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arXiv:2101.01006v1 [q-fin.GN] 4 Jan 2021
Design and analysis of momentum trading strategies
Richard J. Martin∗
January 5, 2021
Abstract
We give a complete description of the third-moment (skewness) characteristics of both linear
and nonlinear momentum trading strategies, the latter being understood as transformations of
a normalised moving-average filter (EMA). We explain in detail why the skewness is generally
positive and has a term structure.
This paper is a synthesis of two papers published by the author in RISK in 2012, with some
updates and comments.
Introduction
Trend-following, or momentum, strategies have the attractive property of generating trading returns
with a positively skewed statistical distribution. Consequently, they tend to hold on to their profits
and are unlikely to have severe ‘drawdowns’. They are very scalable and are employed in most asset
classes—most traditionally in futures, where they are a favourite strategy among CTA (‘commodity
trading advisor’) firms, but also in OTC markets—and by both buy- and sell-side practitioners.
The basic premise behind momentum is to buy what has been going up recently, and sell what
has been going down. In other words, if recent returns have been positive then future ones are
more likely to be positive, and similarly with negative. Systematic strategies formalise this notion
by (i) measuring momentum, essentially by smoothing out recent returns to obtain a signal that
is not too rapidly-varying, and (ii) having a law that turns this signal into a trading position, i.e.
how may contracts or what notional to have on. Put this way, the ideas that only the finest minds
can understand CTA strategies, or how the theory of statistics is of central importance to their
construction, or how one needs to have been steeped in managed futures for many years to build
a workable strategy, are seen to be self-serving and pretentious—a conclusion implicitly arrived at,
even if not thus expressed, by other authors.
Before talking about skewness we may as well deal with the first moment, that is to say the ex-
pected return. it is important to understand that this is an entirely separate matter. Any statement
about this depends on markets exhibiting momentum, i.e. serially-correlated returns. This can be
ascribed to the way information is disseminated into markets or of behavioural characteristics of
market participants. However, it is entirely subjective and is a matter of believing that markets
will continue to behave in the way that they have done in the past. In contrast, as we analyse in
detail here, even if market returns exhibit no serial correlation, during which period the strategy
will produce no average return, the trading returns of a momentum strategy will still have positive
third moment. What is interesting is that the skewness characteristic is a product of the design of
∗Dept. of Mathematics, Imperial College London, South Kensington, London SW1 2AZ, UK. Email:
richard.martin1@imperial.ac.uk
1
the strategy, whereas in long-only equities and credit the (negative) skewness is an intrinsic feature
of the asset class that has to be tolerated and risk-managed.
Positive skewness results from the way positions are taken. Suppose that we look at the trad-
ing returns from one particular instrument (perhaps US Tsy bond futures) of a particular period
(perhaps one week) over a long period of history. Let us group these returns by the size of the
underlying position. The magnitude of P&L will typically be larger when the magnitude of the
position is larger, but crucially it will typically be positive too. This is because momentum strate-
gies typically run bigger positions when they have already made money—as opposed to reversion
strategies that follow the opposite principle. In statistical language the full distribution of P&L is
a mixture of distributions of different mean and variance: the components with a higher variance
have positive mean and that is a recipe for positive skewness.
Studies on the subject have generally been empirical (for a good overview see e.g. [21] and
references therein, and [2] for a general introduction to technical trading). However, there is a
decent literature on quantitative aspects. The first work was by Acar [1] who derived a variety of
results in discrete time using different forecasting models and also ascertained that the distribution
of momentum trading returns has positive skewness. Potters & Bouchaud [20] consider a particular
type of momentum strategy and derive rigorous results about its performance. Bruder et al. [4],
and an even longer extension by Jusselin et al. [11], devote considerable effort to deriving moving-
average filters from an underlying model. In practice, however, this seems to create more problems
than it solves because the assumed model may be wrong: better, we think, is to use a convenient
definition of moving-average filter, here the exponentially-weighted moving average (EMA) as it
is easily calculated by recursion, and design a strategy using those. Then in 2012 two papers
were published in RISK by the author, which considerably broadened the scope of the subject.
Both focused on the third-moment characteristics of momentum models. The first [19] dealt with
linear models, by which we mean a signal proportional to a momentum signal obtained by applying
exponential smoothing to the market returns. The second [15] showed how to deal with strategies
defined as nonlinear transformations of momentum signals, within the same framework. This paper
is a synthesis of these two. More recently Dao [5] focuses on the connection between convexity,
option-like characteristics, and momentum strategies.
In building momentum strategies, two main considerations are important. The first relates to
backtesting, in other words finding what worked best in the past and assuming that it will continue
to do so. Part of the problem with this is that it is too reliant on historical data: if left unchecked,
it wastes an inordinate amount of time in fitting and overfitting, mainly because different models
typically produce almost identical historical performance. The second relates to design: that is to
say, without regard to the past, force the strategy to have certain statistical properties when the
market behaves in predefined ways. One idea, which we consider in depth, is when markets are not
trending (and so market returns are uncorrelated). Another is when the market does trend in a way
that it did in a particular historical scenario, such as gold in the first decade of this century, or in
the first 18 months from January 2019—some designs behave differently from others. There is to an
extent a trade-off between these considerations: better positive skewness and better performance in
certain trending scenarios may be obtained at the expense of worse average historical performance
and vice versa. This necessitates subjective decision, and despite the great effort of systematic
trading firms to claim that there is no discretion in the implementation of their systems—nowadays
assisted by the smoke-screens of statistical theory and ‘machine learning’—inevitably there must
be.
An incidental conclusion from reading [5] is that the SG CTA index is very easily replicated,
giving the lie to the contentions, blithely trotted out by the CTA industry, that barriers to entry are
so high, that the subject can only be understood by those with years of experience, that a cohort of
2
PhDs are required to build strategies, and that proprietary execution algorithms are important—
the last of these is clearly nonsense given the low speed at which the replicating strategy in [5]
trades (see Figure 7 in that paper). In fact, rather than clever trade execution being important
for momentum strategies, it is the reverse that is true: momentum is an important ingredient in
trade execution, as over short time scales many financial time series exhibit momentum. Further,
the relatively poor performance of the SG CTA index since the end of the Global Financial Crisis,
together with the underlying simplicity of momentum trading, should make investors question
whether CTAs’ management fees are justified, as well as how big an asset allocation they should
receive by comparison with standard investments in equities and fixed income.
A consequence of positive skewness is that the proportion of winning trades may well be negative
[20]. Small trading losses are common, but occasional big gains are produced when the strategy
levers itself into a trend. The longevity of trend-following funds suggests that this characteristic
has served them well over the years, pointing to the conclusion that the oft-asked question “What
is your fraction of winning trades?” is misleading. The link between moments and proportion of
winning trades can be formalised with the Gram-Charlier expansion, which estimates for a random
variable Y,
P(Y > E[Y]) ≈1
2−κ3
6√2π
where κ3is the coefficient of skewness of Y(see note1).
We show that the skewness of the returns depends strongly on their period2, so that even if
the one-day returns have no skewness, longer-period returns may be skewed. This may at first
seem curious, and arises because successive daily trading returns are not independent. Thus it is
possible to obtain a skewed distribution by adding non-skewed random variables, if those variables
are appropriately dependent.
This paper is arranged as follows. We begin with linear strategies (§2) and give a complete
exposition of ‘skew theory’ for them. We show how to calculate this skewness as a function of the
return period, by simple application of residue calculus. The skewness of the M-period trading
return depends on M: it rises to a maximum at a period proportional to the typical response-time
of the trending indicator, and then drops as M−1/2(eqs. 15, 17, 18/19). We then test on some real
data and find reasonable correspondence. Finally we analyse a particular hybrid linear strategy
that is not pure momentum and derive a simple condition that ensures positive long-term skewness
of returns.
