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

2. Risk Return & The Capital Asset Pricing Model (CAPM) l To make “good” (i.e., value-maximizing) financial decisions, one must understands the relationship between risk and return l We accept the notion that rational investors like returns and dislike risk l 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 An Example Consider the following return figures for the following year on stock XYZ under three alternative states of the economy P i R i Probability Return in State of Economyof state i state i +1% change in GNP0.25-5% +2% change in GNP0.5015% +3% change in GNP0.2535% where, R i = the return in state i (there are S states) P i = the probability of return i (state i)

4. Q. Calculate the expected return on stock XYZ for the next year A. Expected Returns - An Example Or, use the formula: Use the following table P i R i P i R i Probability Return in State of Economyof state i state i State 1: +1% change in GNP0.25-5% - 1.25% State 2: +2% change in GNP0.5015% 7.50% State 3: +3% change in GNP0.2535% 8.75% Expected Return=15.00%

5. Variance and Standard Deviation of Returns An Example Recall the return figures for the following year on stock XYZ under three alternative states of the economy P i R i Probability Return in State of Economyof state i state i State 1: +1% change in GNP0.25-5% State 2: +2% change in GNP0.5015% State 3: +3% change in GNP0.2535% Expected Return = 15.00% where, R i = the return in state i (there are S states) P i = the probability of return i (state i) and  = the standard deviation of the return:

6. Q. Calculate the variance and standard deviation of returns on stock XYZ A. Variance & Standard Deviation - An Example Or, use the formula: Standard deviation: Use the following table P i X (R i - E[R]) 2. = P i (R i - E[R]) 2 Probability State of Economyof state i +1% change in GNP0.250.04 0.01 +2% change in GNP0.500.00 0.00 +3% change in GNP0.250.04 0.01 Variance of Return =0.02

7. Q. Calculate the expected return on assets A and B for the next year, given the following distribution of returns: A. Expected returns E(R A ) = _____ E(R B ) = _____ State of theProbabilityReturn onReturn on economyof stateasset Aasset B Boom0.4030%-5% Bust0.60-10%25% Portfolio Return and Risk

8. Q. Calculate the variance of the above assets A and B A. Variances Var(R A ) = ____ Var(R B ) = _____ Q. Calculate the standard deviations of the above assets A and B A. Standard Deviations  A = ____  B = ____

9. Expected Return on a Portfolio The Expected Return on Portfolio p with N securities where, E[R i ]= expected return of security i X i = 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(R P ) = X A E(R A ) + X B E(R B ) = (0.50x0.06) + (0.50x0.13) = 0.095 = 9.5% Returns and Risk for Portfolios - 2 Assets

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(R A ) = 0.0384, and Var(R B ) = 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  p = (0.0006) 1/2 = 0.0245 = 2.45%

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

13. Portfolio Diversification Average annual standard deviation (%) Number of stocks in portfolio Diversifiable (nonsystematic) risk Nondiversifiable (systematic) risk 49.2 23.9 19.2 1102030401000 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. This is the risk we care about, as we cannot get rid of it.

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

15. Beta and Market Risk (-4%, 6%) (26%, 18%) Slope =  = 2.5 The Characteristic Line 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 =>

16. Beta and Unique Risk uMarket 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 ()- Sensitivity of a stock’s return to the return on the market portfolio u Covariance of security i’s return with the market return Variance of market return

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

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

19. uLook at the next 3 cases (for the correlation coefficient): uWhere

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

21. Efficient Sets and Diversification  E(R)  = -1 -1 <   = 1

22. The Efficient (Markowitz) Frontier The 2-Asset Case Stock Z Stock Y Standard Deviation Expected Return (%) 75% in Z and 25% in Y  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 Minimum Variance Portfolio (MV) MV

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

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

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

26. Short Selling u Definition The sale of a security that the investor does not own. u How? Borrow the security from your broker and sell it in the open market. u Cash Flow At the initiation of the short sell, your only cash flow, is the proceeds from selling the security. u Closing the Short Eventually you will have to buy the security back in order to return it to the broker. u 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 u 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: 100/1.05 1 = $95.24 u The Short sell -- Borrow the 1-year T-bill from your broker and sell it in the open market for $95.24. u Cash Flow-- The short sell proceeds: $95.24 u 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 u Cash Flow -- The price you have to pay for the T-bill in the open market an instant before maturity (in 1 year): 100/1.05 0 = $100 u 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. A+Lending 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. PP E[R] RfRf A+Borrowing Portfolio Z

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. PP E[R] RfRf What is the optimal strategy for every investor? M

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

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

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. M SML 1

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

34. The Security Market Line (SML) Example: Consider stocks A and B, with:  a = 0.8,  b = 1.2, let E[R m ] = 14% and R f = 4%. By the SML: E[R a ] = 4% + 0.8[14% - 4%] = 12% E[R b ] = 4% + 1.2[14% - 4%] = 16% Consider a portfolio p, with 60% invested in A and 40% invested in B, then: E[R p ] = X a E[R a ] + X b E[R b ] = 0.6x12% + 0.4x16% = 13.6% And, p = X a  a + X b  b = 0.6x0.8 + 0.4x1.2 = 0.96 By the CAPM: E[r p ] = 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.