Investments. MBA teaching notes. João Pedro Pereira

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1 Investments MBA teaching notes João Pedro Pereira Nova School of Business and Economics Universidade Nova de Lisboa jpereira/ April 6, 2016

2 Contents 1 Portfolio Choice Risk and return Holding period return Expected return and standard deviation Risk premium and Sharpe Ratio Exercises Risk Aversion and Capital Allocation Risk Aversion Portfolio rate of return Portfolios of one risky asset and a risk-free asset Risk tolerance and asset allocation Exercises Optimal Risky Portfolios Diversification and portfolio risk Portfolios of two risky assets Portfolios of two risky assets and a risk-free asset The Markowitz portfolio optimization model Exercises The Capital Asset Pricing Model Assumptions and derivation Efficient frontier Capital Market Line Risk aversion and the market risk premium Expected returns on individual securities Security Market Line Interpretation of beta Beta of a portfolio Exercises Arbitrage Pricing Theory and Factor Models Theory Example with 1 factor: the Market model

3 Contents Example with 3 factors: the Fama-French model Pricing equation Details on the factors Portfolio performance evaluation Motivation Adjusting returns for risk Bond Markets Rates of return Effective Annual Rate Annual Percentage Rate Continuous compounding Discounting Bond Prices and Yields Bond Pricing Yield to Maturity Holding-Period Return Default risk and ratings Exercises Term structure of interest rates Spot rates Forward rates Expectations of future interest rates Exercises Bond management Duration Active bond management Exercises The efficient market hypothesis Motivation Random walks and the EMH Implications of the EMH Are markets efficient? Exercises Futures markets Definition Forward contract Futures contract Differences between Futures and Forwards Trading strategies Speculation and leverage Hedging

4 Contents Spread trading Futures prices of stock indices Investment strategies with stock-index futures Creating synthetic stock positions Hedging an equity portfolio from market risk Exercises Solutions to Problems 76 Bibliography 77

5 Chapter 1 Portfolio Choice 1.1 Risk and return Based on chapter 5 of Bodie, Kane, and Marcus (2014) Holding period return Definition 1.1.1: Holding Period Return The Holding Period Return (HPR) on an investment from today (time 0) until T years from now is HPR = P(T) P(0)+Payouts P(0) (1.1) where P(t) is the price of the asset at time t Payouts represent cash income from the asset between 0 and T (dividends for stocks, coupons for bonds), assumed to be paid at the end of the holding period. Instead of HPR, will usually just say rate of return or simply return, and denote it by r. 5

6 1.1. Risk and return Expected return and standard deviation True population moments We want to characterize the probability distribution of returns across future states of nature ( scenarios ). Definition 1.1.2: Mean and Variance For the random rate of return r, µ :=E[r] = σ 2 :=Var[r] = S p(s)r(s) s=1 S p(s)[r(s) µ] 2 where s = 1,...,S are the possible states of nature (scenarios) s=1 p(s) is the probability of state s occuring. The standard-deviation is σ = Var[r]. Example Stock X is trading at $10. You estimate the following scenarios for next year: State of Market Prob Year-end price Dividends Expansion 0.6 $ 13 $ 1 Recession 0.4 $ 8 $ 0 Compute the mean and standard-deviation of returns. Time-series estimation The true moments are not observable, so we have to estimate the inputs to our models

7 1.1. Risk and return 7 Estimation of mean and variance Using a sample of T past observations or realized returns, ˆµ = 1 T T t=1 ˆσ 2 = 1 T 1 r t T [r t ˆµ] 2 t= Risk premium and Sharpe Ratio Terminology for returns in excess of the risk-free rate: 1. Excess return is the difference in any particular period between the actual rate of return on a risky asset and the actual risk-free rate. 2. Risk premium is the difference between the expected HPR on a risky asset and the risk-free rate The Sharpe Ratio, or Reward-to-Volatility Ratio, is an important measure of the trade-off between reward and risk. Definition 1.1.3: Sharpe Ratio Given the forward-looking true population moments and a constant risk-free rate, the Sharpe Ratio (SR) is SR = Risk premium Std-dev of return To estimate the SR from historical returns, SR = Average excess return Std-dev of excess return Example Continuing the previous example for stock X, suppose that the 1-year risk free rate is 3%. Compute the Risk Premium and Sharpe Ratio. Answer: SR =

8 1.2. Risk Aversion and Capital Allocation Exercises Ex. 1 Bodie, Kane, and Marcus (2014) problems at the end of chapter 5: CFA problems 1 through 5 (pages ) 1.2 Risk Aversion and Capital Allocation Based on chapter 6 of Bodie, Kane, and Marcus (2014) Risk Aversion Definition A fair game is a gamble with an expected payoff of zero. Example: toss a coin and double or loose your monthly salary. Definition 1.2.1: Risk aversion An investor is risk averse if he wishes to avoid a fair game. Depending on whether you like a fair game or not, you are either: - Risk Averse - Risk Neutral - Risk Lover We assume that most investors are risk averse: They only take risk if expecting a positive risk premium. Market prices show that most investors are risk averse Modelling risk aversion Different risky portfolios can be ranked with an expected utility function.

9 1.2. Risk Aversion and Capital Allocation 9 Expected utility function If the investor has mean-variance preferences, the expected utility function is: U(r) = E[r] 1 A Var[r] (1.2) 2 Remarks: 1. The parameter A characterizes the investor A > 0 means risk averse A = 0 means risk neutral A < 0 means risk lover (these investors are usually at the casino) 2. This utility score U can be interpreted as a certainty equivalent rate of return. The investor is indifferent between a given risky investment that generates a utility U and a risk-free investment at rate U Example (This is concept check 6.2 in BKM). A portfolio has an expected rate of return of 20% and standard deviation of 30%. T-bills offer a safe rate of return of 7%. 1. Would an investor with risk-aversion parameter A = 4 prefer to invest in T-bills or the risky portfolio? 2. What if A = 2? Mean-Variance Dominance We say that portfolio x mean-variance dominates portfolio y if all risk-averse investors with mean-variance preferences prefer x to y, regardless of their particular value of A (their degree of risk aversion).

10 1.2. Risk Aversion and Capital Allocation 10 Proposition 1.2.1: Mean-Variance Dominance Asset x mean-variance dominates asset y iff: µ x µ y and σ x < σ y or µ x > µ y and σ x σ y where µ i = E[r i ] and σ i = Var[r i ] E[r] σ Portfolio rate of return Definition Definition 1.2.2: Rate of return on a portfolio The return on a portfolio is: r p = I w i r i (1.3) i=1 where I = number of securities in the portfolio w i = proportion of funds invested in security i r i = return on security i

11 1.2. Risk Aversion and Capital Allocation 11 Example Consider the following portfolio: Stock A Stock B Initial Investment $40,000 $60,000 P 0 $20 $10 Initial Number of shares P 1 $24 $11 Compute the portfolio s terminal total value: This implies a portfolio return of /100,000 1 = Now check that (1.3) gives the same number: r p = w a r a +w b r b =... = 0.14 Mean and variance Proposition 1.2.2: Portfolio mean For a portfolio with 2 assets (I=2): [ 2 ] µ p := E[r p ] = E w i r i = w 1 E[r 1 ]+w 2 E[r 2 ] i=1 and Proposition 1.2.3: Portfolio variance For a portfolio with 2 assets (I=2): [ 2 ] σp 2 := Var[r p ] = Var w i r i = w1σ w2σ w 1 w 2 Cov(r 1,r 2 ) i=1

