B6302 Sample Placement Exam Academic Year

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1 Revised June 011 B630 Sample Placement Exam Academic Year Part 1: Multiple Choice Question 1 Consider the following information on three mutual funds (all information is in annualized units). Fund 1 Fund Fund 3 mean volatility If the risk-free rate is 5%, which one of these could be the market portfolio? a. Fund 1 b. Fund c. Fund 3 d. Can t tell Reason: Fund has the highest Sharpe ratio and thus could be the market. Question Suppose that you are Fannie Mae and your duration has increased, because, for example, prepayments slowed or yields decreased. To reduce your duration gap, you could a. Buy long-dated Treasuries b. Sell long-dated Treasuries c. Pay fixed on a long-dated swap d. Pay floating on a long-dated swap e. None of the above Reason: To reduce your duration you need to sell (or short) securities. To do this you could either sell Treasuries or pay fixed on a long-dated SWAP.

2 Question 3 Paying floating on an interest rate swap has cash flows that are very similar to a. Buying a coupon bond of a similar credit quality, maturity and coupon b. Shorting a coupon bond of a similar credit quality, maturity and coupon c. Have returns that are unrelated to coupon bonds Reason: Paying a floating rate implies that you receive a fixed rate. Receiving a fixed rate on a swap is similar to buying a Treasury: if you buy a Treasury you receive a fixed coupon and lose money (mark-to-market basis) if floating rates increase. Question 4 Consider the following term structure: r 1/ = 0.05, r 1 = 0.04, and r 3/ = What is the price of a 1 ½ year coupon bond which pays semi-annual coupons and has an annualized coupon rate of 5% of par (assume $100 par value). a b c d e. none of the above P1.5 = + + = ( ) ( ) Question 5 As diversification increases, the total variance of a portfolio approaches a. Zero b. One c. The variance of the market portfolio d. Infinity e. Can t tell from the given data Reason: greater and greater diversification implies the portfolio better and better approximates the market.

3 Question 6 Consider the following -period binomial model. As usual, over each of the next two periods, the stock can either increase to us 0 or decrease to ds 0. Assume that u = 1.1, d = 0.9 and that r = 0.05 for each period (do not compound). Suppose that the current price of the stock is 10. Find the price of an American put option struck at K = 90. (Round your answer to the nearest 10,000 th.) a b c d e. None of the above. Reason: the stock price in the second period is 13 (upper branch) and 108 (lower branch) and assumes one of 145., 118.8, or 97. in the final period. Clearly the option is never in the money, so its payoff in all states is zero. Thus, its price must be zero. Question 7 The table below provides the time to maturity, coupon rate, par value, yield to maturity and price of two bonds. Time to Maturity (Years) Yield (%) Bond A Coupon Zerocoupon Face Value Price Bond B Spot Rate Calculate the one year spot rate. a. 4.1% b. 5.1% c. 6.03% d. 7.1% e. none of the above Reason: the question wants r 1 where 100 = + 1+ r.5 1+ r1 Solve for r 1 ; r.5 =.04 r 1 =.0603 ( ) ( ) 3

4 Question 8 Suppose that the current term structure is given by: Years Spot Rate 0.5 4% 1 10% A 6-in-6 month s forward rate agreement [FRA] is a swap that will pay in half a year the then-current half-year spot rate and will receive a fixed rate [established today but paid in half a year]. The 6-in-6 FRA s fixed rate should be a % b % c % d % e. none of the above Reason: The question is asking for.5 f.5 where.04.5 f = 1+.5f.5 as computed here does not coincide with a), b), c), or d). Question 9 Two bonds are selling at par value and each has 17 years to maturity. The first bond has a coupon rate of 6% and the second bond has a coupon rate of 13%. Which of the following is always true about the Macaulay durations of these bonds? (Again, recall that Macaulay duration is always positive.) a. The duration of the higher-coupon bond will be higher. b. The duration of the lower-coupon bond will be higher. c. The duration of the higher-coupon bond will equal the duration of the lowercoupon bond. d. There is no consistent statement that can be made about the durations of the bonds. e. The bond s durations cannot be determined without knowing the prices of the bonds. f. none of the above Reason: Duration (risk) is lower when coupons are high or when rates are high. For the coupons, the higher the coupon, the more money you receive early in the bond s life and thus the less risky it is (less sensitive to a rate change). 4

5 Question 10 If CAPM is valid, which of the following situations are possible? (A) Expected Return Beta A 0 1. B 5 1. With identical betas, stocks A, B should have the same expected return by CAPM. They do not. (B) Expected Return Standard Deviation A B 45 5 Possible. (C) Expected Return Standard Deviation Risk-free 10 0 Market 18 4 A 1 1 Possible. (D) Expected Return Standard Deviation Risk-free 10 0 Market 18 4 A 8 48 Er = r +ρ σ A f AM A ( Er r ) M σ M f =.10 +ρam (.8).4 =ρ > 1, AM which is impossible. 5

