Note: There are fewer problems in the actual Final Exam!


 Scarlett Bates
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1 HEC Paris Practice Final Exam Questions Version with Solutions Financial Markets Fall 2013 Note: There are fewer problems in the actual Final Exam! Problem 1. Are the following statements True, False or Ambiguous? Provide a short justification for your answer. (Your score will primarily be determined by your justification.) (a) If CAPM is correct, it follows that the market portfolio is the portfolio with the highest Sharpe ratio of all the possible portfolios. Solution: True: We know that the tangency portfolio has the highest Sharpe ratio of all possible portfolios. CAPM implies that the market porfolio is the tangency portfolio. (b) If one introduces one more asset in the investment opportunity set, then the Sharpe ratio of the new tangency portfolio is at least as high as the Sharpe ratio of the old tangency portfolio. Solution: True. After introducing one more asset, the efficient frontier always moves to the left. This means that the tangent line to the efficient frontier also moves to the left, i.e. its slope (which is the Sharpe ratio) will be the same or higher. (c) Because of the potential for diversification if two stocks are not perfectly correlated, a portfolio that invests 50% in each stock must be less risky that a portfolio invested just in one of the stocks. Solution: False in general. Although there is a benefit to diversification, if one of the stocks is very risky then the portfolio could be riskier than an investment in the low risk stock. For example suppose that σ 1 = 0.1, σ 2 = 0.3 and ρ 1,2 = 0. The variance of the portfolio is: σ 2 P = = Hence σ P = = This has higher risk than investing entirely in stock #1. (d) You are a meanvariance investor and until recently you could not invest in commercial real estate. Today you found out that you can invest in an exchangetraded fund (ETF) which does a good job tracking a broad index of the commercial real estate sector. Then your investment opportunity set should get strictly bigger. Solution: True: It is safe to assume that commercial real estate is not a redundant asset, since it is unlikely to be perfectly correlated with some combination of assets you already own. Therefore, by being allowed to invest in this new sector, your investment opportunity set should get strictly bigger. This means that the efficient frontier moves up and to the left. Thus, for every portfolio on the old efficient frontier, you can find a better one on the new frontier: e.g., you can maintain the same risk as for the old portfolio, but increase your expected return. 1
2 (e) The duration of a (fixedrate) coupon bond is always at least as large as the duration of a zero coupon bond with the same maturity. Solution: False. The duration of a coupon bond is the valueweighted average of the durations of its individual payments, which must be smaller not larger than the duration of its last payment (which is the maturity). (f) (2 points) Consider a coupon bond with a price of 60, and a duration of 5. Then if all the interest rates increased by 1%, the price of the coupon bond would increase approximately by $3. Solution: False. If interest rates increase, then the price should decrease approximately by $3, not increase. (g) The current price of Digital stock is $44 per share. You are offered a forward price for Digital stock to be delivered in one year of $42. The forward price is lower than the spot price because the market anticipates a sharp decline in the price of Digital stock, and the contract offers a way to hedge this risk. (Assume that there is no arbitrage opportunity.) Solution: False. According to spot futures parity, the forward price for Digital should equal F 0 = S 0 (1 + r d), where d is the net benefit of holding the underlying asset (dividends in this case). So the fact that F 0 = 42 is smaller than S 0 = 44 does not mean that the stock price is expected to decline. (Remember that the spot futures parity was determined by arbitrage: if the formula were not true, you could construct an arbitrage.) Besides, if Digital was really expected to have a sharp decline in price, market efficiency would force its price to fall today! (h) The stocks of Merck and Disney are traded at the same price of $37 a share. The historical returns of Merck are more volatile than those of Disney and exhibit higher systematic risk. In addition, given the coming health reform proposal, it is generally believed that Merck may have a negative alpha in the future. It is known that neither Merck nor Disney will pay dividends in a month. Consider now forward contracts on these two stocks with one month to maturity. The forward price of 100 shares of Merck should be lower than the forward price of 100 shares of Disney. Solution: False. Essentially the same argument as in g: the forward price of both stocks should be $37 (1 + r). (i) The current spot price of gold is $300 per ounce and the oneyear futures price is $330 per ounce. (Don t argue with these facts! ) These numbers tell us that the market is expecting at most a 10% increase in the price of gold over the next year. Solution: Uncertain. The futures price does not help us figure out expectations. For example let S 0 be the current price of gold, r be the riskfree rate, and k the per ounce cost of storing gold. Then by no arbitrage, the futures price must be: F 0 = S 0 (1 + r + k) 2
