# Flavors of PMT-based withdrawals, part 1 of 2: Mortality

All spreadsheets (and most financial calculators) have a “PMT” function. It is most commonly used to calculate the payments (“PMT”) for the loan on a mortgage.

`=PMT(5%, 30, 250000, 0)-\$16,262.86`

That’s telling us:

• 5% mortgage
• 30 year payment period
• \$250,000 starting mortgage value
• \$0 final mortgage value (the bank wants it all paid off!)
• You need to pay \$16,262.86 a year. Or, \$1,355 a month.

What does any of that have to do with retirement withdrawals?

A number of people have (as far as I can tell, more or less independently) realised that the same formula can be applied to retirement to create a variable withdrawal schedule.

• longinvest’s Variable Percentage Withdrawal (VPW) on bogleheads.org
• Ken Steiner’s “Actuarial Approach”
• Siegel & Waring’s “Annually Recalculated Virtual Annuity”
• Gordon Pye’s “Retrenchment Rule”

Those are just the ones I know about. There are probably more.

On the surface, the PMT rule is simple & straightforward. Especially compared to something like Guyton’s Decision Rules. You just need four inputs to the function:

1. Current portfolio value. This is trivial to check.
2. Future portfolio value. You should basically always use \$0 for this. Even if you want to leave a bequest, you’re probably better off handling that in another way.
3. The number of years the portfolio needs to last. Okay, now we’re starting to get into complicated territory.
4. The discount rate, or expected returns of the portfolio. Here be dragons.

How do the various implementations of PMT deal with the last two parameters?

VPW just assumes a final end point of 99 years of age. The creator says that you need revisit VPW every few years, especially when you’re getting close to 99, and readjust things. Still, it is basically picking a fixed age that is “really high”.

Steiner says “Until age 95 or life expectancy if longer” and is very explicit that you need to update this (and every other part of the calculation) every single year. He shows nicely the problem with naively using life expectancy from the start; you start at a higher level but need to continually cut spending. (Given Bernicke’s research on spending patterns as people age, I don’t think this is necessarily bad, though. But you’d need to see whether the decline is the same as the declining spending patterns.)

In practice, Steiner’s suggestion of “use 95 or your life expectancy, whichever is greater” means:

• For a man, from age 90 onwards instead of continuing to decrease (5, 4, 3, 2, 1) it stays constant at 3.
• For a woman, from age 91 onwards it stays constant at 4.

It seems like you could simplify and just use

`=max(95 - current_age, 4)`

Waring & Siegel initially say to use age 120 (!!!) since there are at least a few humans who have lived that long. Later on they provide an (optional) adjustment: use the average of 120 (i.e. maximum possible life span) and current remaining life expectancy.

How much of a difference do these different assumptions make?

The results aren’t surprising: the more conservative you are with “number of years”, the lower your pay outs. Steiner is most aggressive (95), followed by VPW (99), then Siegel & Waring’s “average” approach, and finally Siegel & Waring’s vanilla suggestion (120).

That’s a difference between \$61,953 a year and \$51,111 a year; so we’re not talking about trivial differences.

Also interesting to note is how Steiner’s approach struggles in the last few years. That’s a result of your life expectancy staying (mostly) constant at the end of your life. You basically always have “three of four more years left”. Which doesn’t work well with the PMT function.

Likewise, VPW “ends” at age 99. What that means in practice is that once you starting get up there in years you need to revisit the spreadsheet and put in a new final age. Increase it from 99 to something else. From what I can tell, there isn’t a whole lot of guidance on what you should do here, other than “increase it”. Just add 10 and reevaluate in 5 years? Keep it at a constant 10?

(Edit: 7 September 2016: I’ve seen a post by longinvest who suggests the following for VPW. Up to age 80 use a “final age” of 99. From ages 80 to 90, use a “final age” of 110. From age 90 and above, use a “final age” of 120. I haven’t gone back and redone any charts for that.)

Whatever you do, you’re going to have your withdrawals cut a fair amount. That’s not necessarily a bad thing. After all, you front-loaded your income to when you were presumably healthier and more active. Sure, you may have to cut your annual income from \$58,000 to \$35,000 once you hit your mid-90s…but will you still need \$58,000 to maintain your lifestyle?

Of the three, I like Waring & Siegel’s approach the best. It seems crazy to totally disregard the actual data about mortality.

Choosing a fixed date — 95, 99, or 120 — just feels wrong to me. If you’re married, your “joint mortality” is going to be different from a single person. If your spouse is several years younger, you will have a different joint mortality. If you were a smoker, you will have a different mortality.

The Society of Actuaries have a nice page showing how this looks in practice at http://www.longevityillustrator.org/. Look at the difference between a male who smokes and an active female who doesn’t smoke.

How can you not try to take that kind of stark difference into account?

A 65-year old single male who smokes only has a 25% chance of living another 20 years. A 65-year old married male who doesn’t smoke; his wife is 3-years younger and also doesn’t smoke. Now they have a 25% chance of living 34 years.

That’s an extra 14 years you need to plan for.

Using life expectancy definitely has some problems; it is based on mean mortality. No one likes the idea of having a 50% chance of outliving their plan.

Why not using an 80th or 90th percentile life expectancy instead?

Using a 90th percentile life expectancy (from the Society of Actuaries 2012 table) is the purple line. It starts out nearly the same as VPW (a negligible extra \$100 a year). By your late 70s it has dropped a marginal amount below VPW (\$700 a year less). By the time you are 90 you are withdrawing \$50,095 versus VPW’s \$58,814.

Of course, once you hit 90 with VPW you are probably going to replan, which will lower your withdrawals as well.

Using an 80th percentile life expectancy (the red line) makes things worse.

What would things look like if you did some kind of VPW replanning in your 90s? What kind of replanning would you even do?

One possibility: once you hit age 95 change the “final age” from 99 to 105. This has the effect of creating a one-time cut in your withdrawals.

That’s obviously a large drop. The benefit is, you make a one-time adjustment to your lifestyle and then are safe until age 105. (Though, presumably, once you hit 100 or so, you’ll replan once again.)

Another possibility would be to gradually increase your life expectancy, instead of doing it in a lump, painful, sum. That might look like this

That results in you going from \$58,000 a year to \$27,000 a year in 5 years. That’s better than the single year jump but still is a bit precipitous. You can, of course, play around with difference “slopes”. Instead of waiting until 95, maybe replan at 90. Have remaining life expectancy change at faster or slower rates. Of course, at some point you’re just reinventing using mortality directly for over 1/3rd of your retirement span…and I begin to wonder why you wouldn’t just use it for the entire retirement span.

Also note that VPW and Steiner end up being more or less the same thing. Once you get into your 90s they both say “stop using the fixed number and start doing something else to calculate it”.

Of all the above, I like the shape of Waring & Siegel’s “average” approach best. Steiner & VPW’s late-life adjustments feel pretty abrupt and harsh. To make them less harsh, it seems like you would just turn it into one of the other approaches.

Siegel & Waring’s “average” starts out generating \$2,000 less than VPW. By age 75 it is still only about \$3,000 less annually. That feels like a small price to pay for what is, effectively longevity insurance for your portfolio.

The next tricky parameter is the discount rate (aka expected returns)….

[Continue on to part 2…]

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