12 1.2. Risk Aversion and Capital Allocation 12 Remarks about covariance: 1. By definition, Cov(r 1,r 2 ) = E[(r 1 E[r 1 ])(r 2 E[r 2 ])]. 2. We sometimes denote σ 12 := Cov(r 1,r 2 ). 3. The linear correlation coefficient between r 1 and r 2 is a more intuitive measure and is defined as ρ := Cov(r 1,r 2 ) σ 1 σ 2 Always have 1 ρ +1. Example Consider two stocks with the following parameters: Stock µ σ X 10% 20% Y 20% 40% Thecorrelation is0.1. Weformthefollowingportfolio: w x = 0.4,w y = 0.6. Check that the portfolio mean is 0.16 and the standard-deviation is

13 1.2. Risk Aversion and Capital Allocation Portfolios of one risky asset and a risk-free asset Mean and variance Consider a complete or combined portfolio c composed of two assets: A risk-free asset, such as a T-Bill (denote it by f). A risky asset, which may itself be a portfolio of risky assets (denote it by p). The risk-free asset is not random and thus has Var[r f ] = 0 and σ fp = 0. Therefore, where r c = return on the complete portfolio. E[r c ] = w f r f +w p E[r p ] Var[r c ] = w 2 pσ 2 p σ c = w p σ p Capital allocation line Proposition 1.2.4: Capital allocation line All possible combinations of the risk-free asset and the risky portfolio p lie on the following straight line: CAL(p): E[r c ] = r f + E[r p] r f σ p σ c Proof. Using the previous equations, E[r c ] = w f r f +w p E[r p ] E[r c ] = (1 w p )r f +w p E[r p ] E[r c ] = (1 σ c σ p )r f + σ c σ p E[r p ] E[r c ] = r f + E[r p] r f σ p σ c

14 1.2. Risk Aversion and Capital Allocation 14 Example Consider r f = 7% and a portfolio with E[r p ] = 15% and σ p = 22%. All possible combinations of these two assets plot along the Capital Allocation Line: E[r] σ For example, plot the combined portfolio for w p = Plot also a leveraged combined portfolio with w p = 1.5. This requires w f = 0.5 (what does this mean?). The slope of the CAL is E[rp] r f σ p. This slope is called reward-to-variability ratio. It is also called the Sharpe ratio. The higher this ratio, the better Risk tolerance and asset allocation Proposition 1.2.5: Best complete portfolio For an investor with mean-variance preferences, the best combination of the risk-free asset and the risky portfolio p is: w p = E[r p] r f Aσ 2 p and w f = 1 w p (1.4) Proof. The investor gets the following utility from a portfolio on CAL(p): U(r c ) = E[r c ] 0.5 A Var[r c ] = w f r f +w p E[r p ] 0.5 A w 2 pσ 2 p The investor picks the particular portfolio on the CAL that maximizes his utility function. maximize (1 w p )r f +w p E[r p ] 0.5 A w 2 w p pσp 2

15 1.2. Risk Aversion and Capital Allocation 15 The first-order condition is: r f +E[r p ] A σ 2 p w p = 0 w p = E[r p] r f Aσ 2 p Example (Example 6.4 in BKM) Continuing the previous example, check that the optimal portfolio for an investor with A = 4 is wp = Plot the optimal portfolio on the CAL. E[r] σ Exercises Ex. 2 Bodie, Kane, and Marcus (2014) problems at the end of chapter 6 (p. 192): 2, Ex. 3 The return on stock x over the next year will depend on the state of the market as follows: State of Market Prob r x Expansion Recession

16 1.3. Optimal Risky Portfolios 16 The risk-free rate is 2%. What is the Sharpe Ratio of a portfolio that invests 60% in stock x and 40% in the risk-free asset? 1.3 Optimal Risky Portfolios Based on chapter 7 of Bodie, Kane, and Marcus (2014) Diversification and portfolio risk The main idea is Don t put all your eggs in one basket. Different stocks respond differently to economic shocks. Diversifying your funds into several assets reduces the risk of the overall portfolio. The total risk can be decomposed into two parts: 1. The part of the risk that can be easily eliminated through diversification. This is called: (a) unique risk or firm-specific risk; (b) nonsystematic risk; or (c) diversifiable risk 2. The part of the risk that cannot be eliminated and remains even after diversifying. This is called: (a) market risk; (b) systematic risk; or (c) nondiversifiable risk Portfolios of two risky assets Mean and variance Suppose there are just two risky assets (stocks). Recall that the mean and variance of the return on the portfolio formed by these two assets is: E[r p ] = w 1 E[r 1 ]+w 2 E[r 2 ] Var[r p ] = w 2 1σ 2 1 +w 2 2σ w 1 w 2 σ 1 σ 2 ρ where ρ is the correlation coefficient between r 1 and r 2 ( 1 ρ +1).

17 1.3. Optimal Risky Portfolios 17 Figure 1.1: Portfolio diversification (fig 7.2 in BKM) Correlation effects The investment opportunity set depends critically on the correlation between stocks. The smaller the correlation coefficient, the greater the benefits from diversification. Perfect positive correlation (ρ = 1). There is no gain from diversification since the assets are essentially identical (the return on one asset is a linear function of the other). The portfolio standard-deviation is equal to the weighted average of the two standard-deviations σ p = w 1 σ 1 +(1 w 1 )σ 2 which means that all possible portfolios lie on the straight line between the two assets (in σ,µ - space). Imperfect correlation ( 1 < ρ < +1). Now we have the diversification benefit. At each level of µ p, the corresponding σ p is less than in the ρ = 1 case. This is because σ 2 p increases in ρ ( σ 2 p/ ρ = 2w 1 w 2 σ 1 σ 2 > 0). Whereas the expected return on the portfolio is always the weighted average of expected returns on the individual assets, the standard-deviation of the portfolio is now less than the weighted average of the individual standard-deviations. Note that only the portfolios on the upper part of the curve are efficient, that is, they (mean-variance) dominate the ones on the lower part of the curve.

18 1.3. Optimal Risky Portfolios 18 Perfect negative correlation (ρ = 1). For this (theoretical) case we would be able to construct a risk-free asset. Plot all these cases: E[r] σ Minimum variance portfolio The Minimum-Variance Portfolio (MVP) has the smallest possible risk. Proposition 1.3.1: Minimum variance portfolio Given two risky assets, the MVP is given by w 1 = σ 2 2 σ 12 σ 2 1 +σ2 2 2σ 12 and w 2 = 1 w 1 Proof. The problem is : or The foc for w 1 is: minimize w 1,w 2 σ 2 p s.t. w 1 +w 2 = 1 minimize w 1 w 2 1σ 2 1 +(1 w 1 ) 2 σ w 1 (1 w 1 )σ 12 2w 1 σ 2 1 2(1 w 1 )σ (1 w 1 )σ 12 2w 1 σ 12 = 0 σ2 2 w 1 = σ 12 σ1 2 +σ2 2 2σ 12

19 1.3. Optimal Risky Portfolios 19 Example Consider two stocks: with ρ 12 = 0.2. Security µ σ Check that the return on MVP has a mean of 11.31% and a standard deviation of 13.08%. Note that MVP standard deviation is smaller than both individual standard deviations. Example Continuing the previous example, assume instead a negative correlation of ρ 12 = 0.3. Check that the return on MVP has a mean close to the previous case (11.57%), but a much smaller standard deviation of σ MVP = 10.09% Portfolios of two risky assets and a risk-free asset Tangency portfolio The optimal risky portfolio to combine with a risk-free asset is the one that produces the steepest Capital Allocation Line. It is called the Tangency portfolio.

