# Blanchett’s “spending smile” and PMT rates

When we use the PMT calculation to guide our retirement planning, we need to pick a “rate” as one of the parameters. What that rate should be is somewhat subjective, with various philosophies on how it should be chosen. Should we pick the long-term historical market average? Should we use expected future returns? Should we use the TIPS real rate?

Regardless of what rate you pick, however, there’s is always a kind of implicit assumption that, in an ideal world, you’d get constant withdrawals over time. Here’s an example showing what I mean:

If the rate you choose for PMT is exactly equal to the actual return on your portfolio, then you will receive constant withdrawals over your retirement.

If you pick a rate for PMT that is *lower* than the actual return, then your withdrawals will increase over time.

And if you pick a rate for PMT that is *higher* than the actual return, then your withdrawals will decrease over time.

It might seem crazy to **want** decreasing withdrawals over time but the reality is that retiree spending goes down as we age. One paper discussing this is Blanchett’s “Exploring the Retirement Consumption Puzzle”. In that paper he discovers a “spending smile”. Our inflation-adjusted spending drops from ages 65–84 and then starts increasing again. In practice it looks something this:

It is clear that “constant spending” simply isn’t a reality in retirement. But that’s not exactly an actionable insight. Two questions occur to me:

- If we
*planned*on that constant spending, how much of a contingency fund does that leave us? One simplistic way to count this is to just measure the area above the curve in the chart above. - If we want to
*change*our PMT calculation to more accurately reflect this decreased spending as we age…how would we go about doing it?

# Contingency Funds

Let’s start with the easy one. How much contingency do we get for free thanks to this reduced spending as we age? Blanchett provides a formula that describes how spending evolves as we age.

It is dependent on both our current age (*Age*) and on what our expected expenses (*ExpTar*) were. Here’s how it would play out for someone who started retirement at age 65 with expected expenses of $58,164.

We can use Blanchett’s formula to see how much “contingency” builds up over time for various amounts of expected spending.

We can see that for someone who entered retirement at age 65 expecting to spend $40,000 a year would have a contingency fund of just $60,000 by age 85. But someone who expected to spend $100,000 would have a contingency fund of $262,000 by the same age.

In general, the richer you are, the more spending drops off. Which makes intuitive sense, as you probably had more luxuries that would fall away as you age.

If we take the same chart as above but graph it as a ratio of expected spending to actual contingency fund we have:

# Changing PMT to match The Smile

The next question is: what if you don’t *want* that contingency money just sitting around? What if you’d rather *spend* it early in retirement? The PMT calculation — and the related Time Value of Money calculations — offer a lot of flexibility to change the “shape” of our annual withdrawals.

Instead of using PMT we could use the related “present value of a growing annuity” calculation. We just need to arithmetically rearrange that to get “growing PMT”. Except in this case we’d actually want a *decreasing PMT*, which is easy enough. We just use a negative number for *g*. But spreadsheets and financial calculators don’t have built-in support for calculating “growing PMT”. We could obviously code it ourselves but is there an approximation that might be “good enough”?

We noticed above that if we pick a rate for our PMT calculation that is higher than the actual market returns we also get *decreasing payments*. The question then becomes:

*In order to match Blanchett’s “smile curve” of decreasing spending, how much higher than our baseline rate should we use?*

This is just a question of minimizing errors. We can calculate the error between Blanchett’s formula and actual PMT results using *root-mean-square deviation**. *An example of it looks like this:

1,042 is the root-mean-square error (“RMSE”) for this example.

So let’s calculate the RMSE for every pair of “market return” and “pmt rate” between 1% and 10% in 0.1% steps. That is, if we have market returns of 4% and use a PMT rate of 7%, what is the RMSE? Since we are using 0.1% steps it results in 8,464 datapoints. Then when we look at all of that data, how do we minimize the errors?

*delta* is the difference between the market return and our PMT rate. What emerges is a pretty clear picture: errors are minimized (relative to Blanchett’s smile formula) when we pick a PMT rate that is 1-1.5% higher than our expected returns. The lower the market the market return the smaller the delta to minimize errors. If you expect market returns of 2%, then a delta of 1% will minimize errors. But if you expect market returns of 8%, then a delta of 1.5% will minimize errors.

The results are also highly skewed around that optimal point: the left side of the curve is much steeper than the right side of the curve. For our purposes here, picking a rate that is slightly too high is better than picking one that is slightly too low. At the very extreme, picking a rate that is 9% higher than the market return has an RMSE of only about 50,000–75,000. But picking a rate that is 9% *lower* than the market return has an RMSE of almost 250,000…5-times worse.

What do I make of all this?

If you want to account for the reality of declining spending as we age, you should strongly consider bumping up the rate in your PMT calculations by 1%. If you use historical market averages, add 1% to them. If you use TIPS real rates, add 1% to them. If you use 1/CAPE, add 1% to that.

On the other hand, if you *don’t* do that, you will end up with what is likely to be a sizeable contingency fund later in life. How large depends on what you’re expected expenses going into retirement were. But it is pretty likely that this contingency fund will be large enough to cover some, or all, of extended long-term care. That, in turn, means that you don’t need to *explicitly* plan for it. If you *do* explicitly plan for it, you’re making your retirement plan unnecessarily conservative — which means you’re increasing the cost of your retirement, which means you’re working longer than you needed to.