MomentumFinal

Engineered to grow!
: +41 44 586 1222 : hello@acorns.ch : www.acorns.ch
Islisberg 2, 6340 Baar, Switzerland
INSIGHTS
Acorns Investment Research is a Swiss-based
firm specializing in the development of
proprietary trading strategies and software
tools for our clients and our own trading.
Acorns Insights are publicly available reports
that provide non-proprietary disclosures
about our work as well as observations and
commentary on third party published works
that we find interesting and/or useful in our
work. Acorns Insights are meant to offer a brief
synopsis of interesting research topics at a
reasonably high level of technical abstraction.
The reader will be directed to other
documents for further detailed discussion
of the topics under consideration and
additional background information. Acorns
Insights do not provide investment advice or
recommendations to investors of any kind.
ABOUT INSIGHTS
Copyright 2016 Acorns Investment Research,LLC.All rights reserved.This document is distributed for educational and informational purposes only.
Please see the last page of this document for disclaimers and disclosures.
A One Hundred Year Backtest-
Seriously?
Tom Wilson
August 2015

Engineered to grow!
INSIGHTS | AUGUST 2015 | Page 2
Here’s an idea;
• Let’s collect global asset market data (monthly) going back 100 years.
• Then let’s develop a trading strategy and optimize its parameters based on
the first 50 years of data.
• Then let’s test it on the most recent 50 years of out of sample data–from
1964 to 2014.
Seriously?
Would anyone presume to believe that a trading
strategy optimized on 50+ year old data would be
relevant,given all the changes in the market rules,the
world economy,trading strategies,and technology in
the ensuing years? Is it possible for such a strategy to
achieve profitable results? Perhaps surprisingly,the
answers to these two questions appear to be yes and
yes.
In ACenturyofGeneralizedMomentum;FromFlexible
AssetAllocations(FAA)toElasticAssetAllocation(EAA)
published in 2015,authors Wouter J Keller and Adam
Butler undertake the exercise described above and
publish some very impressive results. Their strategy
involves a monthly rebalancing between cash and a
small set of long only investment alternatives. Using
parameters optimized over the market period 1914 to
1964,they provide out of sample results from 1964 to
2014 for three different investment universes;
• Small global asset universe comprised of 7
investment alternatives.
• United States only asset universe comprised of
15 investment alternatives.
• Larger global asset universe commprised of 38
investment alternatives.
In each of these three cases the out of sample (1964
to 2014) trading system results were consistently
and significantly better than the S&P 500 and equal
weighted asset universe models as we will highlight
later in this report.
The paper does not describe in detail the source of
data or data prep/cleanup used for each asset in the
study,but they cite the sources used in aggregate,
which include MSCI,Fama/French,Ibbotson,Barclays,
and Global Financial Data,Yahoo Finance,and
Bloomberg.
Wall Street, early 1900s

Engineered to grow!
In this section I’ll try to summarize the approach and
the findings detailed in the paper. For those with a
strong interest in the subject and/or an interest in the
mathematics,I encourage you to read the full paper.
To perform a parameter optimization on the first
50 years (in-sample data),the authors started with
a simple geometric model to determine the asset
weighting values at each monthly rebalance
period. The equation encompasses return,
volatility,and correlation values for each
asset evaluated over a consistent
lookback period (described
later) as shown below;
wi = (ri)^WR
x (1-ci)^WC /
(vi)^WV for ri > 0,
otherwise wi = 0
where ri,ci,and vi are the return,
volatility,and correlation of asset i over
the prescribed lookback period and WR,
WC,and WV are parameters to be optimized.
For each asset,if the return is less than 0,the
corresponding portfolio weight is set to zero,and a
proportion equal to 1/N (where N is the total number
of assets in the universe) is set aside for cash as a
“crash protection”allocation.
The asset weights are subject to:
wi >= 0 (long only)
sum(wi) = 1.0–WCP (WCP is the“Crash
Protection”weight allocated to cash)
Also,at each monthly rebalance period the money
to be invested in assets other than cash is allocated
to a fixed percentage of the assets that have the top
weighting scores only. For example,if the percentage
(called TopX) is set at 50% and there are 11 assets
with a positive return,then the number of assets
purchased in the rebalance period would be 5
(50% of 11 rounded down). The investment
amount in each funded asset is then
proportional to its weighting score (wi
above) relative to the others.
At each rebalance period
the historical return
(ri),volatility (vi),and
correlation (ci) are each
determined as described
below:
• ri is computed as the average
excess return (actual return minus 13
week TBill yield) over the preceding 1,3,6,
and 12 month periods. By averaging these
four periods they are utilizing results from 12
months of history,but giving greater emphasis
to more recent results.
• vi is computed as the standard deviation of the
previous 12 monthly total returns (not adjusted
for the TBill rate as in the case of ri).
• ci is computed as the correlation between
the previous 12 monthly total returns,and
the previous 12 monthly returns of an equal
weighted portfolio of the full universe of assets.

