# Solutions to Practice Questions (Bonds)

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1 Fuqua Business School Duke University FIN 350 Global Financial Management Solutions to Practice Questions (Bonds). These practice questions are a suplement to the problem sets, and are intended for those of you who want more practice. They are Optional, and are not part of the required material. 2. It is recommended that you look at these problems only after you fully understand how to solve the problem sets, the examples we covered in class, and the ones in the lecture notes. 3. Please note that I have collected these exmples from previous teaching material I have had. As such, while in most cases the notation will match the one used in class, the match is not 00%. 4. Some of these questions are easier than the ones you are expected to know how to solve, while others are above the level of knowledge you are expected to show on quizes and the final. ENJOY!

2 FIN 350 Solutions to Practice Questions 2. Suppose you invest \$,000. You will have \$2,000 in 0 years with this investment. We want to calulate ˆr so that: \$2, 000 = \$, 000( + ˆr) 0. Solving for ˆr, we get 2=(+ˆr) 0 (2) 0 =+ˆr.078 = + ˆr ˆr =7.8% Note: A useful rule of thumb is the Rule of 72. It says that for reasonable rates of return, the time it takes to double your money is approximately r% 72. Note that in this problem, 72 0 years r 7.2%. r% 2. The credit card company claims that the annual interest rate is 2.583% = 8.996%. However, with monthly compounding, the effective annual rate is (.0583) 2 =20.74%. The rate of 8.996% corresponds to an annual rate compounded monthly. 3. First, let us solve for the equivalent monthly rate ˆr: ( + ˆr) 2 =.2 ˆr =(.2) /2 =0.9489%. The present value of what you get is therefore given by PV + = 500 +ˆr = ( + ˆr) 2 ( + ˆr) 60 ˆr [ ] ( + ˆr) 60 =22, The present value of what you will have to pay back is given by [ 500 PV = ( + ˆr) 60 +ˆr ] ( + ˆr) ( + ˆr) [ ] = ( + ˆr) 60 ˆr ( + ˆr) 20 =20, Since the present value of the money you will get is larger than that you will have to pay back (PV + >PV ), you should accept the offer. 4. (a) We are interested in calculating Observe that so that C ( + r) n + C ( + r) 2n + C +. ( + r) 3n PV ( + r) n = C ( + r) 2n + C +, ( + r) 3n PV PV ( + r) n = C ( + r) n

3 FIN 350 Solutions to Practice Questions 3 which, after rearranging, yields C ( + r) n. (b) Let PVP t denote the present value of a t-year deferred perpetuity paying \$ at the beginning of every year. It can be shown that PVP t = ( + r) t +r = r /r ( + r) t. Also, notice that the perpetuity that we are interested in is simply the sum of a -year deferred perpetuity paying \$ at the beginning of every year; a 2-year deferred perpetuity paying \$ at the beginning of every year; a 3-year deferred perpetuity paying \$ at the beginning of every year; Therefore, PVP + PVP 2 + PVP 3 + = /r + /r +r + /r ( + r) 2 + Observe that PV +r = /r +r + /r ( + r) 2 +, so that PV PV +r = r. After rearranging terms, we find that +r r 2. (c) The equivalent semiannual rate ˆr must solve ( + ˆr) 2 =., which implies ˆr =4.88%. The present value of this perpetuity is therefore given by (.0488) 8 500(.0488) =7, (d) Using the formula on slide I..?? (in lecture notes), we have 00(.) =, (a) For no arbitrage to hold, the price of bond B should be three times the price of bond A. This is simply because the face value of bond B is three times the face value of bond A, and both bonds are year zero coupon bonds. Therefore, to make an arbitrage profit you should sell bond B and buy three units of bond A. The cashflow diagram of this strategy is: position Time 0 Time Buy 3 unites of Bond A Sell Bond B Combined 5 0 Thus, you make an arbitrage profit today of \$5.

