Stochastic Present Value with a Floor

As I try to wind down my quant ret-fin stuff to focus on act IV of my V acts of life, I still, every once in a while, have that "I wonder what ___ would look like" moment. Today I wondered, without any real strong goal in mind, what a stochastic present value (spv) of spending would look like with a floor added in.   I had a guess but I wanted to see. In the end it probably doesn't matter since I don't manage a pension fund and as my ret-friend Ken Steiner reminded me sometimes simple is better especially when working with humans. In other words, if present value is a step too far for many retirees, then spv is an alien. But I still had that "wonder" thing.  

Basic Idea

The basic idea is that there is some component of spending that doesn't go away, some physical or psychological minimum that evaporates mostly with death, a floor that could, and in some cases is or should be, hedged out with a closely matched hedging asset. This could be a TIPs ladder or an inflation adjusted annuity (I hear the pure ones are now a magical unicorn) or even social security which I have not even touched today.  In modeling a stochastic present value that means, I guess, that one takes that floor out, it's the hedging asset plus the random spend = the PV distribution for "what is needed at time = 0" to defease the spending plan. I hope I have that right but it doesn't matter. I mean, if I were being paid for any of this, I would endeavor to be very very sure but I am not being paid so take all this with a grain of salt. 

The Math

If I were to throw some math at it, it might look like this 

where spv is our distribution of the estimated present values of spending given all the other parameters. T is some terminal age, say 120 or so, tPx is a conditional survival probability, c(t) is the cash flow in some period of life, a(t) is the annuity cash flow conditional on age x, d is the discount rate that is randomized, and A is the lump sum cost of a(t) at time 0. If I am off we can toss this whole post but I'll proceed anyway.  I kinda don't want to credit D Mindlin for this because he appears to be a bit of a pain sometimes in his written work but I will credit since I did lift the idea from his paper and his content seems otherwise solid. He doesn't do the survival probability thing, though. That's entirely on me. 

Some Parameters

The annuity I priced via this 

But then I double-checked with immediateannuities.com to make sure I was right which is a really weird labor-intensive way of just going to immediateannuities.com for a price in the first place. Idk, I just like to do it myself, I guess.  I was off by a couple hundred dollars so pretty close. 

I am working in real terms so inflation in this spv post is "out." Returns are real as well. Cash flow is a base rate of 40k since every paper in history seems to use 1M in endowment and 4% in spend although there is no endowment here just the spend and it's distribution...to compare to some theoretical set of assets that'd be required to fund the lifestyle.  Age is 62 (me, heh). Discount rate is .04 with a std dev of .12 (arbitrary but whatever, we are just looking for now). Survival probability is calculated via 

where I set x = 62, m = 88 and b = 8 so somewhere in between a SS life table and a SOA annuitant table I think. Spending was not randomized within individual periods though it could have been. Max age was 120. Iterations = 10,000. The variable a(t) was set to 0, 10k, 20k, and 30k just to see what would happen. The value A is derived from a(t) and you can check immediateannuities.com to see roughly what I came up with. 

Lets Fire it Up

When I set the above and then animate it in a simple R program, the distributions look like this: 

and some summary stats since "1 number" is never enough: 

What Does it Mean?

Not sure. I hadn't planned to go into interpretation today.  Mostly the info is in the tails. Of course if one hedges out a portion of spending, the left tail comes in a bit because it is more certain that that is what one will spend.  But the left tail is boring; the right tail is more interesting. Especially, for me, the 95th percentile which is very very close to "fail rate" thresholds used in some Monte Carlo simulations. That tail comes in too, meaning what? I guess that one does not need as big a pile of cash now to dispose of the liability given the uncertain returns plus the certain hedge. Or, alternatively, that the certainty of defeasing the spend plan goes up. But then that is the whole point of annuity hedging. I think. Right? I probably need to play with the other parameters...but won't.