J U N E 2 0 11
Issue #7
W H I T E PA PE R
R I S K- R E T U R N
A N A LY S I S O F D Y N A M I C
I N V E S T M E N T S T R AT E G I E S
Benjamin Bruder
Research & Development
Lyxor Asset Management, Paris
[email protected]
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Nicolas Gaussel
CIO – Quantitative Management
Lyxor Asset Management, Paris
[email protected]
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Issue # 7
Foreword
The investment fund industry has changed dramatically over the last ten years, and we
are now seeing a convergence between hedge funds and traditional asset management. For
example, both institutional and retail investors now have easier access to absolute return
strategies in a mutual fund format. This convergence has accelerated recently with the
emergence of “newcits” and the increasing number of regulated hedge funds. The investment
decision-making process is now more complex as a result, with these dynamic investment
strategies and the number of underlyings and assets growing rapidly. Managing exposure to
risky assets is the main difference between these investment styles and the traditional longonly strategies. This difference is highly significant, however, and is not always understood
by investors and fund managers.
The traditional method for analysing and evaluating a strategy is to use risk-adjusted
performance measurement tools such as the Sharpe ratio (or the information ratio) and
Jensen’s alpha. These financial models were developed to compare long-only strategies, and
are not really suitable for dynamic trading strategies, as they exhibit non-normal returns
and non-linear exposure to risk factors. In the ’90s, practitioners and academics developed
alternative models to take these properties into account. Some extensions of the Sharpe ratio,
such as the Sortino, Kappa and Omega ratios, have now become very popular for analysing
the performance of hedge fund returns. Another way of understanding the risk-return profile
of dynamic strategies has been proposed by Fung and Hsieh (1997) by incorporating nonlinear risk factors in Sharpe-style analysis. These various measures define the empirical
approach in the sense that they are computed on an ex-post basis but are not really suitable
for ex-ante analysis.
All these models are relevant, however they provide only a partial answer to understanding the true nature of a dynamic strategy. Let us consider for example a long exposure
on a call option. From the seminal work of Black and Scholes (1973), we know that this
investment profile is equivalent to a delta-hedging strategy. A long position on a call option is therefore a trend-following strategy with dynamic exposure to the underlying risky
asset. Computing risk-adjusted performance or performing a style regression are certainly
not the obvious tools for analysing this dynamic investment strategy. A better way of understanding the risk and return of such a strategy is to use option theory. In this case, the
strategy’s performance is analysed by investigating both the payoff function and the premium of the option. The latter of these two is split into an intrinsic value component and
a time-value component. Moreover, one generally computes the sensitivity of the premium
to various factors such as volatility, time decay or the price of the underlying asset. This
analytical approach gives a better understanding of the option strategy than the empirical
approach, which consists in analysing the ex-post risk-return profile of the option strategy
by computing some statistics on a real-life investment or on some backtests1 .
1 For example, with the analytical approach, we may show that the trend of the underlying asset only
concerns the payoff function and has no effect on the option premium. Such property could not be derived
from the empirical approach.
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Another interesting example of a dynamic trading strategy is constant proportion portfolio insurance (CPPI), developed by Hayne Leland and Mark Rubinstein in 1976. The
extensive literature on this subject2 is mainly related to the analytical approach, and analysis of CPPI strategies is closer to option theory than the models developed to compute the
performance of traditional mutual funds. However, the CPPI technique is certainly one of
the better-known dynamic strategies used in asset management.
In view of this, we believe that the analytical method could be extended to a large class
of dynamic trading strategies, and not limited to options and CPPI. This seventh white
paper explores this approach. In this white paper, we develop a financial model to better
understand the risk-return profiles of a number of dynamic investment strategies such as
stop-loss, start-gain, doubling, mean-reverting or trend-following strategies. We show that
dynamic trading strategies can be broken down into an option profile and trading impact.
To a certain extent, the option profile can be seen as the payoff function of the strategy,
whereas trading impact can represent the premium for buying such a strategy. In this
context, implementing a trading strategy generally implies a positive cost, which has to be
paid, as explained by Jacobs (2000):
“Momentum traders buy stock (often on margin) as prices rise and sell as prices
fall. In essence, they are trying to obtain the benefits of a call option – upside
participation with limited risk on the downside – without any payment of an
option premium. The strategy appears to offer a chance of huge gains with little
risk and minimal cost, but its real risks and costs become known only when it’s
too late.”
Using this framework, we are also able to answer some interesting questions that are
not addressed by the empirical approach. For example: in which cases is the proportion
of winning bets (or hit ratio) a pertinent measure of the efficiency of a dynamic strategy?
Which dynamic strategies like (or don’t like) volatility? What are the best and the worst
configurations for a given dynamic strategy? What is the impact of the length of a moving
average in a trend-following strategy? What is the theoretical distribution of a strategy’s
returns? Why is long-term CTA different from short-term CTA? What are the risks of a
mean-reverting strategy? By answering all these questions, we provide some insights that
explain why and when some strategies perform or don’t perform, and which metrics should
be used to evaluate their performance. We hope that you will find the results of this paper
both interesting and useful.
Thierry Roncalli
Head of Research and Development
2 in particular the works of Leland and Rubinstein of course but also those of Black, Brennan, Grossman,
Perold, Schwartz, Solanki, Zhou, etc.
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Issue # 7
Executive Summary
Introduction
When building a portfolio, investors have to choose from a wide range of investment styles.
Value investors, trend followers, global-macro or volatility arbitragers, to name just a few,
each offer a different way of generating returns.
Under the reasonable – yet controversial – assumption that markets do work, any extra
return is earned in exchange for a certain degree of risk. Hence, before even measuring
it, it is essential to identify and understand that risk in order to analyse the returns from
certain strategies. Unfortunately, this is a difficult task, especially in the case of dynamic
investment strategies, which are known to generate asymmetric returns. So how should we
proceed?
Since it is well established that options can be replicated using dynamic strategies, the
approach developed in this white paper consists in exploring the extent to which an option
profile can be associated with a given dynamic strategy. To keep things simple, we focus on
strategies running on a single asset. Excluding classical analysis of constant-mix strategies,
some of this paper’s key findings are:
(1) Many dynamic strategies returns can be broken down into an option profile and some
trading impact,
(2) Contrarian strategies on a single asset tend to generate frequent limited gains, in
exchange for infrequent larger losses,
(3) Trend-following strategies on a single asset will perform if the absolute value of the
realised Sharpe ratio is above a certain threshold. The shorter-term the investment
style, the higher this threshold.
Dynamic strategies returns can be broken down into an
option profile and some trading impact
As a prerequisite to our analysis, investment strategies must be described as a function of
the underlying price only. This covers those situations in which a manager can decide how
much he has to invest simply by looking at the price and certain fixed parameters. We show
in this paper that many popular strategies fit into this framework.
In such situations, the holding function can be regarded as a trader’s delta. Hence, its
integral at some point in time corresponds exactly to the option profile associated with this
strategy.
