Momentum Effects: G10 Currency Return Survivals Andrew Clare1, Hartwig Kos2, Natasa Todorovic3
Abstract
This paper analyses momentum effects in G10 currencies by applying survival analysis common in life time statistics to shed a new light on the market efficiency within the currency market. For each of the 90 currency crosses we model the survival probabilities of positive and negative momentums obtained from a wide set of dual crossover moving average combinations. We find strong evidence of inefficiencies: empirical momentums stemming from longer (shorter) moving averages live shorter than (outlive) the theoretical bootstrapped signals. „Enhanced‟ trading strategy based on this finding persistently outperforms the „benchmark‟ trading rule.
1. Introduction
1
Professor in Asset Management, Cass Business School, 106 Bunhill Row, London EC1Y 8TZ, UK, email:
[email protected] 2 PhD Candidate, Cass Business School, 106 Bunhill Row, London EC1Y 8TZ, UK, email:
[email protected] 3 Corresponding author. Senior Lecturer in Investment Management, Cass Business School, 106 Bunhill Row, London EC1Y 8TZ, UK, email:
[email protected] 1
According to Fama (1970) weak form market efficiency suggests that no abnormal return can be earned by applying trading strategies that are purely based on historic price information. Although capital markets are generally regarded as weak and semi strong efficient, currency markets seem to defy the market efficiency model persistently4. The aim of this paper is to analyse data dependencies and patterns in historic currency time series data and implement trading rules that lead to abnormal currency returns that cannot be explained by any systematic risk taking. Following Kos and Todorovic (2008), we analyse positive and negative momentum effects in G10 Currencies5in the period 04/01/1974 to 31/12/2009 via the use of econometric tools common for life time statistics. Specifically, we employ Kaplan and Meier (1958) product limit estimator (PLE) that allows us to identify market inefficiencies by comparing empirical momentum survivorship curves to simulated benchmark ones. This approach is novel in the currency literature. Survival time analysis as a tool for identifying market inefficiencies can be viewed as an extension of „runs rests‟ introduced by Fama (1965). When assessing market inefficiencies stemming from trading rules, researchers often rely on the comparison of empirical trading rule returns and simulated trading rule returns. In many cases these studies are either based on relatively short time windows where actual transaction costs are available or transaction costs are merely assumed to be constant, neither of which is fully satisfactory. Under the specification of survival time analysis market inefficiencies can be detected without any assumption of transaction cost. Practitioners might use the results of survival analysis to improve the performance of technical trading rules as such analysis identifies exact probabilities of survival of trading signals at any given point in time. This information might prove to be very useful in establishing exit points for trading rules. These are only a few examples of possibilities that open up from survival analysis and to the best of our knowledge this is the first study in the currency space that assesses such possibilities. We further examine to which extent currency market inefficiencies identified through survival analysis can be exploited by technical trading rules. To that end, we evaluate the profitability of the „benchmark‟ trading rule, constructed from all of our 39 moving average signals, against that of the „enhanced‟ trading rule. „Enhanced‟ rule is based on a subset of
4
For review of literature, see Froot and Thaler (1990) US Dollar (USD), British Pound (GBP), Japanese Yen (JPY), Euro (EUR), Swiss Franc (CHF), Norwegian Krone (NOK), Swedish Krona (SEK), Canadian Dollar (CAD), Australian Dollar (AUD), New Zealand Dollar (NZD) 5
2
moving average signals, which, according to survival analysis, deviate most from efficiency in the currency market. Our results provide strong evidence of inefficiency in the currency market as empirical momentum signals stemming from crossovers of shorter term (longer term) moving average combinations live longer (shorter) than theoretical benchmark signals. The strength of these deviations, though, diminishes over time. The profitability of the „benchmark‟ trading strategy also deteriorates over time, in line with survivorship analysis, while the „enhanced‟ trading strategy persistently outperforms the benchmark trading strategy over time. The latter opposes the findings of survivorship analysis and indicates that the source of profits of these „enhanced‟ returns does not lie in market inefficiency. The paper is organised as follows: Section 2 reviews the literature; Section 3 describes data and methodology which includes comparison of the survival time methodology with the runs test approach. It highlights not only the similarities, but also the main differences of both approaches. Section 4 presents the results and Section 5 concludes the paper. 