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<<WARNING!!! Long post and nasty algebra to follow!>>

I have only joined The Motley Fool in the past few months, but I already feel Foolish. Primarily, I've learned that I need to start saving NOW for when I retire (I'm only 28). While I've been reading the amusing, enlightening, and enriching stories on TMF, a question has come to my mind: "How much do I need to retire?"

You see, I don't expect that I'll need money after I die, so I don't mind ending up with zero. If I live too long, however, I'll end up broke. Broke and old does not sound very fun. Also, I'm married, and my wife will probably live longer than me. Then there's that whole "leaving stuff to your kids" thing.

Wouldn't it be nice to balance my investment returns against my distributions so I'll never run out of money? That way, I know I'll never be broke. Then again, I kind of want to retire as early as possible to enjoy it as much as possible, so I want to know what a reasonable target balance might be.

Ideally, the amount I take out each year to pay my expenses should match the return I make on my investments. However, each year I'll have to take out more than the year before to cover inflation. To handle that, my returns have to exceed the amount I'll take out. Basically, I want my principle at the end of the year to equal the principle at the beginning of the year, times the rate of inflation (i.e. if inflation is 5%, P(end) = P(begin)*1.05).

If I take my yearly distribution out of my principle at the beginning of the year, and leave the result to grow at some rate (in percent), then the amount of earnings I'll have at the end of the year can be expressed thus:

Earnings = Rate * (Principle(begin) - Distribution) or
E = R * (P(begin) - D)

Now the amount of principle at the end of the year is the amount of principle at the beginning of the year, less the distribution, plus the earnings:

P(end) = P(begin) - D + E
...substitute in for E...
P(end) = (P(begin) - D) + R * (P(begin) - D) or
P(end) = (1 + R) * (P(begin) - D)

Now from our discussion of Inflation we know that P(end) = P(begin) * (1 + I). We substitiute into the above equation:

P(begin) * (1 + I) = (1 + R) * (P(begin) - D)

...from this point we'll just talk about the Principle at the beginning of the year...

P - D = P * (1 + I) / (1 + R)
P - P * (1 + I) / (1 + R) = D
P * [1 - (1 + I) / (1 + R)] = D
P * [(1 + R) - (1 + I)] / (1 + R) = D
P * [R - I] / (1 + R) = D
P = D * (1 + R) / (R - I)

If I wanted to retire this year, and I have $60,000 in household expenses, I expect inflation to average 5%, and I expect to earn 10% on average (near the historical average return for stocks), I can retire if I have $1,320,000 ($60,000 * (1 + .10) / (.10 - .05)). On the other hand, if I expect to earn 22.9% on average (Fool 4 CAGR from 1971-1996), I only need to have $411,955 ($60,000 * (1 + .229) / (.229 - .05)).

If you don't like those nasty fractions, use this formula: P = D * (100 + R) / (R - I), and you don't have to divide your percentages by 100 ($411,955 = $60,000 * (100 + 22.9) / (22.9 - 5)).

Alternatively, if I have $800,000 (5% interest, and 10% returns), my next-to-last formula tells me I can withdraw $36,363 each year ($800,000 * (.10 - .05) / (1 + .10) = $36,363 or $800,000 * (10 - 5) / (100 + 10) = $36,363, whichever).

Really, I should have been able to figure this out months ago, but my algebra is rusty.

PLEASE NOTE!!! One problem with this mechanism is that "average returns" in the stock market never occur; some years are better than average, some years are worse. See, for a discussion of the problems to which this fallacy can lead.

Robert uses the "Rule of Twenty" to avoid this problem, but his example uses the S&P 500 (12% CAGR since 1961) and 4% interest. The formula above would have predicted needing a little below $600,000 for retirement. Obviously, you'll want to give yourself a "cushion" over and above this minimum to handle bad timing.
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