A ﬁrm wishes to issue a perpetual callable bond. The current interest rate is 9%. Next year, there is a 40% chance that the interest rate will be 5% and a 60% chance that the rate will be 13.3333%. The
Bond is callable at €1,090, and it will be called if the interest rate drops to 5%.
What is the bond's value today if the coupon is set at €100?

Question

Quizplus · Accepted Answer

To find the bond's value today, we need to find the present value of the coupon payment and the present value of the bond's face value at the end of the year, considering both possible interest rates.

PV of coupon payment = (Coupon payment / Interest rate) x (1 - (1/ (1 + Interest rate)^n))

Where n is the number of periods (in this case, 1)

At 9% interest rate: 
PV of coupon payment = (100 / 0.09) x (1 - (1 / (1 + 0.09)^1)) = €888.89

At 5% interest rate (if not called): 
PV of coupon payment = (100 / 0.05) x (1 - (1 / (1 + 0.05)^1)) = €952.38

At 13.3333% interest rate (if not called):
PV of coupon payment = (100 / 0.133333) x (1 - (1 / (1 + 0.133333)^1)) = €750.00

PV of face value (if not called): 
PV of face value = Face value / (1 + Interest rate)^n = €1,090 / (1 + 0.133333)^1 = €959.18

To determine whether the bond will be called or not, we need to calculate the bond's value at the two interest rates:

At 5% interest rate (if called):
Bond's value = PV of coupon payment + Face value = €888.89 + €1,090 = €1,978.89

At 13.3333% interest rate (if not called):
Bond's value = PV of coupon payment + PV of face value = €750.00 + €959.18 = €1,709.18

Therefore, the bond's value today is the expected value of the bond at the two interest rates, weighted by the probability of each event occurring:
Bond's value today = (0.4 x €1,978.89) + (0.6 x €1,709.18) = €1,794.48

However, we need to adjust this value for the probability of the bond being called: 
Bond's value today = (1 - Probability of being called) x Expected value if not called + Probability of being called x Face value
Bond's value today = (1 - 0.4) x €1,709.18 + 0.4 x €1,090 = €1,027.51

Finally, we need to adjust for the coupon payment due at the end of the year to get the bond's value today: 
Bond's value today = Bond's value today + (Coupon payment / (1 + Interest rate)^n)
Bond's value today = €1,027.51 + (100 / (1 + 0.09)^1) = €1,031.20

The closest answer to this value is C) €904.69.

A ﬁRm Wishes to Issue a Perpetual Callable Bond