After a number of years of digging through the various
gullies and sewers and back alleys of retirement finance analytics I now find myself, for my
own planning purposes, sticking closer and closer to the management of the continuous present
moment via a household balance sheet than by way of any type of predictions about or projections into the future. The projection
project, to the extent that it is trying to predict anything beyond about 5
minutes from now, is seriously misguided. Even if the results from that effort
are viewed properly as a judgment about risk that is forming in the present
moment -- an insight which is sometimes useful and on which we can or should
sometimes act -- it can even miss the boat there, too. Here is Dirk Cotton on this idea: "the prediction
horizon for retirement portfolio balances is a less than a year, beyond which
the outcomes diverge dramatically… Retirement income studies tend to use
probabilities to focus on long-term sustainability of savings as a function of
market volatility alone. This approach won't catch many quickly developing expense-related
crises, especially since the studies tend to ignore expense uncertainty
altogether. When we say a retiree has a 5% risk of outliving her savings, we
mean a 5% risk of outliving savings due solely to market volatility. But, there
are other risks to those savings that should also be considered… These studies
explain long, slow declines in standard of living, not catastrophic failures,
in a world where market returns are normally distributed and mean-reverting and
no one ever needs to spend more than their 'sustainable withdrawal.' Their
recommendations – diversification and spending adjustments – provide little
help in a spending crisis."
The solution (other than working longer or lottery wins) to problems like the one Dirk is describing can
often, but not always, be found by leaning a little more heavily on something
like a household balance sheet as well as developing more robust systems and
processes and patterns of behavior for monitoring and surveilling the
developing personal and external landscape.
It's here that we might be at our best in terms of tracking judgments
about the impact of spending choices as well as any evolving or impending changes
in our personal lives. On the other
hand, I don't think we have completely eliminated prediction-style cluelessness by going headlong
into the balance sheet, we have just transformed it a bit.
What I mean by that is that while I can (while in HHBS mode anyway) now say with near
certainty what my investment accounts are worth this afternoon, I have a much
harder time saying anything certain this afternoon about the present value of consumption
liabilities, even though they are usually cleverly disguised in the garb of
definitiveness that comes with discounting something to a specific and single
number. The definitiveness of the PV, this "deterministic object"
-- as Dimitry Mindlin calls it in The case for Stochastic Present Values,
a paper (marketing vehicle?) from which I will borrow from heavily going
forward -- has maybe not been earned.
Mostly, it seems, the provenance of this thing is heavily inflected by a history of computational convenience. Dimitry: "Both the supporters of the existing
paradigm and the promoters of “marked-to-market” pension accounting, as diverse
as they are, utilize the following obscure but vital assumption: the present
value of a financial commitment is a deterministic object - a
number…Computational convenience was the primary reason for the acceptance of
the assumption that discount rates and present values must be
deterministic."
But this computational convenience prevents discovery of some fun stuff. Here's Dimitry again:
"The present value is equal to the asset value required at the present to
fund the payments. If the portfolio returns are uncertain, then the present
value—the assets required to fund the payments—is uncertain as well. As a
result, the present value of pension benefit payments funded by a portfolio
of risky assets is uncertain [emphasis in the original]." So then that nod to
convenience can lead to conundrums. Here is
one of them: "Indeed, a riskier portfolio usually has higher expected return,
which is equal to the discount rate. A higher discount rate means lower present
values. The relationship “higher risk means lower liability” is illogical."
And "…the conventional approach employs risk premium without risk, which
makes little sense. One of the unintended consequences of this approach is the
riskier the policy portfolio, the lower the pension “liabilities.”" So that sets up Mr.
Mindlin to make his real case: " One of the main goals of this paper is to
demonstrate that the adoption of a much more realistic assumption—that
present values of pension commitments are stochastic—has numerous
advantages…Overall, the concept of stochastic present values is
well-established and needs no introduction." Maybe not on "needs no introduction," but appealing
nonetheless.
Appealing because he uses this set-up to to make the exact same point
(ok, he beat me by 8 years and has substantially better credentials than me but hey, who's
counting or caring?) I made in one of my last blog posts (that is my self promotion at it's best, by
the way). The point I was trying to make there was that: a) it can be helpful to view the spend liability on
a balance sheet as a distribution rather than "one number," and b) the
distribution -- when one then finds on it the location where the assets available to defease the commitment would sit (i.e., what percentile) -- looks a little like a "probability of success"
calculator. There are probably 2 or 4
people on earth that get jazzed up when this kind of thing appears; I have a
hell of a time explaining why I am one of those to my kids and/or my
girlfriend. But really, this is where his 2009 paper backed up my guess in the last post. I will quote at length (some editing because blogger.com seems to hate math notation):
Note that while A1 is perfectly known, asset values Ak are uncertain for k ³ 2. The funding objective is to make all payments B1 ,…,BN, and, therefore, to achieve nonnegative asset value AN+1 at the beginning of year N+1. Proposition 1 in the Appendix shows that the probability of this event is equal to:
Formula (3) demonstrates that the likelihood of funding the commitment depends on the position of the current asset value A1 within the distribution of RA. In particular, if A1 is equal to the pth percentile of RA, then the probability of funding the commitment is equal to p.[emphasis added] This conclusion is quite intuitive—the commitment is funded if the current assets are greater than or equal to the assets required for funding. As we see from (3), stochastic present value RA is instrumental in the determination of the probability of success in the funding problem.
