × (−3%) + 0.50 × FS + 1%
25 R = R − 2.30 × 5% + 1.50 × (−3%) + 0.50 × FS + 1% Systematic Risk and Betas: Systematic Risk and Betas: Example If it was the case that dollarpound spot exchange rate, S($,£), was expected to increase by 10%, but in fact remained stable during the time period, then FS = Surprise in the exchange rate = actual – expected = 0% 10% = 10% R = R − 2.30 × 5% + 1.50 × (−3%) + 0.50 × (−10%) + 1% 26 Systematic Risk and Betas: Systematic Risk and Betas: Example
R = R − 2.30 × 5% + 1.50 × (−3%) + 0.50 × FS + 1% Finally, if it was the case that the expected return on the stock was 8%, then
R = 8% R = 8% − 2.30 × 5% + 1.50 × (−3%) + 0.50 × (−10%) + 1% R = −12% 27 Now let us consider what happens to portfolios of stocks when each of the stocks follows a onefactor model. We will create portfolios from a list of N stocks and will capture the systematic risk with a 1factor model. The ith stock in the list have returns:
Ri = R i + βi F + εi
28 12.4 Portfolios and Factor 12.4 Portfolios and Factor Models Relationship Between the Return on Relationship Between the Return on the Common Factor & Excess Return
Excess return Ri − R i = βi F + εi
If we assume that there is no unsystematic risk, then ε i = 0 The return on the factor F εi 29 Relationship Between the Return on the Relationship Between the Return on the Common Factor & Excess Return
Excess return Ri − R i = βi F If we assume that there is no unsystematic risk, then ε i = 0 The return on the factor F 30 Relationship Between the Return on the Relationship Between the Return on the Common Factor & Excess Return
Excess return β A = 1.5 β B = 1.0 βC = 0.50 Different securities will have different betas The return on the factor F 31 Portfolios and Diversification Portfolios and Diversification We know that the portfolio return is the weighted average of the returns on the individual assets in the portfolio: RP = X 1 R1 + X 2 R2 + + X i Ri + + X N RN
Ri = R i + βi F + εi RP = X 1 ( R1 + β1 F + ε1 ) + X 2 ( R 2 + β2 F + ε2 ) + + X N ( R N + βN F + εN ) RP = X 1 R1 + X 1 β1 F + X 1ε1 + X 2 R 2 + X 2 β2 F + X 2 ε2 + + X N R N + X N βN F + X N εN
32 Portfolios and Diversification Portfolios and Diversification
1. The weighed average of expected returns. The return on any portfolio is determined by three sets of parameters:
1. The weighted average of the betas times the factor. 1. The weighted average of the unsystematic risks. RP = X 1 R 1 + X 2 R 2 + + X N R N + ( X 1 β1 + X 2 β2 + + X N β N ) F + X 1ε1 + X 2 ε2 + + X N ε N
In a large portfolio, the third row of this equation disappears as the unsystematic risk is diversified away.
33 Portfolios and Diversification Portfolios and Diversification
So the return on a diversified portfolio is determined by two sets of parameters:
1. 2. The weighed average of expected returns. The weighted average of the betas times the factor F. RP = X 1 R 1 + X 2 R 2 + + X N R N + ( X 1 β1 + X 2 β2 + + X N β N ) F In a large portfolio, the only source of uncertainty is the portfolio’s sensitivity to the factor.
34 12.5 Betas and Expected Returns 12.5 Betas and Expected Returns
RP = X 1 R1 + + X N R N + ( X 1 β1 + + X N β N ) F RP Recall that R P = X 1 R1 + + X N R N
and βP β P = X 1 β1 + + X N β N The return on a diversified portfolio is the sum of the expected return plus the sensitivity of the portfolio to the factor.
RP = R P + β P F
35 Relationship Between β & Relationship Between Expected Return The relevant risk in large and welldiversified portfolios is all systematic, because unsystematic risk is diversified away. If shareholders are ignoring unsystematic risk, only the systematic risk of a stock can be related to its expected return. R =R +β F
P P P
36 Expected return Relationship Between β & Relationship Between Expected Return
SML A D B C β RF R = RF + β ( R P − RF )
37 12.6 The Capital Asset Pricing Model and the 12.6 The Capital Asset Pricing Model and the Arbitrage Pricing Theory APT applies to well diversified portfolios and not necessarily to individual stocks. With APT it is possible for some individual stocks to be mispricednot lie on the SML. APT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolio. APT can be extended to multifactor models.
38 Multifactor APT Multifactor APT
R = R + ( R − R ) β + ( R − R ) β + ... + ( R − R ) β
F 1 F 1 2 F 2 k F k Example: A Canadian study (Otuteye, CIR 1991) with five factors: 1. the rate of growth in industrial production 2. the changes in the slope of the term structure of interest rates 3. the default risk premium for bonds 4. inflation 5. The valueweighted return on the market portfolio (TSE 300)
39 Both the CAPM and APT are riskbased models. There are alternatives. Empirical methods are based less on theory and more on looking for some regularities in the historical record. Be aware that correlation does not imply causality. Related to empirical methods is the practice of classifying portfolios by style e.g., Value portfolio Growth portfolio
40 12.7 Empirical Approaches to 12.7 Empirical Approaches to Asset Pricing 12.8 Summary and Conclusions 12.8 Summary and Conclusions The APT assumes that stock returns are generated according to factor models such as:
R = R + βI FI + βGDP FGDP + βS FS + ε
• As securities are added to the portfolio, the unsystematic risks of the individual securities offset each other. A fully diversified portfolio has no unsystematic risk. • The CAPM can be viewed as a special case of the APT. • Empirical models try to capture the relations between returns and stock attributes that can be measured directly from the data without appeal to theory.
41 Review Questions Review Questions Review Questions #: 11.17, 11.24, 11.27, 11.42, 11.45 Review Questions #: 12.2, 12.5, 12.7 What are the advantages and disadvantages of the CAPM and the APT? What is the difference between a kfactor model and the market model? 42 Assignment Questions Assignment Questions Assignment Questions #: 12.1, 12.3, 12,4, 12.6, 12.8 43 ...
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 Spring '09
 MARYHARDY
 Pricing

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