The Capital Asset Pricing Model CAPM: benchmark model of the cost of capital
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1 Finance The Capital Asset Pricing Model CAPM: benchmark model of the cost of capital Finance Fall 2016 Tepper School of Business Carnegie Mellon University c 2016 Chris Telmer. Some content from slides by Bryan Routledge. Used with permission :45
2 Market Cost of Capital Source: Financial Times, February 27, 2015 CAPM 2
3 Market Cost of Capital Source: Financial Times, February 27, 2015 CAPM 3
4 Question/Setup Reminders: Where did the 8.4% cost of capital come from? :: It is the expected return that investors require, given the risk inherent in the cash flows :: It is the discount rate that we use to arrive at PV :: We have used a shortcut: it reflects cash flow risk that is similar to the overall stock market risk? :: We have used less of a shortcut: it reflects the fraction of bonds and stocks that replicate the cash flow CAPM 4
5 Question/Setup Reminders: Where did the 8.4% cost of capital come from? :: It is the expected return that investors require, given the risk inherent in the cash flows :: It is the discount rate that we use to arrive at PV :: We have used a shortcut: it reflects cash flow risk that is similar to the overall stock market risk? :: We have used less of a shortcut: it reflects the fraction of bonds and stocks that replicate the cash flow A puzzle :: Empirical fact: stock market cost of capital (now) is roughly 8% :: So 8.4% for the retailers must mean similar risk... right? Have a look at the stock returns (spreadsheet). Wow! :: Solution: only systematic risk matters CAPM 4
6 What is Risk? CAPM 5
7 What IS Risk? Alternatives :: $110, one year from now costs $100 today. :: Risk-free rate is CAPM 6
8 What IS Risk? Alternatives :: $110, one year from now costs $100 today. :: Risk-free rate is 10%. :: Expected return is (trivially) 10%. CAPM 6
9 What IS Risk? Alternatives :: $110, one year from now costs $100 today. :: Risk-free rate is 10%. :: Expected return is (trivially) 10%. :: A lottery: (a gamble, a risky asset ): $120 w.p. 1/2, $100 w.p. 1/2. :: If cost is $100, expected return is CAPM 6
10 What IS Risk? Alternatives :: $110, one year from now costs $100 today. :: Risk-free rate is 10%. :: Expected return is (trivially) 10%. :: A lottery: (a gamble, a risky asset ): $120 w.p. 1/2, $100 w.p. 1/2. :: If cost is $100, expected return is 10% CAPM 6
11 What IS Risk? Alternatives :: $110, one year from now costs $100 today. :: Risk-free rate is 10%. :: Expected return is (trivially) 10%. :: A lottery: (a gamble, a risky asset ): $120 w.p. 1/2, $100 w.p. 1/2. :: If cost is $100, expected return is 10% :: We call this risk neutrality. CAPM 6
12 What IS Risk? Alternatives :: $110, one year from now costs $100 today. :: Risk-free rate is 10%. :: Expected return is (trivially) 10%. :: A lottery: (a gamble, a risky asset ): $120 w.p. 1/2, $100 w.p. 1/2. :: If cost is $100, expected return is 10% :: We call this risk neutrality. :: If cost is $95, expected return is CAPM 6
13 What IS Risk? Alternatives :: $110, one year from now costs $100 today. :: Risk-free rate is 10%. :: Expected return is (trivially) 10%. :: A lottery: (a gamble, a risky asset ): $120 w.p. 1/2, $100 w.p. 1/2. :: If cost is $100, expected return is 10% :: We call this risk neutrality. :: If cost is $95, expected return is 15.7%. Excess expected return the risk premium is CAPM 6
