A convertible bond matures in 15 years,pays annual coupons,and has a coupon rate of 8 percent.The face value is $1,000 and the conversion ratio is 40.The stock currently sells for $22.80 a share.Similar nonconvertible bonds are priced to yield 9 percent.What is the minimal value of the convertible bond?

Question

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First, calculate the annual coupon payment: 0.08 x $1,000 = $80.
Then, calculate the conversion price: $1,000/40 = $25.
Next, we need to determine whether it is more valuable to hold the convertible bond or a similar nonconvertible bond. To do this, we need to calculate the yield to maturity of the convertible bond. We can use trial and error to approximate the YTM. For example, if we assume a YTM of 8%, the bond would be worth:
($80/1.08) + ($80/1.08^2) + ... + ($80/1.08^15) + $1,000/1.08^15 = $919.39
Since similar nonconvertible bonds are priced to yield 9%, the convertible bond would be more valuable if its YTM were less than 9%. If we repeat our calculation using a YTM of 7%, we get a value of $955.28, which is still less than the minimum value of a similar nonconvertible bond ($1,000). Therefore, the minimal value of the convertible bond is $919.39.

A Convertible Bond Matures in 15 Years,pays Annual Coupons,and Has