3 regardless of expectations. For example suppose that r = 8%, k = 2% and S 0 = 300. Then: F 0 = 300 ( ) = 330 independent of our or the market s expectations about the price of gold in the future. (j) Suppose the current US Dollar/British Pound exchange rate is 2.00 $/, the one year riskfree rate in UK is 4%, and the 1year riskfree rate in the U.S. is 2%. You believe that the following analysis is correct: Because the economy in the UK is stronger than the economy in the US, the Pound is probably strengthening against the Dollar, i.e. the exchange rate $/ will probably increase in the near future. Then this implies that you should be able to lock in a 1year forward exchange rate larger than 2.00 $/. Solution: False. The forward rate is determined by the covered interest parity, which is a noarbitrage relationship: F $/ 0 = E $/ 0 (1 + r $ ) (1 + r ) The US 1year interest rate r $ = 1.5% is smaller than the UK interest rate r = 4%, so F $/ 0 < E $/ 0. Therefore the forward exchange rate must be smaller than 2.00 $/. (k) (2 points) The value of a European call option on a stock is always larger than the value of a futures contract on the same stock. Solution: True. The value of a European call option is always positive, since it gives the right, but not the obligation to buy the underlying stock. The futures contract has a value of zero. This is not to be confused with the futures price, which is positive and is related to the stock price via the spot futures formula. (l) You should only exercise an American put option when the underlying stock price is high enough. Solution: False. If the underlying stock price is high, you gain little from exercising. The statement is actually true if we replace high with low: you should only exercise an American put option when the underlying stock price is low enough. (m) The price of ABC stock is currently $1 per share and you know that ABC will pay no dividends in the next year. You hold an inthemoney American put option on ABC stock, with expiration in oneyear. Because the price is very low today, so it cannot fall much further in the future, you should exercise the option today. Solution: Uncertain. It is possible that exercising the option now is the best thing to do. If you exercise the put today, the advantage is that you would receive the strike 3
4 price (from selling the stock) now rather than later, and therefore you can earn interest on it. However, if you exercise now, you would lose the option value of making an extra $1 when the stock drops further in value. So it depends on whether the time value advantage is large enough to convince you to give up the option value. Typically that happens only when the price of the underlying stock is low enough (so that the option value is low enough), which is the case here. (n) (2 points) Suppose interest rates were negative: r < 0. Then it would not be optimal to exercise early an American put option. Solution: True. By exercising early, one loses both the option value and the time value of the strike price: getting K now is worse than getting K later when interest rates are K negative (in presentvalue terms, one has to compare getting K with getting > K (1+r) τ when r < 0). Also, it does not matter if the stock pays any dividends, since dividends makes the holder of a put option better off (so dividends should not make one exercise early). Problem 2. You are a mean variance investor, and you decide to choose a portfolio with the available assets. You are allowed to invest only in an asset that has a riskfree rate of return r f = 5%, and in two risky assets, GE and KO, which have the following expected return & standard deviation: Asset E( r) σ GE 10% 15% KO 12% 19% The correlation between the two stocks is ρ GE,KO = 0.4. Denote by T the tangency portfolio formed with GE and KO that you want to hold. You know that T invests 50% in GE and 50% in KO. (a) Find the expected return and standard deviation of T. Solution: Recall: If X and Y are random variables with means µ X, µ Y, standard deviations σ X, σ Y and correlation ρ XY, then a linear transformation of X and Y will have mean and standard deviation Z = αx + βy (1) µ Z = αµ X + βµ Y (2) σ Z = α 2 σx 2 + β2 σy 2 + 2αβ cov(x, Y ) (3) where cov(x, Y ) = ρ XY σ X σ Y. In this case, the tangency portfolio is a linear transformation of assets GE and KO T =.5GE +.5KO (4) 4
5 where α =.5 and β =.5. Plugging into the above formula, µ T = =.11 = 11% and σ T = =.143 = 14.3% (b) Find the Sharpe ratio of the tangency portfolio, and show that it is higher than the Sharpe ratio for both GE and KO. Solution: The Sharpe ratio of T is SR T = E T r f σ T = while the Sharpe ratios of GE and KO are: SR GE = SR GE = = = Indeed, SR T =.42 is larger than both.33 and.37. (c) Suppose your utility function is quadratic: U(E, σ) = E 1 2 A σ2 =.42 and your risk aversion coefficient is A = 7. Find your optimal portfolio formed with the riskfree asset, GE, and KO. What percentage of your wealth will you hold in each asset? Solution: Modern Portfolio Theory says that you should only invest in a combination of the riskfree asset and the tangency portfolio T. The weight that you should put on T is given by the following formula: w T = 1 A µ T r f σ 2 T = = So you should hold 42% of your wealth in T, and 58% in the riskfree asset. But T is formed by GE and KO with equal weights, so you will hold 21% in GE, 21% in KO, and 58% in the riskfree asset. Problem 3. A portfolio consists of the following 3 stocks, whose performance depends on the economic environment. Assume the bad economic outcome is twice as likely as the good one. Investment in $(mill.) Good economy Bad economy Stock % 20% Stock 2 1,250 6% 3% Stock % 2% 5