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Qualitatively speaking, the option hedging paradigm can be represented as follows:
KƉƚŝŽŶ
WƌŝĐĞ
/ŶƚƌŝŶƐŝĐ
sĂůƵĞ
dŝŵĞ
sĂůƵĞ
whereas the fund management situation can be represented as follows:
WŽƌƚĨŽůŝŽ
ƐƚƌĂƚĞŐLJ
KƉƚŝŽŶ
WƌŽĨŝůĞ
dƌĂĚŝŶŐ
/ŵƉĂĐƚ
Technically speaking, the option profile is the integral of the holding function while the
trading impact is related to the derivative of the holding function. These observations are
summed up in the following table:
Strategy Type
Option Profile
Trading Impact
Hit Ratio
Convex
Negative
50%
Concave
Positive
50%
Buying when
market goes up
Buying when
market goes down
Average Gain /
Loss Comparison
Average Gain >
Average Loss
Average Loss >
Average Gain
The impact of volatility on directional funds depends on
the leverage of the strategy
Retail networks are used to distribute funds that maintain a constant exposure, typically
investing 20%, 50% or 80% of their wealth in risky markets. On the other hand, some
financial products such as CPPI portfolios implement constant leverage on a given risky
asset. These two strategies belong to the constant-mix category.
When exposure is less than 100%, these strategies will benefit from trading impact.
These strategies will gain even if the return of the underlying asset is zero. On the contrary,
when their leverage is above 100%, constant-mix strategies can be hit badly by trading
impacts. For example, a 3 times leveraged strategy on an equity index with 20% volatility
would loose 12% per annum if the underlying performance is equal to zero, making the
performance attribution difficult in such situation.
Contrarian strategies tend to generate frequent limited
gains in exchange for infrequent larger losses
Some investors base their investment on the principle that they are able to identify an
intrinsic value for certain securities and that markets should eventually converge with their
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Issue # 7
forecasts. For equities, that value is based on forecasts such as the company’s expected
future earnings, their growth rate or the degree of uncertainty surrounding those forecasts.
Since this intrinsic target value changes slowly over time, it can be viewed as an exogenous
parameter of the strategy. As a result, the only remaining variable is the price of the security
itself. These strategies thus fall within the scope of our study. If investors invest in opposite
proportion of the difference between the price and the intrinsic value, the following graph
describes the typical option profile of this strategy.
ϭϭϬй
ϭϬϱй
ϭϬϬй
tĞĂůƚƚŚ
ϵϱй
ϵϬй
ϴϱй
ϴϬй
ϳϱй
KƉƚŝŽŶƉƌŽĨŝůĞ
ϲDŚŽƌŝnjŽŶ
ϭzŚŽƌŝnjŽŶ
ϮzŚŽƌŝnjŽŶ
ϳϬй
ϲϱй
ϲϬй
Ϭй
ϮϬй
ϰϬй
ϲϬй
ϴϬй ϭϬϬй ϭϮϬй ϭϰϬй ϭϲϬй ϭϴϬй ϮϬϬй
ƐƐĞƚƉƌŝĐĞ
As expected, this strategy exhibits a concave profile. The positive trading impact is
illustrated by the fact that if the market continues to quote approximately the same price,
the portfolio value increases. This is due to the numerous buy-at-low sell-at-high trades that
have been made in order to maintain the target proportion of holdings.
Trend-following strategies performances are related to the
square of the realised Sharpe ratio
Trend-following strategies are a specific example of an investment style that emerged as an
industry in its own right. So-called Commodity Trading Advisors are the largest sector of the
Hedge Fund industry. Surprisingly, despite its importance in the investment industry, this
investment style is largely overlooked by standard finance textbooks. Some attempts have
been made to benchmark trend-following strategies against systematic buying of straddles.
This makes sense qualitatively, as the essence of trend-following is to benefit from trends
while accepting that returns will not be generated if markets do not trend enough.
In this white paper, we propose a simple model for trend-following in which we are able to
show that returns can be represented as an option on the square of the realised returns. This
shares some qualitative similarities with the straddle benchmark, but takes the analysis a
step further. For example, we are able to derive a necessary condition on the realised Sharpe
ratio of the underlying asset to obtain positive returns:
1
|Sharpe ratio| > √
2τ
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where τ is the average duration (in years) of the trend estimator. Moreover, the maximum
annual losses due to trading impact are proportional to this threshold multiplied by the
average volatility of the strategy.
Conclusion
In this paper, we review three classes of strategies (directional, contrarian and trendfollowing) and obtain a quantitative breakdown for each of them. This breakdown provides
us with an accurate risk-return analysis of each of these strategies. More specifically it illustrates how the probabilities of making gains and losses have to be analysed together with
the corresponding average amount of gain or losses. High hit ratios are not necessarily a
sign of good strategies, but can reveal exposure to extreme risks.
The analysis of real-life situations would require extending the single asset case to the
multi-asset situations to better understand the result of adding such strategies. It would
also be interesting to build econometric tests to assess whether this model is capable of
providing accurate predictions of the behaviour of specific hedge fund strategies. This has
been left for further research.
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Issue # 7
Table of Contents
1 Introduction
9
2 Breaking investment strategies down into an option profile and
some trading impact
11
2.1 Model and results . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Trading impact associated with the stop-loss overlay . . . . . . . 13
3 Directional strategies: balanced and leveraged funds
15
3.1 Option profile and trading impact . . . . . . . . . . . . . . . . . 16
3.2 Predictions compared to actual backtests . . . . . . . . . . . . . 17
4 Contrarian strategies
19
4.1 Mean-reversion strategies . . . . . . . . . . . . . . . . . . . . . . 19
4.2 Averaging down . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5 Trend-following strategies
25
5.1 Analysis of trend-following strategies in a toy model . . . . . . . 26
5.2 Trend-following strategies as functions of the observed trend . . 28
5.3 Asymmetrical return distribution . . . . . . . . . . . . . . . . . 32
6 Concluding remarks
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Issue # 7
Risk-Return Analysis of
Dynamic Investment Strategies‡
Benjamin Bruder
Research & Development
Lyxor Asset Management, Paris
[email protected]
Nicolas Gaussel
CIO - Quantitative Management
Lyxor Asset Management, Paris
[email protected]
June 2011
Abstract
The investment management industry has developed such a wide range of trading
strategies, that many investors feel lost when they have to choose the investment style
that meets their requirements. Comparing these on a like-for-like basis is a difficult task
about which much has been written. The scope of this paper is restricted to strategies
investing in a single asset, and which are driven by the price of this asset. We show
how those strategies can be fully characterised by two components: an option profile
and some trading impact. The option profile depends solely on the final asset value,
whereas trading impact is driven by the realised volatility. From this analysis, most
of these investment strategies can be categorised in one of three families: directional,
contrarian and trend-following. While directional strategies exhibit the same kind of
behaviour as the underlying, contrarian and trend-following strategies exhibit asymmetric return distributions. Those asymmetric behaviours can be misleading at first
sight, as a seemingly stable strategy may hide large potential losses.
Keywords: Dynamic strategies, option payoff, asymmetric returns, trend-following strategy, contrarian strategy, volatility.
JEL classification: G11, G17, C63.
1
Introduction
We are unable to list here the immense variety of “investment styles” that are used in the
financial industry to generate returns. Value investors, growth investors, trend followers,
global macro analysts, long-short equity managers, special situations specialists or volatility
arbitragers, to name but a few, all rely on different and sometimes opposing views of how
markets work. However, within each style, some people succeed and some do not, preserving
the mystery of what are the determining factors of successful investment strategies. It is
very likely that none of these strategies is able to predominate sufficiently to eliminate the
others, as market forces would disable this very strategy in due course.
‡ We
are grateful to Philippe Dumont, Guillaume Lasserre and Thierry Roncalli for their helpful comments.
