Question

Sammy Jackson just received a graduation gift from her “papa” for graduating college. She wants to buy bonds from Seaside Beach Corporation. Currently, a Seaside bond is available with a 10.00-year maturity, 5.84% APR coupon rate with semi-annual coupons, and $1,000.00 face value. What is the most that Sammy should pay for this bond if the current market rate is 4.08% APR?

Answer 1

To determine the maximum amount Sammy should pay for the bond, we need to calculate the present value of the bond's future cash flows. The bond pays semi-annual coupons, and we will use the current market interest rate to discount those cash flows.

### Given Information:
- Face Value (FV) = $1,000
- Coupon Rate = 5.84% per year
- Maturity = 10.00 years
- Current Market Rate = 4.08% per year
- Coupons are paid semi-annually.

### Step 1: Calculate the semi-annual coupon payment.

The annual coupon payment is calculated as follows:
\[
\text{Annual Coupon Payment} = \text{Face Value} \times \text{Coupon Rate}
\]
\[
\text{Annual Coupon Payment} = 1000 \times 0.0584 = 58.40
\]

Since the coupons are paid semi-annually, the semi-annual coupon payment is:
\[
\text{Semi-Annual Coupon Payment} = \frac{58.40}{2} = 29.20
\]

### Step 2: Determine the number of periods and the semi-annual market rate.

- Number of Years = 10
- Since the coupons are paid semi-annually, total periods (N) = \(10 \times 2 = 20\)
- Semi-annual Market Rate = \( \frac{4.08\%}{2} = 2.04\% = 0.0204\)

### Step 3: Calculate the present value of the coupon payments.

The present value of an annuity (the coupon payments) can be calculated using the formula:
\[
PV(\text{Coupons}) = C \times \left(1 - (1 + r)^{-n}\right) / r
\]
Where:
- \(C\) = Coupon payment per period = $29.20
- \(r\) = Semi-annual market rate = 0.0204
- \(n\) = Total number of periods = 20

Calculating \(PV(\text{Coupons})\):
\[
PV(\text{Coupons}) = 29.20 \times \left(1 - (1 + 0.0204)^{-20}\right) / 0.0204
\]
\[
PV(\text{Coupons}) = 29.20 \times \left(1 - (1.0204)^{-20}\right) / 0.0204
\]

Calculating \( (1.0204)^{-20} \):
\[
(1.0204)^{-20} \approx 0.6667
\]

Thus:
\[
1 - (1.0204)^{-20} \approx 1 - 0.6667 = 0.3333
\]
Now substitute back:
\[
PV(\text{Coupons}) = 29.20 \times \left(0.3333\right) / 0.0204
\]
\[
PV(\text{Coupons}) \approx 29.20 \times 16.333 = 476.24
\]

### Step 4: Calculate the present value of the face value.

The present value of the face value is calculated using the formula:
\[
PV(\text{Face Value}) = \frac{FV}{(1 + r)^n}
\]
Where:
- \(FV\) = Face Value = $1,000
- \(r\) = 0.0204
- \(n\) = 20

Calculating:
\[
PV(\text{Face Value}) = \frac{1000}{(1.0204)^{20}}
\]
Calculating \( (1.0204)^{20} \):
\[
(1.0204)^{20} \approx 1.4859
\]
So:
\[
PV(\text{Face Value}) = \frac{1000}{1.4859} \approx 673.47
\]

### Step 5: Total Present Value of the Bond

Now, add the present value of the coupon payments and the present value of the face value:
\[
PV(\text{Bond}) = PV(\text{Coupons}) + PV(\text{Face Value})
\]
\[
PV(\text{Bond}) = 476.24 + 673.47 \approx 1149.71
\]

### Conclusion

The maximum amount Sammy should pay for the bond is approximately **$1,149.71**.