Question

# How long will you really live for after you retire? Unfortunately, nobody knows the answer to...

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?

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Amount withdrawn each month after retirement = P = \$2000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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