14 What IS Risk? Alternatives :: $110, one year from now costs $100 today. :: Risk-free rate is 10%. :: Expected return is (trivially) 10%. :: A lottery: (a gamble, a risky asset ): $120 w.p. 1/2, $100 w.p. 1/2. :: If cost is $100, expected return is 10% :: We call this risk neutrality. :: If cost is $95, expected return is 15.7%. Excess expected return the risk premium is 5.7%. CAPM 6
15 What IS Risk? Alternatives :: $110, one year from now costs $100 today. :: Risk-free rate is 10%. :: Expected return is (trivially) 10%. :: A lottery: (a gamble, a risky asset ): $120 w.p. 1/2, $100 w.p. 1/2. :: If cost is $100, expected return is 10% :: We call this risk neutrality. :: If cost is $95, expected return is 15.7%. Excess expected return the risk premium is 5.7%. :: We call this risk aversion. CAPM 6
16 What IS Risk? Alternatives :: $110, one year from now costs $100 today. :: Risk-free rate is 10%. :: Expected return is (trivially) 10%. :: A lottery: (a gamble, a risky asset ): $120 w.p. 1/2, $100 w.p. 1/2. :: If cost is $100, expected return is 10% :: We call this risk neutrality. :: If cost is $95, expected return is 15.7%. Excess expected return the risk premium is 5.7%. :: We call this risk aversion. :: What is risk? A positive risk premium. An expected return that is greater than the risk free interest rate. CAPM 6
17 What IS Risk? Alternatives :: $110, one year from now costs $100 today. :: Risk-free rate is 10%. :: Expected return is (trivially) 10%. :: A lottery: (a gamble, a risky asset ): $120 w.p. 1/2, $100 w.p. 1/2. :: If cost is $100, expected return is 10% :: We call this risk neutrality. :: If cost is $95, expected return is 15.7%. Excess expected return the risk premium is 5.7%. :: We call this risk aversion. :: What is risk? A positive risk premium. An expected return that is greater than the risk free interest rate. :: Point: the risk in an asset is reflected in its expected return CAPM 6
18 Risk Risk is manifest in the expected excess return: Risk Premium = E ( r it r ft ) CAPM 7
19 Foundations of the CAPM CAPM 8
20 Remember A firm s cost of capital *is* the expected return on its stock. 1 1 No leverage assumed here. Stay tuned. CAPM 9
21 Basics Given: We can increase the diversification benefit by adding more risky investments :1: Investors prefer to hold well-diversified portfolios Volatility of an equally weighted portfolio vs. the number of stocks :2: There is a limit to diversification: std=40%, correlation=0.28 Point 2. says that it makes sense to write the following CAPM 10
22 Slight Reformulation :: Return on the market portfolio is r mt. This is the market-cap weighted portfolio of all the assets. It is also a well diversified portfolio: no idiosyncratic variation, only common variation. It is the common variation. CAPM 11
23 Slight Reformulation :: Return on the market portfolio is r mt. This is the market-cap weighted portfolio of all the assets. It is also a well diversified portfolio: no idiosyncratic variation, only common variation. It is the common variation. :: Slight reformulation: r it r ft = α i + β i ( rmt r ft ) + εit CAPM 11
24 Slight Reformulation :: Return on the market portfolio is r mt. This is the market-cap weighted portfolio of all the assets. It is also a well diversified portfolio: no idiosyncratic variation, only common variation. It is the common variation. :: Slight reformulation: r it r ft = α i + β i ( rmt r ft ) + εit :: Take expected value E ( r it r ft ) = αi + β i E ( r mt r ft ) CAPM 11
