Introduction to Investments FINAN 3050

Transcription

1 Introduction to Investments FINAN 3050 Week 7: Managing Bond Portfolios (Chp )

2 YTM vs. Spot Rates Definition of Yield to Maturity (YTM): the yield (constant across time) that gives the observed price; denoted by r (without any subscript) P B = T C t + FV t ( 1+ r) ( r) t= 1 1+ T Definition of spot rates: the spot rate s t is the rate of interest for money held from the present time (0) until time t (measured in years); YTM of zerocoupon bonds P B = T C t + t ( 1+ s ) ( s ) t= 1 t 1+ FV T T Slide 2 Week 7

3 YTM vs. Spot Rates: Example Consider a 3-year bond with a coupon rate (annual) of 7%. Determine the price of the bond if the YTM is 5.48%: 7 2 ( ) ( ) ( ) Determine the price of the bond if the 1-year spot rate s 1 =5%, the 2-year spot rate s 2 =5.2% and the 3-year spot rate s 3 =5.5% = ( ) ( ) ( ) = Slide 3 Week 7

4 Extracting Spot Rates from Bond Prices Assume the following information: There is a 1-year zero-coupon bond trading for $98. There is a 2-year, 4%-coupon (annual) bond trading for $102. Determine the 1-year and 2-year spot rate (i.e., s 1 and s 2 ). Slide 4 Week 7

5 Interest Rate Risk: Bond Pricing Relationships (1) Inverse relationship between price and yield (in the following we refer to the yield to maturity) An increase in a bond s yield to maturity results in a smaller price decline than the price gain associated with a decrease in yield relationship between price an yield is non-linear (convex) The price of long-term bonds are more sensitive to yield changes than prices of short-term bonds Price sensitivity is inversely related to a bond s coupon rate: higher coupon rate means lower price sensitivity Price sensitivity is inversely related to the yield to maturity at which the bond is currently selling Slide 5 Week 7

6 Interest Rate Risk: Bond Pricing Relationships (2) Slide 6 Week 7

7 Interest Rate Risk: Duration (1) A measure of the effective maturity of a bond The weighted average of the times until each payment is received, with the weights proportional to the present value of the payment Duration is shorter than maturity for all bonds except zero coupon bonds Duration is equal to maturity for zero coupon bonds A measure of a bond s sensitivity to interest rate risk: higher duration means higher sensitivity to interest rate risk Slide 7 Week 7

8 Interest Rate Risk: Duration (2) Slide 8 Week 7

9 Interest Rate Risk: Duration (3) Calculation: D Duration T Maturity of the bond r...yield to maturity C t Coupon in period t in $ FV Face Value in $ D T = t= 1 t C t FV t ( 1+ r) ( 1+ r) + T Bond Price T Bond Price Slide 9 Week 7

10 Interest Rate Risk: Duration - Example Calculation of the duration for a 3-year 8% coupon bond. The YTM is 10%. 8% Bond Time years Payment PV of CF (10%) Weight C1 X C Sum Slide 10 Week 7

11 Interest Rate Risk: Duration - Exercise Calculate the price and the duration for a 3-year zero-coupon bond. The YTM is 10%. Calculate the price and the duration for a 3-year 8% coupon bond. The YTM is 9%. Calculate the price and the duration for a 3-year zero-coupon bond. The YTM is 9%. Slide 11 Week 7

12 Interest Rate Risk: Duration Price Relationship The duration of a bond can be used to approximate its price change if interest rates change. Calculation: P Change in bond price (i.e., New Price minus Old Price) P/P Relative change in bond price r Change in YTM P P = D r ( + r) 1 ( + r) Modified Duration P= D Modified Duration r P Slide 12 Week 7

13 Interest Rate Risk: Duration Price Relationship - Example Consider a bond with maturity of 30 years, coupon rate of 8% (paid annually) and a YTM of 9%. Its price is $ and its duration is years. Question: What will happen to the bond price if the bond s yield to maturity increases to 9.1%? Calculate the change in YTM: r=9.1%-9%=0.1% Change in price equals -$9.36 P= ( ) $ = $9.36 Slide 13 Week 7

14 Interest Rate Risk: Duration Price Relationship - Exercise Consider a 3-year bond with a coupon rate of 8% and a duration of The bond has a price of $ and a YTM of 10%. Consider an increase in YTM by one basis point (i.e., by or 0.1%) to 10.1% Calculate the resulting percentage change in the bond price Approximate the percentage change in the bond price using the bond s duration Repeat the exercise from before for the following situations: A decrease in YTM by one basis point An increase in YTM by 1% A decrease in YTM by 1% How precise is the approximation using the duration equation in each case? Slide 14 Week 7

15 Interest Rate Risk: Some simple duration rules Rule 1: The duration of a zero-coupon bond equals its maturity. Rule 2: Holding time to maturity and YTM constant, a bond s duration is higher when the coupon rate is lower. Rule 3: Holding the coupon rate constant, a bond s duration increases with time to maturity. Slide 15 Week 7

16 Application of Duration: Passive Bond Management Passive managers take bond prices as fairly set and seek to control only the risk of their fixed-income portfolios. Immunization: strategies that investors use to shield their fixed-income portfolios from exposure to interest rate fluctuations especially important for banks, insurance companies and pension funds that have assets and liabilities, i.e., fixed-income products with different characteristics and different sensitivities to interest rate on both sides of the balance sheets (inflows and outflows) in this case, immunization means to match the duration of assets and liabilities Slide 16 Week 7

17 Application of Duration: Immunization Example (1) Consider the following problem: an insurance company faces an obligation (liability) of $19,487 in seven years; the market interest rate equals 10% p.a. How can you immunize this liability using a combination of 3-year zero coupon bonds and a 15-year bond with coupon rate of 2.7%? Step 1: calculate the duration of the liability Step 2: calculate the duration of the asset portfolio Step 3: find the combination of bonds on the asset side such that the duration of liabilities matches the duration of the assets Step 4: Fully fund the obligation Slide 17 Week 7

18 Application of Duration: Immunization Example (2) Step 1: calculate the duration of the liability Duration of the liability equals 7. Step 2: calculate the duration of the asset portfolio Duration of 3-year zero-coupon equals 3 years Duration of coupon-paying bond equals 11 years Portfolio duration is the weighted average of duration of each component asset w fraction of the portfolio invested in the zero Asset duration = w 3years + (1-w) 11years Slide 18 Week 7

19 Application of Duration: Immunization Example (3) Step 3: find the combination of bonds on the asset side such that the duration of liabilities matches the duration of the assets Asset duration = Liability Duration w 3years + (1-w) 11years = 7 years w=50% Step 4: Fully fund the obligation Present value of the liabilities equals $10,000 = 19,478/(1+0.1) 7 We have to invest $10,000 into the asset side $5,000 into the 3-year zero-coupon bond (we buy 66.5 of these bonds with a face value of $100 each at a price of $75.13 each) $5,000 into the coupon bond (we buy of these bonds with a face value of $100 each at a price of $44.48 each) Slide 19 Week 7

20 Application of Duration: Immunization Example (4) Check if immunization works: consider a drop in interest rates to 9% on the next day (i.e., there are still 7 years left until liability matures) Present value of liability: $10, = $19,478/(1+0.09) 7 Present value of asset portfolio: Zero-coupon bond: P (FV of 100) = $77.2 Coupon-paying bond: P (FV of 100) = $49.2 Portfolio Value = $ $ = $10, Gap of -$10, $10,671.89=$11.4 This strategy ensures that the average duration of assets and liabilities is the same if interest rates change in the short run, both liabilities and assets change to the same extent and the assets will be sufficient to cover the liabilities Slide 20 Week 7

21 Application of Duration: Immunization Example (5) Check if immunization works: consider a drop in interest rates to 9% in one year (i.e., there are only 6 years left until liability matures) Present value of liability: $11,619.5 = $19,478/(1+0.09) 6 Present value of asset portfolio: Zero-coupon bond: P (FV of 100; 2 years left until mat.) = $84.2 Coupon-paying bond: P (FV of 100; 14 years left until mat.) = $50.95 Portfolio Value = $ $ = $11, Gap of -$11,619.5+$11,328.83=-$ This strategy requires rebalancing, as over one year durations of liabilities and assets change present values of liabilities and assets change Slide 21 Week 7

22 Interest Rate Risk: Convexity (1) Remember the non-linear (convex) relationship between yields and bond prices. Approximating bond price changes by duration assumes a linear relationship Convexity considers curvature. Do investor like or dislike convexity? Slide 22 Week 7

23 Interest Rate Risk: Convexity (2) Remember the approximation of bond price changes using duration (it is linear in r ): P P D = r ( + r) 1 23 Modified Duration The same relationship looks as follows if convexity is considered: P P D = r ( 1+ r) 123 Modified Duration CX ( r) 2 Slide 23 Week 7

24 Interest Rate Risk: Calculating Convexity In the spirit of the duration equation similar structure. CX Ct FV T t t 1 ( ) ( 1+ r) ( ) ( 1+ r) = ( + ) t t 1 T T r t= 1 Bond Price BondPrice Example: convexity for a 3-year 8% coupon bond. The YTM is 10%. 8% Bond Time years Payment PV of CF (10%) Weight t (t+1) C Sum /1.1 2 =8.94 Slide 24 Week 7

25 Interest Rate Risk: Convexity - Exercise Consider a 3-year bond with a coupon rate of 8%, a duration of and a convexity of The bond has a price of $ and a YTM of 10%. Consider an increase in YTM by one basis point (i.e., by or 0.1%) to 10.1% Approximate the percentage change in the bond price using the bond s duration AND convexity Repeat the exercise from before for the following situations: A decrease in YTM by one basis point An increase in YTM by 1% A decrease in YTM by 1% Compare your results to the ones from the exercise on slide 14 (approximation using only duration). Slide 25 Week 7

26 Interest Rate Risk: Duration Only and Duration-Convexity Approximations Summary of Results This table summarizes results related to the exercises on slide 14 and slide 25. (Comment: it would be a good exercise to replicate all the results in detail) Duration Appr. Change Rel. Dev. from Approx. D-CX Appr. Chan. Rel. Dev. from Approx. New Change Yield Old Price Relative New Price Change % % % % % % % % Slide 26 Week 7

27 Hmk 1a 1. You purchased a share of stock for $20. One year later you received $1 as dividend and sold the share for $24. Your holding-period return was. 2. An investor invests 40% of his wealth in a risky asset with an expected rate of return of 15% and a variance of 4% and 60% in a treasury bill that pays 6%. Her portfolio's expected rate of return and standard deviation are and 3. Consider the following two investment alternatives. First, a risky portfolio that pays 15% rate of return with a probability of 60% or 5% with a probability of 40%. Second, a treasury bill that pays 6%. The risk premium on the risky investment is. 4. You have $500,000 available to invest. The risk-free rate as well as your borrowing rate is 8%. The return on the risky portfolio is 16%. If you wish to earn a 22% return, you should. 5. Risk that can be eliminated through diversification is called risk 6. Asset A has an expected return of 20% and a standard deviation of 25%. The risk free rate is 10%. What is the reward-to-variability ratio? 7. A portfolio is composed of two stocks, A and B. Stock A has a standard deviation of return of 25% while stock B has a standard deviation of return of 5%. Stock A comprises 20% of the portfolio while stock B comprises 80% of the portfolio. If the variance of return on the portfolio is.0050, the correlation coefficient between the returns on A and B is. 8. The standard deviation of return on investment A is.10 while the standard deviation of return on investment B is.05. If the covariance of returns on A and B is.0030, the correlation coefficient between the returns on A and B is. 9. A portfolio is composed of two stocks, A and B. Stock A has a standard deviation of return of 5% while stock B has a standard deviation of return of 15%. The correlation coefficient between the returns on A and B is.5. Stock A comprises 40% of the portfolio while stock B comprises 60% of the portfolio. The variance of return on the portfolio is. 10. Stocks A, B, C and D have betas of 1.5, 0.4, 0.9 and 1.7 respectively. What is the beta of an equally weighted portfolio of A, B and C? 11. Consider the CAPM. The risk-free rate is 6% and the expected return on the market is 18%. What is the expected return on a stock with a beta of 1.3? 12. Consider the CAPM. The risk-free rate is 5% and the expected return on the market is 15%. What is the beta on a stock with an expected return of 12%? 13. Consider the CAPM. The expected return on the market is 18%. The expected return on a stock with a beta of 1.2 is 20%. What is the risk-free rate? 14. Consider the single factor APT. Portfolio A has a beta of 1.3 and an expected return of 21%. Portfolio B has a beta of 0.7 and an expected return of 17%. The risk-free rate of return is 8%. If you wanted to take advantage of an arbitrage opportunity, you should take a short position in portfolio and a long position in portfolio. 15. Consider the multi-factor APT with two factors. Portfolio A has a beta of 0. 5 on factor 1 and a beta of 1.25 on factor 2. The risk premiums on the factors 1 and 2 portfolios are 1% and 7% respectively. The risk-free rate of return is 7%. The expected return on portfolio A is if no arbitrage opportunities exist. 16. Security X has an expected rate of return of 13% and a beta of The risk-free rate is 5% and the market expected rate of return is 15%. According to the capital asset pricing model, security X is. 17. You invest $600 in security A with a beta of 1.5 and $400 in security B with a beta of.90. The beta of this formed portfolio is. 18. Security A has an expected rate of return of 12% and a beta of The market expected rate of return is 8% and the risk-free rate is 5%. The alpha of the stock is. 19. Use the following to answer questions a-d:

28 15% 10% 5% 0 return 1 2 SML beta a. What is the expected return on the market? b. What is the beta for a portfolio with an expected return of 15% c. What is the expected return for a portfolio with a beta of 0.5? d. What is the alpha of a portfolio with a beta of 2 and actual return of 15%? 20. According to the CAPM, what is the market risk premium given an expected return on a security of 13.6%, a stock beta of 1.2, and a risk free interest rate of 4.0%? 21. Using the index model, the alpha of a stock is 4.0%, the beta if 0.9 and the market return is 10%. What is the residual given an actual return of 15%? 22. The risk premium for exposure to exchange rates is 5% and the firm has a beta relative to exchanges rates of 0.4. The risk premium for exposure to the consumer price index is -6% and the firm has a beta relative to the CPI of 0.8. If the risk free rate is 3.0%, what is the expected return on this stock? 23. The small firm in January effect is strongest. 24. Evidence suggests that there may be momentum and reversal patterns in stock price behavior. 25. In a study of investment behavior of men and women, Barber and Odean find 26. Psychological regret theory says. 27. Mental accounting is a form of framing which is consistent with. Solutions percent E( rp ) =.4(.15) +.6(.06) = σp =.4(.04) = Risk Premium = [.6(.15) +.4(.05)]-.06 = y = = Borrowing = 500,000( ) = 375, unique, firm-specific, diversifiable = (.2) (.25) + (.8) (.05) + 2(.2)(.8)(.25)(.05)Corr Corr= Correlatio n =.0030/[.10(.05)] = σ = (.4) (.05) + (.6) (.15) + 2(.4)(.6)(.05)(.15)(.5) σ 2 = % 12..7