20 1.3. Optimal Risky Portfolios 20 E[r] σ Proposition 1.3.2: Tangency portfolio Given two risky assets and a risk-free rate r f, the tangency portfolio is w 1 = r e 1 σ2 2 re 2 σ 12 r e 1 σ2 2 +re 2 σ2 1 (re 1 +re 2 )σ 12 and w 2 = 1 w 1 with r e j = E[r j] r f, for j = 1,2. Proof. We want to find the portfolio that maximizes the slope of the CAL going through it: E[r p ] r f maximize w 1,w 2 σ p s.t. E[r p ] = w 1 E[r 1 ]+w 2 E[r 2 ] σ p = [w 2 1 σ2 1 +w2 2 σ2 2 +2w 1w 2 σ 12 ] 1/2 w 1 +w 2 = 1 Substitute w 1 = 1 w 2 and solve the foc. Example Continuing the previous example with ρ = 0.2, further assume r f = 6%. Check that the Tangency portfolio is (w 1 = ,w 2 = ). This portfolio has E[r p ] = and σ p =

21 1.3. Optimal Risky Portfolios 21 Optimal complete portfolio Once we have the Tangency portfolio, we can find the optimal complete portfolio that combines T and the risk-free asset. For an investor with mean-variance preferences (equation 1.2), the optimal solution is given by (1.4): w T = E[r T] r f Aσ 2 T Example Continuing the previous example, assume A = 5. Check that the optimal combined portfolio is or equivalently, (w T = ,w f = ) (w 1 = ,w 2 = ,w f = ) The combined portfolio has E[r c ] = and σ c = Plot all these portfolios:

22 1.3. Optimal Risky Portfolios 22 E[r] σ The Markowitz portfolio optimization model The previous concepts can be extended to the case of 1 risk-free asset and N risky stocks. This is called the Markowitz portfolio model. Extension to N risky assets. Intuitively, the analysis can be generalized to 3 risky assets by taking one of the possible two-asset portfolios and a new 3rd asset. Proceeding with these iterations, we could get to N risky assets. Extension to N risky assets plus 1 risk-free asset. The investor will pick one particular portfolio on the mean-variance frontier the tangency portfolio to combine with the risk-free asset. The straight line going through r f and µ T is the efficient frontier. E[r] σ

23 1.3. Optimal Risky Portfolios Exercises Ex. 4 Bodie, Kane, and Marcus (2014) problems at the end of chapter 7 (p. 235): 4 10 Ex. 5 There are two funds available for investment: Fund E[r] σ 2 Bond fund 0.04 (0.1) 2 Equity fund 0.08 (0.2) 2 The correlation between the two funds is 0.3. Additionally, it is possible to invest in a risk-free asset with r f = An investor is willing to tolerate a maximum standard-deviation of What is the expected return on the best portfolio for this investor? Ex. 6 Two stocks have the following means and variances: Stock E[r] σ 2 a 0.08 (0.2) 2 b 0.12 (0.4) 2 The correlation between the two stocks is zero. There is no risk-free asset. Investors have mean-variance preferences given by U(r) = E[r] A 2 Var[r]. Find the optimal portfolio of the two stocks for an investor with A = 3, that is, indicate the optimal weights w a and w b.

24 Chapter 2 The Capital Asset Pricing Model (Based on chapter 9 of Bodie, Kane, and Marcus (2014)) The value of any asset is the present value, or discounted value, of its future cash flows. The CAPM gives us a formula for the discount rate. Hence, it is used everyday by corporations and investors to price investment projects, stocks, mutual funds, etc. The CAPM was developed simultaneously in three papers by Sharpe in 1964, Lintner in 1965, and Mossin in Assumptions and derivation Assumptions: 1. All investors are mean-variance optimizers, i.e., they all use the Markowitz model. 2. Investors have homogeneous expectations. 3. Investors can borrow or lend at a common (exogenous) risk-free rate. 4. All assets are publicly traded and short positions are allowed. 5. The investors planning horizon is a single period. 6. There are no taxes. 7. There are no transaction costs. 24

25 2.1. Assumptions and derivation 25 Derivation: 1. Since all investors estimate the same inputs (means and covariances), the tangency portfolio is the same for every investor. 2. The efficient frontier (namely, the straight line through r f and T) is the same for every investor. 3. Two fund separation: (a) Every investor allocates his wealth between two portfolios: the risk-free asset and the Tangency portfolio. (b) Notethattheweightsintherisk-freeassetandinthetangencyportfolio may differ across investors due to different degrees of risk aversion, but still everybody invests in just those two assets and in no other portfolio. 4. In equilibrium, all risky assets must belong to T. To see this, suppose that IBM is not in T (wibm T = 0). Then, there would be no demand for this stock, (wibm i = wt IBM = 0, for every investor i). We would thus have Demand Supply, which is not equilibrium. Therefore, in equilibrium, wj T > 0, asset j. 5. Furthermore, for every asset, the weight in T must be the same as in the whole market: wj T = Market Cap j n Market Cap n =: w M j, asset j If we all put 2% of our risky money into IBM stock, then IBM will have 2% of all money invested in the stock market, meaning that the market capitalization of IBM will be worth 2% of the whole market capitalization. Note that different investors may put different amounts of money at risk, ie, in the tangency portfolio. But from these amounts, each investor allocates the same 2% to IBM. 6. Hence, the Market portfolio is the Tangency portfolio, M = T. This is the economic content of the CAPM. CAPM The CAPM states that the Market portfolio is mean-variance efficient, that is, Market portfolio = Tangency portfolio

26 2.2. Efficient frontier Efficient frontier Capital Market Line When we use M instead of T, the efficient frontier is called Capital Market Line: E[r] σ Proposition 2.2.1: Capital Market Line Under the CAPM, the efficient frontier is CML : E[r p ] = r f + E[r M] r f σ M σ p where p is an efficient portfolio. Recall that any p CML is a combination of the risk-free and the market portfolio, thus σ p = w M σ M. Example You expect the stock market to go up by 10% over the next year. The standard deviation of the market return is 20%. You can buy1year government bondsyielding4%. You have $100,000 to invest and you are willing to tolerate a risk (standard deviation) of 15%. 1. What is the best allocation of your money?

27 2.2. Efficient frontier How much money do you expect to have one year from now? (Answer: $108,500) Risk aversion and the market risk premium Proposition 2.2.2: Risk premium of the market portfolio E[r M ] r f = Āσ2 M where Ā is the risk-aversion of the representative investor. Proof. Recall that for an investor with risk-aversion parameter A, the best complete portfolio p CML is given by w M = E[r M] r f Aσ 2 M and w f = 1 w M In the aggregate, the amount of borrowing and lending among investors has to net out to zero. Hence, the representative investor will have w f = 0 w M = 1. Denoting by Ā the risk aversion of the representative investor, 1 = E[r M] r f. ĀσM 2 Example (Concept check 9.2 in BKM) Data from the last 8 decades for the S&P500 index yield the following stats: average excess return, 7.9%; standard-deviation, 23.2%. To the extent that these averages approximated investor expectations for the period, what must have been the average coefficient of risk aversion?

28 2.3. Expected returns on individual securities Expected returns on individual securities Security Market Line Proposition 2.3.1: Security Market Line Under the CAPM, the equilibrium expected return for any asset j is given by SML : E[r j ] = r f +β j (E[r M ] r f ) where β j := Cov(r j,r M ) Var[r M ] Different stocks have different betas and thus different expected returns: E[r] β Important remarks about the SML: The SML applies to every single asset or portfolio (not necessarily on the CML). For any asset or portfolio j, the relevant measure of risk is β j, not its variance. 1 1 But for an efficient portfolio on the CML, including the market portfolio, we can use either the SML or CML to get its expected return.

