Jacoby, Stangeland and Wajeeh, 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 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. Chapter 10

2 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 k R k Probability Return in State of Economyof state k state k +1% change in GNP0.25-5% +2% change in GNP0.5015% +3% change in GNP0.2535% 1.00 where, R k = the return in state k (there are S states) P k = the probability of return k (state k)

3 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% State 2: +2% change in GNP0.5015% State 3: +3% change in GNP0.2535% Expected Return =

4 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 k R k Probability Return in State of Economyof state k state k 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 k = the return in state k (there are S states) P k = the probability of return k (state k) and  = the standard deviation of the return:

5 Q. Calculate the variance of and standard deviation of returns on stock XYZ A. Variance & Standard Deviation - An Example Or, use the formula: = Standard deviation: Use the following table State of Economy P k X (R k - E[R]) 2 = P k (R k - E[R]) 2 +1% change in GNP % change in GNP % change in GNP Variance of Return =0.02

Jacoby, Stangeland and Wajeeh, 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 ) = (0.40 % 0.30) + (0.60 % (-0.10)) = 0.06 = 6% E(R B ) = (0.40 % (-0.05)) + (0.60 % 0.25) = 0.13 = 13% State of theProbabilityReturn onReturn on economyof stateasset Aasset B Boom0.4030%-5% Bust %25% Portfolio Return and Risk

Jacoby, Stangeland and Wajeeh, Q. Calculate the variance of the above assets A and B A. Variances Var(R A ) = 0.40 % ( ) % ( ) 2 = Var(R B ) = 0.40 % ( ) % ( ) 2 = Q. Calculate the standard deviations of the above assets A and B A. Standard Deviations  A = (0.0384) 1/2 = = 19.6%  B = (0.0216) 1/2 = = 14.7%

8 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 ) = Returns and Risk for Portfolios - 2 Assets

Jacoby, Stangeland and Wajeeh, 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)

10 Q. Calculate the variance of portfolio p (50% in A and 50% in B) A. Recall: Var(R A ) = , and Var(R B ) = First, we need to calculate the covariance b/w A and B: The variance of portfolio p Q. Calculate the standard deviations of portfolio p A. Standard Deviations  p = (0.0006) 1/2 = = 2.45%

11 Note: uE(R P ) = X A E(R A ) + X B E(R B ) = 9.5%, but uVar(R p ) = < X A Var(R A ) + X B Var(R B ) = (0.50 % ) + (0.50 % ) = 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

Jacoby, Stangeland and Wajeeh, Portfolio Diversification Average annual standard deviation (%) Number of stocks in portfolio Diversifiable (nonsystematic) risk Nondiversifiable (systematic) risk

Jacoby, Stangeland and Wajeeh, 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 nondiversifiable (systematic) market risk uMarket risk is measured by beta - the sensitivity to market changes uExample: Return (%) State of the economy TSE300 BCE Good Poor 6 -4

14 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 => Beta and Market Risk (6%, -4%) (18%, 26%) The Characteristic Line

Jacoby, Stangeland and Wajeeh, Beta and Unique Risk uMarket Portfolio - Portfolio of all assets in the economy. In practice a broad stock market index, such as the TSE300, is used to represent 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

Jacoby, Stangeland and Wajeeh, 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 po rtfolio p (with X A in A and X B in B): Note:, or Thus,

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

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

Jacoby, Stangeland and Wajeeh, The Shape of the Markowitz Frontier - An Example Z. Y  = -1  = 0  = +1

Jacoby, Stangeland and Wajeeh, Efficient Sets and Diversification  = <   = 1

21 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

22 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

Jacoby, Stangeland and Wajeeh, Efficient Frontier Goal is to move UP and LEFT. WHY?  We assume that investors are rational (prefer more to less) and risk averse Expected Return (%) Standard Deviation (Risk)

Jacoby, Stangeland and Wajeeh, Low Risk High Return High Risk High Return Low Risk Low Return High Risk Low Return Which Asset Dominates? Expected Return (%) Standard Deviation (Risk)

Jacoby, Stangeland and Wajeeh, 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

26 Short Selling A Treasury Bill - An Example u The Security u 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 u If the yield on a 1-year T-bill is 5%, then its current price is: 100/ = $95.24 u The Short sell Borrow the 1-year T-bill from your broker and sell it in the open market $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/ = $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

27 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

28 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 - CML - investing in R f and M. Therefore Investors are only concerned withand, 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)

29 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

30 The Security Market Line (SML) u The SML is always linear u CML - just for efficient portfolios SML - for any security and portfolio (efficient or inefficient) 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 ] = E[R b ] = 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 ] = 0.6 $ 12% $ 16% = 13.6%, and  p = X a  a + X b  b = 0.6 $ $ 1.2 = 0.96 By the CAPM: E[R p ] = 4% [14% - 4%] = 13.6% * If A and B are on the SML => P is also on SML