[revised] On the alliance between fail rates and household balance sheets - Part 2

I thought I'd take this past post [On the alliance between fail rates and household balance sheets] a little further.  The post-theme here, in case you want to bail out now, is on the affinity between MC simulation and stochastic present values in some quant terms. My goal here is not really to explain anything to anyone but more to try to consolidate something I didn't understand very well. This self-consolidation may nor may not interest anyone but me. 

Since I started this whole RH enterprise in 2012-ish on the back of what I call pique-fear-curiosity by way of ginning up my own Monte Carlo simulation (MC) in VBA in Excel in order to poke my finger in my banker's eye for trying to charge me 4k -- it ran for 3 hours and locked up my PC, btw -- I've always been interested in the MC construct (though later I mostly abandoned it for better, faster tools).  Then, last year, I got even more interested in the idea of the mathematical "inversion" of MC sim. As a naïf I just assumed that MC was a one way journey -- project out from now to the future (and believe the results, btw) -- and also that it was impossible to un-ring a bell (true in real life). But a year or two ago I saw that in the ordered world of math (maybe less so in real life) it was entirely possible to invert MC and run time backwards. Here I am not necessarily thinking of backward induction though there is that. I am thinking rather of the trick I saw in Robinson and Tahani (R&T) of working a diffusion process backwards towards present value analysis in a way that creates a strong, maybe direct, affinity between the two constructs of present-value and MC-simulation, kinda like this:

Current Assets > SPV[Liability] at t=0   <~>  MC simulation P[W > 0] at some time T     (0)

where SPV stands for stochastic present value and MC for Monte Carlo and the tilde between the arrows means I don't 100% buy eq(0) for reasons I'll try to explain later. 

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Here is how R&T set it up for later discussion.  They assert the following:

  (1)

which is a Geometric Brownian Motion (GBM) diffusion process for cumulative returns or wealth, and where  

 (2)

is a separate GBM diffusion process for spending with a correlation created between (1) and (2) so that spending will change with changes in wealth -- although R&T, exactly like me, make the odd case that spending can be both sensitive to wealth in the conventional sense (wealth down, spending down) and perverse (wealth down, spending up ... which I did in 2008). Alpha here, btw, is a drift-of-spending term and Beta is a spend volatility coefficient and rho is the correlation between spending and returns so Beta of zero is deterministic consumption and Beta > 0 is stochastic.  

So far so good but a couple notes here, first, though:   

• It's nice to see this formulation, and far be-it from me to question academics, but I think this is an academic confection for convenience which I appreciate but it's not really realistic imo.  A fully diffuse spend, with our without correlation, is weird.  Personally I think that considerations of wealth levels, relative status, propriety, conservatism, habit, etc will cause spending to self-correct more often than not, something which is not all that explicit in (2) and might be hard to express, though AR(1) processes might work. TBD, but spending is not really an open diffusion process proper in my opinion with or without the correlation. 
  
  
• I like the R<->C diffusion and correlation here because I rarely see it. Some context, tho for fun: a) Blanchett, in LDI Misapplied postulates the idea of portfolio design anticipating spending (liability-relative optimization) and conforming the portfolio to its related liability undulations, b) a fully responsive spend (% of portfolio) which is portfolio-longevity enhancing but subjects one to severe lifestyle volatility is seen rarely in academic papers but is often discussed elsewhere by amateurs like me, and c) degrees of habit formation (or spend rules, dare I say or the formal academic correlation in (2) ) exist as well (See Mirlees 2000). These "rules" thread the needle between the portfolio and spend undulations (I wrote on this here once). All of these (a-c) are playing with the same idea in a way if one squints carefully. Cooperation between the two processes (R and C) is usually better than not but the extremes (no cooperation and full cooperation) seem to be problematic at times. 

R&T continue later, esp in their appendix, and combine (1) and (2) above into a net wealth process like this:

 (3)

where C(t) is probably really a net consumption construct C(t) =  C*(t) - π(t) and where pi is some kind of income process after retirement (part time job, soc. sec., pension, annuity, etc).  There are other ways to present (3) but this works for now. (3) is, in differential equation form, the net wealth process one would see in a MC sim except that the sim is likely to be quite customized for things like non-normal returns, spend rules or shapes, taxation, etc.  It's just that (3) is more transparent than the typical MC sim black box and a formal way of putting it. 

