Option value of not annuitizing? - another amateur hack attempt

One of the main themes of RH has been that an amateur, armed with nothing other than curiosity and the willingness to embrace some basic math and technology, can gain at least some minimal insight into their otherwise fairly complex retirement problems without the intermediation of "experts" such as potentially conflicted and maybe under-trained financial practitioners or remote, opaque, ungrounded, abstract, and non-skin-in-the-game academics.  I've used that approach often here to get some fairly certain understanding of some core real-world processes that have an impact on retirement.  But maybe here, today, with this post, I have extended myself a little too far out over my skis.


Today I wanted to try a really rude (in the "roughly made or done; lacking subtlety or sophistication" sense), flailing, amateur hack on the option value of not annuitizing all of one's wealth immediately.  It seems like there are plenty of reasons these days to avoid or disdain annuitization: to keep control over one's assets, support a bequest motive, low current interest rates along with high insurance loads make the cost too high, the benefits of risk pooling are underappreciated or misunderstood, full annuitization forecloses on reserves for spend shock risks or variable spending, there might be incomplete markets in the sense of unavailability of products that meet customized needs, etc. etc.  Another reason, according to Milevsky and Young (Optimal Asset Allocation and The Real Option to Delay Annuitization: It’s Not Now-or-Never, 2002) is that there is underappreciated optionality due to the irreversibility of the annuity choice: "we argue that any multi-period framework that ignores the irreversibility of the life annuity purchase does not capture the option value in waiting."  In fact, this point about optionality is maybe a more general problem of financial analysis as a whole, a discipline that perhaps looks too closely and too often at single point NPV choices.  Though things have changed a bit since 1995 here is how Stephen Ross put it back then (Uses, Abuses, and Alternatives to the Net-Present-Value Rule, Financial Management, Vol. 24, No. 3 (Autumn, 1995), pp. 96-102):

"Every project, whether or not it explicitly contains options, always is an option on the movement of capital costs and prices...We must take very seriously the caveat to the NPV that it applies only in cases where an investment does not preclude some alternative investment, because every investment competes with itself delayed in time...when evaluating investments, optionality is ubiquitous and unavoidable. If modem finance is to have a practical and salutary impact on investment-decision making, it is now obliged to treat all major investment decisions as option pricing problems." [emphasis added]

This is an interesting idea when hitched to the annuitization problem and this is also the idea that M&Y undertook in the link above.  So, while continuous form math is mostly beyond me and a relatively simple application of discrete versions of Black-Scholes might be possible but is also beyond me today, simulation approaches to the option pricing problem, on the other hand, aren't beyond me and they open some doors to taking a look at the problem that I might not otherwise have been able to open.  The question then is: can I make some kind of blunt force plausible conclusion here about the option to not annuitize that, in the most general of terms, comports with expert opinion, even if it is by accident or dumb luck?  Maybe I am just embarrassing myself! We'll see. This is one of the advantages of having microscopic blog readership numbers.  If I fall on my face, few are looking.  

Disclaimers and Sandbagging

One of the things I appear to be especially good at lately is hedging or sandbagging so let's get that out of the way first.  

• I can't read or absorb or use the continuous form math of others that have already attacked this problem which limits what I can do,
• There are definitional problems.  I have not defined my terms carefully or at all and I have not looked too closely at what they can and should mean,
• I have limited understanding of the various economic relationships in things like interest rates, risk free or otherwise, and inflation, among other things,
• I have ignored things like: taxes, bequests, utility theory, mortality asymmetries, insurance loads, non-constant spending, a nuanced view of the differences between american and european options, and so forth, 
• There are no doubt unexplored methodological flaws.  The way I am approaching this is terribly naive and may be fatally flawed,
• My approach to the net wealth process and drift is pretty stylized and abstract which is easy to implement, tractable, and transparent but probably limiting. Also since this is in wealth units and uses real returns, I got my head twisted around in confronting the issue of discounting, PVs, annuity pricing and inflation. One might want to assume I stumbled in there somewhere,
• Other, because there is always other.

In other words, what remains below is like sausage making when making sausages with rotten meat. Not only should one not look too closely at how it is done but one also has to ignore the smell. 