29 13. 8% 14. short A, long B; hint: compute the factor s exp return for each stock 15. E ( r A ) = (.01) (.07) = =16.25% 16. E ( r ) would normally be ( ) = 16.5% => overpriced x β p = (1.5) + (.90) = ,000 1, % 19. a. 10% b.? c. 7.5% d.? 20. 8% 21. Residual = 15 ( x10) = 2% % 23. early in the month 24. short-run, long run 25. that men trade more actively than women 26. contends that people have more regret (blame themselves more) when a decision that turned out badly was more unconventional, is consistent with the firm size anomaly, and is consistent with the book-to-market anomaly 27. taking a very risky position with one investment account and a very conservative position with another investment account, investors' irrational preference for stocks with high cash dividends, a tendency to ride losing stock positions for too long

30 Homework 1b From BKM Ch. 5: 9, 10, 11, Ch. 6: 6, 7, 18, 19 Ch. 7: 2, 3, 4, 6, 7, 28 Ch. 8: 6, 16, 24. Solutions to book problems from the Sol Manual: CH E(r X ) = [0.2 ( 20%)] + [0.5 18%] + [0.3 50%)] = 20% E(r Y ) = [0.2 ( 15%)] + [0.5 20%] + [0.3 10%)] = 10% 10. σ X 2 = [0.2 ( 20 20) 2 ] + [0.5 (18 20) 2 ] + [0.3 (50 20) 2 ] = 592 σ X = 24.33% σ Y 2 = [0.2 ( 15 10) 2 ] + [0.5 (20 10) 2 ] + [0.3 (10 10) 2 ] = 175 σ Y = 13.23% 11. E(r) = (0.9 20%) + (0.1 10%) = 19% CH. 6. Think of your CAPM project when solving problems 6 and 7, we will compute means and st deviation for a variety of portfolios, here with diff proportions in stocks and bonds, they will form our frontier. Then we ll use some extra formulas (that apply only in the case of two-asset portfolios) to help us identify the min variance portfolio and the tangent one, but the overall objective is to construct the frontier, and the tangent Capital Market Line. 6. The parameters of the opportunity set are: E(r S ) = 15%, E(r B ) B = 9%, σs = 32%, σ BB = 23%, ρ = 0.15, r f = 5.5% You don t need to know the cov or min-var weight calculations but here they are: From the standard deviations and the correlation coefficient we generate the covariance matrix [note that Cov(r S, r B ) B = ρσsσ BB]: Bonds Stocks Bonds Stocks The min-var portf proportions are: w Min 2 σb Cov(rS,rB ) (S) = = = σ + σ 2Cov(r,r ) ( ) S w Min (B) = This calc will help us add an extra point to the table below: 31.42% stocks/68.58% bond portfolio The mean and standard deviation of the minimum variance portfolio are: E(r Min ) = ( %) + ( %) = 10.89% σ = w σ + w σ + 2w w Cov(r,r [ ] 2 Min S S B B S B S B ) = [( ) + ( ) + ( )] 1/2 = 19.94% % in stocks % in bonds Exp. return Std dev Minimum variance Tangency portfolio B S B

31 Investment opportunity set for stocks and bonds min var B S CAL Standard Deviation (%) 40 The graph approximates the points: E(r) σ Minimum Variance Portfolio 10.89% 19.94% Tangency Portfolio 13.25% 24.57% 18. The expected rate of return on the stock will change by beta times the unanticipated change in the market return: 1.2 (8% 10%) = 2.4% Therefore, the expected rate of return on the stock should be revised to: 12% 2.4% = 9.6% 19. a. The risk of the diversified portfolio consists primarily of systematic risk. Beta measures systematic risk, which is the slope of the security characteristic line (SCL). The two figures depict the stocks' SCLs. Stock B's SCL is steeper, and hence Stock B's systematic risk is greater. The slope of the SCL, and hence the systematic risk, of Stock A is lower. Thus, for this investor, stock B is the riskiest. b. The undiversified investor is exposed primarily to firm-specific risk. Stock A has higher firmspecific risk because the deviations of the observations from the SCL are larger for Stock A than for Stock B. Deviations are measured by the vertical distance of each observation from the SCL. Stock A is therefore riskiest to this investor. Ch a. E(r X ) = 5% + 0.8(14% 5%) = 12.2% " X = 14% 12.2% = 1.8% E(r Y ) = 5% + 1.5(17% 5%) = 18.5% " Y = 17% 18.5% = 1.5% b. i. For an investor who wants to add this stock to a well-diversified equity portfolio, Kay should recommend Stock X because of its positive alpha, while Stock Y has a negative alpha. In graphical terms, Stock X s expected return/risk profile plots above the SML, while Stock Y s profile plots below the SML. Also, depending on the individual risk preferences of Kay s clients, Stock X s lower beta may have a beneficial impact on overall portfolio risk. ii. For an investor who wants to hold this stock as a single-stock portfolio, Kay should recommend Stock Y, because it has higher forecasted return and lower standard deviation than Stock X. Stock Y s Sharpe ratio is: ( )/0.25 = 0.48 Stock X s Sharpe ratio is only: ( )/0.36 = 0.25 The market index has an even more attractive Sharpe ratio: ( )/0.15 = 0.60

32 However, given the choice between Stock X and Y, Y is superior. When a stock is held in isolation, standard deviation is the relevant risk measure. For assets held in isolation, beta as a measure of risk is irrelevant. Although holding a single asset in isolation is not typically a recommended investment strategy, some investors may hold what is essentially a single-asset portfolio (e.g., the stock of their employer company). For such investors, the relevance of standard deviation versus beta is an important issue. 3. E(r P ) = r f + β[e(r M ) r f ] 20% = 5% + β(15% 5%) β = 15/10 = If the beta of the security doubles, then so will its risk premium. The current risk premium for the stock is: (13% - 7%) = 6%, so the new risk premium would be 12%, and the new discount rate for the security would be: 12% + 7% = 19% If the stock pays a constant dividend in perpetuity, then we know from the original data that the dividend (D) must satisfy the equation for a perpetuity: Price = Dividend/Discount rate 40 = D/0.13 D = = $5.20 At the new discount rate of 19%, the stock would be worth: $5.20/0.19 = $27.37 The increase in stock risk has lowered the value of the stock by 31.58%. 6. a. False. β = 0 implies E(r) = r f, not zero. b. False. Investors require a risk premium for bearing systematic (i.e., undiversifiable) risk. c. False. You should invest 0.75 of your portfolio in the market portfolio, and the remainder in T- bills. Then: β P = (0.75 1) + (0.25 0) = a. The beta is the sensitivity of the stock's return to the market return. Call the aggressive stock A and the defensive stock D. Then beta is the change in the stock return per unit change in the market return. We compute each stock's beta by calculating the difference in its return across the two scenarios divided by the difference in market return β A = = 2.00 β D = = b. With the two scenarios equal likely, the expected rate of return is an average of the two possible outcomes: E(r A ) = 0.5 (2% + 32%) = 17% E(r B ) B = 0.5 (3.5% + 14%) = 8.75% c. The SML is determined by the following: T-bill rate = 8% with a beta equal to zero, beta for the market is 1.0, and the expected rate of return for the market is: 0.5 (20% + 5%) = 12.5% E(r) SML A 12.5% M 8% α D D The equation for the security market line is: E(r) = 8% + β(12.5% 8%) d. The aggressive stock has a fair expected rate of return of: E(r A ) = 8% + 2.0(12.5% 8%) = 17% The security analyst s estimate of the expected rate of return is also 17%. Thus the alpha for the aggressive stock is zero. Similarly, the required return for the defensive stock is: E(r D ) = 8% + 0.7(12.5% 8%) = 11.15% The security analyst s estimate of the expected return for D is only 8.75%, and hence: β

33 α D = actual expected return required return predicted by CAPM = 8.75% 11.15% = 2.4% The points for each stock are plotted on the graph above. e. The hurdle rate is determined by the project beta (i.e., 0.7), not by the firm s beta. The correct discount rate is therefore 11.15%, the fair rate of return on stock D. 28. Equation 7.11 applies here: E(r P ) = r f + β P1 [E(r 1 ) r f ] + β P2 [E(r 2 ) r f ] We need to find the risk premium for these two factors: γ 1 = [E(r 1 ) r f ] and γ 2 = [E(r 2 ) r f ] To find these values, we solve the following two equations with two unknowns: 40% = 7% + 1.8γ γ 2 10% = 7% + 2.0γ 1 + ( 0.5)γ 2 The solutions are: γ 1 = 4.47% and γ 2 = 11.86% Thus, the expected return-beta relationship is: E(r P ) = 7% β P β P2 Ch d. 16. a. The grandson is recommending taking advantage of (i) the small firm anomaly and (ii) the January anomaly. In fact, this seems to be one anomaly: the small-firm-in-january anomaly. b. (i) Concentration of one s portfolio in stocks having very similar attributes may expose the portfolio to more risk than is desirable. The strategy limits the potential for diversification. (ii) Even if the study results are correct as described, each such study covers a specific time period. There is no assurance that future time periods would yield similar results. (iii) After the results of the studies became publicly known, investment decisions might nullify these relationships. If these firms in fact offered investment bargains, their prices may be bid up to reflect the now-known opportunity. 24. i. Mental accounting is best illustrated by Statement #3. Sampson s requirement that his income needs be met via interest income and stock dividends is an example of mental accounting. Mental accounting holds that investors segregate funds into mental accounts (e.g., dividends and capital gains), maintain a set of separate mental accounts, and do not combine outcomes; a loss in one account is treated separately from a loss in another account. Mental accounting leads to an investor preference for dividends over capital gains and to an inability or failure to consider total return. ii. Overconfidence (illusion of control) is best illustrated by Statement #6. Sampson s desire to select investments that are inconsistent with his overall strategy indicates overconfidence. Overconfident individuals often exhibit risk-seeking behavior. People are also more confident in the validity of their conclusions than is justified by their success rate. Causes of overconfidence include the illusion of control, self-enhancement tendencies, insensitivity to predictive accuracy, and misconceptions of chance processes. iii. Reference dependence is best illustrated by Statement #5. Sampson s desire to retain poor performing investments and to take quick profits on successful investments suggests reference dependence. Reference dependence holds that investment decisions are critically dependent on the decision-maker s reference point. In this case, the reference point is the original purchase price. Alternatives are evaluated not in terms of final outcomes but rather in terms of gains and losses relative to this reference point. Thus, preferences are susceptible to manipulation simply by changing the reference point. Extra Questions(no solutions): 4. The Fama-French (1996) model has three factors what are they? 6. What are the day of the week effect, the January effect and the small-size firm effect? 7. Provide at least one example for each of the mental errors (forecasting errors, overconfidence, conservatism, representativeness) and behavioral biases (framing, mental accounting, regret, loss aversion)

34 Hmk 2 Ch. 3: 2, 3, 4, 6, 8, 12, 13, 14, 20, 21 Ch. 4: 13 Ch. 12: 3, 5, 8, 18, 19 Ch. 17: 6, 11abcd Ch a. In principle, potential losses are unbounded, growing directly with increases in the price of IBM. b. If the stop-buy order can be filled at $128, the maximum possible loss per share is $8. If the price of IBM shares go above $128, then the stop-buy order would be executed, limiting the losses from the short sale. 3. a. The stock is purchased for: 300 $40 = $12,000 The amount borrowed is $4,000. Therefore, the investor put up equity, or margin, of $8,000. b. If the share price falls to $30, then the value of the stock falls to $9,000. By the end of the year, the amount of the loan owed to the broker grows to: $4, = $4,320 Therefore, the remaining margin in the investor s account is: $9,000 $4,320 = $4,680 The percentage margin is now: $4,680/$9,000 = 0.52 = 52% Therefore, the investor will not receive a margin call. c. The rate of return on the investment over the year is: (Ending equity in the account Initial equity)/initial equity = ($4,680 $8,000)/$8,000 = = 41.5% 4. a. The initial margin was: ,000 $40 = $20,000 As a result of the increase in the stock price Old Economy Traders loses: $10 1,000 = $10,000 Therefore, margin decreases by $10,000. Moreover, Old Economy Traders must pay the dividend of $2 per share to the lender of the shares, so that the margin in the account decreases by an additional $2,000. Therefore, the remaining margin is: $20,000 $10,000 $2,000 = $8,000 b. The percentage margin is: $8,000/$50,000 = 0.16 = 16% So there will be a margin call. c. The equity in the account decreased from $20,000 to $8,000 in one year, for a rate of return of: ( $12,000/$20,000) = 0.60 = 60% 6. a. The buy order will be filled at the best limit-sell order price: $50.25 b. The next market buy order will be filled at the next-best limit-sell order price: $51.50 c. You would want to increase your inventory. There is considerable buying demand at prices just below $50, indicating that downside risk is limited. In contrast, limit sell orders are sparse, indicating that a moderate buy order could result in a substantial price increase. 8. a. Initial margin is 50% of $5,000 or $2,500. b. Total assets are $7,500 ($5,000 from the sale of the stock and $2,500 put up for margin). Liabilities are 100P. Therefore, net worth is ($7, P). A margin call will be issued when:

35 $ 7, P = 0.30 when P = $57.69 or higher 100P 12. The broker is instructed to attempt to sell your Marriott stock as soon as the Marriott stock trades at a bid price of $38 or less. Here, the broker will attempt to execute, but may not be able to sell at $38, since the bid price is now $ The price at which you sell may be more or less than $38 because the stop-loss becomes a market order to sell at current market prices. 13. a b c. The trade will not be executed because the bid price is lower than the price specified in the limit sell order. d. The trade will not be executed because the asked price is greater than the price specified in the limit buy order. 14. a. In an exchange market, there can be price improvement in the two market orders. Brokers for each of the market orders (i.e., the buy and the sell orders) can agree to execute a trade inside the quoted spread. For example, they can trade at $55.37, thus improving the price for both customers by $0.12 or $0.13 relative to the quoted bid and asked prices. The buyer gets the stock for $0.13 less than the quoted asked price, and the seller receives $0.12 more for the stock than the quoted bid price. b. Whereas the limit order to buy at $55.37 would not be executed in a dealer market (since the asked price is $55.50), it could be executed in an exchange market. A broker for another customer with an order to sell at market would view the limit buy order as the best bid price; the two brokers could agree to the trade and bring it to the specialist, who would then execute the trade. 20. (d) The broker will sell, at current market price, after the first transaction at $55 or less. 21. (b) Ch Start of year NAV = $20 Dividends per share = $0.20 End of year NAV is based on the 8% price gain, less the 1% 12b-1 fee: End of year NAV = $ (1 0.01) = $ Rate of return = $ $20 + $0.20 = = 7.92% $20 Ch a. D1 P0 = k g $2 $50 = $2 g = 0.16 = 0.12 = 12% 0.16 g $50 b. D1 $2 P0 = = = $ k g The price falls in response to the more pessimistic forecast of dividend growth. The forecast for current earnings, however, is unchanged. Therefore, the P/E ratio decreases. The lower P/E ratio is evidence of the diminished optimism concerning the firm's growth prospects. 5. a. g = ROE b = = 0.06 = 6.0% D1 = $2(1 b) = $2(1 0.30) = $1.40 D1 $1.40 P = = k g = P/E = $23.33/$2 = $23.33

