Perfect Withdrawal Rates and Random Lifetime

This is another one of my "want to see what it looked like" posts. In this case I wanted to see what a PWR distribution looked like at different ages using a point on a random lifetime distribution as the "horizon."  PWR we've seen before[1]. Instead of holding spending to a number and letting terminal wealth in a net wealth process disperse over a planning horizon, here we hold terminal wealth to zero and let spending be dispersed. The math, as before looks like this:

Perfect Withdrawal Rate[1]

where w is the PWR and is also a random variable because r is random. r is the randomized (normal, sorry) real return and n is the horizon number of periods.

Planning horizon for PWR

Speaking of horizon, here in this post we are using the 95th percentile of a longevity distribution for a conditionally attained age, if I can say it like that. I happen to be using the same horizon-by-age I came up with in On "Post-Modern" Planning Horizons.  The 95th percentile is the grey line below and this data happens to be derived from the SOA IAM table.  Like this:

Figure 1. Planning horizons using random lifetime

So at age 60 the horizon based on a 95th percentile expectation is ~40 years and at 95 it's around 10. The 95th percentile I get from the conditional survival probability which can come from an actuarial table or analytically. I don't know if I have the notation right but it'd be something like this

.95= 1-F(x)t  

where t is the solved-for planning-horizon age that meets the threshold, .95 is the entirely arbitrary longevity risk threshold, and F(x) can be dragged out of an actuarial table or determined via Gompertz math like this:

Gompertz eq for CSP, from Milevsky's Seven Equations

where p is the conditional survival probability, t is the forward year for person aged x, x is the age attained or the age for which the PWR distribution is being created, m is the modal longevity expectation and b is an amount of dispersion in years. I used a table rather than this expression and I then used the number of years (which was rounded to whole years) from the y axis above along the grey line for each age along the x axis. 

Other Assumptions for PWR 

I am using a set of arbitrary placeholder assumptions to run the PWR sim:

Real net return: 4%
Std dev of the return: 12%
Iterations: 20k

Caveat on the parameterization: the resulting spend distribution will look pretty conservative but that is due to: (a) the very conservative planning horizon implied by using the 95th percentile for longevity, (b) what looks like a relatively low real net return which, if you think about it, isn't that low and might actually be a little high. So the spend rates implied by the output charts are probably low because they were always supposed to be low and you just don't see it this way all the time.

The Charts

The main chart shows the cumulative distribution of PWR (the implied z axis or the contour colors with 0->100% going from bottom to top) for each age (y axis) and spend level (y axis).  Age 65 and 4% are white lines just for reference.  In a chart like this the high spend rates are not interesting. The low rates are. Use something like a 10% cumulative probability for PWR that implies that 10% of spend rates in the 20,000 runs had to be lower than that level in order to successfully zero out wealth at the horizon.  That is more or less like a fail rate in a MC sim.  Pick your favorite threshold level for the cumulative dist. for PWR and then follow that for different ages. It looks like this:

Figure 2. PWR cumulative distribution by age - contour

I'll leave interpretation to the reader. 

Then, because who doesn't like 3D, here is the 3D version of the same thing we did above for Figure 2.

Figure 3. PWR cumulative distribution by age - 3D surface

Conclusions

No conclusions. Or maybe "spend less if an early retiree and/or loosen up as you age."  Using these fake parameters, a 4% rate looks risky before about 70-75 with more room to breathe thereafter.  I wouldn't hang too much on this analysis, though.  This was only a quick and dirty look-see.


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[1] Suarez E., Suarez A., Walz D, (2015) The Perfect Withdrawal Amount: A Methodology for Creating Retirement