In Evaluating Retirement Strategies: A Utility-Based Approach - Estrada & Kritzman 2018, Estrada and Kritzman lay out a method for evaluating retirement strategies using a "coverage ratio" that carries a little more information content than a pure "ruin rate" for reasons that are explained in their paper and my linked post. Their method creates a ratio of all the years a portfolio could sustain spending divided by the years to a planning date L that is a fixed longevity estimate. Less than 1 would be a depleted portfolio with some added info on "how much" or "how long" everything worked out ok. Greater than 1 would imply a bequest since, with a ratio > 1, the portfolio lasts beyond L. In general I think that this is a new and useful way to go about retirement analysis.
On the other hand, here is a question: is this coverage ratio thing the same as what I have been writing about recently in terms of "wealth depletion time" (WDT) where wdt is the unknown amount of time t where wealth goes to zero and t is < [T= random lifetime]? Looks similar but no, not even close. But coverage is an easy way, as the authors state, to access some insight on wealth depletion info above and beyond a basic ruin rate percent. As such it is no doubt a good visualization tool that: (a) adds some value to sustainability analysis, and (b) at least strikes at some of the same ideas and issues (i.e., as WDT) about there potentially being a period before L where wealth is gone (with a little info about the "size" of that time) but life still remains.
But that is where the insight and similarity ends for me. Let's here note that WDT is first and foremost an analytical enterprise that supports a random lifetime utility of consumption analysis, without which oversimplifications might misdirect the end user of a utility analysis that has no WDT "game." Second, the big thing to be aware of in WDT world is that the nuances come mostly from the "split utility" where the function is split into two pieces, one part for the time before wealth is depleted and another part for after. While the Estrada paper does go on to employ a utility framework for their ratio, their framework is with respect to "the ratio" rather than to consumption over time so right off the bat this has nothing to do with the "misdirection" I just mentioned. These two analytical tools diverge methodologically, so in the end it's an apples and oranges kind of thing which means maybe you can ignore at least some of the following.
But even if the methods were to be similar, which they aren't, there are some other problems for me in deploying the Estrada "coverage ratio" as a complete and thorough solution for the time being while I think about all of this. For example:
1. Their utility or disutility of the bequest motive (via the ratio) seems odd here. In Estrada, the utility function is applied to the coverage ratio and "it" (the U function) is also, by the way, a type of split function, though not in the way I mentioned above. To the right of "1" (i.e., 1 means spending is "covered" every year of the planning horizon and >1 means spending is covered for more than a lifetime), it is a CRRA risk-aversion-driven concave power function with a positive first derivative and negative second everywhere on the interval. To the left of 1 it is a linear penalty function. I just don't buy this kind of thing (keeping in mind I am an amateur...). A monotonically rising utility of bequest by way of the coverage ratio >1 (to infinity) seems odd to me and, in the extreme, self depriving. This crashes against my intuition (if my intuition even allows utility in the first place!). My amateur intuition, rather, would be that I would separate the bequest from spending first and then try to land the "non-bequest-spending-consumption-portfolio" onto the number zero the day I die (i.e., that is where the coverage ratio would be = 1) rather than fetishize bequest forever all the way up the Estrada utility curve to a coverage ratio of infinity...even if the marginal utility is decreasing. My intuition would perhaps imply a utility function -- depending on how one does this and if we are still using the Estrada "method of U(ratio)" -- that peaks just before or maybe just after 1 and then declines. It'd be a hump with a peak. Then, if time were to be added to this discussion, utility might still be a hump but it would an earlier in life phenomenon over the range of t (and here I mean we are looking at the interval between age x and terminal L, whether L is static or stochastic) rather than later. I suppose this could be done with some type of quadratic U function. Alternatively, in Yaari ('65) he uses a weighting factor (keeping in mind this is a different utility of consumption over time idea rather than something with respect to a ratio). Here is the the exact way Yaari sets it up:
where the left term on the right side is the lifetime utility of consumption and the right is a weighted (the beta term) utility of terminal wealth/savings/bequest that would, I guess, weight itself lightly early in life, weight lightly late in life, but weight heavily in the middle (say around death or maybe middle age)...except that I have no idea where the weights come from. My point here is "the hump" of which Estrada has none and my intuition has a lot.