36 b. PVGO = P0 E 0 $2.00 = $23.33 = $6. 66 k 0.12 c. g = ROE b = = = 0.04 = 4.0% D1 = $2(1 b) = $2(1 0.20) = $1.60 P D1 $1.60 = = k g = P/E = $20/$2 = 10.0 $20.00 E 0 $2.00 PVGO = P0 = $20.00 = $3.33 k a. k = rf + β (km rf) = 6% (14% 6%) = 16% g = (2/3) 9% = 6% D1 = E0 (1 + g) (1 b) = $ (1/3) = $1.06 D1 $1.06 P0 = = = $10.60 k g b. Leading P0/E1 = $10.60/$3.18 = 3.33 Trailing P0/E0 = $10.60/$3.00 = 3.53 E 0 $3 c. PVGO = P0 = $10.60 = $8. 15 k 0.16 The low P/E ratios and negative PVGO are due to a poor ROE (9%) that is less than the market capitalization rate (16%). d. Now, you revise the following: b = 1/3 g = 1/ = 0.03 = 3.0% D1 = E (2/3) = $2.06 D1 $2.06 V = = k g = $15.85 V0 increases because the firm pays out more earnings instead of reinvesting earnings at a poor ROE. This information is not yet known to the rest of the market. 18. a. The value of a share of Rio National equity using the Gordon growth model and the capital asset pricing model is $22.40, as shown below. Calculate the required rate of return using the capital asset pricing model: k = rf + β (km rf) = 4% + 1.8(9% 4%) = 13% Calculate the share value using the Gordon growth model: P D o (1 + g) $0.20 ( ) = = k g = $22.40 b. The sustainable growth rate of Rio National is 9.97%, calculated as follows: g = b ROE = Earnings Retention Rate ROE = (1 Payout Ratio) ROE = Dividends Net Income $3.20 $ = 1 = = 9.97% Net Income Beginning Equity $30.16 $270.35

37 19. a. To obtain free cash flow to equity (FCFE), the two adjustments that Shaar should make to cash flow from operations (CFO) are: 1. Subtract investment in fixed capital: CFO does not take into account the investing activities in long-term assets, particularly plant and equipment. The cash flows corresponding to those necessary expenditures are not available to equity holders and therefore should be subtracted from CFO to obtain FCFE. 2. Add net borrowing: CFO does not take into account the amount of capital supplied to the firm by lenders (e.g., bondholders). The new borrowings, net of debt repayment, are cash flows available to equity holders and should be added to CFO to obtain FCFE. b. Note 1: Rio National had $75 million in capital expenditures during the year. Adjustment: negative $75 million The cash flows required for those capital expenditures ( $75 million) are no longer available to the equity holders and should be subtracted from net income to obtain FCFE. Note 2: A piece of equipment that was originally purchased for $10 million was sold for $7 million at year-end, when it had a net book value of $3 million. Equipment sales are unusual for Rio National. Adjustment: positive $3 million In calculating FCFE, only cash flow investments in fixed capital should be considered. The $7 million sale price of equipment is a cash inflow now available to equity holders and should be added to net income. However, the gain over book value that was realized when selling the equipment ($4 million) is already included in net income. Because the total sale is cash, not just the gain, the $3 million net book value must be added to net income. Therefore, the adjustment calculation is: $7 million in cash received $4 million of gain recorded in net income = $3 million additional cash received that must be added to net income to obtain FCFE. Note 3: The decrease in long-term debt represents an unscheduled principal repayment; there was no new borrowing during the year. Adjustment: negative $5 million The unscheduled debt repayment cash flow ( $5 million) is an amount no longer available to equity holders and should be subtracted from net income to determine FCFE. Note 4: On 1 January 2002, the company received cash from issuing 400,000 shares of common equity at a price of $25.00 per share. No adjustment Transactions between the firm and its shareholders do not affect FCFE. To calculate FCFE, therefore, no adjustment to net income is required with respect to the issuance of new shares. Note 5: A new appraisal during the year increased the estimated market value of land held for investment by $2 million, which was not recognized in 2002 income. No adjustment The increased market value of the land did not generate any cash flow and was not reflected in net income. To calculate FCFE, therefore, no adjustment to net income is required. c. Free cash flow to equity (FCFE) is calculated as follows: FCFE = NI + NCC FCINV WCINV + Net Borrowing where NCC = non-cash charges FCINV = investment in fixed capital WCINV = investment in working capital Million $ Explanation NI = $30.16 From Exhibit 18B NCC = +$67.17 $71.17 (depreciation and amortization from Exhibit 18B)

38 $4.00* (gain on sale from Note 2) FCINV = $68.00 $75.00 (capital expenditures from Note 1) $7.00* (cash on sale from Note 2) WCINV = $24.00 $3.00 (increase in accounts receivable from Exhibit 18A + $20.00 (increase in inventory from Exhibit 18A) + $1.00 (decrease in accounts payable from Exhibit 18A) Net Borrowing = +( $5.00) $5.00 (decrease in long-term debt from Exhibit 18A) FCFE = $0.33 *Supplemental Note 2 in Exhibit 18C affects both NCC and FCINV. Ch a. Actual: ( %) + ( %) + ( %) = 1.65% Bogey: ( %) + ( %) + ( %) = 1.91% Underperformance = 1.91% 1.65% = 0.26% b. Security Selection: Portfolio Index Excess Manager s Market Contribution Performance Performance Performance Portfolio Weight Equity 2.0% 2.5% -0.5% % Bonds 1.0% 1.2% -0.2% % Cash 0.5% 0.5% 0.0% % Contribution of security selection: -0.39% c. Asset Allocation: Market Actual Weight Benchmark Index Return Excess Weight Contribution Weight Minus Bogey Equity % 0.059% Bonds % 0.071% Cash % 0.000% Contribution of asset allocation: 0.130% Summary Security selection -0.39% Asset allocation 0.13% Excess performance -0.26% 11. a. The spreadsheet below displays the monthly returns and excess returns for the Vanguard U.S. Growth Fund, the Vanguard U.S. Value Fund and the S&P 500. Note that the inception date for the Vanguard U.S. Value Fund was 6/29/2000. Monthly Rates of Return: May April 2005 Growth Value Excess Returns Month Fund Fund T-Bills S&P500 Growth Fund Value Fund S&P500 May N/A N/A Jun N/A N/A 1.91 Jul N/A N/A Aug Sep Oct Nov Dec

39 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr Std.Dev b. The standard deviations for the U.S Growth Fund and the U.S. Value Fund are 7.03% and 4.21%, respectively, as shown in the Excel spreadsheet above. c. The betas for the U.S. Growth Fund and the U.S. Value Fund are and 0.799, respectively, as shown in the Excel spreadsheets below:

40 GROWTH FUND SUMMARY OUTPUT OF EXCEL REGRESSION Regression Statistics Multiple R R Square Adj. R Square Standard Error Observations ANOVA df SS MS F Significance F Regression E-25 Residual Total Coefficients Std Error t Stat P-value Intercept S&P E-25 VALUE FUND SUMMARY OUTPUT OF EXCEL REGRESSION Regression Statistics Multiple R R Square Adj. R Square Standard Error Observations 57 ANOVA df SS MS F Significance F Regression E-19 Residual Total Coefficients Std Error t Stat P-value Intercept S&P E-19 d. The formulas for the three measures are: Sharpe: Treynor: r r p σ p p f r r p β f

41 Jensen: α = r { r + β (r r )} p p The values for the three measures are computed as follows: U.S. Growth Fund f p M f U.S. Value Fund Sharpe ( )/7.03 = ( )/4.21 = Treynor ( )/1.432= ( )/0.799 = Jensen 1.16 { ( )} = { ( )} = Note that, in the above calculations, r f and r M are calculated over different time periods for the Growth Fund and the Value Fund because of the 6/29/2000 inception date of the Value Fund. Also, there are some discrepancies due to rounding.

42 Hmk 3 1. Underwriting is one of the services provided by whom? 2. Under firm commitment underwriting, who assumes the full risk that the shares cannot be sold to the public at the stipulated offering price? 3. What is an ECN? 4. The bid-ask spread exists because of the need for dealers to cover expenses and make a modest profit. True/False 5. Consider the following limit order book of a specialist. The last trade in the stock occurred at a price of $40. If a market buy order for 100 shares comes in, at what price will it be filled? Limit Buy Orders Limit Sell Orders Price Shares Price Shares $ $ $ $ Assume you purchased 200 shares of XYZ common stock on margin at $80 per share from your broker. If the initial margin is 60%, the amount you borrowed from the broker is how much? 7. You sold short 200 shares of common stock at $50 per share. The initial margin is 60%. Your initial investment was how much? 8. You short-sell 200 shares of Tuckerton Trading Co., now selling for $50 per share. What is your maximum possible loss? What is your maximum possible gain? 9. You purchased 100 shares of ABC common stock on margin at $50 per share. Assume the initial margin is 50% and the maintenance margin is 30%. Below what stock price would you get a margin call? Assume the stock pays no dividend and ignore interest on margin. 10. Borrowing a security from your broker in order to sell it, with the intention of repurchasing it later when the price is lower, is called what? 12. The margin requirement on a stock purchase is 15%. You fully use the margin allowed to purchase 100 shares of MSFT at $35. If the price drops to $32, what is your percentage loss? 13. What are the difference s between active and passively managed mutual funds? 15. Under SEC rules, the managers of certain funds are allowed to deduct charges for advertising, brokerage commissions, and other sales expenses, directly from the fund assets rather than billing investors. These fees are known as..? 19. The Stone Harbor Fund is a closed-end investment company with a portfolio currently worth $300 million. It has liabilities of $5 million and 9 million shares outstanding. If the fund sells for $30 a share, what is its premium or discount as a percent of NAV? 21. What is a REIT? 22. What is market timing? What funds are likely to be subject to it? 23. You purchased XYZ stock at $50 per share. The stock is currently selling at $65. Your gains could be protected by placing a A) limit-buy order B) limit-sell order C) market order D) stop-loss order 24. The market capitalization rate on the stock of Aberdeen Wholesale Company is 10%. Its expected ROE is 12% and its expected EPS is $5.00. If the firm's plow-back ratio is 40%, its P/E ratio will be. 25. Gagliardi Way Corporation has an expected ROE of 15%. Its dividend growth rate will be if it follows a policy of paying 30% of earning in the form of dividends. 26. Rose Hill Trading Company is expected to have EPS in the upcoming year of $6.00. The expected ROE is 18.0%. An appropriate required return on the stock is 14%. If the firm has a plowback ratio of 60%, its growth rate of dividends should be. 27. Rose Hill Trading Company is expected to have EPS in the upcoming year of $6.00. The expected ROE is 18.0%. An appropriate required return on the stock is 14%. If the firm has a plowback ratio of 70%, its intrinsic value should be. 28. Cache Creek Company is expected to pay a dividend of $3.36 in the upcoming year. Dividends are expected to grow at 8% per year. The riskfree rate of return is 4% and the expected return on the market portfolio is 14%. Investors use the CAPM to compute the market capitalization rate, and the constant growth DDM to determine the value of the stock. The stock's current price is $ Using the constant growth DDM, the market capitalization rate is. 29. Grott and Perrin, Inc. has expected earnings of $3 per share for next year. The firm's ROE is 20% and its earnings retention ratio is 70%. If the firm's market capitalization rate is 15%, what is the

43 present value of its growth opportunities? 30. Annie's Donut Shops, Inc. has expected earnings of $3.00 per share for next year. The firm's ROE is 18% and its earnings retention ratio is 60%. If the firm's market capitalization rate is 12%, what is the value of the firm excluding any growth opportunities? 32. Cache Creek Manufacturing Company is expected to pay a dividend of $4.20 in the upcoming year. Dividends are expected to grow at the rate of 8% per year. The riskfree rate of return is 4% and the expected return on the market portfolio is 14%. Investors use the CAPM to compute the market capitalization rate on the stock, and the constant growth DDM to determine the intrinsic value of the stock. The stock is trading in the market today at $ Using the constant growth DDM and the CAPM, the beta of the stock is. 33. Westsyde Tool Company is expected to pay a dividend of $2.00 in the upcoming year. The risk-free rate of return is 6% and the expected return on the market portfolio is 12%. Analysts expect the price of Westsyde Tool Company shares to be $29 a year from now. The beta of Westsyde Tool Company's stock is Using a one-period valuation model, the intrinsic value of Westsyde Tool Company stock today is 34. Caribou Gold Mining Corporation is expected to pay a dividend of $6 in the upcoming year. Dividends are expected to decline at the rate of 3% per year. The risk-free rate of return is 5% and the expected return on the market portfolio is 13%. The stock of Caribou Gold Mining Corporation has a beta of Using the constant growth DDM, the intrinsic value of the stock is. 35. The Free cash flow to the firm is $300m in perpetuity, the cost of equity equals 14% and the WACC is 10%. If the market value of the debt is $1.0 billion, what is the value of the equity using the free cash flow valuation approach? 36. The free cash flow to the firm is reported as $405 million. The interest expense to the firm is $76 million. If the tax rate is 35% and the net debt of the firm increased by $50, what is the free cash flow to the equity holders of the firm? 37. The free cash flow to the firm is reported as $205 million. The interest expense to the firm is $22 million. If the tax rate is 35% and the net debt of the firm increased by $25, what is the market value of the firm if the FCFE grows at 2% and the cost of equity is 11%? Use the following to answer questions 38-40: In a particular year, Lost Hope Mutual Fund earned a return of 2% by making the following investments in asset classes: 38. The total excess return on the managed portfolio was. 39. The contribution of asset allocation across markets to the total excess return was. 40. The contribution of security selection within asset classes to the total excess return was. 41. A portfolio generates an annual return of 13%, a beta of 0.7 and a standard deviation of 17%. The market index return is 14% and a standard deviation of 21%. What is the Treynor measure of the

44 portfolio if the risk free rate is 5%? 42. A portfolio generates an annual return of 17%, a beta of 1.2 and a standard deviation of 19%. The market index return is 12% and a standard deviation of 16%. What is the Treynor measure of the portfolio if the risk free rate is 4%? 43. A portfolio generates an annual return of 13%, a beta of 0.7 and a standard deviation of 17%. The market index return is 14% and a standard deviation of 21%. What is the Sharpe measure of the portfolio if the risk free rate is 5%? 44. A portfolio generates an annual return of 17%, a beta of 1.2 and a standard deviation of 19%. The market index return is 12% and a standard deviation of 16%. What is the Sharpe measure of the portfolio if the risk free rate is 4%? 45. A portfolio generates an annual return of 13%, a beta of 0.7 and a standard deviation of 17%. The market index return is 14% and a standard deviation of 21%. What is the Jensen measure of the portfolio if the risk free rate is 5%? 46. A portfolio generates an annual return of 17%, a beta of 1.2 and a standard deviation of 19%. The market index return is 12% and a standard deviation of 16%. What is the Jensen measure of the portfolio if the risk free rate is 4%? 47. The portfolio that contains the benchmark asset allocation against which a manager will be measured is often called the.