2. Review of the Literature Upon the findings of DeBondt and Thaler (1985, 1987), Jegadeesh and Titman (1993) and Rouwenhorst (1998), which are the cornerstone of modern behavioural finance work, a school of thought and a vast body of academic literature analysing various aspects of equity momentum has developed. Currency momentum on the other hand was given much less attention in the finance literature. Academic research within the currency space was either focussing on the forward discount bias or the profitability of trading rules, hence Currency momentum. The forward discount bias or “carry” effect has by now been widely accepted as a phenomenon within the currency space. Nonetheless, the sources of that bias remain a subject of academic dispute. For further treatment of the forward discount bias, refer to Fama (1984), Froot and Frankel (1989) or Cavaglia, Verschoor and Wolff (1994). Despite the relative ambiguity of the sources of the forward discount bias, the “carry” phenomenon has very quickly found its way into the finance industry as well as main stream academic research. As opposed to dissecting the sources of carry, more recent papers such as Poljarliev and Levich (2008, 2010) use the carry and other phenomena as distinctive style benchmarks with which they assess the relative performance of active currency managers. Poljarliev and Levich (2008) show that fund
3
managers do not exhibit any alpha persistence; however, they do exhibit style persistence. Poljarliev and Levich (2010) suggest that in the time period before the crash in 2008, carry has been the most crowded currency strategy, which led to a massive unwind in the autumn of 2008. The results after this period allow for less obvious conclusions. Within the area of trading rule research, the most noteworthy early studies are Dooley and Shafer (1976, 1984) and Logue and Sweeney (1977). Both papers suggest very strong returns from applying filter rules. However, Neely and Weller (2011), provide the main criticism of these studies in that their sample periods were quite short and seemed somewhat spurious. They suggest that traditionally three theories have been put forward to explain the success of technical trading rules in the currency arena: 1) the profits are greatest around periods of central banks‟ intervention (see Szakmary and Mathur, 1997 and LeBaron, 1999); 2) the presence of Data snooping when it comes to selecting trading rules which work well on one dataset but may not be profitable on any other and 3) high trading rule returns are a mere reflection of systematic risk taking.
Levich and Thomas (1991) introduce the idea of using re-sampling (bootstrapping simulation) technique to tackle data snooping, which has subsequently become a benchmark methodology to assess the performance of trading rules. Their study investigates a set of five currencies against the US Dollar from 1976 to 1990. Although they show that 25 of the tested filter rules and 14 of 18 tested moving average rules offer results that suggest a statistically significant deviation from normality, their paper still focuses on a fairly narrow range of trading rules. Neely, Weller and Dittmar (1997) and Sullivan, Timmerman and White (1999) introduce more flexible methodologies that allow to control for Data snooping: a genetic program that searches for an optimal trading rule in the former study and comparison of a specific trading rule with a “benchmark” consisted of a large set of trading rules in the latter study. Finally, the evidence is mixed when it comes to investigating whether high returns from currency trading rules could have been generated by systematic risk taking rather than market inefficiency. Kho (1996) evaluates a set of moving average crossover rules with weekly data on foreign currency futures contracts from 1980 to 1991 for five currencies and suggests that high returns have been obtained by systematic risk taking. Later studies, such as Okunev and White (2003) evaluate 354 moving average rules for eight currencies from January 1980 to June 2000 and find that their trading strategy provides an excess returns over the “benchmark” currency basket that is MSCI market cap weighted of roughly 5%-6% per 4
year. To ensure that the results are not due to systematic risk taking, the authors analyse the correlations between the trading rule returns and the “benchmark” currency basket and find them to be close to zero. Chong and Ip (2009) extend Okunev and White‟s (2003) study to emerging market currencies and report 30% plus annualised returns from their strategy, which remain significant even after accounting for transaction costs of 5% per annum.
