Questions: Your company recently issued 10-year bonds at a market price of 1,000 (that is, they were issued at face value). These bonds pay an annual coupon of 95 (47.50 each six months). Due to additional financing needs, the firm wishes to issue new bonds which would have a maturity of 10 years and which would pay 40 in interest every 6 months. Assume that both bonds are of the same risk class (that is, they have the same yield or required rate of return) and that there are no flotation costs. How many bonds, to the nearest whole number, should the firm issue to raise 2,000,000 in cash? - 2,285 - 2,211 - 2,139 - 2,437 - 2,360

Your company recently issued 10-year bonds at a market price of 1,000 (that is, they were issued at face value). These bonds pay an annual coupon of 95 (47.50 each six months). Due to additional financing needs, the firm wishes to issue new bonds which would have a maturity of 10 years and which would pay 40 in interest every 6 months. Assume that both bonds are of the same risk class (that is, they have the same yield or required rate of return) and that there are no flotation costs. How many bonds, to the nearest whole number, should the firm issue to raise 2,000,000 in cash?

- 2,285
- 2,211
- 2,139
- 2,437
- 2,360

Transcript text: Your company recently issued 10-year bonds at a market price of $\$ 1,000$ (that is, they were issued at face value). These bonds pay an annual coupon of $\$ 95$ ( $\$ 47.50$ each six months). Due to additional financing needs, the firm wishes to issue new bonds which would have a maturity of 10 years and which would pay $\$ 40$ in interest every 6 months. Assume that both bonds are of the same risk class (that is, they have the same yield or required rate of return) and that there are no flotation costs. How many bonds, to the nearest whole number, should the firm issue to raise $\$ 2,000,000$ in cash? 2,285 2,211 2,139 2,437 2,360

Solution

Solution Steps

Solution Approach

To determine how many new bonds the firm should issue to raise $2,000,000, we need to calculate the present value of the new bonds' cash flows and equate it to the amount needed. Since the bonds are of the same risk class, they have the same yield. We can use the yield from the existing bonds to find the present value of the new bonds. The yield can be calculated using the existing bond's coupon payment and market price. Once we have the yield, we can calculate the present value of the new bonds' cash flows and determine how many bonds are needed to reach $2,000,000.

Step 1: Determine the Yield of Existing Bonds

The existing bonds have a face value of $ \$1000 $ and pay an annual coupon of $ \$95 $. To find the yield ($ y $), we solve the equation for the present value of the bond:

\[ \frac{95}{2} \left( \frac{1 - (1 + \frac{y}{2})^{-20}}{\frac{y}{2}} \right) + \frac{1000}{(1 + \frac{y}{2})^{20}} = 1000 \]

Solving this equation, we find:

\[ y \approx 0.095 \]

Step 2: Calculate Present Value of New Bonds

The new bonds have a semi-annual coupon payment of $ \$40 $ and the same yield as the existing bonds. The present value ($ PV $) of the new bonds is calculated using:

\[ PV = \frac{40}{2} \left( \frac{1 - (1 + \frac{0.095}{2})^{-20}}{\frac{0.095}{2}} \right) + \frac{1000}{(1 + \frac{0.095}{2})^{20}} \]

This results in:

\[ PV \approx 649.9066 \]

Step 3: Determine Number of Bonds Needed

To raise $ \$2,000,000 $, we need to determine how many new bonds are required. This is calculated by dividing the target amount by the present value of one bond:

\[ \text{Number of Bonds} = \frac{2000000}{649.9066} \approx 3077.365 \]

Rounding to the nearest whole number, we find:

\[ \text{Number of Bonds} = 3077 \]