Unformatted text preview: 1 CAPITAL MARKET THEORY: The Arbitrage Pricing Theory Ross, Westerfield & Jaffe “ Corporate Finance” 7th ed. Chapter 11 From Theory to Data 2 APT assumes that asset returns are generated by a linear factor model: ~
Ri ai bi1 F1 bi 2 F2 ... bik Fk i And the equilibrium condition is: ~
E ( Ri ) R f 1bi1 2bi 2 ... k bik where 1 j - R f ) is the risk premium on factor j (j 1,..., k) APT does not identify the risk factors; and therefore we cannot obtain data on these. How do we estimate the individual factor loadings, the bik ? From Theory to Data 3 We would like to ensure the following: Returns should be linearly related to the factor loadings There should be no added return for bearing risk One method is to hypothesize some factor loadings For example, one can postulate that bi1 is an asset’s beta (like in CAPM), bi2 is the asset’s dividend yield, bi3 is the size of the firm, P/E ratios and so on. Another method is to hypothesize some factors, F1,F2,…,Fk that explain asset returns of all assets These should be macroeconomics variables e.g. inflation, term structure of interest rates, industrial production, and so on. Portfolios and Factor Models 4 Now let us consider what happens to portfolios of stocks when each of the stocks follows a one‐factor model. We will create portfolios from a list of N stocks and will capture the systematic risk with a 1‐factor model. The ith stock in the list have returns: R i R i i F εi Relationship Between the Return on the Common Factor & Excess Return 5 Excess return . 1 5 βB 1.0
C 0. Different securities will have different betas The return on the factor F Relationship Between the Return on the Common Factor & Excess Return 6 Excess return i Ri R i βi F εi
If we assume that there is no unsystematic risk, then i = 0 The return on the factor F Portfolios and Diversification 7 We know that the portfolio return is the weighted average of the returns on the individual assets in the portfolio, where Xi are weights: RP X 1 R1 X 2 R2 X i Ri X N RN Ri R i iF εi RP X 1 ( R1 1 F ε1 ) X 2 ( R 2 2 F ε2 ) X N ( R N N F εN ) RP X 1 R1 X 1 F X 1 ε1 X 2 R 2 X 2 F X 2 ε2 1
2 X N R N X N N F X N εN Portfolios and Diversification 8 The return on any portfolio is determined by three sets of parameters: The weighed average of expected returns. 2. The weighted average of the betas times the factor. 3. 1. The weighted average of the unsystematic risks. RP X 1 R1 X 2 R 2 X N R N ( X 1 β X 2 β X N β ) F 1
N X 1 ε X 2 ε X N ε 1
In a large portfolio, the third row of this equation disappears as the unsystematic risk is diversified away. Portfolios and Diversification 9 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 R1 X 2 R 2 X N R N ( X 1 1 X 2 2 X N N ) F In a large portfolio, the only source of uncertainty is the portfolio’s sensitivity to the factor. Betas and Expected Returns 10 RP X 1 R1 X N R N ( X 1 X N ) F 1
N R P P Recall that and P X 1 1 X N N R P X 1 R1 X N R 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 P F Investors are rewarded for exposure to macroeconomic risks, not firm specific risks Relationship Between & Expected Return Expected return 11 RF SML D A B C Suppose F = market risk premium R RF R M RF ) (P 12 The Capital Asset Pricing Model and the Arbitrage Pricing Theory 1. Like CAPM, bi’s represent the ‘systematic’ component of each security’s risk. Macroeconomic risks cannot be diversified away but affect different assets differently 2. 3.
