Risk and Return and the Capital Asset Pricing Model (CAPM)

1. Risk and Return and the Capital Asset Pricing Model (CAPM) For 9.220, Chapter

2. Risk Return & The Capital Asset Pricing Model (CAPM) • To make “good” (i.e., value-maximizing) financial decisions, one must understands the relationship between risk and return • We accept the notion that rational investors like returns and dislike risk • Consider the following proxies for return and risk: Expected return- weighted average of the distribution of possible returns in the future. Variance of returns- a measure of the dispersion of the distribution of possible returns in the future.

3. Expected (Ex Ante) Return where, Ri = the return in state i (there are S states) Pi = the probability of return i (state i) An Example Consider the following return figures for the following year on stock XYZ under three alternative states of the economy PiRi Probability Return inState of Economy of state i state i +1% change in GNP 0.25 -5% +2% change in GNP 0.50 15% +3% change in GNP 0.25 35%

4. Expected Returns - An Example Q. Calculate the expected return on stock XYZ for the next year A. Use the following table PiRi PiRi Probability Return inState of Economy of state i state i State 1: +1% change in GNP 0.25 -5% - 1.25% State 2: +2% change in GNP 0.50 15% 7.50% State 3: +3% change in GNP 0.25 35% 8.75% Expected Return=15.00% Or, use the formula:

5. Variance and Standard Deviation of Returns where, Ri = the return in state i (there are S states) Pi = the probability of return i (state i) and s = the standard deviation of the return: An Example Recall the return figures for the following year on stock XYZ under three alternative states of the economy Pi Ri Probability Return inState of Economy of state i state i State 1: +1% change in GNP 0.25 -5% State 2: +2% change in GNP 0.50 15% State 3: +3% change in GNP 0.25 35% Expected Return = 15.00%

6. Variance & Standard Deviation - An Example Q. Calculate the variance and standard deviation of returnson stock XYZ A. Use the following table Pi X (Ri - E[R])2.= Pi(Ri - E[R])2 Probability State of Economy of state i +1% change in GNP 0.25 0.04 0.01 +2% change in GNP 0.50 0.00 0.00 +3% change in GNP 0.25 0.04 0.01 Variance of Return =0.02 Or, use the formula: Standard deviation:

7. Portfolio Return and Risk Q. Calculate the expected return on assets A and B for the next year, given the following distribution of returns: A. Expected returns E(RA) = _____ E(RB) = _____ State of the Probability Return on Return oneconomy of state asset A asset B Boom 0.40 30% -5% Bust 0.60 -10% 25%

8. Q. Calculate the variance of the above assets A and B A. Variances Var(RA) = ____ Var(RB) = _____ Q. Calculate the standard deviations of the above assets A and B A. Standard Deviations sA = ____ sB = ____

9. Returns and Risk for Portfolios - 2 Assets Expected Return on a Portfolio The Expected Return on Portfolio p with N securities where, E[Ri]= expected return of security i Xi = proportion of portfolio's initial value invested in security i Example - Consider a portfolio p with 2 assets: 50% invested in asset A and 50% invested in asset B. The Portfolio expected return is given by: E(RP) = XAE(RA) + XBE(RB) = (0.50x0.06) + (0.50x0.13) = 0.095 = 9.5%

10. Variance of a Portfolio The variance of portfolio p with two assets (A and B) where, Standard Deviation of a Portfolio The standard deviation of portfolio p with two assets (A and B)

11. Q. Calculate the variance of portfolio p (50% in A and 50% in B) A. Recall:Var(RA) = 0.0384, and Var(RB) = 0.0216 First, we need to calculate the covariance b/w A and B: = 0.40x(0.30-0.06)(-0.05-0.13) + 0.60x(-0.10-0.06)(0.25- 0.13) = - 0.0288 The variance of portfolio p Q. Calculate the standard deviations of portfolio p A. Standard Deviations sp = (0.0006)1/2 = 0.0245 = 2.45%

12. The Effect of Diversification on Portfolio Risk Note: • E(RP) = XAE(RA) + XBE(RB) = 9.5%, but • Var(Rp) =0.0006 <XAVar(RA) + XBVar(RB) = (0.50 x 0.0384) + (0.50 x 0.0216) = 0.03 • This means that by combining assets A and B into portfolio p, we eliminate some risk (mainly due to the covariance term) • Diversification - Strategy designed to reduce risk by spreading the portfolio across many investments • Two types of Risk: Unsystematic/unique/asset-specific risks - can be diversified away Systematic or “market” risks - can’t be diversified away • In general, a well diversified portfolio can be created by randomly combining 25 risky securities into a portfolio (with little (no) cost).

