# The CAPM (Capital Asset Pricing Model) NPV Dependent on Discount Rate Schedule

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1 The CAPM (Capital Asset Pricing Model) Massachusetts Institute of Technology CAPM Slide 1 of NPV Dependent on Discount Rate Schedule Discussed NPV and time value of money Choice of discount rate influences decisions WACC may be appropriate for average projects What discount rate applies to unique projects? Massachusetts Institute of Technology CAPM Slide 2 of Page 1

2 CAPM: A Basis for Adjusting Discount Rates for Risk Development of the Capital Asset Pricing Model Assumptions about investor s view of risky investments Risk characteristics and components Principle of diversification Beta: a formal metric of risk The Capital Asset Pricing Model relationship between risk and expected return The Security Market Line and expected return for individual investments Use of CAPM principles for project evaluation Comparison of utility theory and CAPM Massachusetts Institute of Technology CAPM Slide 3 of Motivation for CAPM: Investors Prefer Less Risk Consider two investments Deposit \$10 in a savings account with annual yield of 5% Buy stock for \$10 with a chance of selling for \$12 or \$9 in one year Which is more attractive to risk-averse investors? Expected return for savings account = 5% Expected return for stock = (0.5*(12+ 9) -10)/10*100%= 5% For same return, investors prefer less risky savings account What if stock had a 75% chance of selling for \$12? Massachusetts Institute of Technology CAPM Slide 4 of Page 2

3 Motivation for CAPM (2) Bank \$10.00 Stock \$ \$10.50 \$12.00 \$9.00 Massachusetts Institute of Technology CAPM Slide 5 of How do Investors Regard Risk and Return? Two key observations regarding preferences Non-satisfaction For a given level of risk, the preferred alternative is one with the highest expected return (A > C) Risk Aversion For a given level of return, the preferred alternative is one with the lowest level of risk (A > B) Return 0 0 A C Risk B Massachusetts Institute of Technology CAPM Slide 6 of Page 3

4 Risk Metrics: An Empirical Observation Investors expect compensation for variability (risk) Risk-free rate defined as return if no variability More risky securities priced to return premium Correlation between variability and expected return Security Expected Return % Variability: Standard Deviation of Expected Returns (%) Risk free 5 0 U.S Treasuries Fixed Income Domestic Equity International Equity Real Estate Venture Capital Massachusetts Institute of Technology CAPM Slide 7 of Relationship Between Variability and Expected Return An upward trend Argument based on aggregate performance of groups CAPM models expectations for individual investments Expected Return Investment Variability and Expected Return (data from previous table) Standard Deviation of Returns Massachusetts Institute of Technology CAPM Slide 8 of Page 4

5 Components of Risk Finance defines 2 risks (standard deviation) Market Risk (systematic, non-diversifiable) Investments tend to fluctuate with outside markets Declines in the stock market and the price of Microsoft might be correlated Unique or Project Risk (idiosyncratic, diversifiable) Individual characteristics of investments affect return Microsoft might increase in price, despite a decline in the overall stock market Diversify by holding a portfolio of many investments What compensation should investors demand for each type? Massachusetts Institute of Technology CAPM Slide 9 of Role of Diversification Unique risks are reduced by holding an investment portfolio Example: Two Stocks A: Expected return = 20%, Standard Deviation of Expected Returns = 20% B: Expected Return = 20% Standard Deviation of Expected Returns = 20% Consider portfolio with equal amounts of A and B Expected return = 0.5*20% + 0.5*20% = 20% Standard Deviation? Massachusetts Institute of Technology CAPM Slide 10 of Page 5

6 Standard Deviation for a Portfolio Portfolio standard deviation is not a weighted average Portfolio standard deviation σ p = Σ i Σ j x i x j σ I σ j ρ ij for a portfolio of N investments, with i, j = 1 to N x i,x j = Value fraction of portfolio represented by investments i and j σ i, σ j = Standard deviation of investments i and j ρ ij = Correlation between investments i and j ρ jj = 1.0 Massachusetts Institute of Technology CAPM Slide 11 of Example: Standard Deviation for a 2 Stock Portfolio Invest equal amounts in two stocks For both A & B: Expected Return = 20%, Standard Deviation = 20% σ p = (0.5)(0.5)(0.2)(0.2)(1)+ (0.5)(0.5)(0.2)(0.2)(1)+(2)(0.5)(0.5)(0.2)(0.2)ρ a1b Portfolio standard deviation depends on correlation of A, B Correlation Between A & B Portfolio Standard Deviation Portfolio Expected Return % 20% % 20% % % 20% Massachusetts Institute of Technology CAPM Slide 12 of Page 6

7 Example: Standard Deviation for a 2 Stock Portfolio (2) Most investments not perfectly correlated (correlation < 1) Holding portfolio leads to risk reduction With negative correlation, can eliminate all risk Massachusetts Institute of Technology CAPM Slide 13 of Generalization for Portfolio with Many Stocks General formula for standard deviation of portfolio returns σp = Σ i Σ j x i x j σ i σ j ρ ij = portfolio variance For a portfolio of N stocks in equal proportions (x i = x j = 1/N) N weighted variance terms, i = j σ 2 i (Nˆ2-N) weighted cov. terms, i j σ i σ j ρ ij Var(P) = N*(1/N)ˆ2* Average Variance + (Nˆ2-N)* (1/N)ˆ2*Average Covariance Var(P) = (1/N)*Av. Var. + [1-(1/N)*] Av. Cov. Massachusetts Institute of Technology CAPM Slide 14 of Page 7