6 (a) Calculate the expected return of the portfolio. Solution: Investment in $(mill.) Good economy Bad economy Stock % 20% Stock 2 1,250 6% 3% Stock % 2% r P 2,000.25(.13) +.625(.06) +.125(.07).25(.20) +.625(.03) +.125(.02) = = So the expected return of the portfolio is (b) Calculate the variance of the portfolio. E P = 1/3( ) + 2/3( ) = Solution: The variance of the portfolio is σ 2 P = 1/3( )2 + 2/3( ) 2 = (c) What if we add to the portfolio $1,000 million of stock 4, which has a mean return of 4%, has a variance of 0.02, and is uncorrelated with the above portfolio? How will this change the expected return and variance of the total investment? Solution: Call this new portfolio Q. The new expected return and variance are 1, 000 2, 000 E Q = (0.04) + ( ) = , 3, 000 3, 000 ( 2 ( ) σ 2 = (0.02) + (0.0018) = Q 3) 3 Problem 4. The expected return of the S&P 500 index, which turns out to be the tangency portfolio of stocks in the economy, is 16%, with a standard deviation of return of 25% per year. The expected return of Microsoft is unknown, but it has a standard deviation of 20% per year, and a covariance with the S&P 500 index of Assume that the riskfree rate is 6% per year. (a) Compute Microsoft s beta and expected return. Solution: β MSFT = σ MSFT,M = 0.10 σ 2 (0.25) = 1.60, 2 M E MSFT = ( ) =
7 (b) If Intel has half the expected return of Microsoft, then what is Intel s beta? Solution: E INTC = = 0.11, CAPM = 0.11 = β INTC ( ), β INTC = (c) What is the beta and expected return of the following portfolio? 25% in Microsoft 10% in Intel 75% in the S&P 500 index 20% in GM (where β GM = 0.80) 10% in the riskfree asset Solution: β P = = 1.04, E P = ( ) = (d) Is the following statement True, False, or It Depends? If the riskfree rate drops to 5%, but the expected return on the S&P 500 index remains the same, the new expected return on Microsoft increases by 0.6%. In one or two sentences explain or show why this is true or false or why it depends. Solution: It depends, because once the riskfree asset changes, the S&P 500 may no longer be the tangency portfolio. Therefore, we don t know what the new expected return equation looks like. Remember, the S&P 500 is simply some portfolio, which happens to be the tangency portfolio given a riskfree rate of 6%, but it does not necessarily represent the market portfolio in a CAPM context. Problem 5. Assume that the CAPM holds and you have the following information about the returns of Procter & Gamble (PG) and General Motors (GM) for the next year: Asset corr( r, r M ) σ( r) E( r) PG 20% 30%? GM 50% 40%? Market (M) 100% 20% 15% (E.g., PG has a correlation with the market of 20%, and a standard deviation of 30%.) The riskfree asset has a rate of return r f = 5%. 7
8 (a) Calculate the expected returns of PG and GM. (Hint: Calculate first the betas of PG and GM.) Solution: In general, the beta of a stock i can be computed as β i = σ i,m σ 2 M = ρ i,m σ i σ M σ 2 M = ρ i,m σ i σ M. For i both PG and GM we know ρ i,m and σ i, so we can compute their betas: So their expected returns are β PG = β GM = = 0.3 = 1. E PG = r f + β PG (E M r f ) = ( ) = 8% E GM = r f + β GM (E M r f ) = ( ) = 15%. (b) You are now considering a third stock, General Electric (GE). After some analysis, you decide that next year s returns for GE depend only on two possible states for the US economy over the next year: Good and Bad. Each state is equally likely (with probability 0.5). The returns to GE in each state are as follows: State Return to GE Good 35% Bad 15% What is the implied correlation ρ GE,M between the return on GE and the market? (Hint: Calculate the expected return and standard deviation of GE, and then use CAPM to determine GE s beta.) Solution: Compute the expected return, variance, and standard deviation of GE: E GE = ( 0.15) = 10% 2 2 σ 2 GE = 1 ( )2 + 1 ( )2 = σ GE = = 25%. Use the CAPM formula to compute the beta of GE: E GE = r f + β GE (E M r f ) = β GE = E GE r f E M r f = = 0.5. Now use the formula we derived above for β i : β i = ρ i,m σ i σ M = ρ i,m = β i σ M σ i So the correlation between GE and the market is ρ GE,M = = 0.4 = 40%. 8
9 Problem 6. Your firm is evaluating a new risky capital investment. After careful consideration, you decide there are two possible outcomes or states of the world relevant to your project. State 1 represents a mild recovery, and state 2 represents a strong recovery from the recent downturn in the market. You estimate the probabilities of these alternative future states of the world to be 0.7 and 0.3, respectively. You also estimate that the market return will be 10% in the mild recovery state and 30% in the strong recovery case. The project itself has estimated returns of zero in the mild recovery case and 40% in the strong recovery scenario. Assume that the riskfree rate is 10%. (a) What are the expected returns of the market and the project, respectively? Solution: The expected returns of the project P and the market are E P =.7(0) +.3(.40) = 0.12 E M =.7(.10) +.3(.30) = 0.16 (b) What is the covariance between your project and the market? Solution: The covariance between the project and the market is σ P,M =.7(0.12)(.10.16) +.3(.40.12)(.30.16) = (c) Under the CAPM, what is your expected return on the project? Solution: Under CAPM the expected return on P is E P compute β P, compute first market variance =.10 + β P (.16.10). To σ 2 M =.7(.10.16)2 +.3(.30.16) 2 = Then This implies β P = σ P,M σ 2 M = = 2. E P = (.16.10) = (d) Should the project be accepted? Why or Why not? (Be brief.) Solution: No. Since 12% (the actual expected return of the project) < 22% (the CAPM expected return), the project does not offer a high enough expected return to compensate for its risk. Problem 7. You are a financial advisor for high net worth individuals. Your new client Karen just told you that she is willing to invest only in the following three assets: The riskfree asset, which is a TBill with interest rate r f = 5%; A bond mutual fund X, with expected return E X = 8% and standard deviation σ X = 10%. 9
