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Problem Set 1

1

Due: March 30 To be completed in groups of no more than four students

Note: A review of basic discounting concepts could be helpful here. Chapter 4 (in particular, 4.1–4.7) in Hull’s book would be a good place to look. Q1. Suppose that the continuously compounded, annualized yields-to-maturity on one-, two-, three-, four-, and five-year zero-coupon bonds are currently 4.0%, 4.6%, 5.0%, 5.7%, and 6.1%, respectively. a) Compute the current prices of the five zero-coupon bonds. Assume that each bond has a face value of $1000. b) If the two-year bond realizes a continuously compounded return of -5% from time 0 until time t1 = 1 year, what will this bond’s semi-annually compounded return from t1 = 1 year to t2 = 2 years be? c) Compute the current price and the annualized yield-to-maturity of a 5-year, 8% coupon bond with a face value of $1,000. Note: The (continuously-compounded) yield to maturity is defined as the (continuouslycompounded) discount rate y that is independent of term (thus, the term structure is flat) and prices an asset correctly. Thus, a stream of promised deterministic payoffs Xt at time t with price P0 has yield to maturity y defined by X e−ytXt . P0 = t

Q2. a) What is the present value of a stream of cash flows expected to grow at a 10 percent rate per year for 5 years (thus, from year 1 to year 6) and then remain constant thereafter until the final payment in 30 years? The payment at the end of the first year is $1,000 and the discount rate is 3.50 percent. b) You are considering purchasing a share of common stock in an airline. The dividends on this common stock have been growing at a 3 percent rate for the past 20 years, and you expect this to continue indefinitely. Dividends are expected to be $10 per share at the end of the year ahead, and you think 12 percent is the appropriate rate of return on this stock. How much would you be willing to pay for this stock?

Q3. The following table gives the prices of bonds:

Nicolae Gˆarleanu

MFE 230A

Problem Set 1

2

Bond Principal Time to maturity Annual coupon∗ Bond price ($) (years) ($) ($) 100 0.50 0.0 98 100 1.00 0.0 95 100 1.50 6.2 101 100 2.00 8.0 104 ∗ Half the stated coupon is assumed to be paid every six months. For this problem you may use semi-annual compounding or continuous compounding, just make sure to be explicit about it. a) Calculate spot rates for maturities of 6 months, 12 months, 18 months, and 24 months. b) What are the forward rates for the following periods: 6 months to 12 months, 12 months to 18 months, and 18 months to 24 months? c) What are the 6-month, 12-month, 18-month, and 24-month par yields for bonds that provide semiannual payments? [Note: par yield is defined as yield to maturity when the price equals the principal.] d) Suppose that you can trade at the forward rate f0,1,1.5 = 4.80%. Does this constitute an arbitrage? If so, describe clearly the strategy that you would implement and how you would make money without risking any loss. If not, explain why not. e) Estimate the yield to maturity of a 2-year bond providing a semiannual coupon of 7% per annum. Q4. This question is along the lines of work we did in class. Consider a one-period economy featuring two states, g (“good”), and b (“bad”), and two securities, characterized by their returns R1 = (u, d) with u > d and R2 = (R, R) with R > 0. a) No arbitrage requires positivity: u > 0 and R > 0. What other condition must hold on the return values in order to have no arbitrage? [Note: In class we talked mostly in terms of prices and future payoff. A return can also be interpreted as a payoff with a given price; in fact, for any desired price Pi — the “investment” — the associated future payoff is Xi = Pi Ri .] b) Is the market complete? c) Fixing the return values, compute all state price vectors. What are they, in particular, if R1 = (1.1, 0.95) and R = 1.02? d) Compute risk P neutral probabilities; thus, probabilities q1 and q2 such that EQ [R1 ] ≡ j R1s qs = R. e) Consider now a security that pays X3 = 100 × R21 = 100 × (u2 , d 2 ). What price P3 excludes arbitrage? You may keep the numerical values from above.

Nicolae Gˆarleanu

MFE 230A

Problem Set 1

3

f) Using these numerical values, suppose that P3 = 103. Is this an arbitrage? If so describe precisely a trading strategy (i.e., specify portfolio θ) that constitutes an arbitrage. If not, explain why not. Q5. This is also similar to work we did in class. Consider a three-state, one-period world with two securities paying X1 = (1, 2, 3), respectively X2 = (2, 3, 4). The two prices are P1 = 2 and P2 = 3. a) What are the possible state-price vectors and risk-neutral probabilities compatible with securities 1 and 2? b) Consider adding to the market security 3 paying X3 = (2, 0, −2). What prices P3 exclude arbitrage? c) Add now also security 4 with X4 = (1, 1, 2) and P4 = 1.4. What are the possible state-price vectors and risk-neutral probabilities compatible with all securities? d) Consider also security 5 paying X5 = max(4 × X1 , 3 × X3 ) — thus security 5 is a derivative on securities 1 and 3. What prices P5 exclude arbitrage? The following questions are optional. They can be answered based on the class material, but they are slightly more involved, and are not required. e) Given X1 , X2 , and P1 (all as above) only, what prices P2 exclude arbitrage? f) Given X1 , X2 , X4 , P1 , and P2 only, what prices P4 exclude arbitrage?

Nicolae Gˆarleanu

MFE 230A...