But this is where it got interesting (ok, I'm maybe a little odd, right?), and keep in mind I'm pretty much just copy-pasting R&T here...  They continue:

"Using Itô’s Lemma, we can easily show that:"

 (4)

Now, I don't know Ito's lemma from a hole in the ground but this is clearly wealth at some time t as a function of (cumulative) returns in t and wealth at zero net of PV(consumption) based on cumulative returns from zero up to just a tiny bit before or to t; that I understand. They, R&T, go on to say that (4) [this is my numbering system, btw] " Solves the SDE [in (3) and that...] It is obvious [heh] that:"

 (5)

where here I just flipped their (R&T) eq left to right to conform to my interpretation of (0) where, note, (0) is in success terms rather than fail.  To finish out R&T, they go on to say that "This shows that the probability of ruin prior to some time T is equal to:"

 (6)

Ok, so I couldn't Ito-Lemma my way out of a paper bag but I gather this says that the probability of a net wealth fail in the future is more or less the same as a feasibility-test-fail now (starting assets are less than some percentile of stochastic present value of spending at the beginning).  

Cool. But I still needed to understand how this works since I don't have the stochastic calculus (yet...I was thinking of hiring an intern or tutor off the internet). BUT, then, I re-read Dimitri Mindlin's paper on The Case for Stochastic Present Values where he laid all of this out in more arithmetic terms that I could see. Let's follow that idea down his rabbit hole...

Here, let's focus on the Appendix, Proposition 1 of Mindlin 2009.  In Proposition 1, which I will try to translate, he puts it like this "if a random variable RA [required assets, which I will say is w in (5) and (6)] is defined as 

 (7a)

where, in his terms, B is a pension benefit outflow (what I called C) and C is a contribution (which I don't have in my retirement unless you want to paypal me) and R is a series of returns (vs the cumulative thing in R&T).  So, to put this in terms friendly to (1)-(6) we might say this instead:

 (7b)

where C is period consumption, pi is period income, r is period returns, spv is the stochastic present value of consumption and T is the horizon.  Same thing, right? 

He defines something he calls A(k) as the market value of the assets as of the beginning of year k (1<=k<=N+1) so if we revise the notation and flip it around to conform to (0) we could say that he asserts that: 

(8)

(beginning of year so I'd typically call it W(0) = w but maybe I need to be more careful here...TBD; to follow his convention It'd more likely be: W(T+1) >= 0 and W(1) = w I think) and all of which is the same as (5) except we are framing this as success rather than fail and praying that we got the transformation of his notation right which I think I sorta did.  His proof, if I can again transform it correctly into the notation I was using above, goes on like this below but first...let's assert that his proof depends quite a bit on a recursion like this where I am using a 3-period example for this post in my own notation (I might have to change this to W(2...4) and t=1 to T+1 to match his notation...TBD This is why I call myself an amateur): 

(9)

So, while I'm not totally sure on the period zero thing, the recursion is right and in the end this is the same as (3) where r would be a random draw. Mindlin works his equivalent of (9) backwards and also in probability terms like (8). See note [2] for his proof.  My (8) is, I hope, his version here (10)

(10) 

He walks this backwards in Note [2] so let's follow along in my revised notation where I still think I need to adjust the periods; for now I will use his time convention so I don't get confused. I'll come back later, right? heh. 

 (11)

Here (11) is the recursive step from T to T+1 where W(T+1) = f(W(T)). Add some algebra and then backward again...

=(12)

and here (12) is another recursive substitution: W(T) for what created it

=(13)

A little algebra in (13) with the elision on the end which is important to understand how he continues to work this backward all the way to the beginning in (14)....

= (14)

Where the right side is the stochastic present value of consumption which I have simulated before. We end up then with (15) with an asterisk for my presumed mishandling of time and which takes us back to (8):

=(15)

the right side of which (15) I have simulated in the past while describing it as in (16) below:

(16)

except that I weighted consumption with a conditional survival probability tPx and where the time issue I was worrying about becomes moot at about age 115+ since the survival probability approaches zero anyway.  Also, were I to do it again I would start modifying along the lines implied by note[1].

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All of this is a long way of saying that (0) is not a silly proposition in a nicely-ordered math world and that a feasibility check [(15), left side of (0) (5) (8), and the right side of (6) and (10)] is a fairly strong form of retirement analysis. But there are reasons to question the nicely-ordered assumption and therefore to question the post entirely.  Here are some thoughts that undermine my own post: 

1. The right side of (0) is a total fiction. It is based on a fake projection into the future that is likely to never happen except by accident. The left side, on the other hand, has the grace to at least be half-true by which I mean that the assets at t=0 are exactly known. This is why some ret-fin quants love spv or feasibility tests and I am among them. They call it an "observable" and this deserves serious consideration.  I defer to folks like Patrick Collins in work not cited and to Ken Steiner at HowmuchcanIspendinRetirement.com.  
   
   
2. How the spend is modeled out into the future just to be discounted back to spv is not addressed here. Is there random inflation or spending "noise"? any mean-reversion by way of AR(1) processes or self-correction or something else? [not here] Are there "shapes" of spending as in R&T which are quite a dominating feature in a retirement analysis? [not here] Any jumps or chaos? 
    
