Note: as in the previous essays, this is a draft as I hone some of this content. Also, since I view these essays as consolidating and integrating what I've learned about ret-fin so far, I will continue to add to and update this provisional latticework over time in response to new findings or errors.
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This is a continuation of a previous essay on “Five Retirement Process” which can be found here or a blog page summary here..
Process 1 – Multi-period return generation.
It was a little bit of a revelation to me when I started to
realize how little I knew about financial processes at the age of 50. I thought my MBA(finance) had taught me
something and I had naively leaned on that credential confidently for decades
or at least I did in a cocktail party conversation sense. And what I knew about what I didn’t know (or
didn’t know I didn’t know) wasn’t even the worst part. It’s that what I did
(think I) know, I have started to now realize, can mislead. This section of my look at retirement
processes is about how I have misled myself when it comes to “return
generation” over multi-period time. It is very much not about portfolio design
or optimization which is another topic altogether and something I assume as a
precursor or even prerequisite to this topic.
The discussion here is: what happens after
you turn on an optimized “return machine.”
Return generation in this sense has little to do with the
past. It is about expected forthcoming future returns as an engine for healthy
retirement portfolio growth that is, in the end, harnessed in service of a
successful decumulation project. In other words, what are your assets, however
they are allocated, going to produce for you going forward, independent of what
you spend or how long you are going to live? First of all, we need to note that
there is nothing really ever certain about this process. If I say I expect to
earn 5% (let’s ignore taxes fees and inflation for now) then where does that
come from? It’s supposed to come from things like “compensation for risk” or,
I’d like to say and maybe more accurately, from the productive, creative innovation
and risk taking enterprise of the participants in a company or economy. But there are no guarantees that any of this will
be stable or even persist. Think of Japanese equities after about 1989 or the
US stagflation era in the 1970s. But let’s ignore all that and assert that we
have and will continue to have a stable, positive, known expectation for
returns from some source somewhere. Even
then we (ok I) can be misled and it is this mis-direction that I wanted to take
a look at so that we are better informed about return generation processes when
we get into some of the essays to follow, especially #3: portfolio longevity.
There are, no doubt, a multitude of ways I misled myself on this but for this
topic I’ll call out seven in the discussion that follows:
1. Arithmetic single-period returns are not what we earn in the long run,
2. A common estimate for what we earn in the long run (geometric mean) is often a bad estimate,
3. When time and flow come into play, a lower return strategy can sometimes dominate a larger return strategy and do so over relatively short time horizons, breaking the commonplace that more risk means more return…or that waiting will help,
4. The effects of multi period time on realized returns can influence portfolio choice along the efficient frontier. Over the long run, additional risk may not compensate with additional realized return,
5. In a multiplicative return process, the central tendency of terminal wealth over multiple periods of returns can sometimes mislead if one happens to have goals other than maximizing median wealth at the end,
6. Retirees don’t live on the expected geometric mean. They only have one “whack at the cat” so the individual paths they take through the return generation process over time often matters a lot.
7. Non-normal return distributions don’t help us
2. A common estimate for what we earn in the long run (geometric mean) is often a bad estimate,
3. When time and flow come into play, a lower return strategy can sometimes dominate a larger return strategy and do so over relatively short time horizons, breaking the commonplace that more risk means more return…or that waiting will help,
4. The effects of multi period time on realized returns can influence portfolio choice along the efficient frontier. Over the long run, additional risk may not compensate with additional realized return,
5. In a multiplicative return process, the central tendency of terminal wealth over multiple periods of returns can sometimes mislead if one happens to have goals other than maximizing median wealth at the end,
6. Retirees don’t live on the expected geometric mean. They only have one “whack at the cat” so the individual paths they take through the return generation process over time often matters a lot.
7. Non-normal return distributions don’t help us
The starting point…
This essay assumes that we already have the output of a portfolio
optimization process as a starting point.
If we used a Mean-Variance approach to do that, the return and variance
of the return-generating portfolio can be characterized like this, borrowing
the notation from Markowitz (2014):
Where E(p) is the expected (arithmetic, single period) forthcoming return, E(i) is the expected (arithmetic) return of the ith security, X(i) is
the fraction of P invested in the ith security, V(p) is the variance of the
portfolio, and σ(ij) is the covariance between security returns r(i) and
r(j). E(i) can be pulled out of thin air
by judgement or, more often, by using historical data over some look-back period
n with the proviso that even long series of historical data might be an anomaly
in the context of the infinite unfolding future. The arithmetic average return
over look-back period n would be estimated like this:
The key point here is that both the input and output to MV are
arithmetic returns. The output is estimated
expected forthcoming single period MV
optimal returns. There are other
assumptions that probably have to be made such as that the return generation
process as well as the variance and asset covariances are assumed to be stable and
independent over time, something we know is not necessarily always true. E(p), the arithmetic single period estimate
from MVO, then, is the return engine to which we have committed and to which we
will see what happens when time is added to the mix.
