Problems and Solutions CHAPTER I-?problems 1. 1 Problems on Bonds Exercise 1. 1 On 12104/01 consider a fixed-coupon bond Whose features are the following: ; face value: $1 ,OHO coupon rate: 8% ; coupon frequency: semiannual ; maturity: 05/06/04 What are the future cash flows delivered by this bond? Solution 1. 1 l. The coupon cash flow is equal to $40 It is delivered on the following future dates: 05/06/02, 11/06/02, 05/06/03, 1 1/06/03 and 05/06/04. The redemption value is equal to the face value 51,000 and is delivered on maturity date 05/06/04. Coupon = Exercise I . 3
An investor has a cash of at disposal.
He wants to invest in a bond with $1 ,OHO nominal value and whose dirty price is equal to 107. 457%. 1. What is the number of bonds he will buy? 2. Same question if the nominal value and the dirty price of the bond are respectively $100 and 98. 453%. Solution 1. 3 1. The number of bonds he will buy is given by the following formula Number of bonds bought = Cash Nominal Value of the bond x dirty price Here, the number of bonds is equal to 9,306 = 9,306.
048 1,000 x 107. 457% 2. N is equal to 101 ,562 Exercise 1. 4 10,000, coo 100 x 98. 453%
On 10/25/39, consider a fixed-coupon bond whose features are the following: ; face value: Euro 100 coupon rate: 10% coupon frequency: annual ; maturity: 04/15/08 Compute the accrued interest taking into account the four different day-count bases: Actual/Actual, Actual/365, Actual,’360 and 30/360. Solution 1. 4 The last coupon has been delivered on 04/15/99. There are 193 days between 04/15/99 and 10/25/99, and 366 days between the last coupon date (04/15/99) and the next coupon date (04/15/00). The accrued interest with the Actual/Actual day-count basis is equal to Euro 5. 273 x x Euro Euro 5. 273
The accrued interest with the Actual/ASS day-count basis is equal to Euro 5288 x Euro 100 Euro 5.
288 The accrued interest with the Actual/360 day-count basis is equal to Euro 5. 361 x x Euro 100 = Euro 5. 361 There are 15 days between 04/15/99 and 04/30/99, 5 months between May and September, and 25 days between 09/30/99 and 10/25/99, so that there are 190 days between 04/15/99 and 10/25/99 on the 30/360 day-count basis 15 4 (5 * 30) 25- 190 Finally, the accrued interest with the 30/360 day-count basis is equal to Euro 5. 278 1 90 x x Euro 100 = Euro 5. 278 360 Exercise I An investor wants to buy a bullet bond Of the automotive sector.
He has two choices: either invest in a US corporate bond denominated in euros or in a French corporate bond With same maturity and coupon. Are the two bonds comparable? Solution 1. 8 The answer is no. First, the coupon and yield frequency of the US corporate bond is semiannual, while it is annual for the French corporate bond. To compare the yields on the two instruments, you have to convert either the semiannual yield of the LIST bond into an equivalently annual yield or the annual yield of the French bond into an equivalently semiannual yield, Second, the two bonds do not serially have the same rating, that is, the same credit risk.
Third, they do not necessarily have the same liquidity. Exercise 1. 15 What is the price P of the certificate of deposit issued by bank X on 06/06/00, faith maturity 08/25/00, face value an interest rate at issuance of falling at maturity and a yield of 4 as of 07/31 /O? Solution 1. 15 Recall that the price P of such a product is given by P =F 1 + yam x name where F is the face value, c the interest rate at issuance, NC is the number of days between issue and maturity, B is the year. Basis (360 or 365), yam is the yield on money-market basis and NM is the number of days between settlement and maturity.
Then, the price P of the certificate of deposit issued by bank X is equal to P = Indeed, there are 80 calendar days between 06/06/00 and 08/25/00, and 25 calendar days between 07/31/00 and 08/25/00. 2 CHAPTER 2-?Problems Exercise 2. 1 Suppose the I-year continuously compounded interest rate is 12%. What is the effective annual interest rate? Solution 2. 1 The effective annual interest rate is R = e. 12-1 = 12. 75%. Exercise 2. 2 If you deposit $2,500 in a bank account that earns 8% annually on a continuously impounded basis, what will be the account balance in 7,14 years?
Solution 2. 2 The account balance in 7. 14 years Will be 2500. E%xx. 14 = $4,425. 98 Exercise 2. 3 If an investment has a cumulative 63. 45% rate of return over 3. 78 years, what is the annual continuously compounded rate of return? Solution 2. 3 The annual continuously compounded rate of return R is such that 16345 e, BRB we find R c – In(l . 6345)/3. 78- 13%. Problems and Solutions Exercise 2. 7 I _ What is the price Of a S-year bond With a nominal value Of 5100, a yield to maturity of 7% (with annual compounding frequency), a coupon rate and an annual coupon frequency? 2.
Same question for a yield to maturity of 8%, and 10%. Conclude. Solution 2. 7 1. The price P off bond is given by the formula which simplifies into where N , c, y and n are respectively the nominal value, the coupon rate, the yield to maturity and the number of years to maturity of the bond. Here, we obtain for 100 10 P is then equal to 112. 301% of the nominal value or $112. 301. Note that we can also use the Excel function “Price” to obtain P 2. Prices of the bond for different yields to maturity are given in the following table YET Price ($1 107. 985 103. 890 Bond prices decrease as rates increase.
Exercise 2. 14 We consider the following zero-coupon curve: Maturity (years) Zero-Coupon Rate 4. 75 1. What is the price of a 5-year bond with a $100 face value, which delivers a 5% annual coupon rate? 2. What is the yield to maturity of this bond? 3. We suppose that the zero. Coupon curve increases instantaneously and uniformly by 0. 5%. What is the new price and the new yield to maturity of the bond? What is the impact of this rate increase for the bondholder? 4. We suppose now that the zero-coupon remains stable over time. You hold the bond until maturity. What is the annual return rate of your investment?