3. There are layers of spending here, some of which are impervious to change especially on the low or minimum spend side. How are these spend flows modeled and discounted? There are minimum levels of spend that are likely to never disappear and deserve either a zero discount or some kind of separation [1]. Other spending is relatively impervious to risk even with (2) considered. My guess is that different discounts apply to different layers, something to which Mindlin 2009 alludes more than once even though the discount is also connected to the reference/policy portfolio expectations.  I told someone once I think discount rates are more art than science and I still think that is true for reasons in this point. 
   
   
4. We are discounting liabilities, not assets, here. Mindlin says in one spot that they (liab) should be discounted (random draw style) at the policy-portfolio expectation. Ok, but let's caveat point 3 above first and then consider this: when we discount assets at a high rate the PV is lower and this lower asset level reflects, I think, a bit of the difficulty of funding goals; when we discount liabilities at a higher rate (that reflects a policy portfolio) the liability then reflects, counterintuitively, some relative ease at funding the goal. This is nonsense I think unless you have a good objection or if I am misunderstanding...which I might be. This, in my mind, votes for a nudge towards lower discount rates like a pension analyst might do using treasury or more certain rates. We are after all not really discounting an asset-pile here (except abstractly via some policy portfolio...maybe) we are discounting a spend. I will admit as an amateur I don't really get all of this yet, but still...
   
   
5. There has been no talk in my post of longevity yet except as in (16). I thing Mindlin touches on this a bit but me? I'd slap a conditional survival probability onto the spend stream first like I did in (16). 
   
   
6. There is no discussion of exogenous comet strikes from outside the model. This is not really necessary but my point underlines that spending risk can come from ignoring extreme spending randomness and the layers of different types of spending risk we don't understand.  TBD in some other post. 
   
   
7. In Blanchett, he discusses CVaR which is a useful overlay -- untouched here - on spending (or markets for that matter) risk. For much of the distribution of spend risk implicit in spv, if we believe the distribution, we assume that the retiree, ex-new-income or contributions, can recover via the natural "return engine" of the portfolio. But if the horizon is too short and the hit (from markets or spending as in point 6) too big, the only source of recovery or resilience is outside money (jobs, family, gov't) or redundant capital (work longer, lower spend rates, etc) This is for another post on Extreme Value Theory (EVT) but should be noted here. 
   
   
8. This post used, in the main, a paper that was really designed more for pension analysis. Pensions and retirements are pretty analogous in that the liability side is very important and deserves center stage because that is in some sense the whole point of funding a retirement.  On the other hand, Blanchett 2017 points out some key differences, for example 

• Large insurance companies and DB plans have large numbers working in their favor that washes out some idiosyncratic risk and in some respects allows them to create relatively simple models. Retiree cashflows are more complex and can be hit with weird stuff that is hard to model
• For retirees, the liability isn't always a true liability but rather an "aspirational target" that can be adjusted as circumstances change
• Risk aversion can slide into behavioral issues
• Blanchett: "individual retirement cash flows can be expected to be more volatile than those for a large pension fund, they need to be modeled using a higher discount rate." [as an amateur I find this somewhere between dubious and odd as in my point #4. Maybe I just need to spend more time...]

Notes

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[1] On page 16 of Mindlin he did something I didn't notice the last time I read this a few years ago. He takes the certain spend out of the formula and basically turns it into an annuity pmt at what I gather is a 0% real discount. That means that the RA (his term; my spv) is the sum of a purchased annuity and then a stochastic present value of spending less what the annuity cash flow would be. This is sort of like (7b) but his eq is this

where L is the annuity value at t=0 and F(k) is the certain annuity cash flow. I like this and will use this in the next version of my SPV software. 

[2] His A(N+1) is my W(T) and his RA is my spv and his A(1) is my "w." If you can't see that, then all is lost going forward.  

(11)

He is walking all of this back from the end-state N+1. The first line leans on the recursion I reframed in (9), the third line leans on an elision that you need to understand, and the last line in (11) is really (10) which is really (8) and proves (5) ~ (6) = (8) = (10) assuming I didn't miss anything.  

References

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Blanchett & Idsorek, 2017 LDI Misapplied: Income Portfolios and Liability-Driven Investing, Morningstar  

Diamond, Peter and Mirrlees, James (2000), “Adjusting One’s Standard of Living: Two-Period Models,” in Incentives, Organization, and Public Economies, Papers in Honor of Sir James Mirrlees.  

Mindlin, Dimitri, 2009, The case for Stochastic Present Values. SSRN

Robinson and Tahani, 2007. Sustainable Retirement Income for the Socialite, the Gardener and the Uninsured. SSRN