10 An stock mutual fund Y, with expected return E Y σ Y = 30%; The stock and bond indexes are assumed to be uncorrelated: ρ X,Y = 0. = 14% and standard deviation Karen s current portfolio P is invested 25% in X, 25% in Y, and 50% in the TBill. Also, your quants have computed that that the tangency portfolio T is invested 75% in X and 25% in Y. (a) What are the expected return and standard deviation of the return on your client s portfolio P? What are the expected return and standard deviation of the tangency portfolio T? Solution: The portfolio expected returns are E P = 0.25 E X E Y r f = 8% E T = 0.75 E X E Y = 9.5% The standard deviations are (recall that ρ X,Y = 0) σ P = σ T = σ 2 4 X σ 2 4 Y = = 7.91% σ 2 4 X σ 2 4 Y = = 10.61% 4 4 (b) Karen is willing to accept a standard deviation no higher than her portfolio s current standard deviation. Also, she prefers to invest only in the TBill, the bond index X, and the stock index Y. You would like to find another portfolio Q with a higher expected return that Karen s current portfolio. What is the maximum expected return you can achieve? (Hint: recall that any efficient portfolio is a combination of the riskfree asset and the tangency portfolio.) Solution: There are two ways of solving this problem: Solution 1: The standard deviation of your client s current portfolio is σ P = 7.91%. You must suggest Karen an efficient portfolio, which gives you the same standard deviation, but the highest possible expected return. This efficient portfolio, Q, must be on the CML (Capital Market Line). The equation of the CML represents the expected return E of a portfolio as a function of its standard deviation. It has the intercept equals to the riskfree rate r f, and the slope equal to the Sharpe ratio of the tangency portfolio T : E = r f + SR T σ. The Sharpe ratio of the tangency portfolio is So the equation of the CML is SR T = E T r f σ T = E = σ. 10 =
11 Since the portfolio Q that you want to recommend to Karen is on the CML and has σ Q = σ P = 7.91%, the expected return of Q is E Q = = = 8.35%. Note that this is indeed higher than the expected return achieved by Karen s current portfolio: E P = 8%. Solution 2: As discussed before, any efficient portfolio Q must be on the CML, i.e., it must be a combination of the riskfree asset and the tangency portfolio T. We denote by w the weight of Q invested in T. Then the standard deviations should match: σ Q = σ P = wσ T = σ P w 10.61% = 7.91% w = = 74.54%. Then the maximum expected return achievable at the standard deviation of 7.91% is the expected returns of the above portfolio Q: E Q = w E T + (1 w) r f = 8.35%. (c) After some investigation, you decide that Karen has a quadratic utility U(E, σ) = E 1 A 2 σ2, with risk aversion coefficient A = 8. What is her optimal portfolio P formed with X, Y and the riskfree asset? Solution: The optimal portfolio P is a portfolio formed with the riskfree asset and the tangency portfolio T. Denote by w the weight of P invested in T. Then w = 1 A E T σ 2 T r f = = 0.5. The portfolio P invests 0.5 in T and 0.5 in the riskfree asset, i.e. 37.5% in X, 12.5% in Y, and 50% in the TBill. (d) How does the Sharpe ratio of the optimal portfolio P compare with the Sharpe ratio of the portfolio Q you found in (b)? (Hint: There is a qualitative answer, you don t actually have to compute the two Sharpe ratios.) Solution: Both P and Q are on the same capital allocation line between the riskfree asset and the tangency portfolio T (also called the capital market line, CML). Therefore they have the same Sharpe ratio. Problem 8. Consider two stocks: WMT and IBM with the following properties Stock E( r) σ( r) WMT 8% 13% IBM 12% 20% 11
12 The correlation of the two stock returns is and the riskfree rate is ρ WMT,IBM = 9%, r f = 1%. You are advising a client who has $1 million invested. Currently 50% of this money is in WMT and 50% is in IBM. (a) What are the expected return and standard deviation of the return on your client s portfolio? Solution: The portfolio expected return is The standard deviation is σ P = 1 E P = 50% E WMT + 50% E IBM = 10% 2 σ σ WMT σ σ ρ = = 12.41%. IBM 2 2 WMT IBM WMT,IBM (b) You are told that the tangency portfolio formed with WMT and IBM has weights w WMT = 60% and w IBM = 40%. i. What are the expected return and standard deviation of the tangency portfolio? Solution: The expected return and standard deviation of the tangency portfolio are σ T = E T = w WMT E WMT + w IBM E IBM =.6 8% % = 9.6%,.6 2 σ σ σ σ ρ = = 11.66%. WMT IBM WMT IBM WMT,IBM ii. What is its Sharpe ratio? How does it compare with the Sharpe ratio of your client s current portfolio? Explain your finding. Solution: The Sharpe ratio is SR T = E T r f σ T = 9.6% 1% 11.66% = By definition the tangency portfolio must have the highest Sharpe ratio among all the possible portfolios formed with WMT and IBM, so SR T > SR P. We can also check this directly: = SR T > SR P = 10% 1% 12.41% =
13 (c) You wish to match the expected return of your client s current portfolio using only WMT, IBM, and the riskfree security. What is the minimum standard deviation you can achieve? Solution: We can look at the problem in two different ways: Solution 1: The expected return of your client s current portfolio is 10%. Any minimumvariance portfolio P must be a combination of the riskfree security and the tangency portfolio in (b), so denote by w the weight invested in the tangency portfolio, and by 1 w the weight invested in the riskfree asset. Then the expected returns should match: w 9.6% + (1 w) 1% = 10% w 8.6% = 9% w = = %. Then the minimum standard deviation achievable at the expected return of 10% is the standard deviation of the above portfolio P : σ P = w σ T = 12.20%. Solution 2: Since we can also invest in the riskfree asset, the minimumvariance frontier is the capital allocation line from the riskfree asset to the tangency portfolio. Therefore all efficient portfolios have the same Sharpe ratio as the tangency portfolio: Denote by P the portfolio on the CAL which has the same expected return as P (10%). The Sharpe ratio of P is = (10% 1%)/σ P, therefore σ P = = 12.20%. Notice that since E P > E T, in order to achieve the same expected return as P, you want to lever up the tangency portfolio. E P E T r f $1 M = $1.0465M r f of the risky portfolio, and to do this you also need to short $1.0465M $1M = $46, 500 of the risk free security. (d) Suppose the correlation between the two stocks increased to ρ WMT,IBM = 20%. What would happen qualitatively (i.e. don t do any calculations, just tell me the general direction) to the Sharpe Ratio of the optimal portfolio? Justify your answer. Solution: When the two assets are more correlated, the efficient frontier moves to the right. (And in the extreme case when the assets are perfectly correlated, the frontier becomes the line between the two assets.) When the efficient frontier moves to the right, the tangent line (CAL) from the riskfree asset no longer intersects the frontier. This means that the Sharpe ratio must go down. 13
