Question

You want to accumulate $1 million by your retirement date, which is 25 years from now. You will make 25 deposits in your bank, with the first payment occurring today. The bank pays 8% interest, compounded annually. You expect to receive annual raises of 3% which will offset inflation, and you will let the amount you deposit each year also grow by 3% (i.e., your second deposit will be 3% greater than your first, the third will be 3% greater than second, and so on).

How much must your first deposit be if you are to meet your goal of $1 million?

Answer 1

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Answer:

FV of growing annuity

we compute FV as the formula PMT*[{(1+r)^n - (1+g)^n}/(r-g)]*(1+i)
where

PMT =First annual payment
r = rate %
g = growth in payment
n = Time period
(1+i) at end of eqn = This is Annuity Due case = payemnt is made at start of period

So we have $1,000,000 = PMT*[{(1+0.08)^25 - (1+0.03)^25}/(0.08-0.03)]*(1+0.08)
is $1,000,000 = PMT*{4.7547/0.05}*1.08 = 102.70*PMT
So PMT = 1,000,000/102.70 = $9736.96