Question

Freiburg Faucets Incorporated issued 20.00-year bonds 4.00 years ago with a coupon rate of 9.25% APR. The bonds pay semi-annual coupons, have a face value of $1,000 each and were issued at par value. What is the price of the Freiburg bond’s today if investors want a 7.16% APR return for bonds of similar risk and maturity?

Answer 1

To determine the price of Freiburg Faucets Incorporated's bond today, we can use the present value of the bond's cash flows approach. The cash flows consist of the semi-annual coupon payments and the face value received at maturity.

Here are the steps to calculate the price:

1. **Gather the bond information**:
- Face value (FV) = $1,000
- Coupon rate = 9.25% per annum
- Coupons per year = 2 (since they are paid semi-annually)
- Coupon payment = (Coupon rate / Coupons per year) * Face value
- Years remaining = 20.00 - 4.00 = 16.00 years

2. **Calculate the annual coupon payment**:
\[
\text{Coupon payment} = \frac{9.25\%}{2} \times 1000 = 0.04625 \times 1000 = 46.25
\]

3. **Calculate the total number of coupon payments remaining**:
\[
\text{Total payments} = 16.00 \text{ years} \times 2 = 32 \text{ payments}
\]

4. **Calculate the required return per period**:
- The required return (APR) is 7.16%, so the semi-annual required return is:
\[
\text{Semi-annual return} = \frac{7.16\%}{2} = 0.0358
\]

5. **Determine the present value of the coupon payments**:
The present value of an annuity formula is used here:
\[
PV = C \times \left(1 - (1 + r)^{-n}\right) / r
\]
- Where \( C \) is the coupon payment, \( r \) is the semi-annual required return, and \( n \) is the total number of payments.
\[
PV_{\text{coupons}} = 46.25 \times \left(1 - (1 + 0.0358)^{-32}\right) / 0.0358
\]

Let's calculate it step-by-step:
\[
PV_{\text{coupons}} = 46.25 \times \left(1 - (1 + 0.0358)^{-32}\right) / 0.0358
\]

First calculate \( (1 + 0.0358)^{-32} \):
\[
(1 + 0.0358)^{-32} \approx 0.3688
\]
Therefore:
\[
PV_{\text{coupons}} = 46.25 \times \left(1 - 0.3688\right) / 0.0358 = 46.25 \times 0.6312 / 0.0358 \approx 46.25 \times 17.6357 \approx 816.52
\]

6. **Determine the present value of the face value**:
We also need the present value of the face value:
\[
PV_{\text{face value}} = \frac{FV}{(1+r)^n} = \frac{1000}{(1+0.0358)^{32}}
\]
Now calculate \( (1 + 0.0358)^{32} \):
\[
(1 + 0.0358)^{32} \approx 3.4112
\]
Thus:
\[
PV_{\text{face value}} = \frac{1000}{3.4112} \approx 292.68
\]

7. **Calculate the total price of the bond**:
Finally, combine the present values of the coupons and the face value:
\[
\text{Bond Price} = PV_{\text{coupons}} + PV_{\text{face value}} = 816.52 + 292.68 \approx 1109.20
\]

The price of the Freiburg bond today, given the required return of 7.16%, is approximately **$1,109.20**.