## An Illusion of Numbers

Research on Safe Withdrawal Rates

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JWR1945
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An Illusion of Numbers
I have often read that we should not be concerned about stock prices if we are investing for the long-term. That isn't really true. In some cases it is helpful. In others it is reckless. It is one thing to be overly concerned about getting the best possible prices. It is another thing to disregard prices entirely.

One year among many.

Suppose that you have an initial investment that grows by 10% each year. At the end of forty years, the balance is (1.10)^40 times initial balance.

That is, your balance increases by 10% in the first year and that is 1.10 times the initial balance. And it is increased by 10% in the second year and that is 1.10 times the first year's balance or (1.10)^2 times the initial balance. Each year ends with 1.10 times the previous year's balance. After 40 years, your balance is (1.10)^40 times your initial balance. Using my scientific calculator, I find that (1.10)^40 = 45.259. If you had started with \$10 000, you would end up with \$452 590. That's not bad.

Now let us insert one really bad year. Your investments lose 50%. If you have left your money in the account, you now have (0.50)*(1.10)^40 or 22.630 times your initial balance or \$226 300, assuming that you had started with \$10 000. That's a lousy result, but it is still a whole lot better than when you had started. If you had placed your money on the sidelines for that single year, you would still have your \$452 590.

If you have no ability at all to measure risk and reward, you would have to accept your lumps. Since you would have no ability at all to weigh the upside potential against the downside risk, you would just have to look at long-term numbers to see which is more likely. You would see that there are lots of years with 10% increases and very few with a 50% loss. You would choose to accept the risk, being that you had no ability to estimate it.

And this is how the numbers would back you up. Your final balance after 41 years would be (0.50)*(1.10)^40 = [0.50/1.10]*(1.10)^41. Your annualized return would be found by taking the 41st root of this. That is, you would calculate the 41st root of [0.50/1.10] and multiply it by 1.10 (which is the 41st root of (1.10)^41). That product is 0.98095*1.10 = 1.07904. You would report your annualized percentage return as 7.904% (which is 1.07904 - 1 times 100%).

That single year, a disaster that cut your final balance in half, doesn't seem so bad when expressed as an annualized return. Your investments returned 7.9% instead of 10.0%.

None of the numbers change if the bad year occurs early, late or somewhere in the middle. You still end up with \$226 300 instead of \$452 590.

Investing every year.

Suppose that you invest \$1000 at the end of each year for 40 years and that you get a return of 10% every year. Your final balance is \$442 593. (There is a formula for calculating this. It is a little bit more complicated than before. In essence, it is the same formula that is used with mortgages.)

In this case you had to invest \$40 000 total to get something similar to the previous return (based on \$10000 up front).

Now let us see what happens if you have that single bad year. If it happens in year 41, your balance is cut in half and you end up with \$221 296. But if it happened right at the beginning, it would have had much less of an effect.

That first \$1000, when invested at 10% for 40 years, grew to \$45 259. [The formula is \$1000*(1.10)^40 = \$45 259.] Your first \$1000 contributed that much to the \$442 593 final balance. If it were cut in half, it would have contributed \$22 630 to the final balance. It would have been \$22 630 [actually, \$22629.50] less than \$442 593. The balance would have been \$419 963.

[As a detail, I will point out that this is for 40 years, having a loss in the first year. I have compared this to having 40 good years followed by one bad year, which would be 41 years.]

If you have a loss right at the beginning, it does not hurt you very much. If you have a loss at the very end, it hurts you severely.

Withdrawing funds in retirement.

Now let us look at the distribution stage of investing. During retirement, you withdraw from your account every year instead of adding to your account. This is very similar to the second example. The difference is that a loss at the beginning is a true disaster. Its effects are minimal later on.

Suppose that there is an amount that you can withdraw safely throughout your entire retirement period. Suppose that you can estimate that amount reasonably well. If you lose 50% of your balance in the first year, you must reduce your withdrawals right away. Your initial calculations have been optimistic.

Now suppose that the loss occurs at the very end of your retirement period. It only affects what remains in the account. Your final withdrawal amount is cut in half. The total dollar loss is much less.

Similar, yet different.

These three conditions are similar in many respects. But they are quite different.

In the first case, there were no contributions and no withdrawals. In the second, there were only contributions. In the third, there were only withdrawals.

Looking at the second and third examples, we see that when the bad year happens is tremendously important. Its importance is greatest when the balance is greatest. Its importance is smallest when the portfolio balance is smallest.

Thus, we can say that having a bad year does not matter too much when you have no money (or when you have very little money).

We have seen that if you do not touch your money, a single bad year does not seem quite as bad when you report your total return as an annualized percentage. In the example, the annualized percentage fell from 10.0% to 7.9%.

From these examples we see an illusion of numbers.

This illusion hides the fact that your money was cut in half. When you have money (at the end of accumulation or early during distribution), those losses are important. They hurt. In the example, you must cut your retirement income in half.

Have fun.

John R.