Question

Assume a 10% annual interest rate.

(i) What is the present value of a 25 year, $900 annuity if the first payment does not occur until 7 years from today?

(ii) What would be the present value of a perpetuity with the same characteristics ($900, first payment in 7 six years).

Answer 1

Interest rate = 10%

(i) Annuity, n=25 years

no of payment years for annuity = 25 years

Cash flow = $900

Since the first Cashflow occurs at 7 years from now. Hence we can calculate the value of this annuity at the 6th year and then we can calculate the value of the annuity now using discounting.

The formula to calculate the value of annuity one year before the first cashflow occurs is:

$PV = \frac{C}{r}[1-\frac{1}{(1+r)^{n}}]$

Therefore value of annuity at 6th year is:

P 900

PV₆ = 9000*0.907704 = 8169.3360

The formula to calculate the present value of a cashflow C_n i.e., value at t=0 is given by:

On PV=(1+r)n

Value of annuity now i.e., at t=0 = 8169.3360/(1+0.1)⁶ = 4611.377207

Answer -> 4611.377207

(ii) Perpetuity

The formula to calculate the present value of perpetuity is:

$PV =\frac{C}{r}$

PV = 900/0.1 = 9000. This is the value of perpetuity at 6 years from now i.e., t=6. We can calculate the value at t=0 using the discounting formula used above:

PV(t=0) = 9000/(1+0.1)⁶ = 5080.26537

Answer -> 5080.26537