More recent studies such as Burnside, Eichenbaum, and Rebelo (2011) and Menkhoff, Sarno, Schmeling and Schrimpf (2011) follow the cross sectional approach introduced by Okunev and White (2003). The study by Menkhoff, Sarno, Schmeling and Schrimpf (2011) is noteworthy for its connection to the traditional momentum literature. The study implements the Jegadeesh and Tittman (1993) approach within the currency space. Their sample consists of cross sectional data of 48 countries over a time period from January 1976 to January 2010. Whereby not all of the 48 markets have the full data history form 1976, they are included in the cross sectional sample as they become available. In the spirit of Jegadeesh and Tittman‟s (1993) work, they create winner and loser portfolios and find that some of the combinations earn unconditional average excess returns of up to 10% per year. The authors also suggest that momentum returns are different from the carry element in currency markets and as a result they are not well captured in earlier academic research.
The motivation of this paper is to analyse data dependencies and patterns in historic currency time-series data using survival analysis of currency momentum, with the aim to implement trading rules that lead to abnormal currency returns that cannot be explained by any systematic risk taking. We follow Jochum (2000) and Kos and Todorovic (2008) closely. Both papers apply survival analysis to equity market. The main focus of the latter paper is based on the idea that under the notion of weak market efficiency, empirical equity returns should follow random pattern. Hence, positive or negative returns of an empirical return time series should not systematically outlive positive or negative returns created from a random return time series. In that sense, Kos and Todorovic (2008) study can be seen as an extension of the „runs test‟ introduced by Fama (1965). Whilst Fama (1965) compares the ratio of positive to negative returns with some theoretically derived value, Kos and Todorovic (2008) utilise the Kaplan-Meier (1958) PLE, which allows them to compare empirical survivorship curves to Monte Carlo simulated survivorship curves, using daily data for S&P global sector indices from 1998 to 2006. The results of the study suggest that various sectors show significant deviations from normality that can be exploited by even simple trading rules. 5
Albeit the fact that Kos and Todorovic (2008) approach offers an attractive alternative to the traditional Jegadeesh and Titman (1993) methodology, it comes with some deficiencies in the implementation. Firstly, their paper analyses simple return time series causing all their momentum signals to be very short lived and quite difficult to interpret. Further, no subsampling of the data was done. Additionally, they utilise a Monte Carlo simulation that defines Random walk or ARMA (1,1) processes as an appropriate benchmark processes. The main problem of this implementation is the fact that one has to make assumptions about the distributional characteristics of the underlying data time series. Given the fact that the PLE is non-parametric and the benchmark process is calculated based on a parametric model, the simulation process suffers from an inherent estimation error. What is more, Kos and Todorovic (2008) evaluate the deviation of empirical survivorship curves from benchmark survivorship curves, by comparing the average survival times. This is a very crude way of measuring differences between survivorship curves.
In this paper, we improve the shortcomings of Kos and Todorovic (2008) methodology and apply it to the currency space. Specifically, the improvements consist of: the analysis of considerably longer time period, which allows for sub-sampling; the use of the set of moving average pairs as momentum signals which enables momentum to live longer and be more interpretable; the use of re-sampling as a simulation methodology (which can be constructed in a non-parametric framework as the PLE) to establish a set of benchmark survivorship curves. Finally, we estimate the significance of the difference between empirical and benchmark survivorship curves by applying the Wilcoxon Log-rank test that is a standard test in survival time statistics.