25 Slight Reformulation :: Return on the market portfolio is r mt. This is the market-cap weighted portfolio of all the assets. It is also a well diversified portfolio: no idiosyncratic variation, only common variation. It is the common variation. :: Slight reformulation: r it r ft = α i + β i ( rmt r ft ) + εit :: Take expected value E ( r it r ft ) = αi + β i E ( r mt r ft ) :: CAPM follows from α i = 0 E ( r it ) = rft + β i E ( r mt r ft ) CAPM 11
26 Graphically CAPM 12
27 Capital Market Line Source: Burton Hollifield 35 CAPM 13
28 CAPM E ( r it ) = rft + β i E ( r mt r ft ) :: Beta is all that matters for one firm s cost of capital relative to another s. CAPM 14
29 CAPM E ( r it ) = rft + β i E ( r mt r ft ) :: Beta is all that matters for one firm s cost of capital relative to another s. :: More sophisticated versions of this? Multi-factor models (more than one beta) CAPM 14
30 Beta is the OLS Regression Coefficient Mathematically it must be the case that β i = Cov( ) r i, r m Var ( ) r m = Cov( r i, r m ) σ 2 m = Cov( r i, r m ) σ m σ i σ i σ i σ i σ m = Corr ( r i, r m ) σ i σ m = ϕ i,m σ i σ m CAPM 15
31 CAPM Quantity and price of risk in asset i E ( r i r f ) = βi E ( r M r f ) Security Market Line (SML) CAPM 16
32 Applying the CAPM CAPM 17
33
34 Value Weighting Works for Betas Recall, expected returns for portfolios: µ p = γ 1µ 1 + γ 2µ 2 CAPM 18
35 Value Weighting Works for Betas Recall, expected returns for portfolios: µ p = γ 1µ 1 + γ 2µ 2 Sub-in CAPM, recall that γ 1 + γ 2 = 1, and then simplify: r f + β p E ( ) [ r m r f = γ 1 r f + β 1 E ( ) ] [ r m r f + γ 2 r f + β 2 E ( ) ] r m r f β p E ( ) r m r f = [γ 1β 1 + γ 2β 2 ]E ( ) r m r f CAPM 18
36 Value Weighting Works for Betas Recall, expected returns for portfolios: µ p = γ 1µ 1 + γ 2µ 2 Sub-in CAPM, recall that γ 1 + γ 2 = 1, and then simplify: r f + β p E ( ) [ r m r f = γ 1 r f + β 1 E ( ) ] [ r m r f + γ 2 r f + β 2 E ( ) ] r m r f β p E ( ) r m r f = [γ 1β 1 + γ 2β 2 ]E ( ) r m r f Cancel the market risk premium: β p = γ 1 β 1 + γ 2 β 2 CAPM 18
37 Value Weighting Works for Betas Recall, expected returns for portfolios: µ p = γ 1µ 1 + γ 2µ 2 Sub-in CAPM, recall that γ 1 + γ 2 = 1, and then simplify: r f + β p E ( ) [ r m r f = γ 1 r f + β 1 E ( ) ] [ r m r f + γ 2 r f + β 2 E ( ) ] r m r f β p E ( ) r m r f = [γ 1β 1 + γ 2β 2 ]E ( ) r m r f Cancel the market risk premium: β p = γ 1 β 1 + γ 2 β 2 Portfolio betas are value-weighted averages of the betas of the things in the portfolio CAPM 18
38 Value-Weighting r p = N γ i r i i=1 µ p = N γ i µ i i=1 β p = N γ i β i i=1 CAPM 19
39 Expected Returns on the Retailers The date is January You have 1,200 shares of AEO and 2,300 shares of GPS. What is the expected return over the next year on each of the stocks? Use the betas that appear below. Calculate expected returns based on pair of betas. What is the annual expected return on your portfolio? How much money do you expect your portfolio to be worth in one year? What is your portfolio beta? What is the expected return on your portfolio, based on its beta? CAPM 20
40 Exercise Spreadsheet-based exercise CAPM 21
41 A Capital Budgeting Problem American Eagle is considering a new marketing campaign. :: Cost = 400 :: Payoffs = 220 in one year and 225 in two years :: Compute NPV Based on sample betas Based on industry betas CAPM 22
42 Exercise Spreadsheet-based exercise CAPM 23