45 Solutions 1. ib 2. u 3. el com net 4. t or less 6. Borrowing = 200 (80)(1.60) = 6, Investment = 200(50)(.60) = 6, Ans: There is no upper limit to the price of a share of stock, therefore no upper limit the price you will have to pay to replace the 200 shares of Tuckerton. Tuckerton could go bankrupt with a share price of $0. You could keep the entire proceeds from the short sale. Maximum gain = proceeds minimum possible replacement cost = 200 ( $50 ) 200 ( $0 ) = $10, Ans: P = 50 = or 2500=.7x100xX 10. sh s Ans: Margin = 35 x 100 x.15 = 525 Stock value at start = 100 x 35 = 3500 Stock value at end = 100 x 32 = 3200 Decrease in value = 300 Percentage loss = 300 / 525 = 57% b % d 23. B % 27. $ % 29. $ $ $ $ bn m 37. 2, Excess Return = = (.10.50)(.1000) + (.90.50)(.0500) = ( ).10 + ( ).90 = Sh= Bogey

46 Ch 9 Problems: 4, 9, 10, 11, 13, 14, 15, 18, 19, 24, 28, 31, 32, 33, 34, 39bcd Legend: PVIF(n, r) = 1 / (1+r) n PVAF(n, r) = 1/r 1/[r(1+r) n ] FVAF(n,r)= [(1+r) n -1]/r 4. The bond price will be lower. As time passes, the bond price, which is now above par value, will approach par. 9. a. The bond pays $50 every six months. Current price: [$50 PVAF(4%, 6)] + [$1000 PVIF(4%, 6)] = $1, Assuming the market interest rate remains 8% semi=4% per half year, price six months from now: b. Rate of return = [$50 PVAF(4%, 5)] + [$1000 PVIF(4%, 5)] = $1, $50 + ($1, $1,052.42) $1, % semi $50 $7.90 = $1, = = 4.00% per six months = 10. a. From [$40 PVAF(x%, 40)] + [$1000 PVIF(x%, 40)] = $950 solve for x. You will find that the yield to maturity per 6 months is x= 4.26%. This implies a bond equivalent yield to maturity of: 4.26% 2 = 8.52% semi Effective annual yield to maturity = (1.0426) 2 1 = = 8.70% b. Since the bond is selling at par, the yield per 6 months is the same as the semi-annual coupon, 4%. The bond equivalent yield to maturity is 8%. Effective annual yield to maturity = (1.04) 2 1 = = 8.16% c. Keeping other inputs unchanged except P = 1050, we find a bond equivalent yield to maturity of 7.52% semi, or 3.76% per 6 months. Effective annual yield to maturity = (1.0376) 2 1 = = 7.66% 11. Since the bond payments are now made annually instead of semi-annually, the bond equivalent yield to maturity is the same as the effective annual yield to maturity. The equation is [$80 PVAF(x%, 20)] + [$1000 PVIF(x%, 20)] = P The resulting yields for the three bonds are: Bond equivalent yield = Bond Price Effective annual yield $ % $ % $ % 9-1

47 The yields computed in this case are lower than the yields calculated with semi-annual coupon payments. All else equal, bonds with annual payments are less attractive to investors because more time elapses before payments are received. If the bond price is the same with annual payments, then the bond's yield to maturity is lower. 13. Remember that the convention is to use semi-annual periods: Maturity Maturity Per 6 months Bond equivalent Price (years) (half-years) YTM semi YTM $ % 4.634% $ % 3.496% $ % 7.052% $ % % $ % 8.000% $ % 8.000% 14. Zero 8% coupon 10% coupon a. Current prices $ $1000 $ b. Price one year from now $ $1000 $ Price increase $ $ $ 9.26 Coupon income $ 0.00 $80.00 $ Income $ $80.00 $ Rate of Return 8.00% 8.00% 8.00% 15. The reported bond price is: 100 2/32 percent of par = $1, However, 15 days have passed since the last semiannual coupon was paid, so accrued interest equals: $35 (15/182) = $2.885 The invoice price is the reported price plus accrued interest: $ The solution is obtained using Excel: A B C D E % coupon bond, 2 maturing March 15, Formula in Column B 4 Settlement date 2/22/2006 DATE(2006,2,22) 5 Maturity date 3/15/2014 DATE(2014,3,15) 6 Annual coupon rate Yield to maturity Redemption value (% of face value) Coupon payments per year Flat price (% of par) PRICE(B4,B5,B6,B7,B8,B9) 13 Days since last coupon 160 COUPDAYBS(B4,B5,2,1) 9-2

48 14 Days in coupon period 181 COUPDAYS(B4,B5,2,1) 15 Accrued interest (B13/B14)*B6*100/2 16 Invoice price B12+B The solution is obtained using Excel: A B C D E F G 1 Semiannual Annual 2 coupons coupons 3 2/22/200 2/22/200 4 Settlement date Maturity date 3/15/ /15/ Annual coupon rate Bond price Redemption value (% of face value) Coupon payments per year Yield to maturity (decimal) Formula in cell E11: YIELD(E4,E5,E6,E7,E8,E9) 24. a. The bond sells for P=$1, based on the 3.5% yield to maturity: [$40 PVAF(3.5%, 60)] + [$1000 PVIF(3.5%, 60)] = $1, [n = 60; i = 3.5; FV = 1000; PMT = 40] Therefore, yield to call is x=3.368% per 6 months, 6.736% semi: [$40 PVAF(x%, 10)] + [$1100 PVIF(x%, 10)] = $1, [n = 10; PV = ; FV = 1100; PMT = 40] b. If the call price were $1050, we would set FV = 1050 and redo part (a) to find that yield to call is x=2.976% per 6 months, 5.952% semi. With a lower call price, the yield to call is lower. c. Yield to call is x=3.031% per 6 months, 6.062% semi: [$40 PVAF(x%, 4)] + [$1100 PVIF(x%, 4)] = $1, [n = 4; PV = ; FV = 1100; PMT = 40] 28. April 15 is midway through the semi-annual coupon period. Therefore, the invoice price will be higher than the stated ask price by an amount equal to one-half of the semiannual coupon. The ask price is percent of par, so the invoice price is: $1, (1/2 $50) = $1,

49 31. a. (1) Current yield = Coupon/Price = 70/960 = = 7.29% (2) YTM = 3.993% per 6 months or 7.986% semi bond equivalent yield (3) Realized compound yield is 4.166% (per 6 months), or 8.332% semi bond equivalent yield. To obtain this value, first calculate the future value of reinvested coupons. There will be six payments of $35 each, reinvested semiannually at a per period rate of 3%: 35 x FVAF(6, 3%) = $ [PV = 0; PMT = $35; n = 6; i = 3%] The bond will be selling at par value of $1,000 in three years, since coupon is forecast to equal yield to maturity. Therefore, total proceeds in three years will be $1, To find realized compound yield on a semiannual basis (i.e., for six half-year periods), we solve: $960 (1 + x realized ) 6 = $1, x realized = 4.166% (per 6 months) b. Shortcomings of each measure: (1) Current yield does not account for capital gains or losses on bonds bought at prices other than par value. It also does not account for reinvestment income on coupon payments. (2) Yield to maturity assumes that the bond is held to maturity and that all coupon income can be reinvested at a rate equal to the yield to maturity. (3) Realized compound yield (horizon yield) is affected by the forecast of reinvestment rates, holding period, and yield of the bond at the end of the investor's holding period. 32. a. The yield to maturity of the par bond equals its coupon rate, 8.75%. All else equal, the 4% coupon bond would be more attractive because its coupon rate is far below current market yields, and its price is far below the call price. Therefore, if yields fall, capital gains on the bond will not be limited by the call price. In contrast, the 8.75% coupon bond can increase in value to at most $1050, offering a maximum possible gain of only 5%. The disadvantage of the 8.75% coupon bond in terms of vulnerability to a call shows up in its higher promised yield to maturity. b. If an investor expects rates to fall substantially, the 4% bond offers a greater expected return. c. Implicit call protection is offered in the sense that any likely fall in yields would not be nearly enough to make the firm consider calling the bond. In this sense, the call feature is almost irrelevant. 33. Market conversion value = value if converted into stock = $28 = $ Conversion premium = Bond value market conversion value = $775 $ = $

50 34. a. The call provision requires the firm to offer a higher coupon (or higher promised yield to maturity) on the bond in order to compensate the investor for the firm's option to call back the bond at a specified call price if interest rates fall sufficiently. Investors are willing to grant this valuable option to the issuer, but only for a price that reflects the possibility that the bond will be called. That price is the higher promised yield at which they are willing to buy the bond. b. The call option reduces the expected life of the bond. If interest rates fall substantially so that the likelihood of call increases, investors will treat the bond as if it will "mature" and be paid off at the call date, not at the stated maturity date. On the other hand if rates rise, the bond must be paid off at the maturity date, not later. This asymmetry means that the expected life of the bond will be less than the stated maturity. 39. c. The advantage of a callable bond is the higher coupon (and higher promised yield to maturity) when the bond is issued. If the bond is never called, then an investor will earn a higher realized compound yield on a callable bond issued at par than on a non-callable bond issued at par on the same date. The disadvantage of the callable bond is the risk of call. If rates fall and the bond is called, then the investor receives the call price and will have to reinvest the proceeds at interest rates that are lower than the yield to maturity at which the bond was originally issued. In this event, the firm's savings in interest payments is the investor's loss. b. (3) The yield on the callable bond must compensate the investor for the risk of call. Choice (1) is wrong because, although the owner of a callable bond receives principal plus a premium in the event of a call, the interest rate at which he can subsequently reinvest will be low. The low interest rate that makes it profitable for the issuer to call the bond makes it a bad deal for the bond s holder. Choice (2) is wrong because a bond is more apt to be called when interest rates are low. There will be an interest saving for the issuer only if rates are low. c. (3) d. (2) 9-5

51 Ch 14 Problems: 1, 2, 4, 5, 6, 7, 8, 24 Ch 15 Problems 1, 3, 5, 6, 7, 8, 11, 14, 16, 17 CH14 1. c is false. This is the description of the payoff to a put, not a call. 2. c is the only correct statement. 4. Cost Payoff Profit Call option, X = Put option, X = Call option, X = Put option, X = Call option, X = Put option, X = In terms of dollar returns: Price of Stock Six Months From Now Stock price: $80 $100 $110 $120 All stocks (100 shares) 8,000 10,000 11,000 12,000 All options (1,000 shares) ,000 20,000 Bills options 9,360 9,360 10,360 11,360 In terms of rate of return, based on a $10,000 investment: Price of Stock Six Months From Now Stock price: $80 $100 $110 $120 All stocks (100 shares) -20% 0% 10% 20% All options (1,000 shares) -100% -100% 0% 100% Bills options -6.4% -6.4% 3.6% 13.6% Rate of return (%) All options 110 All stocks Bills plus options S T a. Purchase a straddle, i.e., both a put and a call on the stock. The total cost of the straddle would be: $10 + $7 = $17 b. Since the straddle costs $17, this is the amount by which the stock would have to move in either direction for the profit on either the call or the put to cover the investment cost (not including time value of money considerations). 7. a. Sell a straddle, i.e., sell a call and a put to realize premium income of: $4 + $7 = $11

52 b. If the stock ends up at $50, both of the options will be worthless and your profit will be $11. This is your maximum possible profit since, at any other stock price, you will have to pay off on either the call or the put. The stock price can move by $11 (your initial revenue from writing the two at-themoney options) in either direction before your profits become negative. c. Buy the call, sell (write) the put, lend the present value of $50. The payoff is as follows: Final Payoff Outlay S T X Position Long call Initial C = 7 < 0 S T > X S T 50 Short put -P = -4 -(50 S T ) 0 Lending 50/(1 + r) (1/4) Total [50/(1 + r) (1/4) ] S T S T The initial outlay equals: (the present value of $50) + $3 In either scenario, you end up with the same payoff as you would if you bought the stock itself. 8. a. By writing covered call options, Jones receives premium income of $30,000. If, in January, the price of the stock is less than or equal to $45, he will keep the stock plus the premium income. Since the stock will be called away from him if its price exceeds $45 per share, the most he can have is: $450,000 + $30,000 = $480,000 (We are ignoring interest earned on the premium income from writing the option over this short time period.) The payoff structure is: Stock price Portfolio value Less than $45 (10,000 times stock price) + $30,000 Greater than $45 $450,000 + $30,000 = $480,000 This strategy offers some premium income but leaves the investor with substantial downside risk. At the extreme, if the stock price falls to zero, Jones would be left with only $30,000. This strategy also puts a cap on the final value at $480,000, but this is more than sufficient to purchase the house. b. By buying put options with a $35 strike price, Jones will be paying $30,000 in premiums in order to insure a minimum level for the final value of his position. That minimum value is: ($35 10,000) $30,000 = $320,000 This strategy allows for upside gain, but exposes Jones to the possibility of a moderate loss equal to the cost of the puts. The payoff structure is: Stock price Portfolio value Less than $35 $350,000 $30,000 = $320,000 Greater than $35 (10,000 times stock price) $30,000 c. The net cost of the collar is zero. The value of the portfolio will be as follows: Stock price Portfolio value Less than $35 $350,000 Between $35 and $45 10,000 times stock price Greater than $45 $450,000

53 If the stock price is less than or equal to $35, then the collar preserves the $350,000 in principal. If the price exceeds $45, then Jones gains up to a cap of $450,000. In between $35 and $45, his proceeds equal 10,000 times the stock price. The best strategy in this case is (c) since it satisfies the two requirements of preserving the $350,000 in principal while offering a chance of getting $450,000. Strategy (a) should be ruled out because it leaves Jones exposed to the risk of substantial loss of principal. Our ranking is: (1) c (2) b (3) a 24. a. Joe s strategy Final Payoff Position Initial Outlay S T < 1200 S T > 1200 Stock index 1200 S T S T Long put (X = 1200) S T 0 Total S T Profit = payoff S T 1260 Sally s Strategy Final Payoff Position Initial Outlay S T < 1170 S T > 1170 Stock index 1200 S T S T Long put (X = 1170) S T 0 Total S T Profit = payoff S T 1245 Profit Sally Joe S T b. Sally does better when the stock price is high, but worse when the stock price is low. (The break-even point occurs at S = $1185, when both positions provide losses of $60.) c. Sally s strategy has greater systematic risk. Profits are more sensitive to the value of the stock index. 25. This strategy is a bear spread. The initial proceeds are: $9 $3 = $6 The payoff is either negative or zero: Position S T < < S T < 60 S T > 60 Long call (X = 60) 0 0 S T 60 Short call (X = 50) 0 (S T 50) (S T 50) Total 0 (S T 50) 10

