Question

How long will you *really* live for after you retire?

Unfortunately, nobody knows the answer to this question.

One approach to answering this age-old question is to consider this: how much should you have in your retirement account upon retiring so that you can withdraw a fixed sum forever? This may seem odd, but it is indeed possible.

Suppose that you invest $PMT into your retirement account for 30 years at an average monthly APR of 12.5% (very possible with mutual funds, stocks, and the correct portfolio balance). Once you retire, you move your lump sum of money into a low risk account offering you an average yield of 2.5% APR compounded monthly.

How big should PMT be so that you can withdraw $2,000 from your retirement account (upon retiring) without the account ever depleting?

Answer #1

Let the amount accumulated after investing $PMT each month for 30 years be X

Amount withdrawn each month after retirement = P = $2000

Interest earned after retirement = r = 2.5% annual = 0.025/12 monthly

Hence, X = P/(1+r) + P/(1+r)^{2} + ..... = P/r =
2000/(0.025/12) = $960000

Value of $PMT invested each month for n = 360 months (30*12 months) = $960000

Rate of return on investment i = 12.5% = 0.125/12 monthly

=> 960000 = PMT(1+i)^{n-1} +....+
PMT(1+i)^{2} + PMT(1+i) + PMT = PMT[(1+i)^{n}
-1]/i

=> 960000 = PMT[(1+0.125/12)^{360} -1]/(0.125/12)

=> PMT = 960000*(0.125/12) / [(1+0.125/12)^{360} -1]
= 245.67

Hence, $245.67 should be deposited monthly for 30 years

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