Engineered to grow!
universes,they find a couple of combinations that
produce strong performance across the board,which
they label the Golden Defensive EAA model and the
Golden Offensive EAA model. These correspond to
optimizations of the Calmar 5% goal and Calmar 10%
goal,and are defined as below;
• wi = SQRT[ri x (1–ci)] for ri > 0,else wi = 0
(Golden Defensive EAA)
• wi = (1–ci) x ri ^ 2 for ri > 0,else wi = 0 (Golden
Offensive EAA)
Note that the vi term is missing.During early in-
sample testing it was determined that the volatility
factor had very minimal impact on the strategy
performance given ri and ci,so it was set to 0 for all
subsequent tests.
They also found that the best TopX value is a
decreasing function of N (number of assets in
universe) as N grows,but that at small values of N,a
TopX value around 50% is better. They propose that
the TopX value used for out of sample testing is set to
the minimum of;
• 1 + [SQRT(N) rounded up to the nearest integer]
• N/2 rounded down to the nearest integer
For out-of-sample testing of the three asset universes
where N = 7,15,and 38,the TopX values are 3,5,and 8
respectively.
Next we can apply the Golden Defensive EAA and
Golden Offensive EAA models to the 50 years of
The goal of the in-sample optimization is to determine
the“best”values of WR,WV,WC,and TopX. But in
order to optimize we must have an optimization
goal we are trying to achieve. The Sharpe Ratio is
often used as a goal for optimization,but the authors
argue that over a long time horizon the“fat tails”of
the return distribution (as evidenced by the Great
Depression in 1929 and the Global Financial Crisis of
2008) can have a huge impact on the trading results
,and these are not accounted for sufficiently in
the Sharpe Ratio calculation. They suggest that an
optimization goal that uses the maximum drawdown
as its risk dimension is more appropriate,and they
choose to utilize the Calmar Ratio which is defined as:
CRt = (R-t)/D
Where R is the annualized return,t is the threshold
return that the investor is seeking,and D is the
maximum drawdown. In their optimizations they use
CR5 (5% target return) and CR10 (10% target return)
as goals to optimize for a defensive and offensive
investor respectively.
So now with the optimization parameters defined,
the optimization goals set,and fifty years of historical
data between 1914 and 1964 for in-sample testing,
the authors proceed to analyze the data. The paper
does a good job of describing the process and many
observations along the way which I won’t duplicate
here,but the results are very interesting. After testing
49 combinations of parameters across all three asset

Engineered to grow!
out-of-sample data we’ve reserved–1964 to 2014–to
see if these golden models are really golden. Below
I’ve attached a summary of the major results directly
from the source paper. The measurements used for
comparison purposes in the tables below are;
• R (CAGR)–annual compound growth rate over
the 50 year period
• V-annualized volatility (standard deviation)
• D-maximum drawdown
• CR5-the Calmar Ratio based on a 5% target
• Sharpe Ratio–the sharpe ratio adjusted for a
riskfree rate assumed to be the 13 week TBill
rate
• 3y (Roll > EW)–the percentage of time that
the 3 year rolling returns using the strategy are
greater than the three year rolling returns of
an equal weighted portfolio of the same assets
rebalanced monthly.
• Costs (annual)-for analysis purposes,the
authors assume a one way transaction cost of
0.1% on each transaction to cover commissions,
slippage,etc. The costs in the table are then the
annualized cost of all transactions.
And the benchmarks for performance comparison are;
• EW-an equal weighted portfolio of the same
assets rebalanced monthly.
• SP500-S&P 500 performance
Wall Street, early 1900s

