We need to calculate the Yield to Maturity (YTM) of a bond given the following information: Face Value (FV) = $1000 Market Price (PV) = $960 Coupon Rate = 5% Maturity = 15 years We need to calculate the YTM in two scenarios: once annually and semiannually. Since the exact formula is iterative, we will use an approximation formula.

1. Problem Description

We need to calculate the Yield to Maturity (YTM) of a bond given the following information:

Face Value (FV) = $1000

Market Price (PV) = $960

Coupon Rate = 5%

Maturity = 15 years

We need to calculate the YTM in two scenarios: once annually and semiannually. Since the exact formula is iterative, we will use an approximation formula.

2. Solution Steps

First, let's calculate the annual coupon payment (C).

$C = Coupon Rate * Face Value = 0.05 * 1000 = 50$

Now, we can approximate the YTM for the annual case. The formula for approximate YTM is:

$YTM_{annual} = \frac{C + \frac{FV - PV}{n}}{\frac{FV + PV}{2}}$

Where:

C = Annual coupon payment

FV = Face Value

PV = Present Value (Market Price)

n = Number of years to maturity

Plugging in the values:

$YTM_{annual} = \frac{50 + \frac{1000 - 960}{15}}{\frac{1000 + 960}{2}}$

$YTM_{annual} = \frac{50 + \frac{40}{15}}{\frac{1960}{2}}$

$YTM_{annual} = \frac{50 + 2.67}{980}$

$YTM_{annual} = \frac{52.67}{980} \approx 0.0537$

$YTM_{annual} \approx 5.37 \%$

Now, let's calculate the YTM for the semiannual case. In this case, we adjust the inputs as follows:

Semiannual coupon payment: $C_{semiannual} = \frac{C}{2} = \frac{50}{2} = 25$

Number of periods: $n_{semiannual} = n * 2 = 15 * 2 = 30$

$YTM_{semiannual} = \frac{C_{semiannual} + \frac{FV - PV}{n_{semiannual}}}{\frac{FV + PV}{2}}$

$YTM_{semiannual} = \frac{25 + \frac{1000 - 960}{30}}{\frac{1000 + 960}{2}}$

$YTM_{semiannual} = \frac{25 + \frac{40}{30}}{\frac{1960}{2}}$

$YTM_{semiannual} = \frac{25 + 1.33}{980}$

$YTM_{semiannual} = \frac{26.33}{980} \approx 0.0269$

This result (0.0269) is the semiannual yield. To annualize it, we multiply by 2:

Annualized $YTM_{semiannual} = 2 * 0.0269 = 0.0538$

Annualized $YTM_{semiannual} \approx 5.38 \%$

3. Final Answer

Annual YTM: approximately 5.37%

Semiannual YTM: approximately 5.38%