Question

Whipple Corporation just issued 310,000 bonds with a coupon rate of 6.20 percent paid semiannually that mature in 15 years. The bonds have a YTM of 6.64 percent and have a par value of $2,000. How much money was raised from the sale of the bonds?

Note: Round your intermediate calculations to two decimal places and final answer to the nearest whole dollar amount.

Answer 1

To calculate how much money was raised from the sale of the bonds, we need to determine the present value (PV) of the bonds given their characteristics.

The bonds pay semiannual coupon payments, which we can calculate as follows:

1. **Coupon Payment Calculation:**
- Coupon rate = 6.20%
- Par value = $2,000
- Coupon payment (semiannual) = (Coupon rate/2) x Par value = (6.20% / 2) * $2,000 = 0.031 * $2,000 = $124.

2. **Number of Periods Calculation:**
- Maturity = 15 years
- Since the coupons are paid semiannually, the total number of periods = 15 years * 2 = 30 periods.

3. **Yield to Maturity (YTM) Calculation:**
- YTM = 6.64%
- Since the bonds pay semiannual interest, we need to divide the YTM by 2: YTM (semiannual) = 6.64% / 2 = 0.0332 or 3.32%.

4. **Present Value Calculations:**
- We need to calculate the present value of the coupon payments and the present value of the par value (face value) paid at maturity.

- **Present Value of the Coupon Payments:**
PV_coupons = C × [1 - (1 + r)^(-n)] / r
where
C = coupon payment = $124,
r = YTM (semiannual) = 0.0332,
n = number of periods = 30.

\[
PV_{coupons} = 124 \times \left[1 - (1 + 0.0332)^{-30}\right] / 0.0332
\]

\[
PV_{coupons} = 124 \times \left[1 - (1.0332)^{-30}\right] / 0.0332
\]
Calculating \((1.0332)^{-30}\):
\[
(1.0332)^{-30} \approx 0.41113
\]

Thus,
\[
PV_{coupons} = 124 \times \left[1 - 0.41113\right] / 0.0332
\]
\[
PV_{coupons} = 124 \times \left[0.58887\right] / 0.0332
\]
\[
PV_{coupons} \approx 124 \times 17.7244 \approx 2191.8576
\]

So,
\[
PV_{coupons} \approx 2191.86
\]

- **Present Value of the Par Value:**
\[
PV_{par} = F / (1 + r)^n
\]
where
F = par value = $2,000.

\[
PV_{par} = 2000 / (1.0332)^{30}
\]

Calculating \((1.0332)^{30}\):
\[
(1.0332)^{30} \approx 2.61017
\]

Thus,
\[
PV_{par} = 2000 / 2.61017 \approx 766.41
\]

5. **Total Present Value of One Bond:**
\[
PV_{total} = PV_{coupons} + PV_{par}
\]
\[
PV_{total} = 2191.86 + 766.41 \approx 2958.27
\]

6. **Total Amount Raised from Sale of Bonds:**
- Number of bonds = 310,000.

\[
Total\ Amount\ Raised = PV_{total} \times \text{Number of Bonds}
\]
\[
Total\ Amount\ Raised \approx 2958.27 \times 310,000 \approx 916,569,700
\]

Therefore, the total amount of money raised from the sale of the bonds is approximately **$916,570,000** (rounded to the nearest whole dollar).