…and the Step-off Point
At this point, we’ll start going into an examination of what
happens to the “return engine” after it
is up and running. We’ll do this by taking a look at each of the seven points I
mentioned above, one at a time.
1. Arithmetic single-period returns are not what we earn in the long run.
This is the step-off point to where it can become easy, even
though most of what is to follow is reasonably well known, to be misled. To
start with, even though we use arithmetic average returns to optimize the
portfolio, we don’t earn the arithmetic return over the long run. It decays in
the flow of time due to the variability. Markowitz: “in the long run, one gets the
geometric mean return, not the arithmetic mean return.” That’s because the
return generation process is multiplicative in multi-period time with the multiplicativity
being represented by the geometric mean return at some end-period N:
The outcome of this multi-period process will always be less
than the arithmetic return. It will also, for a portfolio with more than one
asset, since the geometric return of the average arithmetic returns is not
equal to the average of the geometric returns, be more than the weighted
average of the geometric returns of the individual assets. For that matter, it
will be unlikely as well to match one of the more common estimators for geometric
return at long horizons. More on that later.
To help visualize this we can resort to simulation, the full
image of which is deferred until item 6 below. Let’s take a look at three
assets: a “low risk” asset with a return of .035 and a standard deviation of
.04, normally distributed for modeling convenience; a risky asset with a return
of .08 and a standard deviation of .18; and a 50/50 rebalanced allocation[1] of
each of the first two in a portfolio with a correlation of -.10 between the two
return series. We’ll ignore taxes, inflation and fees. Then let’s run it out
(with a limited run of 3000 iterations) over 30 periods and, while doing so, we’ll
calculate the (expected) geometric return for each asset for each period N for
N = 1:30. When we do that it looks like
this:
Here, you can see in one chart pretty much everything we
came to see: the volatility-induced decay, the inequality at n=30 between
arithmetic mean and my simulated estimation of the geometric mean, the inequality
between the geometric mean of the average and the average of the individual geometric
means, the tendency of the geometric mean to approach a limit at long horizons,
and the trickiness of trying to estimate the geometric mean at planning horizons
that are more than one period and less than a typical lifetime.
The process problem in the previous section is well known
and the approximate solutions to estimation are reasonably well known, too. Given how important the relationship is and how
pernicious the effects of return decay over time, most reputable finance
textbooks will provide what I will call the “common estimator” for the long-term
expectation for the geometric mean:
where G(∞)
is the estimated geometric mean at an infinite horizon, µ is the arithmetic return
estimate and σ is the standard deviation of returns. And then, if the textbooks are any good, they
will also provide inverted versions of this and any related formulas so that if
one has only compound returns one can work back to arithmetic in order to have
the approximately correct inputs for MV optimization (or retirement simulation
for that matter).
What is less well known is that this is just one of many estimation
tools and an imperfect one at that. Bernstein (1997) helpfully provides some context
that shows the common estimator is better framed as
Which is itself evidently a special case of
where when beta is zero or the arithmetic return is small
things reduce back to the common estimator.
He also inverts this to solve for the arithmetic return given G.
Michaud (2003) provides three variations on estimation of
the geometric mean, one of which is the common estimator we’ve already seen, and
another of which has the grace to consider intermediate time horizons,
something most people forget about in geometric mean estimation. The third he considers more accurate than the
others if the number of periods gets large but I have not shown it here.
Mindlin (2012) gives us four different formulas for estimation
of G (not shown) including the common estimator. He also helpfully shows where the derivation
of that common formula comes from, discusses the pros and cons of the various
estimates, tests them against historical data and then provides a concluding
point of view: “Relationship (A1) [common estimator] is the simplest,
popularized in many publications, but usually suboptimal and tends to
underestimate the geometric return.” You can read his paper to see which one he
likes better.
Not surprisingly, now that I think about it, the best cover
of this estimation topic that I’ve seen comes from Markowitz and Blay (2014).
