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

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

----------------------------

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

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