UNIVERSITY OF TORONTO Joseph L. Rotman School of Management SOLUTIONS. C (1 + r 2. 1 (1 + r. PV = C r. we have that C = PV r = $40,000(0.10) = $4,000.

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1 UNIVERSITY OF TORONTO Joseph L. Rotman School of Management RSM332 PROBLEM SET #2 SOLUTIONS 1. (a) The present value of a single cash flow: PV = C (1 + r 2 $60,000 = = $25, )2T (1.055) 16 (b) The present value of an annuity is given by PV = [ ] 1 C 1 (1 + r 12 )12T r 12 $250,000 = (C/0.01)[1 (1.01) 144 ]. Solving for C we get C = $3, (c) From the valuation of a perpetuity, PV = C r. Therefore, r = C PV = $6,000 $40,000 = 15%. Since PV = C r we have that C = PV r = $40,000(0.10) = $4,000. 1

2 From the annuity formula, PV = C r [ ] 1 1 (1 + r) T We can manipulate this formula to obtain the result that T = ln ( [ ] ) 1 PV r ln 1 40,000(0.1) C 6,000 = = 11.5 years. ln(1 + r) ln(1.1) 2. (a) The term structure of one-year to three-year spot interest rates (r 1 to r 3 ) are the solutions to the following system of equations: = 1000 (1 + r 3 ) = r (1 + r 2 ) = r (1 + r 2 ) (1 + r 3 ) 3 Solving the above yields r 1 = 5%, r 2 = 8%, and r 3 = 12%. The term structure of forward rates follows from the following equations: f 1 = r 1 = 5%, f 2 = (1 + r 2 ) 2 /(1 + r 1 ) 1 = 11.09%, and f 3 = (1 + r 3 ) 3 /(1 + r 2 ) 2 1 = 20.45%. (b) The yield to maturity of the three-year zero coupon bond, y 3, solves: = 1000 (1 + y 3 ) 3, is the same as the 3-year spot rate, r 3 = To compute the yield of the annuity, y 2, we need to solve = y (1 + y 2 ) 2. for y 2. To do this, first let z = 1/(1 + y 2 ). The equation then becomes = 1000z z 2. The solution is z = , which implies y 2 = 6.93% (the other solution to this equation is negative and thus cannot be a proper YTM). To get YTM of the third instrument, y 3, we need to solve = y (1 + y 3 ) (1 + y 3 ) 3. Solving this equation by hand is not straightforward, so you could use the function IRR in Excel to solve it. The YTM is 10.14%. 2

3 (c) Suppose we hold x 1 units of the three-year zero coupon bond, x 2 units of the annuity, and x 3 units of the third bond. For this portfolio to mimic the payoff of the two-year zero coupon bond, we need to ensure that 0 = 0x x x = 0x x x 3 0 = 1000x 1 + 0x x 3 The solution (x 1 = 3, x 2 = 1, x 3 = 1) indicates that to replicate the two-year zero coupon bond, we need to short sell 3 units of the three-year zero, short sell one unit of the annuity, and buy one unit of the third bond. The amount of money we need to spend to construct this portfolio is exactly equal to the price of the two-year zero coupon bond, $857.34, and thus no arbitrage is possible here. (d) The yield to maturity of the two-year zero coupon bond is equal to the two-year spot rate, r 2 = 8% (this is because the bond is priced correctly and there is no arbitrage opportunity here). Since the replicating portfolio from part (c) has exactly the same cash flows and the same price as the two-year zero coupon bond, it must have the same YTM of 8%. (e) The replicating portfolio from part (c) still costs $ Thus, we have two strategies that lead to the same pattern of cash flows, but have different prices. If we buy the replicating portfolio and, at the same time, short sell one unit of the two-year zero-coupon bond, this will make us an instant profit of $42.66 and set all future cash flows equal to zero. We will get a payoff today, and will not need to make any payment in the future it is a free lunch. Of course, we would not stop at trading just one bond, but would repeat the above strategy as much as we could. 3. (a) Setting prices of the bonds equal to the present value of their cash flows, we get = = = r 1 (1 + r 2 ) 2, r 1 (1 + r 2 ) 2, 1000 (1 + r 3 ) 3. Solving this gives r 1 = 3%, r 2 = 5%, and r 3 = 7%, which implies that f 1 = 3%, f 2 = (1 + r 2 ) 2 /(1 + r 1 ) 1 = 7.04%, and f 3 = (1 + r 3 ) 3 /(1 + r 2 ) 2 1 = 11.12%. (b) We know that in the world with no arbitrage, the forward rate for year 3 should be f 3 = 11.12%. However, in this case we have a quoted forward rate of 9%. This means that the quoted forward rate is too low, and we want to borrow, say $1000, using the forward contract. In order to obtain an arbitrage profit, we need to create a synthetic forward contract that allows us to earn a forward rate of 11.12% using a portfolio of 3

