Question

Assume the government issues a semi-annual bond that matures in 5 years with a face value of $1,000 a coupon yield of 10 percent. What would be the price if the yield to maturity (semi-annual compounding) on similar government bonds were 8%?

*Make sure to input all currency answers without any currency symbols or commas, and use two decimal places of precision.

Answer 1

To calculate the price of the bond, we will use the present value of the bond's future cash flows, which consist of the coupon payments and the face value at maturity.

1. **Determine the bond's parameters**:
- Face value (FV) = $1,000
- Coupon rate = 10%, so the annual coupon payment = 10% of $1,000 = $100
- Since the bond pays semi-annually, the semi-annual coupon payment (C) = $100 / 2 = $50
- Number of years to maturity = 5, thus the total number of periods (N) = 5 * 2 = 10
- Yield to maturity (YTM) for similar bonds = 8% annual, so semi-annual YTM (r) = 8% / 2 = 4% = 0.04

2. **Calculate the present value of the semi-annual coupon payments**:
\[
PV_{coupons} = C \times \left(1 - (1 + r)^{-N} \right) / r
\]
Substituting the values:
\[
PV_{coupons} = 50 \times \left(1 - (1 + 0.04)^{-10}\right) / 0.04
\]

First, calculate \((1 + 0.04)^{-10}\):
\[
(1 + 0.04)^{-10} = (1.04)^{-10} \approx 0.675564
\]

Now substitute this back:
\[
PV_{coupons} = 50 \times \left(1 - 0.675564\right) / 0.04
\]
\[
PV_{coupons} = 50 \times \left(0.324436\right) / 0.04
\]
\[
PV_{coupons} = 50 \times 8.1109 \approx 405.54
\]

3. **Calculate the present value of the face value**:
\[
PV_{FV} = \frac{FV}{(1 + r)^N}
\]
Substituting the values:
\[
PV_{FV} = \frac{1000}{(1 + 0.04)^{10}}
\]
Calculate \((1 + 0.04)^{10}\):
\[
(1.04)^{10} \approx 1.48024
\]

Now substitute this back:
\[
PV_{FV} = \frac{1000}{1.48024} \approx 675.56
\]

4. **Calculate the total price of the bond**:
\[
Price = PV_{coupons} + PV_{FV}
\]
\[
Price = 405.54 + 675.56 \approx 1081.10
\]

Therefore, the price of the bond is **1081.10**.