Bond Valuation and Yield Calculations

Posted on Oct 13, 2024 in Mathematics

1. Calculating Bond Price with Annual Coupon Payments
   A German company issues a bond with a par value of €1,000, 8 years to maturity, and an 8% annual coupon rate. If the yield to maturity (YTM) is 10%, what is the current bond price in euros?

2. Calculating Yield to Maturity
   A Japanese company has a bond selling for 107.218% of its ¥100,000 par value. The bond has a 6.8% annual coupon rate and matures in 20 years. What is the yield to maturity of this bond?

3. Calculating Coupon Rate
   Nikita Enterprises has bonds on the market making annual payments, with 17 years to maturity, a par value of $1,000, and selling for $969. At this price, the bonds yield 8.1%. What is the coupon rate?

Coupon rate = $77.58 / $1,000 = 0.0776, or 7.76%

4. Calculating Bond Price with Semiannual Payments
   Westco Company issued 14-year bonds one year ago at a 6.9% coupon rate. The bonds make semiannual payments and have a par value of $1,000. If the YTM is 5.5%, what is the current bond price?

5. Calculating YTM with Semiannual Payments
   Ashburn Company issued 14-year bonds two years ago at a 9.7% coupon rate with semiannual payments. If these bonds currently sell for 102% of par value, what is the YTM?

YTM = 4.709% * 2 = 9.42%

6. Calculating Coupon Rate with Semiannual Payments
   Draiman Corporation has bonds with 12 years to maturity, a YTM of 9.7%, a par value of $1,000, and a current price of $948. The bonds make semiannual payments. What is the coupon rate?

Annual Coupon = $44.79 * 2 = $89.58
Coupon Rate = $89.58 / $1,000 = 8.96%

7. Calculating Real Rate of Interest
   If Treasury bills are currently paying 6.15% and the inflation rate is 2%, what are the approximate and exact real rates of interest?

Approximate Real Rate:

R ≈ r + h

r ≈ 0.0615 – 0.020 = 0.0415, or 4.15%

Exact Real Rate (Fisher Equation):

(1 + R) = (1 + r)(1 + h)

(1 + 0.0615) = (1 + r)(1 + 0.020)

r = [(1 + 0.0615) / (1 + 0.020)] – 1 = 0.0407, or 4.07%

8. Bond Price Changes Over Time
   Bond X is a premium bond with an 8.9% coupon rate, a 6.9% YTM, and 14 years to maturity. Bond Y is a discount bond with a 6.9% coupon rate, an 8.9% YTM, and also 14 years to maturity. Both bonds make semiannual payments, have a $1,000 par value, and interest rates remain unchanged. Calculate the bond prices today, in one year, three years, eight years, twelve years, and fourteen years.

Calculations are omitted for brevity. Use the provided formulas and adjust N for each time period. Bond prices converge to par value as maturity approaches.

9. Impact of Interest Rate Changes
   Bond Sam and Bond Dave have 9.6% coupon rates, make semiannual payments, are priced at par value ($1,000), and have 6 and 23 years to maturity, respectively. Calculate the percentage price change if interest rates suddenly rise or fall by 3%.

Calculations are omitted for brevity. Use the provided formulas and adjust the YTM. Longer-maturity bonds are more sensitive to interest rate changes.

10. Coupon vs. Zero-Coupon Bonds
    Your company needs to raise $41.2 million and wants to issue 30-year bonds. The required return is 6.2%, and the tax rate is 22%. You are evaluating two options: a semiannual coupon bond with a 6.2% coupon rate and a zero-coupon bond. Both bonds have a $1,000 par value.

1. How many of each bond would you need to issue?

Coupon Bonds:
Number of bonds = $41,200,000 / $1,000 = 41,200

Zero-Coupon Bonds:
Price of zero-coupon bond = $1,000 / (1.031)^60 = $160.13
Number of bonds = $41,200,000 / $160.13 = 257,287

2. What will be the repayment in 30 years for each bond?

Coupon Bonds:
Repayment = 41,200 * ($1,000 + (0.062/2)*$1,000) = $42,477,200

Zero-Coupon Bonds:
Repayment = 257,287 * $1,000 = $257,287,000

3. Calculate the first-year after-tax cash outflows under both scenarios, assuming IRS amortization rules apply to zero-coupon bonds.

Calculations are omitted for brevity. For coupon bonds, the after-tax cash outflow is the coupon payment less tax savings from the interest expense. For zero-coupon bonds, the after-tax cash outflow is the change in the bond’s value due to amortization multiplied by the tax rate.

11. Calculating YTM from Bond Quotes
    (Problem and table omitted for brevity. Use provided formulas and adjust for semiannual payments.)

12. Calculating Bond Price and Current Yield
    (Problem and table omitted for brevity. Use provided formulas and adjust for semiannual payments.)

13. Calculating Coupon Rate from Bond Quotes
    (Problem and table omitted for brevity. Use provided formulas and adjust for semiannual payments.)

14. Calculating Current Yield and Capital Gains Yield
    Bond P is a premium bond with a 9.1% coupon rate. Bond D is a discount bond with a 5.1% coupon rate. Both bonds make annual payments, have a 7.1% YTM, a $1,000 par value, and six years to maturity. Calculate the current yield and expected capital gains yield for each bond.

Calculations are omitted for brevity. Use provided formulas.

15. Calculating Holding Period Yield (HPY)
    You buy an 8.3% annual coupon bond for $915. The bond has 10 years to maturity and a $1,000 par value. Two years later, the YTM declines by 1%, and you sell. Calculate the expected YTM, the selling price, and the HPY.

Calculations are omitted for brevity. Use provided formulas.