Notes in a 2001 German paper about ruin estimation math

Albrecht and Maurer

I just (11/14/17) ran into the following in a 2001 German paper (Self-annuitization, ruin risk in retirement and asset allocation : the annuity benchmark by Albrecht and Maurer, Univ. of Mannheim; end of Appendix B).  It is precisely the same as what I often use here and accurately describes what I built earlier this year in my FRET algorithm...except that I am not using German actuarial tables from 1994.

Excerpt from the paper:

My FRET tool from 2017

The notation I used in my Flexible Ruin Estimation Tool (or Fret ;-) ) is like this:  

where: 

• LPR is the "lifetime probability of ruin" and the
• gw(t) term is the function representing the process for building a pdf for portfolio longevity under some set of randomized assumptions and is, in fact, simulated, and the
• tPx  is the conditional mortality term 

The Germans use an actuarial table for mortality while I usually use a Gompertz related function that I lifted from Milevsky's 7 Equations book like this: 

where p is the conditional survival probability and is represented in tPx above in LPR or "the probability that a person aged x will survive to time t."  I have also used Social Security and Society of Actuaries Individual Annuitant Mortality tables as needed in different circumstances or other simulations. 

Therefore the LPR calc can be considered taking the probability distribution (or mass in my case) of "portfolio longevity in years" (or survival or ruin etc) along the path of start-age-whatever to infinity and then weighting all those probabilities by the conditional survival probability along each year of the path, conditioned on the start age. 

Some Misc Considerations re LPR

This flexible LPR approach has some nice advantages, imo:

1. Simpler than the Kolmogorov PDE.  LPR(fret) satisfies the Kolmogorov partial differential equation for LPR without one having to know differential equations or knowing how to "solve" them by way of a finite differences algo (FD) which I call simulation by other means (it's hard. I transcribed the code for that once). The PDE looks like this:  

This equation is mostly inscrutable to me and I do not have the calculus for it. On the other hand I once taped this to my fridge for a year. I still didn't understand it until: a) I started to get how diffusion equations work and relate to simulation, and b) I had an odd dream once where the coefficient (uW-1) looped in the dream and I woke up and I said "I get it," walked three feet to my computer, and coded the sim that now drives the gw(t) term above in the LPR algo. Lambda in the PDE, by the way, is the mortality thing. The rest on the right side of the equality I won't bother with for now.  

I once ran a bake-off between LPR and a FD routine for the PDE along multiple test dimensions. Except for minor variations coming from the wobbles of simulation, the two aligned almost exactly.  

2. LPR has a "Roundness" to it.  By roundness I mean that the LPR has some robust or comprehensive advantages over some of the rudimentary sims you might see from an advisor.  Typically (not always) an advisor Monte Carlo simulation (MC) for "retirement ruin risk" will use a fixed horizon for mortality (30 years) i.e., the sim is run for 30 intervals and the number of times it runs out of money is the fail rate. It is not just that this (usually) ignores the magnitude (failing at year 29 is qualitatively different than failing at year 3 if you have to live it in real life) or the ability for humans to see the fail coming and adapt.  It (MC) is also potentially impoverished in couple other under-considered ways: 

a. It (MC) does not consider what Albrecht and Maurer call the full "constellation" of asset exhaustion possibilities. MC goes to, say, 30 years. The LPR g(t) function considers portfolio longevity all the way to infinity (or at least 120 years or so net the start age). 

 

b. It (MC) does not typically consider what Milevsky has called the full "term structure" of mortality. MC to 30, LPR: start age to infinity (or age 120 or so) by way of a vector of conditional survival probabilities conditioned on the start age. 

This means that, for me, the LPR I get of, say, 12% feels different than a 12% I might get from a fixed horizon MC. This is very subjective of course but I always feel like I want to ask what did the MC miss? where I kind of take the LPR result and run with it. This "trust" is very much a personal bias but then again I also don't trust either LPR or MC fully and me? I use a methodology of triangulation using many tools way beyond these two in trying to understand my own retirement. 

3. LPR has a little Flexibility. LPR shares this feature with MC, of course, in the sense that the nature of the return distribution can be pushed towards anything: fat tailed, discontinuous, custom, other where the Kolmogorov PDE assumes a normal distribution. Most MC (vs LPR) will also have a slight advantage over both LPR and the PDE in that they can be customized towards the extreme in terms of things like spending patterns, taxes, fees etc etc.  I personally keep my LPR or MC sims simple for reasons of speed and/or transparency or at least as much transparency as one can coax from a black box sim.  

Fin