4 FIN 350 Solutions to Practice Questions 4 (b) If you buy bond A you get a return of approximately %, which is clearly greater than 2%. As such you will borrow from the bank at 2% and buy bond A. More specificaly, you can borrow from the bank = \$98.04 dollars and buy unit of bond A for \$90. This leaves you with an arbitrage profit of \$8.04 today. The arbitrage table is as follows: Position Time 0 Time Borrow From Bank Buy Bond B Combined (a) First, let s find the discount factors for and 2 years: DF, and DF 2. We know that the present value of each of the bonds is given by C DF + C 2 DF 2. This implies that the following two equations for the two bonds: 85.0 = 3DF + 03DF = 0DF + 0DF 2 Solving these two equations for the two unknowns DF anddf 2 yields: DF =0.9, DF 2 =0.8. Therefore, the price of a 2 year zero coupon bond with a face of \$00 should be F DF 2 = 00(0.8) = \$80. (b) r = DF = 0.9 =0. and r 2 = = =0.2. DF (c) For a zero coupon bond the yield is always equal to the relevant spot rate. Thus, y = r 2 =0.2. (d) The yield to maturity of a two year r%(annual) coupon bond solves the equation Thus for bond A the yield to maturity solves 85. = C +y + C + F ( + y) y A ( + y A ) 2, and for bond B it solves 97 = y B ( + y B ) 2. Solving these two equations either directly or by trial and error gives y A = y B = Comment: In an exam a solution stating that each of the two yields is between 0. and 0.2 would suffice.

5 FIN 350 Solutions to Practice Questions 5 7. (a) The one year zero coupon bond s price satisfies PV A = F DF, which implies that the one year discount factor is DF = PV A F = =0.9. On the other hand, the price of bond B satisfies Pluging in the relevant known values yields PV B = C DF +(C + F ) DF = 0DF + 0DF 2 = 0(0.9) + 0DF 2 = 9 + 0DF 2. Therefore, DF 2 = (02.5 9)/0 = 0.85, which implies that the price of a two year zero coupon bond with face of value of \$00 is F DF 2 = 00(0.85) = \$85. (b) If the 2 year zero coupon bond is \$95 dollars an easy way to make an arbitrage profit is to sell the 2 year zero coupon bond and buy the one year zero coupon bond. This will generate a profit of = \$5 dollars today. At the end of the first year you will receive \$00, since you are long the one year zero coupon bond. You can use this \$00 in order to cover the \$00 you need to pay at the end of the second year, as you are short the two year zero coupon bond. (c) If there were no arbitrage opportunities the price of a 2 year zero coupon bond with face value of \$00 would have to be F DF 2 = 00(0.85) = \$85. This implies that the 2 year zero coupon bond is underpriced relative to a replicating portfolio of a 2 year zero coupon bond, which uses bonds A and B. First find the portfolio that would replicate a 2 year zero coupon bonds using bonds A and B: { 00nA +0n B = 0 0n B = 00 { na = n B = Since the bond is cheaper than the replicating portfolio we buy the bond and sell the replicating portfolio. This implies that we buy the bond, buy units of bond A and sell units of bond B. Notice: The replicating portfolio of the 2 year zero coupon bonds consist of short position of units of bond A and long position units of bond B. Since in the arbitrage strategy we are selling the replicating portfolio we need to put a minus in front of the relevant quantities (i.e, under the arbitrage strategy we buy units of bond A and sell units of bond B). This gives an arbitrage profit of \$85 \$80 = \$5

6 FIN 350 Solutions to Practice Questions 6 at time 0. At time, the replicating portfolio generates a cash (in)flow of At time 2 it generates a cash (in)flow of (00) (0) = \$ (0) = \$00. This is enough to cover the \$00 cash (out)flow which is required in order to pay the person who bought the 2-year zero coupon bond from us. Here is the complete arbitrage table: Strategy C 0 C C 2 Buy 0.09 bonds A Sell 0.9 bonds B Buy 2-year ZCB Total 5 0 0

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