Question

Assume a par value of $1,000. Caspian Sea plans to issue a 20.00 year, annual pay bond that has a coupon rate of 16.00%. If the yield to maturity for the bond is 16.0%, what will the price of the bond be?

Answer 1

To calculate the price of a bond, we can use the present value formula for the cash flows associated with the bond. The present value of a bond is the sum of the present value of its future coupon payments and the present value of the par value repaid at maturity.

1. **Bond Details:**
- Par value (Face value, \( F \)) = $1,000
- Coupon rate = 16.00%
- Annual coupon payment (\( C \)) = Coupon rate × Par value = 0.16 × $1,000 = $160
- Years to maturity (\( N \)) = 20
- Yield to maturity (YTM, \( r \)) = 16.00% or 0.16

2. **Calculate the Present Value of Coupon Payments:**
The present value of the coupon payments is calculated using the formula for the present value of an annuity:
\[
PV_c = C \times \left( \frac{1 - (1 + r)^{-N}}{r} \right)
\]
Substituting the values:
\[
PV_c = 160 \times \left( \frac{1 - (1 + 0.16)^{-20}}{0.16} \right)
\]

3. **Calculate the Present Value of the Par Value:**
The present value of the par value is calculated using the formula for the present value of a lump sum:
\[
PV_F = \frac{F}{(1 + r)^N}
\]
Substituting the values:
\[
PV_F = \frac{1,000}{(1 + 0.16)^{20}}
\]

4. **Calculate the Total Present Value (Price of the Bond):**
The price of the bond \( P \) is the sum of the present values of the coupon payments and the par value:
\[
P = PV_c + PV_F
\]

Let's do the calculations step-by-step:

### Step 1: Calculate Present Value of Coupon Payments

\[
PV_c = 160 \times \left( \frac{1 - (1 + 0.16)^{-20}}{0.16} \right)
\]

Calculating \( (1 + 0.16)^{-20} \):
\[
(1 + 0.16)^{-20} \approx (1.16)^{-20} \approx 0.0807
\]
Therefore,
\[
PV_c = 160 \times \left( \frac{1 - 0.0807}{0.16} \right) \approx 160 \times \left( \frac{0.9193}{0.16} \right) \approx 160 \times 5.7463 \approx 919.04
\]

### Step 2: Calculate Present Value of Par Value

\[
PV_F = \frac{1,000}{(1 + 0.16)^{20}} \approx \frac{1,000}{(1.16)^{20}}
\]

Calculating \( (1 + 0.16)^{20} \):
\[
(1.16)^{20} \approx 6.883
\]
Therefore,
\[
PV_F = \frac{1,000}{6.883} \approx 145.66
\]

### Step 3: Calculate the Price of the Bond

Now add the two present values together to find the price of the bond:
\[
P = PV_c + PV_F \approx 919.04 + 145.66 \approx 1,064.70
\]

### Final Answer

The price of the bond is approximately **$1,064.70**.