4. Unlike CAPM, APT allows a number of systematic risk factors, not only the risk due to the market. APT does not require we all hold the market portfolio APT is based on the ‘no‐arbitrage’ argument, CAPM is based on equilibrium arguments. The Capital Asset Pricing Model and the Arbitrage Pricing Theory 13 5. 6. 7. APT applies to well diversified portfolios and not necessarily to individual stocks. With APT it is possible for some individual stocks to be mispriced ‐ not lie on the SML. APT can be extended to multifactor models. 14 Multifactor APT: Which factors to use? Chen, Roll and Ross (1986): found the following six macroeconomic variables to have high explanatory power changes in expected inflation, unanticipated inflation, industrial production; unanticipated excess of commercial bonds over government bonds (long term) and unanticipated excess of long term government bonds over treasury bills. Fama and French’s (1993) three factor model: Fama and French started with the observation that two classes of stocks have tended to do better than the market as a whole: (i) small caps and (ii) stocks with a high book‐to‐market ratio (customarily called value stocks). Therefore they propose the following model: rit = ai + βMRMt + βSMBSMBt + βHMLHMLt + eit 15 Multifactor APT: How many factors to use? Roll and Ross (1980) 1260 NYSE listed companies Daily returns, 1962‐1972 Shares were grouped into groups of 30 in the factor analysis Results: In over 75% of the groups there was a 50% probability that 5 factors were sufficient In over 40% of the groups, there was less than 10% probability that a sixth factor had explanatory power Summary 16 The APT assumes that stock returns are generated according to factor models such as: R R β F β F β F ε II
SS 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 ofthe 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. 17 RISK, COST OF CAPITAL AND CAPITAL BUDGETING Ross, Westerfield & Jaffe “Corporate Finance” 7th ed. Chapter 12 Outline 18 The Cost of Equity Capital Estimation of Beta Determinants of Beta Extensions of the Basic Model The Costs of Debt and Preferred Stock The Weighted Average Cost of Capital Divisional and Project Costs of Capital Cost of Capital 19 The Cost of Capital depends on the use of funds, not the source! We need to earn at least the required return to compensate our investors for the financing that they provide. The return to an investor is the same as the cost to the company Our cost of capital provides us with an indication of how the market views the risk of our assets The Cost of Equity Capital 20 Firm with excess cash Pay cash dividend Shareholder invests in financial asset A firm with excess cash can either pay a dividend or make a capital investment Invest in project Shareholder’s Terminal Value Cost of Equity 21 The return required by equity holders given the risk of cash flows from the firm. There are two major methods for determining the cost of equity The Dividend growth model The SML or CAPM approach The Dividend Growth Model 22 For a dividend paying stock: PV (Cashflows ) D1 D2 D3 ...
(1 Re ) (1 Re ) 2 (1 Re ) 3 If we assume a constant growth rate: D1 P0 Re Re g D1 g
P0 The Dividend Growth Model 23 Requires P0, D0 and g. g can be estimated using: Analysts estimates of growth rates Average historical growth rate of dividends. g= ROE x retention ratio Dividend Growth Rate Model Example 24 A public utility is currently trading at $60 per share, paid a dividend of $4 last year and it is estimated that the dividend will grow steadily at 6% per year into the indefinite future. According to the DGM, the cost of equity is: D1
g (0.06) 13.07% Re P0
60 The SML Approach 25 Considers risk and degree of uncertainty Recall the SML: RE R f E ( E ( RM ) R f )
where risk‐free rate is denoted by Rf , market risk premium by E(RM) – Rf and is a measure of the systematic risk of asset. The Cost of Equity 26 To estimate a firm’s cost of equity capital, we need to know three things: The risk free rate, RF The market risk premium RM – RF Cov( RE , RM ) E , M The company equity beta E 2
Var ( RM )
