Question

Caspian Sea Drinks needs to raise $42.00 million by issuing bonds. It plans to issue a 15.00 year semi-annual pay bond that has a coupon rate of 5.11%. The yield to maturity on the bond is expected to be 4.80%. How many bonds must Caspian Sea issue? (Note: Your answer may not be a whole number. In reality, a company would not issue part of a bond.)

Answer 1

To determine how many bonds Caspian Sea Drinks needs to issue, we first need to calculate the price of one bond. The price of a bond can be calculated using the present value of its future cash flows, which consist of periodic coupon payments and the face value of the bond repaid at maturity.

1. **Bond Parameters**:
- Face Value (FV): $1,000 (assuming standard face value since it's not specified)
- Coupon Rate = 5.11%
- Coupon Payment (C) = \( \frac{5.11\% \times 1000}{2} = 25.55 \) (since it's semi-annual)
- Yield to Maturity (YTM) = 4.80%
- Number of Years = 15 years
- Number of Periods (N) = \( 15 \times 2 = 30 \) (since it's semi-annual)
- Yield per Period = \( \frac{4.80\%}{2}= 2.40\% = 0.024 \)

2. **Price Calculation**:
The price of the bond can be calculated using the following formula:
\[
P = \left( \sum_{t=1}^{N} \frac{C}{(1 + r)^t} \right) + \frac{FV}{(1 + r)^N}
\]
where:
- \( P \) = price of the bond
- \( C \) = coupon payment
- \( r \) = yield per period
- \( N \) = total number of periods
- \( FV \) = face value of the bond

Plugging the values into the formula:

**Calculating Present Value of Coupon Payments**:
\[
PV_{\text{coupons}} = \sum_{t=1}^{30} \frac{25.55}{(1 + 0.024)^t}
\]
This is a geometric series and can be calculated as:
\[
PV_{\text{coupons}} = C \times \left( \frac{1 - (1 + r)^{-N}}{r} \right)
\]
\[
= 25.55 \times \left( \frac{1 - (1 + 0.024)^{-30}}{0.024} \right)
\]

**Calculating Present Value of Face Value**:
\[
PV_{\text{FV}} = \frac{1000}{(1 + 0.024)^{30}}
\]

**Calculation**:
\[
PV_{\text{coupons}} \approx 25.55 \times 23.034
\]
\[
PV_{\text{coupons}} \approx 588.15
\]

\[
PV_{\text{FV}} \approx \frac{1000}{(1.024)^{30}} \approx \frac{1000}{2.025}
\]
\[
PV_{\text{FV}} \approx 493.80
\]

Adding both values together:
\[
P \approx 588.15 + 493.80 = 1081.95
\]

3. **Number of Bonds Required**:
The total amount Caspian Sea Drinks wants to raise is $42.00 million.
\[
\text{Number of Bonds} = \frac{42,000,000}{\text{Price per Bond}}
\]
\[
\text{Number of Bonds} = \frac{42,000,000}{1081.95} \approx 38763.98
\]

4. **Final Answer**:
Since you cannot issue a fraction of a bond, Caspian Sea Drinks would issue **38,764 bonds** (rounding up to the nearest whole bond).