29 2.3. Expected returns on individual securities 29 Example Industry-type application of the CAPM.Suppose E[r M ] = 10% and r f = 4%. You estimate stock a will pay a dividend of $2 one year from now. After that, you expect dividends to grow at 5% per year. You also estimate the beta of the stock to be β a = 0.9. What is the equilibrium price of the stock? Note: recall that the present value of a stream of dividends growing at rate g is P 0 = D 1 r g, where r is the discount rate. Thus, you just need to use the CAPM to estimate the required discount for stock a. Answer: P a = $ Interpretation of beta Why do betas explain differences in expected returns? 1. What matters in β j is Cov(r j,r M ) (the market variance is the same for all stocks) 2. Hence, high beta means high covariance with the market. 3. Thecontributionofanindividualsecurityj toσ 2 M isproportionalto Cov(r j,r M ). Intuitively, high Cov(r j,r M ) means that j wins/looses in the same states as the market wins/looses, which increases the variance of the market. See BKM for a formal proof. 4. Since all investors hold the market portfolio (in some combination with the risk-free rate), investors do not like stocks with high betas (why?) 5. Therefore, investors require high expected returns to hold high-beta stocks Example Consider the following two companies: Duke Energy Corporation (DUK): an energy company based in North Carolina, with a significant part of its operations in regulated markets. Ralph Lauren Corporation (RL): sells lifestyle products (clothing, accessories, fragrances). Which one is likely to have a higher beta? Why? On , β DUK = 0.28,β RL = 0.93

30 2.3. Expected returns on individual securities Beta of a portfolio Proposition 2.3.2: Portfolio β For a portfolio p with N securities, β p = N w i β i i=1 where w i are the portfolio weights. Proof. From the definition of beta, N β p := Cov(r p,r M )/Var(r M ) = Cov( w i r i,r M )/Var(r M ) i=1 N N = w i Cov(r i,r M )/Var(r M ) = w i β i i=1 i=1 Example (This is concept check 9.3 in BKM). Suppose that the risk premium on the market portfolio is estimated at 8% with a std of 22%. What is the risk premium on a portfolio p invested 25% in Toyota and 75% in Ford, if they have betas of 1.10 and 1.25, respectively? Note that p in the previous example is not efficient: it will surely have too much σp 2. However, the CAPM does not reward all of σ2 p ; it only rewards the part that is nondiversifiable, which depends on Cov(r p,r M ) or β p.

31 2.4. Exercises 31 Example Continuing the previous example: 1. Construct an efficient portfolio q on the CML with β q = Since the SML applies to any portfolio and the betas are the same, the risk premium on q is the same as p: 2 E[r q ] r f =... This means that E[r q ] = E[r p ]. 3. Plot the CML and the approximate location of p and q 2.4 Exercises Ex. 7 Bodie, Kane, and Marcus (2014) problems at the end of chapter 9 (p. 317): 3 7, 17, Alternatively, to compute the risk premium on q without using the SML, Note that r q r f = w M r M +w f r f r f = w M r M +w f r f (w M +w f )r f = w M (r M r f ).

32 Chapter 3 Arbitrage Pricing Theory and Factor Models Based (somewhat) on chapters 10 and 24 of Bodie, Kane, and Marcus (2014), with additional materials from other sources. 3.1 Theory The APT was developed by Ross (1976). 1. Factor model. The APT starts by assuming that stock returns are generated by K factors: r j = a j + K β jk F k +ε j, j = 1,2,...,N (3.1) k=1 with the following assumptions: A1: E[ε j ] = 0, j A2: Cov(F k,ε j ) = 0, k,j A3: Cov(ε j,ε i ) = 0, j i The point is to find a small number of factors (K << N) that satisfy this model. 1 1 Equation (3.1) is sometimes stated as deviations from means. Take expectations on both sides to get E[r j ] = a j + K k=1 β jke[f k ]. Plug the resulting value for a j into (3.1) 32

33 3.1. Theory Diversification. Under (3.1), the return on a portfolio p with N stocks is and its risk is r p = a p + Var[r p ] = Var }{{} total risk K β pk F k +ε p k=1 [ K ] β pk F k k=1 } {{ } systematic risk + Var[ε p ] }{{} nonsystematic risk to get It can be shown that the second term goes to zero in a large, well-diversified portfolio (w j = 1/N): Var[ε p ] N 0 Hence, ε p 0 in a very large portfolio (in the limit), and r p = a p + K β pk F k 3. No arbitrage. If we can trade the factors to hedge the random part of r p, then it can be shown that there are no arbitrage opportunities only if ( K a p = 1 β pk )r f Then, the expected return on a well-diversified portfolio must be: E[r p ] = r f + when F k are traded portfolios. k=1 k=1 K β pk (E[F k ] r f ) (3.2) k=1 For an individual security (which is not a well-diversified portfolio), we only get an approximation. r j = E[r j ]+ K β jk ˆFk +ε j k=1 with ˆF k := F k E[F k ] and thus E[ˆF k ] = 0. Stock returns deviate from their means as a result of unexpected realizations of risk factors. We can further subtract r f from both sides to get K r j r f = E[r j r f ]+ β jk ˆFk +ε j which is the version presented in BKM. Note that this is just a mathematical manipulation of (3.1); it is still not saying anything about what variables explain E[r j ]. k=1

34 3.1. Theory 34 In summary, Definition 3.1.1: Arbitrage Pricing Theory (APT) If there are no arbitrage opportunities, the expected return on an individual security j is K E[r j ] r f + β jk FRP k (3.3) k=1 where: FRP k, the Factor Risk Premium, is: If factor k is a traded positive-investment portfolio, FRP k = E[F k ] r f = E[F k r f ] If factor k is a traded zero-investment long-short portfolio, FRP k = E[r long r short ] If factor k is not a traded portfolio, FRP k is a free parameter to be estimated. β jk is the loading of asset j on factor k. In practice, the loadings are estimated through a time-series regression of excess returns for asset j on the factors. Remarks: In practical applications we typically ignore the nuances and replace the with = The different alternatives for the FRP is advanced material not covered in BKM, you will not be tested on this, and we will only apply the tradedfactors version (Fama-French model below). 2 For empirical applications with traded factors, the following modification of (3.2) is more useful: 2 In any case, the intuition is the following. The general statement of APT, for any type of factor, is just that E[rp] e = K k=1 β pkfrp k, with FRP k 0, where rp e is the excess return on a well-diversified portfolio. But for traded factors, we can be more specific. If, for example, factor 1 is traded, we can put F 1 on the LHS of the equations above. The factor model would obviously have β 11 = 1 and β 1k = 0,k = 2,...,K. Then, the pricing equation with F 1 on the LHS would give E[F1 e] = FRP 1, thus pinning down the FRP for factor 1.

35 3.2. Example with 1 factor: the Market model 35 Proposition 3.1.1: Alternative alpha statement of the APT The risk premium on a well-diversified portfolio p is E[r p r f ] = α p + K β pk FRP k (3.4) If all factors are traded and there are no arbitrage opportunities, we must have α p = 0 k=1 Advantages and disadvantages of the APT: The main advantage of the APT is that it can accommodate several sources of systematic risk. Natural macroeconomic factor candidates are: interest rates, GDP growth, inflation, energy prices, etc. The main disadvantage is that the theory does not specify which are the true factors. It has been and empirical quest, with the current number of factors in the literature at over 300 and counting... The horse that is leading the race at this point is the FF Example with 1 factor: the Market model The Market Model states that there is just one factor: the market. The return generating process is: r j = a j +β j r M +ε j, j (3.5) If there are no arbitrage opportunities, expected returns are given by: E[r j ] = r f +β j (E[r M ] r f ), j which is the twin brother of the CAPM s SML. Example Market neutral strategy. This investment strategy is typical of many hedge funds. It is is from BKM (old 2005 ed, sec 10.4).

