Calculations and measures
There
are different similar datasets that can help with this analysis. We used one
from economist Robert Shiller data [who in reading my book has suggested his
own repository]. For interpretation ease, we look only at the
stock [with dividends] return, where for each date we then see the subsequent 10-year
return. We’ve geometrically converted.
Now
that series would run through 2012 of course, where the final 10-year forward
returns meet up with today. We also
recreated a second modified series looking at 2.5-year returns to better see the
recent coronavirus-cyclical turn; along with my own Bayesian interpretations
that any “underlying” depressed decades are known some-way through that lengthy
period.
There
is no better alternative for us. Right now,
we are drowned out by the finance industry popularizing the arbitrary calculation
“as of” the top of the previous bull market, or the yet unparameterized
subsequent bear market. Even the notion
of using 10-years from the start is a bit dubious, and something we have
discussed with Professor Shiller when I ran the TARP analytics team at the U.S. Department of the
Treasury [given the promotion of the CAPE valuation metric].
Now
using the total stock return, we see the series that has a reasonably high 5%
standard deviation [more on this later] for a 10-year annualized. Most practitioners would venture to say this
would equate to ~5%x[√10 years] or 16% annualized returns in any given 1 -year horizon. Or even higher if using a crude, arithmetic series.
Nevertheless
utilizing this most generic formulae, there would be a 5% chance of total stock
returns for any given decade would be < 0%.
Low, but also implying an uncertain 30% chance of being < 6%
[incidentally a measure most felt would be needed to
justify balance sheet debt].
As
noted above, these are good return distributions except when you fall well
below 6% and still have large household balance sheet debts. This is critical time to know the risk and
the precious time squandered being levered [by implication] with the
markets. We discuss more of these
implications below.
Cycles and math
There
are multiple contextual changes over time, which make these lengthy time series
difficult to gauge using straight rules.
For example, <X% or for >Y years, etc.
We
use a technique familiar to our followers, laying at the crossroads of
probability theory and machine learning.
Looking at the series in terms of return zones/clusters, which we see
oscillate below. Always averaging less
than 8% [particularly in the late-20th century following a uniquely
famous depression era in the early century prior to that]. But we are now in the lowly -2% to +3% range,
to be considered at the higher-end of a lost regime [notice we are no longer
thinking in terms of decades].
In
truth the time length concept doesn’t matter as much as the fact that the
cycles are in clear clusters not based on said levels [e.g., <0%].
Once
we leave the lost regime we enter the yellow-colored transitionary regime, and
then [after some time] can only enter the positive regime.
The
same cycle works in reverse. There is
no erratic jumping about over the transitionary regime: once in a regime such
as the lost regime the market tends to stay there for a while. Almost a year, minimum!
It
is important to not have spurious or rash signals to fit your insights around. What if we instead had popularly looked
exclusively at the same 24% of historical time, statistically ill-defining the weakest
24% of returns [<5.5%] as “lost decade”.
See what those contemporaneous erratic signals would be, below.
If
unclear, there are wide, overlapping swaths [eg in the 1930s] where regimes
were constantly flip-flopping over the “transitionary decade” and making
investment insights error-ridden and impossible. A low quality use of statistics!
But
when we data-intermixed on the probability theory and machine learning we can
use our evolving clusters to create beautiful, large data contemplations. And the fact that we are looking right now at
trending into the low returns is meaningful.
These returns qualify [unofficial since the dividend payment payments
have not been fully calculated yet by the market] to be in the “transitionary
regime”; adding further fuel that there is likely a somewhat greater-than-appreciated
risk upon shareholders in the near the future.
Perhaps not forever, but the direction of the signal is, here, now
known.
We
see that there is a much higher likelihood of seeing the low-end of the full
return distribution potential, which we will solve for below. Again this is not meant to conjure false
doom, but the risks are distinctly higher [and likelier for sure] so that one
would be foolish to not be prepared as they speculate with their capital in the
near future.
Noise decomposition
Here
we will explore both probability theory and data science mathematics in their separate
ways. For example looking at anything
sub-3% annual as low returns, but also anything super-8% as potentially an aggressive
omen. When looking at these charts think
about the stories we allow ourselves to be told, to explain the noise we [more
often than not] experience.
Looking
at the long view we notice that the depth of a market crash, instead of the
duration of the same crash, suggests a slightly lengthier period of lost
regime ahead. This is not well
understood by most speculators and investors who assume, particularly in recent
decades, that any large crash is [if anything] must at once be followed by a
late-20th century style recovery. Such a
generation must learn.
