   SEARCH HOME Math Central Quandaries & Queries  Question from Polly, a student: I expect to retire in 12 yearsI expect to retire in 12 yearsQuandaries & Queries. Beginning one month after my retirement, I would like to receive $500 a month for 20 years. How much must I deposit into a fund today to be able to do so if the rate of interest on the deposits is 12% compounded monthly? Hi Polly. Here is a two step method for answering the question: Step 1: Find the value of the annuity at the time it commences payments to you. That is, how much will it have to be worth in 12 years in order to be depleted in 32 years at a rate of$500 per month with the balance compounded monthly at 12% (annualized)?

Clearly the answer to this is not just 12 x 20 x 500, because that ignores the compounding balance. The last $500 payment has 240 months of interest. Since each month earns 1% interest (that's 12% divided by twelve payments per year), we have$500 = A(1 + 0.01)^240, where A is the "original" amount 20 years earlier which grew into this $500 payment. A (1.01)240 = 500 A = 500 (1.01)-240 (my calculator works this out to about$45.90, but we'll use the exact form above instead).

The previous payment (this is the second last month) has one less month of interest, so we'd have a -239 instead of -240 there.

And so on for each payment. The first payment is after one month of interest, so it is just 500 (1.01)-1.

If we add these all together from the first to the last, we get the total principal (I'll call it P) 20 years earlier (when we stopped paying for the annuity and it started paying back):

P = 500 (1.01)-1 + 500(1.01)-2 + ... + 500(1.01)^-240

Now I can factor out the 500 itself:
P = 500 [ (1.01)-1 + (1.01)-2 + ... + (1.01)-240 ]

Now I will make that series of exponents go from 1 up to some positive value:
P = [ 500 (1.01)-240 ] [ (1.01)-239 + (1.01)-238 + ... + (1.01)0 ]
P = [ 500 (1.01)-240 ] [ 1 + 1(1.01) + 1(1.01)2 + 1(1.01)3 + ... + 1(1.01)239]

Notice that this is a geometric progression where the scale factor is the [500 (1.01)^-240], the common ratio is 1.01 and runs from exponent 0 to 239. The well-known formula for the sum of a geometric progression is

S = a[rn+1 - 1] / (r - 1)

where S is the sum, a is the scale factor, r is the ratio and n is the highest exponent. So for us, this becomes:

S = [500 (1.01)-240 ] [ (1.01)-240 - 1 ] / 0.01

which simplifies:

S = 50000 [ 1 - (1.01)-240 ]

Using my calculator, that works out to S = $45,409.71. Compare that to 12 x 20 x 500 =$120,000 for payments from a non-interest-bearing (perhaps under a mattress? :) account. Clearly, you need to save much less in the next 12 years to get to $45,409.71 than to$120,000!

By the way, there is of course a formula for all this too - but I didn't want to spoil your exploration of the mathematics. It is this:

P = R [ ( 1 - 1/ [1 + in] ) / i ]

where P is the "Present" value (at the start of the payment process; in our case this is what we want to find), i is the interest per payment period (1% in our case), and n is the number of payments (240 in our case). And R is the payment size ($500). That's the same as what we worked out earlier, by the way. Now that we know the amount of money we need to start our 20 year payment period, we are ready for step two. Step 2: How big should we make a single lump sum investment today in order for it to be worth$45,409.71 in 12 years based on 12% interest compounded monthly?

Let L be the lump sum we are trying to find. Each month, we are earning 1% interest.

Initially we have L.
After one month we have L(1 + 0.01).
After two months we have L(1 + 0.01)(1 + 0.01)
After n months we have L (1.01)^n.

So after 144 months, we have
$45,409.71 = L (1.01)144. And we solve for L as follows: L = 45409.71 (1.01)-144 My calculator tells me this makes L =$10,836.05

So this means that if you deposit $10,836.05 today in an account that earns 12% interest paid monthly, it will be worth$45,409.71 in 12 years. Then if you start withdrawing \$500 per month from that same account, the account will be emptied 20 years later.

These days, it is very hard to find funds that pay 12% interest guaranteed over 32 years, but I hope you can find one.

Cheers,
Stephen La Rocque.     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.