36 3.2. Example with 1 factor: the Market model 36 A portfolio manager has identified an underpriced portfolio p with the following characteristics: (r p r f ) = (r SP500 r f )+ǫ p Ifwetake expectations onbothsidesofthisequation, weseethat the APT in (3.4) says that we should not be seing this constant value of α p = Nevertheless, the manager is very confident about its alpha of 4%. However, even if the manager is right, he may loose money if the whole market turns down. He would like to explore the relative mispricing of p, regardless of what happens to the market. Solution: 1. Thesolution is to constructatracking portfolio (T) that matches the systematic component of p. (a) T must have a beta of 1.4, which requires w SP500 = 1.4 and w f = 0.4. (b) The return on the tracking portfolio is thus r T = 1.4r SP r f (r T r f ) = 1.4(r SP500 r f ) 2. The investment strategy is to go long (buy) on p and go short (sell) on T. 3. The combined portfolio C thus has a return of r C = r p r T = (r p r f ) (r T r f ) = 0.04+ǫ p This combined position is thus market neutral. Regardless of what happens to the market, the manager earns 4%. 3 Remark. This example also illustrates the arbitrage process that restores equilibrium. Investors will want to follow this strategy as much as possible. However, going long on p bids up its price, until alpha disappears. 3 Note that there is still some residual risk, ǫ p. This will be small if the single market factor explains r p well and p is a large portfolio.

37 3.3. Example with 3 factors: the Fama-French model Example with 3 factors: the Fama-French model Pricing equation Fama and French (1993) propose the following 3-factor asset pricing model: Definition 3.3.1: Fama and French 3-factor model The expected return on stock j is E[r j ] r f = β jm (E[r M ] r f )+β js E[SMB]+β jh E[HML] (3.6) wherethe loadings (β jm,β js,β jh ) are theslopes in the time-series regression (r j r f ) t = a j +β jm (r M r f ) t +β js SMB t +β jh HML t +ε jt (3.7) Details on the factors Toformthetwo newfactors, FFdivideall firmsintosixbuckets dependingontheir size (market equity, ME) and the ratio of book equity to market equity (BE/ME): 4 50th ME prct Small Value Big Value > 70th BE/ME prct Small Neutral Big Neutral Small Growth Big Growth < 30th BE/ME prct Small stocks have ME smaller than the median ME. Typically, small stocks perform better than what the CAPM predicts (this is a so called anomaly). Value stocks have BE/ME higher than the 70th BE/ME percentile; their book-to-market ratio is High. Growth stocks have BE/ME lower than the 30th BE/ME percentile; their book-to-market ratio is Low. Typically, BE/ME is high when the ME (denominator) is low. This happens when the firm has had low returns and is now near financial distress. Nonetheless, most of these firms usually rebound and thus, if you hold a large portfolio of these firms, you end up making more money than their CAPM beta would suggest (another CAPM anomaly). Each month, the factors are computed in the following way: 4 See the details at

38 3.4. Portfolio performance evaluation 38 SMB (Small Minus Big) is the average return on the three small portfolios minus the average return on the three big portfolios, SMB = 1/3 (Small Value + Small Neutral + Small Growth) - 1/3 (Big Value + Big Neutral + Big Growth) Historically, the SMB portfolio generated an annual return somewhere between 1.5% and 3%. This is the size premium. HML (High Minus Low) is the average return on the two value portfolios minus the average return on the two growth portfolios, HML = 1/2 (Small Value + Big Value) - 1/2 (Small Growth + Big Growth) Historically, the HML portfolio generated an annual return somewhere between 3.5% and 5%. This is the value premium. This model has had considerable empirical success in explaining CAPM anomalies (portfolios that don t plot on the SML) and in capturing the variation in the cross-section of expected returns. Thus, Fama and French (1996) argue that SMB and HML mimic combinations of two underlying risk factors of special concern to investors. 3.4 Portfolio performance evaluation Motivation 1. One important question in finance is: How to assess the performance of a fund manager? 2. We cannot just look at raw realized returns because we want to distinguish stock-picking skills from simple risk taking. If we see a big return, was it because the manager was able to identify mispriced stocks or was it because he took large risks and got lucky? 3. Therefore, we need to compute risk adjusted returns, that is, we need to measure the difference between the empirical realized returns and the returns appropriate for the risk of the fund Adjusting returns for risk The most widely used performance measure is Jensen s alpha. It is the empirical counterpart to the APT equation (3.4).

39 3.4. Portfolio performance evaluation 39 Definition 3.4.1: Jensen s alpha Jensen s α for fund p is the intercept in the time-series regression K (r p r f ) t = α p + β pk Fkt e +ε pt, t = 1,...,T k=1 where F e kt is the excess return on the traded factor k at time t: If factor k is a traded positive-investment portfolio, F e kt = (F k r f ) t If factor k is a traded zero-investment long-short portfolio, F e kt = (r long r short ) t We now have two models to adjust returns for risk. CAPM To evaluate the performance of fund p, estimate the following time-series regression: (r p r f ) t = α p +β p (r M r f ) t +ε pt (3.8) According to the CAPM, or the 1-factor Market Model APT, we should find α p = 0. If instead we find that α p is (statistically significantly) positive, we can conclude that the fund returns are higher than what its level of risk would require (according to the CAPM). In other words, the manager has skill. Graphically, a positive Jensen s alpha implies that the portfolio lies above the SML: E[r] β

40 3.4. Portfolio performance evaluation 40 Remark. Model (3.8) is the standard regression to estimate the CAPM beta. BKM call this the single-index model. The market model equation (3.5) is sometimes also used in the industry to estimate the CAPM beta, but this is only equivalent to (3.8) when the interest rate is constant (which is not the case in reality). FF3 If we don t believe that CAPM is a good model to adjust returns for risk, we can use the Fama-French model. Run the regression (r p r f ) t = α p +β pm (r M r f ) t +β ps SMB t +β ph HML t +ε pt Again, if α p > 0 (statistically), the manager has skill. Note that theβ pm estimator that comes out of this regression is not thecapm beta (due to the presence of other regressors).

41 Chapter 4 Bond Markets Based on chapters 5, of Bodie, Kane, and Marcus (2014). 4.1 Rates of return Returns are usually expressed in annual terms. There are 3 alternative representations of the same underlying Holding Period Return (HPR, as defined in equation 1.1) Effective Annual Rate Definition 4.1.1: Effective Annual Rate The Effective or Equivalent Annual Rate (EAR) corresponding to an investment with a given HPR over T years is EAR : 1+EAR = (1+HPR) n where n = 1/T is the number of compounding periods per year Example A 1-month T-bill (zero coupon bond) is trading at 99.75% of par value. Check that: HPR(T = 1/12) =... = % 41

42 4.1. Rates of return 42 and EAR =... = % Annual Percentage Rate Definition 4.1.2: Annual Percentage Rate The Annual Percentage Rate (APR) with compounding frequency n, denoted r n, corresponding to an investment with a given HPR over T < 1 years is r n = HPR n where n = 1/T is the number of compounding periods per year Example Continuing the previous example, check that the APR with monthly compounding is r 12 =... = % Remarks: A complete specification of an APR always needs to state n, unless it is clear from the security specifications. When n = 1, the APR is also the EAR: r 1 = EAR Continuous compounding Definition 4.1.3: Annual rate with continuous compounding The annual rate with continuous compounding, denoted r, corresponding to an investment with a given HPR over T years is a e r = 1+EAR r = ln(1+ear) = 1 T ln(1+hpr) a Recall that lim n (1+ y n )n = e y Example r =... = % Continuing the previous example, check that

43 4.2. Bond Prices and Yields Discounting Proposition 4.1.1: Discounting The value at time t of a cash flow C(T) to be received in T years is P(t) = [ C(T) 1+ rn(t,t) n ] n (T t), discrete compnd. (n < ); C(T) e r (t,t) (T t) = C(T)e r (t,t) (T t), cts. compnd. (n = ) where r n (t,t) is the interest rate between t and T, with compounding frequency n. Example Do the previous examples in reverse, i.e., start from the interest rates and compute the T-bill price. Example Continuing the previous examples, assume that in the future you will be able to reinvest at the same T-bill rate that is available now. How much money do you need to invest today to have $ in 3 months? Answer: Bond Prices and Yields Bond Pricing Price units Bond prices are typically expressed as a percentage of face value. To understand its meaning, note that: Total Investment (in $) = Price (in % of FV) Face Value (in $) Think of FV as a quantity.