An interlude on the option-like nature of trend-following ensues (§3) which is a natural conse-
quence of §2. This has since been treated in an excellent paper by Dao et al. [5] who point out
that in effect one is buying long-dated options and selling short-dated ones. This explains neatly
why momentum strategies suffer badly from whipsawing, when short-date volatility is high, and
perform well when the market moves steadily in one direction. The effect is important because it
causes momentum strategies to hold on to most of their previous profits during the periods where
they are not making money. As pointed out by Till & Eagleeye [21], this “long-option behaviour”
distinguishes them from other strategies that tend to have higher Sharpe ratio, the implication
being that the higher Sharpe is a form of remuneration for negative skewness. By the same token,
1Take for example the exponential distribution: the exact probability of exceeding the mean is e−1≈0.368; the
Gram-Charlier approximation gives 1
2−2/(6√2π)≈0.367, which is very close. On the other hand there may be
considerable divergence for other distributions: for example the third moment may not even exist, or one may have
a right-skewed distribution with negative third moment but positive higher odd moments. Nonetheless, the above
approximation works ‘better than one might suppose’.
2By the M-period trading return we mean the gain in P&L between date nand date n+M. This is to be
distinguished very carefully from the market return which simply means the change in price of the traded asset over
that period. We are assuming a discrete time model with the time increment being 1 day.
3
as pointed out in a different context in [17], positive skewness can result in longer drawdown times
than strategies with zero skew, depending on what period of history is being considered—so time
spent in drawdown is not necessarily a good measure of strategy performance.
We then move on to nonlinear strategies in §4. An arbitrary nonlinear function of several mo-
mentum factors (of different speeds) would be very difficult to analyse, so we opt for nonlinearly
transforming each momentum factor first, and then the position is a weighted sum of the trans-
formed factors. We extend the work on skewness of trading returns, studying the effect of the
nonlinear transformation. This analysis is primarily a matter of algebra, and we derive new results
(27;28,29) for the term structure of the skewness of trading returns. For some useful instances of
the model (30,31,32) we can evaluate these expressions in closed form, making for easy compu-
tation. It turns out that the nature of the transformation is very important and can cause the
positive skewness to disappear or even become negative. For example, one simple transformation is
the binary construction with a position of +1 or −1 according as the momentum factor is positive
or negative—but, as we show here, there are good reasons to suppose that it is not optimal. We
conclude with some general remarks about the optimal design.
1 Basics and model setup
We have already mentioned returns and now formalise this notion. Simply, the return is either
the change in price, or the relative change in price, the former being Xn+1 −Xnand the latter
(Xn+1 −Xn)/Xnwhich is approximately the difference in ln Xbetween the two time points. The
latter can only be applied when prices are positive and is therefore inapplicable to asset classes
such as interest-rate swaps3, but it is the most natural definition for equities, bond futures and
most commodities. We say ‘most’ commodities because a major upset occurred in the front WTI
oil contract in April 2020 when it went negative [16]. On the other hand the former definition is
most natural for interest-rate futures. We use the former definition in the ensuing algebra but this
choice is not critical to the theoretical development.
We define Un+1 to be the return per unit volatility4for the asset Xbetween time nand n+ 1,
i.e.
Un+1 =Xn+1 −Xn
ˆσn
,ˆσn=En[(Xn+1 −Xn)2]1/2(1)
with Endenoting an expectation conditional on Fn, the information known up to and including
time n. The reason for dividing by ˆσis that we wish Unto be nondimensional and appropriately
normalised.
Following the general principle that one should bet a number of contracts (or contract notional)
inversely proportional to the contract volatility5, we define the position in the asset to be ϕn/ˆσn
at time n, where ϕnis to be a function of any or all of the U’s up to and including Un. One must
3As a swap at inception has zero PV. If one wants to consider the momentum of the underlying swap rate, rather
than the contract PV, then this is quite sensible, though not quite the same as the contract PV includes the effects
of carry and rolldown, not just the changes in par swap rate. Nonetheless one should still use absolute changes for
the obvious reason that interest rates can go negative; also the absolute variation in EUR and USD rates as examples
has not been strongly spot-dependent over the last 25 years or so.
4Also known as the risk-adjusted return.
5Essentially to keep a reasonably constant level of risk on: if the same position is held while the volatility rises
substantially, one is likely to break one’s market risk limits. More formally this follows from stochastic control theory,
see e.g. [3, §14]. Writing µfor the drift and σfor the volatility of the traded asset, the optimal position always emerges
as µ/σ2multiplied by a few factors that pertain to the exact setup (utility, etc). If the dimensionless trading signal
Sis representative of the risk-adjusted return µ/σ, the position is S/σ.
4
0.5
1
1.5
2
2.5
3
3.5
4
2000 2005 2010 2015 2020
CTA
TSY
SPX
−1.5
−1
−0.5
0
0.5
1
1.5
0 50 100 150 200 250
skewness
return period /days
CTA
UST
SPX
Figure 1: Performance, or ‘total return’, and skewness of market returns, for three indices: SG CTA index;
US Treasuries (7–10y bucket); equities (SPX). Data source for top plot: Bloomberg.
have ϕn∈ Fn, otherwise the strategy can cheat by looking ahead. Clearly the P&L arising from
the period between time nand time n+ 1 is ϕnUn+1 .
For much of this paper we assume that the risk-adjusted returns Unare i.i.d.6and of zero
mean. Thus, we are studying the behaviour of the strategy under the assumption that it is not
generating any expected return (Potters & Bouchaud do the same). We also assume that Unhas
zero third moment, so that its first three moments are 0,1,0. Thus although Unhas no skewness, we
are about to show that the trading return may have skewness. We make no further distributional
assumptions about the (Un).
6The raw returns need not be, because of stochastic volatility, which is another reason for dividing off by an
estimate of the stdev of the asset return.
5
The period-Mtrading return is defined as
Y(M)
n=
M−1
X
k=0
ϕn+kUn+k+1.(2)
The first moment of the trading return is clearly zero. The second moment is given by7
(ϕ0U1+···+ϕM−1UM)2.
Let us consider this expectation on expansion as a product. The cross-terms all vanish because
each contains a Un+1 term multiplied by a term in Fn. This leaves the squared terms, which give
simply
Mhϕ2i
(as hU2i= 1). The proportionality in Mis a consequence of the trading returns being uncorrelated
(note that we have not said ‘independent’).
The third moment of the trading return is given by
(ϕ0U1+···+ϕM−1UM)3.
Expanding this as a product, we obtain four types of terms:
(i) ϕn1Un1+1ϕn2Un2+1ϕn3Un3+1 with n1< n2< n3;
(ii) ϕ2
nU2
n+1ϕmUm+1 with n < m;
(iii) ϕ2
nU2
n+1ϕmUm+1 with n > m;
(iv) ϕ3
nU3
n+1.
The independence of the U’s and the assumptions about their moments show that (i), (ii) and (iv)
all vanish. We are therefore left with (iii), which can be written
3X
0≤m<n<Mhϕ2
nϕmUm+1i.(3)
As ϕ2
nmay depend on Um+1, this expression is not necessarily zero, though it must be zero when
M= 1 (as the sum in (3) is empty: the cause is symmetry of the market returns). The factor of 3
comes from the three ways of permuting the indices in (iii).
It is worth mentioning that if the third moment of the market returns is nonzero then this is
likely to influence the skewness of the trading returns. ‘Risk markets’ such as equities and credit
typically have returns that have positive expected return but negative skewness. Positivity of
the expected return will make a momentum signal have on average a positive allocation, that is,
E[ϕn]>0. The terms listed as type (iv) above will then cause the skewness of the short-term
trading returns to be negative. In fact this is visible in Figure 1.
7h·i denotes realisation average.