8 Generalization for Portfolio with Many Stocks (2) σ p = (1/N)* Average Variance + (1-(1/N)) Average Covariance For large N, 1/N 0 Average variance term associated with unique risks becomes irrelevant Average covariance term associated with market risk remains Massachusetts Institute of Technology CAPM Slide 15 of Defining a Formal Measure of Risk Investors expect compensation for systematic. undiversifiable (market) risk Standard deviation of returns reflects market & unique risks Need method to extract market portion of risk Define a reference point: the market portfolio The full set of available securities r m = Expected return for market portfolio σ m = Standard deviation of expected returns on market portfolio Beta: index of investment risk compared to market portfolio β i = ρ i, m σ i /σ m Massachusetts Institute of Technology CAPM Slide 16 of Page 8

9 What Does Beta Imply? By definition, the market portfolio has beta = 1.0 Beta describes the relative variability of returns Concerned with correlated (systematic) portion of returns If investment amplifies movements in market portfolio beta > 1 If attenuates, movements in market portfolio beta < 1 Beta reflects market risk of an investment Investors expect higher returns for increased market risk Thus, higher returns for investments with higher betas Can be calculated for individual investments or portfolios Portfolio beta = weighted average of individual betas Massachusetts Institute of Technology CAPM Slide 17 of Investment Portfolios and the Efficient Frontier Example demonstrated role of diversification Combinations of many securities result in optimum Maximum return for given risk level Minimum risk for given level of return Assumes no borrowing or lending Sub-optimal combinations lie below, to right of frontier Risk-free Return (%) Standard Deviation Massachusetts Institute of Technology CAPM Slide 18 of Page 9

10 Combining Risk-Free and Risky Investments For any combination of risk-free and risky investing Investor can mix investments in portfolio and risk-free to achieve desired return Expected return is weighted average of risk-free (Rf) and portfolio return (Rp) Standard deviation of Rf = 0 σ mix = x p σ p Rp Return (0%) Rf Lend Borrow σ p Standard Deviation Massachusetts Institute of Technology CAPM Slide 19 of CAPM: Selecting a Portfolio to Maximize Returns for Risk Infinite number of portfolios, even on efficient frontier Tangent point yields optimum CAPM shows expected return for investment combinations Borrowing Return (%) Rp Rf Lending σ p Standard Deviation Massachusetts Institute of Technology CAPM Slide 20 of Page 10

11 Determining Expected Return for Individual Investments CAPM models maximized expected return Beta indexes risk of individual investment to market portfolio Market portfolio is tangent point in CAPM Relation between beta and individual expected return results Return (%) Rp Rf Market Portfolio Lending Borrowing Return (0%) Rm Rf Market Portfolio Borrowing σ p Standard Deviation Bm Beta Massachusetts Institute of Technology CAPM Slide 21 of How Does Expected Return Relate to Beta? Security Market Line (SML) Rp = Rf + Bp*(Rm - Rf) Rm - Rf is the market risk premium Bp is the beta of the portfolio or investment to be evaluated For the market portfolio, Bm = Bp = 1 Total expected return is Rm For other investments, expected return scales with Bp Return (0%) Rm Rf Borrowing Market Portfolio Lending Bm Beta Massachusetts Institute of Technology CAPM Slide 22 of Page 11

12 Differences Between Borrowing and Lending Rates Not typical to have same rate for borrowing and lending Risk-free rate generally unattainable for small investors Adjustments to model possible, minor, illustrated below Point of tangency shifts Return (%) Borrowing Lending Standard Deviation Massachusetts Institute of Technology CAPM Slide 23 of Implementing the CAPM: From Theory to Project Evaluation Relation between market risk and expected return Investments have market risk and unique risk components Market risk commands premium over risk-free rate Unique risk is managed (averaged out) by diversification Project discount rate should be based on project beta Investors can diversify away unique project risks Adjustment apparent if project is carbon-copy of firm (McDonald s #10,001) ==> WACC applies Proper adjustment not trivial on most projects Consider past experiences, returns in comparable industries Detail unique aspects of specific project Apply information to adjust discount rate Massachusetts Institute of Technology CAPM Slide 24 of Page 12

13 A General Rule for Managers Portfolio theory translates to a simple rule for managers: Use risk adjusted discount rate to calculate NPV for projects, Accept all positive NPV projects to maximize value Shareholders capable of diversifying unique risks by holding multiple assets Positive NPV implies market risk in projects is expected to be compensated If projects are properly valued, shareholder wealth is maximized Massachusetts Institute of Technology CAPM Slide 25 of Limitations and Conflicts in Practice Estimating project beta may not trivial Budget constraints conflict with positive NPV rule Employees worry about unique project risks Career can be adversely affected by bad outcomes Not always in a position to diversify (limited to few projects) Issue might be addressed through proper incentives Reliance on past performance to dictate future choices Individuals and companies are often risk positive Entrepreneurs Sometimes only choice is bet the company Massachusetts Institute of Technology CAPM Slide 26 of Page 13

14 How does Utility Theory Compare with CAPM? Utility Applies a single discount rate for time value Adjusts for risk preference of decision-maker Utility is bottom-up and focused on individual preferences CAPM Adjusts discount rate for overall aversion to market risk No adjustment for risk preferences of decision-maker Based on top-down, aggregate perspectives Utility and CAPM Both value risky opportunities, accounting for risk aversion Under the right circumstances, should give same results No double counting Massachusetts Institute of Technology CAPM Slide 27 of Summary CAPM adjusts discount rates for risk Models maximum expected return for level of risk Based on observations of securities markets Unique risks can be diversified Investors expect compensation for market risk Standard deviation of returns reflects both market & unique Beta is index of market part of investment risk Security Market Line relates expected return to beta Rp = Rf + Bp*(Rm - Rf) Moving from theory to practice can be problematic Massachusetts Institute of Technology CAPM Slide of Page 14

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