3. Data and Methodology 3.1. Data description and transformations This paper uses two datasets compiled from Factset, Datastream and Bloomberg databases. Dataset I contains daily New York closing mid-values for G10 currencies, as well as threemonth cash rates for corresponding countries. It spans from 04/01/1974 to 31/12/2009 and contains 9025 trading days. G10 currencies are selected as they stand for the most liquid ones. All exchange rates are expressed as units of domestic currency versus one unit of foreign currency. Each of the historic currency price time series is rebased to 100 as of the 04/01/1974. A total of 90 currency pairs are analysed. Given the long history of this dataset,
6
since the Euro (EUR) rate does not date back to the mid seventies, the sample is backfilled with the historic Deutschmark (DEM) rate. The original EUR fixing rate of 1.95583 DEM per 1 EUR (as of 1 January 1999) is applied. Dataset I is split into nine sub samples almost equal in length6. Due to the 25 year span, this dataset is intended for the analysis of the long term behaviour of moving average rules.
Dataset II includes bid/ask spreads for each of the currency crosses in addition to daily New York closing mid-values for G10 currencies for the period 27/03/2002 to 31/12/20097. Overnight or one week interbank rates for the respective currency blocks are also included. All the bid/ask spreads of the non-dollar crosses have been synthetically created from dollar crosses. This dataset will facilitate the evaluation of the trading profitability of moving average rules.
For the trading rule implementation in later parts of the paper, currency returns as opposed to prices are used. However, various currencies have had significant interest rate differentials. To control for this natural bias, we adjust currency returns for any interest differential as follows:
(1)
Where the first term represents the daily interest rate differential between foreign ( domestic
and
currencies and the second is the currency return. In Dataset I (Dataset II) the
calculation of the interest rate differential is based upon the three month T-Bill rate (overnight or one-week rates) for each of the respective currencies. The adjusted return time series obtained from Equation (1) results in approximate currency returns that can be earned by following a futures based investment strategy. In order to incorporate the interest rate adjustment in the survivorship analysis itself, the historic price time series, from which the moving averages are calculated, is recalculated on the basis of interest rate adjusted returns, as per Equation 1. 6
The first eight sub samples consist of exactly 1000 observations and the ninth sub sample consists of 1025 observations. The reason for the almost equal split is the fact that each of the sub samples will show similar levels of statistical confidence, given the equal amount of data analysed . 7
This time period coincides with the last two sub samples of the first dataset. 7
3.2. Defining momentum and moving average combinations To define a momentum signal, we utilize a simple price moving average filter as it is one of the most widely used technical trading rules. A positive momentum signal is observed when the short term moving average is above the long term moving average and is given by:
(2)
Conversely, if the short term moving average is below the long term moving average then a negative momentum signal is observed. It is defined as:
(3)
There is no unified rule as to which moving average combination should be used to generate trading signal. Whilst Levich and Thomas (1991) apply short term focused trading signals, practitioners such as Elder (2002) suggest moving average ranges starting from 10, 20 and up to 50 days. We define the range of short term moving averages (SR) as 1 to 5 days as well as 10, 15, 20 and 25 days. Long term moving averages (LR) are defined as 5, 10, 15, 20, 25 and 30 days. Note that in any combination a SR has to be shorter than a LR. This leads to 39 moving average combinations upon which the survivorship analysis is based, as shown in Table 1. TABLE 1 Moving average combinations SR 1 SR 2 SR 3 SR 4 SR 5 SR 10 SR 15 SR 20 SR 25
LR 5 1/5 2/5 3/5 4/5
LR 10 1/10 2/10 3/10 4/10 5/10
LR 15 1/15 2/15 3/15 4/15 5/15 10/15
LR 20 1/20 2/20 3/20 4/20 5/20 10/20 15/20
LR 25 1/25 2/25 3/25 4/25 5/25 10/25 15/25 20/25
LR 30 1/30 2/30 3/30 4/30 5/30 10/30 15/30 20/30 25/30
Note: LR denotes long term moving averages and SR denotes short term moving averages.