13. Diversifiable risk is also called unique risk, firm-specific risk, or unsystematic risk. Since we can get rid of this risk through portfolio diversification, we don’t care too much about it. Portfolio Diversification Average annualstandard deviation (%) 49.2 Diversifiable (nonsystematic) risk 23.9 19.2 Nondiversifiable(systematic) risk This is the risk we care about, as we cannot get rid of it. Number of stocksin portfolio 1 10 20 30 40 1000

14. Beta and Unique Risk • Total risk = diversifiable risk + market risk • We assume that diversification is costless, thus diversifiable (nonsystematic) risk is irrelevant • Investors should only care about non-diversifiable (systematic) market risk • Market risk is measured by beta - the sensitivity to market changes • Example: Return (%) State of the economy TSE300 BCE Good 18 26 Poor 6 -4

15. Beta and Market Risk Interpretation: Following a change of +1% (-1%) in the market return, the return on BCE will change by +2.5% (-2.5%) NOTE: If the security has a -ve cov w/ TSE 300 => The Characteristic Line • (26%, 18%) Slope = b = 2.5 • (-4%, 6%)

16. Beta and Unique Risk • Market Portfolio - Portfolio of all assets in the economy. In practice a broad stock market index, such as the S&P/TSX, is used to represent (proxy) the market • Beta (b )- Sensitivity of a stock’s return to the return on the market portfolio Covariance of security i’s return with the market return Variance of market return

17. Markowitz Portfolio Theory • We saw that combining stocks into portfolios can reduce standard deviation • Covariance, or the correlation coefficient, make this possible: The standard deviation of portfolio p (with XA in A and XB in B): Note: , or Thus,

18. Markowitz Portfolio Theory - An Example • Consider assets Y and Z, with • Consider portfolio p consisting of both Y and X. Then, we have: Expected Return of p Standard Deviation of p

19. Look at the next 3 cases (for the correlation coefficient): • Where

20. The Shape of the Markowitz Frontier - An Example Rho = -1 . . Y . 20.0% . 18.6% Rho = +1 15.8% 17.2% . 14.4% Z Rho = 0 1.2% 2.72% 5.16% 7.69% 10.247%

21. Efficient Sets and Diversification r E(R) = -1 r < 1 -1 < r = 1 s

22. The Efficient (Markowitz) FrontierThe 2-Asset Case • Expected Returns and Standard Deviations vary given different weighted combinations of the two stocks • The Feasible Set is on the curve Z-Y • The Efficient Set is on the MV-Y segment only Expected Return (%) Stock Y Minimum Variance Portfolio (MV) 75% in Z and 25% in Y MV Stock Z Standard Deviation

23. The Efficient (Markowitz) FrontierThe Multi-Asset Case • Each half egg shell represents the possible weighted combinations for two assets • The Feasible Set is onand inside the envelope curve • The composite of all asset sets (envelope), and in particular the segment MV-U constitutes the efficient frontier Expected Return (%) U Minimum Variance Portfolio (MV) MV Standard Deviation

24. Efficient Frontier • We assume that investors are rational (prefer more to less) and risk averse Goal is to move UPWARD and to the LEFT. Return Risk

25. Which Asset Dominates? Return Low Risk High Return High Risk High Return Low Risk Low Return High Risk Low Return Risk

26. Short Selling • Definition The sale of a security that the investor does not own. • How? Borrow the security from your broker and sell it in the open market. • Cash Flow At the initiation of the short sell, your only cash flow, is the proceeds from selling the security. • Closing the Short Eventually you will have to buy the security back in order to return it to the broker. • Cash Flow At the elimination of the short sell, your only cash flow, is the price you have to pay for the security in the open market.

