B6302 B7302 Sample Placement Exam Answer Sheet (answers are indicated in bold)

B6302 B7302 Sample Placement Exam Answer Sheet (answers are indicated in bold) Part 1: Multiple Choice Question 1 Consider the following information on three mutual funds (all information is in annualized units). Fund 1 Fund 2 Fund 3 mean 0.45 0.35 0.10 volatility 0.80 0.50 0.10 If the risk-free rate is 5%, which one of these could be the market portfolio? a. Fund 1 b. Fund 2 c. Fund 3 d. Can t tell Question 2 Suppose you have two call options struck on the same stock and the strike (exercise price) of the first option (K 1 ) is less than the strike of the second option (K 2 ), that is, K 2 >K 1 >0. In a 1-period binomial model, assuming that us>k>ds and that u>1+r>d, then a. The delta for the first option (Δ 1 ) is always greater than the delta for the second option (Δ 2 ). b. The delta for the first option (Δ 1 ) is always less than the delta for the second option (Δ 2 ). c. Not enough information to answer the question

Question 3 An investor invests 30 percent of his wealth in a risky asset with an expected rate of return of 0.15 and a variance of 0.04 and 70 percent in a T-bill that pays 6 percent. His portfolio's expected return and standard deviation are Er p σ p a. 0.114; 0.12 b. 0.087; 0.06 c. 0.295; 0.12 d. 0.087; 0.12 e. none of the above Question 4 Suppose that u=1.5, d=0.5, r f =0.10, S=100. What is the price of a call option struck at K=0? a. 150.00 b. 100.00 c. 50.00 d. 90.909 e. None of the above. Question 5 The delta of an in-the-money put option is always less than -0.5. a. Always true b. Always false c. Could be, but does not have to be.

Question 6 Consider a two-period binomial model. The current stock price is $50 and in each of the next two years, the stock price can increase or decrease by 20%. This implies that u=1.20 and d=0.80. Assume that the interest rate factor for each period is 1+r f =1.051. Suppose that the prices of a European put struck at K=52 at each node are given by: P 0 =4.1923; P d =9.4636; P u =1.4147; P uu =0; P ud =4 and P dd =20. You can round your answer to the nearest penny. Then, the price, at time zero, of an American put struck at K=52 is a. $4.19 b. $4.77 c. $5.09 d. $6.01 e. None of the above. Question 7 Assume the correlation coefficient between Intel and the market is 0.40. What percentage of Intel s total risk is idiosyncratic (i.e., nonsystematic) (round your answer to the nearest $)? a. 40% b. 60% c. 16% d. 84% e. None of the above. Question 8 Consider an at-the-money call option on Google, one-year time-to-maturity, European style. Assume Google doesn t pay dividends. The current stock price S=$500. The annual risk free rate is 5%. Which of the following could be the call option price? a. $25 b. $23 c. $21 d. $19 e. None of the above f. Can t tell from the given data

Question 9 One should never invest in the following type of stocks a. Stocks with negative expected return b. Stocks with negative beta c. Stocks with negative Sharpe ratio d. None of the above Question 10 Consider an asset allocation problem with a single risky asset with expected return μ and volatility σ and assume the risk-free rate is r. Let r p denote the risky return on a portfolio consisting of w% of your wealth in the risky asset and the remainder in the risk free asset. The Sharpe ratio of a a. leveraged portfolio is greater than the Sharpe ratio of an unleveraged portfolio b. leveraged portfolio is less than the Sharpe ratio of an unleveraged portfolio c. leveraged portfolio is the same as the Sharpe ratio of an unleveraged portfolio d. can t tell from the data. Question 11 Suppose you have a portfolio with $200 invested in a stock with a beta of 1.4 and $100 short a stock with a beta of 0.8, and $100 short a stock with a beta of 0.6. The equity premium is 8%. What is the expected excess return on your portfolio? a. 5.6% b. 0.0% c. 7.5% d. Not enough information

Question 12 Suppose that duration of Fannie Mae s assets has recently increased, because, while the duration of Fannie Mae s liabilities did not change [?]... To reduce the firm s duration gap, Fannie Mae could a. Buy long-dated Treasuries b. Sell long-dated Treasuries c. Pay floating on a long-dated swap d. None of the above Question 13 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 and borrowing at floating rate to finance the purchase b. Shorting a coupon bond of a similar credit quality, maturity and coupon and investing at floating rate the proceeds c. Receiving returns that are unrelated to coupon bonds d. None of the above Question 14 Consider the following term structure: r 1/2 = 0.05, r 1 = 0.04 and r 3/2 = 0.025. 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. 106.89 b. 99.87 c. 110.84 d. 103.60 e. none of the above

Question 15 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 0.5 4 Coupon Zerocoupon Face Value Price 100 98.04 Bond B 1 6 6 100 100 Spot Rate Calculate the one year spot rate. a. 4.12% b. 5.12% c. 6.03% d. 7.12% e. none of the above. Question 16 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. 7.088% b. 8.088% c. 9.088% d. 10.088% e. none of the above

Question 17 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 Question 18 Consider a 1-year-to-maturity bond with face value of $10,000 and 5% coupon rate. Assume the market yield curve is flat at 3%. Calculate the DV01, for this bond. a. $0.99 b. $1.14 c. 0.88 year d. $10.08 e. none of the above

