Find the lump sum deposited today that will yield the same t

Find the lump sum deposited today that will yield the same total amount as payments of $5,000 at the end of each year for 8 years, at an interest rate of 6% compounded annually.

Solution

Let the amount deposited today be $P. The formula for the maturity value F, in case of compound interest is P( 1 +r/100)ⁿ , where r is the rate of interest for the period and n is the number of periods so that, in the 1^st case, F = P( 1+ 0.06)⁸ = 1.593848075 P. In the 2^nd case, $5,000 are deposited at the end of each year for 8 years, at an interest rate of 6% compounded annually which is the same as $5,000 deposited at the beginning of each year for 7 years, at an interest rate of 6% compounded annually. The formula for an annuity is F= p [{(1 + r)ⁿ – 1} / r] where p is the periodic payment, r is the rate of interest per period in decimals and n is the number of terms. Here, p = $5000, r = 6/100 = 0.06 and n = 7. Thus, F = 5000[ (1.06)⁷ -1}/ 0.06 = 5000(1.503630259 – 1)/ 0.06 = 41969.19. Therefore, 1.593848075 P = 41969.19 so that P = 41969.19/1.593848075 = $ 26331.99. The answer is $26331.99