In chapter 3 (Mean-Variance Approximations to the Geometric Mean) of volume 1
of Risk-Return Analysis, he proposes
six approximations to the geometric mean each of which are then carefully considered
in terms of their usefulness, judged against the estimation error given
different parameterization, and tested against historical data. Note that across
all of the sources I’ve mentioned so far, the different estimation formulas are
generally consistent because all are working off the same math and principles.
The “common estimator,” for what it’s worth, was quickly dismissed by Markowitz
in his first pass.
His purpose in evaluating six different approaches was this:
“The approximations tested in the present chapter support practice in three
ways. First, they are further tests of the efficacy of MV approximations to
expected utility… Second, they show…whether the arithmetic mean and variance
outputs of an MV analysis can be used to estimate the portfolio geometric mean
with reasonable accuracy. Third, they consider which of the six MV
approximations proposed for g have worked the best historically.” His conclusions point to a group of three estimators
the use of which depend on the circumstances of the study being done. Apparently,
none are perfect in all situations. I’ll leave it to the reader to read the
source and deconstruct the nuance. Recommended reading if you like this kind of
thing.
Finally, most papers that dig into this concept will often point
out that these estimators are related to limits and that, given large numbers,
long series, and independent draws of returns, the following will be the core
relationship of interest (Markowitz(2014)):
Or maybe I could have just used the common estimator and
accepted the consequences now that I know the pros and cons. Since I do not run a pension fund or price
insurance products or manage client money, I find that some combination of the
simple common estimator and/or simulation to be more than adequate for me. At a minimum it means that I am at least paying
attention to the effects of time and volatility on the output of single period
mean-variance optimization. As long as I
remember that: 1) it’s a rough estimate not truth, 2) it is not typically very
accurate as time gets long and/or volatility gets high, 3) it is an estimate of
the geometric mean of rebalanced portfolio and not the average of individual
asset compound returns, 3) it is wrong for short horizons, and 4) it may be good enough most of the time.
Using the same math and method for the geometric mean as
above let’s compare two different strategies over 30 periods. Again, we will
skip over distractions like taxes, fees, and inflation:
a. Risky: ~N[r=.08, sd=.18]
b. Higher return but even riskier: ~N [r=.09, sd=.25]
b. Higher return but even riskier: ~N [r=.09, sd=.25]
More return is better, just riskier, right? Not so
fast. When we look at the (simulated,
expected) geometric mean over multiple periods a new view emerges. Here is the
expected geometric mean of the two strategies calculated for each period N, N =
1:30…
It’s not just that the lower return strategy dominates in the long run, it’s that it does it by the 4th period. If one were to be an institution, subtleties like this may or may not matter but for retirees with limited lifespans and tactical considerations like optimal age to reallocate into annuities, short planning horizons might make this type of analysis more germane. In other words, the long run dominant multi-period strategy might not really be so for shorter horizons. It’s worth thinking about. Fortunately, Michaud (2003) helps us out. He provides a rough estimator for intermediate horizons that looks like this:
This particular N-period estimate of the geometric mean, presented
in terms of the single period mean, is not perfect nor is it always entirely
accurate but it is in fact useful for understanding the multiple-period but non-infinite
consequences of different strategies. There may be better ways (say simulation)
or estimation formulas (which I don’t have) for this task. Let’s look at the N-period estimation formula
vs. simulation. This is the same chart
as above but with now the N-period estimation overlaid on top:
It’s hard to say whether the simulation does not have enough
iterations to drive the average to expected value or whether the estimation formula
is imprecise. Maybe both. What we do see is that both illustrate the same
multiperiod consequences, both have a similar fade-to-limit, and both have a
similar (earlier in this case) crossover point.
In other words, we are taught the same story and analytically they both get
to vaguely the same conclusion.
Useful.
Note that there is not always a crossover. This example was
reverse engineered to create one but seems plausible, nonetheless. There are
other reasons to find this type of analysis useful. Michaud (2003) points out
that it can be used in evaluating tax effects of going from a concentrated
position to diversified and also that the internal organs of a Monte Carlo
simulator have the same process as the N-period geometric mean except that the GM
is sometimes more efficient for professional reasons and perhaps more
transparent.
4. The effects of multi period time on realized returns can influence portfolio choice along the efficient frontier. Over the long run, additional risk may not compensate with additional realized return.
