Simulated stochastic present value of spending meets an efficient frontier

This post falls in into that category of "let's check out what it looks like before we have any idea what it means."  That "let's check out..." thing also appears to be a recurring theme here.

In this case I wanted to take the stochastic present value idea -- i.e., take a deterministic spending present value calc and animate it with random variables for the numerator (spending, inflation), the denominator (discount rates representing the return generating forces at play), and duration in terms of longevity and use it to create a distribution of spending NPVs for use with a household balance sheet -- and then see what happens if we connect it to an efficient frontier of asset allocations[3] for a given set of arbitrary assumptions.  That is a mouthful.  Let's take a look.


First, if I have two assets with (expected forthcoming) arithmetic returns of 9% and 2%, standard deviations of 18% and 4%, and a correlation of -.05, the efficient frontier might look like this where the blue dots are the allocations that represent unit steps in standard deviation (.04, .05, .06 etc):

Figure 1

Now, I want to create a series of stochastic present value (spv) distributions at the various points on the frontier to see if there is a minimum spv (at some mean or pth percentile) since it might (???) make sense to look for the lowest cost spend commitment in PV terms for an efficient set of portfolio return and sd pairs for a given risk posture with respect to future spending[1].  The spv, by the way, is calculated like this -- and forgive my false notation since I know not what I do:

where Ct is a custom spend path (in this post it is 100k inflated at a randomized 3% with a spending variance above and beyond inflation of 8% in each period t and downward spending deflections of 20% at 68 and another 20% at 85 ... assuming we start with a 60 year old), L here is 120, N is a randomized end-age using Gompertz math with mode/dispersion of 90/8.5 for a 60 year old, and D is the randomized discount rate using return and variance as follows.  I will calculate the spv[2] based on randomizing D using a return and sd assumption that come from each of the blue dots in figure 1 which represent unit steps in standard dev (makes it easier to chart).  Note that this also means that while the efficient frontier is rendered in single-period/arithmetic terms, the spv will in effect be a geometric multi-period thing since the randomized return series are chained in the sim that is required to derive the spv.   Given all that I will chart the mean and the 95th percentile spv-s.  I could have/should have rendered this in 3D since the spv that I calculated along that EF line would make a third dimension but I did not have the wherewithal to pull it off this afternoon. You'll have to imagine the spv data as sitting in a third dimension exactly above the EF line that is in 2D.  Yes, you, all three of my readers, will have to imagine this on your own.

Before we chart what happens I have to point out the following:

1. You have to buy into the stochastic present value concept, which you might not want to do. See some of my past posts or check out Mindlin on spv at SSRN or CDIadvisors and decide for yourself if this is a "buyable" proposition,
2. You have to realize that the output of a stochastic present value calc/sim is a distribution,
3. You don't have to, but you might want to, pretend that spv is somehow not simulation by other means, which, in my opinion, it most certainly is,
4. You have to ignore, perhaps, that this is just one unique, arbitrary and maybe not totally rational set of assumptions, and
5. This post totally ignores rational human behavioral effects like people adjusting spending in the face of bad financial environments along the way which certainly would happen in real life.

Then with that out of the way we can chart it out like this:

Figure 2

GREY -- The grey dotted line is the deterministic version of the spv calc for the Ct I described above. It is more or less an annuity style calc but here with a custom cash flow. The calculation is rendered, if I have it right, like this where d is a fixed .028, Ct is as above, and tPx is the conditional survival prob for a 60 year old male using annuitant mortality assumptions:

BLUE -- The blue line is the mean spv at each return/sd pair along the EF.  This would, in 3d mode, sit above the EF in figure 1 in a third dimension.

RED -- The red line is the spv 95th percentile at each pair along the EF.  This would, in 3d mode, sit above the EF in figure 1 in a third dimension.

Conclusions?

As in past posts I am loathe to make conclusions on sketchy stuff like this but since I have gone to the effort of throwing this out into the world, and since I have bothered to provide some of the disclaimers above, I'll proffer the following but with cringing uncertainty:

• The deterministic calc is a head-fake.  PV calcs were always a convenience due to past constraints on complexity and computing power.  Distributions and probability-based thinking, even though I do not have formal training there, is always a better approach. For example, a higher discount rate in the deterministic calc would lower the NPV but that is like taking all the benefits of risk while ignoring the actual real risk as it is typically articulated in variance terms. That would be a neat trick, eh? 
• It seems like a bond-heavy approach (or alternatively we could maybe also propose another construction: "certainty-heavy" for when instead of going left on the variance we go "up" on the spv distribution) would likely -- even though I do not have the info or good visualizations here to support the claim -- be dominated by a fairly priced annuity. This is something I should dig into more than I have here, 
• It looks like the "mean expectation for the spv," which is a cousin to the deterministic calc by the way, likes heavy allocations[3] to equities...but that mean is more or less in the middle of the distribution and the whole point of the spv analysis is to create an uncertain distribution in order to evaluate degrees of risk out in the tails, and
• Higher percentiles on the spv distribution -- say the 95th percentile...which is more expensive (i.e., if funded) but more certain of being achieved when funded since very few spending scenarios remain on the right side of 95%  -- look less friendly to equity given the volatility of the return generating processes. I know I have not explained that well but maybe in another post... 

In the end all of this feels like it is more or less an asset allocation analysis in disguise.  In that guise, the conclusions here are not revolutionary and are consistent with other posts I've done and other papers I've read where equity allocations between half and just shy of 100% seem to be the "zone" when put into a retirement context and within which there is not a ton of sensitivity to variations in the allocations[3].  Or at least this might be true before the allocation decision gets thrust into a dynamic-mode-over-time i.e., a glide path.  But that concept I do not touch here.    

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[1] That is a really big assumption or goal that I do not substantiate or support here. Nor do I define what I call risk posture except to say that it would be where one wants to pull data from the spv distribution. The mainstream alternative way to do this, and an easier way I think, is to do traditional MC simulation.  Whether this (spv) is better or more transparent somehow is highly debatable.

[2] this is still a spreadsheet sim with a rather thin 3000 iterations.  This is because I have not had time to code this into R where it is easier to step up my simulation game.  It is winter "ski break" and I have kids to feed...

[3] Here is a big fat obvious disclaimer: do NOT take any investment or allocation advice from me. This is a naive amateur analysis that is just for fun and just for me.  I am just playing with my own learning process.  Ask your real advisor for help on this stuff.  I still do.