A Glance at the SWR Equation

Research on Safe Withdrawal Rates

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JWR1945
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A Glance at the SWR Equation

Post by JWR1945 »

Refer to my earlier post Looking deeply into the SWR Equation from Wed Jul 16, 2003 at 10:07 am CDT.
http://nofeeboards.com/boards/viewtopic.php?t=1117

NOTE: In it I present gummy's formula for the Safe Withdrawal Rate. It has the form: [Balance at year (N) / Initial Balance] = Gain_Product_Term*(1 - w*gMS). The Gain_Product_Term is gain multiplier when there are no withdrawals. It is the product of the gain multipliers for individual years. The Gain_Product_Term = g1*g2*g3*... where g1 is the gain multiplier for the first year, g2 is the gain multiplier for the second year, g3 is the gain multiplier for the third year and so on. The gain multiplier for any year is the ratio of your portfolio balance to that of the previous year before making any withdrawals. If your portfolio gained 20%, the gain multiplier would be 1.20. If it lost 8%, your gain multiplier would be 0.92. Withdrawals are given as a fraction of the Initial Balance. For example, a withdrawal of 5% of the Initial Balance would correspond to w = 0.05. The term gMS is gummy's Magic Sum of gain multipliers. It is gMS = ( 1/[g1] + 1/[g1*g2] + 1/[g1*g2*g2] + ... ).

Here are some observations concerning the formula. They are not especially difficult. You have probably known them all along. This just brings them together.

1. Although gummy's Magic Sum formula gMS always increases (while making withdrawals), this does not mean that a portfolio must eventually go bankrupt. I offer this example to appeal to your common sense: if your portfolio always gains 5% and you always withdraw 2%, your portfolio never runs out of money. In fact, it always grows.
2. The effects of the first year are also seen in all subsequent years. The effects of the second year are also seen in all of the years that follow. And so forth.
3. Having a gain in the first year is the equivalent of reducing your withdrawal rate to [w/g1] in all of the years that follow. Having a loss in the first year is the equivalent of increasing your withdrawal rate to [w/g1] in all of the years that follow. In the first instance, the gain multiplier g1 is larger than one (i.e., you had a gain) so that [w/g1] is smaller than [w]. In the second case, the gain multiplier g1 is less than one (since you had a loss) so that [w/g1] is bigger than [w]. In this sense the gains or losses in subsequent years are decoupled when it comes to portfolio safety.
4. To a good first approximation, you can act as if each year is the start of a brand new retirement sequence. The formula shows this. But you must keep in mind that this is never completely true. The sequence of gain multipliers (or returns) is not quite independent. There is memory. gummy has determined that inflation introduces a memory effect in the short-term and he includes that effect in his Monte Carlo model. raddr has determined that mean reversion really does exist and that it results in a memory effect in the long-term. raddr has defined mean reversion precisely and he includes its effects in his Monte Carlo model.
5. One important element in preventing portfolio failure is to limit losses in any single year. Looking at gMS, if any year has an exceedingly high loss, then its gain multiplier is very close to zero. The value of [1/(the gain multiplier)] is exceedingly high. That can overcome the effects from all of the previous years. Having a cash component can eliminate that possibility. It places a floor on any year's gain multiplier term. Diversifying can decrease the likelihood of a heavy loss. Leverage can magnify the effect of losses and cause bankruptcy.
6. The Safe Withdrawal Rate formula can be used during accumulation. Deposits are the same mathematically as negative withdrawals. Simply replace the minus sign with a plus sign.
7. When the Safe Withdrawal Rate formula is used during accumulation, it is best to multiply everything out. In that form it becomes apparent that each year's deposit grows only according to all of the years that follow. Your return in the first year affects only that year's return. But your return in the tenth year affects the entire portfolio up until the tenth year.
8. This highlights the difference between making deposits and making withdrawals. During accumulation, the latest gain or loss dominates. During distribution, the earliest gain or loss dominates. The reason is that balances grow with deposits and decrease with withdrawals. A balance of zero is possible when making withdrawals. That is not possible when making deposits. (That is excluding leverage. With leverage it is possible to have negative gain multipliers.)

Of all of these observations, I think that decoupling is the most important. A good year early means good things in all of the years that follow. An especially bad year early on spells disaster.

Have fun.

John R.
JWR1945
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Joined: Tue Nov 26, 2002 3:59 am
Location: Crestview, Florida

Post by JWR1945 »

Quoting myself:
8. This highlights the difference between making deposits and making withdrawals. During accumulation, the latest gain or loss dominates. During distribution, the earliest gain or loss dominates.
This may explain the reason why so many people have thought that valuations don't matter.

When you are accumulating assets, the performance during your first few years does not matter nearly as much as later on. Your initial purchase prices affect only a small fraction of your eventual holdings. Your gains and losses in later years affect your entire portfolio.

For people who are accumulating assets, initial valuations don't matter all that much. Because they continually add assets, early mistakes are not too painful.

It is exactly the opposite during distribution. Your initial valuation affects everything that happens later on. It is difficult for people who have been thinking about building up a portfolio for decades to realize that retirement is different. You have had to figure that out for yourself. No one has been telling you otherwise. Until now.

Have fun.

John R.
peteyperson
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Post by peteyperson »

Hi John,

This highlights the need to be well diversified with plenty of cash/bonds in order to reduce the overall volatility and the cashflow problems over multiple years that an up and down market can create.

Petey
JWR1945 wrote:Quoting myself:
8. This highlights the difference between making deposits and making withdrawals. During accumulation, the latest gain or loss dominates. During distribution, the earliest gain or loss dominates.
This may explain the reason why so many people have thought that valuations don't matter.

When you are accumulating assets, the performance during your first few years does not matter nearly as much as later on. Your initial purchase prices affect only a small fraction of your eventual holdings. Your gains and losses in later years affect your entire portfolio.

For people who are accumulating assets, initial valuations don't matter all that much. Because they continually add assets, early mistakes are not too painful.

It is exactly the opposite during distribution. Your initial valuation affects everything that happens later on. It is difficult for people who have been thinking about building up a portfolio for decades to realize that retirement is different. You have had to figure that out for yourself. No one has been telling you otherwise. Until now.

Have fun.

John R.
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