# Income Smoothing survey

In my last essay I talked about how you can think of your retirement portfolio as a pension and use the PMT formula to gauge how over- or under-funded that “pension” is.

Our personal pension can become mis-funded (I’m getting tired of writing “over- or under-funded” constantly) when we try to smooth our annual withdrawals.

Those large jumps in withdrawals are why we want to smooth things out. 1950 was not an especially noteworthy year to retire and we still see a half-dozen large changes in our real income. Out of that 30 year sequence there are 15 years where your income changes by greater than 5%. That includes years when it goes up by 26% and years when it goes down by 23%.

## Inflation-adjusted versus Nominal

When we actually make withdrawals from the bank to fund our living expenses we think in nominal terms. This is especially true when it comes to cuts. A cut in nominal withdrawals is a lot more painful than when inflation silently eats away.

I’ll use nominal withdrawals for most of the discussion but try to come back to inflation-adjusted numbers in a future blog.

The first thing we could try is just to average things out. Let’s try a 3-year rolling average.

This naive approach looks (to the naked eye) like an improvement. We’ve smoothed out some of the herky-jerky changes in withdrawals, where we would give ourselves a big raise one year and big cut soon after.

The red boxes show the areas where smoothing helped us avoid unnecessary temporary cuts.

There are other ways you could do income smoothing, though.

Ken Steiner’s suggestion is to start with last year’s withdrawal, inflation-adjust it, and then see if it fits within a 10% corridor around the raw PMT calculation. In practice it looks like this:

- Last year you withdrew $45,000.
- Inflation was 3%, so the inflation-adjusted amount is $46,350
- Raw PMT tells you to withdraw $55,000 this year.
- The lower bound is $49,500 (55000 × .9) and the upper bound is $60,500 (55000 × 1.1).
- Last year’s inflation-adjusted amount is below the lower bound. So use the lower-bound of $49,500 for this year’s withdrawal instead.

And on bogleheads longinvest suggests a 30% band with 10% adjustments for those who really must smooth their incomes.

Putting them side-by-side makes it easier to see the differences.

longinvest’s suggestion is the “smoothest”. Steiner’s technique of looking at inflation-adjustments to the previous years leads to an interesting “hump” around year 23.

1921 is an especially good showing for longinvest smoothing throughout the Great Depression!

One problem all of the above suffer from: arbitrary numbers. Why a rolling average over 3 years instead of 5? Why a 10% corridor for Steiner instead of a 15% corridor? Why a 30% band for longinvest instead of a 20% band? And how would you pick between the three?

The first thing we can do is come up with a metric for “smoothness”. That’s pretty easy: just use the standard deviation of the annual change.

If your withdrawals are:

- $40,000
- $41,000
- $47,000
- $52,000

Then the annual changes are:

- $1,000
- $6,000
- $5,000

And the standard deviation is 2645.

For the 1950 retiree we’ve been looking at so far the standard deviations over a 30-year period are:

`Raw PMT: 8,255`

Steiner: 4,953

3yr avg: 3,352

longinv: 2,005

But we don’t just want “low volatility”. After all, you can have the *lowest* volatility by never varying your withdrawals. What we really want is low volatility while keeping our Rainy Day Fund *(**read this** to see what this is all about)* balance close to $0.

That means we’re not building lots of unhedged risk but we’re also not being overly cautious.

At the end of 30 years the Rainy Day Fund for each of the three looks like:

`Steiner: $37,235`

longinv:$139,550

3yr avg: $ 1,459

The longinvest method had the lowest standard deviation but it comes at the cost of a large Rainy Day Fund.

By using a ratio of standard deviation divided by Rainy Day Fund we can compare them. A bigger number means a lower standard deviation or a lower Rainy Day Fund. Bigger is better.

`longinv: 3.57`

Steiner: 5.42

3yr avg: 204.39

At least for the year 1950, the 3-year rolling average is the strongest performer by a large amount. It has the second lowest standard deviation of withdrawals and the smallest Rainy Day Fund.

Instead of looking at just a single retirement year, we can run it for every retirement year from 1871 to 1985 (which allows our simulations to end 30 years later in 2015). Then we can take the median of all the results. This helps us make sure we’re not cherry picking scenarios that overly favourable to any one algorithm.

It looks like simply taking a 3-year rolling average works substantially better than either Steiner or longinvest’s algorithm.

## WTF Steiner?!?

You may have noticed that Steiner’s mean and standard deviation are eye-popping. A mean of 192 and a standard deviation of 1707. What’s going on there?

It is just a data artifact. In one backtesting year (1983) the Steiner algorithm got extraordinarily lucky. At the end of 30 years it’s Rainy Day Fund was only $7 and it has accomplished that with much lower volatility than raw PMT. That’s an artifact of the end point. If the simulation ends a few years earlier or later the anomaly largely goes away.

## Why 3 year rolling averages?

Simple 3-year rolling averages seem to perform well. But why 3 years? Why not 2 or 5 or 7?

I chose 3 somewhat arbitrary to start with. Let’s try with different timespans and see the impact.

The 2-year rolling average has the highest mean and median, though the standard deviation is a bit worrying. It is caused by another lucky end point. For a 1913 retiree, at the end of 30 years, the Rainy Day Fund has a minuscule $3.50 in it.

Let’s try removing these extreme outliers caused by endpoint timing and get a less distracting picture.

(Notice that the median has barely changed from above; that’s why median is a better metric to look at for this than mean or standard deviation.)

That’s better. It looks like we’ve narrowed the field to two clear choices: the 2-year rolling average or the 3-year rolling average. One has a better median, one has a better standard deviation. One gives you smoother income, one is more efficient (tracks PMT closer; builds up less risk; leaves less income on the table).

Here’s the difference between the two in animated form.

To my mind, the improved smoothing of the 3-year rolling average makes me like it better.