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As heterogeneous as those strategies might be, there is a need for final investors to
compare them in order to understand which are the determining factors in their performance,
how they compare and whether it makes sense to pay fees to a professional investment
manager.
Some metrics such as Sharpe ratio, factor analysis and specialised index benchmarking
have become widely accepted tools for analysing investment performance in the mutual fund
universe. However, this multi-factor analysis fails to account for more complex strategies
such as the ones used by hedge funds. Those funds use dynamic strategies that generate
returns that are difficult to link to the behaviour of standard factors such as the main equity
indexes. Imagine a stylised situation in which one has to compare two investment strategies.
The first would systematically sell short puts on the S&P index while investing the proceeds
in short-term bonds. The second would consist in investing eighty percent of its assets in
short-term bonds and using the remaining cash to invest in quarterly call options on the
S&P index. Which is the better strategy? How can they be compared? It is quite clear
that a Sharpe ratio or linear factor analysis would not provide enough information to assess
their quality.
Some attempts have been made to provide answers to those questions, some of which
we will now review. Addressing non-linearities in returns both in pricing models and in
fund performance is not a new topic. To quote only a few, Harvey and Siddique (2000)
propose a factor model incorporating not only the returns but the square of the returns
to explain hedge fund returns. Elsewhere, Agarwal and Naik (2004) propose a similar
approach for analysing equity-oriented hedge fund performance. However, instead of using
the square of returns, they create synthetic factors that mimic the performance of call
options, introducing different kinds of non-linearities. Fung and Hsieh (2001) focus on trendfollowing strategies. Roughly, trend-followers invest in proportion to the past performance
of a specific market, whether it is positive or negative. As a result, those strategies resemble
the delta of a straddle option in qualitative terms. Fung and Hsieh (2001) are thus testing
the hypothesis that the performance of trend-following strategies can reliably be compared
to the simulated performance of a strategy rolling lookback straddles on the MSCI World
index. In the professional world, many analysts classify strategies depending on whether
they are convergent or divergent, depending on their tendency to go against the market or
to follow it (see Chung et al. (2004), for instance).
The common feature of these approaches is that the design of the non-linear relevant
factor is mostly based on qualitative considerations and is used to provide econometric tests
of those assumptions. In this paper, we follow a slightly different route. Instead of trying
to analyse real-life hedge fund returns, we aim to constructively identify the exact payoff
generated by popular dynamic strategies.
The main idea for identifying this payoff is borrowed from option pricing literature. It is
well known that the strategy for obtaining a call option payoff consists in investing its delta
in the market on a day-to-day basis, this delta being the derivative of the price of the call. In
so doing, one obtains the payoff of a call, less the so-called gamma costs. Imagine now that
a portfolio strategy can be defined as a function of a given underlying asset price. It is very
likely that the payoff generated by such a strategy is the primitive of this strategy exposure,
plus or minus some trading impact. This covers those situations in which a manager can
decide how much to invest merely by looking at the price and some fixed parameters. We
show in this paper that many popular strategies fit this description.
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This paper is thus structured as follows. In the second section, we show how simple
strategy performances can be broken down into an option profile and some trading impact.
Section 3 is dedicated to studying the properties of simple directional strategies consisting
in holding a constant fraction of one’s wealth in a given asset. In section 4, we develop
an analysis of two popular contrarian strategies: the “return to average” strategy and the
averaging down strategy and we list their common features. Lastly, section 5 is devoted to
trend-following strategies, where an original result illustrates their typical convex behaviour.
2
Breaking investment strategies down into an option
profile and some trading impact
First of all, let us emphasise that the aim of this white paper is not to insist on the mathematics of the results but on the financial messages that are obtained. As a result we often
omit mathematical aspects such as filtrations, continuity of functions or detailed properties
of processes that would be necessary for a rigorous presentation. We hope that what has
been lost in terms of accuracy and rigour will be offset by the gain in simplicity and legibility.
2.1
Model and results
Consider a simplified situation in which an asset can be traded at each date t at price St ,
without any friction of any kind. St is supposed to be governed by an ordinary diffusion
model:
dSt
S0
= St × (μt dt + σt dWt )
= s
An investor is running an investment strategy consisting in holding a number f (St ) of
securities at any time. This strategy is supposed to depend only on the price itself and some
parameters, but not on time or on other state variables. For the sake of simplicity, it is
assumed that interest rates are zero1 . The wealth of the investor at each date t is denoted
Xt . On a day-to-day basis, or between t and t + dt, variation in wealth is written as:
dXt = f (St ) dSt
(1)
If St was deterministic and non-stochastic, it would be clear that:
XT − X0
=
ST
S0
f (St ) dSt
= F (ST ) − F (S0 )
where:
F (S) ≡
a
S
f (x) dx,
whatever the a chosen. If it is not, Ito’s lemma applied to the function F yields the important
property we want to emphasise.
1 To recover results in situation with interest rates, S and X will have to be replaced respectively by
t
t
St e−rt and Xt e−rt in the different equations.
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Proposition 1 Any portfolio strategy of the type (1) described above can be broken down
into an option profile plus some trading impact as follows:
1 T XT − X0 = F (ST ) − F (S0 ) + −
f (St ) St2 σt2 dt
(2)
2 0
option profile
trading impact
While simple, the above proposition yields some interesting qualitative properties. Regarding model assumptions, it is worth noticing the robustness of this property, which can
be obtained whenever Ito’s lemma can be used with continuous price processes. This covers
a wide variety of models and does not rely on special probabilistic assumptions. These include Black-Scholes, of course, but also local and stochastic volatility models. On the other
hand, the option profile obtained is European. If the strategy is richer in terms of variable
states, the option profile will be a function of those different states.
Let us now elaborate on these two terms. Trading impact depends on the variation in
the number of holdings in relation to market fluctuations. If the strategy involves buying
when the market goes up (f > 0) then the trading impact will necessarily be negative,
illustrating the “buy high sell low” curse of trend followers. Conversely, strategies that play
against the market will always have positive trading impact, benefiting from the opposite
effect. This trading impact increases with the volatility.
Interestingly, the sign of trading impact is directly related to the convexity of the option
profile. Positive trading impact is necessarily associated with concave profile. This is no
surprise to those familiar with option hedging, where it is well known that hedging a convex
profile will generate positive gamma gains, which explains the difference between option
prices and their intrinsic value.
All strategies with positive trading impact share similar characteristics in terms of winning probability. When trading impact is positive, using Proposition 1 leads to:
Pr {XT ≥ X0 } > Pr {F (ST ) ≥ F (S0 )}
In a situation where F is non-decreasing, we get:
Pr {XT ≥ X0 } > Pr {ST ≥ S0 }
The probability of showing a profit is therefore higher than the probability of the underlying
asset going up. On a weekly basis, most financial assets have a near 50/50 probability of
going up or down, which means that those strategies have more chance of showing a profit
than a loss. The higher the trading impact or the more concave the option profile, the
higher the probability of showing a profit. This effect is offset by higher potential losses. A
concave profile will therefore always exhibit negative skewness. Using a plain realised Sharpe
ratio to assess future fund performance is very likely to be flawed, as it would be inflated
artificially by the frequent positive gains. Following this analysis, rather than indicating a
good portfolio manager, frequent positive gains may be symptomatic of strategies with high
possible losses. The reverse holds for strategies with negative trading impact.