Note here that RA: "Required Assets (RA) is defined as the stochastic
present value of the benefit payments less contributions...Random variable RA is the asset value required at the present to fund the commitment." And: "Let Ak be the market value of the plan assets at the beginning of year k...Note that ... A1 is perfectly known."
Ah, well, isn't it fun to be validated. The point here is that in present-time balance sheet world (vs future time simulation), it
looks like that if current assets are greater than some threshold point on the spend
distribution, we're good…otherwise
not. I think this is a type of
simulation and ruin risk calculation by another name and method. This was my point in the previous
post.
Speaking of jazzed up, this line of thinking got me all
analytical, sometimes a real joy of a response to those around me but that's another story. So I re-wrote my excel "spend sim" in R to get
a better grip on this and then I ran a whole bunch of more rigorous scenarios
to illustrate the concept to myself. The
problem was that I hit a wall almost immediately. I can run as much data and scenarios as I
want but on the other hand I can now prescribe exactly nothing to no one because I couldn't even
reconcile to myself the right way to do this.
Because there are so many judgment paths one can follow on how one might model a
stochastic present value of a spending commitment -- how much and what type
uncertainty to put into the numerator…and/or denominator, how to model and
implement inflation and discount rates in real and/or nominal (or stochastic, which I have not done) terms, which terms and parameters are important, etc. etc. -- it becomes more
likely than not that there are significant policy choices about how one goes
about doing this. I was trying to go down
a path of rigor but right now that is probably premature. So, rather than giving the detailed analysis that I thought I was going to do I'll just visually show what some of the variations I was
working on looked like. In the chart below:
- GREY DOTTED - the grey line is the baseline deterministic calc for a custom spend liability. I did this using annuity calc math and a conditional survival probability extracted from a SOA IAM table. Spend = 100k in 3% inflated income, stepped down 20% at 68 and 85, and discounted at ~2.8% which generally synchronizes me with a similar calc at aacalc.com,
- BLACK/BLUE - the black line has relatively low uncertainty in the numerator of the stochastic PV. Mostly this means longevity. Black uses low/conservative discount rates that are perhaps (??) consistent with a desire to fund the liability with low return assets like bonds or annuities. On the other hand this might be backwards...The mean of black is within a few dollars of GREY. BLUE adds historical inflation vol to BLACK (no serial correlation, though),
- RED/GREEN - Red is like black but now more heavily loaded with uncertainty in the numerator (historical inflation draws, spend variance, etc) and it also has, like GREY, a low treasury/annuity-range discount rate (note: red has a normally distributed spend variance; green is the same as red but with a variance of spending that is skewed to the upside in spending terms…just like mine),
I wasn't going to do this but just for fun let's run a little test to see how it looks in practice. We'll run a version of purple but with a higher discount rate (7%, since the spend is in terms of inflation-adjusted nominal spending using draws on US inflation history). Since the start spend (though later it is more customized and varied than a typical constant spending approach) is 100k and people like the multiple 25 we can locate that "initial asset value" on the spend-distribution as if we are just starting a retirement. In the 7% distribution (not shown) 2.5M would be at about the 94.2nd percentile. If we believe Mindlin's math above and proofs in the paper, that is the estimated success rate. Just to double check I put the same custom spend path into my MC sim and tried to tune (that might be a red flag) the assumptions to the task at hand [1]. I got a success rate of 94.6%. Then to triple check I put some similar assumptions into my FRET (flexible ruin estimation tool , a ruin calc using a joint probability, and yes the word "fret" is a joke)[2]. I got a success rate of 94.9%. That's pretty cool. Now I'll be the first to say that I could blow up that coincidence pretty quickly by changing the spend volatility just a little tiny bit and I also don't think I have all the rates set up coherently. Nor have I randomized the discounting yet like Mindlin does. But I'll let all of this stand for now.
And like I said, remember that this post is not prescriptive or any type of tutorial. I am just trying to get my arms around this myself to see if any of this is useful. My guess is that
it is and will continue to be a useful addition. First, I like the balance
sheet approach in general and second, I also now like using/seeing the blown out version of the present value of
spending (distribution). It expands my ability to see
what is going on and it ties together (I think) concepts like balance sheets and simulated success rates. It also (importantly for retirees) fosters a hyper-focus on spending and evolving or even discontinuous judgments about spending risk.
What's not to like. Here we give the
last word to Mr Mindlin:
"Various measurements of stochastic present values—the mean, standard deviation, percentiles, etc. —“know” something about the magnitude and volatility of the commitment as well as the risk/return properties of the policy portfolio and its ability to fund the commitment. The utilization of these measurements makes possible the analysis of the multitude of risks retirement plans face (e.g. capital markets, inflation, interest rates, longevity, etc.) in one all-inclusive framework. Moreover, as we see in the next section, these measurements may provide a quantitative framework for the optimization of asset allocation, contribution and benefit design decisions for various retirement programs."
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[1] a reasonable set of assumptions: age 59, 2.5M endowment, stochastic longevity with same mode and dispersion as the spending sim, 60/40 allocation, no fees or taxes, etc.
[2] age 59, same mode and dispersion on the longevity distribution assumptions, and a weighted average spend rate adjusted from 1/25 to closer to 1/29 to reflect the custom spend weighted by time duration, etc.
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