14 Problem 9. (a) The price of a 7year strip is given: B 7 = 73 (face value F = 100). What is its yield to maturity? Solution: ( ) 1/7 100 YTM 7 = 1 = B 7 ( ) 1/ = 4.60% 73 (b) The yield to maturity of a 10year strip is given: YTM 10 = 7%. What is its price B 10? Solution: B 10 = F 100 = = (1 + YTM 10 ) Problem 10. You are told that the current yield curve for years 1, 2 and 3 is the following: Maturity YTM 0.5% 1% 2% The 3year Treasury note with face value $1000 and annual coupon rate 5% trades at $ (a) Is the price of the bond consistent with the given list of yields? Solution: The annual coupon is 5% $1000 = $50. So the coupon bond pays $50 in years 1 and 2, and in year 3 it pays both the coupon ($50) and the face value ($1000), so $1050 in total. The fair price of the coupon bond is given by the Present Value formula: P = $ $ $ = $ Notice that the fair price is smaller than the actual price at which the bond trades: P = $ Therefore, the coupon bond is not fairly priced. (b) If the price of the coupon bond is not consistent with the given list of yields, what would you do to in order to take advantage of this situation? Be explicit in describing your strategy. You are allowed to invest in the given coupon bond, and in 1year, 2year, and 3year strips. Solution: The coupon bond trades at a higher price (P = $1090.0) than the fair price ($1088.2), thus the coupon bond is overpriced. You should do the following: Short the expensive asset, the coupon bond, for $1090.0, and Buy the cheap asset, the replicating portfolio of strips. What quantity of each strip should you buy? * * * = year strips, = year strips, and = year strips. 14
15 What are the cash flows from this strategy? Let s first calculate the price of each individual strip. For example, the price of the 3year strip is We get the following table B 3 = 100 (1 + YTM 3 ) = = Maturity Strip price So you buy 50/100 = 0.5 of 1year strips, 50/100 = 0.5 of 2year strips, and 1050/100 = 10.5 of 3year strips (we assume that you can buy and sell fractions of a strip). The cash flow at time zero you get from selling the strips is CF 0 = 0.5 $ $ $94.23 = $ Note that this is the same as the fair price P. The cash flows from buying the strips (buying the replicating portfolio) and shorting the coupon bond are: Buy Strips Time Cash Flow Short Bond Time Cash Flow So the cash flow today is positive: $ $ = $1.80, while the cash flows in the future all add up to zero. You can just pocket the difference today and the cash flows from the long position will take care of the cash flows from the short position. You have just performed what is called an arbitrage. As we saw in other examples (e.g., the red dollar / blue dollar case), we know that this type of arbitrage is not really riskfree. Problem 11. Suppose that the term structure is currently flat so that bonds of all maturities have yields to maturity of 10%. Currently a 5year coupon bond with annual coupons (with the first one due in 1 year) and face value of $1,000 is selling at par. (a) What is the current price of the 5year bond? What are the annual coupons? Solution: The price of the 5 year bond must be $1,000. Denote the annual coupons by C, ans solve for C in the asset pricing formula. P 0 = 1000 = C C C C C
16 This is one linear equation in one unknown C. The solution is easily computed: C = 100. Notice that this means that the coupon rate C equals 10%, which is exactly the yield 1000 to maturity. The result is more general, i.e. when the coupon rate equals the yield, then the bond price is equal to par value. We already mentioned this result in class. (b) A year from now interest rates will be depend on the stance of monetary policy. If monetary policy is tight the yields to maturity on all bonds will be 12%. If monetary policy is loose the yields to maturity on all bonds will be 8%. If you sell the bond a year from now when monetary policy is tight what will be the return to your investment over the year? If you sell the bond a year from now when monetary policy is loose what will be the return to your investment over the year? Solution: The price of the bond when monetary policy is tight will be: P 1,tight = = In this case the net return will be r 1,tight = P 1,tight + C P 0 P 0 = The price of the bond when monetary policy is loose will be: = 3.92%. P 1,loose = = In this case the net return will be r 1,loose = P 1,loose + C P 0 P 0 = = 16.62%. Problem 12. We see the following yield curve for discount (or zero coupon) bonds. Maturity Yield to Maturity 1 year 6% 2 years 7% 3 years 8% If the fair price for a 4year annuity paying $100 per year is $334.57, what is the yield to maturity on a four year zero coupon bond? Solution: The fair price of a 4year annuity paying 100 per year should satisfy: P = (1 + YTM 4 ) 4. 16