This breakdown may be worth bearing in mind from a qualitative point of view. In some
cases, it might be tempting to focus on one term and to neglect the other, but in general
they are of equal importance. A typical example of such bias is the stop-loss overlay, which
is commonly used to protect against losses in a portfolio.
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2.2
Trading impact associated with the stop-loss overlay
Imagine a situation where an investor runs an investment strategy which value is denoted St .
In order to limit its losses, this investor wants to add a separate overlay to the strategy, which
would consist in doing nothing while the strategy is above a certain threshold level Sstop , and
going short when it is below. In the following example, we will consider the initial strategy
as an underlying asset, and focus on the stop-loss overlay analysis. This overlay is purely
price dependent and should satisfy the assumptions of proposition 1. The corresponding
function f , representing the number of risky asset shares in the overlay portfolio, would be:
0
when St > Sstop
f (St ) =
−1 when St ≤ Sstop
Along the lines of Proposition 1, the option profile is a put option of strike Sstop . Trading
impact is more difficult to assess since f cannot be differentiated in a traditional manner.
Technically, one should use another version of Ito’s formula, sometimes referred as Tanaka’s
formula. Interested readers can refer to Carr and Jarrow (1990) for a detailed analysis of
that strategy. Qualitatively, f can somehow be differentiated and its differential is equal
to zero everywhere except at Sstop , where it has an infinite positive value. Hence, trading
impact will necessarily be negative, proportional to the volatility and to the time spent by
the underlying asset around Sstop . In a risk-neutral world, the average trading impact of
this stop-loss strategy is equal to the cost of the put. Interestingly, the stop-loss strategy
which could appear to be a free lunch as compared to buying a put option, generates some
trading impact, which is equal in average to the price of the put itself.
To confirm this effect, this strategy is simulated in Figure 1, with a stop loss level Sstop
equal to 90% of the original price. The resulting wealth is well below the asset price. Indeed,
this policy has a very strong trading impact around the strike level Sstop . Each time this level
is crossed, the strategy suffers from significant trading costs (see Figure 2 for a description).
These costs cause the wealth level to deviate from the target profile. Each time the stop
Figure 1: Stop loss/start gain strategy trajectory
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711635_214664_ white paper 7 Dynamic Strategy Risk Return.indd Sec1:13
ĞůƚĂ;ƌŝŐŚƚƐĐĂůĞͿ
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Figure 2: Trading impact appearance when crossing the strike
Asset move
Wealth
Hedged
profile
Losses
Stop loss level
Asset Price
loss level is crossed, the loss with respect to the target profile is proportional to the size of
the price movement. Total trading cost is therefore proportional to the volatility and to the
number of times the asset price crosses the strike Sstop . In the following, we calculate the
average trading cost, supposing that the asset price has a trend equal to the risk-free rate.
In that case, the average trading cost is exactly equal to the price of the put option.
2.2.1
A tree-based approach
One might wonder whether this is an artefact of continuous time or not. To answer this
legitimate question, we provide a discrete time analysis and show how a similar result can
be obtained. We use a discrete tree-based approach to estimate trading costs, encountered
each time the asset price crosses the stop-loss barrier. In this tree model, the asset price can
increase or decrease by ±h at every time step. Each time step typically represents a business
day. Thus√the typical time step is δt = 1/260 years and the asset price variation size should
be h = σ δt to obtain an annualised price volatility equal to σ. We obtain the tree-based
representation of Figure 3, where the stop loss level Sstop = 1 − h2 is represented by the red
line, just below the initial price. Each time the price crosses this red line (from above or
below), the investor loses h2 with respect to the target payoff max (St , Sstop ). Therefore, the
trading costs, i.e. the losses with respect to the target payoff, will be h2 multiplied by the
number of times thatthe level Sstop is crossed. The average number of times the barrier is
T
, where T is the total investment period, and δt is the time step2 .
crossed behaves like π2 δt
Thus the average trading cost is given by:
T
h 2T
L≈
→σ
2 π δt
2π
This is exactly the price of the put option, in a Gaussian framework. This equality between
the average trading cost and the price of the put option holds in any risk neutral model.
2 This
is the central limit probability for the asset value to quote at that value.
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Figure 3: Tree based representation of the asset price
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Compared to the protection based on the put option, the stop loss/start gain strategy
has approximately the same average cost. But this cost is very uncertain, while the cost of
the put option is known at inception. Even if the price of a put option is too high compared
to the expected future volatility, a hedging policy can be applied to replicate this option
(with the Black-Scholes formula). In this case, the cost of protection will depend on the
realised volatility alone, irrespective of the number of times the strike is crossed.
A profit-taking strategy is the exact opposite of the stop loss strategy. Diametrically
opposite results thus apply. As for the stop loss, investor can choose between selling a call
option and a profit-taking strategy. The call option seller abandons returns above Sstop
in exchange for receiving a fixed premium P . The definitive profit-taking strategy also
abandons returns above the same threshold, but is not exposed to market drawdowns once
the profits have been locked in. The profit-taking strategy that re-invests in the asset when
the price is below the threshold abandons high asset performance, but receives trading gains
each time the strike is crossed.
3
Directional strategies: balanced and leveraged funds
A common way to capture the risk premia yielded by equity markets consists in running
investment strategies that invest a stable proportion of one’s assets in risky markets. Merton
(1971) shows that this strategy is indeed optimal for investors having a logarithmic utility
and constant assumptions on expected returns and risks. Those strategies are sometimes
referred to as constant-mix strategies. Retail networks are used to distribute products tagged
as conservative, balanced or agressive. They often implement constant-mix type of strategies,
typically investing 20%, 50% or 80% of their wealth in risky markets. The popular 130/30
leveraged strategies are another example of constant-mix. Leveraged strategies whereby an
investor maintains a constant leverage of 2 or 3 on some asset also belong to the constantmix category. Constant proportion portfolio insurance (CPPI) strategies are a combination
of constant mix strategies plus some zero-coupon bond. Eventually, a strategy consisting in
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being short the market is a constant mix strategy with a leverage equal to −1.
A widespread proxy to understand the behaviour of constant-mix is the equivalent buy
and hold strategy. We will show in the following that, even if satisfactory for a short-term
horizon, this approximation may turn to be misleading over an horizon of more than a few
months.
3.1
Option profile and trading impact
For the sake of simplicity, we shall focus on a single asset case, similar to the one described
in section 2. The constant proportion of wealth invested in the risky asset, exposure, is
denoted by e. Locally, the relative variation of wealth is proportional to the return of the
risky asset:
dXt
dSt
=e
(3)
Xt
St
dSt
t
where dX
Xt is the return of the strategy between t and t + dt and St is the return of the
risky asset. This equation is slightly different from equation (1) as it involves exposure rather
than number of shares. However, we can follow exactly the same route, and by applying
Ito’s lemma to ln Xt we get a similar proposition.
Proposition 2 In the case of constant exposure e, the portfolio strategy can be broken down
into an option profile multiplied by the exponential of the trading impact in the following way:
⎛
⎞
e
⎜1 T 2 ⎟
ST
⎜
⎟
2
XT = X0
exp ⎜ e − e
σt dt⎟
(4)
S0
⎝2
⎠
0
option profile
trading impact
T
where 0 σt2 dt is the cumulated variance of the risky asset between times 0 and T . If
volatility is constant over time, this quantity is equal to σ 2 T .