M The SML Approach: Example I 27 Suppose your company has an equity beta, βe, of 0.58 and the current risk‐free rate is 6.1%. If the expected market risk premium is 8.6%, what is your cost of equity capital? Using SML : Re = 6.1 + .58(8.6) = 11.1% SML : Example II 28 Suppose the stock of Stansfield Enterprises, a publisher of PowerPoint presentations, has a beta of 2.5. The firm is 100‐percent equity financed. Assume a risk‐free rate of 5‐percent and a market risk premium of 10‐percent. What is the appropriate discount rate for an expansion of this firm? SML: Example III 29 Suppose company A has a beta of 1.2. The market risk premium is 8% and the risk free rate is 6%. Company A’s last dividend was $2 per share and is expected to grow at a stable rate of 8% indefinitely. The stock currently sells for $30. What is A’s cost of equity? SML : Example IV 30 Suppose Stansfield Enterprises is evaluating the following non‐
mutually exclusive projects. Each costs $100 and lasts one year. Project Estimated CF for next year IRR NPV @ 30% A 2.5 $ 150 50% $15.38 B 2.5 $ 130 30% $0 C 2.5 $ 110 10% ‐$15.38 SML : Example IV IRR Project 31 30% 5% Good project A SML B C Bad project Firm’s risk (beta) 2.5 An all‐equity firm should accept a project whose IRR exceeds the cost of equity capital and reject projects whose IRRs fall short of the cost of capital. Advantages and Disadvantages of SML 32 Advantages: Explicitly adjusts for systematic risk Applicable to all companies, as long as we can compute beta Advantages and Disadvantages of SML 33 Disadvantages Have to estimate the expected market risk premium, which can vary over time Have to estimate beta, which can also vary over time Using the past to predict the future Estimation of Beta 34 Theoretically, the calculation of beta is straightforward: Cov( Ri , RM ) σ i2 β
Var ( RM )
σM Problems 1. Betas may vary over time. 2. The sample size may be inadequate. 3. Betas are influenced by changing financial leverage and business risk. Solutions Problems 1 and 2 (above) can be moderated by more sophisticated statistical techniques. Problem 3 can be lessened by adjusting for changes in business and financial risk. Look at average beta estimates of comparable firms in the industry. Stability of Beta 35 Most analysts argue that betas are generally stable for firms remaining in the same industry. That’s not to say that a firm’s beta can’t change. Changes in product line Changes in technology Deregulation Changes in financial leverage Using an Industry Beta 36 It is frequently argued that one can better estimate a firm’s beta by involving the whole industry. If you believe that the operations of the firm are similar to the operations of the rest of the industry, you should use the industry beta. If you believe that the operations of the firm are fundamentally different from the operations of the rest of the industry, you should use the firm’s beta. Don’t forget about adjustments for financial leverage. Determinants of Beta 37 Business Risk Cyclicity of Revenues Operating Leverage Financial Risk Financial Leverage Cyclicality of Revenues 38 Highly cyclical stocks have high betas. Empirical evidence suggests that retailers and automotive firms fluctuate with the business cycle. Transportation firms and utilities are less dependent upon the business cycle. Note that cyclicality is not the same as variability—
stocks with high standard deviations need not have high betas. Movie studios have revenues that are variable, depending upon whether they produce “hits” or “flops”, but their revenues are not especially dependent upon the business cycle. Operating Leverage 39 The degree of operating leverage measures how sensitive a firm (or project) is to its fixed costs. Operating leverage increases as fixed costs rise and variable costs fall. The degree of operating leverage is given by: Sales EBIT DOL = × EBIT Sales Operating Leverage 40 $ Total costs Fixed costs EBIT Volume Fixed costs Volume Operating leverage increases as fixed costs rise and variable costs fall. Financial Leverage and Beta 41 Operating leverage refers to the sensitivity to the firm’s fixed costs of production. Financial leverage is the sensitivity of a firm’s fixed costs of financing. The relationship between the betas of the firm’s debt, equity, and assets is given by: Debt Equity Asset = × Debt + × Equity Debt + Equity Debt + Equity Financial leverage always increases the equity beta relative to the asset beta. Financial Leverage and Beta: Example 42 Consider Grand Sport, Inc., which is currently all‐equity and has a beta of 0.90. The firm has decided to lever up to a capital structure of 1 part debt to 1 part equity. Since the firm will remain in the same industry, its asset beta should remain 0.90. However, assuming a zero beta for its debt, its equity beta would become twice as large: Asset = 0.90 = 1 1 + 1 × Equity Equity = 2 × 0.90 = 1.80 ...
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- Ri, Equity Capital