54 Breakeven occurs when the payoff offsets the initial proceeds of $6, which occurs at a stock price of S T = $ S T - 4 Profit -10 Payoff 26. Buy a share of stock, write a call with X = 50, write a call with X = 60, and buy a call with X = 110. Position S T < < S T < < S T < 110 S T > 110 Buy stock S T S T S T S T Short call (X = 50) 0 (S T 50) (S T 50) (S T 50) Short call (X = 60) 0 0 (S T 60) (S T 60) Long call (X = 110) S T 110 Total S T S T 0 The investor is making a volatility bet. Profits will be highest when volatility is low so that the stock price ends up in the interval between $50 and $60. CH15 1. Put values also increase as the volatility of the underlying stock increases. We see this from the parity relationship as follows: C = P + S 0 PV(X) PV(Dividends) Given a value of S and a risk-free interest rate, if C increases because of an increase in volatility, so must P in order to keep the parity equation in balance. Numerical example: Suppose you have a put with exercise price 100, and that the stock price can take on one of three values: 90, 100, 110. The payoff to the put for each stock price is: Stock price Put value Now suppose the stock price can take on one of three alternate values also centered around 100, but with less volatility: 95, 100, 105. The payoff to the put for each stock price is: Stock price Put value 5 0 0

55 The payoff to the put in the low volatility example has one-half the expected value of the payoff in the high volatility example. 3. Note that, as the option becomes progressively more in the money, its hedge ratio increases to a maximum of 1.0: X Hedge ratio X Hedge ratio /150 = /150 = /150 = /150 = /150 = /150 = a. When S = 130, then P = 0. When S = 80, then P = 30. The hedge ratio is: [(P u P d )/(us 0 ds 0 ) = [(0 30)/(130 80)] = 3/5 b. Riskless portfolio S =80 S = shares puts Total Present value = $390/1.10 = $ c. Portfolio cost = 3S + 5P = $ P = $ Therefore 5P = $ P = $54.545/5 = $ The hedge ratio for the call is: [(C u C d )/(us 0 ds 0 )] = (20 0)/(130 80) = 2/5 Riskless portfolio S =80 S = shares Short 5 calls Total C = $160/1.10 = $ C = $10.91 Put-call parity relationship: P = C S 0 + PV(X) $10.91 = $ ($110/1.10) $100 = $ d 1 = N(d 1 ) = d 2 = N(d 2 ) = Xe rt = $47.56 C = S 0 N(d 1 ) Xe rt N(d 2 ) = $ P = $5.69 This value is from our Black-Scholes spreadsheet, but note that we could have derived the value from put-call parity: P = C S 0 + PV(X) = $8.13 $50 + $47.56 = $ The call price will decrease by less than $1. The change in the call price would be $1 only if: (i) there were a 100% probability that the call would be exercised; and (ii) the interest rate were zero. 14. The call option with a high exercise price has a lower hedge ratio. The call option is less in the money. Both d 1 and N(d 1 ) are lower when X is higher. 16. The call option s implied volatility has increased. If this were not the case, then the call price would have fallen. 17. The put option s implied volatility has increased. If this were not the case, then the put price would have fallen.

56 Introduction to Investments FINAN 3050 Week 7: Equity Valuation (Chp )

57 Intrinsic Value and Market Price Problem: determine the value of equity (i.e., of common shares) Market Price (P 0,MP ): consensus value of all potential traders Intrinsic Value (P 0,IV ): Self assigned value Variety of models are used for estimation Trading Signal P 0,IV > P 0,MP : Buy P 0,IV < P 0,MP : Sell or Short Sell P 0,IV = P 0,MP : Hold or Fairly Priced Slide 2 Week 7

58 Realized vs. Expected Holding Period Return (1) Remember our definition of HPR (slide 10 of week 2): r t = P t t, MP P0, MP+ i= 1 P 0, MP D i Useful to evaluate an investment ex-post (i.e., when it is done and all cashflows and prices are observable and known). If we think about doing an investment and want to evaluate it ex-ante we need to look at the Expected HPR! We know the current price P 0. We must estimate the expected dividend payments and the expected price at the end of the holding period. E ( r) t = E t ( P ) P + E( D) t, MP 0, MP i= 1 P 0, MP i Slide 3 Week 7

59 Realized vs. Expected Holding Period Return (2) Again, two ways to come up with the expected HPR: Market-Oriented: use the CAPM to estimated the return of a stock required by the market (we refer to this rate as E(k)) Look at the firm, estimate future dividends and estimate the expected HPR (we refer to this rate as E(r)) Before, we called the difference between E(r) and E(k) the alpha of a stock. Trading Signal E(r) > E(k): Buy (positive alpha case) E(r) < E(k): Sell or Short Sell (negative alpha case) E(r) = E(k): Hold or Fairly Priced (zero alpha case) Comparing P 0,IV and P 0,MP or E(r) and E(k) always gives the same signal! Slide 4 Week 7

60 For a 1-Year Holding Period (1) Consider the definition of the Expected HPR: E ( P ) P E( D ) ( E 1, MP 0, MP r ) + 1 = P 0, MP 1 Assume that we know P 0,MP, the required rate of return E(k) and have estimates for E(P 1 ) and E(D 1 ). Goal: determine if we should buy or sell this stock Two ways to solve this problem: 1. Assume that E(k)=E(r) (i.e., the expected rate of return equals the required rate of return) and check whether P 0,IV =P 0,MP (i.e., the intrinsic value of the stock equals its market price) 2. Assume that P 0,IV =P 0,MP (i.e., the intrinsic value of the stock equals its market price) and check whether E(k)=E(r) (i.e., the expected rate of return equals the required rate of return) Slide 5 Week 7

61 For a 1-Year Holding Period (2) Solution 1 Solution 1: we assume that E(k)=E(r) Then, we calculate the IV out of the equation for the Expected HPR: E( k) = E ( P ) P + E( D ) 1, MP P 0, IV 0, IV 1 P 0, IV = E ( D ) + E( P ) 1 1+ E( k) 1, MP Then we compare the quoted price P 0,MP and the calculated intrinsic value P 0,IV : P 0,IV > P 0,MP : buy P 0,IV < P 0,MP : sell P 0,IV = P 0,MP : fairly priced Slide 6 Week 7

62 For a 1-Year Holding Period (3) Solution 2 Solution 1: we assume that P 0,MP =P 0,IV The, we calculate the expected rate of return using the equation for the Expected HPR: E( r) = E ( P ) P + E( D ) 1, MP P 0, MP 0, MP 1 Then we compare the expected rate of return E(r) and the required rate of return E(k) (could come from the CAPM): E(r) > E(k): Buy (positive alpha case) E(r) < E(k): Sell or Short Sell (negative alpha case) E(r) = E(k): Hold or Fairly Priced (zero alpha case) Slide 7 Week 7

63 For a 1-Year Holding Period (4) Example You expect the price of IBX stock to be E(P 1,MP )=$59.77 per share a year from now. Its current market price is P 0,MP =$50, and you expect it to pay a dividend one year from now of E(D 1 )=$2.15 per share. The stock has a beta of 1.15, the risk-free rate is 6% per year, and the expected rate of return on the market portfolio is 14% per year. Determine if this stock is a good investment! Solution 1: Determine E(k): E(k) = 6%+1.15 (14%-6%) = 15.2% Determine P 0,IV = ($2.15+$59.77)/1.152 = $53.75 Compare P 0,IV and P 0,MP : $53.75 > $50 BUY! Slide 8 Week 7

64 For a 1-Year Holding Period (5) Example You expect the price of IBX stock to be E(P 1,MP )=$59.77 per share a year from now. Its current market price is P 0,MP =$50, and you expect it to pay a dividend one year from now of E(D 1 )=$2.15 per share. The stock has a beta of 1.15, the risk-free rate is 6% per year, and the expected rate of return on the market portfolio is 14% per year. Determine if this stock is a good investment! (same problem as on the previous slide) Solution 2: Determine E(r): E(r) = ($59.77-$50+$2.15)/$50 = Determine E(k): E(k) = 6%+1.15 (14%-6%) = 15.2% Compare E(r) and E(k): 23.84% > 15.2% BUY! Slide 9 Week 7

65 For a 1-Year Holding Period (6) Exercise Consider again the problem discussed on the previous two slides. What does the expected price of the stock in one year, i.e. E(P 1,MP ), has to be to make the stock be priced fairly? Slide 10 Week 7

66 A More General Model: Dividend Discount Model (1) Motivation Consider the 1-year definition of the intrinsic value: P 0, IV = E ( D ) + E( P ) 1 1+ E( k) 1. MP What is E(P 1,MP )? Assume that the stock sells for its intrinsic value in one year, i.e. in one year P 1,IV =P 1,MP. We can estimate P 1,IV and plug it into the equation for P 0,IV. E ( P ) = E( P ) 1, MP 1, IV = E ( D ) + E( P ) 2 1+ E( k) 2. MP P 0, IV ( D1) E( k) ( D2) + E( P2MP) ( 1+ E( )) 2 E E = + 1+ k Slide 11 Week 7

67 A More General Model: Dividend Discount Model (2) Equation If you continue to substitute for expected future market prices (i.e., for E(P 2,MP ), E(P 3,MP ), etc.) indefinitely you get: Description: E 1+ ( D1) E( k) E( D2) ( 1+ E( k) ) P 0,IV Intrinsic value of stock at t=0 E(k)...required return/discount rate E( D3) ( 1+ E( k) ) P0, IV = E(D t ) expected dividend payment in period t... Slide 12 Week 7

68 A More General Model: Dividend Discount Model (3) Equation How to use this model? we need more assumptions on future dividends Assumption 1: constant dividend E(D 1 )=E(D 2 )= =E(D) P IV = 0, E( D) E( k) Assumption 2: dividend that grows at the constant rate g ( D ) = D ( 1 g) E ( D ) = D ( ) 2 E g P0, IV E( D ) ( E( k) g) = 1 Slide 13 Week 7

69 A More General Model: Dividend Discount Model (4) Example High Flyer Industries has just paid its annual dividend of D 0 =$3 per share. The dividend is expected to grow at a constant rate of g=8% until infinity. The beta of High Flyer stock is 1.0, the risk-free rate is 6%, and the market risk premium is 8%. What is the intrinsic value of the stock? E(k)=6%+1.0 8%=14% P 0,IV =$54 P ( ) ( ) $3 0, IV = = $54 What is the intrinsic value if the beta changes to 1.25? E(k)=6% %=16% P 0,IV =$40.50 Slide 14 Week 7

70 A More General Model: Dividend Discount Model (5) Exercise IBX s stock dividend at the end of this year is expected to be E(D 1 )=$2.15, and it is expected to grow at g=11.2% per year forever. If the required rate of return on IBX stock is E(k)=15.2% per year, what is its current intrinsic value, P 0,IV? What is IBX next year s expected intrinsic value, P 1,IV? If an investor were to buy IBX stock now and sell it after receiving the $2.15 dividend a year from now, what is the expected holding-period return? Slide 15 Week 7

71 A More General Model: Dividend Discount Model (6) Some More Remarks Constant Growth Dividend Discount Model: Only valid if g is less than k What happens if g is larger than k? Why can this not be the case? Implications of the constant growth DDM: a stock s value will be greater The larger the expected dividend per share The lower the market capitalization rate, k The higher the expected growth rate of dividends, g Slide 16 Week 7

72 Multistage Growth Dividend Discount Models (1) Consider a firm that quickly grows in its early years and then, once it matures, yields a dividend with constant growth until infinity. Dividends in the early years: $.32, $.41, $.50, $.60 Constant growth of dividends thereafter: g=9.375% The market capitalization rate (i.e., E(k)) equals 13.1%. The current intrinsic value is ( P ) $. 32 $.41 $.50 $.60 4, MP P 0, IV = E Slide 17 Week 7

73 Multistage Growth Dividend Discount Models (2) Determine E(P 4,MP ): assume that the stock trades for its intrinsic value in four years; thus we must calculate the intrinsic value of a constant growth dividend discount model in four years E ( P ) E( P ) ( ) ( ) $.60 4, MP = 4, IV = = $17.62 Plug E(P 4,MP ) back into our equation for P 0,IV : P $.32 $.41 = $ $ $ , IV + = $12.08 Slide 18 Week 7

74 Multistage Growth Dividend Discount Models (3) Exercise Consider the firm of the previous example. Now assume that its beta equals 1.0 (the risk-free rate is 5% and the market risk premium is 6%). Calculate the new discount rate. Determine the current intrinsic value of this company. Slide 19 Week 7

75 Determination of Dividend Growth Rate (1) Problem: how to come up with an estimate of dividend growth rate g Solution: look at the firm s earnings (rather profits) and investment strategy; basically a firm has two choices to spend its profits 1. Reinvest them into the firm 2. Pay them out as dividends 3. A combination of (1) and (2) Slide 20 Week 7

76 Determination of Dividend Growth Rate (2) Equation The dividend growth rate depends on the reinvestment rate and on the return generated on these reinvestments. Description: b earnings reinvestment rate, earnings retention rate ROE return on equity; rate of return earned on reinvestments g...dividend growth rate g = ROE b Slide 21 Week 7

77 Determination of Dividend Growth Rate (3) Example Consider a firm with $100 million of assets (all equity-financed) and 3 million shares outstanding. The Return on Equity (ROE) equals 15%. Earnings account for $15 million or $5 per share. Management decides to reinvest b=60%. i.e., = $9 million The capital stock of the firm increases by $9 million, i.e., 9% (given the initial capital of $100 million). Endowed with 9% more capital the firm earns 9% more income per year and dividends grow by 9% per year. g = ROE b = 15 % 0.60= 9% Slide 22 Week 7

78 Determination of Dividend Growth Rate (4) Different Retention Rates What is the influence of different retention rates on the firm value? If two firms are identical but one reinvests 60% and the other one reinvests 0% which one is going to have a higher firm value? Answer: this depends on the profitability of the reinvestment, i.e., compare the ROE to the required rate of return as implied by the CAPM. Example: Consider a firm with $100 million of assets (all equity-financed) and 3 million shares outstanding. The Return on Equity (ROE) equals 15%. The risk-adjusted discount rate implied by the CAPM is k=12.5%. Earnings account for $15 million or $5 per share. Management decides to reinvest b=60%. Slide 23 Week 7

79 Determination of Dividend Growth Rate (5) Example From slide 22 we know that if b=60% then g=9%. The value of the firm in this case is: ( 1 0.6) $ 2 D1 = $5 = D $2 1 P0, IV = = = ( k g) ( ) $57.14 If the firm does not reinvest, there is no dividend growth and the value is: P $ = = = k D $40 Slide 24 Week 7

80 Determination of Dividend Growth Rate (6) Exercise Consider the firm from before. Now assume that the ROE equals 12.5%. Does an earnings retention ratio (i.e., b or reinvestment ratio) of 60% increase firm value relative to the case of 0% earnings retention ratio? Explain your result. Consider the firm from before. Now assume that the ROE equals 10%. Does an earnings retention ratio (i.e., b or reinvestment ratio) of 60% increase firm value relative to the case of 0% earnings retention ratio? Explain your result. Slide 25 Week 7

81 Finance 3050 Homework #1 Spring 2007 Consider the following 3 investments: Investment E(R) F A Cov(a,b) = B Cov(a,c) = C Cov(b,c) = On graph paper, plot the three investments. Use F on the x-axis, and E(R) on the y-axis. 2. Calculate the portfolio E(R) and F for each of the following combinations: 10%A - 90%B 10%A - 90%C 10%B - 90%C 50%A - 50%B 50%A - 50%C 50%B - 50%C 90%A - 10%B 90%A - 10%C 90%B - 10%C Plot each point on the graph 3. Find the minimum variance portfolio proportions for each two asset portfolio, (A,B); (A,C); (B,C). The minimum variance portfolio can be found by the formula Where W 1 = proportion of the portfolio in investment 1, and (1- W 1 )= proportion of the portfolio in investment 2 Calculate the E(R) and F for each minimum variance portfolio. Plot each variance- minimizing portfolio on the graph. Draw a curve showing the portfolio possibilities for each of the three pairs (A,B); (A,C); (B,C). Use a different color for each curve. 4. Suppose investors could only invest in A,B,C or some combination of any two (but not all three). a) Identify on your graph a segment where all investors would want to select a portfolio. b) Might any investor want to invest in A alone? What about B? What about C? c) Would investors want to invest in any combination of A&B?, A&C?, B&C?