The Setup and Simulation as a Proxy for Option Pricing

Thanks, btw, to Corey Hoffstein at Newfound for sending me the links to material with which I could tutorialize myself on this subject.  While formal BS math is mostly impenetrable to me and even the simplified versions require a supple mind to directly interpret the normal distributions N(d1) and N(d2), simulation is an easy way to access the intuition on the underlying problem.  For example I lifted a simple BS options pricing example from a Stern school work on Options pricing (chapter 5 Options Pricing Theory and Models) and was able to reproduce the formal BS result to within a few cents via a simple spreadsheet sim. In over-summarized form it looked like this with the equation on the x-axis being the option pricing mechanism:

I'll use the same approach for the annuitization question here. The term or tenure will be in increments at each future age for a 60 year old 61:90. I'll simulate wealth outcomes using a simplified net wealth process in wealth unit terms like this: mW - 1 where mu is the drift, W is the portfolio in unit terms from the prior period and 1 is a spend; dispersion comes from the simulation and randomized m. This net-wealth process has no weights related to longevity since it is independent of that. The drift m could maybe be anything between the risk free rate and a plausible net real rate for a reference portfolio. The BS example above uses the risk free rate as it should. Since I did not want think too hard about the relationships between things like Rf and inflation and a usable drift (when in real return wealth-unit mode) I just picked three .01, .02, and .03[1]. The standard deviation is a faked .10. The strike price, and this is where I might have also gone wrong, is variable depending on the tenure. I set the strike equal to an annuity price for the remaining consumption at that tenure age based on weighting by a conditional survival probability for that age from the SOA IAM table with a G2 extension. Discount rates are, well, they are a bit fuzzy to me; see bullet six above. Since we are in wealth units using real rates and we have a constant spend of 1 that is the same today as it was yesterday and will be the same tomorrow I feel like the discount has already been applied or at least with respect to inflation. That is unsatisfactory but in this "draft 1" of this post I'm going to go with it. I will take feedback by the way. I am easily coached.

The Experiment 

To summarize, the basic idea that I'm experimenting with here is that: 1) an annuity price for constant consumption out at some tenure t might be considered a "strike" price, 2) the simulated distribution of future net-wealth values could be used as a weighting mechanism, and 3) the sum of the differences between the net-wealth value and the "strike" above the strike (intrinsic value below strike = 0) when probability weighted and discounted might be considered an option value. The question then might be, if I got it right and have not embarrassed myself yet: what tenures have the highest option value (before it starts to decline)? I am hoping this would answer "how long might I wait" (in today's terms, anyway) or maybe "at what age does the value of the option to not annuitize start to decline (in today's terms)?" Does any of this make sense? I hope so. I haven't really convinced myself yet that this is a legit way to do it but if we can suspend my disbelief for a moment, the result would look like this:

This kinda breaks my common sense since it feels like the option value should be high today and declining over time but I am guessing that since we are really looking at different tenures for a fixed age today, rather than some life process, the optimal age/tenure needs to be the focus of the chart. I think that if I were to animate this rolling forward for different start ages or use continuous math my guess is that we'd see the former intuition play out which is in fact what I think you see in the Milevsky and Young paper if I am reading it right.

On the other hand this chart -- whether by accident, luck or design -- gives insight that is plausible and referenced elsewhere by others. The common wisdom that comes from people more serious and rigorous than me is that mid 70s to mid 80s is roughly the range for optimal annuitization. Depending on one's assumptions about the drift term -- and here we might lean towards the lower values to stay closer than not to options pricing theory to the extent that I understand it -- the chart gives answers that appear to be in the same range as the received wisdom. Also, and this is probably just by happenstance, the lowest of the three drift parameters gives a peak value at a tenure around 76 to 78. In the hyper-sophisticated M&Y paper, the optimal age for annuitization (male, since I am using male SOA data for me) looks like it is somewhere between 73 and 80 depending on a risk aversion parameter between 1 and 2. For gamma=1.5 the optimal age is 76. I think, without looking, one of their tables also used age 60 as a base. Maybe I am lucky like a broken clock that gets the time right twice a day but at least my numbers look about right here. If I really am either naive[2] or dead wrong I'll post whatever someone wants to send me and then either pull my post or print a bigger disclaimer at the top.

   

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[1] As a side note, Pragmatic Capitalism has a good case in a post that the real return for equities, if we ex out US experience post world war II, might be closer to .03 than when using the possibly aberrant US data from that period (north of .06). A blended portfolio might be expected to be lower still. For option pricing we could go even lower or to zero or some spread between Rf and inflation. Pick a number or if you are an economist, write a guest post for me.  In the end the (net real) drift term, for the option, for today's purposes, is unknown...at least to me.  

[2] Here is another example of an untouched topic: next year, when everything has changed, the calculation could be (will be) different.  Also, this line of thinking assumes some degree of balance sheet feasibility now or in the very near future.  To have the option to annuitize means having the option..not being totally under water now and for the rest of the planning horizon.  Another untouched thing is that the future distributions of a net wealth process includes the idea of negative wealth which is an abstract concept not typically seen in real life although it has its uses and interpretations in modeling (see the footnote here that takes a stab at it simulated distribution of future net-wealth values ).