Question 19 If CAPM is valid, which of the following situations are possible? Expected Return Beta (A) A 20 1.2 B 25 1.2 Expected Return Standard Deviation (B) A 30 35 B 45 25 Expected Return Standard Deviation (C) Risk-free 10 0 Market 18 24 A 12 12 Expected Return Standard Deviation (D) Risk-free 10 0 Market 18 24 A 28 48 Expected Return Beta (E) Risk-free 10 0 Market 18 1.0 A 30 1.5 The following situations are possible: a. A and B and E b. B and C c. C and D d. B only e. none of the above

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 1.12 0.0225.0225 Risk Free rate 0 0.02 0 0 Stock 1 2 0.22 0.25 0.045 Stock 2 0.5.07 0.09.01125 Solution: 1. The beta of the market is 1. 2. The beta, variance and covariance of the risk-free rate are 0. 3. To find the beta of stock 1, compute the covariance divided by the market variance to get 0.045/.0225 = 2. 4. This in turn implies that since the expected return of stock 1 is 0.22, ( ) ( ) E(r) rf = β1 E(rm r f) 0.22 0.02 = 0.2 = 2 E(rm r f) E(r ) r =.01 m f Which implies that the market return is 12%. 5. Given 4, the excess expected return on stock 2 is 5%, which implies the expected return is 7%. 6. Finally, the covariance of stock 2 with the market is cov = (.5)(.0225) = 0.01125.

Long Question 2 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, where 1 d =. u 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 2%. 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). Solution: Put-call parity implies the put is worth K C0 + S0 = 9.02 1+ r Since the put price is higher than $9.02, one can write a put at the price $9.50 and buy a portfolio of one call option C, risk free assets of value K/(1+r) and short one share of the underlying stock. The cash flow today is $9.50 $9.02 = $0.48 The future cash flow is always zero as implied by the put-call parity. This $.48 is an arbitrage profit.

Long Question 3 Consider a 2-yr-to-maturity European put option with a strike price of $48 on a stock whose current price is $50. We assume there are 2 time steps, each covering one year, and in each time interval, the price moves up to us or down to ds with u = 1.2 and The annual risk free rate of interest is 2%, i.e., 1 + r t = 1.02 per period. Price the European put option today by constructing the replicating portfolio at each node, and working backwards. You may alternatively use risk neutral valuation methods. 1 d =. u Solution: P u = 0 $60 S T $72 PT max{0, E S } = 0 = P T uu S 0 = $50 u = 1.2 P 0 = $3.22 d =.83 $41.66 P d = $6.71 $50 $34.58 0 = P ud max{0, 48 34.58} = 13.42= P dd P r d d RN f = u 1.02.83 = 1.2.83.19 = =.51.37.51(0) +.49(13.42) Pd = = 6.71 1.02.51(0) +.49(6.71) P 0 = = $3.22 1.02 (Here we have used risk neutral valuation by first computing the risk neutral probabilities and working backwards through the tree.)

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. Solution: Strategy 1: Buy a zero bond which matures m+n 6 month periods from now. Value of this strategy today is: Strategy 1: $1/ 1 m+ n rm+ n + = In our case, m = n = 1; therefore, the value of the strategy today is equal to δ 2 = 0.96. Strategy 2: Buy a zero bond which matures in m periods and then, at maturity, reinvest all the proceeds in a new zero bond which matures in n periods (using the forward rate, generically, f, for n-periods). The value of the forward rate leg, after m periods is: 2 δ m+ n n f $1/ 1 + (n six-month periods) 2 then, to buy a bond with a principal equal to the above value, we have to pay today: 2 2 2 m n n rm f f at time 0: $1/ 1+ 1+ = δ m / 1+ In our case, δ m = δ 1 = 0.98, and n = 1. Now since both of these strategies are riskless (the spot and forward rates are contractually enforceable) and they had the same cost, we have, by no arbitrage, that the values must be the same, which implies: n f δ f δ 1 + =, or 1+ = 2 δ 2 δ m 1 m+ n 2

Solving for the forward rate, we have that: 1/n m 1 δ δ 0.98 f = 2 1 = 2 1 = 2 1 = 4.16% δm+ n δ 2 0.96. (B) Suppose your broker quoted the forward rate of 4.1%. Construct a trade exploiting this opportunity. You would want to lend at a higher forward rate and borrow at a lower rate. So, you call up your broker and take a borrowing forward position. Now you have to replicate a forward lending position by trading in bonds. Lending position means outlay of cash one period from today, and inflow of cash two periods from today. This means that you have to be short the 6-month bond and long the 1-year bond. The long/short quantities should be such that the cost of setting up the position today is zero, i.e., δ 0.98 49 = = = ; i.e., you need to buy 49 1-year bonds and 1 N δ2 δ1 0 N δ 2 0.96 48 short 48 6-month bonds. As a result, you pay nothing at time 0, at time 1 you have to pay $48 to cover your short position in the 6-month bond, and at time zero you will receive $49 from the one-year bond. Your return is (49-48) / 48 = 2.08% in 6 months, or, equivalently, 4.16% on an annual scale, as required.