6 (E) Expected Return Beta Risk-free 10 0 Market A by CAPM, Er A = rf +βa( Er M rf ) = (.18.10) =..30 (the actual expected return is inconsistent with the risk adjusted expected return and thus inconsistent with CAPM) Answer: Clearly, B and C are the only possibilities remaining. Question 11 In setting the Federal Funds Rate the Fed: a. Can primarily influence the long-term rates, since there is much more volatility in the short term. b. Has much control over spot rates at short maturities. c. Tries to flatten the yield curve. d. Will always set the rates as low as politically feasible in order to stimulate growth. Reason: See the course text (if you have not had a macroeconomics course) where this is discussed. Question 1 When immunizing your portfolio of $150,000 spent entirely on 5-year zero coupon bonds from parallel shifts using 10-year zero coupon bonds: a. You will have to short more than $150,000 worth of 10-year zero-coupon bonds because their duration is higher. b. You will have to short less than $150,000 worth of 10-year zero-coupon bonds because their duration is higher. c. You will have to long $150,000 worth of 10-year zero-coupon bonds. d. You cannot immunize your portfolio from parallel shifts using only 10-year bonds. You require at least a barbell hedge. Reason: Self-explanatory 6

7 Question 13 Suppose there are two scenarios, A and B, that can happen in year 1 with probabilities 95% and 5%, respectively. A CDS that pays $1 only if scenario A happens is priced at 90 cents. A CDS that pays $1 only if scenario B happens is priced at 6 cents. What is the price of a one-year (treasury) zero with $100 par? a. $9 b. $94 c. $96 d. $98 e. This cannot be determined without knowing the one-year spot rate. Reason: The price is just the sum of the two state prices. The probabilities are irrelevant: they are incorporated in the state prices. Question 14 To price swaps: a. We use the Libor spot rates to compute the forward rates in order to price the pay floating side. b. We compute the cost today to the pay fixed side of entering a swap contract. c. We can use the term structure of treasuries to compute the discount factors in the numerator and denominator, since any difference with the Libor market cancels out. d. We back out the Libor term structure from the Libor forward rates. e. We need to compute the term structure 6-months forward, since floating payments on swaps are typically computed based on the spot rates used in the preceding 6- month period. Reason: Libor forward rates are quoted. A complete set of Libor forward rates allows the construction of the Libor term structure and vice versa. In this sense it is no different than the Treasury term structure. Question 15 Suppose the YTMs on 0.5-year, 1-year, and 1.5-year (Treasury) Zeroes are all 0%. If the price of a -year % T-note is $10, can you bootstrap the -year Zero YTM? If so, what is it? a. Yes, the -year Zero YTM is 0%. b. Yes, the -year Zero YTM is 1%. 7

8 c. Yes, the -year Zero YTM is %. d. Yes, the -year Zero YTM is 3%. e. No, the -year Zero YTM cannot be found unless we know the price of a -year Zero. Reason: We can bootstrap the -year Zero YTM (r) from the price of the - year % T-note, which can be written as: $10 = $1 + $1 + $1 + $101/(1+r/) 4 r = * [(101 / 99) 1/4 1] = 1.00% Question 16 Suppose you expect the (initially flat) yield curve to develop a positive hump at the 5- year maturity. Which investment strategy below could you use to profit from this belief, while simultaneously immunizing your portfolio against parallel shifts in yield curve today? a. Buy $1M each of 1-year and 9-year Zeros and short $M in 5-year Zeros b. Buy $1M each of 5-year and 9-year Zeros and short $M in 1-year Zeros c. Short $1M each of 1-year and 9-year Zeros and buy $M in 5-year Zeros d. Short $1M each of 5-year and 9-year Zeros and buy $M in 1-year Zeros Reason: A positive hump in the yield curve occurs when short-term and long-term bond yields fall, so that their prices increase. Thus, you would want to buy the 1-year and 9-year Zeros. To immunize your portfolio, you could short intermediate-term bonds, such as 5-year Zeros. Note: It turns out that equal total amounts of these bonds in a long-short portfolio will create an immunized portfolio because the durations offset when you hold 50% 1-year and 50% 9-year bonds and short 100% 5-year bonds because 0.5 * * 9 = 5. The strategy above uses exactly $M in the long and short positions with these weights. 8

9 Long Question 1: CAPM Use the CAPM model to fill in the blanks in the following table: Market Portfolio beta Expected return variance covariance with the market Risk Free rate Stock Stock By construction, the beta of the market is 1;. The beta, variance, and covariance of the risk free rate are all 0; 3. The beta of stock 1 is cov( r1, r M ).045 = = σ To compute Er M, observe: Er r =β Er r ( ) 1 f 1 M f M ( ).0.0 = Er.0 M Thus Er =.1 5. Er = r +β ( Er r ) f M f ( ) = =.07 or 7% cov r, r =β σ =.5x.05 = ( ) M M 9