6
2 Linear strategies
2.1 Notation and definitions
We now specialise these results to linear strategies, by which we mean
ϕn=∞
X
j=0
ajUn−j.(4)
Linear strategies have several advantages. They are easily constructed, for example through EMAs
(aj∝αj) which can be implemented recursively. They are also easily added, so that one can
combine momentum of different periods (or even have negative weights on momentum of certain
periods, so that one may capture counter-trending behaviour). Finally, analysis is reasonably
straightforward, and the moments of the trading returns can be captured using the coefficients (aj)
alone.
We shall need the autocovariance function of the impulse response:
Ra
k=∞
X
j=0
ajaj+k, k ≥0 (5)
and also the system function, i.e. the z-transform of the weights:
A(z) = ∞
X
j=0
ajz−j, z ∈C.(6)
This is bounded for |z| ≥ 1. A linear combination of EMAs always has a rational system function
and its poles are usually a key part of the design and analysis. For a general account, refer to [9].
The simplest example is the single-EMA case, which we will call ‘EMA1’: aj=αj+1 , and
A(z) = α
1−αz−1. This arises as the difference between the spot price and an EMA of past prices,
and is used by Potters & Bouchaud in [20]. The decay-factor αis linked to the effective period
of the EMA, N, by α= 1 −N−1, so that the EMA becomes progressively slower, or more highly
smoothed, as α→1.
Another important example is the difference of two expressions of this form, which we call
‘EMA2’: aj=αj+1−βj+1
α−βand A(z) = 1
(1−αz−1)(1−βz−1). This arises as the difference between two
EMAs of prices, a common device in technical analysis8. It has less day-to-day variation than
EMA1, on account of being the difference of two smoothed prices, or equivalently a double (rather
than single) EMA of the returns.
It is convenient to define a class of models that we call SPRZ (‘simple poles, regular at zero’).
The precise conditions are: A(z) bounded in |z|>1−εfor some ε > 0; the only singularities to
be simple poles; and A(z) regular at the origin. These should be thought of as mild analytical
conditions that enable the ready application of residue calculus; models with multiple poles can be
understood as limiting cases of models with simple poles as the poles coalesce.
2.2 Note on continuous time formulation
This subsection may be omitted at a first reading.
8See for example [10, §9] and also in many online articles on tea-leaf reading, e.g. www.stockcharts.com/school.
7
The definitions of the EMA relate neatly to a continuous-time setting. For a process Xtwe can
define any time-invariant linear system as
K[X]t=Zt
−∞
K(t−s)dXs
where Kis commonly known as the kernel. If Xtis a unit Brownian motion then the variance of
K[X]tis
kKk2=Z∞
0
K(t)2dt,
which we call the square-norm.
An EMA1 is then the difference between Xand its exponentially-weighted moving average,
which is an exponential smooth of the returns:
Xt−Zt
−∞
˙αe ˙α(s−t)Xsds =Zt
−∞
e˙α(s−t)dXs(7)
and its square-norm is 1/2 ˙α(of course ˙α > 0).
An EMA2 is the difference of two of these, with ˙
β > ˙α:
Zt
−∞ e˙α(s−t)−e˙
β(s−t)dXs(8)
and its square-norm is ( ˙α−˙
β)22 ˙α˙
β( ˙α+˙
β). The limit ˙
β→˙αobviously only makes sense if we
divide by ˙
β−˙αfirst, giving Zt
−∞
(t−s)e˙α(s−t)dXs(9)
and its square-norm is 1/4 ˙α3. This can also be written as the EMA of Xminus the double-EMA
(EMA of the EMA) of X. We call this ‘EMA2=’.
An important practical aspect of continuous-time signals is the notion of path-length, defined
for a process Ytto be
1
TZT
0|dYt|.
This is a concern because it relates to the rate at which money is lost in proportional transaction
costs. Infinite path length results in an infinite rate of loss, unless the problem is obviated. For a
Brownian motion the path length is infinite as |dYt|=O(√dt). Now if Xtis a Brownian motion
and we pass it through a linear system of kernel K, then what is the path length of K[X]t? We see
that
Zt+dt
−∞
K(t+dt −s)dXs−Zt
−∞
K(t−s)dXs=K(0) dXt+dt ·Zt
−∞
K′(t−s)dXs
and the first term will generate infinite path length unless K(0) = 0. The second term (without
the dt) is Normally distributed of zero mean and variance kK′k2, so its expected absolute value
is (2kK′k2/π)1/2. The conclusion is that the path length is finite for EMA2, infinite for
EMA1. This does not rule out the use of EMA1, as the theory of trading under proportional
transaction costs is reasonably well-established (see e.g. [18, 13, 14] and references therein), but it
does suggest that in trading systems EMA2 is preferable.
Now it may be advantageous to minimise the path-length, subject to two conditions: (i) the
variance of the output is to be unity (as otherwise the solution would be K≡0), and (ii) the
8
average lookback period is to be fixed (as otherwise we could take a normalised EMA2 and allow
both speeds to tend to zero). The latter constraint can be implemented in various ways, but a
felicitous one turns out to be
Minimise Z∞
0
K′(t)2dt s.t. Z∞
0
K(t)2dt = 1 and Z∞
0
t−1K(t)2dt = 1/τ (10)
where τ, of units time, is a given parameter. One boundary condition is K(0) = 0, and we require
sensible behaviour at ∞. This is a standard type of variational calculus problem and gives rise to
the ODE
K′′(t) + (λ+µt−1)K(t) = 0 (11)
where λ, µ are Lagrange multipliers. It is an easy exercise to see that te−˙αt is one solution of this9,
and so in this particular sense EMA2= is an optimal choice of momentum filter.
2.3 Second and third moments
It is immediate that the second moment of the M-period trading return is M Ra
0. For the third
moment, we have to find the Um+1Ujterm (j6=m+ 1) in ϕ2
nin the expression (3). This is
2∞
X
k=0,k6=n−m
an−makUm+1Un−k+1.
This now has to be multiplied by 3ϕmUm+1 and the expectation taken. Thus it is necessary to
look for any overlap between Un−k+1 and ϕm, and so in the k-summation we only need terms with
k≥n−m, and exclude the others. The resulting expression emerges as
3X
0≤m<n<M
2∞
X
k=n−m
an−m−1akam−n+k= 6 X
0≤m<n<M
an−m−1Ra
n−m= 6
M−1
X
k=1
(M−k)ak−1Ra
k.(12)
If we understand a pure momentum, or trend-following, strategy to be one in which all the (aj)
are positive, then by (12) the trading returns must be positively skewed. By the same token a
counter-trending strategy, with the a’s negative, has a negatively skewed return distribution even
if the market returns are symmetrical.
We can z-transform (12), i.e. multiply by z−Mand sum from M= 1 to ∞, to get
G3(z) = 6z
(z−1)2
∞
X
k=1
ak−1Ra
kz−k.(13)
From the presence of a double pole in G3(z) at z= 1 we deduce that the third moment is asymptot-
ically 6MP∞
k=1 ak−1Ra
kas M→ ∞. Recalling that the second moment is linear in M, we deduce
that as a function of the return period Mthe skewness starts from zero, reaches an extremum
somewhere and decays as M−1/2.
The M−1/2asymptotic is, intriguingly, the same as that observed in L´evy processes. However,
the origin of the skewness is completely different. With a L´evy process, it arises because the one-
period returns are asymmetrical but independent. Here, they are symmetrical but not independent!
9Depending on the Lagrange multipliers. There an infinity of solutions, and the ‘sensible’ ones are of the form
polynomial ×exponential. The polynomials in question are related to the Laguerre polynomials.
9
2.4 Further analysis of the third moment
The asymptotic third moment (without the 6Mprefactor) is P∞
k=1 ak−1Ra
kwhich is also equal to
1
2πiI|z|=1
A(z)A(z−1)2dz
(to see this, write the integrand as a product of Taylor series; the integral pulls out the z−1term).
This expression can be calculated using residue calculus, if we restrict ourselves to the SPRZ case,
as X
j
ρjA(α−1
j)2.
