A Helpful Theorem

To calculate withdrawal rates for various final balances, it is sufficient to know the withdrawal rates for final balances of zero and 100% of the initial balance. Rates for intermediate balances maintain the same proportions.

For example, the withdrawal rate for a final balance equal to 50% of the initial balance is one-half way between the Safe Withdrawal Rate (a final balance of zero) and the Constant Terminal Value Rate (a final balance equal to the initial balance).

This is true for all combinations of fixed withdrawal rates based on a percentage of the initial balance and on a percentage of the current balance.

Gummy’s Safe Withdrawal Rate Equation

Gummy (retired Professor Peter Ponzo) derived the Safe Withdrawal Rate equation.

Gummy's (Peter Ponzo's) web site
Gummy's Sensible Withdrawal Rates
Gummy's (Peter Ponzo's) Equation

Gummy’s Safe Withdrawal Rate equation can be written in this form:

current balance/initial balance = (overall ratio without deposits or withdrawals GN)*(1 – [withdrawal rate/WFAIL])

where WFAIL is the withdrawal rate that produces a balance of zero for the particular sequence being examined. In Gummy’s notation, 1/WFAIL = gMS.

In terms of Gain Multipliers GN = g1*g2*..*gN and g1 = (1+return in year 1) = balance at the end of year 1/balance at the beginning of year 1, g2 = (1+return in year 2) = balance at the end of year 2/balance at the beginning of year 2,..,gN = (1+return in year N) = balance at the end of year N/balance at the beginning of year N. Observe that the final balance/initial balance after one year is G1. After two years it is G2 since the balance at the end of year 1 equals the balance at the beginning of year two. And so on. After N years, GN = the balance at the end of year N/initial balance.

The Theorem

Define W0% as the withdrawal rate that produces a balance of exactly zero after N years. This is the Historical Surviving Withdrawal Rate for that particular sequence at year N. Here, w0% = WFAIL.

Define W100% as the withdrawal rate that makes the final balance equal to the original balance. In this case, 1 = GN*(1 – [W100%/WFAIL]) or 1/GN = 1 – [W100%/WFAIL].

Let Wx be the withdrawal rate that produces an intermediate balance x*(the initial balance). Then,

x = final balance/initial balance = GN*(1 – [Wx/WFAIL])

1 – [Wx/WFAIL] = x/GN = x*(1 – [W100%/WFAIL])

WFAIL – Wx = WFAIL*(1 – [Wx/WFAIL]) = WFAIL*x*(1 – [W100%/WFAIL]) = x*(WFAIL – W100%)

WFAIL – Wx = x*(WFAIL – W100%)

Wx = WFAIL - x*(WFAIL – W100%) = WFAIL + x*(W100%-WFAIL)

Generalizing the Theorem

Suppose that withdrawals are based on the portfolio’s current balance as opposed to its initial balance. So long as the percentage remains fixed, the theorem still holds. Since it is fixed, all that happens is that the annual gain multipliers are reduced by a common factor. Where Gummy’s formula uses gk, we now substitute hk = gk*(1-withdrawal fraction). The withdrawal rates WFAIL (which equals W0%) and W100% change. So does GN. Everything else has the same form. All of the steps in the proof remain the same.

Now apply the theorem to aggregates: collections of Historical Surviving Withdrawal Rates for a specified number of years N (typically, 30 years). Since each individual Historical Surviving Withdrawal Rate has the same proportionality factor x, all linear combinations share the same scale factor. Regression equations retain the same proportional scaling behavior. Standard deviations, which are important for determining Safe Withdrawal Rates, also scale by the common factor.

The theorem holds for negative balances and balances greater than the original balance. However, once a balance has become negative, it is meaningless. Do you really intend for a stock market gain of 20% to multiply a negative balance by 1.20? What meaning would you assign to such an outcome?

For increasing balances, the theorem makes sense. Negative withdrawal rates make sense as well. When a withdrawal is negative, it means that you make a deposit. Gummy’s formula applies during accumulation. It is known as dollar cost averaging.

Application

I have incorporated these findings into Super Variable Terminal Value Rate SVTVR calculator L. It is a simplified version of SVTVR K.

Have fun.

John Walter Russell
August 19, 2006