This is an exploration with untested and un-reality-checked ideas on "retirement as a real option." This is just for fun and self-learning and the conclusions are going to be thin. Skepticism is warranted.
What feedback I do have from this effort looks a little similar to other work I've seen that tells me that, up to a point, holding a risky portfolio has some option value that is accretive to undefined bequest plans and maybe to an undefined and unscheduled decision on when (what age) to eliminate risk in favor of something else...like lifetime income. But that is a pretty indirect inference. And also well known already.
The Point of this Post
In a past post I tried to look at retirement as a "real option" by using an annuity boundary as a strike price. This was in order to do (via simulation, not Black Scholes) an amateur reproduction of some work by smarter ret-fin guys than me so that I could learn something new, especially about the benefits of not annuitizing "just yet." In this post I wanted to look at the idea again but now without the annuity boundary and this time with some variation in spend rates and return assumptions. In addition I wanted to also try to discount the intrinsic-value results at expiration with a conditional survival probability. The goal here, by the way, is "self-learning" and just to "see what it looks like." At this point all of this is pure play and I'm not sure if my assumptions and approach are on solid enough ground yet to make any conclusions if they are even meaningfully interpretable in the first place. More on that later.
The other reason I'm here on this is that so many of my posts seem to focus on the negative: sequence risk, ruin, wealth depletion, disutility, mortality, etc. Dark stuff. It's easy (for me) to forget that one of the many reasons to hold a risky portfolio -- and not 100% bonds or life income annuities or TIPS ladders with a side portfolio -- is that it represents a big upside option above and beyond providing what it is supposed to provide: reasonable probabilities of funding consumption. It is an option to get access to additional lifestyle growth, bequest motives, etc. that might otherwise be foregone. That means the option has value, up to a point, in the context of finite, random life. This is a second attempt to try to see and understand the process and "the point." My guess is that it'll take a while and more than this preliminary and flawed post for me to get there.
The Model
Net Wealth Process
The model starts with a simple net wealth process (NWP) that runs from time zero to an "expiration" at time T. The formal form for this NWP might look like this stochastic differential equation [1]:
 |
| real net wealth process - formal |
Where W is wealth in time t, c is fixed real consumption per unit of time, mu and sigma are portfolio return and standard deviation respectively, Bt is a Brownian Motion process, and w is initial wealth. The drift term can hit zero, btw, so not quite "classic" GBM, I hear [2]. I can't personally handle SDEs so in simulation mode I have the equivalent to the above equation that looks more like this, in non-standard notation:
Wt = µW(t-1)-1, µ≈N[rp,σp], via simulation
real net worth process - casual
real net worth process - casual
noting that this process is rendered here in wealth units rather than dollar values.
The Options Model - Baseline
Black Scholes is more or less inpenetrable to me so the model in this post uses simulation to get to the same place. The insight comes from "The Intuition Behind Option Valuation: A Teaching Note" Grossman et al, Dartmouth and Haskayne school of Business. Friends at NewFound Research sent this over to RH earlier this year. From that link, with some mods to synch it up with my notation we have this: "the option value is the sum of the probability-weighted present values of intrinsic value" represented by
C is a call option, WT is the wealth at "expiration" (this is typically a stock price and is so in the link) coming out of the simulated wealth process from above, and P[WT] is the probability distribution extracted from the sim for the end wealth states. k is the strike. This was instantiated in a rudimentary spreadsheet model with an entirely inadequate 5000 iteration runtime framework.
C = ∑ { P(WT) * PV(max[0, WT - k]) }
C is a call option, WT is the wealth at "expiration" (this is typically a stock price and is so in the link) coming out of the simulated wealth process from above, and P[WT] is the probability distribution extracted from the sim for the end wealth states. k is the strike. This was instantiated in a rudimentary spreadsheet model with an entirely inadequate 5000 iteration runtime framework.