3.3. Construction of survivorship curves
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The basic idea in creating survivorship curves is to model the probability of the persistence of some pre-specified signal within a given data sample. To illustrate this concept, Table 2 shows hypothetical trading signals that have been created from a dual crossover moving average trading rule. The trading rule generates a positive („buy‟) signals if the short term moving average is above the long term moving average (SR>LR) and vice versa. This gives a series of trading signals of different lengths scattered along the empirical time series. In Table 2 we can identify two positive momentum signals that survive one day, three signals that survive two days, two signals that live for four days and one signal that lasts seven days. The survivorship analysis aims to analyse the survival characteristics of the trading signals that have been created by the moving average crossover rules. This cannot be estimated at a single point in time because such observations occur randomly within the sample. Survivorship and Hazard curves which will be described in this section, as laid out by Kaplan and Meier (1958), overcome the problem of analysing uncensored datasets. By constructing the PLE, Kaplan and Meier (1958) found a way of ordering data such that survivorship probabilities can be calculated and inferences can be made. Originally, this methodology has been used in biomedical research to investigate the effectiveness of medical treatment on patient groups. However, over time, the methodology has found its use in analysing economic problems, such as the analysis of unemployment rates or the estimation of credit default rates as suggested by Kiefer (1988) and more recently in equity space as seen in Jochum (2000) and Kos and Todorovic (2008). TABLE 2 Graphical description of duration data
7
1
2
1
4
2
4
SR>LR
SR>LR
SRLR
SR>LR
SR>LR
SR>LR
SRLR
SRLR
SR>LR
SR>LR
SR>LR
SR>LR
SRLR
SRLR
SRLR
SR>LR
SR>LR
SR>LR
SR>LR
SR>LR
+ + + + + + + 0 0 + + + + 0 + 0 0 0 + + + + 0 + + 0 0 + + + + 0 + +
2
Note: A ‘+’ indicates a positive momentum signal generated if SR>LR and ‘O’ otherwise.
We will now outline the statistical principals of survival time analysis. The probability of failure or survival (in our case of a momentum signal) for a pre-specified time horizon can be written as follows:
9
(4) (5) (6)
Function F(t) in Equation (4) is defined as the probability of failure, i.e. probability of T being smaller than a time t, whereby T is a random variable denoting the time a momentum seized to exist and time t is pre-specified time. Hence, this is the probability that the time at which a momentum stops is before the end of the pre-specified time. Function S(t) in Equation (5)denotes the probability of success, i.e. the probability of T to be after the prespecified time t. Given the fact that F(t) and S(t) are mutually exclusive events, the link between both can be summarized in Equation (6). Taking the derivative of F(t) and S(t) produces the corresponding density functions for the two probabilities. Both measures in Equations (7) and (8) can be seen as the rate of either failure, or survival per unit of time. Equation (9) shows the linkage between both density functions.
(7) (8) =
(9)
Another concept in the subject of lifetime statistic that will help understanding of construction and the interpretation of the Kaplan-Meier PLE is the Hazard function, presented in Equations (10) and (11). Given the linkage between failure and survival probabilities, there are obviously many ways to express the Hazard function. For simplicity, we apply the standard definition of the concept. For a more in-depth treatment please refer to Kiefer (1988) and Lawless (2003). (10) (11) Equation (10) shows the general definition of the Hazard curve while Equation (11) gives the precise definition in terms of probabilities. A Hazard curve denotes the conditional probability of an observation ceasing to exist within a pre-defined time horizon from t to t+dt, given that it has survived until t. Interpreting this measure in terms of the momentum 10
signal, it would allow to calculate the conditional probability of a momentum seizing to exist in the time period between, say, day 10 and day 11, given it has survived 10 days.
Let us focus now on the construction of the Kaplan-Meier (1958) PLE. The PLE is a nonparametric measure; hence it does not rely on any assumption of distributional characteristics of the underlying data. This is particularly useful for the analysis of financial time series. Let ni represent the number of momentum signals in the sample immediately before failures occur in the given time interval, i.e. the number of observations „at risk‟ for that time interval. Further, let ni’ represent the number of momentums in that time interval immediately after failures occur and di the number of momentums that seize to exist (failures) during that same interval. The ratio between ni’ and ni represents a conditional probability of momentum survival for that time interval. The conditional probability of survival is shown in Equation (12) and its link to the hazard curve is shown in equation (13). Taking the product of the periodical survival probabilities, one can obtain the cumulative survival probability, which is in effect the PLE, as given in Equation (14).
