A custom spend liability calc vs. a fixed annuity

An annuity is one way to step off the longevity risk treadmill by shifting that late age risk to a third party that can pool it over a large population (something we cannot replicate ourselves, btw).  That makes the price of an annuity that approximates the net present value of a forthcoming retirement spending plan a good proxy for a "free boundary" i.e., a boundary below which is -- when a retirement balance sheet is initially funded and valued with non-annuity assets -- an infeasible retirement and above which is a retirement that can be feasible with or perhaps without annuitization depending on, well, depending on a bunch of stuff.  Ignore for the moment that: a) that means annuities might be relevant to only a very tiny cohort, those at or near or quickly descending towards the "boundary" from above, and b) in practice we'd look at an entire household balance sheet rather than just the spending plan in isolation.  The focus here rather is the idea that the spending plan, whether by design or by the nature of normal random variance in spending will never, ever match the cash flow from an annuity.  And that's before we factor in something like spending "shocks." This could be a problem don't you think?

A simple fixed (non inflation adjusted) annuity price is relatively simple to estimate.  A spending plan is also relatively simple to "price" as well and in theory it can be done with the same or similar math.  The problem is that the spending plan can vary greatly from the annuity path before we even get to the idea of random variance in spending.  That means that if annuitization ever does occur then it seems like there must be periods of cash flow under-funding or over-funding which also means, in simple terms, that one has to deal with this somehow.  I have no idea how the pros like CFAs or retirement specialists handle this kind of thing but I presume that one would have to have some kind of a side account that would reserve overfunding and either put it in market assets or into incremental annuities (or TIPS) to satisfy some of the future liability and especially those future liabilities that end up underfunded.  One could also imagine a borrowing facility to deal with some amount of underfunding when needed.

To price the annuity we can use standard formulas from any intro finance book.  Here I'll use the annuity formula from Milevsky's 7 Equations book because I happen to have it on my desk.  From the first page of chapter 3 it looks like this:

iPx is a conditional survival probability estimable from actuarial tables = tPx below

R is a discount rate or return rate assumption, more or less = r below

To price the custom spend liability (ignoring random variance for now) I'll use a fake amateur formula I made up late last year.  I recently sent this to someone in the industry that knows this kind of thing way better than I do. Embarrassingly, I didn't realize until after I sent it that it is in fact the same as the annuity formula except with some unnecessary complications that add almost no value.  I was feeling pretty clever at the time; not so much now.  On the other hand I have habituated myself to various forms of embarrassment over the last 59 years.  The formula I sent looked like this:

s is a custom cash flow

tPx is the same as above, the probability that someone age x will survive t years

r is the same R as above but could be different…

alpha and beta are un-needed hyperbolic discounting factors that can modify the discount rate over time.

If you look closely, you can see that these two formulas are exactly the same.  The only differences are that: 1) the spend liability formula has a custom cash flow "s" but if one replaces "s" with $1 it's the same, 2) the sum symbol in the spend liability is not to-infinity but since the conditional survival probability goes to zero around age 120 it's effectively the same thing and the notation should probably be infinity anyway, and 3) I have a totally unnecessary coefficient in the csl that makes the discount rate 'hyperbolic' so that I can throw some time asymmetry in there if I want i.e., perhaps I might want to weight the late age more (lower discount rate) than the younger ages (higher discount rate) or vice versa.  That means that with the exception of #3, which as I mentioned is absolutely totally unnecessary and which I threw in "just for fun" or "just in case," we have the same formula.  The same but different.  Let's run these as "different" and see what happens. 

The basic setup is to first design a custom spend liability. For that we'll take a 59 year old male and project 100k in spending inflated at 3% into perpetuity.  To make it feel like my own plan (all the numbers are different, though...) and to make it vary a bit we'll step this spending down 20% at 69 and another 20% at age 86 [1]. The conditional survival probability (CSP) is derived from the SOA Individual Annuitant Mortality table with the G2 extension for 2018, if I got that right (either way it's a conservative mortality assumption). The discount rate is somewhere near a blended treasury security rate.  For that I use an unsubstantiated 2.8% in this example which I will also use to price the annuity. This rate could be justified (for the spend liability anyway) to be a little higher but that complicates things a bit and requires a discussion of the art of discount rates.  The alpha and beta coefficients for the weighted discount rate can be somewhere between 1,0 and "something else." I use an entirely arbitrary and baseless 1.3,.13 which does, however, overweight old age and underweight younger ages a little bit. We'll see if that works later. 

For the annuity, the conditional survival probability is the same and the discount rate is the same as that used in spend liability above. $1 cash flow is assumed in the formula but this will be solved here for the cash flow that makes the annuity npv equal to the spend liability. Note that for simplicity we are ignoring money's worth ratios and the overhead loads of the insurance concept. We can add those later.