Now
for the current bear market that we are now in, it has been only approximately a
year in length so far, and of medium depth [per asset class]. So does not predict much more of anything
other than we are [particularly given the near 1-year duration] in a likely
transitionary stage and one that we claim will linger for a while. And then also jointly seeing a lost regime
probabilistically, though again not necessarily in for a worse or longer than
expected one. Sad for some, but that’s
the current reality [which may get better; may get much worse].
We also show a few proofs below for these expected results of balancing variance from model versus variance from error [here, or click my university teaching page at top of here which may temporarily require Google sign-in] where the larger standard deviation of 5% for total stock returns can be decomposed into more modest: 2.0% for positive, 2.1% for transitionary, 2.5% for lost.
Proof
1
From moment generation functions we know Variance[X] = E[X2]-E[X]2.
If this initiation is unclear, contact me for added resources. Y=regime
∴ Variance[X|Y]
= E[X2|Y] - E[X|Y]2
E[Variance[X|Y]]
= E[E[X2|Y]] – E[E[X|Y]2]
= E[X2|Y] – E[E[X|Y]2]
Expand terms, and rearranging:
= Variance[X] – Variance[E[X|Y]
∴ Variance[X] = Variance[E[X|Y] + E[Variance[X|Y]]
Proof
2
y – β0 - β1x
= ε
SSyy - β1SSxy = SSE
SSyy = Covariance(y,y)*n = (σy)2*n = SST
β1SSxy =
β1Covariance(x,y)*n = SSR
β1 = SSxy/SSxx =
Covariance(x,y)/(σx) 2
Note that SST-SSR=SSE, and therefore SST=SSR+SSE
Proof
3
Consider error to be the difference between the estimated θ^ of the θ regime variable.
Error2 [θ^]
= E[[θ ^-θ] 2]
= Variance[θ ^-θ] + [E[θ ^-θ]] 2
= Variance[θ ^] + Bias[θ ^]2
What
about for the regime-agnostic variation in either <0% returns, or <5.5% returns [i.e., the lowest 24% of times]? As expected, but wrong per Proof 3 above, it
is a lower 1.0% for the <0% returns.
And 2.1% for the <5.5% returns.
And
as a reminder we don’t need to solve for other higher return variations. The mathematical formulaes already evidences
that there will be continuous higher [even if we tamp it down in our overall
analysis] bias considerations.
Everything
considered then, our own probability analysis of the data is that we are seeing
the following long-term returns:
15% chance returns above +12%
25% chance returns between +6 and +12%
40% chance returns between +1 and +6%
20% chance returns below +1%
Perhaps
not extreme [despite 20% of mostly down returns is several times higher than the regime-agnostic reports we discussed earlier]. And we’ll explore what these return ranges really mean, towards
the end of this article.
Age and life dependency
The
question of risk is amplified by circumstance, as shown in our actuarial
article [page 17]. Consider two people, both contributing as
follows and neither withdraws over the decade.
|
Year |
Person 1 |
Person 2 |
|
1 |
$Z |
$Z/10 |
|
2 |
|
$Z/10 |
|
3 |
|
$Z/10 |
|
4 |
|
$Z/10 |
|
5 |
|
$Z/10 |
|
6 |
|
$Z/10 |
|
7 |
|
$Z/10 |
|
8 |
|
$Z/10 |
|
9 |
|
$Z/10 |
|
10 |
|
$Z/10 |
|
Total |
$Z |
$Z |
An
early-stage person has more of a retirement contribution profile pattern 2, and
a late-stage person has more of profile pattern 1. Say this is a lost decade where returns
simply randomly spin up ~W% or down ~-W% in any
given year. Persons 1 and 2 will
undoubtedly have quite different results.
A
lost regime therefore has different consequences for distinct people. Add unexpected liquidity concerns [solvency
crisis at any cycle-end]. Or the
inability to have perfectly passively indexed portfolios. And then we have convexity drip risk
augmented on.
We
see this in practical terms as retirement-dated funds, and 60/40 funds, have gone nowhere [i.e., 0] over the past few
years. But those who continued to blindly
rebalance and contribute are doing even worse than 0.
The
key is to be attentive with life’s risks.
Simply assuming there is an auto-pilot optimal outcome leads to the
types of “shocking” joint long-term risks we see today.
