A bond with unlimited maturity (a perpetuity bond) has a face value of $1,000, and a coupon rate of 5%. If the market interest rate is 10%, what is the current market price of this bond?

Question

BlackTom AI · Accepted Answer

Consider the structure of a perpetuity bond: it pays a constant annual coupon forever and has a face value that is not repaid as a lump sum at maturity since there is no maturity. The annual coupon here is 5% of 1000, which equals 50 dollars per year.

Option $1000: This would be the price if the coupon rate equaled the market rate and the bond priced at par. Since the coupon rate (5%) is lower than the market rate (10%), investors require a discount, so the price must be below 1000, not equal to it.

Option $400: This is too low given the cash flow of 50 per year and a 10% discount rate. Using the perpetuity formula, price = annual coupon / discount rate = 50 / 0.10 = 500. A price of 400 would imply a 50 divided by 0.125, i.e., a 12.5% discount rate, which does not match the given 10% market rate.

Option $600: While closer than 400, this still misaligns with the fundamental perpetuity pricing rule. If the market rate is 10%, the present value of a perpetual 50-dollar stream is 50 / 0.10 = 500, not 600. The extra 100 would correspond to a lower required return than 10% or a higher coupon, neither of which applies here.

Option $500: This matches the standard formula for a perpetuity: price = annual coupon / market rate. With 50 dollars of annual coupon and a 10% market rate, 50 / 0.10 equals 500. This result also aligns with intuition: since the coupon rate is half the market rate, the price is about half the face value, i.e., 500 of 1000.

A bond with unlimited maturity (a perpetuity bond) has a face value of $1,000, and a coupon rate of 5%. If the market interest rate is 10%, what is the current market price of this bond?单项选择题

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