Meanwhile the second moment is MRa
0, and
Ra
0=1
2πiI|z|=1
A(z)A(z−1)z−1dz =A(0)2+X
j
ρjα−1
jA(α−1
j).(14)
Collecting the results together, we deduce that the skewness of M-period trading returns, for large
M, is
κ(M)
3∼6PjρjA(α−1
j)2
A(0)2+Pjρjα−1
jA(α−1
j)3/2M1/2
.(15)
In the EMA1 case we immediately obtain
κ(M)
3∼6α
(1 −α2)1/2M1/2∼3√2N
M1/2
where the right-hand expression is obtained by assuming that N= (1 −α)−1is not small. In the
EMA2 case, the poles are at α,βand are of residue α2,−β2, and the result is, after a little algebra,
κ(M)
3∼6(α+β)(1 −αβ)1/2
(1 −α2)1/2(1 −β2)1/2(1 + αβ)1/2M1/2∼3√2Nα+Nβ
M1/2
.
We can also return to (12) to get the exact third moment, not just the long-term asymptotic.
To do this, we write (12) in terms of A(z), as
6
(2πi)3
M−1
X
k=1
(M−k)∞
X
j=0 IIIA(y)yj−1A(z)zj+k−1A(w)wk−2dw dy dz
in which the contours for w- and y-integrals are of radius 1 −εand the contour for zis just |z|= 1
(the need for this will presently become apparent). The j-summation and the y-integral can be done
immediately (the placement of the contours causes |yz|<1, which is necessary for convergence of
the sum; in doing the y-integral, expand the contour out to ∞and pick up the residue at y= 1/z
on the way). Next do the k-summation using the identity
m−1
X
k=1
(m−k)rk−1≡rm−1 + m(1 −r)
(1 −r)2
to arrive at
6
(2πi)2I I A(z−1)A(z)A(w)(wz)M−1 +
∗
z}| {
M(1 −wz)
(1 −wz)2w−1dw dz.
10
The marked term exactly generates the large-Mresult we have already obtained, once the w-
integral is done (again, by expanding the contour out to ∞and picking up the residue at w= 1/z
on the way). The remaining part can be calculated by collapsing the w-contour around all the
singularities inside the unit circle (note that no singularity arises from the (1 −wz)2term in the
denominator, as |wz|<1), and then collapsing the z-contour. In the SPRZ case, we finally obtain
the third moment as
6MX
j
ρjA(α−1
j)2−6A(0) X
j
ρjA(α−1
j)−6X
j,k
ρjρkα−1
jA(α−1
k)1−αM
jαM
k
(1 −αjαk)2.(16)
For EMA1, the exact expression for the skewness is therefore
κ(M)
3=6α
(1 −α2)1/2M1/21−1−α2M
1−α2M−1.(17)
This rises from zero to a peak and then rolls off as O(M−1/2) (see Figure 2a). The maximum
skew10 is roughly 2.1–2.4, and occurs for period M≈1.1N(recall α= 1 −N−1).
For EMA2, the exact expressions for the second and third moments are
µ(M)
2=M(1 + αβ)
(1 −αβ)(1 −α2)(1 −β2)(18)
µ(M)
3=6M(α+β)(1 + αβ)
(1 −αβ)(1 −α2)2(1 −β2)2(19)
+6α3(1 −α2M)
(α−β)2(1 −α2)3(1 −αβ)+6β3(1 −β2M)
(α−β)2(1 −β2)3(1 −αβ)
−6αβ2(1 −αMβM)
(α−β)2(1 −β2)(1 −αβ)3−6α2β(1 −αMβM)
(α−β)2(1 −α2)(1 −αβ)3
and κ(M)
3=µ(M)
3µ(M)
23/2as usual. This is qualitatively similar to EMA1 (see Figure 2b). The
maximum skew is around 2.1, and occurs at period M≈1.7(Nα+Nβ), provided Nαand Nβare
not too far apart. In the extreme case where either of the N’s is equal to 1, we recover EMA1. The
limit β→αis well-behaved, but algebraically messy and omitted here.
In essence, (16) telescopes the various geometric series that are implicit in the calculation of
(12), and allows it to be done with an amount of computational effort independent of M.
2.5 Empirical results
For a demonstration using real data we use two datasets: the CHFUSD futures and the S&P500
futures11. For risk-adjusting the returns we use a 20-day EMA of squared price changes to estimate
the volatility (ˆσnin the definition of Un). We are using an EMA2 with N= 20,40.
It is worth recalling the assumptions that we made in deriving our formulae: (i) independence
of the risk-adjusted returns (Un); (ii) symmetry of their distribution up to the third moment. In
practice the first clearly does not hold, because it implies that momentum strategies do not generate
positive expected return, whereas the evidence is that on average they do. That means that when
we examine real data, the observed skewness of returns may well not equal the theoretical result,
10Unless Nis small we can approximate (17) as (3/√2x3)(e−2x−1 + 2x), with x=M/N; the maximum of this
function is ≈2.41 and occurs at x≈1.07.
11Bloomberg: SF1 Curncy and SP1 Index. These are the front contracts, rolled 10 days before expiry to create
generic series. Data range: 01-Jan-90 to 31-Dec-09.
11
(a)
0
0.5
1
1.5
2
2.5
0 50 100 150 200 250
skewness
return period /days
N=10
N=20
N=40
(b)
0
0.5
1
1.5
2
2.5
0 50 100 150 200 250
skewness
return period /days
N=5,10
N=10,20
N=20,40
Figure 2: Skewness of trading returns, as a function of period, for (a) EMA1 type model, (b) EMA2 type
model. Note the characteristic shape.
by virtue of the mean being different. We therefore plot the central skew (third central12 moment
divided by 3
2power of the second central moment—the usual definition) and also the ‘non-central’
skew (third moment about zero divided by 3
2power of the second moment about zero). If the effect
of trending is to generate a slightly positive expected return but keep the other moments roughly
equal, then the non-central skew will be fractionally higher than the central skew. As to (ii), we
know that equity markets occasionally have very negative returns.
Figure 3 shows the results for the two markets, superimposing also the theoretical result from
Figure 2b. In spite of the deficiencies in the modelling assumptions the agreement is not bad and
the general shape is right. The short-term skewness for the equity market is nonzero because of the
asymmetry of the market returns; the higher long-term skewness is best ascribed to the particularly
good trending behaviour in the mid-1990s generating high trading returns. The skewness of the
12Central moment = moment about the mean.
12
(a)
-0.5
0
0.5
1
1.5
2
2.5
0 50 100 150 200 250
skewness
return period /days
central skew
non-central skew
theoretical
(b)
-1
-0.5
0
0.5
1
1.5
2
2.5
3
0 50 100 150 200 250
skewness
return period /days
central skew
non-central skew
theoretical
Figure 3: Skewness of trading returns, as a function of period, for (a) CHFUSD, (b) S&P500 futures;
theoretical result also shown. N= 20,40.
trading returns is far higher than that of the underlying markets (i.e. of the Un’s): the latter is (to
within 0.1) typically about 0.0 for CHFUSD and −0.2 for S&P500. This shows that the skewness
comes entirely from the momentum strategy. The Gram-Charlier formula for the probability of
exceeding zero is modified to Φ(r)−κ3/(6√2π) in the presence of nonzero first cumulant (expected
return); here r=κ1κ1/2
2is the Sharpe ratio and Φ is the Normal c.d.f. For horizons Min the
range 100–200 days the Sharpe ratio of each is roughly +0.2 and the skewness is around 2, so this
gives the probability of exceeding zero as about 0.45, which corresponds well with the empirical
value—note that it is less than one-half.
2.6 Hybrid linear models
Suppose that a strategy has a trend-following and a counter-trending characteristic, as would
happen if its weights were obtained from a linear combination of EMA2’s, with opposite signs.