Engineered to grow!
(N=7, OS, Golden) EAA Def EAA Off EW SP500
R (CAGR) 13,1% 14,5% 9,5% 9,9%
V (An. Volatility 9,7% 11,4% 11,3% 15,0%
D (Max DDown) 19,1% 25,3% 43,4% 50,8%
CR5 (Calmar 5%) 42,4% 37,6% 10,4% 9,6%
SR (Sharpe Rf) 83,3% 82,6% 39,7% 32,1%
3y (Roll > EW) 68,5% 79,0% 0,0% ?
Costs (Annual) 0,5% 0,6% 0,0% 0,0%
TABLE 1 / FIGURE 1: Results for the small global asset universe (N=7)

Engineered to grow!
R (CAGR) 10,3% 11,7% 9,6% 9,9%
V (An. Volatility 8,8% 10,1% 10,7% 15,0%
D (Max DDown) 12,5% 12,5% 34,8% 50,8%
CR5 (Calmar 5%) 42,6% 53,4% 13,4% 9,6%
SR (Sharpe Rf) 59,9% 65,6% 43,1% 32,1%
3y (Roll > EW) 50,5% 64,4% 0,0% ?
Costs (Annual) 0,6% 0,7% 0,0% 0,0%
TABLE 2 / FIGURE 2: Results for the United States asset universe (N=15)

Engineered to grow!
R (CAGR) 12,8% 14,3% 9,6% 9,9%
V (An. Volatility 7,9% 10,1% 9,6% 15,0%
D (Max DDown) 8,6% 14,1% 36,7% 50,8%
CR5 (Calmar 5%) 90,2% 65,7% 12,6% 9,6%
SR (Sharpe Rf) 97,4% 91,0% 47,8% 32,1%
3y (Roll > EW) 76,0% 79,3% 0,0% ?
Costs (Annual) 0,6% 0,6% 0,0% 0,0%
TABLE 3 / FIGURE 3: Results for the larger global asset universe (N=38)

Engineered to grow!
As you can see the results are consistently impressive
across all three asset classes under study. In reviewing
the methodology and results,there are several new or
unusual characteristics that I would like to highlight;
• The use of the Calmar Ratio as an optimization
goal results in strategies that tend to minimize
drawdowns as opposed to maximizing
efficiency (i.e.return/volatility risk) as
characterized by the Sharpe Ratio. It could be
viewed as a better global optimization over a
long time horizon than the local optimization
afforded by the Sharpe Ratio. It is commonly
seen in practice that long periods of steady
gains can be wiped out by a single short
term negative event (aka fat tail,black swan,
etc). This study shows that by avoiding large
drawdowns the long term gains are improved.
• The correlation variable,ci,is based on
an unusual definition.It is measured as
the correlation between the asset under
consideration and an equal weighted portfolio
of all of the assets in the universe. As such it
is a single number for each asset,rather than
a matrix of cross correlations between each
pair of portfolio constituents. By defining the
correlation in this manner it gives a strong
indication of the impact of the individual asset
on the portfolio as a whole.
• The“crash protection”allocation to cash
is based on each asset’s 12 month return
momentum,not the actual near-term return
value,and is also adjusted for the risk free
alternative. Recall that if the average“net
return”(actual return minus risk free rate) over
the past 12,6,3,and 1 month lookback periods
is negative,then the weight of the asset is set to
zero and the cash allocation weight is increased
by 1/N. As a result of this approach,only assets
with long term negative performance are
converted to cash in any rebalance period.
• The inclusion of global asset classes appears
to improve the performance of the EAA
strategies relative to the benchmarks. This is
not surprising,but is a revalidation of lessons
learned in many other previous studies that
evaluate long term (10+ years) performance.
• The optimized Golden Defensive and Offensive
EAA models turn out to be deceptively simple
equations that were derived from a fairly
complex analysis. One wonders if there might
be an underlying“simple explanation”for these
results. It seems counterintuitive to me that a
trading model developed by optimizing generic
parameters over a fifty year time span would
continue to“work”over a subsequent fifty year
time period–unless there are some underlying
principles involved.
I hope you find this summary interesting and
encourage you to review the original paper,available
at the Social Science Research Network,http://papers.
ssrn.com/sol3/papers.cfm?abstract_id=2543979.

