I'll use two tables: 1) the Social Security 2013 Life table, and 2) the Society of Actuaries Individual Annuity Mortality table (a healthier, self-selecting, longer living cohort I gather). Depending on the need I either access the data directly or in some cases I borrow the work done at AAcalc.com where he uses the SOA 2012 IAM table with projection scale G2. I will also look at several different risk/expectation categories: mean, 80th percentile, and 95th percentile.
The tools I'll use are: 1) Excel PMT with type set to zero (end of period), PV set to $1, FV=0, and Nper set to "the mortality table expectation for age minus age", 2) the RH40 formula or [ Age / (40 - Age/3) ] as a proxy for an age and risk adjusted spend rate, and 3) later on I'll look again at the Blanchett simple rule based on simulations but here with different term assumptions for mortality.
The goal: a) see what PMT looks like but with variable longevity based on real table data, b) see where RH40 falls into this soup-mix, and c) look again at what Blanchett looks like with real variations in longevity rather than the fake-ish linear approximation I used in the past.
The result is illustrated in Figure 1. The colored solid lines are the Excel PMT function using different longevity expectations. The dotted line is the RH40 formula. I'm hoping the chart is self documenting but it may not be.
Figure 1. Excel PMT vs RH40 for different longevity expectations.
Then, since RH40 and the SOA 95th percentile approach seem to have an affinity, I took just that data only and then varied the PMT function that uses the SOA 95th percentile expectation by changing the real return +/-1% either way to see how much it moved. Figure 2 is the result. Green is +1% real return and Blue is -1%. Again, I hope this self-documents.
Figure 2. RH40 v Excel PMT (SOA 95 percentile) for diff return expectations
Then I re-ran the Blanchett simple rule with different longevity assumptions. I had done this in the past by either using a fixed age assumption (95, say) or alternatively with a linear approximation to actuarially changing longevity (by changing the expectation from 95 to 100 in a straight line between age 80 and 100). Here I do both of those but then also plug in the terminal age "duration" expectation using the same tables as above. The result is as in Figure 3.
Figure 3. Blanchett Simple Rule with variable longevity
Conclusions?
With respect to the goals I set above:
a) I had been skeptical of the PMT function because it seems like it was a little too generous when I used it in the past and I felt that it under-reflected risk. Here, though, when I put in hyper conservative longevity assumptions it looks ok. And by OK I mean I have no scientific meaning for that.
b) RH40 and a very conservative interpretation (longevity and rates) of the PMT function line up nicely which pleases me. RH40 was kind of a well-informed joke with myself, created so that I'd have my own formula but it seems to do the job more often than not. It appears to be itself a very very conservative and easy to remember proxy for other peoples math, one that I could remember, apply, and kinda defend in a pinch[2]. While there is no mathematically defensible logic (yet) for why RH40 and PMT (using the SOA 95th percentile) diverge at early ages (e.g., 58 - ~70) it would be very easy, since we are only looking at longevity here, to construct a plausible narrative for that divergence. Here I am thinking about risks that are more likely to afflict the young than the old, things like sequence of returns risk and expectations about future spend shocks, among other things. Certainly RH40 could be a good seat-of-the-pants starting point to which I could then bend the results based on either real inputs at a given age or maybe just variations in optimism or risk aversion. Who knows? Future post...
c) I have not absorbed yet what I have charted for the Blanchett comparison. I'll leave any detailed interpretation to later posts. In general, I think RH40 hangs in there here as well and at late ages, when some bets are off and other calcs (like RMD style calc described in a prior post) come into play, RH40 might be better than the Blanchett simple rule. But I've mentioned that before. For now, for the next 20 years to age ~80, I think that any one of these approaches -- whether PMT, Blanchett, RH40, or anything else that adjusts with age and changing longevity and risk assumptions -- is probably good enough if used dynamically. Certainly these would be better than sticking with one assumption made at the beginning and then held forever come hell or high water.
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[1] Mortality table data used. Mostly from AAcalc.com but also some from the tables directly.
Ok, I guess I lean pretty heavily on AAcalc.com here but in my defense I'll say a couple things: 1) this guy I would trust with the math of retirement any day. He's got the goods and he has a very constructively correct point of view and he knows Merton math, 2) he helped me more than he probably should have with me trying to recreate what he did in terms of stochastic dynamic programming and backward induction...a programming and analytic breakthru for me that I will appreciate for a long time, and 3) I went to the effort of -- for a few sample ages -- recreating what he did with the percentile results in the SOA table. Only after I was able to precisely recreate his results for a few ages like 58 70 80... etc was I able to with abandon just accept what he had at aacalc.com. It was tedious to capture but faster, at the time, than having me run the table at each age and get 50th 80th 95th percentile and so forth. So aacalc. For now. I trust.
[2] Easy to remember is the key. I seriously doubt I could remember off the top of my head the math for the PMT formula P = ( Pv * r ) / [ 1 - (1 + r)-n ] though its not that hard. And even if I could and even though most hand calculators can do this I also seriously doubt I would be able to consistently remember values from a longevity table. This is why people create and use rules of thumb. Sure, if I am at my computer and if I am so inclined, it's pretty easy. But that is not always the case.
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