27. Short Selling A Treasury Bill - An Example • The Security -- A Treasury bill is a zero-coupon bond issued by the Government, with a face value of $100, and with a maturity no longer than one year. If the yield on a 1-year T-bill is 5%, then its current price is: 100/1.051 = $95.24 • The Short sell -- Borrow the 1-year T-bill from your broker and sell it in the open market for $95.24. • Cash Flow-- The short sell proceeds: $95.24 • Closing the Short -- At the end of the year - buy the T-bill back (an instant before it matures) in order to return it to the broker • Cash Flow -- The price you have to pay for the T-bill in the open market an instant before maturity (in 1 year): 100/1.050 = $100 • Risk-Free Borrowing -- This transaction is equivalent to borrowing $95.24 for one year, and paying back $ 100 in a year. The interest rate is: (100/95.24) -1 = 5% = the 1-year T-bill yield

28. Risk-free borrowing and lending • Consider combinations of the risk-free asset with a portfolio, Z, on the Efficient Frontier. • With a risk-free asset available, taking a long f position (positive portfolio weight in f) gives us risk-free lending combined with A. • Taking a short f position (negative portfolio weight in f) gives us risk-free borrowing combined with A. E[R] Portfolio Z A+Borrowing A+Lending Rf P

29. Risk-free borrowing and lending • Which combination of f and portfolios on the Efficient Frontier are best? • Portfolios along the line tangent to the Efficient Frontier dominate everything else. Now, the only efficient risky portfolio on the Markowitz Efficient Frontier is Portfolio M. E[R] M Rf P What is the optimal strategy for every investor?

30. The Capital Market Line (CML)The Efficient Frontier With Risk-Free Borrowing and Lending • Lending or Borrowing at the risk free rate (Rf) allows us to exist outside the Markowitz frontier. • We can create portfolio A by investing in both Rf (lending money) and M • We can create portfolio B by short selling Rf (borrowing money) and holding M . Expected returnof portfolio CML CML is the new efficient frontier B . M . A Risk-freerate (Rf ) Standarddeviation ofportfolio’s return.

31. The Capital Asset Pricing Model (CAPM) Note • all securities are in M, and • all investors have M in their portfolios since they are all on the new efficient frontier - CLM - investing in Rf and M. Therefore Investors are only concerned with and , and with the contribution of each security i to M, in terms of • contribution of systematic risk (measured by beta) • contribution of expected return According to the CAPM: where,

32. The Security Market Line (SML) The Capital Asset Pricing Model (SML): Note: (1) -> entire risk of i is diversified away in M (2) -> security i contributes the average risk of M SML . M 1

33. The Security Market Line (SML) • The SML is always linear • CML - just for efficient portfolios • SML - for any securityand portfolio (efficient or inefficient)

34. The Security Market Line (SML) Example: Consider stocks A and B, with: ba = 0.8,bb = 1.2, let E[Rm] = 14% and Rf = 4%. By the SML: E[Ra] = 4% + 0.8[14% - 4%] = 12% E[Rb] = 4% + 1.2[14% - 4%] = 16% Consider a portfolio p, with 60% invested in A and 40% invested in B, then: E[Rp] = XaE[Ra] + XbE[Rb] = 0.6x12% + 0.4x16% = 13.6% And, bp = Xa ba + Xb bb = 0.6x0.8 + 0.4x1.2 = 0.96 By the CAPM:E[rp] = 4% + 0.96[14% - 4%] = 13.6% * If A and B are on the SML => p is also on SML

35. Summary and Conclusions • The CAPM is a theory that provides a relation between expected return and an asset’s risk. • It is based on investors being well-diversified and choosing non-dominated portfolios that consist of combinations of f (risk free security) and M. • While the CAPM is useful for considering the risk/return tradeoff, and it is still used by many practitioners, it is but one of many theories relating return to risk (and other factors) so it should not be regarded as a universal truth.