As in Proposition 1, the wealth can be split at a certain date into an option profile
and some trading impact. However, here the two terms are multiplied rather than added
together. The option profile is a power option, whose power is exposure e, while the trading
impact depends on realised volatility alone. Since both terms are positive, constant-mix
strategies always ensure positive wealth in a market that trades continuously without any
gap.
The e − e2 term is a variation of holdings as in Proposition 1. For confirmation of this,
let us consider the discrete case where a fraction of the wealth, e, is invested in a risky asset.
t
Let f denote the number of securities held in the portfolio. Initially ft = eX
St . If the risky
asset moves by x% then we have:
St+δt
Xt+δt
Δf
= St × (1 + x)
= Xt × (1 + ex)
Xt+δt
Xt
= e
−e
St+δt
St
The variation of holdings can then be computed as
e (e − 1)
Δf
= Xt ×
ΔS
St2
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Figure 4: One year option profile adjusted from trading impact, with 20% volatility
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The trading impact in equation (4) appears to be of the same kind as in equation (1) of
Proposition 1 and related to the variation of number of holdings with respect to the variation
of the risky asset. The option profile is not linear with respect to the asset price, except in
the obvious cases of a delta one product (e = 1) or a portfolio fully invested in cash (e = 0).
In the case of balanced funds, with positive exposure and no leverage, the option profile
is concave and trading impact is positive (see Figure 4). In terms of indexation to the
underlying market, the strategy is less indexed to the risky asset when its value is high, and
more indexed to this asset when its value is low. On the other hand, leveraged strategies
(e > 100%) and short selling strategies (e < 0) exhibit a convex profile. Those profiles offer
potentially very high returns at the cost of more frequent losses. Trading impact increases
rapidly as e grows (see Figure 5). For example, the influence of volatility is 3 times larger3
for e = 3 than for e = 2. Figure 4 describes the effet of those strategies on final wealth,
taking both target payoff and trading impact into account for 20% volatility and a 5-year
horizon.
3.2
Predictions compared to actual backtests
All these formulas are derived from continuous time mathematical models, but they are quite
accurate in practical situations. We compare backtested results of constant mix strategies
combining the DJ Eurostoxx 50 index and cash. For each simulation, constant mix strategies
start with an initial value equal to 1. Then, the value of the strategy after one year is compared to the relative value of the Eurostoxx 50 with respect to the starting date. Simulations
are started on each business day between January 1987 and December 2010. Figures 6 and
7 plot the values of each backtest with respect to the relative value of the Eurostoxx 50.
These values are compared to the prediction obtained with formula (4), where the volatility
parameter is set equal to 20%. Figure 6 illustrates the balanced strategy (50% invested
`
´
expression 12 e − e2 is equal to −3 when e = 3, and is equal to −1 if e = 2, while the term σ 2 T
remains unchanged for both exposures.
3 The
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Figure 5: Annualised trading impact as a proportion of the initial wealth
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in Eurostoxx and 50% invested in cash), while Figure 7 illustrates the 4 times leveraged
Eurostoxx 50 strategy.
The cash rate is still taken to be 0. As shown in Figure 6, the prediction is very accurate
for the 50/50 constant mix strategy. On the other hand, the 4 times leveraged strategy has
a larger prediction error. This is because the prediction is computed with fixed volatility of
20%. The effective 1-year volatility of the Eurostoxx can be very different from this value.
When e = 50%, sensitivity to the realised variance is very low and equal to 12.5% of the
Figure 6: 50/50 constant mix strategy on the Eurostoxx 50
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R I S K - R E T U R N A N A LY S I S O F D Y N A M I C I N V E S T M E N T S T R AT E G I E S
Issue # 7
Figure 7: 4 times leveraged strategy on the Eurostoxx 50
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variance4 .
It is much higher for leveraged strategies. When e = 4 for instance, it is equal to −600%
of the variance. The final value of the leveraged strategy thus depends heavily on the realised
variance. This explains the differences with respect to the 20% volatility formula. As bullish
markets are generally less volatile than bearish markets, backtested portfolios are generally
higher than the formula for high returns of the Eurostoxx 50 index (right side of Figure 7),
and lower than the formula for most negative Eurostoxx returns (left side of Figure 7).
4
Contrarian strategies
Let us now focus on contrarian strategies. From Proposition 1, we expect that, going against
the market, those strategies will have a tendency to exhibit frequent small gains and less
frequent large losses. In particular, this section is devoted to the study of two popular
strategies: the mean-reversion strategy and the averaging down strategy.
4.1
Mean-reversion strategies
4.1.1
Strategy definition
In some situations, investors state that an underlying should quote close to a price denoted
Starget in the sequel. Starget can be obtained as the fundamental value of a stock obtained
using financial analysis. It can also be a kind of average value around which an asset is
supposed to exhibit some mean-reversion behaviour. Mean-reversion rationales are frequent
in financial markets, as certain ratios are supposed to remain within a certain absolute range,
outside which the situation is deemed abnormal. Price/earnings, volatility levels, spreads
between stocks or indexes among others are indicators that are commonly used as a basis
for mean-reversion analysis.
4 Sensitivity
e = 50%.
to the cumulated variance σ 2 T is equal to
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1
2
`
´
e − e2 . This expression is equal to 12.5% when
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Figure 8: Mean-reverting investment policy
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All these strategies can be summarised in a simple guideline: buy the asset when its price
is below its target Starget , and sell it when the price is above. The simplest corresponding
rule consists in holding an amount proportional to the distance between the price and its
target level, Starget . In the same framework as Proposition 1 this can be written as:
f (St ) =
m (Starget − St )
St
(5)
f (St )×St is the total amount of the risky asset bought at time t and m is a scaling coefficient.
This investment rule is illustrated in Figure 8. Obviously, this investor has a short position
when the asset price is above the average, and a long one if the asset price is below the
average. The number of risky asset shares in the portfolio decreases with respect to the
asset price. f is effectively a decreasing function of S, as shown in Figure 8. The investor
thus takes advantage of volatility when the price oscillates around a given level (even if this
level is not the target level Starget ).
In terms of risks, this exposure policy would be unlimited if the asset price rose to infinity.
Moreover, the number of asset shares is unlimited when the asset price goes to 0. Note that
this framework makes it possible to set limits on the amount and/or number of shares, by
using a different definition of f . This would lead to more acceptable maximum risks for this
strategy. However, to keep things simple, this is not done here. In this situation, proposition
1 applies straightforwardly. The option profile and the trading impact are equal to:
Option profile = m × (Starget ln (ST ) − ST )
T
1
Trading impact =
σt2 dt
mStarget
2
0
As expected, the trading impact is always positive and proportional to the realised variance
of the asset during the investment period. Thus, for a given final value of the asset, the
final wealth increases with the realised variance. Conversely, the option profile is concave
and can potentially lead to unlimited losses.
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Figure 9: Investor wealth (S0 = 80%, Starget = 100%)
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Analysis of the option profile
Suppose that the asset price at initial date is not equal to the average price. A significant
profit can be made if the asset price is closer to the average at maturity than it was at
inception date. Figure 9 shows the final wealth of an investor as a percentage of the initial
wealth. Suppose that the initial price of the asset is equal to 80% of the long term average Starget = 100%. Over a six-month time horizon, a significant mid-term profit can be
generated if the asset price moves closer to the average. If the price increases from 80% to
100%, the realised gain is 3.3%. Conversely, significant losses are incurred if the asset price
decreases. If the price falls from 80% to 60%, realised losses are equal to 7.7%. Nevertheless,
investors may accept this risk if they strongly believe that the asset price will converge to
Starget in the near future.