10 Long Question Consider the setting of 1-period binomial tree model for a stock that is currently trading at S. Over the next time interval the price can either go up to us or down to ds. There is a European call and put option currently struck at K=$50 that are both expiring in 1 year. The current price of the call option is $10 and the current price of the put is $9.50. The current stock price is $50. The 1-year spot rate is %. Use simple interest for all of the discounting calculations. a. According to put-call parity, what is the cost of replicating the put? b. Using put-call parity to replicate the put, explicitly construct an arbitrage trade. (By explicitly I mean construct a sequence of trades that results in a sure profit, either now or in the future with no obligations at the other time periods.) a) K C + S = 9.0 ( 1+ r ) 0 0 f Thus P 0 = price of the put must exceed $9.0, which $9.50 does. b) Thus: (1) write a put at a price of $9.50 () buy a call (3) invest K 1+ r f in risk free assets (4) short one share of the underlying stock Profit: $9.50 $9.0 = $.48 Future cash flow liability is zero. 10

11 Long Question 3 Consider a -yr-to-maturity European put option with a strike price of $48 on a stock whose current price is $50. We assume there are time steps, each covering one year, and in each time interval the price moves up or down by 0%. The annual risk free rate of interest is %; i.e., 1+r t = 1.0 per period. Price the two-year, European put option today by constructing the replicating portfolio at each node, and working backwards. You may alternatively use risk neutral valuation methods. Risk neutral valuation is simpler: P S uu = $60 0 u S u = (1.) ($50) = $60 S ud = $50 0 S 0 = 50 d S u = (1 -.) ($50) = $40 S du = $50 0 S dd = $3 $16 rf d Π RN = risk neutral probabilities = = =.55 u d 1..8 P u = 0 1 P d = (.55(0) +.45(16)) = P 0 = (.55( 0 ) +.45(7.0588)) =

12 Long Question 4 The following securities are trading in the market: (a) a 6-M zero coupon bond which is trading at 0.98 (per $1 of par value); and (b) a 1-year zero coupon bond which is trading at 0.96 (per $1 par). Assume that your broker quoted the half-a-year in half-a-year forward at 4.1%. Using the two assets above, construct a trade exploiting the arbitrage opportunity that presented itself. [You cannot invest at the risk-free rate for 6-M or 1-year, but can only trade the bonds.] Explain carefully how you will make money by describing trades today, in half a year, and in a year. The cash flows from the various securities are (assuming long positions): t = Long 1, 6 mo zero Long 1, 1 yr zero Long one forward contract where.98 = Let us suppose you want to create the cash flow t= with x y z 6 mo. zeros 1 yr zeros forward contracts The cash flow will be: 1

13 t = x 1.00x -.96y 1.00y z 1.00z where you want -.98x -.96y = 100 x -.98z = 0 y + z = 0 => -y = z; x =.98 z =.98 (-y) x = -.98 y Thus, -.98x -.96y = (-.98y) -.98z = y -.96y = y = 100 y = 50,000 Thus, z = - 50,000 y =.98 (-50,000) = - 45,000 Check the solution: it satisfies the equations and you earn an arbitrage profit immediately. This works because the rate implicit in the forward contract is inconsistent with the forward rate implied by r.5 and r 1. 13

14 Long Question 5 (i) (ii) You have a portfolio of $10,000,000 of duration 5 bonds and $8,000,000 of duration 3 bonds. You wish to perfectly hedge this portfolio against interest rate change using an 8-year T-bond of duration 6.4. What position in the 8- year bonds should you take? A friend of yours is trying to persuade you to invest in a particular (one-period) project. He has convinced you that the project's return pattern is independent of the pattern of market returns. You are the owner of a well diversified portfolio. There is a 5% probability, however, that you will lose everything. What rate of return should you require if the project is successful (75% prob.)? r f = 10%, ERM = 15%. That is, what ROR would you require in the good state? (i) Let D denote duration; P, price; n, the number of bonds, etc. You want D Portfolio V P = n 5 P 5 D 5 + n 3 P 3 D 3 + n 8 P 8 D 8. You want D P = 0 n 5 P 5 = $10 M (duration 5 bonds) n 3 P 3 = $8 M (duration 3 bonds) Solve for n 8 P 8 = V 8, the value of the 8 year duration 6.4 bonds: 0 = 10M (5) + 8M (3) + V 8 (6.4) - 74 = V 8 (6.4) V 8 = M -- short MM of the 8 year bonds. (ii) by CAPM ρ project, M = 0 so β p = 0 Thus Er project = r f =.5 (- 1) +.75 (r success ) =.10 r success =.4666 or 46.66%. 14