13
-1.5
-1
-0.5
0
0.5
1
1.5
0 50 100 150 200 250
skewness
return period /days
weights -1,1
weights 1.476,-1
Figure 4: Skewness of trading returns, as a function of period, for hybrid model with both trending and
counter-trending behaviour, in two cases.
It may be desirable to ensure that the long-term skewness remains positive, as this is associated
with longevity of the strategy. There are two situations in which this arises. In what is basically a
trend-following strategy, it is desired:
(i) to make small bets on short-term reversion without this upsetting the behaviour if a longer-
term trend occurs;
(ii) to make a small bet against very long-term trends on the supposition that what goes up must
eventually come down (or vice versa), provided this bet is not too large.
In the first case the weights on most recent returns will be negative; in the second, it is the weights
on the distant past that will be negative. The idea is to make sure that they are not too negative,
in a sense to be made precise.
We have a model of the form
A(z) = λF(αF−βF)
(1 −αFz−1)(1 −βFz−1)+λS(αS−βS)
(1 −αSz−1)(1 −βSz−1)
where λFand λSare the multipliers on the fast and slow components. Positive asymptotic skewness
is ensured by (15): X
j
ρjA(α−1
j)2>0 (20)
where there are now four poles α1=αF,α2=βF,α3=αS,α4=βS. Thus
ρ1=α2
FλF, A(α−1
1) = λF(αF−βF)
(1 −α2
F)(1 −βFαF)+λS(αS−βS)
(1 −αSαF)(1 −βSαF)
and similarly for the other three. The LHS of (20) is a homogeneous cubic in λF, λS, which will
factorise as
P(λF, λS) = (λF−ζ1λS)(λF−ζ2λS)(λF−ζ3λS)
14
where the ζ’s are functions of the four poles. It is possible to identify the coefficients of λ3
F,λ2
FλS,
λFλ2
S,λ3
Sas functions of the poles, then evaluate them and factorise the cubic by the Cardano-
Tartaglia formula. However for practical purposes one might just as well write a numerical routine
for LHS(20) and find the roots ζinumerically. One root ζ1has to be real, and the other two are
likely to be complex because we expect Pto be strictly increasing in λFand in λS: raising either
weight should enhance the trending behaviour and hence the asymptotic skewness.
As a particular example, let NαF= 5, NβF= 10, NαS= 20, NβS= 40. Then ζ1≈ −1.476, and
the condition for positive asymptotic skewness is simply13
λF+ 1.476 λS>0.
This being so, it is easily incorporated into an optimisation as a ‘style constraint’. Figure 4 shows
the results for two examples, (i) λF=−1, λS= 1, so the short-term behaviour is counter-trending
and generates negative skewness; (ii) the critical case λF= 1.476, λS=−1, where now just enough
long-term counter-trending behaviour is added to make the asymptotic skew zero at leading order.
These exemplify the cases (i), (ii) discussed above. The results were obtained using (16) again,
which is not laborious despite there being four poles (so that the double summation has sixteen
terms): it is preferable to Monte Carlo simulation, which even with a few hundred thousand
simulations generates noticeable uncertainty.
3 Option-like nature of trend-following
As pointed out in [21], trend-following strategies are often thought to have a long-option-type payoff
on account of the positive skewness. For linear strategies this can be formalised as follows. The
M-period trading return is
Y(M)
n=u′Γu,u=Un+MUn+M−1···′
where the symmetric matrix Γis given by
Γ=1
2
0a0a1a2···
a0...............
a1...0a0a1a2···
a2...a0000···
.
.
....a1000···
...a2000···
.
.
..
.
..
.
..
.
....
Mrows
.(21)
The moments of Y(M)
nrelate to the spectrum of Γ, and direct calculation reveals
DY(M)
nE= tr(Γ) = 0,DY(M)
n2E= 2 tr(Γ2),DY(M)
n3E= 8 tr(Γ3).
Writing Γin terms of its eigenvalues γjand normalised eigenvectors ej, we have an expression
that is a weighted sum (weights adding to zero) of ‘orthogonal quadratic bets’, i.e. squared linear
combinations of returns, which are like straddle payoffs but have constant convexity:
Y(M)
n=X
j
γj(ej·u)2.(22)
13Because the other two terms in Pmultiply to give a quadratic that is always positive, and hence of no consequence.
15
Now tr(Γr) = Pjγr
j, so the moments of Y(M)
nrelate to the moments of the eigenvalue distribution.
It is easy to see the rank of Γis ≤2M(and is M+ 1 in the EMA1 case as then the rows after
the Mth are linear multiples of each other), which limits the number of nonzero eigenvalues to
2M. The interpretation of all this is that a positively skewed strategy has a small number of large
positive eigenvalues and a larger number of smaller negative ones. This generates a small number
of large positive-convexity bets and a larger number of smaller negative-convexity bets, which is
where the positive skewness comes from. Dao et al. [5] make the same point, but emphasise the
important point that the positive-convexity bets are long-dated options and the negative-convexity
bets short-dated ones. Thus in situations where the long-term volatility is elevated and the short-
term volatility is low, momentum strategies work well.
3.1 *Full distribution of trading returns
This subsection may be omitted at a first reading.
If we want to know the full distribution of trading returns, we need to make an assumption
about the full distribution of the market returns, whereas until now we have only used the first
three moments.
We can use the ideas of the previous section to compute the full distribution of trading returns,
exactly as is done by Acar [1, Ch.3] using generating functions. The moment-generating function
of the M-period return is
FM(s) := Dexp(sY (M))E=exp s(ϕ0U1+···+ϕM−1UM).
Let us assume that the (Un) are Normally distributed. Then for a linear model
FM(s) = [det(I−2sΓ)]−1/2
with Γas above. (See e.g. [7] for details on quadratic transformations of Normal variables.)
Before proceeding further we should note that the above expression is not very helpful because
it requires the manipulation of Γwhich is an infinite matrix. Let us therefore evaluate ab initio
the expression exp s(ϕ0U1+···+ϕM−1UM).
Conditioning on (U1,...,UM), effectively fixing those values, we have (inside the exponential) a
linear combination of U0, U−1,..., added to another expression that is a function of (Uj)M
j=1 only.
In the first part the coefficients are
U0:s(a0U1+a1U2+···+aM−1UM)
U−1:s(a1U1+a2U2+···+aM−2UM)
and so on. These variables can then be integrated out to give the expression
exp (1
2s2∞
X
l=−1 M
X
k=1
ak+lUk!2)= exp (1
2s2∞
X
l=−1
M
X
j,k=1
aj+lak+lUjUk).
The other part of the expression depends on the U’s through pairwise products, i.e. U1U2, etc, and
is easily seen to be
exp (sX
1≤j<k≤M
ak−j−1UjUk).
16
Next we multiply these two expressions to obtain
FM(s) = Dexp u′b
G(s)uE,u=U1U2··· UM′,
with
b
Gjk (s) = 1
2s2∞
X
l=−1
aj+lak+l+1
2s1j6=ka|k−j|−1.(23)
Finally integrate uout, to get the more manageable
FM(s) = [det(I−2b
G(s))]−1/2(24)
instead—the dimension of b
Gis just M. Again the infinite summation in the expression for b
Gjk
can be done easily enough by residue calculus in the SPRZ case.
The moments and cumulants can now be obtained by performing a Taylor series of, respectively,
FM(s) and log FM(s) around the origin, using standard identities for expanding the determinants.
The derivation is laborious, however. To obtain an approximation to the distribution of Y(M), we
may use Fourier transform inversion or, as discussed in [7], saddlepoint methods, which work well
on this problem (see also [12] for a general discussion).