Both Bernstein (1997) and Michaud (2003) point out that
transforming an efficient frontier into one that reflects the geometric mean
can reveal that there may be
inflection points where an additional increment of risk delivers no additional
or even less return. This is more or
less the same process that drove what we saw in the previous section for strategy
comparison over time but now thrust into two-dimensional MV space by way of the
common estimator along with allocation choice and all the implications that
come from that. Using the same low-risk
and risky asset from above and then using 11 allocation steps from 0% risky to
100% risky this is what the efficient frontier looks like when rendered as both
the traditional arithmetic output (blue and green) as well as the same frontier
with the arithmetic return re-rendered as geometric using the “common
estimator” (orange). An intermediate N-period horizon would be somewhere in between
the two lines and may not have an inflection point for this example.
You can see that an allocation, given these particular
parameters, of greater than 90% to the risk asset results in an inflection
point (in red; might be hard to see that it is starting to go down) beyond
which there is lower expected return for an additional unit of risk. This would
likely be avoided, all else being equal.
One can begin to see why alternative risk strategies, the ones that help
tamp down portfolio risk for similar levels of return (say, maybe, by adding a
trend following allocation), might start to look attractive over the long run
when designing portfolios in retirement. Note that there may be no inflection
point at all or it may lie on the end points.
At first glance this entire exercise might seem to
contradict or vitiate some of the main conclusions of MPT and MV optimization.
But Markowitz himself has already addressed this in his own literature. Not
just the quote above about how “in the long run, one gets the geometric mean return, not
the arithmetic mean return” but also when he makes a direct link between the
geometric mean and MV frontiers in his recent work (2014, 2016). For example:
“The Markowitz (1959) Chapter 6 position was that a cautious investor should not choose a mean-variance efficient portfolio with a higher arithmetic mean (and therefore higher variance) than that of the efficient mean-variance combination that approximately maximizes expected log [think multi-period geometric returns]. A portfolio that is higher on the frontier subjects the investor to more volatility in the short run with no greater return in the long run.” [his emphasis], and
“Markowitz (1959) does not recommend that an investor choose the MEL [max expected log] portfolio. Rather it recommends that the investor not select a portfolio that is higher on the efficient frontier, since such a portfolio has greater short-run volatility but less long-run return—in the Kelly sense—than the MEL portfolio.”
One of the fun things about multiplicative (compounding)
processes is that they can produce wildly skewed results on the right side of
the terminal distribution. That’s another way of saying that one has a remote
possibility of getting very rich at the end of a compounding return process. That’s cool but there are prices to pay for
that kind of possibility. Let’s take a look.
If we ignore spending and taxes and so forth and we look at
the terminal wealth distribution of $1 growing within a return process over
time we start to see a fair amount of dispersion as time passes. It also goes towards a lognormal shape since
we can’t crash through zero but we can reach for infinity. The two arbitrary return processes we’ll use
for the illustration are from the same examples above. The normal distribution
assumptions are for my convenience:
b. High risk: N[r=.09, sd=.25]
The chart that follows is a representation of the unfolding
wealth distribution for return process “a” as “t” goes from 1 to 10. Note that
the image is rotated to push the 10th period towards you and the
first period towards the back. This makes the positive outcomes stretch out to
the left not the right. This is rendered as a probability mass based on the
empirical frequencies coming out of a mini-simulation.
This, on the other hand, is the unfolding wealth
distribution for the high-risk return process “b” as t goes from 1 to 10. Note that the image
has a similar orientation as the above.
I’ve tried to keep the scale and perspective roughly the same so that
the charts are comparable. I’ve just rotated
it a little bit so we can see it better.
The point of these illustrations is to show: a) the widening
dispersion over time, b) the tendency of the dispersion to skew and expand to the
upside, and c) the way that the high risk asset can create extreme outcomes
compared to the balanced asset.
But this is a distracting exercise. What we are distracted
from is seeing what is going on at the lower end of the distribution which means we need
to take our eyes off the extreme outcomes.
