Endowments, Retirement and Spending

I was recently following a twitter thread on a question about endowment targets.  The proximal question at the top of the thread was "Is 4% real a reasonable target for an endowment?" I didn't jump in on that because (a) I am a fin-twit coward and newbie, and (b) at first I was confused about whether they were talking about returns or spend rates, but it did get me thinking.  My first thought was that it doesn't matter whether it was spending or returns because those issues are necessarily and importantly intertwined (a point often forgotten or ignored) and the second thought was that they were asking the same question about endowments that retirees do when they talk about sustainability of their retirement portfolios over time.

Of course there are differences.  Taxation is different, endowments are generally perpetual, there is often a dedicated and experienced staff, and, importantly, they have contribution inflows (that's what I need! gofundme...). In addition I guess there are some other soft issues like politics to consider. No one (yet) is asking me to divest from energy or tobacco.


But maybe to look at this we can call an endowment that has no contribution inflows a "tax-advantaged immortal retiree." Then we can look at the question from my side of the lake.  In that situation of immortality we need to fund ourselves forever.   Ed Thorpe, in his latest book, suggested that the rate a for perpetual spend is somewhere around 2%. I think he is about right on this given his genius and my deference to genius. Also, I've had a chance to simulate a lot of stuff and the "around 2" seems to be more or less where you get to a perpetuating process.  It may well be that Armageddon scenarios could torpedo the 2% process but "around 2" is a good enough place to start. We'll assert it...without proof.

Now, it is also entirely possible, in the absence of donations that is, that there are policy issues or exigencies that would demand a higher rate of consumption than 2. That's fine but the minute you go from "around 2" to "above around 2" you pretty much go from perpetual to not-perpetual.  At that point, portfolio longevity (PL) becomes a topic that needs to be discussed and understood.  And this is where retirees sometimes have a perverse advantage.  They, contra endowments, have an entirely separate process called longevity; they often die before portfolio longevity ever becomes an issue.  It's just that in this particular post we are ignoring that mortality thing.

Portfolio longevity is relatively easy to analyse, given simplifying assumptions, and it's a useful discussion to have. I'm sure endowments pay for painfully expensive optimization software and painfully expensive consultants but spending just a little bit of inexpensive time understanding  portfolio longevity -- not as a thing but as the result of a multi-period net wealth process (NWP) -- would at least buttress the basic concept of and visualization of and discussion of sustainability over infinite horizons, especially since PL is a distribution not a point estimate (see note 1).

For retirees, sustainability is often modeled by Monte Carlo simulation which is typically denominated in fail rates. This is a start, of course, but the problem here is that MC is often modeled with a fixed 30 year duration which is not our problem here today; we need infinity.  Another way sustainability is modeled for a retiree is via a lifetime probability of ruin (LPR) which factors in random lifetime rather than fixed.  But this isn't helpful either because the LPR is bounded by the separate, independent process for human mortality (HM).  But I think LPR can help us because it is made up of two pieces: PL and HM. Ditch the HM and you get PL which is what we need to know for the immortals.

Just for LPR background reference, here is Kolmogorov's differential equation for LPR, a formula I clearly cannot read (this is from Milevsky's 7-Equations book):

On the other hand, a more tractable way to do something like the PDE is via simulation by way of an expression like this, which happens to satisfy the PDE and is also extremely easy to simulate[2]:

In this version, g is some function of a net wealth process (it's portfolio longevity probability) at time t while the term on the right is HM by way of a conditional survival probability at time t for someone aged x.  This produces almost the exact same result as the PDE. I say almost because the g() term comes from simulation and will have some variability.  I ran it head to head with a finite differences approximation to the PDE and came up with basically the same answers across a large number of test conditions.  But now let's ditch the right side (of the right side).  The left side (of the right side) is a function whose output is a probability distribution called "portfolio longevity in years" (PLY). The distribution is extracted indirectly from a net wealth process that we've seen before:

            W_t = µW_(t-1)-1,  µ≈N[r_p,σ_p], via simulation

Which happens to be, if you'll notice, more or less the coefficient of the 2nd term in the PDE which is why it works.  gw() is a distribution pulled from the process of simulating W(t) by way of counting the frequency of years survived across the iterations.

This, if you haven't gathered or if I haven't mentioned it, is easily simulable[3]. And when simulated, now unconstrained by HM, we get, baring perpetuality, a distribution of PL likelihoods...and a defective distribution at that[1].  Almost all NWPs will have a "defective" cumulative longevity probability (CLP) at infinity:  0 < CLP < 1. The hidden "other" distribution, the simulated portfolios that last forever, when added back to CLP, will make it sum to 1. 

