7: Solution
Chapter Interest Rates and Bond Valuation Page 226 Questions: 2, 3, 4, 5, 6, 7, 8 2. Price and yield move in opposite directions; if interest rates rise, the price of the bond will fall. This is because the fixed coupon payments determined by the fixed coupon rate are not as valuable when interest rates risehence, the price of the bond decreases.
NOTE: Most problems do not explicitly list a par value for bonds. Even though a bond can have any par value, in general, corporate bonds in the United States will have a par value of $1,000. We will use this par value in all problems unless a different par value is explicitly stated. 3. The price of any bond is the PV of the interest payment, plus the PV of the par value. Notice this problem assumes an annual coupon. The price of the bond will be: P = $75({1 [1/(1 + .0875)]10 } / .0875) + $1,000[1 / (1 + .0875)10] = $918.89 We would like to introduce shorthand notation here. Rather than write (or type, as the case may be) the entire equation for the PV of a lump sum, or the PVA equation, it is common to abbreviate the equations as: PVIFR,t = 1 / (1 + r)t which stands for Present Value Interest Factor PVIFAR,t = ({1 [1/(1 + r)]t } / r ) which stands for Present Value Interest Factor of an Annuity These abbreviations are short hand notation for the equations in which the interest rate and the number of periods are substituted into the equation and solved. We will use this shorthand notation in remainder of the solutions key. 4. Here we need to find the YTM of a bond. The equation for the bond price is: P = $934 = $90(PVIFAR%,9) + $1,000(PVIFR%,9) Notice the equation cannot be solved directly for R. Using a spreadsheet, a financial calculator, or trial and error, we find: R = YTM = 10.15%
1
If you are using trial and error to find the YTM of the bond, you might be wondering how to pick an interest rate to start the process. First, we know the YTM has to be higher than the coupon rate since the bond is a discount bond. That still leaves a lot of interest rates to check. One way to get a starting is point to use the following equation, which will give you an approximation of the YTM: Approximate YTM = [Annual interest payment + (Price difference from par / Years to maturity)] / [(Price + Par value) / 2] Solving for this problem, we get: Approximate YTM = [$90 + ($64 / 9] / [($934 + 1,000) / 2] = 10.04% This is not the exact YTM, but it is close, and it will give you a place to start. 5. Here we need to find the coupon rate of the bond. All we need to do is to set up the bond pricing equation and solve for the coupon payment as follows: P = $1,045 = C(PVIFA7.5%,13) + $1,000(PVIF7.5%,36) Solving for the coupon payment, we get: C = $80.54 The coupon payment is the coupon rate times par value. Using this relationship, we get: Coupon rate = $80.54 / $1,000 = .0805 or 8.05% 6. To find the price of this bond, we need to realize that the maturity of the bond is 10 years. The bond was issued one year ago, with 11 years to maturity, so there are 10 years left on the bond. Also, the coupons are semiannual, so we need to use the semiannual interest rate and the number of semiannual periods. The price of the bond is: P = $34.50(PVIFA3.7%,20) + $1,000(PVIF3.7%,20) = $965.10 7. Here we are finding the YTM of a semiannual coupon bond. The bond price equation is: P = $1,050 = $42(PVIFAR%,20) + $1,000(PVIFR%,20)
2
Since we cannot solve the equation directly for R, using a spreadsheet, a financial calculator, or trial and error, we find: R = 3.837% Since the coupon payments are semiannual, this is the semiannual interest rate. The YTM is the APR of the bond, so: YTM = 2 3.837% = 7.67% 8. Here we need to find the coupon rate of the bond. All we need to do is to set up the bond pricing equation and solve for the coupon payment as follows: P = $924 = C(PVIFA3.4%,29) + $1,000(PVIF3.4%,29) Solving for the coupon payment, we get: C = $29.84 Since this is the semiannual payment, the annual coupon payment is: 2 $29.84 = $59.68 And the coupon rate is the annual coupon payment divided by par value, so: Coupon rate = $59.68 / $1,000 Coupon rate = .0597 or 5.97%
3