In a previous post I pointed out that attempting to reduce
return volatility is probably a constructive activity for a retiree due to the
pernicious effects of sequence of returns risk. What I failed to mention is that it's not just about
volatility it is also about either the asymmetry [1] of volatile returns with
respect to a threshold return level and/or about mean-variance efficiency in terms of seeking the
same level of returns but with lower volatility. Sometimes those two are the same
thing. Either way, in retirement the
game is usually to mitigate the bad (bad returns, fail rates, bankruptcy risk, etc)
and stay open to the good (returns, median terminal wealth, increased spending,
max geometric return…whatever). That's a
pretty neat trick, especially if you can pull it off without options where
asymmetry is the main game. And I'll say
"especially without options" because long options are generally
pretty expensive. In fact sometimes the demand for options, both for upside
capture and downside hedging, can be so intense that I want to sell
options to try to capture some of that hope and fear. That's been a good business for me for at
least the last five years. No point in giving that up. But I think there are
ways to both reduce standard return volatility and also to wind it up a bit
asymmetrically so that it leans a little bit in the direction we want. I'm not saying 100% of investors are going to
do that because I don't think that necessarily sounds like a healthy "market"
in the aggregate. But I do think it can
be done within certain boundaries.
I only have a little more than 60 months of data and that is
only during a bull market Dec 2011 to 2017, so untested by things like '87 and
'08. But that is good enough for
now. To test that series for asymmetry
and related effects -- while I'm sure there are many ways to do it
(Sortinos, semi-variance, etc) -- one of the best ways I can think of for my purposes is to use
what is called the Omega ratio[2].
The ratio, of course, has it's pros and cons and it is uniquely
sensitive to one of its parameters (the return hurdle or "minimum acceptable
return") but also does a yeoman's job of sussing out up/down asymmetry
especially since, contra mean-variance analysis, it looks at more moments of
the return distribution. Let's take a
look. According to Mr Wiki: "The Omega Ratio is a risk-return performance
measure of an investment asset, portfolio, or strategy. It was devised by
Keating & Shadwick in 2002 and is defined as the probability weighted ratio
of gains versus losses for some threshold return target. The ratio is an
alternative for the widely used Sharpe ratio and is based on information the
Sharpe ratio discards. Omega is calculated by creating a partition in the
cumulative return distribution in order to create an area of losses and an area
for gains relative to this threshold.
The ratio is calculated as:
where F is the cumulative distribution function of the returns
and r is the target return threshold defining what is considered a gain versus
a loss. A larger ratio indicates that the asset provides more gains relative to
losses for some threshold r and so would be preferred by an investor. When r is
set to zero the Gain-Loss-Ratio by Bernardo and Ledoit arises as a special
case.[3]"
Since the cumulative distribution of returns is used in this
game, let's look at the CDF (using monthly returns) of the following four return series:
- SPY (S&P
500)
- AGG (US aggregate bonds)
- AOM (40/60 to 50/50 allocation ETF
depending on when you ask iShares)
- my data (systematic rules-base alt-risk
strategy)
Figure 1. Omega Ratio and the Cumulative Distribution
Figure 1 shows the CDF of the four series along with a threshold
of zero. The Omega ratio would be calculated as: a) the area between the
threshold "r" (0 in the x axis) and the CDF when the CDF is above r
(right side) divided by b) the area from r to the CDF on the left when the cdf
is below r…or green divided by red in the chart legend. The idea here is that the ratio of upside to downside with respect
to a required return tells us a little more than a Sharpe or Sortino. It's a tricky metric with a lot of pros and
cons and a few ways to cheat but in general I like it. The main thing to remember here is that the ratio depends
heavily on the Minimum Acceptable Return (MAR) "threshold." Different levels for the MAR can be used to
evaluate hedge fund strategies side by side…but that is not really what we are
doing here. We are evaluating the impact on retirement success rates. For retirement evaluation I like (opinion!)
using a MAR of 0 because we are usually thinking about consumption against a
variable portfolio so that negative returns (<0) at the wrong time and in
the wrong sequence have a big impact.
Evaluate hedge funds and maybe MAR can be > 0. Evaluate retirement theory and I'll stick
with a MAR = 0. So for MAR = 0 and using
a monthly return series over the 60 month timeframe 2/2012-2/2017[4], Omega might
look like this:
Table 1 - Omega ratios for four strategies
SPY 2.8
AGG 1.7
AOM 2.2
Systematic 3.5
For some additional context before we go on, here is the probability density
function for the 4 assets so that one can see a PDF view of variance with
respect to the zero "r" (these are monthly returns again, btw):
Figure 2. Probability Density and the Four Return
Distributions
Table 1 used a MAR of zero. Using the same monthly series of data and now
varying MAR from 0 to .01 in steps of .001 (or 0 up to 12.7% if annualized)
it gets more complex for the different threshold values:
Figure 3. Omega Ratio At Different MAR Threshold Levels.
