To calculate how many bonds Caspian Sea Drinks needs to issue to raise $49 million, we first need to determine the price of each bond.
Step 1: Calculate the price of the bond.
The price of a bond can be calculated using the formula for the present value of the bond's future cash flows, which includes periodic coupon payments and the final redemption of the face value at maturity.
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Coupon Payment Calculation: The bond has a coupon rate of 5.20%, and since it pays semi-annually, the semi-annual coupon payment is calculated as follows: \[ \text{Coupon Payment} = \frac{0.052 \times 1000}{2} = 26.00 \text{ dollars} \] (Assuming the face value of the bond is $1,000)
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Number of Periods: The bond has a maturity of 18 years and pays interest semi-annually. Therefore, the number of periods is: \[ n = 18 \times 2 = 36 \text{ periods} \]
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Yield to Maturity (YTM): The yield to maturity is 4.85% annually, or: \[ r = \frac{0.0485}{2} = 0.02425 \text{ per period} \]
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Present Value of Coupon Payments: The present value of the coupon payments can be calculated using the formula for the present value of an annuity: \[ PV_{\text{coupons}} = C \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \] where \(C\) is the coupon payment per period. Plugging in the values: \[ PV_{\text{coupons}} = 26 \times \left( \frac{1 - (1 + 0.02425)^{-36}}{0.02425} \right) \]
Calculating the above: \[ PV_{\text{coupons}} \approx 26 \times 25.1326 \approx 654.46 \]
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Present Value of Face Value: The present value of the face value is given by: \[ PV_{\text{face value}} = \frac{F}{(1 + r)^n} = \frac{1000}{(1 + 0.02425)^{36}} \] Calculating this gives: \[ PV_{\text{face value}} \approx \frac{1000}{2.3995} \approx 416.68 \]
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Total Price of the Bond: Finally, the price of the bond is: \[ P = PV_{\text{coupons}} + PV_{\text{face value}} \approx 654.46 + 416.68 \approx 1071.14 \]
Step 2: Calculate the number of bonds needed to raise $49 million. The total amount needed is $49,000,000. Therefore, the number of bonds needed is: \[ \text{Number of Bonds} = \frac{49,000,000}{P} = \frac{49,000,000}{1071.14} \approx 45,703.97 \]
Since a company cannot issue a fraction of a bond in practice, Caspian Sea Drinks would need to round up to the nearest whole number: \[ \text{Number of Bonds} = 45,704 \]
Thus, Caspian Sea Drinks must issue 45,704 bonds to raise $49 million.
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