44 4.2. Bond Prices and Yields 44 Example A Bond with 5% annual coupons is priced at %. This means that: If we invest 1 M$, we are able to buy a face value ( quantity ) of FV = 1M$/ =...$ Our next coupon will be C =...$ At maturity, we will receive a total of $1,028, Remarks: Interpret the symbol % as meaning 0.01 to avoid messing up the units in your calculations. BKM still show examples of the old days when bonds had $1000 or $100 denominations. Price of fixed-coupon bonds Definition 4.2.1: Price of fixed-coupon bond where P = m i=1 c/n [1+r(0,t i )] t + 100% (4.1) i [1+r(0,t m )] tm P is price of the bond today n is the number of coupon payments per year c is the annual coupon rate with compounding frequency n t 1,t 2,...,t m = T are the years until the coupon payment dates (with T denoting the maturity) r(0,t i ) is the discount rate or required return from today (t = 0) until t i years, expressed as an Equivalent Annual Rate (EAR) Important special case: zero-coupon bonds (aka zeroes, or pure discount bonds).

45 4.2. Bond Prices and Yields 45 Proposition 4.2.1: Price of a Zero-Coupon Bond (ZCB) The price of a ZCB maturing in T years is P = 100% [1+r(0,T)] T Quoting conventions 1. The Quoted (or clean, or flat) price quoted by a trader does not include interest that accrued since the last coupon date. 2. The Invoice (or dirty, or full, or cash) price is the price that the buyer actually has to pay the seller: Invoice price = Quoted price + Accrued interest This is the price given by (4.1). 3. The accrued interest (AI) is: AI = Day-count conventions: N. days since last coupon Interest due in full period N. days between coupons actual / actual(in period) for T-Notes and T-Bonds. actual/360 used in T-Bills and money markets. 30/360 used in corporate bonds. Example Consider a 15/Jul/2020 4% Bond, paying semi-annual coupons. The settlement date is 10/April/2012 (note that 2012 is a leap year). With actual/actual day count, With 30/360 day count, AI = = % of FV 182 AI = = % of FV 180 In Excel, use the functions COUPDAYBS() and COUPDAYS() to count days.

46 4.2. Bond Prices and Yields Yield to Maturity Definition 4.2.2: Yield to Maturity Given the price of a fixed-coupon bond, the YTM (y, as an EAR) is the constant discount rate that solves y : P = m i=1 c/n [1+y] t + 100% (4.2) i [1+y] tm Example A one-month T-Bill sells for % (of par value). Check that the one-month yield is y = 4% (EAR). Plot Price versus Yield and note the inverse relation. Remarks: In general, we need Excel (YIELD or IRR function, or Solver) or a financial calculator to compute the YTM of a bond with multiple coupons. YTM are sometimes quoted as APR, aka bond equivalent yields. Example Price = 95%, 10-yr Maturity, Coupon rate = 7% with semiannual coupons. The IRR() fn in Excel outputs %, which is a 6-month rate. Then, we would express the YTM in one of the following ways: Bond Equivalent Yield (APR). Effective Annual Yield (EAR). y APR 2 = 3.86% 2 = 7.72% y EAR 1 = (1.0386) 2 1 = 7.88% Traders also talk about a Current Yield = Annual interest / market price. For the previous example, y CY = 7/95 = 7.37%.

47 4.2. Bond Prices and Yields 47 Interpretation. The YTM will be the realized yield if: you hold the bond until maturity; and all coupons are reinvested at the ytm. Example year 8% bond with semiannual coupons, trading at a ytm of y 2 = 8% and a price of 100%. Check that under the two conditions specified above, the realized yield at the end of 2 years will be r 2 = 8%. Remark. From the previous example, note that at the ex-coupon date, Y T M = Coupon rate P = 100%. Check that YTM > Coupon rate P < 100% and YTM < Coupon rate P > 100% Holding-Period Return The Realized Yield, or Holding-Period Return, or Realized Compound Return depends on: Selling price of the bond; Reinvestment rate for the coupons. Example year 8% bond with semiannual coupons, trading at a ytm of y 2 = 8% and a price of 100%. In six months the ytm falls to 7%. 1. Check that P 6m = 101.4% 2. Check that the HPR for 6 months is r 2 = % (APR). Example Consider a 2-year, 10% bond with annual coupons, trading at a ytm of 10%. You hold the bond until maturity, but are only able to reinvest the coupons at 8%. Check that the HPR is 9.9% (EAR).

48 4.3. Term structure of interest rates Default risk and ratings Corporate bonds trade at an higher yield than risk-free government bonds because corporations may default. Default risk is measured by a credit rating. Long Term Obligation Ratings S&P Moody s Meaning AAA Aaa Highest quality, minimal credit risk AA Aa A A BBB Baa Adequate protection, moderate credit risk BB Ba Speculative, significant risk B B CCC Caa CC Ca C C Nonpayment highly likely; in default (Moody s) D in Default Obligors rated BBB or better are called investment-grade ; investors rated BB or worse are called speculative-grade. Each category can also be appended witha+or-sign(s&p)orwith1,2,3numbers(moody s) toshowrelativestanding within the category Exercises Ex. 8 Bodie, Kane, and Marcus (2014) problems at the end of chapter 14 (p. 480): 6, Term structure of interest rates Spot rates Definition 4.3.1: Spot rate The spot rate r(0,t) is the yield to maturity on a zero-coupon bond with maturity T.

49 4.3. Term structure of interest rates 49 Definition 4.3.2: Term structure of interest rates The term structure of interest rates, or zero yield curve, or simply yield curve, is the set of spot rates r(0,t) for different maturities T. Example The following Zero-Coupon bonds are trading in the market: Time to Maturity Price (%) YTM % % % Hence, the term structure of interest rates is: r(0,1) = 5% r(0,2) = 6% r(0,3) = 7%. The spot rates are related to the price of coupon-bearing bonds through (4.1). Example Consider a bond with 5% annual coupons and 2 years to maturity. The total value must be the sum of the present values of all cash flows. Using the spot rates from the previous example: P =... = % Remarks: The term yield curve is also used to denote the set of YTM. In addition to the spot rate curve defined above, we can also have forward rate curves (using the fwd rates defined below) Forward rates A forward rate applies to a time period in the future. Definition 4.3.3: Forward rate Given a set of spot rates, the forward rate between t 1 and t 2 (0 < t 1 < t 2 ), expressed as an EAR, is given by f(t 1,t 2 ) : [1+r(0,t 1 )] t 1 [1+f(t 1,t 2 )] (t 2 t 1 ) = [1+r(0,t 2 )] t 2

50 4.3. Term structure of interest rates 50 Interpretation. The fwd(1,2) is the rate of return on a future investment that will make me indifferent between the following alternatives: 1. Invest in a 2-year zero coupon bond. 2. Invest in a 1-year zero coupon bond. After 1 year reinvest the proceeds in another 1-year bond. Example Continuing the previous example, check that f(1, 2) = % and f(2,3) = % Expectations of future interest rates Term structure under certainty If all investors know for sure the path of future interest rates (i.e, there is no risk), then: 1. The forward rate computed today is the value that the spot rate will take on in the future. 2. All bonds provide the same return over any given holding period. Example Consider the following bonds(all have annual coupons): Bond A B C Maturity (yrs) Coupon rate 4% 6% Price (%) Compute the spot rates (this procedure is called Bootstrap) 2. Compute the sequence of 1-yr forward rates. 3. Check that the YTM for bond A, B and C are respectively 10%, 10.98%, and 11.91%. 4. Plot the YTM curve, the spot-rate curve, and the forward-rate curve. 5. Check that if we invest for 1 year, all bonds produce the same return (recall that under certainty we know the future value of interest rates).