4.1.3
Trading Impact
Now, suppose that the investor starts with initial wealth X0 = 100%. Suppose also that
the initial price S0 of the asset is equal to the average price Starget . Figure 10 shows the
wealth of the investor at time T as a function of ST . We also assume that the annualised
volatility of the risky asset is a constant 20%. In this figure, the option profile is always
lower that the initial wealth of the investor. Mathematically, F (ST ) is negative for all ST .
The option profile always makes a negative contribution to the performance. Naturally, this
loss increases when the asset price moves away from the average. Profits come from the
trading gains. These gains increase when the strategy is performed for a longer time.
This strategy thus delivers performance when, after a volatile trajectory, the asset price
finishes near the average. On the other hand, significant short-term losses can occur if the
price moves away from the average. This is therefore an asymmetric strategy, involving
small and slow gains with high probability and large and quick losses with low probability.
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Figure 10: Investor wealth (S0 = Starget = 100%)
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Averaging down
In this section, we will show that this type of strategy is highly likely to deliver gains,
balanced by a significant bankruptcy risk.
4.2.1
A miracle recipe for recovering losses?
Let us start with an example. Suppose that an investor buys a stock at $100 price, and
that the stock price drops to $90. The difference between the average entry price and the
current price is $10. If the investor buys another share in the same stock (i.e. doubles his
position), the average buying price is now $95. The difference between the current price
and the average entry price is now only $5. Of course this new figure does not correspond
to any actual loss reduction. The $10 losses are just diluted into a larger position. As the
exposure is larger, a small $5 increase of the stock price is now sufficient to cancel out all
previous losses. But the larger exposure to the asset may also exacerbate future price falls.
A $10 asset price decrease will now lead to a $20 loss. Some investors may be tempted to
average down once again, in order to take advantage of a potential rebound. On the other
hand, the investor may face severe risks after doubling his exposure a few times.
This strategy can be related to the martingale gambling technique. It was originally
designed for a game in which a gambler doubles his stake if his bet is successful, or else
loses it. The martingale strategy is to double the bet after every loss, so that the first win
would recover all previous losses, plus a profit equal to the original stake. The gambler will
almost surely win if he is allowed to bet an infinite number of times (see Harrison and Kreps
(1979) for a detailed analysis). Unfortunately, he may go bust before his first win in real life
situations, as his stakes double at each loss. Indeed, after 5 consecutive losses, the gambler
has to bet 32 times his original stake, which is unacceptable in most real life situations. The
losses in the most adverse scenario (going bust) are of several orders of magnitude over the
expected gains. For the same reasons, averaging down strategies can recover losses if a small
price increase occurs, but may result in the investor going bust if this increase happens too
late.
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Figure 11: Exposure policy as a function of the wealth level
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Clearly, averaging down could absorb losses provided the asset price stays above a certain
limit. Beyond this limit, the strategy may lead to huge losses due to the increasing exposure
to the asset. The goal of this section is to identify this threshold.
4.2.2
A fine line between objective achievement and severe losses
Let us formalise this strategy. As previously, Xt stands for the investor’s wealth at time
t, and St for the price of the risky asset. Suppose that an investor has an initial wealth
X0 = 100%, and wants to obtain a target wealth of Xtarget = 110% whenever the asset
price increases by R = 10%. Initially, this objective can be attained by investing 100% of
the wealth in the risky asset. Now, suppose this investor starts from a wealth level of Xt .
The exposure needed to obtain target wealth of Xtarget if the underlying asset moves by
10% is described by the following relationship:
Xtarget = Xt (1 + R × e (Xt ))
or equivalently:
1
(6)
(Xtarget − Xt )
R
The variation of wealth will be governed by equation (3) but with variable exposure. The
determining factor in this strategy is the distance between current wealth and targeted
wealth. It shares some similarities with contrarian strategies in which the determining
factor is the distance between the current asset value and a target asset value. This exposure
policy is decreasing with respect to current wealth, as shown in Figure 11. Intuitively, it
is an increasing function of the objective, as more risks must be taken to achieve higher
objectives.
e (Xt ) =
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Proposition 3 Consider a portfolio strategy that follows the exposure rule (6). The distance
to the target wealth can be broken down into an option profile and some exponential trading
impact:
⎛
⎞
T
− R1
⎜ 1
1
⎟
ST
1
⎜
⎟
Xtarget − XT = (Xtarget − X0 )
(7)
exp ⎜−
σt2 dt⎟
+ 2
S0
⎝ 2 R R
⎠
0
option profile
trading impact
First of all, the objective is only attained asymptotically, as exposure converges to 0 as
the strategy approaches the objective. The option profile is always below the static strategy
with similar initial exposure, as shown on Figure 12. The averaging down overperformance
may only come from trading impact (i.e. volatility). The longer the investment period, the
higher this trading impact will be. In this example, volatility is presumed to be constant
and equal to 20%.
Figure 12: Averaging down strategy outcome
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While the strategy profile stays close to the objective when performance is positive, it
decreases very quickly for negative returns. In this example, the profile leads to bankruptcy
when the asset price decreases by 20%. Indeed, the risky asset return objective R is typically
−1/R
low compared to 100%. Therefore, the term SST0
may be a very large negative power
of the spot price5 . This large power explains the sudden loss behaviour of the option profile.
From a financial point of view, this is explained by the increasing exposure when the
strategy moves away from the objective. Over a six month horizon, the strategy delivers a
positive P&L as soon as the risky asset has a return greater than −10%. If asset volatility
is equal to 20%, the probability of a positive P&L after 6 months is 76%. Unfortunately,
5 In Figure 11, we take R = 10%. Therefore the option profile involves a power − 1 = −10 of the
R
underlying asset.
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Figure 13: 1Y return of the average down strategy
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this attractive hit ratio is counter-balanced by potentially very large losses. The probability
of going broke within six months is 2%. This is low, but not negligible, as this single event
could absorb years of profits plus the initial capital.
4.2.3
Backtesting
This asymmetry can be shown in a backtest of this strategy on the DJ Eurostoxx 50 index.
We provide a series of one-year simulations, with starting dates ranging from January 1987
to December 2010. The return objective is set at 10%. The resulting one-year returns
are displayed in Figure 13. One-year returns meet the objective for long time periods. For
example, these returns are very stable from 1991 to 2000, providing comforting results during
those nearly ten years. On the other hand, it leads to severe collapses, or even going broke
during the market downturns of 2002 and 2008.
5
Trend-following strategies
Trend-following strategies are a specific example of an investment style that emerged as an
industry in its own right. So-called Commodity Trading Advisors are the largest sector of
the Hedge Fund industry. According to BarclayHedge6 , the total AUM managed by CTA as
of Q1 2011 was $290Bn AUM, equivalent to 15% of total Hedge Fund AUMs at that date.
This said, trend-following styles are not restricted to CTA funds, as they have been used
by many other investment managers for a long time. They were mentioned, for instance, in
Graham (1949):
“In this respect the famous Dow Theory7 for timing purchases and sales has
had an unusual history. Briefly, this technique takes its signal to buy from
a special kind of break-through of the stock averages on the up side, and its
6 See
7 An
www.barclayhedge.com/research/indices/cta/Money_Under_Management.html.
outmoded denomination of momentum or trend-following techniques.
