Questions: You own investment A and 10 bonds of bond B. The total value of your holdings is 14,633.4. Investment A is expected to make annual payments forever. The first payment for investment A is expected to be 227 and all subsequent payments are expected to increase by 2.19 percent per year forever. The expected return for investment A is 11.47 percent per year and the first payment will be made in 1 year. Bond B has a coupon rate of 19.30 percent, par value of 1000, YTM of X percent, 17 years until maturity, and semiannual coupons with the next coupon expected in 6 months. What is X ? Input instructions: Input your answer as the number that appears before the percentage sign. For example, enter 9.86 for %, 86 % (do not enter .0986 or 9.86 %). Round your answer to at least 2 decimal places.

You own investment A and 10 bonds of bond B. The total value of your holdings is 14,633.4. Investment A is expected to make annual payments forever. The first payment for investment A is expected to be 227 and all subsequent payments are expected to increase by 2.19 percent per year forever. The expected return for investment A is 11.47 percent per year and the first payment will be made in 1 year. Bond B has a coupon rate of 19.30 percent, par value of 1000, YTM of X percent, 17 years until maturity, and semiannual coupons with the next coupon expected in 6 months. What is X ? Input instructions: Input your answer as the number that appears before the percentage sign. For example, enter 9.86 for %, 86 % (do not enter .0986 or 9.86 %). Round your answer to at least 2 decimal places.
Transcript text: You own investment $A$ and 10 bonds of bond $B$. The total value of your holdings is $\$ 14,633.4$. Investment $A$ is expected to make annual payments forever. The first payment for investment $A$ is expected to be $\$ 227$ and all subsequent payments are expected to increase by 2.19 percent per year forever. The expected return for investment $A$ is 11.47 percent per year and the first payment vatil be made in 1 year. Bond $B$ has a coupon rate of 19.30 percent, par value of $\$ 1000$, YTM of $X$ percent, 17 years until maturity, and semiannual coupons with the next coupon expected in 6 months. What is $X$ ? Input instructions: Input your answer as the number that appears before the percentage sign. For example, enter $9.86 \mathrm{for} \%, 86 \%$ (do not enter .0986 or $9.86 \%$ ). Round your answer to at least 2 decimal places. $\square$ percent
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Solution

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Solution Steps

To solve this problem, we need to determine the yield to maturity (YTM) of bond B, denoted as \( X \). First, calculate the present value of investment A using the formula for a growing perpetuity. Then, subtract this value from the total holdings to find the value of the 10 bonds. Finally, use the bond pricing formula to solve for the YTM of bond B.

Step 1: Calculate the Present Value of Investment A

The present value of investment \( A \) can be calculated using the formula for a growing perpetuity:

\[ PV_A = \frac{C}{r - g} \]

where:

  • \( C = 227 \) (the first payment),
  • \( r = 0.1147 \) (the expected return),
  • \( g = 0.0219 \) (the growth rate).

Substituting the values, we find:

\[ PV_A = \frac{227}{0.1147 - 0.0219} \approx 2446.1207 \]

Step 2: Calculate the Value of Bonds B

The total value of holdings is given as \( 14633.4 \). The value of the 10 bonds \( B \) can be calculated as:

\[ \text{Value of Bonds B} = \text{Total Value} - PV_A \]

Substituting the values:

\[ \text{Value of Bonds B} = 14633.4 - 2446.1207 \approx 12187.2793 \]

Step 3: Calculate the Yield to Maturity (YTM) of Bond B

The yield to maturity \( X \) can be found by solving the bond pricing equation. The price of a bond is given by:

\[ P = \sum_{t=1}^{n} \frac{C}{(1 + \frac{X}{2})^t} + \frac{F}{(1 + \frac{X}{2})^n} \]

where:

  • \( C = 0.193 \times 1000 / 2 = 96.5 \) (semiannual coupon payment),
  • \( F = 1000 \) (par value),
  • \( n = 17 \times 2 = 34 \) (total periods).

Setting the bond price equal to the calculated value of bonds \( B \):

\[ 10 \times P = 12187.2793 \]

Solving for \( X \) yields:

\[ X \approx 0.15599996214117556 \implies X \approx 15.6\% \]

Final Answer

The yield to maturity \( X \) for bond \( B \) is

\[ \boxed{15.6} \]

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