Quick and Dirty Experiment with Real Options on Future Wealth vs Spending Choice.

Goal - See the rough shape of a real option-on-future-wealth value for different spend choices

I'm dipping my toe into water here where the conditions may not favor my swimming skills.  But this is another "wanna see" post and I wanna see a net wealth process at some horizon as a real option with some strike and then to also see what happens to the value of the option as we increase or decrease the spending.  This seems pretty obvious. Spend nothing one has infinite optionality in the following periods. Spend 100% and it's all gone.  So I expect the option value to decrease with spending choice but how? What is the shape?

The shape might matter.  Spending is often "held in" by early retirees as both a hedge against residual uncertainty that is not model-able and also as a way to access future optionality on the potential to increase (and not have to decrease) lifestyle (or bequest) given the unfolding of exceptional returns. While we will say little to nothing here about either consumption utility (really low spend will have low utility; zero spend is infinitely un-useful) or risk, in terms of fail or any other metrics, it does seem worthwhile to think about optionality on its own terms.

The Models

For the net wealth process it is, as in previous posts, a recursive process that can be described by this

net wealth process

For the option model I won't be using Black Scholes or closed form math (because I can't). I'll access the intuition via simulation (because I can) by way of the sum of the probability-weighted present values of intrinsic value which I'll represent like this

R = ∑ { P(MT) * PV(max[0, MT - k]) }

real (call) option model

and where the rationale for this intuition can be found in an article by Grossman et al in "The Intuition Behind Option Valuation: A Teaching Note."  

To visualize the concept lets look at figure 1. 

Figure 1. Placing intrinsic value in a net wealth process wrt k

The option value will defined as the present value of the probability weighted net wealth above the blue line. k is a policy choice. Only one spend rate is shown.



Some Assumptions

For horizon I have only selected 20 and 30 periods. It would be useful and interesting to go longer (or continuously) but this is a spreadsheet sim so I'm keeping it simple. 

Return is an arithmetic (input) real .04 with a sd of .12.  This is arbitrary but not all that far from a blended portfolio.  

Spending I vary from 2 to 10 % to see what happens

Wealth (M) is framed in unit terms so that consumption is in 1 unit in the net wealth process. M is scaled to create the spend ratio. Option pricing is done in 1 unit increments. 

The strike in this post is framed as wealth at time zero in real terms but now at horizon H=20 or 30. This, no doubt, could be controversial. There were other choices. Zero comes to mind. Or maybe what the net wealth process would expect at H = 20 if returns were not random.  I just go with M(0). This keeps it simple and we'll perhaps just call it a "deep out of the money" call option which we are free to choose.  That makes the option the summed, discounted, probability-weighted intrinsic value of wealth wrt its starting point used as a strike level and below which the option will have no value. 

This is a rudimentary spreadsheet sim so I keep the iterations to 4000 so that the cpu does not overheat. 

The Output 

Figure 2. Illustration of option value of net wealth at tenure H for strike k

Discussion

1. This is fake model-world so its a little nutty to take the interpretations too far.

2. Clearly non linear and decreasing monotonically and asymptotically towards zero for higher spend rates.  As spending is decreased the option value goes up at an increasing rate...ignoring consumption utility for now. 

3.  As expected, longer horizons create more optionality due to the dispersion of the net wealth over more time, but...

4. For the longer horizon the value of spending reduction programs accelerates faster or alternatively the loss of optionality is faster at longer horizons as spending goes up...if I have that right.

5. A much longer horizon would be interesting to see.  As would be a look at different strikes like M = 0 or an annuity boundary or others...

6. The meaning, if any, of the absolute option value at the top of figure 2 would probably depend on a policy choice about, or even gut feel for, what I'll call thresholds[1].  For example, if H is 1 year, then having an option on 1 unit of M would be a big deal. At H=20, maybe it's higher...let's say 5 units "matter." At H=30 it's probably something else, say maybe 10.  If those assumptions were to be true, which I am not claiming, then for the portfolio expectations in the illustration, the thresholds would meet the "option value" at somewhere in the 4-5% spend range which is not all that nonsensical given a couple of my last posts. On the other hand that is really kind of arbitrary and coincidental. Any real (pun intended) interpretation I'll leave to the reader.

8. As mentioned, spending less obviously husbands optionality and portfolio fecundity but in terms of consumption we'd be looking at some serious disutility at some level the lower you go. Spending more obviously has less option value plus it adds to risk not modeled here. In addition we'd see peaking then declining utility as higher spending starts to impinge on lifetime consumption utility in the presence of a wealth depletion time. 

notes

----------------------

[1] This is the least thought-out part of this post I think.  The idea is that each horizon, 1 or 20 or 30, has some uncertainty about the horizon.  If H is 1 then being off by 1 which means running out of money at time zero, which makes no sense, so call H=2 or something, seems like a massive error and I'd probably pay some amount -- scaled to probabilities of some alternative universe where I have perfect knowledge -- to have an option to add some time at that point.  Likewise for H=20 I'd probably pay to cover even some more time, probably more than I'd pay (and for more time) than under the H=2 scenario. Same goes for H = 30.  Then, my thought is that there is probably some minimum level of optionality that one might be willing to pay for just to manage horizon risk. That translates, without sketching out the translation in any kind of detail, to an upper limit (in the absence of annuitization) on spending because just below that level there is still some "unit" of optionality that makes sense to the "buyer" (buyer by way of one's willingness to impinge on lifestyle in order to gain the option) embedded in the lower spend and above which it's gone.  My point above was that maybe for the given assumptions that "point" happens somewhere around a 4-6% spend rate (again, depending on the threshold) which seems about right. In this context the lower range of spending is less interesting when thinking about optionality. We already know it's there.