Answered

AMany years ago, Castles in the Sand Incorporated issued bonds at face value at a yield to maturity of $6.6\%$. Now, with six years left until the maturity of the bonds, the company has run into hard times and the yield to maturity on the bonds has increased to 13%. What is now the price of the bond? (Assume semiannual coupon payments.)

Note: Do not round intermediate calculations. Round your answer to 2 decimal places.


BSuppose that investors believe that Castles can make good on the promised coupon payments but that the company will go bankrupt when the bond matures and the principal comes due. The expectation is that investors will receive only 86% of face value at maturity. If they buy the bond today, what yield to maturity do they expect to receive?

Note: Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.



a. Bond price

b. Yield to maturity		\%

Verified Answer :

  • Use the bond pricing formula \(P = C \times \frac{1-(1+r)^{-n}}{r} + \frac{F}{(1+r)^n}\) with \(C=33\), \(F=1000\), \(n=12\), and \(r=0.065\) to find the price: \(\boxed{\$738.92}\).
  • Determine the new yield by solving for \(r_b\) in the equation \(P_a = 33 \times \frac{1-(1+r_b)^{-12}}{r_b} + \frac{860}{(1+r_b)^{12}}\), yielding an annual YTM of \(\boxed{10.94\%}\).

Explanation

  1. Analysis of Bond Conditions

The goal is to calculate the price of a bond and its expected yield to maturity under certain conditions. We assume a standard face value \(F = \$1000\). The bond has an annual coupon rate of \(6.6\%\), resulting in semiannual coupon payments of \(C = \frac{0.066 \times \$1000}{2} = \$33\). There are \(6\) years remaining to maturity, which corresponds to \(n = 12\) semiannual periods. For part (a), the current yield to maturity (YTM) is \(13\%\), so the periodic yield is \(r_a = \frac{0.13}{2} = 0.065\). For part (b), we consider the case where the investor receives only \(86\%\) of the face value at maturity, i.e., \(F_m = \$860\), while the price remains the same as in part (a).

  1. Calculating the Current Bond Price

To find the price of the bond under the current yield, we use the present value formula for bond pricing:
\[P = \sum_{t=1}^{n} \frac{C}{(1+r_a)^t} + \frac{F}{(1+r_a)^n}\]
Using the formula for the sum of a geometric series, this simplifies to:
\[P_a = C \times \left( \frac{1 - (1+r_a)^{-n}}{r_a} \right) + \frac{F}{(1+r_a)^n}\]
Substituting the values \(C = \$33\), \(n = 12\), \(r_a = 0.065\), and \(F = \$1000\):
\[P_a = \$33 \times \left( \frac{1 - (1.065)^{-12}}{0.065} \right) + \frac{\$1000}{(1.065)^{12}}\]
Calculating these terms, we find the price \(P_a \approx \$738.92078985\). Rounding to two decimal places, the current price is \(\$738.92\).

  1. Calculating the Expected Yield to Maturity

Now we find the yield to maturity \(r_b\) expected by investors who purchase the bond at price \(P_a \approx \$738.92078985\) knowing they will only receive \(F_m = 0.86 \times \$1000 = \$860\) at maturity. The equation is:
\[P_a = \sum_{t=1}^{12} \frac{33}{(1+r_b)^t} + \frac{860}{(1+r_b)^{12}}\]
Solving this for \(r_b\), we find the periodic rate \(r_b \approx 0.0546786\). The annual yield to maturity is \(YTM_b = 2 \times r_b \times 100 \approx 10.9357\%\). Rounding to two decimal places, the expected yield to maturity is \(10.94\%\).

Examples

Understanding bond pricing is crucial for financial analysts and investors evaluating corporate health. When a company faces financial distress, the market discounts the bond's value, reflecting the higher risk of default or reduced recovery at maturity. This exercise demonstrates how shifts in market expectations directly impact both current market prices and prospective yields, which is a fundamental concept in managing fixed-income portfolios and assessing credit risk.

Related Questions