Properties of power series

Question

Calculate the present value P of an annuity in which $9,000 is to be paid out annually perpetually, assuming an interest rate of 0.03. Round to the nearest dollar.

Answer 1

Properties of power series: Calculate the present value P of an annuity in which $9,000 is to be paid out annually perpetually, assuming an interest rate of 0.03. Round to the nearest dollar.

Detailed Solution:

Given/Known: P = ?? R = $9,000 i = 0.03 = 3% Perpetually: t = ∞ n = ∞

P = (R)(1 − (1 + i)⁻ⁿ)/(i)

= (9000)(1 − (1 + 0.03)^-∞)/(0.03)

= (9000)(1 − (1.03)^-∞)/(0.03)

= (9000)(1 − 0)/(0.03)

= (9000)(1)/(0.03)

= 9000/0.03

= $300,000

Present Value P = $300,000

Verify using TI-84 Plus TVM Solver Finance Apps:

* On TI-84 Plus: Select Key "Apps" ==> Finance ==> TVM Solver

* Fill the known: N = 1000 = ∞, I% = 3. R = PMT = −9000, P/Y = C/Y = 1

N = 1000 (For N, use any large number − as small as N = 762, gives P = $300,000 )

I% = 3

PV = 300000 (To Calculate Present Value: P, Press Key: ALPHA + ENTER (SOLVE))

PMT = −9000 (Annual Payments = -$9,000)

FV = 0 (Future Value: no change - leave it "0")

P/Y = 1

C/Y = 1

PMT: END (Ordinary Annuity)

Present Value P = $300,000