I was playing around with a new formula [ mW - 1 ] today for several reasons: 1) it is the net-wealth-process coefficient of the second term of the Kolmogorov PDE profiled earlier on the blog; I thought I should get to know it better, 2) using it in a simple version makes it relatively easy to do a spreadsheet sim of a wealth depletion process without getting into a lot of programming, and 3) it gave me an easy way to see some stuff on sequence of returns risk I had been meaning to check out. m here is a real return/growth rate most likely net of fees and taxes, W is a "wealth units" thing (a 4% spend rate on $1 means 1/.04 = 25 wealth units), and the "1" -- I finally got through my thick skull -- is 1 wealth unit spent (really, it took me a month to figure that out?). I'm assuming I can use this the way I will below so I will. The way I'll use it is in this form:
 |
| equation 1 |
Sequence risk - static
With the equation in this form it seems relatively easy to see (to me I think) that bad returns early in a sequence mean spending takes a relatively bigger chunk of whatever wealth is present which then has to catch up with higher returns later...but we know that's hard and and why geometric returns with vol present are lower than arithmetic returns over time. In figure 1 I'll use a 60 year old and a real return of .06-.03 = .03 and no vol (except in the sequence trend scenarios). I'll run it three times: steady growth of 1.03, an average of 1.03 but with a trend going from 1.02 to 1.04 (adverse sequence risk), and an average of 1.03 but going from 1.04 to 1.02. This has been done a bunch of times by others so it shouldn't be too surprising.
I don't think we need to quantify the dispersion. The point is to illustrate visually that there is a material amount of difference between the different paths which is also a way of saying that volatility matters, especially when spending is present. Which reminds me. I was going to sneak this into another post but forgot. The clever out there, and I admit I have tried to do this myself, will often point out that volatility is not risk, permanent loss of capital is risk. But here -- since the net wealth process of equation 1 can go to and through zero -- what they, and I, forget is that the two statements are the same in the presence of spending. Volatility does create permanent loss of capital, in this case in relative terms in that the permanent loss is earlier rather than later when returns are in the "wrong" sequence.
Sequence risk - dynamic version
Ok, now let's breathe a little vol into this. In the following effort, we'll skip age and just look at portfolio longevity (how long is the portfolio greater than zero; we could set another threshold too if we wanted). The real return will be .05-.03 = .02 which maybe seems low but it is not meant to be a forecast, just a low rate for the illustration. Wealth units will be set to 25 which means an implicit spend rate of 4%. Vol, in std dev terms, for lack of any other number I called .10. This is normally distributed, by the way, which no one seems to like (except maybe Marlena I Lee [2013] in the last post's footnote), but that's the best I can come up with today. I'll then do the following:
1) sim it 3000 times over 150 periods using a simple excel sheet (I'd do more but it's slow)
2) take some measure of what we have with some summary stats
3) for comparison, use some deterministic formulas to see if we are close
4) form an opinion on risk if it makes sense to do so
The deterministic formulas I'll use for calibration are:
1. Moshe Milevsky's "how long will it last" formula from his 7 Equations book which looks like:
EL estimated portfolio longevity
M nest egg (money)
w withdrawal rate in dollars per year
g annual real growth rate of portfolio
2. Michael Zwecher's "annuity number of periods" from his "Retirement Portfolios" book which looks like:
 |
i(e) expected inflation
lifestyle spend rate
wealth portfolio or endowment
For each sim iteration I'll count the number of non-zero wealth years and at the end I'll sweep the 3000 of them into 31 bins from zero to 150 years in breaks of 5 (the 150+ bin will catch everything with more than 150 years). After I do that, plus some summary stats, plus the two formulas above, it looks like this:
Discussion -- First of all what I notice right off the bat is that the mean of the sim distribution is pretty darn close to the results from the Zwecher formula (with Milevsky close behind)[1]. That's pretty cool, so that formula might be a good rule of thumb equation to use in the absence of simulation tools. But once that is said, it's pretty obvious that (if using Milevsky's advice) matching a portfolio longevity "number" to an average age expectation using a simple formula does not really convey the true risk (in my opinion ... but then again he did also say in his paper on tempering one's enthusiasm for ruin stats: start with deterministic 1.0 conversations before getting to stochastic 2.0 stuff).
If I were to use a single number for life expectancy ( maybe mid to late 90s depending on the life table used, let's call it 95) for planning purposes then I think that a single number for portfolio longevity (= ~38 years or to age ~98) is probably not enough. If the minimum sim years above for portfolio longevity is to age 71 and the 5th percentile is 78 and the mode is 84 and the median is 91 then all of those are quite a bit less than a planning-life-expectancy of 95 (and much lower than the mean of the portfolio longevity distribution itself for that matter) and I need to have a much closer examination of risk and things like spending and allocation choices. This is the effect of unknown future volatility which is also similar to but not the same as saying there is sequence of returns risk[2]. That means that the strategy discussion that one might have with oneself or an advisor should be relatively long and hard and focus on the left side of that distribution. (hint: spending has a pretty big impact here as it always does)
If, on the other hand, I were to use a distribution for life expectancy, which I probably should, then we are really into more advanced simulation and ruin risk probability estimation which is another topic altogether.
-----------------------
[1] I need to be fact-checked sometime. I'm never sure if I get these formulas exactly right...
[2] the random draws may just be a whole series of bad return draws but I'll assume based on the static analysis that some of it comes from bad (but average) sequences as well. We'd have to dig into the paths to see.
No comments:
Post a Comment