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Jim Hunter decided to retire to Florida in 10 years. What amount should Jim invest today so that he will be able to withdraw $25,000 at the end of each year for 30 years after he retires. Assume he can invest money at 9% interest compounded annually. (Using the Tables)
The present value of an ordinary annuity is the sum of the present values of the future periodic payments at the point in time one period before the first payment.
What is the amount that must be paid (Present Value) for an annuity with a periodic payment of R dollars to be made at the end of each year for N years, at an interest rate of I% compounded annually? For this scenario, we must compute the present value of each future payment at compound interest, at a time one period before the initial payment, and adding all the present values of all the payments.
Remember that the compounded amount of $P is S = P(1 + i)^n. Here, P is the present value, S = the compounded amount (or in this case, the periodic payment). Considering a simple example, find the present value of of a series of $200 payments made at the end of each year for 3 years in a fund that pays 4% interest compounded annually.
With the first periodic payment being made at the end of the first year, its present value is $200/(1 + .04)^1 = $200/1.04.
With the second payment being made at the end of the second year, its present value is $200/(1.04)^2.
With the third payment being made at the end of the third year, its present value is $200/(1.04)^3.
The total present value of the three payments is therefore, $200/(1.04) + $200/(1.04)^2 + $200/(1.04)^3.
In general terms, we can write this as P = R/(1+i)^1 + R/(1+i)^2 _ R/(1+i)^3.......+R/(1+i)^(n-1) + R/(1+i)^n.
It is clear that the series of present valuse form a geometric progression with first term a = R/(1+i), the common factor r = 1/(1+i) and n = the number of terms, or payments. WIth the sum of such a progression being S = a(r^n - 1)/(r - 1), substitution leads us to
.................................P = R[1 - (1 + i)^(-n)]/i
where P = the Present Value, R = the periodic payment (or rent), n = the number of payment periods, and i = I/100n.
Example: What is the present value of an annuity that must pay out $12,000 per year for 20 years with an annual interest rate of 6%? Here, R = 12,000, n = 20, and i = .06 resulting in
P = 12000[1-(1.06)^-20]/.06 = $137,639
Therefore, the purchase of an annuity for $137,639 bearing an annual interest of 6%, will enable the $12,000 annual payment over a 20 year period, for a total payout of $240,000.
This should enable you to solve your problem.