Impact of Longevity Expectations on the Stochastic PV of Spending

The stochastic present value (SPV) of consumption -- from what I can tell an inverted way to do Monte Carlo as a feasibility check -- typically depends on a "horizon" (i.e., fixed to, say, 30 years) to do its thing. I thought I'd instead throw some random longevity at SPV and see what happens to a couple different summary thresholds(median and 95th percentile, with a nod to the absolute value for the starting endowment) for evaluating the distribution.


1. Longevity

I am using a simplified expression here using Gompertz math I borrowed from a book by Milevsky. The expression for the conditional survival probability goes like this:

p is the modeled survival probability
x is the age of the person doing the analysis
m is the mode of the distribution
b is a factor for the dispersion
t is the year into the future for the person aged x

When I fit this to a table like the SS life table (average population) or the SOA table for annuitants (healthy self selected population supposedly) and then extend it out to hypothetically shorter and longer lived people I come up with four different scenarios for longevity. Like this below where the x axis is the number of years past 60 (my arbitrary start age in this post) and the Y is the conditional survival probability for a 60 year old in each of those years along X:

Scenario B is the equation fit to the SS table (60 yo male) and C is the SOA table. A and D are hypothetical cases for ill health and hyper long lived.

Mode and dispersion for longevity scenarios A-D:

     M         b 
A  80        12 
B  85.5     10.5
C  90         8.5
D  95         7

2. The Simulation Set-up and Some Assumptions

I have a rudimentary SPV simulator written in R. The core function for the expected value of the SPV might look like this though I never really calculate the expected value. I just inspect the distributions:

where

i = the number of sim iterations
y = ~ 120 years so T is ~ 60 but could be infinity
tPx = the conditional survival probability at time t for a person aged x
c* = a variable consumption plan with random inflation and random spend variance
d = a randomized discount rate chained multiplicatively over all t

Within the sim we have a number of arbitrary, non-realistic, illustrative-only assumptions that include:

- 100k start spend
- Inflation randomly sampled from 1900+ US data with no autocorrelation or mean reversions.
- Discount rate of .04 and .10 standard deviation
- Start age = 60
- 10000 iterations
- a spend variance in-period of .10*
- The endowment, if we need to know it, we'll say is $3M

Note: these assumptions are not yet supposed to be meaningful in a real-world sense. There are some other assumptions and biases baked into the sim that are not seen here.


3. The Simulation Output


The four longevity scenarios combined with the above assumptions produce these probability weighted distributions for the NPV of spending over more or less infinite time. I've been told these are reciprocal gamma distributions but I would not personally know.

Scenario key:
A - black
B - blue
C - red
D - dark green
the arbitrary endowment of 3M is the vertical grey line

(left to right if the colors are hard to see)

Some summary stats for median and 95th percentile. We'll skip the mean:

Which, if charted, look like:

Conclusions

This shows me why I don't do SPV for myself too much. The reconciliation of the subjectivity inherent in generating the distributions with the current endowment with a policy choice about what statistics to use is hard. For me personally it is not worth it but the visual might be helpful in a long conversation with a well informed retiree.

The other conclusion is that living a long time is expensive.


* this might be modeled wrong and I have a plan to go back and look at this...


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Follow-up: Simple reality check vs. complex SPV