2. ALL of the last 5 or 6 or 7 papers I've read imply that an Estrada ratio < 1 is optimal where the utility of the Estrada "coverage ratio" rises forever above 1. Some of the points below will cover this but the basic idea is that a utility function that rises with a coverage ratio > 1 is at odds with other retirement research. Not at odds, really, it is a direct contradiction. I realize the methods are different here but the retiree outcomes aren't. Estrada is rewarding behavior that a ton of other research doesn't. Go read some of the papers here. As always, I might have mis-understood.
3. The "Coverage Ratio" idea at least looks like it ignores late age income and/or insured income. The reason that point #2 above is salient is that the reason that a depletion of wealth that is before death (or the planning horizon) is potentially optimal is that many retirees have income from at least some source. This income can even be implicit (e.g., family support or soc security not applied to the analysis) rather than explicit (e.g., explicit SS, pensions, annuities). The presence of the "income," whether pensionized or not, can create a medium-to-strong case for depleting wealth before death. Here is how I framed the idea of implicit income in a past post using Leung and Yaari for references:
The most important feature of equations (46) and (47) is that the former does not involve S* and the latter does not involve c*. This means that when insurance is available the consumer can separate the consumption decision from the bequest decision, in the sense that the saving plan S affects the consumption plan c only globally, but not locally. We have seen in Case B that in the absence of insurance such a separation is not possible. p 149My point here is that the "coverage ratio" is an income-less, insurance-less analytical tool where income, insured or otherwise, at the end of life seems to change the game by a lot and income seems to be a part of life's game whether it is admitted directly in a model or not.
4. Estrada and Kritzman should figure out how to again split their split utility function. Given #2 and #3 above as well as the work on WDT that I've referenced before, the coverage ratio might be salvaged a bit with not only a weighting feature related to bequests but also splitting the function again for ratios < 1 (if I had courage I would work all of this out but I have no courage today) . For example, an 80/20 coverage ratio (.80) would not be put into the E&K "utility penalty world" but it would somehow acknowledge that the 20% of the interval L that is not "covered" really has some utility related to the explicit or implicit income that might be available. I think the problem might be that once you go down this analytical path you might as well go all the way and just DO the full WDT utility analysis. That path would defy the "easy" idea in the title of this post, though.
5. Estrada and Kritzman Have a Fixed Horizon "L." This "L" thing should be randomized. This is, and seems to always have been, the weakness of an awful lot of retirement papers. I mean, if the coverage ratio is something that is easily discovered via simulation then it is not all that hard to also throw in a stochastic L that obeys mortality logic, whether from actuarial tables or analytic frameworks like Gompertz-Makeham. If an amateur like me can do it, then well....... But once this is admitted we are back to the last two sentences in #4.
6. There may be other "easy" ways to access the intuition on "coverage" (and WDT) -- other than using a coverage ratio -- that might be useful, too. I haven't thought about this very carefully yet but what I'm thinking about here is the process I went through to decode the Kolmogorov PDE for lifetime probability of ruin (LPR) via methods that, as a non-academic amateur, I could understand and access myself. In that particular case I sim-ed a net wealth process to extract the longevity of a portfolio in years for each sim-ed lifetime and then from that pile I extracted a frequency distribution of portfolio longevity. From that I could then infer a probability distribution of "portfolio longevity in years" for a given setup of assumptions. Of course when that pdf is joined with a conditional survival probability for age(x)[t:T] one gets a great ruin risk estimate that maps to Kolmogorov's LPR as it might be expressed in Milevsky's finite differences solution that satisfies the PDE. But my real point here is that I'm thinking that by backing off of the summary LPR "ruin percentage" and working with one of its precursors (cumulative probability of portfolio longevity...independent of lifetime process at this point) over an interval x:t (where x is current age and t is some planning horizon for the given age) could provide us as much or more info than "coverage ratio" and then also be useful in calculating ruin/sustainability, too. Just a thought. I'm not sure how I'd layer on stuff like utility and insured income but that's another day...
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