Question

Suppose the prices of one-year, two-year, and three-year zero coupon bonds each with a par value of $100 are $90,$80, and $70, respectively. Determine the arbitrage-free price of the annuity

Answer 1

Let us say that one each of the zero coupon bonds are bought today. Total investment = $90 + $80 + $70 = $240

This would payoff $100 at the end of year 1, $100 at the end of year 2, and $100 at the end of year 3. This is exactly like an annuity in which a lump sum is invested now, and a fixed payoff is received in the future.

Since we know the prices of the zero coupon bonds today, the total investment required to buy the bonds is the arbitrage-free price of the annuity.

The arbitrage-free price of the annuity is $240

This can be verified as below :

The annual compounded rate of return on the zero coupon bond investment is calculated using RATE function in Excel :

nper = 3 (3 year investment)

pmt = 100 (annual payment)

pv = -240 (initial investment)

RATE is calculated to be 12.04%

The present value of annuity of $100 for 3 years is calculated using PV function in Excel :

rate = 12.04%

nper = 3

pmt = 100

PV is calculated to be $240

The arbitrate-free price of the annuity = present value of