4 Nonlinear strategies
As in the first part of the paper we write the position as ϕn/ˆσn. Clearly the P&L arising from
the period between time nand time n+ 1 is ϕnUn+1. We define a momentum factor Vnto be a
moving average of the U’s, with weights (aj)∞
j=0. The position, however, will no longer be simply
proportional to Vn, but instead transformed using a nonlinear ‘activation14 function’ ψ:
Vn=∞
X
j=0
ajUn−j;ϕn=ψ(Vn).(25)
We use the autocovariance function Ra
kas before but now for convenience we stipulate that Vnhave
unit variance, which is to say Ra
0= 1. Then |Ra
k| ≤ 1 (by Cauchy-Schwarz), and the inequality will
be strict in all practical examples. Importantly, when designing the activation function ψwe know
that the typical scale of variation of its input is unity.
In the EMA1 model, we have
aj= (1 −α2)1/2αj, Ra
k=αk,
differing trivially from what we had in §2 by a scaling factor so that Ra
0= 1. For EMA2,
aj=(1 −α2)(1 −β2)(1 −αβ)
1 + αβ 1/2αj+1 −βj+1
α−β, Ra
k=αk+1(1 −β2)−βk+1(1 −α2)
(α−β)(1 + αβ).
If β=αthen this is well-defined and is simply a double-EMA of the returns.
This completes our description of the model setup: in summary, the position taken is a nonlinear
transformation of smoothed vol-adjusted returns, divided by the volatility of the underlying. It is
specified by the EMA parameter(s) and the function ψ. A general system can then be built by
taking a linear combination of these models with different speeds.
14Following neural network parlance.
17
4.1 Skewness of nonlinear models
We make the same assumptions as before about the nature of the risk-adjusted returns Un, that
is they are i.i.d. and their first three moments are 0,1,0. Then the second moment of the trading
return is DY(M)
n2E=Mhϕ2i(26)
and the third moment is
DY(M)
n3E= 3
M−1
X
k=1
(M−k)Hk, Hn−m=hϕ2
nϕmUm+1i.(27)
Whereas in §2 we evaluated this summation directly, we can no longer do this and therefore take
a slightly more roundabout route. Note first that the triple (ϕ2
n, ϕm, Um+1) is a simple function
of the triple (Vn, Vm, Um+1) = (Z1, Z2, Z3) say, which has a trivariate Normal distribution with
covariance matrix
Σ=
1Ra
n−man−m−1
Ra
n−m1 0
an−m−10 1
.
The determinant and inverse of Σare
∆ = 1 −a2
n−m−1−R2
n−m,Σ−1=1
∆
1−Ra
n−m−an−m−1
−Ra
n−m1−a2
n−m−1an−m−1Ra
n−m
−an−m−1an−m−1Ra
n−m1−(Ra
n−m)2
.
Writing the expectation hϕ2
nϕmUm+1ias an integral we get15
Hn−m=1
(2π)3/2∆1/2
∞
ZZZ
−∞
exp(−1
2z′Σ−1z)ψ(z1)2ψ(z2)z3dz1dz2dz3
and then integrate over z3(effectively, integrating Um+1 out). After a fair amount of algebra, the
whole lot tidies up to give an expression as an expectation over the distribution of (Z1, Z2) which
is bivariate Normal with unit marginals and correlation ρ=Ra
k:
Hk= 2ak−1ψ(Z1)2ψ(Z2)Z1−ρZ2
2(1 −ρ2)ρ=Ra
k
(28)
= 2ak−1ψ(Z1)ψ′(Z1)ψ(Z2)ρ=Ra
k
,(29)
where the last line follows from integrating by parts w.r.t. Z1, assuming differentiability of ψ.
In the linear case, ψ(z) = z, we have Hk= 2ak−1Ra
k, and hence the third moment is 6 PM
k=1(M−
k)ak−1Ra
kas we saw earlier. When ψis not linear, we have to calculate Hkfor each kand then
do the sum by explicit calculation. That said, it is possible to find functions ψof the ‘right
shape’ for which the double integral implicit in (29) can be done in closed form. We discuss some
next. The integration is done over Z2first conditionally on Z1, so that Z2∼N(ρZ1,1−ρ2),
and then Z1is integrated out. Some helpful identities used in the calculations are given in the
Appendix. Furthermore, the summations can be evaluated by recursion in M. In detail, writing
ΘM= 3 PM−1
k=1 (M−k)Hkand SM=PM−1
k=1 Hk, we initialise S1= 0, Θ1= 0, and for M≥2 we
have SM=SM−1+HM−1and ΘM= ΘM−1+ 3SM.
Remember that in obtaining the skewness we need to divide the third moment by the 3
2power
of the second, so it is convenient to deal with functions normalised to ψ(Z)2= 1.
15Notation zjust means the column vector (z1, z2, z3)′.
18
(c)
-3
-2
-1
0
1
2
3
-3 -2 -1 0 1 2 3
transformed, ψ(z)
raw signal, z
linear
λ=0.5
λ=1.0
λ=1.5
(d)
0
1
2
0 50 100 150 200 250
skewness
return period /days
linear
λ=0.5
λ=1.0
λ=1.5
20
(e)
-3
-2
-1
0
1
2
3
-3 -2 -1 0 1 2 3
transformed, ψ(z)
raw signal, z
ε=0.3
ε=0.6
ε=0.9
(f)
0
1
2
0 50 100 150 200 250
skewness
return period /days
linear
ε=0.3
ε=0.6
ε=0.9
Figure 5: Activation functions and their associated term structure of skewness: (a,b) sigmoids with
λ= 1.0,2.0; (c,d) reverting sigmoids with λ= 0.5,1.0,1.5; (e,f) double-step with ε= 0.3,0.6,0.9.
Using EMA2 type momentum signal throughout, with N= 20,40.
21
4.2.1 Simple sigmoid, ψ(z) = cλ·2Φ(λz)−1
In effect, this caps the position when the magnitude of the the momentum signal is large. We have
Hk= 2ak−1c3
λ
(2/π)3/2λ
√1 + λ2arctan λ2ρ√1 + λ2
p1 + 3λ2+ 2(1 −ρ2)λ4!ρ=Ra
k
(30)
and for normalisation we require cλ=2
πarctan λ2
√1+2λ2−1/2. We obtain in the limit λ→0 that
Hk= 2ak−1Ra
kwhich is as expected the linear result.
4.2.2 Reverting sigmoid, ψ(z) = cλ·ze−λ2z2/2
The behaviour of this one is more nonlinear in the sense that it begins reducing the position
when the momentum gets too high, ultimately to zero if the momentum is strong enough. The
rationale for this is that a very strong trend might be more susceptible to reversing (market
overbought/oversold), justifying a reduction in position. The maximum positions are held when
z=±λ−1. We have
Hk= 2ak−1c3
λ
ρ1−(1 −ρ2)λ4
1 + 3λ2+ 2(1 −ρ2)λ45/2ρ=Ra
k
(31)
and for normalisation we require cλ= (1 + 2λ2)3/4. Notice that Hkbecomes negative if λis high
enough: this is not surprising because ψ′(Z) is negative for |Z|> λ−1, wherein the model is betting
against the trend. As expected λ→0 gives back the linear result again.
4.2.3 Double-step, ψ(z) = cλ·(1z>ε −1z<−ε)
This is +1 if the momentum is positive enough, −1 if negative enough, and zero at intermediate
levels, with the width of the ‘dead zone’ being 2ε > 0. This is similar to the one considered by
Potters & Bouchaud. We have16
Hk= 2ak−1c3
λφ(ε) Φ
εr1 + ρ
1−ρ!−Φ
εr1−ρ
1 + ρ!!ρ=Ra
k
(32)
and for normalisation we require cλ=2Φ(−ε)−1/2.