82 Finance 3050 Homework #2 Spring 2007 I. Visit the website and gather the following information for Pfizer, Inc, (ticker symbol: PFE): 1. The closing price for PFE s common stock on Tuesday. 4/10/07 2. PFE s current annual dividend 3. PFE s EPS for last year 4. The current (as of 4/10/07) analysts consensus estimate of PFE s EPS growth for the next five years (per annum) II. Assuming the following information: E(r m ) = % R f = 5.25% PFE s beta = 0.70* III. Calculate the following: a. Based on part II, what rate of return should PFE s shareholders require on their investment? b. If PFE retains 62% of its earnings, and can earn a return of 10% on reinvested funds, at what rate can dividends grow indefinitely? c. Based on your answer to (a) and information obtained in part I, if PFE s dividend (Part I, item 2) is expected to grow indefinitely at the rate found in part (b), what is a share of PFE stock worth today? (Assume that the dividend you found in part I is D 0 ) d. By how much is PFE stock overpriced or underpriced when comparing your answer to part (c) with the current price (part I, item 1)? e. What is PFE s dividend yield? f. If PFE s earnings grow at the estimated growth rate for 5 years (item I, part 5), what will EPS be at the end of 2011 (begin with the EPS for 06 [part 1, item 3])? g. Using your answer to (f), and assuming that PFE s (trailing) P/E ratio is 11 at the end of 2011, what should a share of stock sell for at that time? h. Assume that dividends will not grow for the next 5 years. If you buy a share of PFE stock at the current price (part I, item 1), receive the current dividend (part I, item 2) for each of the next 5 years, then sell the stock for the price found in (g), what will be your dollarweighted average annual rate of return (Calculate IRR using P 0, D 1 -D 5 and P 5 )? * Ignore any actual beta for PFE that you may find in yahoo or any other source.

83 Finance 3050 Midterm #1 Part I - Problem Solving 1.a. Suppose you sell short 300 shares of JNJ at $55. You put in $8,250 of your own money for margin purposes. If the price of JNJ falls to $47, how much do you gain or lose as a percentage of your original investment? b. If the price of JNJ falls to $45 and a dividend of $1/share is paid, what is the % margin in your account? 2. Municipal bond A pays interest at the rate of 3.5%, while Corporate bond B pays interest at the rate of 5.50%. If Lynette pays tax at a marginal rate of 36%, which bond should she choose to invest in? Support your answer with figures. 3. Suppose you buy 400 shares of CSCO at $25.00 using $5,000 of your own money and borrowing the other $5,000 on margin. If the maintenance margin must be 25%, how low can the price of CSCO go before a margin call is issued? (Ignore interest) Midterm #2 1. If inflation in a certain economy is expected to be 150%, what nominal rate of interest must be charged to earn a real rate of return of 8%? 2 Douglas earns the following rates over four years of investing: r 1 =.22, r 2 =.12, r 3 = -.08, r 4 =.09. What is the geometric average of these returns? 3. Suppose you manage a portfolio with E(r) = 16% and F = 30% The risk-free rate is 2%. a. If a client of yours chooses to invest 65% of her wealth in your portfolio and the rest in t- Bills, what would be the E(r) and F of her complete portfolio? b. Ignoring part (a) above, if your client wanted to achieve an E(r) of 10% on her allocation portfolio, what proportion of her wealth would she need to invest in your portfolio? Final Exam 1. A stock is expected to pay a dividend of $1.48/share at the end of each of the next three years. At that time, the stock is expected to have year-end earnings of $5.95 and a P/E ratio of 19. If investors require a return of 11% on this stock, how much should a share be worth today? 2. Fairview Industrial Products Inc. Is expected to have year-end earnings (E 1 )of $7.85/share. Fairview intends to plowback 60% of earnings on which a return (ROE) of 12% is expected. What is the PVGO for Fairview if shareholders require a 10% return on their investment?

84 3. BMC equipment inc. has just paid a dividend (D 0 ) of $3.28. Dividends are expected to grow at a rate of 5.2% indefinitely. If BMC is correctly (fairly) priced at $31.95, what is the market capitalization rate (k) for this stock? 4. Suppose you buy one put option contract (100 shares) on Whitbury co. with an exercise price of $40 and six months to expiration. The current price of the stock is $44.35 and the premium on the option is $.68. If the price of the stock is $38.75 on the expiration date of the option, what is your profit (or loss) in dollars and as a percent of your original investment (all prices given are per share)? 5. Find the value of a call option with the following characteristics: Today s price of the underlying stock (S 0 ): $50 Price of the stock in one year (S 1 ): $65 or $37 Exercise price of the call option (X): $45 Risk-free interest rate (r f ): 3.5%

85 1 st Homework Assignment Due on September 16 th before class Important information (read carefully): a. The first page of the homework that you turn in must be the standardized Cover Page (i.e., the first page of this document). Clearly indicate on your cover sheet which exercises you solved and are ready to present in class (these are the exercises which are graded). b. Make a copy of your solutions before you turn them in. c. Show your calculations. You might get (partial) credit for wrong solutions if your way of approaching the question makes sense. No credit will be given to correct solutions that are not accompanied by the appropriate calculations. Problem 1 A T-bond expires in 4 years (assume that the bond was just issued). It has a coupon rate of 8%, a face value of $1000 and pays coupons annually. The asked price is quoted as 102:10 while the bid price is 101:20. a. What is the asked price expressed in dollar terms? b. What is the bid price expressed in dollar terms? c. If you hold the bond until maturity, what is going to be your holding period return? What is going to be the annualized holding period return? d. What is the internal rate of return if you invest into the bond? Solution Problem 1 a. The asked price is P(asked)=(102+10/32)% of $1000=$ b. The bid price is (101+20/32)% of $1000 = $ c. Price at maturity = 1000; intermediate CFs are 4*80=320; price at which you bought the bond is the ask price: HPR=( )/ =0.29; the annualized HPR is (1+0.29)^ = d. The bond pays 4 payments of $80 every year in addition to the par of $1000 in the end of the fourth year. The price of the bond is your answer to part a., $ You need to find the interest rate that this cash flow stream provides (N=4, PV= , FV=1000, PMT=80, P/YR=1, I/YR=?). The answer is Alternatively, you can specify the CFs explicitly: CFj= , CFj=80, CFj=80, CFj=80, CFj=1080, IRR/YR=???. The answer is

86 Problem 2 a. You own $2847. The interest rate is 6.43% p.a. and compounding happens daily (assume that there are 250 business days per year). What is your future value after 3 years? b. Now again you own $2847 and you want to invest them for 20 years. What must be the interest rate such that you end up with exactly $5000 after 20 years if compounding happens yearly? c. Again you start out with $2847. The interest rate is 4.65% p.a.. How long do you have to invest your money in order to get $3500 if there is monthly compounding? Solution Problem 2 a. $2847*(1+6.43%/250)^(250*3)=$ (N=250*3=750; PV=-2847; I/YR=6.43 (enter like this); P/YR=250; FV=?) b. I/YR=2.856% (N=20; P/YR=1; FV=5000; I/YR=?) or 2847*(1+r)^20=5000; r=(5000/2847)^(1/20)-1= c. N=53.4 (I/YR=4.65; FV=3500; P/YR=12; N=?); 2847*( /12)^N=3500; N=ln(3500/2847)/ln( /12)=53.4 months Problem 3 to 4: The following information is relevant for solving Problem 3 to 4. Consider the following price and dividend history for stock XYZ: Year Beginning-of-Year Price Dividend Paid at Year-End 2002 $100 $ $110 $ $90 $ $95 $4 An investor buys three shares of XYZ at the beginning of 2002, buys another two shares at the beginning of 2003, sells one share at the beginning of 2004, and sells all four remaining shares at the beginning of Problem 3: a. What are the arithmetic and geometric average 1-year holding period returns of this investment? b. What is the 2-years holding period return in 2004 (i.e., the 2-years holding period return in the period 2002 to 2004)? c. What is the 3-years holding period return of this investment in 2005? Solution Problem 3-2 -

87 (A) The time-weighted average returns are based on year-by-year rates of return. Year Return = (Capital Gain+Dividend)/price ( )/100=14.00% ( )/110=-14.55% ( )/90=10.00% Arithmetic Mean: (14%-14.55%+10%)/3=3.15% Geometric Mean: [(1+0.14)*( )*(1+0.1)]^(1/3)-1=2.33% (B) ( )/100 = (C) ( )/100 = 0.07 Problem 4: What is the investor s internal rate of return? (Hint: Carefully prepare a table with all the cash flows for the four dates.) Solution Problem 4 Prepare a table of Cash-Flows Time Cash-Flow Explanation Buy three shares at Buy another two shares at 110 plus dividend income on three shares held *4=110 Dividends on 5 shares plus sale of one share at $90 3 4*95+4*4=396 Dividend on four shares, plus sale of four shares at 95 Calculate the internal rate of return: /(1+r)^1+110/(1+r)^2+396/(1+r)^3=0 (P/YR=1; CFj=-300; CFj=-208; CFj=110; CFj=396; IRR/YR=?) IRR=-0.166% Problem 5 to 8: The following information is relevant for solving Problem 5 to 8. Assume that you manage a risky portfolio, called Your Fund, with an expected rate of return of 17% and a standard deviation of 27%. The T-Bill rate (i.e., the risk-free rate) is 7%. The risky portfolio includes the following investments in the given proportions: Problem 5: Stock A 27% Stock B 33% Stock C 40% - 3 -

88 a. Your client chooses to invest 70% of a portfolio in Your Fund and 30% in the T-Bill. What is the expected return and standard deviation of your client s portfolio? b. What are the investment proportions of your client s overall portfolio, i.e. your client s position in stock A, stock B, stock C and the T-Bill? c. What is the reward-to-variability ratio (S) of Your Fund and your client s overall portfolio? d. Draw the CAL of Your Fund on an expected return/standard deviation diagram. What is the slope of the CAL? Show the position of your client on your fund s CAL. Solution Problem 5 a. E(rp)=0.3*7%+0.7*17%=14%; SDp=0.7*27%=18.9% b. T-Bills = 30%; Stock A = 0.7*27%=18.9%; Stock B = 0.7*33%=23.1%; Stock C = 0.7*40%=28.0% c. Your S = (17-7)/27=0.3704; your client s S = (14-7)/18.0 = E(r) 17 % P CAL ( slope=.3704) 14 client 7 d % σ Problem 6: Suppose the same client from the previous problem decides to invest in Your Fund a proportion y of his total investment budget so that his overall portfolio will have an expected rate of return of 15%. a. What is the proportion y? b. What are your client s investment proportions in your three stocks and the T-Bill? c. What is the standard deviation of the rate of return on your client s portfolio? - 4 -

89 Solution Problem 6 a. E(rp)=(1-y)*7%+y*17%=15%; 15%-7%=y*(-7%+17%); y=(15-7)/10=0.8 b. T-Bills = 20%; Stock A = 0.8*27%=21.6%; Stock B = 0.8*33%=26.4%; Stock C = 0.8*40%=32.0% c. SDp=0.8*27%=21.6% Problem 7: Suppose the same client from the previous problem decides this time to invest in Your Fund a proportion y of his total investment budget so that the rate of return of his overall portfolio will have a standard deviation of 20%. a. What is the investment proportion y? b. What is the expected rate of return on your client s portfolio? Solution Problem 7 a. 20%=y*27%; y=20%/27%; y= b. E(rp)=( )*7% *17%=14.407% Problem 8: You estimate that a passive portfolio invested to mimic the S&P 500 stock index yields an expected rate of return of 13% with a standard deviation of 25%. Draw the CML and Your Fund s CAL on an expected return/standard deviation diagram. a. What is the slope of the CML? b. Characterize in one short paragraph the advantage of Your Fund over the passive fund. Solution Problem 8 a. Slope of the CML: (13-7)/25=0.24 (scan the graph from the solution manual) b. My fund allows an investor to achieve a higher expected rate of return for any given standard deviation than would a passive strategy. Problem 9 to 10: The following information is relevant for solving Problem 9 to 10. Imagine that the returns of the two stocks, Amgen and Coca Cola, which caught your attention when you last flipped through the WSJ depend on the overall market conditions in the following way: Amgen would make a return of -20% (i.e. a loss) if markets are bearish, 18%, if they are normal, and 50%, if markets are bullish. The corresponding numbers are -15%, 20%, and 10% for Coca Cola. The probability of Bear Markets is 20%, of Normal Markets is 50%, and of Bull Markets is 30%. This information is summarized in the table below: - 5 -

