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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.)

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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).