4. Math break: Safe Withdrawal Rate

This weekly blog is a little early, as I'm packing for a vacation in St John. The snow will have to wait until my return!

In my last blog, I talked about the formula that connects the expected rate of return, inflation, time period and SWR - the Safe Withdrawal Rate. To recap, the SWR reflects the annual amount you can withdraw from your account (adjusted to inflation) without running out of funds.

In this blog I'll show you how I derived the formula.

Let's recall the variables in the picture:

• N is the time period, in years
• G is the expected annual growth factor (for example, if you expect 8% average growth, G = 1.08)
• F is the inflation factor (for example, a 3% inflation corresponds to F = 1.03)
• R is the net growth, R = G/F

Our goal is to show that SWR is (1 - 1/R) / (1 - 1/R^N).

To prove the formula, the first step is to realize that by using the real rate of return R we can forget about inflation and assume you withdraw the same amount of dollars every year - call it W. Our goal is to start with S dollars, withdraw W every year and end up after N years with exactly 0 dollars. Since we use inflation-adjusted terms, we need to assume that every year, the balance of the portfolio grows by a factor R. 

Each withdrawal has a different amount of time to grow before withdrawn. The first one had no time at all: it cost us exactly W. The second one could grow for a year before it was taken from the account, meaning it actually cost us only W/R out of the original sum we had. The third one grew for two years, costing us W/R², etc. The last withdrawal grew for N-1 years, costing us a meager W/R^N-1.

Denoting P=1/R we see that the total cost of the withdrawals was

S = W + W*P + W*P² + ... + W*P^N-1

Dividing by W  we get

S/W = 1 + P + P² + ... + P^N-1 = (1 - P^N) / (1 - P)

(using geometric summation formula for the last equality.)

But S must also be equal to the original sum with which we started, since we assume that after we finished all the withdrawals we were left with $0. Hence, the Safe Withdrawal ratio SWR is W/S. From here we get:

SWR = W/S = (1 - P) / (1 - P^N) = (1 - 1/R) / (1 - 1/R^N)

QED

PS  bonus exercise: show that when the growth and the inflation converge to the same value (i.e., R tends to 1), SWR converges to 1/N. Explain the result.