90 Bear Market Normal Market Bull Market Probability Return of Amgen -20% 18% 50% Return of Coca Cola -15% 20% 10% Problem 9 What are the expected returns and standard deviations of the returns of Amgen and Coca Cola? Solution Problem 9: expected return Expected Return of Amgen=E[rA] = (Probability of Bear Market)*(Return of Amgen in Bear Market) + (Probability of Normal Market)*(Return of Amgen in Normal Market) + (Probability of Bull Market)*(Return of Amgen in Bull Market) = = 0.2 *(-20%)+0.5*(18%)+0.3*(50%)=0.2 *(-0.2)+0.5*(0.18)+0.3*(0.50)= 20% = 0.2 Expected Return of Coca Cola=E[rCC]=(Probability of Bear Market)*(Return of Coca Cola in Bear Market) + (Probability of Normal Market)*(Return of Coca Cola in Normal Market) + (Probability of Bull Market)*(Return of Coca Cola in Bull Market) = = 0.2 *(-15%)+0.5*(20%)+0.3*(10%) = 0.2 *(-0.15)+0.5*(0.20)+0.3*(0.10) = 10% = 0.1 Solution Problem 9: standard deviations Variance[rA] = (Probability of Bear Market) * ( Return of Amgen in Bear Market - E[rA]) 2 + (Probability of Normal Market) * ( Return of Amgen in Normal Market - E[rA]) 2 + (Probability of Bull Market) * ( Return of Amgen in Bull Market -E[rA]) 2 = = (0.2) * ( ) 2 + (0.5)*( ) 2 +(0.3)*( ) 2 = (0.2)*(-0.4) 2 + (0.5)*(-0.02) 2 +(0.3)*(0.3) 2 = (in decimals) or 592 if you used percentages SD is the square root of the variance: ^(1/2)= (or 24.33%) Variance[rCC] = (Probability of Bear Market)*( Return of Coca Cola in Bear Market - E[rCC] ) 2 + (Probability of Normal Market)*( Return of Coca Cola in Normal Market -E[rCC]) 2 + (Probability of Bull Market)*( Return of Coca Cola in Bull Market -E[rCC]) 2 = (0.2)*( ) 2 + (0.5)*( ) 2 +(0.3)*( ) 2 = (0.2)*(-0.25) 2 + (0.5)*(0.1) 2 +(0.3)*(0) 2 = (in decimals) or 175 if you used percentages SD is the square root of the variance: ^(1/2)= (or 13.23%) Problem 10 Assume that of your $10,000 portfolio you invest $9000 in Stock Amgen and $1000 in Coca Cola

91 a. Calculate the returns on your portfolio for every market condition, i.e., Bear Market, Normal Market and Bull Market. b. What is the expected return on your portfolio? c. What is the standard deviation of the returns on your newly formed portfolio? Solution Problem 10 The weight on Amgen in this portfolio is w(amgen) = 9/10 = 0.9. The weight on Coca Cola is w(coca Cola) = 1/10 = 0.1. a. The returns in the different market conditions are: i. Bear Market: -19.5% ii. Normal Market: 18.2% iii. Bull Market: 46% b. E[rP]= w(amgen)*e[ra] + w(coca Cola)*E[rCC] = 0.9* *0.1 = = 0.19 (in decimals); alternatively, you can calculate 0.2*(-19.5) + 0.5* *46 = 19 c. Variance[rP] = (Probability of Bear Market)*(Return of Portfolio in Bear Market -E[rP]) 2 + (Probability of Normal Market)*( Return of Portfolio in Normal Market-E[rP]) 2 + (Probability of Bull Market)*(Return of Portfolio in Bull Market -E[rP]) 2 = = (0.2)*( ) 2 + (0.5)*( ) 2 +(0.3)*( ) 2 = (0.2)*( ) 2 + (0.5)*(-0.008) 2 +(0.3)*(0.27) 2 = SD = ^(1/2)=

92 FINAN 3050 Introduction to Investments Cover Page 2 nd Homework Assignment Student name: Student ID: Mark exercises solved: Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 Problem 6 Problem 7 Problem 8 Problem 9 Problem 10 Comments: - 1 -

93 2 nd Homework Assignment Important information (read carefully): a. The first page of the homework that you turn in must be the standardized Cover Page (i.e., the first page of this document). Clearly indicate on your cover sheet which exercises you solved and are ready to present in class (these are the exercises which are graded). b. Make a copy of your solutions before you turn them in. c. Show your calculations. You might get (partial) credit for wrong solutions if your way of approaching the question makes sense. No credit will be given to correct solutions that are not accompanied by the appropriate calculations. Problem 1-3: Abigail Grace has a $900,000 fully diversified portfolio. She subsequently inherits common stock of company MyStock worth $100,000. Her financial advisor provided her with the following forecasted information: Expected Monthly Standard Deviation Returns of Monthly Returns Original Portfolio 0.67% 2.37% MyStock 1.25% 2.95% The correlation coefficient of MyStock stock returns with the original portfolio returns is Problem 1: Assuming Grace keeps the MyStock stock, calculate the: a. Expected monthly return of the new portfolio which includes the Mystock stock b. Covariance of MyStock stock returns with the original portfolio returns. c. Standard deviation of her new portfolio which includes the MyStock stock. Solution Problem 1: - 2 -

94 a. ( ) + ( ) = 0.728% b. COV = = c. SD = [(0.9^2 2.37^2) + (0.1^2 2.95^2) + ( )]^0.5 = % Problem 2: Assuming Grace sells the MyStock stock and invests the proceeds in risk-free government securities yielding 0.42% per month, calculate the: a. Expected monthly return of the new portfolio which includes the government securities b. Covariance of the government security returns with the original portfolio returns. [Hint: think about how much covariation there can be between a risky and a riskfree asset?] c. Standard deviation of her new portfolio which includes the government securities. Solution Problem 2: a. ( ) + ( ) = 0.645% b. COV = = 0 c. SD = [(0.9^2 2.37^2) + (0.1^2 0) + ( )]^0.5 = 2.133% Problem 3: Based on conversations with her husband, Grace is considering selling the $100,000 of MyStock stock and acquiring $100,000 of OtherStock common stock instead. OtherStock stock has the same expected return and standard deviation as MyStock stock. Her husband comments, It does not matter whether you keep all of the MyStock stock or replace it with $100,000 of OtherStock stock. State whether her husband s comment is correct or incorrect. Justify your response. Solution Problem 3: The comment is not correct. It also depends on the covariances between each security and the original portfolio

95 Problem 4: An investor can design a risky portfolio based on two stocks, A and B. Stock A has an expected return of 18% and a standard deviation of return of 20%. Stock B has an expected return of 14% and a standard deviation of return of 5%. The correlation coefficient between the returns of A and B is 0.5. The risk-free rate of return is 10%. The proportion of the optimal risky portfolio that should be invested in stock A is? A. 40% B. 100% C. 0% D. 60% [Hint: think about the condition that defines the optimal risky portfolio slide 4 of week 4 and check this condition for the four choices.] Solution Problem 4: A. 0.4* *14 = 15.6; Variance = 97 SD = 9.8; S = ( )/9.8 = 0.57 B. 1.0* *14 = 18.0; SD = 20; S = (18-10)/20 = 0.4 C. 0.0* *14 = 14.0; SD = 5; S = (14-10)/5 = 0.8 D. 0.6* *14 = 16.4; Variance = 172 SD = 13.1; S = ( )/13.1 = 0.5 Problem 5: Karen Kay, a portfolio manager at Collins Asset Management, is using the Capital Asset Pricing Model for making recommendations to her clients. Her research department has developed the information shown in the following table: Return Forecast by Research Dep. Standard Deviation Beta Stock X 14.0% 36% 0.8 Stock Y 17.0% 25% 1.5 Market Index 14.0% 15% 1.0 Risk-Free Asset 5.0% - 4 -

96 Calculate the expected return and alpha for each stock. Solution Problem 5: For stock X: E(r) = 5% (14%-5%) = 12.2% Alpha(x) = 14% % = 1.8% For stock Y: E(r) = 5% (14%-5%) = 18.5% Alpha(y) = 17% % = -1.5% Problem 6: Imagine that this time you have two stocks named ABC and XYZ whose returns depend on the state of the world economy. The market portfolio constructed of the shares of firms (including the ones of ABC and XYZ) has a return that also depends on the state of the world economy. Recession Regular Boom Probability Return of ABC 10% 20% -10% Return of XYZ -20% 25% 10% Return of Market Portfolio 0% 5% 10% Return of Riskless Asset 5% 5% 5% a. What is asset ABC s market Beta? b. What is asset XYZ s market Beta? c. Imagine that you invest half of your funds in ABC, and the other half in XYZ, what is the market Beta of this portfolio? Solution Problem 6: (a) For asset ABC: Expected Return of ABC: (-10) = 10% - 5 -

97 Expected Return of Market: = 5% (b) For asset XYZ: Expected Return of XYZ: 0.25 (-20) = 10% Expected Return of Market: = 5% (c) For a portfolio: Beta(p) = (-2) = 0.5 Problem 7: Biopharma is a pharmaceutical company. Biopharma s annual stock returns have a CAPM beta of 1.25 (i.e. β=1.25). The market portfolio s return is 13%, and the risk-free rate is 5%. a. What is the required expected return for Biopharma according to the CAPM? b. The firm has the opportunity to develop a new drug. This project requires an initial outlay of $400,000 and it will bring expected revenues of $100,000 in each of the next 6 years. The riskiness of this project is the same as the overall riskiness of Biopharma. Should the management team of Biopharma do the project? Or should it rather not do it? Explain your answer. Solution Problem 7: (a) Required expected return: - 6 -

98 (b) Investment decisions: The PV equals 378, and the Net Present Value (NPV) is, as a consequence, negative (-400, ,448.27). Alternatively you can calculate the project s IRR (CFj -400,000; then six times the CFj 100,000) IRR = 12.98% IRR < 15% do not undertake the project. Problem 8: If the simple CAPM is valid, is the following situation possible? Explain your answer. Portfolio Expected Return Beta A 20% 1.4 B 25% 1.2 Solution Problem 8: Not possible. Portfolio A has a higher beta than Portfolio B, but the expected return for portfolio A is lower. Problem 9: If the simple CAPM is valid, is the following situation possible? You also know that the Market Risk Premium (E(r M )-r f ) equals 20%. Explain your answer. [Hint: plug everything you know into the CAPM equation slide 14 of week 4 and verify that the implied riskfree rate makes sense.] Portfolio Expected Return Beta A 25% 1.5 B 5% 0.5 Solution Problem 8: Not possible. Plug the values into the CAPM equation and solve for the risk-free rate: 25 = r f + 1.5*20 and 5 = r f + 0.5*20 the risk-free rate would have to be minus 5% to make these relationships hold this does not make sense! - 7 -

99 Problem 10: If the simple CAPM is valid, is the following situation possible? Explain your answer. [Hint: Think about the optimality of the Market portfolio (refer to slide 4 and slide 7 of week 4). Verify that the Market portfolio is optimal in that sense.] Portfolio Expected Return Standard Deviation Risk-Free 10% 0% Market 18% 24% A 16% 12% Solution Problem 10: Not possible. The reward-to-variability ratio for Portfolio A is better than that of the market, which is not possible according to the CAPM, since the CAPM predicts that the market portfolio is the most efficient portfolio. S(A) = (16-10)/12 = 0.5 S(M) = (18-10)/12 =

100 FINAN 3050 Introduction to Investments Cover Page 3 rd Homework Assignment Student name: Student ID: Mark exercises solved: Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 Problem 6 Problem 7 Problem 8 Problem 9 Problem 10 Problem 11 Problem 12 Comments: - 1 -

101 Problem 1: 3 rd Homework Assignment Find the price of a coupon bond (par value of $1000) which pays 12% coupon semiannually, matures in 5 years, and has a yield to maturity of 10%. Solution: FV=1000, PMT=60, i=5% (per 6-month period), n=10 PV= Problem 2: A coupon bond which pays coupon of $100 annually has a par value of $1000, matures in 5 years, and is selling today at a $72 discount from par value (i.e., at $928). Find the yield to maturity of this bond. Solution: FV=1000, PMT=100, n=5, PV= -928 i=12% Problem 3: Rank the following bonds in order of descending duration (note that you should not actually calculate the duration). Bond Coupon Time to Maturity Yield to Maturity A 15% 20 years 10% B 15% 15 years 10% C 0% 20 years 10% D 8% 20 years 10% E 15% 15 years 15% Solution: C: Highest maturity, zero coupon D: Highest maturity, next-lowest coupon A: Highest maturity, same coupon as remaining bonds B: Lower yield to maturity than bond E E: Highest coupon, shortest maturity, highest yield of all bonds

102 Problem 4: Consider a zero-coupon bond (face value of $1000) that matures in 3 years. The YTM is 10%. Find the price, the duration and the convexity of this bond. Solution: 1. Price: 1000/1.1 3 = Duration = 3 3. Convexity = 1/ (3+1) 1 = 9.92 Problem 5: Consider a 15%-coupon bond that matures in 3 years (face value of $1000). The YTM is 10%. Find the price, the duration and the convexity of this bond. Solution: Convexity: 1/1.1 2 ( ) = Problem 6: You purchased a 5-year annual interest coupon bond one year ago. Its coupon interest rate was 6% and its par value was $1,000. At the time you purchased the bond, the yield to maturity was 4% (and the time to maturity was 5 years). If you sold the bond after receiving the first interest payment and the bond's yield to maturity had changed to 3% (and the time to maturity is now 4 years), your annual total rate of return on holding the bond for that year would have been

103 Solution: 7.57% 1. Buy price (one year ago; 5 years to maturity; YTM 4%): Sell price (4 years to maturity; YTM 3%): Capital Gain: Interest Income: Holding Period Return = ( )/ = Problem 7: Consider the following $1,000 par value zero-coupon bonds: Calculate the expected one-year interest rates one year (from year 1 to 2), two years (from year 2 to 3), three years (from year 3 to 4) and four years (from year 4 to 5) from now. Solution: One Year: (1+0.06) 1 (1+f1) = (1+0.07) 2 r = Two Years: (1+0.07) 2 (1+f2) = ( ) 3 r = 0.10 Three Years: ( ) 3 (1+f3) = ( ) 4 r = Four Years: ( ) 4 (1+r) = ( ) 5 r = Problem 8: You will be paying $10,000 a year in tuition expenses at the end of the next two years. Bonds currently yield 8%. 1. What is the present value and duration of your obligation? 2. If you want to invest only in one zero-coupon bond to immunize your obligation, what does the maturity of this zero-coupon bond have to be? - 4 -

104 3. Verify that your answer to (2.) really immunizes your obligation. For that purpose, suppose the rates immediately increase to 9%. What happens to your net position, that is, to the difference between the value of the bond and that of your tuition obligation? What happens if rates fall to 7%? Solution: 1. The present value of the obligation is $17, and the duration is years. 2. To immunize the obligation, invest in a zero-coupon bond maturing in years. Since the present value of the zero-coupon bond must be $ , the face value must be: = If the interest rate increases to 9%, the zero-coupon bond would fall in value to $ The PV of the tuition obligation would fall to $ , so that the net position changes by $ If the interest rate decreases to 7%, the zero-coupon bond would rise in value to $ The PV of the tuition obligation would rise to $ , so that the net position changes by $0.19. Problem 9: The risk-free rate of return is 10%, the required rate of return on the market is 15%, and High-Flyer stock has a beta coefficient of 1.5. a. If the dividend per share expected during the coming year, D 1, is $2.50 and g=5%, at what price should a share sell? b. If the share sells at $25, what must be the market s expectation of the growth rate of dividends (keeping all other information the same)? Solution: (a) K=10% (15%-10%)=17.5% Therefore, P 0 = $2.50/( ) = $20 (b) $25 = $2.50/(0.175-g) g = 7.5% - 5 -