(12) (13)
with k = 1 and ni‟ = ni - dj
(14)
The PLE estimator will approach the true survival function, when a large enough sample is taken. In order to make inferences of the validity of the estimator, the variance of the estimator has to be calculated:
(15)
Besides the calculation of the PLE estimator itself, this study relies on the comparison of any empirical survival curve with a theoretical benchmark survival curve to assess the presence of momentum effects. In order to facilitate such comparison, we use the Wilcoxon
11
specification the Cox-Mantel Log Rank test8, to verify potential differences between the empirical model and benchmark process. The Log Rank Test is based on the premise that every observation point on the survivorship curve can be seen as a contest between the two survival samples (empirical and benchmark).The test defines the null hypothesis as the probability of failure being equal for both survival samples at any given time.
3.4. Survivorship analysis versus runs test The survivorship analysis represents an extension of the concept of runs test, which was introduced by Fama in 1965. By marking a positive return as „+‟ and a negative return as ‟–„ in the runs test, one can count sequences of the same sign (the „runs‟) and assess whether they are in line with what is expected. For the purpose of illustration, one could imagine a sequence of positive and negative returns as:
. Dooley and Shafer (1976)
apply runs test to currencies, their analysis suggests that there is some degree of deviation form normality within currency returns. The problem with the runs test is that it works appropriately for establishing a total number of runs, however when it comes to analyzing the lifetime characteristics of runs, it fails to perform9. Specifically, when looking at the estimates for the average life of runs, Fama‟s (1965) calculations systematically underestimate the empirical results. Contrary to that, the proposed survivorship methodology gives results that do not show any systematic bias and are closer to the empirical observations; hence they are more conservative when it comes to testing for market efficiency. What is more, the proposed methodology is sufficiently flexible to test more complicated trading signals than Fama‟s (1965) runs test. Original specification of the runs test bases its methodology on the stochastic characteristics of Markov type processes. It assumes independence between return observations and it assigns probabilities of transition between states (between positive returns, negative returns and zero returns), for the respective time increments. This assumption limits the runs test specification to the analysis of single return observations only and it cannot be used for momentum signals directly due to correlations between them. Hence, the trading rules that are proposed later in this paper cannot be tested directly using Fama‟s framework. Additionally, survival analysis allows for differing benchmark model assumptions, while Fama‟s runs test assumes just normal distribution and if an empirical returns stream contains higher moment, the test is not able to capture this. Finally, another key advantage of the proposed methodology over the runs test is 8 9
For details on the log rank test see Lawless (2003) The results of comparison between runs tests and survival analysis are available from authors. 12
that the former facilitates hypothesis testing as opposed to the latter. Fama proposes a test design that allows for hypothesis testing of the total number of runs, but not the life of runs, while log rank tests we apply in the survival analysis allow for highly accurate hypothesis testing of simple signals (e.g. return runs), or more complicated trading signals (e.g. momentum). 3.5. Creating the Theoretical Benchmark Process One central aim of this paper is to evaluate whether empirical survivorship functions have unusual pattern when compared to some theoretical benchmark. Therefore, a benchmark process that comprises a fair representation of the return generating process of the various currency pairs has to be defined. As noted earlier, the PLE is non-parametric. Hence, the use of a benchmark process that does not require any assumptions about distributional characteristics of the underlying data is the most appropriate simulation setup. For this reason we base the benchmark process on re-sampling simulation without replacement, i.e. the permutation technique rather than general bootstrapping (re-sampling with replacement). Whilst Karolyi and Kho (2004) point out that the majority of finance studies employ resampling with replacement, general statistics literature gives only limited guidance as to which simulation methodology is preferable10. Given the lack of academic consensus as to which technique is preferable, our rationale for choosing permutation is twofold: 1) generally, permutation will produce consistent estimates of a distribution of a statistic even under very weak condition, as shown by Politis and Romano (1994) and 2) the test conducted in this study is a hypothesis test, as to whether the empirical survivorship curve is a fair representation of survivorship pattern under the assumption of market efficiency or not. For hypothesis test setups, permutation is regarded as the most appropriate re-sampling approach. After every permutation, a log rank test between the empirical survivorship curve and the simulated survivorship curve is calculated. It should be noted that for large datasets the log rank test statistic follows a standard normal distribution. Hence, repeatedly recalculating the test statistic will yield a distribution of the test statistic from which inferences can be made. The number of iterations is chosen to be 500.