At MAR = 0 the systematic risk approach clearly dominates
(yea me). But at MAR >.02 and beyond
the S&P dominates systematic. But
then again for those who, like me, do not allocate 100% to equities at all
times, note that a systematic alt risk approach (mine anyway) seems to dominate
pretty far into MAR territory. So how big a deal is this MAR thing? I think it's
probably easiest to see the outsize role the threshold level plays in this type
of evaluation by looking back at the PDF in Figure 2. If, for example, the threshold were to be set
above a .05 monthly return, pretty much only equities will win in this artificial
world…ever. Below .05 and equities will certainly lose…always. But, to reiterate a previous point, while
MARs > 0 (or < 0?) might be fun for comparative hedge fund strategy
evaluation, a threshold of zero (in a no inflation world!) seems important to
me when we think about sequence risk for retirees. With inflation considered, it's maybe a
little bit of a different story, but that is not told here.
OK so let's now say that an alt risk approach has the at
least the potential for "good" asymmetry (omega version) when referenced to a 0%
MAR hurdle. Now what…or so what? Well, now let's simulate a retirement to see
what happens with our new "optimal strategy." But first let's look at the simulator. It uses annual returns rather than
monthly so the comparative Omega evaluation gets a little weird if not a little
bit off the rails since I am not intimately familiar with how the formula works
in all situations especially with annual data. But if we were to
sanction using annual data the ratio might look like this using mis-match
historical data[5][6]:
If one were to allow us to roll with the caveats in the footnotes above, the
systematic approach with annual data, when the threshold is zero, looks like (even
if I'm off by a little bit) a pretty dominant strategy, all else equal. So, assuming that fake-sim-world might have something meaningful to say to us, let's try it out. Again, if I haven't already mentioned it, the
proviso here is that setting future [systematic] return expectation based on a
5 year sample is wee bit suspect but that is the game we have today[8].
The base case sim was run with a generic assumptions[6] using historical
data and a 50/50 allocation (mean return = .076, standard deviation .097): the fail rate was .169 with a mean fail duration of 6 years. I forgot to save
median terminal wealth which might have been useful.
The alternative case, keeping our proviso in mind, was to
use a return distribution roughly shaped to my alt-risk systematic rules-based data
series (mean ≈ .07, standard
deviation ≈ .04, some minor negative skew). The fail
rate in this case turned out to be about .119 with a median fail duration of 4 years. Without getting into confidence intervals,
that's about a 30% reduction in fail risk. Not too shabby. Most of that, though, comes not from any kind of skew
asymmetries[7] but from the left tail of the return distribution not extending too far beyond zero and the overall variance being narrow rather than wide. We mostly already knew (and could have predicted) most of this though from the prior post where we concluded lower vol probably trumps higher vol in portfolios with a retirement consumption
constraint. Just for fun, here are several other "strategies" I ran before I got tired of running sims:
Conclusions?
I'm not sure how many solid conclusions can be made here
given that setting forward expectations is a little hard to do in general as well as in particular with the short back-history of the systematic series. I was also playing fast and loose with some assumptions here and there while not being terribly rigorous with the simulations in the final analysis.
On the other hand, it looks like: 1) strategies that seek mean-variance
efficiency with respect to volatility (same return lower vol) probably have a
role to play in retirement, and 2) evaluating strategies with respect to the probability
weighted ratio of gains versus losses for some threshold return target,
especially since it captures more moments of the return distribution than MVO,
is probably something to at least pay attention to.
-------------------------------------
[1] I should be careful about using the word asymmetry. In this post I'm referring to threshold
asymmetry (statistical artifact of a ratio selected by way of an arbitrary
threshold) rather than distribution skew.
In fact, when I looked at distribution skew and played around with it,
it contributed less than I thought. Taking the negatively skewed distribution
that comes from a 50/50 allocation, I
fabricated a mirrored positive skew similar in measure to the negative. With
that a simulated fail rate only changed by 60 basis points or probably
statistically still within the domain of "noise."
[2] Keating and Shadwick, 2002. https://faculty.fuqua.duke.edu/~charvey/Teaching/BA453_2006/Keating_An_introduction_to.pdf
[3] https://en.wikipedia.org/wiki/Omega_ratio
[4] The easiest way to calculate Omega is either in Excel by
doing Riemann sums (definite integral using the sum of a bunch of little
rectangles with the area calculated as [change in y (height)] x [(distance to
the CDF along x) (length)] relative to the threshold level) or by using a
built-in function in something like R (PerformanceAnalytics package). This is
helpful because I don't know calculus, or at least my last class was in 1977…
[5] the first two strategies' historical data and return
distributions are based on 88 years of history 1928-2016. The systematic
strategy, however, only has 5 years of data so for "historical" I
took a mean of a bunch of rolling 12 month periods as a proxy. For the R function I used a distribution
shaped to what I know from 5 years of data.
I wouldn't typically recommend setting forward expectations based on 5
years but I have what I have and we are playing a "what if" game
anyway so maybe I can be a little fast and loose…
[8] I point out in a couple places that using 5 years of data
for evaluate forward moving strategies is bad, and it is. Most commentators lean towards 40-50 or maybe
100 years. Here is Cliff Asness on time
frame effects on Mean Variance: https://www.aqr.com/cliffs-perspective/efficient-frontier-theory-for-the-long-run
[6] standard assemblage: think $1M, 4% spend, 50/50 alloc,
age 60 etc.
[7] there is a little bit of distribution asymmetry. It doesn't measure well but you can see visually
in both the PDF and CDF how the left tail for AOM, the asset allocation
"passive" baseline, has a little fatter downside tail than the systematic
strategy.
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