From this it can be seen that as ε→0, creating a binary response ψ(z) = sgn(z), the skewness
vanishes17. One way of understanding this is to see that as the position is always of magnitude 1, it
does not have the characteristic—associated with positive momentum—of increasing the position
when the P&L is positive. A less precise explanation is that during periods in which the market
is not trending, the strategy loses money rather quickly because it is buying and selling the same
size of position as it holds when a trend has been detected. By contrast, the other two (sigmoidal)
functions that we have just examined only trade a small size until a trend is established, resulting
in P&L distribution with more, but smaller, losses, and fewer, but bigger, gains: that is where
their positive skewness comes from. A recent piece on FX strategy [6] makes this point somewhat
16Do not use these results for ε < 0.
17This can also be seen from the sigmoidal case when λ→ ∞, as the argument in the arctan() term goes to zero.
The result can also be seen directly from (28), because ψ(Z1)2= 1 a.s. and and Z1−ρZ2is independent of Z2, so
the expectation decouples into a product of two expectations each of which is zero. Technical point: The behaviour
as ρ→ ±1 is awkward. In fact, this limit is irregular and the result as ρ→1 is not the same as that for ρ= 1;
similarly for −1. However, these two values of ρcannot occur in our problem.
22
tangentially. The authors use the simple step function and note that the performance is excellent
when the market is trending but very poor otherwise. In the light of what we have just said, we
are not surprised by this observation.
As εis raised, notice that the skewness rises without limit, which seems rather good: however,
if εis too high then the algorithm hardly ever trades, so practicalities dictate ε.1.5.
0
0.1
0.2
0.3
0.4
0.5
0 0.5 1 1.5 2
0
1
2
3
SR
skewness
width parameter, ε
SR
skewness
Figure 6: Sharpe ratio and skewness as a function of εfor double-step type activation function,
showing performance dropoff for ε > 0.6.
wR
wS\λ0.25 0.50 0.75 1.00 1.25 1.50 1.75
∞0.39; 1.77 0.45; 1.17 0.48; 0.69 0.48; 0.34 0.45; 0.09 0.40; −0.09 0.35; −0.23
5.0 0.38; 1.74 0.44; 1.19 0.48; 0.75 0.48; 0.43 0.46; 0.21 0.43; 0.04 0.39; −0.09
2.4 0.38; 1.73 0.43; 1.21 0.47; 0.80 0.48; 0.50 0.47; 0.29 0.45; 0.14 0.42; 0.02
1.5 0.38; 1.71 0.43; 1.22 0.46; 0.83 0.48; 0.56 0.47; 0.37 0.46; 0.22 0.43; 0.11
10.38; 1.70 0.42; 1.23 0.46; 0.87 0.47; 0.62 0.47; 0.43 0.46; 0.30 0.45; 0.19
0.67 0.38; 1.69 0.41; 1.24 0.45; 0.91 0.47; 0.67 0.47; 0.50 0.46; 0.37 0.45; 0.28
0.4 0.37; 1.67 0.41; 1.25 0.44; 0.94 0.46; 0.73 0.46; 0.57 0.46; 0.46 0.46; 0.37
0.2 0.37; 1.65 0.40; 1.27 0.43; 0.99 0.45; 0.80 0.46; 0.66 0.46; 0.56 0.46; 0.48
00.37; 1.63 0.39; 1.29 0.41; 1.05 0.43; 0.89 0.44; 0.77 0.45; 0.69 0.45; 0.62
Figure 7: Sharpe ratio and skewness as a function of λand the ratio wR/wSfor compound sigmoidal
activation function. In each pair of numbers, the first is the SR, and the second is the skewness.
4.3 Skewness computation
We show in Figure 5 the term structure of skewness for the different examples given above: sigmoid
(a,b), reverting sigmoid (c,d), double-step (e,f). The linear result is overlaid for comparison. The
precise choice of momentum crossover does not affect the main conclusion, and we have used
EMA2 with N= 20,40 throughout. Using a faster or slower momentum measure simply stretches
or compresses the graph in a horizontal direction, as it did in the linear models.
23
-3
-2
-1
0
1
2
3
-3 -2 -1 0 1 2 3
transformed, ψ(z)
raw signal, z
Figure 8: Sketch of the activation function highlighted in Figure 7 (parameters: wR= 0.71,
wS= 0.30, λ= 0.75).
0
20
40
60
80
100
120
2012 2014 2016 2018 2020
−50
−40
−30
−20
−10
0
10
20
30
40
50
Oil price
P&L
Oil
P&L (Sigmoid) − RHS
P&L (Rev. sigmoid) − RHS
Figure 9: Trend-following oil over a decade, using the sigmoid and reverting sigmoid for activation
function (λ= 0.75, N1= 10, N2= 20). The reverting sigmoid fails to capitalise on the full selloff
in late 2014.
It is apparent from the results that as the activation function becomes progressively less linear,
the main effect is to compress the graph in a vertical direction, so that the maximum skewness
is reduced. With the reverting sigmoid, the graph can be affected much more, to the extent of
becoming negative when λis high enough: we predicted this earlier when remarking that Hkcould
24
become negative as a result of the activation function being decreasing over much of its domain,
so the model spends a lot of time incrementally trading against the trend rather than with it. (In
fact for ψ(z) = ze−λ2z2/2the critical λ, above which the skewness is no longer everywhere positive,
is around 1.3. An explanation is in the Appendix.)
The general conclusion so far is that any capping effect in the activation function will cause the
trading returns to be less positively skewed, and any reverting effect will exacerbate this reduction
in skewness. From the perspective of skewness alone, these effects should be avoided as much
as possible. However, they may well be justified by reason of risk management and/or expected
return, so we consider these next.
4.4 Empirical analysis and expected return
Analysis of the expected return is a totally different proposition because there are no theoretical
guidelines at all. One can only adopt an empirical approach, seeing what has worked in the past,
and relying on it continuing to do so.
We need to decide what objective function is to be maximised, and the most natural thing
to do is to maximise the Sharpe ratio (SR) of the trading strategy, i.e. use an objective function
that directly relates to trading model performance. As the SR is the expected return divided by
the volatility, we will be penalising any effect that increases volatility without generating enough
extra return. Taking a range of futures contracts across different asset classes (stocks, bonds, FX,
commodities) and a range of EMA2 periods (5 vs 10 days, 10 vs 20 days, etc.), we have run trading
simulations over the available history, which is typically 20 years or more, and calculated the Sharpe
ratio; this gives a list of Sharpe ratios, one for each contract and speed. For simplicity we are going
to use the same activation function across all contracts and speeds. We then average the list of
Sharpe ratios and use this as our performance indicator, to be maximised18 .
We first examine the double-step activation function. Here there is only one parameter to
adjust, namely ε, the half-width of the ‘dead zone’. Figure 6 shows the performance as a function
of ε. It is not surprising that the SR drops off as εbecomes large, because the strategy hardly
ever has a position on and can never make any money. What is interesting is that the performance
for ε < 0.6 is so flat. Thus from the perspective of expected return, one may as well choose any
ε < 0.6.
However, when we overlay the conclusion about skewness, we can sharpen this deduction. As the
skewness has a term structure, we look at one return-period (M) throughout: we choose M= 100
for convenience, this being the top of the curve for a linear activation function when the EMA
periods are 20,40 (see Figure 5(b,d,f )). The skewness is also shown in Figure 6, and clearly it
increases with increasing ε, so from that perspective alone we prefer εas high as possible. If we
can push εup to about 0.6 without decreasing the SR, and in doing so can have positive skewness
as well, then we should do just that. So this is our first conclusion about design of nonlinear
momentum strategies: the blue line in Figure 5(e), ε= 0.6, is a good construction.
Next we turn to the sigmoidal functions that we introduced earlier, and take a linear combination
of them:
ψ(z) = wS2
πarctan λ2
√1+2λ2−1/22Φ(λz)−1+wR(1 + 2λ2)3/4ze−λ2z2/2
with weights wR,S >0. To normalise the weighted function we enforce the elliptical constraint
w2
S+ 2wSwRcos δ+w2
R= 1,
18The degree of temporal and cross-asset-class diversification that can be obtained is governed by what is commonly
known as ‘breadth’ and explained in detail by Grinold [8].