By
my actuarial estimates there is nearly a 20% or so financial advantage in your retirement
portfolio, by taking advantage of a highly-probable lost regimes [and fundamentally
higher still, for early-career individuals considering household formation].
Not
by staying in markets, but by tilting to paying down the shouldering debt
expenses. Timely avoiding the period
where there is anyway a reasonably higher chance to know of sustained, sub-debt
returns.
And
paradoxically freeing yourself up convexively sooner helps build physically
enduring habits. To confidently steer
towards achieving even loftier life and retirement dreams, currently delayed by
high debt costs.
Broad cross-asset, financial wealth globally
We
also look at lost decades in bonds, similar to the total stock returns used
above.
There
would be a 15% chance of total bond returns in any given decade to be <
0%. But given the smaller 3% standard
deviation in these 10-year annualized returns, this also implies a 70% chance
of bonds providing < 6%.
One
of these trade-offs, expressed here as excess returns, is negative [between
stocks and bonds] more than 14% percent of the time. And at ~ 1.5% excess returns demanded we mostly
align with the historic, lost regime periods we have studied in this
article. Also the periods that are
concentrated and heavily-overlapping with the lowest total stock return
periods.
What
about for joint-portfolios? Stocks
dominate the variance aspect of a portfolio, even when bonds are known to at
least buffer the more volatile swings [though clearly not in 2022]. For a 50/50 fund for example the returns were
negative [less than a percent of the time], and below 4.5 percent [a quarter of
the time].
This
is far less than would be expected by random covariation among asset classes [inverse concordance in fact, where there is
no data quality for copula modeling.] Again,
suggestive proof that more than “extreme exclusive” diversification occurs with
a broad basket of assets.
This
broad portfolio still has a 3% standard deviation [slightly higher than bond
investments alone]. While this seems
appealing, it is much higher to justify solely owning just a portfolio of risk
at the expense of high-yielding interest burdens.
Also
a note that while Professor Shiller’s data only focused on US markets, our
statistical analogies can be applied to the broader global and private market risks. Where similar magnitude volatility, more
questionable asset returns, shortened time history for risk analysis, all
meet-up [and all in a typically mediocre way].
Ignored decisions from history
Despite
the typical feeling working through TARP, and the ever more recherché
article that the run-up is most different post-global financial crisis. The results of the cyclical regimes show it
closer to a much longer run. Say, multiple
decades.
Some
of this difference is mathematical, a combination of actuarial smoothing for
the most recent time period, as well as the exhaustion of inflation and then
spike in the most recent period. It
makes the most recent, and at time unofficial, results appear better than the real
risk. Very unlikely to continue, and we
know the risk is to the downside particularly with high inflation
dampening real returns [perhaps only tamping down on high nominal growth rate].
Market
risk tends to expose victims that are the least prepared [see the randomness chase portion of our colors and numbers book], in the market’s
view. There is always someone who is the
weakest and only capable of tolerating an aggregate loss at the critical
level. A level which will [by definition]
get tested in at least isolation, and hopefully not in contagion – or convex
with other risk variables.
We
don’t know how the lost regime would unfold.
How much of it will only be measured in stock and bond markets. Likely it will be the combination that
unfolds across societal/political regimes, labor markets, and housing. Traditional measures of risk that worked well
in the past [inflation metrics, unemployment, nominal savings, etc] may fail to
adequately capture demographic and aggregate financial distortions this go
around.
To
summarize, all is not negative to be clear.
But there are still clear risks particularly depending on your
opportunity costs. Don’t assume for example that when we stated mid-article
that there was a 40% chance of returns [between +1% and +6%], that this meant
just +6%. That’s just being both hopelessly
foolish. Being positively biased in
recent years has had a cost.
As
it has did during this popular video in February 2022. And as it likely will continue to be in the
years and decades ahead. Be prepared to
rise, and importantly to flourish, on the other side of whatever
current regime is in store.
Appendix
I
am grateful for the many interactions in person and online over the course of this
writing. It has been an exciting and mixed
group on both extremes of either overly negative about markets and the economy [and
on the other extreme, overly upbeat]. As
a statistician I draw on everything possible to shape factual context, glean from the underemphasized [the missing data concept of our colors and numbers book], and the message.
Let’s
highlight a few interesting accounts or comments that have been useful in
refining analysis [excluding the many who reached out but did not want acknowledgement!]:
purchase colors and numbers, or statistics topics
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