Question

You have your choice of two investment accounts. Investment A is a 9-year annuity that features end-of-month $2,180 payments and has an interest rate of 8 percent compounded monthly. Investment B is an annually compounded lump-sum investment with an interest rate of 10 percent, also good for 9 years.

How much money would you need to invest in B today for it to be worth as much as Investment A 9 years from now? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

Answer 1

Computation of FV of Investment A:

Future value of annuity can be computed as:

FV_A = P [{(1+r) ⁿ -1}/r]

P = Periodic deposit = $ 2,180

r = Periodic rate = 8 % p.a. or 0.08/12 = 0.00666666667 p.m.

n = Number of periods = 9 years x 12 months = 108

FV _A = $ 2,180 x [{(1+0.00666666667) ¹⁰⁸ -1}/0.00666666667]

     = $ 2,180 x [{(1.00666666667) ¹⁰⁸ -1}/0.00666666667]

     = $ 2,180 x [(2.04953023652023-1)/0.00666666667]

     = $ 2,180 x (1.04953023652023/0.00666666667)

     = $ 2,180 x 157.42953539932

     = $ 343,196.387170518 or $ 343,196.39

Computation of PV of Investment B:

FV of lump-sum investment i.e. Investment B should be same as investment A which is $ 343,196.387170518

PV and FV of a single sum are related as:

PV = FV/(1+r) ⁿ

r = Periodic rate = 10 % p.a.

n = Number of periods = 9

PV = $ 343,196.387170518 / (1+ 0.1) ⁹

         = $ 343,196.387170518 / (1.1) ⁹

     = $ 343,196.387170518 / 2.357947691

      = $ 145,548.770433058 or $ 145,548.77

We need to invest $ 145,548.77 in investment B today for it to be worth as much as Investment A 9 years from now.