43 American Eagle Diversifies Suppose that American Eagle buys 20% of First Energy s (FE) regulated utility business. What is the new firm s cost of capital? Utility beta = 0.3, retailer beta = FE EV = $36b, so a 20% piece is worth $7.2b. AEO EV = $3.1b. AEO is now 30% retailer, 70% investor-owned utility. CAPM 24
44 Where the Numbers Came From Open spreadsheet capm.xlsx What to watch for: :: Calculation and summary statistics for total returns :: Portfolio analysis How well did your portfolio from above workout over the 13.5 year time period? :: Calculation of betas and implied expected returns CAPM 25
45 Further Applications CAPM 26
46 CAPM: Estimating β on Industry Portfolios CAPM: Estimating β on Industry Portfolios Industry: Gold Beta=0.44 Daily Return Market all NYSE, AMEX, and NASDAQ [FF] FindIndustryBeta.do Daily data: 1980 : 2012 Gold Industry Portfolio Excess Return fit Expected Returns 27 CAPM 27
47 CAPM: Estimating β on Industry Portfolios CAPM: Estimating β on Industry Portfolios Industry: Guns Beta=0.70 Daily Return Market all NYSE, AMEX, and NASDAQ [FF] FindIndustryBeta.do Daily data: 1980 : 2012 Guns Industry Portfolio Excess Return fit Expected Returns 27 CAPM 28
48 CAPM: Estimating β on Industry Portfolios CAPM: Estimating β on Industry Portfolios Industry: Rtail Beta=0.95 Daily Return Market all NYSE, AMEX, and NASDAQ [FF] FindIndustryBeta.do Daily data: 1980 : 2012 Rtail Industry Portfolio Excess Return fit Expected Returns 27 CAPM 29
49 CAPM: Estimating β on Industry Portfolios CAPM: Estimating β on Industry Portfolios Industry: Aero Beta=0.99 Daily Return Market all NYSE, AMEX, and NASDAQ [FF] FindIndustryBeta.do Daily data: 1980 : 2012 Aero Industry Portfolio Excess Return fit Expected Returns 27 CAPM 30
50 CAPM: Estimating β on Industry Portfolios CAPM: Estimating β on Industry Portfolios Industry: Fin Beta=1.28 Daily Return Market all NYSE, AMEX, and NASDAQ [FF] FindIndustryBeta.do Daily data: 1980 : 2012 Fin Industry Portfolio Excess Return fit Expected Returns 27 CAPM 31
51 Industry Betas Aswath Damodaran s Website CAPM 32
52 Time Varying Risk Free Rate :: It s important to use today s risk-free rate :: Cost of capital must change with level of interest rates See spreadsheet CAPM 33
53 Empirical Performance of CAPM :: We ll leave this for more advanced classes :: Good discussion in text book. :: CAPM is foundational. More sophisticated models are important in practice, but knowing/understanding/using CAPM is a pre-requisite. :: Optional exercise with Fama-French factors: (basic, with simulation CAPM 34
54 Takeaways CAPM 35
55 Takeaways :: CAPM: cost of capital depends only on beta: E ( r i ) = rf + β i E ( r m r f ) :: Portfolio Calculations: Portfolio expected returns and betas are value-weighted averages of the expected returns and betas inside the portfolio :: Coming in Finance II: Portfolio choice More rigorous treatment of CAPM More state-of-the-art versions of CAPM (e.g., Fama-French 3-factor model) CAPM 36
56 Opportunity Cost of Capital Where Did the 8.4% Come From? Expected Returns 28 CAPM 37
57 Implications The return on Microsoft has :: A higher standard deviation that the return on the S&P500 :: A beta close to 1.0 Is Microsoft riskier than the market? CAPM 38
58 Implications Security Market Line Consider assets A, B and C: :: Which asset would you hold? :: Would you hold any of asset C? :: What are the betas: β A, β B, β C? CAPM 39
59 Jargon Capital Market Line (µ against σ) Security Market Line (µ against β) CAPM 40
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