A type of heat map of ruin risk arising from: age, allocation, spending, and return distributions

Since I have been playing around with ruin risk math lately (I feel compelled to asterisk this with links to why ruin risk is often misused and misunderstood, but won't) I thought I'd try this post.  I did this for no other reason than I wanted to see what it looked like and because I can.  What I am trying to do here can be achieved by other means -- like simulation or my new joint probability tool or other analytic techniques -- of course but I chose to use the spreadsheet version of the kolmogorov equation I got from Prof. Milevsky because it is simple and fast to do the exercise.  Simulation would have taken me a couple days at least.  What I lose in customization of circumstances I gain in time spent with my kids.

The thing I am trying to do.

The thing I am trying to do here to see what it looks like to map out a color-coded ruin risk calc based on combinations of parameters for: age, returns, volatility, asset allocation, and spend rate.  I'm sure something like this has been done before but what the heck, I had a spare couple of hours.


The background allocation guesstimate.

Since I am not simulating or combining asset classes for a more realistic view I pondered how to generate my assumptions for return and vol that would mimic various allocations to risk.  The best I could do was come up with a return profile for two assets, risky and less risky.   For this I winged it with the first being 7% real return and 18% std dev. and the second being 2% real return and 4% std dev. This seems arbitrary and lowish which it might be but let's say an expectation for equities (risky) is 11% and inflation is 4%. Before we discuss fees and taxes we might easily be at 7% or lower.  For the less risky asset (What? treas bonds or aggregate us corp? I don't know) I gathered that using the same logic could get me to 2% or even negative if I wanted to be realistic about it.  We'll go with 7 and 2 for now.  For correlation I am going with -.10 which is also arbitrary but because correlations swing around from positive to negative over time I had to pick something.  A look at the last 30  years or so show a tendency towards negative so I "arbitrarily" picked a negative correlation that was somewhere in the mix.  As you can tell I am not a retirement finance scientist. I am an amateur hack, a label I wear proudly.

So using these parameters it is easy to generate a mean-variance frontier like this...

...where I'll use the return and vol results for the allocations to risk of 0% to 100% taken in 11 steps, or... 

risk
alloc	rr	sd
0.0	.020	.0400
0.1	.025	.0386
0.2	.030	.0457
0.3	.035	.0583
0.4	.040	.0736
0.5	.045	.0902
0.6	.050	.1076
0.7	.055	.1254
0.8	.060	.1434
0.9	.065	.1616
1.0	.070	.1800

Fwiw, I'll assume, depending on how one looks at these things, that the tangency portfolio is somewhere in that 10 - 30% allocation to risk.  That may be irrelevant or useful later.  TBD.

Also fwiw, the assumption here appears to be independent returns so it looks like that means we have forsworn things like dynamic volatility, dynamic serial correlation, trends, mean reversion, etc etc. The academic types can work on that. Also, don't forget that this is very very generalized. Individual paths in real life can be extreme.

The other assumptions.

The other assumptions revolve around what I have been personally working with recently such as:

• ages: 60 70 80 90 
• spend rates: .03 .04 .05 .06 .07
• longevity: mode 90 and dispersion 8.5 which sorta syncs up with SOA annuitant mortality est.

The "heat map."

When I take all of that above and blindly plug it into the equation for risk it looks like this below.  First up is the color legend where the ruin risk for everything below 5% risk is green, the ruin risk below 10% is blue (down to the 5% level), and so forth. The lowest risk is outlined in a dark green line.

the legend, in terms of lifetime probability of ruin...

... and the map. There is a little "significant digit" abuse going on here; past the first couple it probably doesn't matter much, but...

Any Conclusions?

I'm a little reluctant to make conclusions for two reasons: 1) I mainly did this to see what it looks like, a goal that has already been achieved, and 2) it is way too easy to over-interpret and mis-use fail rate analysis.  However, if I were forced to say something it might be with comments like this:

• younger ages are, not surprisingly, not a good match for aggressive spending. That is a time to be a little circumspect (don't listen to the people around you -- I won't say this is from personal experience but one can infer -- that say that you are acting like a life-denying soul-sucking tightwad. You really do still have future risk that needs to be managed)
• Older ages -- contra some advice I see...and then there are those age based allocation rules of thumb -- seem to be highly tolerant of high allocation risk (and spending for that matter).  This does not surprise me. There are fewer years to go and economic theory, from what I've read so far, seems to support this idea of late-age risk tolerance.  That's my plan by the way.
• The allocations with the lowest ruin risk estimate, in those scenarios where the pressure of spending over time is less intense (e.g., old age, or early ages with the lower spend rates), look like they represent something close to the mean variance tangency portfolio given the assumptions we set up.  I guess that is not too surprising.
• Where the spending intensity is high (e.g. young ages and/or high spend rates) not only is the ruin risk high, the allocations shift much higher on the EF.  Maybe I could have guessed that but it always seems counter-intuitive.  That result, no doubt, is from what I want to call the "lottery ticket effect" of volatility for situations when one is challenged by some combination of low wealth, high spending, long time frames, or maybe carelessness.  That'd be another post though.  
• We still haven't addressed or solved or even thought about the issue of having left too much money on the table at end of life if we choose to be hyper-risk-averse in early retirement.