On the alliance between fail rates and household balance sheets

In a previous post I riffed, among other things, on my shift, over 10 years, from simulated fail rates to the household balance sheet. The latter comment implies, but did not make explicit, that my move was from an accounting balance sheet to an actuarial one, and from a deterministic or point estimate of spending as part of the A/L calc to a 'distribution' of spending via a stochastic present value calculation.  

What I learned later in my blog-career, and have posted on before, is that these concepts are not completely incompatible. In Robinson and Tahani (2007) they gave me the key even though I didn't (don't) really know the math.  Either way, let's set it up like this.  First, almost all simulation actualizes some kind of stochastic differential equation of flow and time. Simulation just happens to be a blunt-force tool of proles like me used to get to the same place with a little more customization and less academic ambition and maybe a little less transparency.  And some humility I might add.

The epiphany in R&T was that the probability of a future fail state in a net wealth process (keep in mind I don't really 100% believe in fail rates anymore) is basically the same as asking whether a balance sheet is "feasible" in year zero.  Feasible in this case would mean that one has enough assets at the beginning of a plan in order to defease the estimate of the liability for spending.  This is not dissimilar to the machinations of pension plans and endowments, btw. These are interesting topics. 

In R&T, they first set up -- correctly I think, and in a stark, healthy contrast to other papers on the topic -- a set of stochastic differential equations for the processes in play. Like this:

where the first line is an absolute wealth process in brownian motion terms, the second is consumption in it's Brownian motion stochastic mode (god bless them for doing that for consumption), and the third is the correlation between the first two. Re that last comment, they even -- omg I have made this point before and I have felt like the only person on earth -- described in a footnote that spending could be "perversely" correlated with a wealth process, ie if wealth goes down one might go on some kind of weird destructive shopping spree. That was exactly me in 2007-8. Don't be me.  [geeezus. It took me 10 years to even see what those equations are trying to say...not sure I even get it now]

The net-wealth process that these equations imply would look like this (and here we have stepped into the appendix):

Almost all papers on quant retirement finance will have a version of this equation to describe a stochastic net wealth process; it took me a decade to be able to see it. That equation. plus the others, esp A.2, if u are not aware, are very, very descriptive of the Monte Carlo simulation one might get from one's bank or financial advisors. Let's say they are the same for the sake of argument.  Now, R&T go on somewhere in their paper (read it so I don't have to spoon feed it) to make the case that mathematically these processes, the forward simulation ones and the backward SPV ones, are the same (via Ito calc in the appendix). That means that the stochastic present value of spending vis-a-vis the initial wealth is the same - sorta - as some net-wealth stopping time.   Here, btw, is the SPV of spending in R&T terms: 

where basically that expression is the present value of spending over a random lifetime and where C is consumption and R^-1 is a discount rate. On the right that is an indicator function that operates on is/isn't alive at time t. Then, because they are deft with their stochastic calculus, they proceed like this (in the appendix):

and so if u didn't get it, and I am assuming I did get it but u can correct me on this, that means that the probability of a future fail is ~the same as the probability (did I get that right?) of the initial wealth being less than the present value (stochastic or otherwise) of consumption.  I mean that's the math, anyway. I don't really think that's true in a fundamental future sense, because no one knows the future, but it's true enough for this post. That means that I don't really believe in a real world sense that forward sims and backward SPVs are exactly the same in my gut, I just think they are well affiliated based on A.4 and A.5. Plus, if I had to pick a favorite child, I might have to go with SPV for its manifest attractions: the presence, when integrating with a household balance sheet (HHBS), of the direct present observables like investment account values.  Sims at time = t, by contrast, have almost no truth, only conjecture. 

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Robinson, C., Tahani, N. (2007) Sustainable Retirement Income for the Socialite, the Gardener and the Uninsured. York U