A convertible bond has an 8% annual coupon and 15 years to maturity. The face value is $1,000 and the conversion ratio is 40. The stock currently sells for $20.875 per share. Similar nonconvertible bonds are priced to yield 9%. The value of the convertible bond is at least:

Question

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The annual coupon payment is $80 ($1,000 x 8%). The bond will pay a total of 15 x $80 = $1,200 in coupon payments over its life.

To calculate the bond's value, we need to discount the bond's future cash flows back to the present. Since similar nonconvertible bonds are priced to yield 9%, we will use a discount rate of 9% to calculate the bond's value.

Using the formula for the present value of an annuity, we can calculate the present value of the bond's coupon payments:

PV of coupon payments = $80 x [(1 - (1 / (1 + 0.09)^15)) / 0.09] = $753.86

To calculate the present value of the bond's principal, we need to consider two scenarios:

Scenario 1: The bond is not converted into stock. In this case, the bondholder will receive the face value of the bond at maturity, which is $1,000.

Scenario 2: The bond is converted into stock. In this case, the bondholder will receive 40 shares of stock (since the conversion ratio is 40) at a price of $20.875 per share. Therefore, the conversion value of the bond is 40 x $20.875 = $835.

To calculate the bond's value, we need to determine which scenario will result in a higher value. We can do this by calculating the present value of each scenario:

PV of scenario 1 = $1,000 / (1 + 0.09)^15 = $279.39

PV of scenario 2 = $835 / (1 + 0.09)^15 = $919.39

Since the PV of scenario 2 is higher, the bond's value is $919.39.

A Convertible Bond Has an 8% Annual Coupon and 15