First Inc. is facing a liquidity problem and needs to issue a perpetual bond that is callable only at the end of year 2. The call price is $1,000 (face value) plus an additional coupon payment. The current interest rate is 8% per year. At the end of each year, interest rate will either increase or decrease by 1% with equal probability. a. What is the bond's value today if the coupon rate is set at 7%? b. What is the value of the call provision?
To calculate the bond's value today and the value of the call provision, we can use the present value formula. The present value of a bond is the summation of the present value of its cash flows, which include the coupon payments and the face value.
a. To calculate the bond's value today with a coupon rate of 7%, we need to determine the present value of the bond's future cash flows. Each year, there is a 50% chance that the interest rate will either increase or decrease by 1%.
The coupon payment can be calculated by multiplying the coupon rate (7%) by the face value ($1,000). So, the coupon payment is $70.
To determine the present value of each cash flow, we need to discount the cash flows at the appropriate interest rate. Since the interest rate can change each year, we will discount each cash flow using the expected interest rate for that year. We calculate the expected interest rate as the average of the possible interest rates.
For example, in year 1, the expected interest rate is (8% + 9%) / 2 = 8.5%. The expected interest rates for year 2 and year 3 are 8.5% and 7.5% respectively.
The present value of the coupon payments and face value can be calculated as follows:
Year 1: PV(coupon payment) = $70 / (1 + 0.085)
Year 2: PV(coupon payment) = $70 / (1 + 0.085)^2
Year 3: PV(face value) = $1,000 / (1 + 0.075)^3
Next, we sum up the present values of the cash flows to calculate the bond's value today:
Bond's value today = PV(coupon payment) + PV(coupon payment) + PV(face value)
b. To calculate the value of the call provision, we need to determine the expected value of the call option at the end of year 2.
The call provision allows the issuer to call back the bond at the end of year 2 for the call price, which is $1,000 plus an additional coupon payment.
To calculate the expected value of the call option, we need to consider the probability of the bond being called. Since the interest rate can increase or decrease by 1% with equal probability, the bond will be called if the interest rate decreases to the point where the coupon rate (7%) becomes attractive for the issuer.
So, for each possible interest rate in year 2 (8% or 9%), we compare the coupon rate (7%) to determine if the bond will be called. If the interest rate decreases to 8% or below, the bond will be called.
The value of the call provision can be calculated as follows:
Value of call provision = Probability of bond being called × Expected call price
The probability of the bond being called can be determined as the average of the probabilities of interest rates being 8% or lower in year 2.
For example, for the interest rate of 8% with 50% probability, the bond will be called. For the interest rate of 9%, the bond will not be called.
The expected call price is the sum of the face value ($1,000) and the additional coupon payment ($70).
By calculating the value of the call provision using these formulas, we can determine its value.