Question

Today is your cousin’s 12th birthday. Her parents are preparing to save for her college tuition. They decide that they will save the equal amount of $10,000 into a savings account, starting today and continuing every birthday up to and including her 20th birthday. Assume the savings account pays 7% interest compounded annually. If instead they gave a lump-sum amount TODAY, how much money will your cousin’s parents need to deposit in the account to give her to give the same value gift?

Answer 1

Computation of Future Value of Annuity

Yearly installment = $ 10000

No.ofYears = 9

Rateof interest = 7% Compounded Annually

We know that Future Value of Annuity Due = C[ {( 1+i)^n-1}/i](1+i)

Here C = Cash flow per period

I = Rate of interest per period

n =No.of payments

Future Value of Annuity due = $ 10000[ { ( 1+0.07)^9 -1}/0.07]( 1+0.07)

= $ 10000[ { ( 1.07)^9-1}/0.07] ( 1.07)

= $ 10000[ { 1.838459-1}/0.07] ( 1.07)

= $ 10000[ { 0.838459/0.07} ] ( 1.07)

= $ 10000*11.97799*1.07

= $ 128164.49

Hence the Future Value of Annuity due is $128164.49

Computation of lumpsum amount deposited to be today

We know thaat Present value = Future Value / ( 1+i)^n

Here I = Rate of interest

n = No.of payments

Present value = $ 128164.49/ ( 1+0.07)^9

= $ 128164.49/( 1.07)^9

= $ 128164.49/1.838459

= $ 69713

Hence the lumpsum amount deosited to be today is $ 69173 to give her the same value of gift.