This subject of projecting future spend-liability-NPVs came to mind for two reasons. First I remembered that I created an NPV model (not stochastic) back in 2003 for the education liability I keep on my household balance sheet. It was complex and with three kids in independent schools -- and one would presume decent colleges later, for a total scope of around 25 years under the yoke -- the total cost in either absolute or present value terms was eye opening. I thought I had a handle on it (and I did, it has been remarkably accurate and useful) but I was irritated that the NPV went up each year even as an old year fell off the front end. It finally dawned on me that I had huge lumps in my cash flow and that when they were distant they were discounted exponentially and had a small impact but as years fell off the front end, that mass of lumpy liability moved closer to me out of the discounting fog and started to dominate the math. That meant that the NPV rose each year to a critical point a couple years ago. It now thankfully declines each year and will continue to do so all the way to zero around 2026-2027 barring 5 year programs or graduate programs I agree to help with (knock on wood for no). The same effect would presumably obtain for lumpy and/or long retirements when estimating (projecting) a spend liability.
Second, I think I've mentioned that it sometimes bothers me that the vast majority of retirement finance lit seems like it ignores or blows by or even condescends to early retirement. Sometimes this is implicit or accidental. Sometimes it is explicit. Here is an example I recently ran into. In a piece that I otherwise greatly enjoyed and found fairly useful (Determining Discount Rates Required to Fund Defined Benefit Plans, Turner et. al 2015...I know...I need another hobby) they address early retirement thusly:
Although a major factor affecting uncertainty in liabilities is uncertainty in mortality, which spreads liabilities over many years, we separate the issues of uncertain duration of liabilities from uncertain magnitude of liabilities by presenting a simple two-period model – the present funding period and the future payment period D years from present. Early retirement also affects both the duration and magnitude of liabilities, but is not considered here. [emphasis added]It's kind of like the experience of online dating in Florida in one's 50s...one gets either silence or blunt rejection.
In considering the idea of projecting a stochastic present value analysis [I encourage you to see note1] of a spend liability into future time from a platform of today -- we'll do it from the perspective of a 50 year old -- I think we can show both of the two ideas I mentioned above in action: 1) NPVs of the liability estimate can rise (nominal) before they fall as mortality probabilities start to take over, and 2) early retirement can have a different point of view than traditional in ways you probably haven't thought about or that might be invisible to a 65 or 70 year old.
Here is the setup:
- We use a 50 year old with a mortality distribution generated using Gompertz math with a mode of 90 and a dispersion of 8.5. This vaguely comports with a SOA table for annuitants but not exactly. Close enough,
- The spend is 100k per year at age 50. At 50 the SPV model[2] simulates spending out into the future by chain-inflating it using random draws of US inflation history with no attempt at serial correlation. Spending has an additional variance (stdev) of .12 added on. There are no discontinuities like stepping down spending at a later age though I did that in past posts. The spend sim is run 20,000 times at each projection age,
- For the "platform age" of 50, but now projecting the estimate for a new start-age of 51 or 85, the 100k spend is projected to 51 or 85 at a fixed 3% inflation "budget" so that we have a new "start spend" at the given projection age,
- At projection age 51 or 85, the mortality distribution is recast using the same mode and dispersion but with the new "survived to" projection age as the start age. This means that while terminal longevity doesn't change all that much the mean sure does(goes up),
- At the projection age of 51 or 85 the SPV model is re run for that age 20k times. The mean of the SPV distribution is calculated. It is not discounted in any way at that point. I could probably re run this for, say, 80th or 95th percentile,
- This whole process is done once with a discount rate inside the SPV model of 3% and a second time with 7%. Those might be reasonable boundaries for me, not sure about you. The link above makes a good case for using discounts above bond/treasury/annuity and below market/portfolio rates. It wouldn't be unreasonable for 3-7% to be at least a decent range for this post's purposes.
With all that, we get a chart that shows what I might estimate for a spend liability next year or the year after that or the year after that...but with factors for both changing mortality probabilities and random variation in both spending and inflation. A point of view on discounting is needed here but we at least have some reasonable choices for a range. It looks like this:
It looks like using higher discount rates over longer time frames has the nominal SPV result rising. I didn't show it but it rises less than average inflation so that's good but it does rise in absolute dollars which might be counter to expectations if one hasn't thought about it. This phenomenon would only be visible or meaningful for an early retiree. At 65, depending on discount rate assumptions, it would be less visible or meaningful to the extent that one would have even cared about it in the first place. Also, I have to say that this is just a budgetary game. In real life I know that in my education model the projection did in fact rise for a few years but there was a lot of volatility from many things as I re-estimated each year in real-time: realized tuition differed from projected, discount rate estimates moved around, education growth rates both realized and expected moved around, the model changed (added books and travel for college), marginal and average tax rates changed, etc. The same type of moving around would be true for this type of retirement estimation: mean mortality due to survival, broad longevity expectations changing, personal health, realized and expected spending, inflation estimates, etc. Or as they put it in the link that I shared above:
In the short run, mortality is unpredictable for individuals, but mortality rates are highly predictable for large groups. In the long run, even the rates are uncertain and everything about the uncertainty is uncertain – its growth rate, its variance, its distribution, and its impact on liabilities.
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[1] see past posts on SPV here, here, and here. Note that I am not really doing SPV since I have not randomized or geometricized (if that's even a word) the discount rate in the denominator. The extra spend vol sorta kinda stands as a type of proxy as does the use of ranges of discount rates...for now...while I try to figure out how to do it better.
[2] my fake math for the spend sim model in a past post looked like this where n is random duration based on longevity math, x is random inflation, y is something for discontinuous spending steps, z is random spend variance and (1+r)^t is the discount but in the future should probably be more like [(1+R1)*...*(1+RN)] where R is a random variable that has a mean and standard deviation of ____ :
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