Much like a neighborhood with a bunch of 1-bedroom homes and one 40 room
mansion, the extreme house distracts and makes the average neighborhood home-price
meaningless. In that kind of situation,
the median makes more sense as a descriptive central statistic. The same idea is true for terminal wealth in
a multiplicative process. The median will
make more sense. And in fact, the median
of terminal wealth and the geometric mean are related under certain assumptions
about the distribution. In Michaud
(2003) he puts it simply: “the medians of terminal wealth and of the geometric
mean, GM, are related according to the formula
Median of terminal
wealth = (1 + GM)N (5)
and goes on to say: “…the expected geometric mean is a
consistent and convenient estimate of the median terminal wealth via (5). Since
the multiperiod terminal wealth distribution is typically highly right-skewed,
the median of terminal wealth, rather than the mean, represents the more
practical investment criterion for many institutional asset managers, trustees
of financial institutions, and sophisticated investors. As a consequence, the expected geometric mean
is a useful and convenient tool for understanding the multiperiod consequences
of single-period investment decisions on the median of terminal wealth.”
So far so good.
Median wealth both matters and makes sense mathematically. But even the median obscures some things a
little bit when we compare two different risk-strategies over time. Let’s look again more closely at strategies a
and b above at the 10-period mark. Same
assumptions as above and we’ll also add a risk-free asset that has an arbitrary
3% return and no variance. Would that I could find such a thing. When charted out as a mass function based on
the frequencies coming out of the simulation and then overlaid one on top of
the other, it looks like this. The 50/50 strategy is blue, the high-risk strategy
is orange, and the risk free asset is the grey zone.
This is interesting. First, though, remember that we
stipulated that the average is meaningless. Then, we already demonstrated that
per Michaud (2003), and pretty much everyone else, that the median matters. And
in this case, yes, the median wealth of the risky process is in fact higher
than the balanced portfolio and the
risk-free rate which we might expect from the higher return and volatility
profile. But, as a retiree, we might sometimes
feel like we also need a full dose of “subjective relative likelihood.”
For that we have to look at the emerging mass of probability in addition to the
extreme outcomes on the right side (the fun ones)…and even in addition to the
median. For that full dose we can use
the mode of the distributions. But we can also use our eyes to literally see
“the mass” of the probability, of which the mode is a proxy. In the case and chart above we can see that
the mode of the “balanced” portfolio is still, at t=10, above (to the right of)
the risk-free outcome (still plenty to worry about on the left…) while the mode
of the high risk asset – even while the terminal outcomes are stretching to the
lottery-like far right horizon – is starting to slide uncomfortably to the
left. So, for the high-risk asset the fun
and good outcomes are extreme and getting more so, the median is still a worthy
statistic, but the mass of likelihood looks like it is moving in a rough
direction. If, for example, I happened
to want to annuitize 100% of my wealth in 10 years for some reason, I might
think twice about the high-risk proposition and I might even be a little less
sanguine about the balanced approach than I might have otherwise been. The
likelihood of reaching a goal has now been compromised a bit by accepting risk
that otherwise shows positive outcomes by common measures of central tendency.
The point here is that some of our normal assumptions and
common analytic frameworks can mislead. Arithmetic returns are not what we
earn; the geometric mean is what we realize. Median terminal wealth is a better
statistic than the average. But even simulated median wealth outcomes can
distract us from the underlying risk in the ending distribution of
possibilities.
The expected geometric mean is a useful and important
concept and enhances our understanding of the return generation process over
and above arithmetic expectations but the geometric mean itself obscures the
reality a bit. The geometric mean is an expectation or an average over many
possible lives or in my case the iterations of a simulation. We (I) don’t have the luxury of many lives,
we (I) have one. That means that the path we are dealt in life not only matters,
it can matter greatly. Here is Michael
Zwecher:
“Each client gets only one chance at retirement, and they want to preserve a lifestyle. Statements about portfolios that contain phrases like ‘on average’ or ‘expected returns’ or ‘expectations’ or any such terms should trigger a wary posture if they omit hard stops on losses. If I were going to be able to play a game repeatedly I’d be happy to know that the odds were on my side and if I lost a round I could come back. But if I only get one turn, one whack at the cat, then I want to make sure that losing doesn’t break me, no matter how favorable the odds. To put it another way, even a single round of Russian roulette is not for me, no matter what the payoff.”