51 4.3. Term structure of interest rates 51 Uncertainty and expectations In reality we don t know the future value of interest rates. What do forward rates tell us about future spot rates? Expectations Hypothesis The forward rate equals the market consensus expectation of the future spot interest rate. That is, forward 0 (t 1,t 2 ) = E 0 [spot t1 (t 1,t 2 )] Liquidity Preference Hypothesis Most investors are short-term and therefore will only hold long-term bonds if they receive a premium. Thus, the forward rate exceeds the expected spot rate by a liquidity premium: forward 0 (t 1,t 2 ) = E 0 [spot t1 (t 1,t 2 )]+ liquidity premium Implications: 1. Expectations of increases in future interest rates must always result in a rising yield curve. 2. However, the converse is not necessarily true. Even if the yield, spot, and forward curves are all rising, the market may be expecting constant or even decreasing future interest rates (why? liquidity premium). 3. In practice, interest rates are very hard to predict... Relation to the business cycle The slope of the yield curve is typically a good leading indicator of the business cycle: An upward sloping curve usually forecasts good times ahead A flat or downward sloping curve usually forecasts bad times ahead For example, the ECB estimates the yield curve for the euro area every day. Check

52 4.4. Bond management Exercises Ex. 9 Bodie, Kane, and Marcus (2014) problems at the end of chapter 15 (p. 507): 11, Bond management Duration How to measure the sensitivity of the bond price to changes in interest rates, i.e., the risk of a bond? Definition 4.4.1: Macaulay Duration The Macaulay Duration is where y is the YTM (EAR) D := 1 P m i=1 t i C ti (1+y) t i (4.3) t i are the coupon payment dates (in years) C ti is the total cash flow at date t i : C ti = c/n for i < m, and C ti = c/n+1 for i = m, where n is the coupon payment frequency. Example Consider a 3-year 5% bond paying annual coupons. The bond is trading at y = 4%. Check that P = and D =... = Proposition 4.4.1: Price sensitivity to interest rate changes For a small change in YTM ( y), the bond price change is well approximated by P P = D 1 y (4.4) 1+y

53 4.4. Bond management 53 Proof. The change in price due to a change in interest rates is P P(y + y) P(y). If the change in interest rates is small and instantaneous (i.e., time is standing still), the change in price can be approximated by taking a Taylor series expansion of P around y: P(y + y) = P(y)+ dp dy y +1/2 d2 P dy 2 ( y) Taking derivatives and dividing through by P, we get P P = D 1 1+y y C( y) where D is as defined in (4.3) and C, denoted Convexity, is C := m i=1 t i(t i +1) P C ti (1+y) t i 1 (1+y) 2 Ignoring term of order higher than y produces the approximation in (4.4). Meaning of Duration. Bonds with higher Duration are more sensitive to interest rate changes. Hence, Duration is a measure of the riskiness of a given bond. Example Continuing the previous example, assume that YTM increases by 20 basis points. 1. The proportional price change for the previous bond is P P =... = % 2. Now consider a 3-year ZCB also trading at an initial YTM of 4%. Since D =... the proportional price change in this bond will be P P =... = % Remarks: The Modified Duration is D := D/(1 + y). Using D instead of D, (4.4) becomes P P = D y.

54 4.4. Bond management 54 From (4.3), we can also think of duration as a weighted average of time, i.e., D = m t i w ti t=i where the weights are w ti = Ct i /P. Thus, the units of duration are (1+y) t i years. In this sense duration measures how fast a bond generates cash flows. For the same maturity, a bond with higher coupons will have lower duration. This second interpretation of duration is easier to understand, but it is nearly worthless for practical applications Active bond management If a manager is able to forecast unanticipated interest rate movements better than the rest of the market, then equation (4.4) suggests an investment rule: Active bond management with Duration Expect interest rates to decrease Increase the portfolio duration Expect interest rates to increase Decrease the portfolio duration This rule implies that we readjust the bond portfolio, before interest rates change, according to the following formula: Proposition 4.4.2: Macaulay Duration of a portfolio If the term structure is flat, i.e., all bonds have the same YTM, then the Macaulay duration of a bond portfolio is: D p = N D i w i (4.5) i=1 where N is the number of bonds in the portfolio w i = V i /V p istheweight inbondi, wherev i istheamount($) invested in bond i and V p is the total value of the portfolio, V p = N i=1 V i. Caveat:

55 4.4. Bond management Strictly speaking, this procedure is only correct under the following assumptions: (a) the term structure is flat (all bonds have the same YTM); (b) the change in yield is small and instantaneous. 2. Nevertheless, (4.4) is still a good guide for bond management in most real-life cases Exercises Ex. 10 Bodie, Kane, and Marcus (2014) problems at the end of chapter 16 (p. 547): 3, 4.

56 Chapter 5 The efficient market hypothesis Based on chapter 11 of Bodie, Kane, and Marcus (2014) 5.1 Motivation Do security prices reflect the available information? Important implications for Investment decisions Efficient resource allocation 5.2 Random walks and the EMH EMH statement: 1. The Efficient Market Hypothesis (EMH) is that stock prices reflect all the available information. 2. If EFM is true, then prices change only in response to new information, which is by definition unpredictable 3. The mathematical model for random price changes is a random walk. 1 1 To be precise, investors expect to be rewarded for risk, so the expected price change is positive over time (this is called a submartingale). Hence, the common model for prices is a positive trend with random fluctuations about the trend. 56

57 5.3. Implications of the EMH 57 EMH and Competition: 1. Once information becomes available, market participants quickly analyze it to trade on it. 2. Hence, competition ensures prices reflect information 3. However, there is a Catch-22: Stock analysis ensures markets are efficient Stock analysts must be compensated for their work, which can only happen if markets are not efficient Equilibrium? Benefit = Cost Versions of the EMH 1. Weak-form. Prices already reflect all information in past trading data (stock prices and volumes) 2. Semistrong-form. Prices already reflect all public information (past prices, plus accounting, macro, media, etc, information) 3. Strong-from. Prices already reflect ALL information public and private related to the firm. Example (This is concept check 11.1 in BKM) 1. Suppose high-level managers make superior returns on investments in their company s stock. Would this be a violation of weak-form market efficiency? Would it be a violation of strongform market efficiency? 2. If the weak form of the EMH is valid, must the strong form also hold? Conversely, does strong-form efficiency imply weak-form efficiency? 5.3 Implications of the EMH Some implications of the EMH: Weak-form efficiency implies that Technical analysis is worthless

58 5.4. Are markets efficient? 58 Semistrong-form efficiency implies that Fundamental analysis is worthless Semistrong-form efficiency implies that the best investment strategy is Passive Management (buy and hold an index fund), rather than Active Management (stock picking). Example (This is concept check 11.3 in BKM). What would happen to market efficiency if all investors attempted to follow a passive strategy? 5.4 Are markets efficient? Evidence on Mutual Fund Performance is mixed Some evidence of persistent positive (superstar or hot-hand managers) and negative performance (due to too much trading). But most managers are not consistent. How to correctly adjust returns for risk? Some apparent anomalies may represent poorly understood risk premia. Case in point: rise and fall of LTCM in late 1990s. Conclusion: Don t try this at home! Individual investors (with other day jobs) should not engage in stock picking. Its better to buy a diversified portfolio. Professional analysts are probably making an extra return at least equal to the cost of gathering information. Caveat. Two economists are walking down the street. They spot a $20 bill on the sidewalk. One starts to pick it up, but the other one says, Don t bother; if the bill were real someone would have picked it up already.

