Questions-Bond Valuation and Interest Rates

Q1) Microhard has issued a bond with the following characteristics: Par=$1,000 Time to Maturity: 13 years Coupon Rate:8% Semiannual payments Calculate the price of this bond if the YTM is: 8% 10% 6% a. P = $45({1 − [1 / (1 + .045)42]} / .045) + $1,000[1 / (1 + .045)42]P = $1,000.00 When the YTM and the coupon rate are equal, the bond will sell at par.   b. P = $45({1 − [1 / (1 + .060)42]} / .060) + $1,000[1 / (1 + .060)42]P = $771.63 When the YTM is greater than the coupon rate, the bond will sell at a discount.   c. P = $45({1 − [1 / (1 + .030)42]} / .030) + $1,000[1 / (1 + .030)42]P = $1,355.52

Q2) Watters Umbrella Corp. issued 15-year bonds 2 years ago at a coupon rate of 7 percent. The bonds make semiannual payments. If these bonds currently sell for 90 percent of par value, what is the YTM? Here we are finding the YTM of a semiannual coupon bond. The bond price equation is:    P = $900 = $42(PVIFAR%,36) + $1,000(PVIFR%,36)  Since we cannot solve the equation directly for R, using a spreadsheet, a financial calculator, or trial and error, we find:  R = 4.788% Since the coupon payments are semiannual, this is the semiannual interest rate. The YTM is the APR of the bond, so:  YTM = 2 × 4.788%YTM = 9.58%

Q3) Rhiannon Corporation has bonds on the market with 14.5 years to maturity, a YTM of 7.5 percent, and a current price of $1,061. The bonds make semiannual payments. What must be the coupon rate on these bonds? 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 = $1,061 = C(PVIFA3.75%,29) + $1,000(PVIF3.75%,29) Solving for the coupon payment, we get: C = $40.99 Since this is the semiannual payment, the annual coupon payment is: 2 × $40.99 = $81.97 And the coupon rate is the annual coupon payment divided by par value, so: Coupon rate = $81.97 / $1,000Coupon rate = .0820, or 8.20%

Q4) Locate the Treasury bond in the following table maturing in August 2027. Assume a par value of $5000. Is this a premium or discount bond? What is its current yield? What is its yield to maturity? This is a premium bond because it sells for more than 100 percent of face value. The dollar asked price is: Price = (133.3359 / 100)($10,000)Price = $13,333.59 The current yield is the annual coupon payment divided by the price, so: Current yield = Annual coupon payment / Asked priceCurrent yield = $525.00 / $13,333.59Current yield = .0394, or 3.94%

Q5) The Faulk Corp. has a 4 percent coupon bond outstanding. The Gonas Company has a 10 percent bond outstanding. Both bonds have 13 years to maturity, make semiannual payments, and have a YTM of 7 percent. If interest rates suddenly rise by 2 percent, what is the percentage change in the price of these bonds?  Any bond that sells at par has a YTM equal to the coupon rate. Both bonds sell at par, so the initial YTM on both bonds is the coupon rate, 10 percent. If the YTM suddenly rises to 12 percent:   PLaurel= $50(PVIFA6.0%,8)+ $1,000(PVIF6.0%,8)= $937.90PHardy= $50(PVIFA6.0%,34)+ $1,000(PVIF6.0%,34)= $856.32  The percentage change in price is calculated as: Percentage change in price = (New price – Original price) / Original price ΔPLaurel%= ($937.90 – 1,000) / $1,000= –.0621, or –6.21%ΔPHardy%= ($856.32 – 1,000) / $1,000= –.1437, or –14.37%  If the YTM suddenly falls to 8 percent: PLaurel= $50(PVIFA4.0%,8)+ $1,000(PVIF4.0%,8)= $1,067.33PHardy= $50(PVIFA4.0%,34)+ $1,000(PVIF4.0%,34)= $1,184.11  ΔPLaurel%= ($1,067.33 – 1,000) / $1,000= +.0673, or +6.73%ΔPHardy%= ($1,184.11 – 1,000) / $1,000= +.1841, or +18.41%  All else the same, the longer the maturity of a bond, the greater is its price sensitivity to changes in interest rates. Notice also that for the same interest rate change, the gain from a decline in interest rates is larger than the loss from the same magnitude change. For a plain vanilla bond, this is always true.

Q6) The Faulk Corp. has a 5 percent coupon bond outstanding. The Gonas Company has a 11 percent bond outstanding. Both bonds have 14 years to maturity, make semiannual payments, and have a YTM of 8 percent. If interest rates suddenly rise by 2 percent, what is the percentage change in the price of these bonds? Initially, at a YTM of 8 percent, the prices of the two bonds are:   PFaulk= $25(PVIFA4%,28) + $1,000(PVIF4%,28) = $750.05 PGonas= $55(PVIFA4%,28) + $1,000(PVIF4%,28) = $1,249.95   If the YTM rises from 8 percent to 10 percent: PFaulk= $25(PVIFA5%,28) + $1,000(PVIF5%,28) = $627.55 PGonas= $55(PVIFA5%,28) + $1,000(PVIF5%,28) = $1,074.49   The percentage change in price is calculated as: Percentage change in price = (New price – Original price) / Original price ΔPFaulk%= ($627.55 – 750.05) / $750.05 = –.1633, or –16.33%ΔPGonas%= ($1,074.49 – 1,249.95) / $1,249.95 = –.1404, or –14.04%  If the YTM declines from 8 percent to 6 percent: PFaulk= $25(PVIFA3%,28) + $1,000(PVIF3%,28) = $906.18PGonas= $55(PVIFA3%,28) + $1,000(PVIF3%,28) = $1,469.10  ΔPFaulk%= ($906.18 – 750.05) / $750.05 = +.2082, or  + 20.82%ΔPGonas%= ($1,469.10 – 1,249.95) / $1,249.95 = +.1753, or  + 17.53%  All else the same, the lower the coupon rate on a bond, the greater is its price sensitivity to changes in interest rates.

Q7) Hacker Software has 9.8 percent coupon bonds on the market with 18 years to maturity. The bonds make semiannual payments and currently sell for 107.7 percent of par. What is the current yield on the bonds? What is the YTM?  What is the effective annual yield? The bond price equation for this bond is:   P0 = $1,077 = $49(PVIFAR%,36) + $1,000(PVIFR%,36) Using a spreadsheet, financial calculator, or trial and error we find: R = 4.466% This is the semiannual interest rate, so the YTM is: YTM = 2 × 4.466%  YTM = 8.93% The current yield is: Current yield = Annual coupon payment / PriceCurrent yield = $98 / $1,077Current yield = .0910, or 9.10%  The effective annual yield is the same as the EAR, so using the EAR equation from the previous chapter: Effective annual yield = (1 + .04466)2 – 1Effective annual yield = .0913, or 9.13%