4 three bonds. Let x be the number of units of 2-year 5% coupon bond in the portfolio, y be the number of units of 2-year 1% coupon bond in the portfolio, and z be the number of units of 3-year zero coupon bond in the portfolio. In order for this portfolio to have the same cashflow as a forward contract, we need the cashflows of the portfolio to be 0 at t = 0 and t = 1, and 1000 at t = 2. This requires us to solve the following system of equations: 0 = x y z, 0 = 50x + 10y, 1000 = 1050x y. Solving the equations, we obtain x = 1/4, y = 5/4, and z = Combining these positions, we have the following cashflows Cashflows Positions t = 0 t = 1 t = 2 t = 3 Borrow $1000 from the bank using a forward rate of f 3 = 9% Buy 1/4 unit of 2-year 5% coupon bond Sell 5/4 units of 2-year 1% coupon bond Buy units of 3-year zero coupon bond Total Note that the three bond positions together create a synthetic forward contract that allows us to lend 1000 at t = 2 and receive at t = 3 (i.e., a forward rate of 11.12%). Alternatively, we can buy 1.09 units of 3-year zero coupon bond. The combined cashflows will then be Cashflows Positions t = 0 t = 1 t = 2 t = 3 Borrow $1000 from the bank using a forward rate of f 3 = 9% Buy 1/4 unit of 2-year 5% coupon bond Sell 5/4 units of 2-year 1% coupon bond Buy 1.09 units of 3-year zero coupon bond Total which gives us an arbitrage profit of today. (c) The price of the 1-year zero coupon bond with face value of 1000 is given by 1000/(1 + r 1 ) = 1000/1.03 = The price of the 2-year zero coupon bond with face value of 1000 is given by 1000/(1 + r 2 ) 2 = 1000/(1.05) 2 =

5 (d) A portfolio that buys 0.1 unit of the 1-year zero-coupon bond, 0.1 unit of the 2- year zero-coupon bond, and 1.1 units of the 3-year zero-coupon bond will give us the same cash flows as the level-coupon bond (100, 100, and 1,100 in years 1, 2, and 3, respectively). The price of the level-coupon bond should therefore be P = = , which means that the current price of 1000 is cheap and we want to buy the actual bond and sell the portfolio of zero-coupon bonds that replicates the cash flows of the level-coupon bond. This will give us a cash flow of = today, and cash flows of = 0 in one year, = 0 in two years, and = 0 in three years. Thus we earn an arbitrage profit of today. 4. (a) To figure out the 1-year spot rate, we buy a 1-year deferred perpetuity and sell a 2-year deferred perpetuity. This effectively will give us a 1-year pure discount bond with face value $100 and it costs = Therefore, we have r 1 = = 3%. To figure out the forward rate for the second year, we buy a 2-year deferred perpetuity and sell a 3-year deferred perpetuity. This effectively will give us a 2-year pure discount bond with face value $100 and it costs = It follows that the forward rate for the second year is f 2 = DF 1 DF 2 1 = = 4% To figure out the forward rate for the third year, we buy a (b) Let y 1, y 2 and y 3 be the yield-to-maturity of the three deferred perpetuities with first payment starting at t = 1, t = 2, and t = 3, respectively. We have = 100 y 1 y 1 = 4.415%, = y 2 (1 + y 2 ) y 2 = 4.418%, = y 3 (1 + y 3 ) 2 y 3 = 4.421%. For y 2 we need to solve the following quadratic equation y y = 0, which can be easily done. To solve for y 3, we need to solve a cubic equation y y y = 0. 5

6 For this equation, we can use Solver tools in Excel (under Data, Solver) to obtain the solution. Although y 3 is highest, it does not mean that the deferred perpetuity that starts at t = 3 is underpriced. When the term structure of interest rates is increasing, recent cashflows are discounted at a lower rate than distant cashflows. As a result, the deferred maturity that starts paying earlier will have a lower yield-to-maturity even though they are all properly underpriced. Whether you prefer one deferred maturity to the other depends on your preference, and you may still want to buy the one with lower yield if you have preference for near term cashflows. 6