Question

ssume 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%?

Answer 1

To calculate the price of a bond, you can use the present value formula for the cash flows generated by the bond, which includes the coupon payments and the face value at maturity.

### Given Information:
- Face Value (FV) = $1,000
- Coupon Rate = 10%
- Yield to Maturity (YTM) = 8%
- Maturity = 5 years
- Coupons are paid semi-annually

### Step 1: Calculate the coupon payment
The coupon payment (C) can be calculated as follows:
\[
C = \frac{\text{Coupon Rate} \times \text{Face Value}}{n}
\]
where \( n \) is the number of payments per year (2 for semi-annual).

\[
C = \frac{0.10 \times 1000}{2} = \frac{100}{2} = 50
\]
So, the semi-annual coupon payment is $50.

### Step 2: Determine the total number of periods
Since the bond matures in 5 years and pays semi-annually:
\[
\text{Total Number of Periods} = n \times \text{Maturity} = 2 \times 5 = 10 \text{ periods}
\]

### Step 3: Calculate the semi-annual YTM
The semi-annual yield is:
\[
\text{YTM}_{\text{semi-annual}} = \frac{8\%}{2} = 4\% = 0.04
\]

### Step 4: Calculate the present value of the cash flows
The price of the bond (P) can be calculated by summing the present values of the coupon payments and the face value at maturity:

\[
P = \sum_{t=1}^{10} \frac{C}{(1 + r)^t} + \frac{FV}{(1 + r)^{10}}
\]
Where:
- \( r \) = semi-annual YTM = 0.04
- \( C \) = semi-annual coupon payment = $50
- \( FV \) = face value = $1,000

Using the formula:
\[
P = \sum_{t=1}^{10} \frac{50}{(1.04)^t} + \frac{1000}{(1.04)^{10}}
\]

### Step 5: Calculate the present value of coupon payments
Calculating the present value of the coupon payments (a standard annuity formula):

\[
PV_{\text{coupons}} = C \times \left( \frac{1 - (1 + r)^{-n}}{r} \right)
\]
\[
PV_{\text{coupons}} = 50 \times \left( \frac{1 - (1.04)^{-10}}{0.04} \right)
\]

Calculating part of it:
\[
1 - (1.04)^{-10} \approx 1 - 0.675564 \approx 0.324436
\]
\[
PV_{\text{coupons}} = 50 \times \left( \frac{0.324436}{0.04} \right) \approx 50 \times 8.1109 \approx 405.545
\]

### Step 6: Calculate the present value of the face value
\[
PV_{\text{face value}} = \frac{1000}{(1.04)^{10}} \approx \frac{1000}{1.48024} \approx 675.564
\]

### Step 7: Combine both present values to get the price
\[
P \approx PV_{\text{coupons}} + PV_{\text{face value}}
\]
\[
P \approx 405.545 + 675.564 \approx 1081.11
\]

Thus, **the price of the bond would be approximately \$1,081.11**.