To illustrate this point, let’s go back to item 1 above
where I first laid out the geometric mean for a balanced portfolio by way of
simulation over 30 periods. Now we’ll pull back the curtain a bit and show some
of the underlying simulation data from which the mean was extracted. This is only
a simple average done with few iterations and fewer periods but we can at least
note that the geometric mean gets closer and closer to its own truth (expected
value) when the iterations get large and the periods N get long. Here we have
only 3000 iterations and N=30 but it will do to make the point. There are two
charts, one for the 50/50 portfolio we used before and one for the high risk
portfolio, also from before. The Y scale
is kept the same on each for comparison. Only the first 50-100 iterations are
shown to keep it clean. The black line is the average of the iterations and
stands in for the expected geometric mean for each N=1:30. [3]
So, yes, the expected N-period geometric mean is a very nice,
convex, and monotonically downward sloping thing that limits-out towards the
eventual true expected geometric mean at infinity with increasing certainty on
either side of the mean. But the individual paths look like they are one heck
of a hay-ride when viewed on a human-life-scale. That scale is us. Note that in the charts, the areas of interest
(stress) are probably the region at or below where G(m) = 0 and also the left-ish
region where N is less than or equal to either expected lifetime or some other
planning horizon.
This kind of image that I’ve drawn is no doubt one of the
reasons that portfolio optimization criteria like geometric mean maximization
or alternatively Max(E[log(1+r)], in a Kelly or Hakanson sense, don’t sit perfectly
well with people living real lives. The geo
mean max idea is mathematically elegant and intellectually convincing but only
over very very long time horizons for a lot of different "tries" while for finite N (say that range of N = 10
or 20 or 30) it looks pretty risky if not downright jarring. Here’s Michaud
(2003): “For many investors and institutions, the Hakanson proposal is often
not a practical investment objective.” My
amateur guess is that this perspective is at least indirectly the basis for critiques
like the ones from Samuelson or Merton in that the geo-mean does seem a little
unhinged from rationality, at least in the early time intervals. On the other
hand, Markowitz (2016) took a shot at getting in the last word on this:
Theorem: If Harry repeatedly invests in a portfolio whose E log(1+R) is greater than that of Paul, then --with probability 1.0—there will come a time (T(0)) when Harry’s wealth exceeds Paul’s and remains so forever thereafter.” [his emphasis; note that “come a time” might take a while]
----
While we’re at it here, there is one other way that the individual paths can matter. We haven’t come to the spending process essay yet but since we are discussing multiplicative returns series -- and the individual “paths” through them -- I feel like we need to bring up the topic of sequence of returns risk now rather than later.
Sequence risk is well known in the planning industry, or at least it should be, and is easily found in the literature. The basic idea is that poor returns early in retirement can impinge on success rates over finite horizons which is easily illustrated with numerical examples or monte carlo simulation. I’m going to come at sequence risk in a slightly different way via what is called the perfect withdrawal rate (PWR). PWR is the constant consumption rate that will deplete a portfolio to zero at the end of the planning horizon given perfect hindsight on the return series realized. It is represented mathematically [4] like this if the initial endowment is $1 and the planned ending endowment is zero:
But first, I’m going to rename PWR (or w on the left of equality) as “spend capacity” because “w”, with perfect
hindsight, is what the return generation process will allow us to spend. The elegant part of describing the spend
capacity of a portfolio this way (right side) is that it is rendered entirely in terms of a “an individual,
realized, return-generation series” (i.e., a path) which is exactly what we are
discussing in this section. This also
allows us to discuss return sequences – and their influence on spend capacity along
with the intuition of the link between the two -- without having to resort to numerical
examples. But before we do that, I’ll re-render the formulation above so that,
even for those that can read the notation, it might be easier to access the
intuition directly. Here is the equation
deconstructed:
When you look at this this way you can see, mechanically
speaking and without needing a numerical example, that lower (the absence of higher) returns early in the interval (where there are fewer (1+r)s ) rather than late
(more (1+r)s ) will create a bigger denominator and therefore smaller spend
capacity “w” and vice versa. And, in
fact, Suarez (2014) says this: “the denominator...can be interpreted as
a measure of sequencing risk.“ That risk
can be present in one path vs another even when the N-period compound means
are otherwise the same. One more reason
to have respect for the nuances of a return-generation process.
Last but not least, we need to acknowledge that return
distributions are typically not normally distributed. This characteristic does
not help us with the multi-period consequences of investment decisions. While the effect on the multi-period geometric
mean may not always be large, it is there.