25
where cos δ, the correlation between the R- and S- signals, is given by
cos δ=λ(1 + 2λ2)1/4
1 + λ2arctan λ2
√1 + 2λ2−1/2
.
The effective number of parameters is now two: the horizontal scaling λand the ratio wR/wS. The
results are shown numerically in the table of Figure 7.
The general picture is that the performance surface is rather flat. Provided one avoids the far
left (where the function is too linear and suffers from putting on too much risk when momentum
is high) or the top right (where it reverts to zero too quickly when momentum is high), any of the
pairs (wR
wS, λ) would do reasonably well.
Again, we overlay the conclusions about skewness, simplifying as before by using the skewness
of M= 100-day returns. We know that higher skewness arises from a low value of λand from
wR/wSsmall i.e. little reversive behaviour. This means going as far as possible to the left of the
table, and steering well clear of the top right. Going to the left does lose performance (SR), so
some trade-off is required. One particular example is highlighted (wR= 0.71, wS= 0.30, λ= 0.75)
and sketched in Figure 8. It is seen that this does not yield the highest SR, but it is close to the
maximum. It does not revert as strongly as the pure ze−λ2z2/2does (top row of table), and is likely
to be preferable on account of its better third-moment characteristics.
We now return to the discussion at the outset about designing strategies that perform well in
specific scenarios. Figure 9 shows the results for the sigmoid and reverting sigmoid, with λ= 0.75,
N1= 10, N2= 20, over ten years, for the oil market. It is clear that the reverting sigmoid
does substantially less well. This is because the trend is strong and persists for a long time, so
the reverting behaviour of the activation function causes severe underperformance. Yet the third
column of Figure 7 suggests that the reverting sigmoid (top row) on average performs better than
the sigmoid (bottom row). Part of the selling-point of CTA strategies is their ability to produce
‘alpha’ in scenarios such as the selloff in oil (and other commodities, and associated equities) in
late 2014. It follows that one should make sure that the strategy does well in such scenarios, rather
than simply relying on what has produced the best historical SR. This example also corroborates
our earlier remark about calibration being sensitive to data history and therefore subjective. Were
the oil selloff in late 2014 absent from the calibration, one would arrive at different conclusions
about the optimal model.
As a final comment, we see that the smoother activation functions offer only a small improve-
ment in Sharpe ratio over the double-step. However, we have not considered transaction costs, and
models that generate sudden large trades can be more difficult to run a large amount of money on.
Models that take positions more gradually are therefore easier to handle in trading. They are also
easier to handle in backtesting, because a slight change in the definition of the momentum oscil-
lator, which is the input to the activation function, can for a discontinuous function make a huge
difference to the simulated position: even minor changes to the strategy can produce unpredictable
results.
That said, the characteristic of the double-step function, that it waits until the momentum is
above a certain level before trading, may be worthy of further investigation. Thus one aims for a
function that is zero for 0 <|x|< ε, then rises smoothly until a maximum is reached, and then
rolls off slowly and asymptotes to a level above zero.
26
5 Conclusions and final remarks
We have shown how to analyse the behaviour of a variety of trend-following models by particular
reference to the skewness of the distribution of trading returns. To do this we have needed only the
first three moments of the market returns, thus keeping the modelling quite general. As regards
linear models the most important formulae are (14,16) giving the second and third moments of the
trading returns in an elegant application of residue calculus. Pure momentum (trending) strategies
generate positive skewness even though the market returns might be totally symmetrical. The
skewness depends on the return period and has a characteristic term structure which we have
derived, illustrated and verified with real data. Hybrid strategies, with trending and counter-
trending behaviour, may exhibit a more complex term structure of skewness, and we have shown
how to analyse a general linear model.
We have investigated ‘nonlinear momentum strategies’ from different angles, understanding the
Sharpe ratio and the skewness—in essence, the first and third moments—of their trading returns.
The former was investigated empirically, and the latter mathematically. We have also pointed out
that it may be wise to consider the behaviour of a strategy in specific scenarios, especially if they
are a raison d’ˆetre of momentum trading. Specific conclusions about optimal design are given in
the text, but two salient ones are repeated here.
First, the common practice of forming a momentum signal from moving averages and then
making a ‘binary bet’ on it, +1 or −1 according as momentum is positive or negative, is not a
good construction; we make other comments about discontinuous models in the next paragraph. It
can be improved by waiting until the signal reaches a threshold before trading. Secondly, although
it is a good idea to reduce—not just cap—the position when the momentum is very high, it is
not advisable to reduce it too much. If too aggressive, the reducing effect can undesirably cause
negatively-skewed trading returns, and much extra trading: when the momentum finally begins
to fall, one ends up buying back the position and then dumping it again as the momentum goes
back to zero. Also, the reducing effect loses profit badly when the trend is large and prolonged
(Figure 9).
One area that we have not touched upon is the design and analysis of the opposite type of
strategy, i.e. mean reversion. We can obtain a reversion model from a momentum model simply
by multiplying by −1. This of course makes the skewness negative, but we do not want it to be as
negative as possible. Therefore the optimal design will not simply be the reverse of what we have
done here. Instead, something like the reverting sigmoid, now written as ψ(x) = −cλxe−λ2x2/2, is
likely to be a good idea. When the market deviates from the reversion level a long way, some risk
is taken off, which is likely to be beneficial.
A common explanation for the positive skewness is that it arises from the strategy having
positive convexity, as mentioned for example in [5]. This is partly true, and we have explained its
origin in §3, but in fact there is more to it than that, as will be discussed in forthcoming work.
27
A Formulary
A.1 Expectation formulae
If (Z1, Z2)∼N2(ρ) (the bivariate Normal distribution with N(0,1) marginals and correlation ρ)
then
Ef(Z1, Z2)e−a2
1Z2
1/2e−a2
2Z2
2/2=1
√Db
Ef(pD2/DZ1,pD1/DZ2)
where under b
E, (Z1, Z2)∼N2(bρ), with
bρ=ρ
√D1D2
, Di= (1 −ρ2)a2
i+ 1, D = (1 −ρ2)a2
1a2
2+a2
1+a2
2+ 1
The following results are of use in obtaining (30–32). For Z∼N(0,1),
Z2ne−b2Z2/2= (2n−1)!!(1 + b2)−(2n+1)/2
hφ(a+bZ)i=1
√1 + b2φa
√1 + b2
hΦ(a+bZ)i= Φa
√1 + b2
hZφ(a+bZ)i=−ab
(1 + b2)3/2φa
√1 + b2
hZΦ(a+bZ)i=b
√1 + b2φa
√1 + b2
hΦ(aZ)Φ(bZ)i=1
2πarctanab
√1 + a2+b2+1
4
The last result follows from differentiating both sides w.r.t. a, and using integration by parts on
hZφ(aZ)Φ(bZ)i.
A.2 Skewness of reverting sigmoid activation function
We justify why the skewness is always positive for |λ|.1.3. It was derived for linear models by
means of z-transforms that for large M,
DY(M)
n3E∼3M∞
X
k=1
Hk.
We are going to calculate this infinite sum, at least approximately, in the EMA1 case. By (31) the
condition for positivity of the above expression is
∞
X
k=1
2αkαk1−(1 −α2k)λ4
1 + 3λ2+ 2(1 −α2k)λ45/2>0.
Write α2k=uand approximate the sum as an integral over u, to give
Z1
0
u1−(1 −u)λ4
1 + 3λ2+ 2(1 −u)λ45/2
du
u>0.
Upon doing the integral and tidying up, one ends up with
2 + 9λ2+ 7λ4−8λ6>0;
so λ2<1.65. This does not explain rigorously what goes on in the pre-asymptotic region when M
is not large, and it uses EMA1 rather than EMA2, but the above analysis seems sufficient.
28
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