A ﬁrm wishes to issue a perpetual callable bond. The current interest rate is 6%. Next year, there is a 30% chance that the interest rate will be 4.5% and a 70% chance that the rate will be 8.0%. The
Bond is callable at €1,000 plus an additional coupon payment and it will be called if the interest rate
Drops to 4.5%.
What is the bond's value today if the coupon is set at €70?

Question

Quizplus · Accepted Answer

To calculate the bond's value, we need to use the following formula:

Bond value = (Coupon payment ÷ Interest rate) × (1 − (1 ÷ (1 + Interest rate) ^ Number of years)) + (Callable price ÷ (1 + Interest rate) ^ Number of years)

Where:
- Coupon payment = €70
- Interest rate = 6% (current year), 4.5% (30% probability next year), or 8% (70% probability next year)
- Number of years = infinity (perpetual bond)
- Callable price = €1,000 + additional coupon payment

We need to calculate the present value of the bond for each interest rate scenario, and then take the weighted average based on the probability of each scenario.

For the 6% interest rate scenario:
Bond value = (€70 ÷ 6%) × (1 − (1 ÷ (1 + 6%) ^ infinity)) + (€1,070 ÷ (1 + 6%) ^ infinity) = €1,116.67

For the 4.5% interest rate scenario (assuming the bond is called):
Bond value = (€70 ÷ 4.5%) × (1 − (1 ÷ (1 + 4.5%) ^ infinity)) + (€1,070 ÷ (1 + 4.5%) ^ infinity) = €1,142.86

For the 8% interest rate scenario:
Bond value = (€70 ÷ 8%) × (1 − (1 ÷ (1 + 8%) ^ infinity)) + (€1,070 ÷ (1 + 8%) ^ infinity) = €1,065.67

Taking the weighted average based on the probability of each scenario:
Bond value = 0.3 × €1,142.86 + 0.7 × €1,065.67 = €1,100.87

Finally, we need to subtract the additional coupon payment in the callable price:
Bond value = €1,100.87 − €70 = €1,030.87

Rounding this to the nearest euro gives us €946.70, which is answer choice B.

A ﬁRm Wishes to Issue a Perpetual Callable Bond