The fat tailed distributions observed in some market returns
at some time intervals can sometimes be pronounced. To model this effect in
simulation, one can approximate it by designing and combining multiple return
regimes in some proportion, for example: one common one with high returns and
narrow variance and one less common one with low return and wide variance. More
than two does not seem to add much to the modeling but then again, this kind of
thing is at the edge of my skills. This
idea of “regimes” is equivalent, in my own attempts to model returns, to a
gaussian mix of multiple normal return distributions.[5]
To test the effects on multi-period geometric returns using
this method, here is a quick, arbitrary, simulated example. I worked backwards by first creating a
gaussian mix based on a 80/20 combination of two normal distributions and then
I modeled a normal distribution using the first two moments of the mix. I
didn’t care about the exact level of the return and variance as much as I did
just the idea of doing it at all. These are the return distributions I came up
with. You should be able to see the fat tail on the left in blue. This is not
as dramatic as I might have wanted but it’ll do:
Then, when we take these two distributions (the mix) and use them to
generate returns over 30 periods (50,000 iterations) and render the results in
terms of the expected geometric mean at each period N for N=1:30, this is what
happens (at least for this example):
The difference isn’t huge in this particular
parameterization and example, but it is there. One more thing to worry about in
a return generation process.
Recap
The main thrust of this essay is that after the task of
designing and optimizing a portfolio is complete, you are still not done
because, if you have not already factored in some of the issues above, the
portfolio may not live up to its expectation for a variety of reasons that are
linked to time and the volatility of returns.
Taking higher risk is supposed to lead to higher return and better
outcomes but when that higher risk return-generation process is subjected to
the flow of time you may end up with less than you thought and less even than a
lower return strategy. It should be clear now
that time subjects a process to change and risk. One should know this not only at the point
the portfolio is designed but one should also be prepared to monitor the
situation as time unfolds. To recap the main points:
- Over the long run you don’t get the arithmetic return that is the output of MVO
- The long run geometric mean of the average return is not the average of the geo means
- A common estimator for the long run geometric mean is imperfect but usable
- A low return strategy can dominate a high return/high volatility strategy fairly quickly
- Retiree planning time-frames may be less than infinity making geometric mean estimation hard
- In portfolio design a unit of risk may, at some point, not reward with more long term return
- Skewed terminal wealth dispersion means median wealth is relevant and related to the geometric mean
- High volatility strategies may be challenged in terms of likelihood of helping reach goals
- Maximizing geometric means sounds like a good idea but may not be on a scale of one life
- The individual lives of retirees do not earn the geometric mean, only their own “path”
- Two paths that that end in the same place may still have different capacities for spending
- Non-normal distributions don’t help with multi-period returns. Call it an uncertainty tax.
Notes
-----------------------------------
[1] Note that a rebalanced portfolio in the presence of a
continuous trend or a trend with few and/or small reversions will underperform
one that is not rebalanced. In a permanent uptrend, continually selling a 7%
return asset and buying a 4% asset will underperform a buy and hold, knowing
though that your allocation will be wildly out of whack at some point.
[2] I picked up that expression from something by Michael
Zwecher, whom I highly recommend in terms of his retirement writing. He
attributes the phrase to the Midwest, somewhere around Wisconsin. Wisconsin is
also where instead of saying “six of one, half dozen…” they say “a horse a
piece.”
[3] Note that this illustration might have more illustrative
power if the higher risk portfolio had a higher limit for geometric mean than
the balanced portfolio. Then it might be easier to see that there may be higher
long term return both arithmetically and geometrically but also that the short
term Geo mean outcomes would be hair raising. Easy enough to contrive the next
time I do this.
[4] Suarez and Suarez (2014)
[5] There may be some analytic ways to do this kind of
thing. Right now, I only know how to simulate this effect.
Reference
-----------------------------------
Markowitz,
H., Blay, K. (2014) Risk Return Analysis, Volume 1. The Theory and
Practice of rational Investing. McGraw
Hill.
Markowitz,
H. (2016) Risk Return Analysis, Volume 2. The Theory and Practice of
rational Investing. McGraw Hill.
McCullogh, B. (2003), Geometric Return and Portfolio Analysis. New Zealand Treasury Working Paper.
McCullogh, B. (2003), Geometric Return and Portfolio Analysis. New Zealand Treasury Working Paper.
Michaud, Richard (2003, 2015) A
Practical Framework for Portfolio Choice
Mindlin, D (2012) On the Relationship between Arithmetic and Geometric Returns,
Geometric Mean, Wikipedia
Mindlin, D (2012) On the Relationship between Arithmetic and Geometric Returns,
Geometric Mean, Wikipedia
Suarez, E., Suarez, A., Waltz, D; (2014) The Perfect
Withdrawal Amount: A Methodology for Creating Retirement Account Distribution
Strategies.
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