17 Thus the yield to maturity on a four year zero coupon bond must satisfy: or which implies = = YTM 4 = 100 (1 + YTM 4 ) 4 ( ) 1/ = 8% (1 + YTM 4 ) 4 Problem 13. You have the following information about the prices of a 1year strip and a 2year coupon bond. The 1year strip pays a face value of $100 in one year, and currently sells for $96. The 2year coupon bond has a face value of $1000 and an annual coupon of $60. The bond currently sells for a price of $1050. (a) What are the implied yields to maturity on 1 and 2year strips? Solution: We first compute YTM 2 satisfies: 1050 = YTM 1 = = 4.17% (1 + YTM 2 ) 2 = (1 + YTM 2 ) From this, we get = = (1+YTM 2, and so (1 + YTM ) 2 2 ) 2 = We solve for YTM 2 : ( ) 1/ YTM 2 = 1 = 3.35% (b) Consider a 2year annuity with annual payments of $500. What is the fair price for this annuity? Solution: The fair price of this annuity should be: =
18 Problem 14. Many institutions have fixed future liabilities (such as pension payments) to meet and they fund these future liabilities using defaultfree fixed income securities. When discount bonds of all maturities are available, these institutions can simply buy discount bonds to fund their liabilities. For example, if there is a fixed liability equal to 1 million dollars five years from now, an institution can buy a discount bond maturing in five years with a face value of 1 million dollars. Unfortunately, there may not be the right discount bonds for a fixed future liability and coupon bonds must be used. Then an institution faces reinvestment risk of the coupons. For example, suppose that the yield curve is flat at 10% and we have the following coupon bonds (with annual coupons): Bond Principal Coupon Years to Maturity A B Also, suppose we have a 1 million liability five years from now. (a) What are the prices of the two bonds? Use 4 decimal digits. Solution: At a discount rate of 10% the price of the five year is and the price of the ten year is P 5 = = P 10 = = (b) Suppose that the yield curve will remain unchanged for the following five years and you have decided to use bond A to fund the liability. That is, you want to invest in bond A and invest the coupons at the prevailing interest rates to produce a future value at the end of year five of 1 million. How much should you invest in bond A? Use 2 decimal digits. Hint: To produce a future value of $1 million in 5 years, today you should invest the present value of $1 million (at 10% interest rate). Since in (a) you computed the price of bond A, you can find out how many bonds of type A you should buy today. Solution: At a discount rate of 10% the present value of $1,000,000 in 5 years is $620, To have $1,000,000 in 5 years you need to invest $620, in bond A. This means that you will have to purchase 620,921.32/ = 5, bonds (note that in these calculations I am carrying many decimal digits). The cash flows created by this will be: 18
19 Year Cash Flow 1 78, , , , , (c) Now suppose that right after you invested in bond A, the yield curve makes a parallel move down by 1% to 9%. What is the future value five years from now of your investment? What is the future value if the yield curve moves up by 1% to 11%? Please explain why the future value changes differently depending on the direction of the change in the yield curve. Solution: At an interest rate of r the future value of the above cash flows in 5 years will be: FV = 78, (1 + r) , (1 + r) + 600, The future value at various interest rates will be: r Future Value 9% 990, % 1,000, % 1,009, (d) Part (c) shows that the future value of your investment is sensitive to interest rate fluctuations and you face the risk that your future liabilities may not be met. You should try to immunize this interest rate risk. But how? Do the following: i. Compute the duration of bonds A and B. Use 2 decimal digits. Solution: We can use the formula from class. Recall that the price of A is P A = The duration of bond A is: D A = 15/1.10 P A /1.102 P A /1.105 P A 5 = 3.95 Similarly the duration of bond B is D B = ii. You want to construct a portfolio of the two coupon bonds A and B so that the future value of the portfolio is 1 million and the duration of this portfolio is equal to 5 years, assuming that the yield curve will remain flat at 10%. Find the percentage of your wealth that you want to invest in A and B. Use 2 decimal digits (e.g., 37.54%). Solution: Let x be the proportion invested in bond A and (1 x) be the amount invested in bond B, then we need the duration of the portfolio to equal 5: x (1 x) 6.28 = 5. 19
20 Solve this equation: x = (5 6.28)/( ) = , or x = 54.94%. Hence the amount invested in bond A is $620, = $341, and the amount invested in bond B is $620, = $279, You should buy 341,134.17/ = 2, units of bond A and 279,787.15/ = 2, units of bond B. iii. Show that if immediately after you purchased this portfolio the yield curve makes a permanent parallel downward or upward move of 1%, the future value of this portfolio at the end of year 5 will still be approximately 1 million. You have immunized the portfolio of the risk associated with parallel movements of the yield curve by buying a portfolio of coupon bonds so that the duration of the portfolio matches the number of years to the payment of the fixed liability. Solution: We have the following cash flows from the portfolio: Year Cash from A Cash from B Total Cash 1 43, , , , , , , , , , , , , , , , , , , , , , , , , At the end of year 5 the accumulated value of these cash flows will be (including accumulated investment and sale of 10 year bond): Interest Rate Value 9% 1,000,339 10% 1,000,000 11% 1,000,291 Problem 15. (a) On March 3, 2008 the closing spot price of the S&P 500 index was $ , while the closing futures price for the June contract was $ Compute the dollar value of the stocks traded on one contract on the S&P 500 index (recall that the actual value comes with a multiplier of 250). If the margin requirement is 10%, how much must you deposit with your broker to trade the June contract? Solution: The closing spot price was $ The dollar value of stocks traded is: 250 $ = $332, 835 The closing futures price for the June contract was $ , which has a dollar value of: 250 $ = $333, 500 Therefore, the required margin deposit is 10% $333, 500 = $33,