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selling signal from a similar break-through on the down side. The calculated-not
necessarily actual-results of using this method show an unbroken series of profits
in operations from 1897 to 1946. On the basis of this presentation the practical
value of the Dow Theory would appear firmly established; the doubt, if any,
would apply to the dependability of this published record as a picture of what a
Dow theorist would actually have done in the market.”
Surprisingly, despite its importance in the investment industry, this investment style is
largely overlooked by standard finance textbooks. Most available documents about trendfollowing techniques consist of a collection of testimonials of how successful traders managed
to make money out of their trading rules. It is difficult to identify a clear direction among
those publications. Of these, let us quote Turtles (2003), which offers a valuable introduction
to CTA systems8 . This text summarises the trading principles advocated by Richard Dennis,
a pioneer of systematic CTA trading.
From a very different perspective, as mentioned in the introduction, Fung and Hsieh
(2001) propose to benchmark trend-following strategies against the returns of a lookback
straddle on the MSCI World. They follow a line of thought similar to Proposition 1 and
infer this profile from the qualitative properties of the investment process.
This section is devoted to providing a simplified analysis of those strategies. While diversification across a large number of assets greatly improves the efficiency of trend-following
strategies, to keep things simple we are going to focus on the single asset case here. First,
trend-following strategies are analysed within a simple tree model, which identifies the qualitative properties of CTA strategies. In a second section, a more refined model is proposed,
which gives a precise representation of trend-following strategies in terms of option profile
and trading impact.
5.1
Analysis of trend-following strategies in a toy model
This section provides stylised facts about trend-following strategies in a binomial tree model.
It is borrowed from the first part of Potters and Bouchaud (2005). In this model, the
risky asset starts with an initial price of $100. This price increases or decreases by $1 at
each period, with a similar probability. The period would typically be one day. Figure 14
describes the cumulated P&L of a corresponding long investment in this asset for the next
three periods.
In this framework, the simplest trend-following strategy would be the following. Suppose
that an investor observed a positive trend before the start date. His strategy is to invest $100
in the asset, and to sell this position after the first negative return is observed (which can be
interpreted as a negative trend). Figure 15 represents the P&L of this strategy depending
on the risky asset trajectory. We also compute the probability of each outcome, assuming
that the positive and negative return probabilities are both equal to 50%. The final P&L is
represented in light blue boxes.
In this simple model, the loss probability is 50%, while the gain probability is only 25%
(the remaining 25% corresponding to neither profits or losses). However, while losses are
limited to $1, the gains can go up to any value. The average gain is $2, which offsets the
smaller gain probability. Thus, the returns distribution of this strategy is positively skewed:
small losses are frequent, while gains occur rarely, but with a larger amplitude. This is
8 The
text “The original Turtle Trading Rules” may be found on the web site www.originalturtles.org.
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Figure 14: P&L of a risky asset investment in the binomial model
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Figure 15: P&L of the trend-following strategy
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1
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16
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Figure 16: P&L distribution of the trend-following strategy
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illustrated in Figure 16. With Proposition 1 in mind, this is very consistent with a strategy
exhibiting a convex option profile and negative trading impact. Let us now be more precise
and obtain the corresponding breakdown in a continuous time model.
5.2
5.2.1
Trend-following strategies as functions of the observed trend
A model of a trend-following strategy
Trend-following estimators use past returns to predict future price changes. In the previous
example, the trend was estimated using only the last return: the investor forecasts a positive
(resp. negative) return if the last one was positive (resp. negative). In reality, longer period
returns, such as one month or one year past returns, are used for estimation. Many trend
followers use moving averages of prices or returns in order to provide more stable predictions.
For example, some consider the difference between a short-term (e.g. one month) moving
average of the asset price, and a long-term average (e.g. six month). This difference is
positive when past returns are mostly positive during the last six month period. It is
therefore considered to be a valid estimator of past trends. In real life situations, investors
use more complex combinations of such indicators.
Our purpose here is not to find the best estimate, but to provide clear analysis. We will
therefore consider the simple example of a moving average of past daily returns. We choose
a moving average with exponential weights:
μ̂t =
n
1 − iδt St−iδt − St−(i+1)δt
e τ
τ i=0
St−(i+1)δt
Such an estimator has interesting properties. It depends on a single parameter τ , which
represents the average duration of estimation. Due to exponential weighting, recent returns
have a larger impact than older ones. It does not suffer from “threshold effect”, that is,
changes of regimes due to a past observation that exits the averaging window. Moreover,
this estimator has theoretical foundations, as it can be interpreted as the Kalman filter for an
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unobservable trend. Lastly, it produces simple formulas for the performance of the related
strategy. In particular, one can derive the dynamics of the moving average depending on
the asset returns:
1 St+δt − St
δt
μ̂t +
μ̂t+δt = 1 −
τ
τ
St
or, in a continuous time framework:
1
1 dSt
dμ̂t = − μ̂t dt +
τ
τ St
(8)
In the following, we suppose that the investor considers that the best returns estimation
between times t and t + dt is μ̂t dt. Based on this assumption, the investor can simply apply
the optimal Markowitz/Merton strategy9 . Here, we will still consider that the risk-free rate
is 0. In this case, its exposure to the risky asset will be:
et = m
μ̂t
σ2
(9)
where μ is the trend estimator, σ is the annualised volatility of the underlying, and m is a risk
tolerance parameter. Interestingly, this exposure strategy is qualitatively consistent with the
principles described in the Turtles Rules. The exposure to the risky asset is proportional
to this risk tolerance. This parameter depends on the investor’s risk profile. This exposure
is also proportional to the trend estimator value μ̂t and inversely proportional to the risk.
Note that it is not capped and is almost never 0.
Following this strategy, the investor’s wealth evolves as equation (3):
dXt
Xt
dSt
St
μ̂t dSt
= m 2
σ St
= et
(10)
Now, by inverting equation (8), the dynamic of the wealth can be written as a function of
the trend μ̂t . We can then follow the steps of Proposition 2 where the variable is no longer
the asset price St but the asset trend μ̂t .
Proposition 4 The logarithmic return of a trend-following strategy that follows the exposure
function (9) can be broken down into an option profile and some trading impact:
⎛
⎞
⎟
T 2
⎜ τ XT
μ̂t
1
1
⎜
⎟
dt⎟
ln
1
−
= m ⎜ 2 μ̂2T − μ̂20 +
m
−
(11)
2
X0
σ
2
2τ
⎝2σ
⎠
0
option profile
trading impact
It is worth remembering that the trend μ̂t is not a model assumption but the actual
measured trend.
5.2.2
Option profile
The option profile is related to the square of the observed trend:
τ Option profile = m 2 μ̂2T − μ̂20
2σ
9 See
Merton (1971) for details.
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This is similar in concept to the straddle profile suggested by Fung and Hsieh (2001). It is
also perfectly in line with regressions including squared returns factors performed by Harvey
and Siddique (2000). This option profile part evolves quickly, and has a short term memory.
It vanishes along with the measured trend μ̂. Interestingly, the worst case scenario is that
the measured trend falls from its initial value μ̂0 to 0, which would lead to an upper limit
mτ 2
of 2σ
2 μ̂0 . This quantity depends largely on the value of the measured trend at the start.
This is natural, as the strategy is more exposed to the risky asset when the trend is high.