To compare these two and to see what it looks like on paper, ignoring things like the obvious disclaimers[2], we can chart it out like this where the solid lines are the spend liability and the annuity with normal discounting and the dotted lines use the spend liability with a hyperbolic discount as the baseline:

In table format we can state it like this table below.  The first row is the "annuity = custom spend liability(csl)" math without the (unnecessary) hyperbolic twist in the discount rate. The second row includes the twist.  The first column is the npv of the annuity which is set equal to (solved for using the cash flow as the variable) the npv of the spend calc.   The second column is an annuity price estimate that comes from aacalc.com for the age and cash flow parameters used. This is a site and method I trust (and this also shows, I guess, that I have not totally lost my mind in annuity pricing).  The third column is the constant cash flow of the annuity (col 1) that we solved for. 

	npv	aacalc	annual cf
A=CSL	2,324,304	2,319,219	126,101
A=CSL(h)	2,417,188	2,411,913	131,141

Clearly the annuity will require a high initial cost and a payout higher than the initial stage-1 spend rate to reflect the inflation assumptions embedded in the spend liability that are not in the annuity (but could be).  On the other hand the annuity is quite a bit cheaper than it might have otherwise been because of the attempt to match it to the custom step-downs at 69 and 86. Whether the annuity would precisely defease spending in a real life process is highly doubtful especially since we did not really specify much yet about the real-world process of what look like over-payments in the early years and underpayments later.

So, now let's play out the management of the over/under in my usual naïve amateur-hack way.  We'll put any early over-funding into a side account and then put that at risk earning an after tax 5% with (gasp!) no volatility.  As long as I can ignore that little vol distraction I can probably also ignore other frictions and realities like variable (geometric) returns, taxes, behavioral considerations, changes in risk aversion, changes in treasury-linked discount assumptions, etc.  Now we'll see how far we can go before we hit some level of under-fundedness in any given future projected year. Note that the real pros often suggest laddering into more annuities (or TIPS), a subtlety I ignore here. 

But what about longevity?  In the math above we used a vector of conditional survival probabilities to weight the cash flows and price both spending and annuities.  Now here we should pay closer attention to more meaningful and discrete guideposts. For guideposts we'll say that the cohort mean (age 59 male) per aacalc.com is 81.74 and the 95th percentile is 96.89.  These are both a hair's breadth from my own calc based on the SOA data. Close enough.  

  

If I do the naïve side-account thing, I can see (not shown) that the annuity plus the side-reserve (no borrowing assumed) should defease the spend liability cash flow estimate successfully to about age 93 (hope I got that right!) after which it looks like there is some degree of shortfall.  Recall that average life expectancy was said to be ~82 and the 95th percentile was ~97. Not sure if defeasement to 93 is reassuring or not under these assumptions.  If, on the other hand, I spend more on an annuity based on the spend liability calculated with a bent discount assumption using the hyperbolic coefficient [see note 3!] with the arbitrary assumptions I used, we can survive shortfall-free until age 102.  That probably is at least a little more reassuring.  And expensive (about 4% more). 

The most I can say here, I guess, is that if I, personally, happen to have an unusually odd point of view about my future spending plan (let's say I think it is radically discontinuous or I have weird views about time preferences) I can maybe, in very general terms anyway, at least get a sense for what that might look like in a real-world product (annuity) that could plausibly fund the plan.  We showed however that there might be some defeasement uncertainty with our simple modeling. The real problem though is that even if one were to have a smooth spending plan and perfect foresight of inflation no annuity will ever really perfectly defease a spending plan because we have not even touched on random spending variance yet.  That means that in the end the problem -- all of this, even though I took a naïve shot at it -- is one of cash flow process management which is really a separate topic altogether.  Oh, and don't forget, we'd be recalculating the liability (and the annuity) more or less continuously.  Things change.   

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[1] there appears to be some evidence that supports this notion of declining spending. See this post.  Health care costs and long term care costs might come into play but since we are operating in the context of a household balance sheet in this post that might be quantified as a separate liability vs. normal operational spending.

[2] The "obvious" (to me) disclaimer being: a) I'm not sure if I am standing on solid theoretical or even practical ground, b) I am just beginning to explore an idea rather than expounding or tutorializing, and c) I am unwilling to do any work beyond the "game's" parameters to see how it might really work under different conditions.  In other words this is all a form of sand bagging, right?

[3] This is a trick!  One could just as easily do the same thing by using an overall lower discount rate.  In fact, when solved for discount rate that makes the CSL = annuity the rate was around 2.5%.  On the other hand, that affects all years equally.  The hyperbolic discount is thrown in there just for fun to show an asymmetrical view of the effect of discount rates in future time.  The hyperbolic thing adds little value and probably muddies the water way more than it helps.