The Options Model - With a Longevity Discount
This possibly unnecessary complication takes me down an interpretation and meaning rabbit hole. The idea here is that at expiration there is a probability < 1 of still being alive to enjoy the fruits of the option. I wanted to use a conditional survival probability to not so much "weight" results as to discount the value of long expiring options for that risk of not being alive. Is this legit? No idea. It certainly ignores the concept of estate planning. But I'm going to do it anyway. In correspondence, Prof. Milevsky had an emphatic yes for my question on whether to throw it in there and told me this kind of thing had been considered by Boyle and Schwartz as far back as 1977 [3] but that paper is a little dense for me and I'm not sure if what I am doing is similar or being applied in a reasonable way. It's a start, anyway. I used to call this kind of post a game. Let's call it that.
In the model (game), at each expiration, the intrinsic values calcs are probability-weighted and PV-discounted and then they are discounted again for a survival probability for a person age x at time T (expiration) and then summed. I use SOA IAM mortality data for longevity probabilities. Then some algebra gets us to this expression which I think is correct or at least correct enough for now:
where RC is what I am calling a "retirement real option (call)," TPx is the conditional survival probability for a person age x at expiration, k is the strike, W-k is the intrinsic value, P(WT) is the probability of WT extracted from simulation, d is a discount rate, and T is the tenure or expiration date.
Other Model Assumptions
In the model (game), at each expiration, the intrinsic values calcs are probability-weighted and PV-discounted and then they are discounted again for a survival probability for a person age x at time T (expiration) and then summed. I use SOA IAM mortality data for longevity probabilities. Then some algebra gets us to this expression which I think is correct or at least correct enough for now:
Other Model Assumptions
Age 60, wealth is in wealth units (25 for 4% spend, 20 for 5%), returns are in real terms 1->5%, sigma is .15, the discount is a real risk free rate (I used 2 to exaggerate results; this is a flawed assumption), strike prices as described below, expirations are at each age from 61->95. The model is run in limited sets of spendrate[4,5] & return[1:5] because it was tedious to do more.
The Strikes
Building the model was easier than figuring out what to do with the strike price. And I still don't think I'm even close. The rabbit hole here is even deeper and more topsy-turvey than when I was adding survival probability. I'm not sure the way I did it will stand scrutiny but I wasn't sure what else to come up with for this preliminary effort. The original intuition was to use zero for a strike thinking that 0 is a level for ruin and we want optionality above that . But I didn't like that. The problem there is that if I am 60 I would not use a strike of zero for an age 61 expiration based on the expectation that a wealth process under no risk would not go to zero in one period. I want to price a call that represents the upside of taking risk but without paying for optionality on wealth I already/still have.
So for each of the 10 sets (spendrate[4,5] & return[1:5]) I ran a separate NWP with vol set = 0 [4]. That represents an hypothetical and idealized NWP run at that return and spend rate but with no volatility whatsoever. What is that idealized NWP, really? Unknown! But it maybe represents the lower boundary of decreasing volatility (let's call that risk for fun) and above which exists the optionality of taking risk. Right? Isn't it? I think so...maybe... Then, for those sets and NWPs that tend to head towards zero, I set a min strike of 1 wealth unit. That was arbitrary but it made sense at the time when I made it up. Could be zero, could be 10. I'll have to think about this some more the next time I do this. Charted out for the 10 test sets, the strikes could be represented like this:
Note: this is where I could use a little reader engagement (except for David who is almost my only correspondent). It's entirely possible that I have totally lost my mind on this and have detached from reality if not merely financial theory. Could always use a little feedback. I have no real academic or professional background on this kind of thing and no access to professors, assistants, or colleagues. Splendid isolation as they say.
The Output
When I do all this above (using the longevity discounted version) for the 10 sets, this is what I get:
Possible Interpretations
Uh, well, this was fun and all, and not too terribly difficult, but I have no idea how to interpret this. I thought a good long while on this and then punted. When in doubt on retirement-quant questions, where else does one go? Prof. Milevsky, of course. I asked him how I'd interpret the "retirement option." His response was quick: "it is the hedging cost." I think he's right but I also think, if I am not mistaken (and I may be), that I didn't tell him that I was working the call side and not the put side. So that means I have to come up with something on my own for my private alice-in-wonderland world of retirement that I've built. Or figure out if I'm wrong.
