Question

i hope i can have explanation about how to find the ans.8

8. Valuing Callable Bonds Canton Industries has decided to borrow money by issuing perpetual bonds with a coupon rate of 7 percent payable annually. The one-year interest rate is 7 percent. Next year, there is a 35 percent probability that interest rates will increase to 9 percent, and there is a 65 percent probability that they will fall to 6 percent a. What will the market value of these boods be if they are noncallable? h. If the company decides instead to make the bonds callable in one year, what cou- pon will be demanded by the boodbolders for the bonds to sell at par? Assume that the bonds will be called if interest rates rise and that the call premium is. equal to the annual coupon. c. What will be the value of the call provision to the company?

Answer 1

Part (a)

Let's shift ourself at t = 1, i.e. 1 year ahead.

As a bond holder i will receive the coupon due at the end of year 1 + I will have the bond with certain value.

The coupon or interest I will receive at the end of year 1 = Coupon rate x Par value of a bond = 7% x 1,000 = $ 70

Value of the perpetual bond at the end of year 1 = Annual coupon at the end of year 2 / interest rate for year 2

Interest rate for year 2 = 9% with 35% probability and in that case value of the bond at the end of year 1 = 70 / 9% = 777.78

Interest rate for year 2 = 6% with 65% probability and in that case value of the bond at the end of year 1 = 70 / 6% = 1,166.67

Hence, expected value of bond at the end of year 1 = 777.78 x 35% + 1,166.67 x 65% = $ 1,030.56

Bond holder's cash flows at the end of yer 1 = 70 + $ 1,030.56 = $ 1,100.56

This is the value at t = 1

Hence, value today = Value at the end of t = 1 discounted by 1 year = 1,100.56 / (1 + 7%) = $ 1,028.56

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Part (b)

Let C be the desired annual coupon

If interest rate rises: the bonds will not be called and then the price at the end of year 1 will be = C + C / 9% = C + C / 0.09

If interest rate falls: the bonds will be called and then the price at the end of year 1 will be = C + Par value + Call premium = C + 1,000 + C (as the call premium is same as the coupon payment) = 1,000 + 2C

Expected cash flows at the end of year 1 = 35% x (C + C / 0.09) + 65% x (1,000 + 2C) = 5.54C + 650

Price today = Par value = Present value of expected cash flows = (5.54C + 650) / (1 + 7%) = 5.18C + 607.48

This should be equal to the par value

Hence, 5.18C + 607.48 = 1,000

Hence, C = (1,000 - 607.48) / 5.18 = 75.83

Hence the coupon demanded = $ 75.83 and the coupon rate demanded = coupon / par value = 75.83 / 1,000 = 7.583%

Part (c)

At t = 1

Value of the non callable bond with this coupon rate = 75.83 / 6% = 1,263.79

Value of the callable bond = Par value + Call value = 1,000 + 75.83 = 1,075.83

value of the call provision = value of non callable bond - value of the callable bond = 1,263.79 - 1,075.83 = 187.96

Probability of the bond getting called = probability of interest rate fall = 65%

Hence, value of the call provision today = 65% x 187.96 discounted by one period = 65% x 187.96 / (1 + 7%) = $ 114.18