59 5.5. Exercises Exercises Ex. 11 Bodie, Kane, and Marcus (2014) problems at the end of chapter 11 (p. 381): 1, 2, 5, 9, 12. Ex. 12 Bodie, Kane, and Marcus (2014) CFA problems at the end of chapter 11 (p. 384): 1, 2, 3, 4.

60 Chapter 6 Futures markets Based on chapter 22 of Bodie, Kane, and Marcus (2014). 6.1 Definition Forward contract Definition 6.1.1: Forward contract A Forward contract is an OTC agreement between two counterparties whereby: 1. One counterparty ( long forward position ) agrees to buy N units of a given asset, at a future date T, for a forward price F(0,T) defined today 2. The other counterparty ( short forward position ) agrees to sell the asset under the same conditions. Payments: 1. At inception there is no payment. 2. At the settlement date T, there are two alternatives according to what the contract specifies: 60

61 6.1. Definition 61 Physical settlement. The short delivers the physical asset and the long pays the price initially agreed upon, F(0,T). Cash settlement. The payoff to the long forward counterparty is CF to long forward at T = N [S(T) F(0,T)] (6.1) where S(T) is the spot price of the underlying asset at the settlement date T. The payoff to the short is the symmetric of (6.1). In either case, the effective price paid for the asset at time T is F(0,T), rather than S(T). Example Consider a Treasury Bill that matures in 9 months. A bank quotes a 6-month forward contract on this ZCB at F(0,0.5) = , with cash settlement. 1. Afirm buystheforward or buysthebondforward, i.e., takes the long position in the forward. No payments exchange hands today. 2. Suppose that 6 months from now, r (0,0.25) = 3%. (a) The forward payoff to the firm is CF to long forward at T =... = N (b) The effective price paid for the bond is Eff Price = N [S(0.5) CF from forward] = N... (c) This guarantees that the firm can invest at an effective rate of r (0,0.25) = 4% (check this), rather than at the r (0,0.25) = 3% currently available in the market. Draw the Forward contract profit profile at maturity for each position: Profit F(0,T) S(T)

62 6.1. Definition Futures contract Definition 6.1.2: Futures contract A Futures contract is an Exchange-traded contract between two counterparties whereby: 1. One counterparty ( long futures position ) agrees to buy N units of an underlying asset, at a future date T, for a future price F(0,T) defined today 2. The other counterparty ( short futures position ) agrees to sell the asset under the same conditions. Figure 6.1 lists the most common futures contracts. Figure 6.1: List of typical futures contracts (from BKM) One of the most important Futures exchanges is Remarks:

63 6.1. Definition 63 Settlement type. Some contracts are only cash settled (eg, stock index futures), whereas others permit physical delivery (eg, crude oil). However, in practice most contracts are closed out, or reversed, before maturity, so that actual physical delivery only happens in a small fraction of contracts (1% 3%). Convergence. The futures price and the spot price must converge at maturity; otherwise, there is an arbitrage opportunity. In other words, the basis must go to zero: 1 [F(t,T) S(t)] t T Differences between Futures and Forwards Futures and forward contracts are very similar. However, there are some important differences: 1. Futures are exchange-traded. In a Futures, the counterparty to every trader is the exchange clearinghouse. 2. Futures are standardized. We can only trade the contracts for the underlying securities, maturities, and quantities that the exchange specifies. This makes futures more liquid than forward contracts, but less adaptable to specific needs of traders. 3. Futures are marked-to-market daily. Profits or losses accrue to traders with daily frequency. If we buy a futures contract today (i.e., enter into the long position), our cash flow tomorrow is: CF to long futures at day 1 = N [F(1 day,t) F(0,T)] This mark-to-market is repeated every day until we close the position. If we hold the futures until maturity, CF to long position over life of the futures = N [F(1 day,t) F(0,T) +F(2 days,t) F(1 day,t) F(T,T) F(T 1 day,t)] 1 The basis can be defined as either F S or S F. = N [S(T) F(0,T)]

64 6.2. Trading strategies 64 The last line uses the fact that the futures price must converge to the underlying security price at the futures maturity date: F(T,T) = S(T). Hence, the total payoff is similar to a forward contract (equation 6.1), despite the timing of the cash flows being different. 4. Margins. Futures traders need to post an initial margin with the exchange (5% 15% of the total value of the contract) and must replenish the account whenever the daily losses drive the balance below a given maintenance margin. Margins and daily mark-to-market virtually eliminate the counterparty credit risk in futures contracts. 6.2 Trading strategies Speculation and leverage Speculation trading rule Expect futures price to increase Long futures Expect futures price to decrease Short futures Example (Example 22.3 in BKM) Crude oil futures are trading at $/bbl. Each contract is on 1000 barrels. You believe that oil prices are going to increase. 1. Trading strategy(today): (long/short) futures 2. Result (some days latter): suppose that the futures price increases by $2 and you decide to close the position. Profit =... = 2000 $/contract Trading in futures allows for much higher leverage than trading in the underlying spot asset. Example example, (Example 22.4 in BKM) Continuing the previous

65 6.2. Trading strategies 65 Suppose the initial margin for the oil contract is 10%. 1. Initial margin =... = 9186 $/contract 2. The $2 increase represents a percentage gain of Return on posted margin = $/ctt = 21.77% If instead we had invested in spot oil, the return for the same $2 increase would have been only Return on spot oil = $/bbl = % A 10-to-1 ratio! Hedging Hedging trading rule To hedge an underlying cash exposure, take a position in futures such that the gains/losses in the futures offset the losses/gains in the cash position. Short spot position (ie, will purchase later) or investment sometime in the future (ie, loose if price increases) Long futures Long spot position (ie, will sell later) or borrowing sometime in the future (ie, loose if price decreases) Short futures Example (Example 22.5 in BKM) Consider an oil producer planning to sell barrels of oil in February that wishes to hedge against a possible decline in oil prices. February futures are trading at $/barrel. Each contract is on 1000 barrels. 1. To hedge its production, the company should (buy/sell) (number) contracts. 2. Whatever the spot oil price in February, the profit/loss in the futures exactly offsets the change in the cash position, such that the effective sale price is $/barrel. Figure 6.2 illustrates this.

66 6.2. Trading strategies 66 Figure 6.2: Hedging with futures (Fig 22.4 in BKM) Spread trading Definition: a spread trade consists of a long position in one futures contract and a short position in another futures contract. Goal: Speculate or hedge a possible price divergence/convergence Typical spreads: Different but related commodities Different delivery months of the same commodity Advantages: Only relative price changes matter, i.e., do not have to guess direction of the market Margin requirements are lower because spreads are less volatile than absolute levels. Hence, can get more leverage.