My best guess goes something like this... This feels like the other side of the "hedge cost" idea. If there were idealized return vehicles with no vol, this option value might represent some kind of an opportunity cost thing of sticking to that no vol approach vs taking on something like a little sequence and (even more) ruin risk in exchange for the upside. It might also represent the opportunity cost of "open-ended directed-bequest potential at expiration" in exchange for taking on some risk. It might also represent a decision framework where at, say, the early expirations we might still be able to rationalize "rolling" the option into the future expirations but where also, at some point, we wouldn't. That'd mean what at that point? "Exercise" and then go to cash...or annuitize maybe? Maybe. The analysis certainly shows, to my eye anyway, that we buy bequest optionality and "mid-late age roll" value by spending less early.
Conclusions
Conclusions would be quite risky at this point. The most I can say is something like this:
My best guess goes something like this... This feels like the other side of the "hedge cost" idea. If there were idealized return vehicles with no vol, this option value might represent some kind of an opportunity cost thing of sticking to that no vol approach vs taking on something like a little sequence and (even more) ruin risk in exchange for the upside. It might also represent the opportunity cost of "open-ended directed-bequest potential at expiration" in exchange for taking on some risk. It might also represent a decision framework where at, say, the early expirations we might still be able to rationalize "rolling" the option into the future expirations but where also, at some point, we wouldn't. That'd mean what at that point? "Exercise" and then go to cash...or annuitize maybe? Maybe. The analysis certainly shows, to my eye anyway, that we buy bequest optionality and "mid-late age roll" value by spending less early.
Conclusions
Conclusions would be quite risky at this point. The most I can say is something like this:
- Holding "the option" maybe makes sense, depending on expectations about forthcoming returns and stable spend rates, up to a point and it looks like, without over-interpreting, that the "point" somewhere in one's mid 70s to mid 80s is where that holding game gets less interesting. On the other hand, this concept is well known already.
- Spending less early enhances optionality (for bequest, for a later age to no longer hold the option). But, again, this is not only known but common sense.
- As in a point I've made past posts: return (i.e., we are indirectly talking about allocation choices here...though we have not really explored vol effects yet) matters (of course) but spending looks like it matters more.
- The "annuity boundary" concept (not discussed here) still makes more sense to me. Once wealth falls through that level, the opportunity for lifetime income is foregone maybe forever. That seems like a big deal.
- Whether any of this type of thinking represents anything better or more interesting than a Floor and upside approach (depending on resources) is an untouched issue. TBD...
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[1] this is from "Retirement Needs Framework -- Is Your Standard of Living Sustainable during Retirement? Ruin Probabilities, Asian Options, and Life Annuities", Milevsky and Robinson. 2000(?)
[2] My sometimes minor issue with the econo-blogo-twito-sphere is that this equation is the beating heart of retirement and the implications get forgotten or brushed off. All those smart portfolio wonks forget the "c." They forget that "c" exists at all or that it is what (a) makes portfolio management a multi period problem rather than single, and (b) can make wealth go to zero or negative.
[3] Equilibrium Prices of Guarantees under Equity-Linked Contracts Author(s): Phelim P. Boyle and Eduardo S. Schwartz Source: The Journal of Risk and Insurance, Vol. 44, No. 4 (Dec., 1977), pp. 639-660 American Risk and Insurance Association
[4] Actually it was more of a limit as sigma->0.
[2] My sometimes minor issue with the econo-blogo-twito-sphere is that this equation is the beating heart of retirement and the implications get forgotten or brushed off. All those smart portfolio wonks forget the "c." They forget that "c" exists at all or that it is what (a) makes portfolio management a multi period problem rather than single, and (b) can make wealth go to zero or negative.
[3] Equilibrium Prices of Guarantees under Equity-Linked Contracts Author(s): Phelim P. Boyle and Eduardo S. Schwartz Source: The Journal of Risk and Insurance, Vol. 44, No. 4 (Dec., 1977), pp. 639-660 American Risk and Insurance Association
[4] Actually it was more of a limit as sigma->0.
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