105 Problem 10: Investment firm SmartInvest wants to value firm SkiUtah. The following information is available for SkiUtah: risk-free rate = 5.0%, expected market return = 12%, beta = SmartInvest expects that SkiUtah s dividends are going to grow by 15% in the first 3 years and by 8% thereafter. The current dividend (that has just been paid out) is $2.10. Find the intrinsic value of SkiUtah using the multistage DDM and the CAPM. Solution: Market capitalization rate: *( ) = Next 3-years dividends are: 2.10*1.15 = 2.415; 2.415*1.15 = 2.777; 2.777*1.15 = Thereafter there is a constant growth rate of 8%: 3.194*1.08/( ) = 74.5 The intrinsic value is: 2.415/1.1263^ /1.1263^ /1.1263^ /1.1263^3 = Problem 11: MF Corp. has an ROE of 16% and a retention rate of 50%. The market capitalization rate k equals 12%. If the coming year s earnings are expected to be $2 per share, at what price will the stock sell? What price do you expect MF shares to sell for in three years? Solution: g=roe b= =0.08 D 1 =$2 (1-0.50)=$1.00 P 0 =$1.00/( )=$25 P 3 =P 0 (1+g) 3 =$31.49 Problem 12: Your preliminary analysis of two stocks has yielded the information set forth below. The market capitalization rate k for both stock A and stock B is 10% per year. Stock A Stock B - 6 -

106 Expected return on equity, ROE 14% 12% Estimated earnings per share $2.00 $1.65 Estimated dividends per share $1.00 $1.00 Current market price per share $27.00 $25.00 Determine the intrinsic value of each stock and decide in which, if either, of the two stocks would you choose to invest. Solution: Stock A: b=50%; g = = 7.0%; V0 = 1/( ) = $33.33 Stock B: b=1-$1/$1.65=0.394; g = = 4.728%; V0 = 1/( )=$18.97 You would choose to invest in Stock A since its intrinsic value exceeds its price. You might choose to sell short stock B

107 FINAN 3050 Introduction to Investments Cover Page 4 th Homework Assignment Student name: Student ID: Mark exercises solved: Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 Problem 6 Problem 7 Problem 8 Problem 9 Problem 10 Comments: - 1 -

108 4 th Homework Assignment Important information (read carefully): a. The first page of the homework that you turn in must be the standardized Cover Page (i.e., the first page of this document). Clearly indicate on your cover sheet which exercises you solved and are ready to present in class (these are the exercises which are graded). b. Make a copy of your solutions before you turn them in. c. Show your calculations. You might get (partial) credit for wrong solutions if your way of approaching the question makes sense. No credit will be given to correct solutions that are not accompanied by the appropriate calculations. Problem 1: Suppose you think Wal-Mart stock is going to appreciate substantially in value in the next year. Say the stock s current price, S 0, is $100, and the call option expiring in one year has an exercise price, X, of $100 and is selling at a price, C, of $10. With $10,000 to invest, you are considering three alternatives: a. Invest all $10,000 in the stock, buying 100 shares. b. Invest all $10,000 in options, buying 1000 options. c. Buy 100 options for $1,000 and invest the remaining $9,000 in a money market fund paying 4% interest annually. What is your rate of return for each alternative for four stock prices one year from now? Price of stock 1 year from now $80 $100 $110 $120 a. All Stocks b. All Options c. Money market fund + options Solution: Price of stock 1 year from now $80 $100 $110 $

109 a. All Stocks b. All Options c. Money market fund + options Price of stock 1 year from now $80 $100 $110 $120 a. All Stocks -20% 0% 10% 20% b. All Options -100% -100% 0% 100% c. Money market fund + options -6.4% -6.4% 3.6% 13.6% Problem 2: The common stock of the P.U.T.T. Corporation has been trading in a narrow price range for the past month, and you are convinced it is going to break far out of that range in the next three months. You do not know whether it will go up or down, however. The current price of the stock is $100 per share, the price of a three-month call option with an exercise price of $100 is $10, and a put with the same expiration date and exercise price costs $7. Consider buying a straddle. Draw a diagram showing the payoff and profit of this strategy at maturity. How far would the price have to move in either direction for you to make a profit on your initial investment? Solution: The straddle costs $17 stock price has to move by at least $17 in either direction. Problem 3: A vertical combination is the purchase of a call with exercise price X 2 and a put with exercise price X 1, with X 2 greater than X 1. Graph the payoff to this strategy (do not graph the profit, i.e., ignore the costs to establish the positions)

110 Solution: See solution graph in the manual on page 14-4 Problem 4: You write a call option with X = $50 and buy a call with X = $60. The options are on the same stock and have the same maturity date. One of the calls sells for $3; the other sells for $9. Draw a graph including the payoff and profit of this strategy at the option maturity date. What is the break-even point for this strategy? Solution: Solution graph is in the manual on page Breakeven occurs when the payoff offsets the initial proceeds of $6, which occurs at a stock price of $56. Problem 5: Joseph Jones, a manager at Computer Science, Inc. (CSI), received shares of company stock as part of his compensation package. The stock currently sells at $40 a share. Joseph would like to defer selling the stock until the next tax year. In January, however, he will need to sell all his holdings to provide for a down payment on his new house. Joseph is worried about the price risk involved in keeping his shares. At current prices, he would receive $400,000 for the stock. If the value of his stock holdings falls below $350,000, his ability to come up with the necessary down payment would be jeopardized. On the other hand, if the stock value rises to $450,000 he would be able to maintain a small cash reserve even after making the down payment. Joseph considers three investment strategies: 1. Strategy A is to write January call options on the CSI shares with a strike price of $45. These calls are currently selling for $3 each. 2. Strategy B is to buy January put options on CSI with strike price $35. These options also sell for $3 each. 3. Strategy C is to write the January call options and buy the January put options (this strategy is called a zero-cost collar)

111 Evaluate each of these strategies with respect to Joseph s investment goals. What are the advantages and disadvantages of each? Which would you recommend? Solution: By writing covered call options, Jones receives premium income of $30,000. If, in January, the price of the stock is less than or equal to $45, he will keep the stock plus the premium income. Since the stock will be called away from him if its price exceeds $45 per share, the most he can have is: $450,000 + $30,000 = $480,000 (We are ignoring interest earned on the premium income from writing the option over this short time period.) The payoff structure is: Stock price Portfolio value Less than $45 (10,000 times stock price) + $30,000 Greater than $45 $450,000 + $30,000 = $480,000 This strategy offers some premium income but leaves the investor with substantial downside risk. At the extreme, if the stock price falls to zero, Jones would be left with only $30,000. This strategy also puts a cap on the final value at $480,000, but this is more than sufficient to purchase the house. By buying put options with a $35 strike price, Jones will be paying $30,000 in premiums in order to insure a minimum level for the final value of his position. That minimum value is: ($35 10,000) $30,000 = $320,000 This strategy allows for upside gain, but exposes Jones to the possibility of a moderate loss equal to the cost of the puts. The payoff structure is: Stock price Portfolio value Less than $35 $350,000 $30,000 = $320,000 Greater than $35 (10,000 times stock price) $30,000 The net cost of the collar is zero. The value of the portfolio will be as follows: Stock price Portfolio value Less than $35 $350,

112 Between $35 and $45 Greater than $45 $450,000 10,000 times stock price If the stock price is less than or equal to $35, then the collar preserves the $350,000 in principal. If the price exceeds $45, then Jones gains up to a cap of $450,000. In between $35 and $45, his proceeds equal 10,000 times the stock price. The best strategy in this case is (c) since it satisfies the two requirements of preserving the $350,000 in principal while offering a chance of getting $450,000. Strategy (a) should be ruled out because it leaves Jones exposed to the risk of substantial loss of principal. Our ranking is: (1) c (2) b (3) a Problem 6: Draw the payoff diagram for the following investment strategy using ABC share and call options on ABC shares: 1. Buy a share of ABC 2. Write a call on ABC with X=50 3. Write a call on ABC with X=60 4. Buy a call on ABC with X=110 Solution: Devise a portfolio using only call options and shares of stock with the following Buy a share of stock, write a call with X = 50, write a call with X = 60, and buy a call with X = 110. Position ST < < ST < < ST < 110 ST > 110 Buy stock ST ST ST ST Short call (X = 50) 0 (ST 50) (ST 50) (ST 50) Short call (X = 60) 0 0 (ST 60) (ST 60) Long call (X = 110) ST 110 Total ST ST 0 The investor is making a volatility bet. Profits will be highest when volatility is low so that the stock price ends up in the interval between $50 and Look at the picture in the book at page

113 Problem 7: A put option with strike price $60 currently sells for $2. To your amazement, a put on the same firm with same expiration date but with a strike price of $62 also sells for $2. How can you exploit this obvious mispricing using a simple zero-net-investment strategy (i.e., a strategy that has zero net investment costs) that gives you a non-negative payoff at expiration of the options with certainty? Also draw the payoff diagram at expiration in order to verify that your strategy yields a non-negative payoff for every stock price S T. Solution: Buy the X = 62 put (which should cost more than it does) and write the X = 60 put. Since the options have the same price, the net outlay is zero. Your proceeds at maturity may be positive, but cannot be negative. Position ST < < ST < 62 ST > 62 Long put (X = 62) 62 ST 62 ST 0 Short put (X = 60) (60 ST) 0 0 Total 2 62 ST 0 Payoff = Profit (because net investment = 0) S T Problem 8: Assume the current stock price, S 0, equals $100. In one year the stock price can either go up to $200 or down to $50. The risk-free interest rate equals 8%. Determine the price of a call with strike price, X, being equal to $

114 Solution: H=0.833 buy 5 stocks and write 6 options Risk-free payoff equals: = 250 and 5 50 = C=250/1.08 C=44.75 Problem 9: Assume the current stock price, S 0, equals $100. In one year the stock price can either go up to $300 or down to $25. The risk-free interest rate equals 8%. Determine the price of a call with strike price, X, being equal to $75. Solution: H=0.8181= buy 9 stocks and write 11 options Risk-free payoff equals: = 225 and 9 25 = C=225/1.08 C=62.88 Problem 10: Consider the call options given in Problem 8 and Problem 9. Determine the appropriate prices of put options with a strike price of X=$75 using the put-call parity relationship. Solution: P = C-S 0 +PV(X) Given the data in Problem 9: P = /1.08 = Given the data in Problem10: P = /1.08 =

115 Introduction to Investments FINAN 3050 Week 3: Asset allocation: risky vs. risk-free assets (Chp. 5.5,5.6) Asset allocation: optimal risky portfolios (Chp. 6.1,6.2)

116 Allocating Capital between Risky and Risk-Free Assets Overview Goal: split investment between safe and risky assets Safe asset (risk-free asset): T-bills Risky asset: portfolio of stocks Issues: Examine the risk-return tradeoff Influence of investor s risk aversion on the allocation Slide 2 Week 3

117 Allocating Capital between Risky and Risk-Free Assets Equation: Expected Returns and Standard Deviation Notation Input: r f risk-free rate E(r p ) expected return of the risky asset y...share invested in risky asset (in %); (1-y) share invested in risk-free asset (in %) σ rf/p standard deviation of the risk-free asset (rf) or the risky asset (p) Notation Output: E(r c ) expected return of the combined portfolio σ c standard deviation of the combined portfolio (c) ( r ) ( ) c y Erp + ( y) rf E = 1 σ c = y σp Slide 3 Week 3

118 Allocating Capital between Risky and Risk-Free Assets Example: Expected Returns and Standard Deviation Consider the following data: Risk-free asset: Rate of return: r f = 7% Standard deviation of return (SD): σ rf = 0% Risky asset: Expected rate of return: E(r p ) = 15% Standard deviation of return (SD): σ p = 22% y=50% ( ) = ( ) 7= 11 E = = 11 r c σ c Slide 4 Week 3

119 Allocating Capital between Risky and Risk-Free Assets Investment Opportunity Set Slide 5 Week 3

120 Allocating Capital between Risky and Risk-Free Assets Equation: Slope of the Capital Allocation Line The slope S can be calculated in the following way: S = E ( ) r c r f σ c The slope has a nice interpretation: It represents the ratio of the risk premium to the standard deviation It measures the additional return for taking on one more unit of risk It is called the reward-to-variability ratio or Sharpe Ratio Slide 6 Week 3

121 Allocating Capital between Risky and Risk-Free Assets Exercise For each of the following combinations determine the expected rate of return, the risk premium, the standard deviation, the reward-to-variability ratio and the point in the graphic on slide 5. Case 1: y=1.0 Case 2: y=0.0 Case 3: y=0.75 Slide 7 Week 3

122 Allocating Capital between Risky and Risk-Free Assets Exercise Assume that you manage a risky portfolio with an expected rate of return of 14% and a standard deviation of 22%. The risk-free rate is 5%. The risky portfolio includes the following investments in the given proportions: 60% in a U.S. Stock Portfolio and 40% in an Emerging Market Stock Portfolio. Suppose a client would like to invest in your risky portfolio a proportion y of his total investment budget so that his overall portfolio will have an expected rate of return of 10%. What is the proportion y? What are your client s investment proportions in the U.S. Stock Portfolio, the Emerging Market Stock Portfolio and the T-Bill? Slide 8 Week 3

123 Allocating Capital between Risky and Risk-Free Assets Investing more than 100% into the risky asset (1) If we invest 100% into the risky asset, we get to point P on the CAL. How can we invest even more in the risky asset? Answer: by borrowing money at the risk-free rate (leverage) and invest it into the risky asset Slide 9 Week 3

124 Allocating Capital between Risky and Risk-Free Assets Investing more than 100% into the risky asset (2) Example: assume we want to invest 125% into the risky asset Portfolio weights: y = 125% and (1-y) = -25% (i.e., borrowing) Expected return of the combined portfolio: E ( ) = ( ) 7= 17 r c Standard deviation of the combined portfolio: σ c = = 27.5 Reward-to-variability ratio: S = = 0.36 Slide 10 Week 3

125 Allocating Capital between Risky and Risk-Free Assets Investing more than 100% into the risky asset (3) Usually investors cannot borrow at the risk-free rate. Consider again the example where you borrow 25%. But now you can only borrow at a rate of 9%: r B = 9% Portfolio weights: y = 125% and (1-y) = -25% Expected return of the combined portfolio: E ( ) = ( ) 9= r c Standard deviation of the combined portfolio: σ c = = 27.5 Reward-to-variability ratio: S = = 0.27 Slide 11 Week 3

126 Allocating Capital between Risky and Risk-Free Assets Investing more than 100% into the risky asset (4) Slide 12 Week 3

127 Allocating Capital between Risky and Risk-Free Assets Optimal allocation between risky asset and risk-free asset How to determine the optimal allocation between risky asset and risk-free asset? The investor s risk aversion determines the optimal point on the CAL. Greater levels of risk aversion lead to larger proportions of the risk-free rate. Lower levels of risk aversion lead to larger proportions of the risky asset. Slide 13 Week 3

128 Allocating Capital between Risky and Risk-Free Assets Capital Market Line Capital Market Line: is the Capital Allocation Line using the market index portfolio as the risky asset. Historical evidence: Slide 14 Week 3