21 (b) If the June futures price were to increase to $1340, by what amount would your margin account change if you entered the long side of the contract at the price given in (a)? Solution: The futures price increases by: $1340 $1334 = $6. The credit to your margin account would be: $6 $250 = $1, 500. (c) What percentage return would you earn on your net investment? Solution: The gain in the margin account is a percent gain of: $1, 500/$33, 350 = = 4.5%. Note that the futures price itself increased by only 0.45%. (d) If the June futures price falls by 1%, what is your percentage return? Solution: Following the reasoning in part (c), any change in the futures price is magnified by a ratio of (1/margin requirement). This is the leverage effect. The return will be 10%. Problem 16. Suppose the 1year futures price on the S&P500 is below the current value of the index multiplied by (1 + r), where r is the one year Tbill rate. Then investors expect the index to decline. True or false? Solution: False. The index has also benefits (dividends), which do not accrue to the futures contract owner. Therefore the futures price is F 0 = S 0 (1 + r d) (spotfutures parity with costs and benefits). Then F 0 will always be below S 0 (1+r), provided that the dividend yield is positive (which it usually is). This has nothing to do with what investors expect the future value of the index to be. Problem 17. Fill in the missing numbers in this table of stock index futures prices. Country Index Index Dividend Spot interest Futures Futures Price Name Value Yield rate Term (same unit as (annualized) (annualized) (months) index value) Japan Nikkei % 0.57 % 3??? US S&P ??? 1.77 % Solution: The formula for the futures price on the Nikkei 225 index is: F Nik 0 = S 0 (1 + r d) 3/12 = ( ) 3/12 =
22 Similarly, for the S&P 500, = F SP 0 = S 0 (1 + r d) 9/12 = ( d) 9/12. Therefore the (annualized) dividend yield on the S&P 500 is d = ( ) 12/ = = 1.38% Problem 18. Suppose it is now the end of December. The current yield curve is flat at 3%. The June futures price for gold is $988.80, whereas the next December futures price is $ Is there an arbitrage opportunity here? How would you exploit it? Solution: In what follows we will assume no costs of carry, and no convenience yield. Suppose the spot price of gold is S 0 (we don t need to know what it is). According to the parity relation, the proper prices for the June and December futures are: F June = S 0 (1 + r f ) 1/2 F Dec = S 0 (1 + r f ). If we divide the second equation by the first, we obtain F Dec = F June (1 + r f ) 1/2 = $ /2 = $ The actual futures price for December is too high relative to the June price. You should short the December contract and take a long position in the June contract. This would imply cash flows as in the following table Position t = 0 June December Long June contract $ receive gold Short December contract +$ give away gold Notice that the cash flows: $ after 6 months, and +$ after 12 months represent a long position in a forward loan. To cover this loan, one can buy a 6month strip and short a 12month strip: 22
23 Position t = 0 June December Long June contract receive gold Short December contract Buy 6month strip (F = ) /2 = deliver gold Short 12month strip (F = ) = Net You make $5.42 arbitrage profit every time you make this trade. Problem 19. Suppose oneyear interest rates are 3% in the US and 5.5% in the UK, and that the pound is currently worth $2.00. (a) What should the 12month forward exchange rate for the pound be to rule out arbitrage opportunities? Solution: By covered interest rate parity: 0 = E $/ 1 + r $ 0, 1 + r F $/ where E $/ 0 is the spot exchange rate from dollars into pound, and F $/ 0 is the pound futures price. Therefore F $/ 0 = = $1.95/. (b) Suppose you read in the paper that the 12month forward is actually $1.90/. Explain in detail how to exploit the mispricing using only the following instruments: buying or selling US or British 12month bills, and buying or selling the pound forward. Ignore transaction costs. If you could only borrow the equivalent of $100,000, how much money would you make? Solution: Since the actual forward price $1.90/ is less than the correct price $1.95/, you should buy the forward and short the pound. The strategy is as follows: Borrow in London Convert immediately to $$, lend in the US Enter into a futures contract to buy (1 + r ) in one year 23
24 Position Today 12 Months Borrow pounds in London + 50, , = 52, 750 Exchange pounds to dollars 50, 000 in the spot market +$100, 000 Invest in 12 month Tbills $100, 000 +$103, 000 Enter long forward contract + 52, , = $100, 225 Net 0 $2, 775 The only limit is the total amount you could borrow in London to set up the arbitrage (presumably you d want to borrow an infinite amount if you could). Problem 20. (a) The following data from the Wall Street Journal give you the spot and forward exchange rates for the Yen on March 8, 1995: Japan Yen spot... $ day forward... $ day forward... $ day forward... $ March 8, 1995 (Reuter)  Treasury Bill rates are 5.56% for onemonth, 5.81% for three months, 6.04% for six months. Copyright 1995, Reuters News Service Construct the zerocoupon term structure for Japanese spot interest rates. Solution: Recall covered interest rate parity: F $/ 0,T = E$/ r $,T 1 + r,t where E $/ 0 is the spot exchange rate from dollars into Yen, and F $/ 0,T is the price of the Yen future with term T. For example (interest rates are all annualized): ( ) 30/365 F $/ 1 + 0,30 days = r$,30 E$/ r,30 ( ) 30/ = r,30 ( ) 365/ r,30 = ( ) 1 = 1.159%