Engineered to grow!
DISCLOSURES AND DISCLAIMERS
This document has been prepared by the author(s) listed on the cover page,and is provided for informational and
educational purposes only.The views expressed herein represent the current opinions of the authors of this document
only,which may be different from,or inconsistent with,the views of Acorns Investment Research.Such views may be
historic or forward-looking in nature and may reflect the subjective assumptions and judgments of the author(s) of
this document.The views expressed in this document are subject to change without notice.
While the information contained herein was obtained from or based upon sources believed by the author(s) to be
reliable,Acorns Investment Management has not independently verified the information and provides no assurance as
to its accuracy,reliability,suitability or completeness.Acorns Investment Management and/or its staff and employees
may have market views or opinions that differ from those discussed,and may have a significant financial interest in (or
against) one or more of such positions or theses and/or any financial instruments mentioned.
This document may employ data derived from third-party sources.No representation is made as to the accuracy of
such information and the use of such information in no way implies an endorsement of the source of such information
or its validity.All information is provided as of the date of this document,and Acorns Investment Management
undertakes no obligation to update the information at any point in the future.
Under no circumstances should this document or any information contained herein be construed as investment advice
or as an offer to sell or the solicitation of an offer to buy any securities or other financial instruments.This document is
not,and shall not,be construed as an advertisement or an offer or solicitation for any brokerage or investment advisory
services.Acorns Investment Research is not an investment management firm,and offers no specific recommendations
or advice on the purchase of any financial instrument of portfolio.
Any discussion of past performance is not necessarily indicative of future results,and Acorns Investment Management
makes no representation or warranty,express or implied,regarding future performance or events.Any statements
regarding future events constitute only the subjective views or beliefs of the author(s).Words like“believe,”“expect,”
“anticipate,”“promise,”“plan,”and other expressions or words of similar meanings,as well as future or conditional verbs
such as“will,”“would,”“should,”“could,”or“may”are intended to identify forward-looking statements.
Certain assumptions have been made in the course of preparing this document.Acorns Investment Management
makes no representations or warranties that these assumptions are accurate.Any changes to assumptions made in the
preparation of this document could have a material impact on the information presented.
The information contained herein is not intended to provide,and should not be relied upon for,investment,
accounting,legal or tax advice.This document does not claim to advise the reader concerning the nature,potential,
value or suitability of any particular sector,geographic region,security,portfolio of securities,transaction,investment
strategy or other matter and the information provided is not intended to provide a basis upon which to make an
investment decision.The reader should make his or her own independent decision regarding whether to enter into any
transaction,and the reader is solely responsible for his or her investment or trading decisions.
In no event shall the author(s),Acorns Investment Management or any of its officers,employees or representatives,
be liable for any claims,losses,costs or damages of any kind,including direct,indirect,punitive,exemplary,incidental,
special or,consequential damages,arising out of or in any way connected with any information contained herein.This
limitation of liability applies regardless of any negligence or gross negligence of the author(s),Acorns Investment
Management,its affiliates or any of their respective officers,employees or representatives.The reader accepts all risks
in relying on this document for any purpose whatsoever.
No part of this material may be reproduced in any form,or referred to in any other publication,without express written
permission.

MomentumFinal

Recommended

Recommended

More Related Content

What's hot

What's hot (19)

Viewers also liked

Viewers also liked (20)

Similar to MomentumFinal

Similar to MomentumFinal (20)

MomentumFinal