Let's take a look.  In a past post (A thousand years of the 4% rule: the naked, ugly glory of it all) I illustrated a 4% rule over 1000 years like this:

The x axis is years; y probability. The green(?) line is the cumulative probability of a net wealth fail. The net wealth fail is really just the integration of the pdf for PLY (that pdf happens to be the distribution that is actually used in LPR[1]). You can see in the chart above that for these assumptions the CLP (the fail part of the cdf) limits itself out at around a 60% fail rate starting at somewhere around 150 years and then thereon all the way out to infinity. The other 40% of simulated portfolios (not shown) lasted forever. 

The only reason this kind of bet (60% expected fail) can be sold to a retiree is that the probability that they will live forever is constrained by the black line at somewhere around 88 years. The black line, when multiplied by the green line (in it's pdf form), gives you LPR for a human...LPR is thus the raw portfolio longevity probability distribution weighted by conditional survival probablility which tends towards zero at around age 120, or somewhere around 60 years for me.  This is, in my opinion, why the 4% rule "works," or rather "worked" for  Bengen using historical data. 30 years was just short enough.  But this is also why it is sometimes a bad bet since history and aggressive longevity assumptions are maybe the wrong platform for making mission-critical wagers.  Here, for example, is a wager viewed on a scale of human time for some set of assumptions I can't remember:

An LPR here of 23% might be acceptable for a human but note that it's only 23% because of the presence of mortality (black), otherwise it looks like the cumulative distribution for PLY (green) is going to limit out at around an 80% fail and a 20% succeed at infinity...but it's also getting pretty close at the 60 year mark which is small change for a perpetuity (or an immortal).

As a side note, if ginning up a distribution is too much work, the PLY can be estimated analytically by this equation in Milevsky's 7-Equations book which is explained there:

But let's get back to the immortal, tax-free retiree-endowment.  I think that a professionally run endowment could certainly spend millions on staff and optimization software and consultants but I also think that understanding the endowment as a net wealth process over eternity might be an easier play especially for a cheapskate individual immortal. The task, once seen for what it is, would then be to reduce the risk of fail and to increase the effective longevity of the capital by bending the green curve out and lower.  Essentially we are saying to ourselves to come up with a policy that points the CDF of the PLY at the right side of the chart as low as one wants to go given any trade-offs.  As I mentioned, a 2% spend probably get's you pretty far (presence of contributions/donations will change the discussion, of course...) to infinite (but not guaranteed) sustainability.  Anything above 2 is then a policy choice, likely with some difficult trade-offs involved.  If the pain threshold is high, one could target a perpetual success rate of 95% or something, in which case the spend rate is likely pretty low, I want to say 2.5% or thereabouts. That's a guess; in years past I would have run the sim's and done the distributions and figured it out but I have bigger fish to fry right now...  But, any way you cut it, it goes like this: high-return+low-spend = flatter curve with later approach to limit over the years, low-return+high-spend = steep and tall curve with an earlier approach to limit over the years. Vol, of course, has its own weird effects but  higher is generally worse for the NWP than lower.  For help with vol effects: think efficient frontiers, trend following, alternative risk allocations, spending control, etc...

The point is that the shape of the PLY distribution and the time and level and speed as it approaches the fail limit are controlled by at least three levers that should be pretty obvious: return expectations & volatility (allocation choice) and the spend rate. All of these three interact in terms of how and when and where the limit is found.  Again, all this is a policy choice.  This may explain why I didn't care about whether fintwit was talking about spend rates or return expectations. They are intertwined.  For both endowments and retirees there is almost no separability between the allocation and spend choice given the time-based net process involved.  This is why I find myself surprised when I hear about rumors of endowments that make a hard separation -- as did all of my ex-advisors, as do many retirees, and as do no small number of fin-twit-blogger types -- between spending and portfolio choice.


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[1] from a past post...the "defective" freq distrb (call it a pdf) for PLY might look like this where x is in years and in reality the end point of x is not 200 it is infinity; same idea:

[2] ...and, as Prof. Milevsky pointed out to me when he was helping me decode this problem, the distributions in the PDE are "normal" where in simulation the distributions can be shaped in any way the designer wishes...

[3] Notice that blogger sleight of hand?  I distracted you with the "MC sim won't work here" and then, in the end, slipped in what is effectively a MC sim, just with an infinite horizon and a statistical transform.  I felt like I had to take a detour to get there to make a point.