This option profile is displayed in Figure 17, where we consider a six month moving average,
with 15% risky asset volatility, and a risk tolerance parameter m = 5% (see the next section
for details about this set of parameters). Furthermore, we assume that the initial measured
trend is equal to μ̂0 = 0. As a consequence, the option profile is always positive.
Figure 17: Short term option profile, as a function of the risky asset return
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On the other hand, trading impact is obtained by a cumulated function of the realised trend:
T 2
1
μ̂t
1
dt
(12)
m 2 1− m −
Trading impact =
σ
2
2τ
0
This trading impact changes slowly, and has a long-run memory due to its cumulative nature.
μ̂2
The evolution of this long term P&L depends only on the ratio σt2 which is the square of
the measured short term Sharpe ratio. Trading impact is negative on the long term if this
1
ratio is below τ (2−m)
, and otherwise positive.
The√ risk tolerance m can be related to the target volatility σtarget by setting m =
σtarget 2τ . In practical examples, m (around 10%) will be small with respect to 1. Thus,
μ̂2
1
a good estimate for long term trading impact per annum is given by σt2 − 2τ
. This provides
a necessary condition for the trend-following strategy to generate profits:
1
|Sharpe ratio| > √
2τ
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For a 6 month moving average (i.e. τ = 12 ), the long-term profit increases if the annualised
measured Sharpe ratio is above one in absolute terms.
Eventually, the worst-case scenario for the long term P&L is that μ̂t = 0, which is no
σ
m
√
surprise! In that case, annual losses due to trading impact are capped at 2τ
= target
. In
2τ
the case of a six-month moving average, annual maximum long-term losses are equal to the
target volatility of the strategy.
5.2.4
Backtested example
In this section, the trend-following strategy is backtested on the DJ Eurostoxx 50 index.
Instead of performing the trend-following strategy directly on the index, it is run on a
volatility targeted index which constantly adjusts its exposure to the Eurostoxx 50 in order
to keep its volatility equal to σ = 15%. The strategy itself uses a six-month moving average
(τ = 12 ), with target volatility of σtarget = 5%. This involves a parameter m = 5%, with an
average absolute exposure above 33% (the average backtested absolute exposure is around
50%).
Figure 18: Backtest of a trend-following strategy on the vol-target Eurostoxx 50
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The backtest is performed between January 1987 and May 2011. Figure 18 shows the
breakdown of the strategy value between the option profile and the trading impact. Long
term trading impact is computed from formula (12), independently of the strategy value
calculations. This component is smoother than the strategy NAV. The short-term option
profile is the difference between the strategy NAV and the long term P&L. This component
is always positive, as the measured trend at the start is set at 0. The strategy NAV is
therefore always above the long term value. By construction, this long term cannot decrease
more than 5% p.a. (the instantaneous slope of the long term P&L, i.e. the daily trading
impact, is given in Figure 19). The worst-case scenario thus appears fairly acceptable.
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Figure 19: Annualised daily contribution to trading impact
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5.3
Asymmetrical return distribution
The return distribution of the strategy depends heavily on the distribution on the trend
estimator μ̂, as shown in equation (11). Of course, the behaviour of μ̂ depends of the model
assumptions on the asset price dynamics. Let us suppose a standard Gaussian dynamic for
St with fixed volatility σ and a fixed trend μ. In this case, the long term distribution of the
trend estimation is Gaussian, with an average of μ and volatility of √σ2τ .
In the case of the six-month exponential moving average, the standard deviation of the
measured trend is equal to the standard deviation of the yearly return of the underlying asset.
The long term returns of the strategy are driven by the square of the measured trend μ2
and are then highly asymmetric. The stationary distribution of the annualised contribution
to the long term P&L is given in Figure 20, in the same conditions as in the backtest of the
previous section. This distribution is computed under two sets of assumptions: in the first
one, the underlying asset has no drift (risk neutral probability), while in the second one the
Sharpe ratio is constant and equal to 1 (this involves a constant drift μ = 15%) .
This distribution looks very similar to the toy model version of Figure 16. The general
result remains unchanged. The probability of losing is higher than the probability of gaining,
but the average gain is much higher than the average loss. As expected, the trend-following
strategy works poorly in trendless models (the average P&L is equal to 0), and much better
in models with a positive trend. For an easier comparison of the winning and losing probabilities, the cumulative version of the distribution is illustrated in Figure 21. When there is
no trend, the positive P&L probability is around 30% as opposed to 50% in the case where
the Sharpe ratio is equal to 1.
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Figure 20: Stationary distribution of the annualised daily trading impact
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6
Concluding remarks
In this paper, we have shown how the returns on many popular strategies can be broken down
into an option profile and some trading impact, both being of similar importance, on average.
We reviewed three classes of strategies (directional, contrarian and trend-following), and
obtained a quantitative breakdown for each of these. This breakdown allows for an accurate
risk-return analysis of each of those strategies. More specifically, it illustrates how the
probabilities of making gains or losses have to be analysed together with the corresponding
average amount of gain or losses. High hit ratios are not necessarily signs of good strategies,
in fact they can reveal exposure to extreme risks.
Analysis of real-life situations would involve extending the single asset scenario to multiasset situations, for a better understanding of the impact of adding such strategies. It
would also be interesting to build econometric tests to assess whether our model can provide
accurate predictions of the behaviour of specific hedge fund strategies. This has been left
for further research.
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References
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Market Overreaction, Review of Financial Studies, 3(2), pp. 175-205.
[22] Merton R.C. (1971), Optimal Consumption and Portfolio Rules in a Continuous-Time
Model, Journal of Economic Theory, 3(4), pp. 373-413.
[23] Merton R.C. (1981), Timing and Investment Performance: I. An Equilibrium Theory
of Value for Market Forecast, Journal of Business, 54(3), pp. 363-406.
[24] Perold A.F. (1986), Constant Proportion Portfolio Insurance, Harvard Business
School, Manuscript.
[25] Potters M. and Bouchaud J-P. (2005), Trend Followers Lose More Often Than They
Gain, arxiv.org/abs/physics/0508104v1.
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R I S K - R E T U R N A N A LY S I S O F D Y N A M I C I N V E S T M E N T S T R AT E G I E S
Issue # 7
Lyxor White Paper Series
List of Issues
• Issue #1 – Risk-Based Indexation.
Paul Demey, Sébastien Maillard and Thierry Roncalli, March 2010.
• Issue #2 – Beyond Liability-Driven Investment: New Perspectives on
Defined Benefit Pension Fund Management.
Benjamin Bruder, Guillaume Jamet and Guillaume Lasserre, March 2010.
• Issue #3 – Mutual Fund Ratings and Performance Persistence.
Pierre Hereil, Philippe Mitaine, Nicolas Moussavi and Thierry Roncalli, June 2010.
• Issue #4 – Time Varying Risk Premiums & Business Cycles: A Survey.
Serge Darolles, Karl Eychenne and Stéphane Martinetti, September 2010.
• Issue #5 – Portfolio Allocation of Hedge Funds.
Benjamin Bruder, Serge Darolles, Abdul Koudiraty and Thierry Roncalli, January
2011.
• Issue #6 – Strategic Asset Allocation.
Karl Eychenne, Stéphane Martinetti and Thierry Roncalli, March 2011.
Q U A N T R E S E A R C H B Y LY X O R
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Issue # 7
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The Lyxor White Paper Series is a quarterly publication providing
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