Prototype of an "annuity price as a proxy for a free boundary" concept

Inspired by a recent reading of a Patrick Collins paper[1] on the concept in the title above (he's got some good stuff out there, by the way. Another paper of his on total return trusts[2] is one of the better descriptions of the "retirement problem" I've read out there) I decided to prototype a "free boundary" visualizer similar to something he had in his paper.  The idea here is that rather than wait around for ruin and bankruptcy, which is no fun at all, one might rather consider (if one has the capacity to do so) locking in a pooled risk lifestyle (by way of a single premium immediate annuity, say) if one is approaching the boundary where a portfolio might fall below the cost to "lock-in" a preferred lifestyle and also if one is uncertain, if the portfolio were to fall below the lock-in price and no lock-in were to take place, whether the portfolio would ever recover enough to maintain a tolerable lifestyle.  Since that boundary will move around with age and prevailing interest rates among other things this would be more of an exercise in ongoing portfolio monitoring rather than a specific concrete answer to a specific question but probably something worth knowing along the way.



Using the same code I have in my FRET (flexible ruin estimation) tool I tried to chart out both: a) the dispersion of wealth-unit outcomes in the mini-sim I do for portfolio longevity by year (display: min, max, median and 5th and 95th quantiles), and b) an estimate of an annuity price for the 1 unit of spend in the sim (some new code for that[3]).

Here is that basic idea charted out.  The black lines are the portfolio dispersion and the red line is my attempt at an annuity "boundary" price over time using the same hazard math I have in my FRET tool and a set of assumptions about mortality and interest rates. In this illustration, the black lines assume: age 60, a net real return of .05, sd = .10, and spend = .035 (wealth units = 28.6).   The red line assumptions: start age = 60, max age = 120, discount rate = .025, modal peak = 90, dispersion = 8.5, load factor = .10.  There were 5000 iterations and the portfolio was run for 200 periods.

I'm pretty sure I have not ironed this out very well so we'll call this illustrative or a prototype[4].  The longer term purpose here, for me, is to add another tool to my tool box in order to augment my sense of sustainability as I work my way through an early retirement and also to get some type of early warning if I think I might be headed for trouble.  I could also use this, I suppose, to compare strategies but I don't usually sit around contemplating big switches in my investment approach.  But the "real" purpose was just curiosity, to see if I could come up with something when I am working in wealth units rather than actual dollars in a traditional sim.  In other words, just for fun.  I'll figure out the usability later.

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[1] Monitoring and Managing a Retirement Income Portfolio, Copyright: Patrick J. Collins, Ph.D., CFA, Huy Lam, CFA, Josh Stampfli, MS EESOR [2015]

[2] Promises and Pitfalls of Total Return Trusts, Collins, American College of Trust and Estate Counsel 2001. Part 1 and Part 2

[3] basically I am stepping through each age level and using Gompertz-Makeham math and a variation on simple annuity math like this, if I have the notation right. Whether this would be meaningful in the real world of insurance company pricing is another question altogether:

[4] for example, since interest rates vary, I'm thinking having a view of a range might help. Here is a version with a second boundary (dotted red) using a 4% rate vs 2.5.