25 Similarly, ( ) 365/ r,90 = ( ) 1 = 1.377% ( ) 365/ r,180 = ( ) 1 = 1.367% (b) What would happen if the Japanese Central Bank raised the spot interest rates? Would that increase the forward exchange rate? Solution: No. To increase the forward exchange rate, the opposite should happen: Japanese rates go down and/or US rates go up. Problem 21. Suppose that you feel that market volatility is going to rise because of increased uncertainty due to the Federal Reserve s actions. You do not have a prediction, however, about where the market will go. Can you devise an investment strategy using options to take a bet on your feelings? Describe the strategy. Suppose instead that you think market volatility is going to fall, what strategy would you follow in this case? Can you devise this strategy in a way that you are not exposed to significant downside risk? Solution: If you feel that the market s expectation of volatility is too low you could bet on this using a straddle where you simultaneously buy a put and a call with the same strike price. Suppose, for example, that the S&P500 is currently at 1,400 and call and puts are available with strike prices of 1,400. Then the payoff and profit from these would be 200 Payoff and Profit from Straddle 150 C+P 100 Payoff and Profit Payoff Profit Stock Price Notice that you only make a profit if the index moves far enough from 1,400. If you believe that volatility is going to fall compared to the market s expectations you could write a straddle, but this exposes you to large potential losses. In addition to writing the straddle (a call and a put) you could buy a call with a strike price of 1500 (for example) and buy a put with a strike price of 1300 (for example). 25
26 100 Payoff Profit 50 Payoff and Profit Stock Price Problem 22. Consider the following option strategy: Long one call with $100 strike price, bought for $6 Long one call with $90 strike price, bought for $20 Short one call with $105 strike price, sold for $8 Short one call with $95 strike price, sold for $16 (a) Draw a picture of the payoff of the option strategy at expiration as a function of the stock price. Solution: The payoff of this option strategy at maturity is Range of stock price S T < S T < S T < S T < S T S T at expiration Payoff of long 0 S T 90 S T 90 S T 90 S T 90 Call with K = 90 Payoff of long S T 100 S T 100 Call with K = 100 Payoff of short 0 0 S T + 95 S T + 95 S T + 95 Call with K = 95 Payoff of short S T Call with K = 105 Total payoff of 0 S T 90 5 S T option strategy The payoff to this position at expiration looks like 26
27 12 Option Payoff Payoff Stock Price (b) Draw a picture of the investor s profit as a function of the stock price. Solution: The cost of this position is: = 2. Hence the investor s profit as a function of the stock price shifts down by 2. It looks like 12 Option Profit Profit Stock Price Problem 23. Stock ABC trades today at $100, and in the past 6 months its volatility was σ = 30% (annualized). The 3month Tbill rate is r = 2.75% (annualized). Consider the following table of European option prices which all expire in 90 days: Strike Call Price Put Price
28 (a) What is the price of a straddle with strike price of 100? Draw the graph of the profit of the straddle. Solution: A straddle is created by buying an ATM (atthemoney) call and an ATM put on the stock. The price therefore is $ $5.59 = $ The graph is (b) You think that the volatility of ABC will increase, but you are not completely sure and feel that buying a straddle would be too aggressive (and too expensive). Can you create an option that would do what you want? Solution: You essentially want to flatten the tails of the straddle distribution. To this end, you can sell an OTM call with strike K 1 = 110 and an OTM put with strike K 2 = 90. From the table, the price of the new option strategy is $11.85 $2.65 $1.84 = $7.36. The graph is Looks familiar? It is the profit of a short position in a butterfly option 28
29 And it makes sense, because a long position in a butterfly is a bet that volatility will decrease. Therefore a short position is a bet that volatility will increase, but it is a more moderate bet than a straddle We can also do this more formally, and write the payoff and profit tables: Range of stock price S T < S T < S T < S T S T at expiration Payoff of long 0 0 S T 100 S T 100 Call with K = 100 Payoff of long 100 S T 100 S T 0 0 Put with K = 100 Payoff of short (S T 110) Call with K = 110 Payoff of short (90 S T ) Put with K = 90 Total payoff of S T S T option strategy Range of stock price S T < S T < S T < S T S T at expiration Profit of long S T S T Call with K = 100 Profit of long S T S T Put with K = 100 Profit of short S T Call with K = 110 Profit of short S T Put with K = 90 Total profit of S T S T option strategy Problem 24. Today it is November 29, 2010, and you are contemplating the following table of April 2011 options. 29
30 Calls Strike Puts Last Chg Bid Ask Vol Open Int. Price Last Chg Bid Ask Vol Open Int , , , , , , , , , , ,649 LAST TRADE as of 29Nov :03 AM , , , , , We are told that the annualized interest riskfree rate over the life of the options is r = 0.185% per year. (a) On what date will all these options expire? Solution: On the third Friday of April 2011, which falls on April 15, (b) How much would you get if you exercised the K = 130 call today? (We are talking about the call option with strike price K = 130.) How much would you get if you sold the K = 130 call today at the market (bid) price? Recall that today is November 29, Solution: I would get the difference between today s stock price, S t = , and the strike price, K = 130, that is S t K = = If I sold the call option today, I would get (c) Should you exercise today the K = 130 call? Why or why not? Is there any reason why you might want to exercise the call option before it expires in April? Solution: I would not want to exercise the call option today, because I would get only for it, while if I sold it at the market price, I would get more for it: In general, you should not exercise a call option before it expires: in general you should get more if you sold it. Another way of thinking about that is that by exercising the 30
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