10
See Horowitz (2001) for a survey of the literature.
13
The disadvantage of the simple permutation approach is the fact that a mere re-shuffling of returns may break the volatility structure of a time series. To correct for this, following Pasqual, Romo and Ruiz (2005), we implement the GARCH (1,1) based re-sampling and obtain results that are not qualitatively different from the simple permutation. Hence, we only report the results based on simple permutation re-sapling in the remainder of this paper11.
4. Empirical Results 4.1.
Survivorship curves and log rank test results
The trading signal can be chosen arbitrarily as long as it is applied to both the empirical series and the simulations. This paper defines any positive or negative momentum signal by applying a filter rule, which is based on the moving average calculations introduced in section two. If the short term return moving average is above (or below) the long term return moving average, then a positive (or negative) signal is obtained. Panel A of Table 3 shows the empirical survival curve for positive momentum signals from the 1/10 day short term/long term moving average combination for the USDGBP exchange rate. The results suggest that during the sample period there have been 4445 observations where the one day price was above the ten day average price. Out of these 4445 observations, 3686 observations survived one further day. Hence, the PLE from day one today two is 82.92%. The probability of survival beyond three days is 60% and it diminishes to less than 10% after 14 days. The average survival time can be calculated as the sum of the periodical survival probabilities multiplied by the respective time increment (in our case 1 day), as in Equation (16):
(16)
The average survival time of the positive moving average curve, amounts to approximately 6 days. To assess whether a survival time of 6 days can be reasonably expected for this moving average combination, we simulate a benchmark survival curve by random permutation resampling based on 500 iterations on the same time series and the same filter rule. Panel B of Table 3 displays the results.
11
Results based on GARCH (1,1) bootstrap are available on request from authors. 14
TABLE 3: Empirical and Simulated PLE curve for USDGPB_1/10_Positive Momentum PANEL A
PANEL B
Survival Function of POSITIVE Market Momentum
j 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56
Ordered failure time
intact before t
ending at contributi KM Variance time t on to KM estimator estimator
t(j)
nj
dj
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57
4445 3686 3139 2695 2316 1982 1693 1450 1234 1058 909 784 681 588 507 441 385 335 296 262 232 207 187 168 150 133 118 104 93 85 77 70 64 58 52 47 44 41 38 35 32 29 26 23 20 17 15 13 11 9 7 5 4 3 2 1
759 547 444 379 334 289 243 216 176 149 125 103 93 81 66 56 50 39 34 30 25 20 19 18 17 15 14 11 8 8 7 6 6 6 5 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 1 1 1 1 1
(nj'/nj)) S(t) VAR(S(t)) 82.92% 82.92% 0.00003 85.16% 70.62% 0.00002 85.86% 60.63% 0.00002 85.94% 52.10% 0.00002 85.58% 44.59% 0.00001 85.42% 38.09% 0.00001 85.65% 32.62% 0.00001 85.10% 27.76% 0.00001 85.74% 23.80% 0.00001 85.92% 20.45% 0.00001 86.25% 17.64% 0.00001 86.86% 15.32% 0.00000 86.34% 13.23% 0.00000 86.22% 11.41% 0.00000 86.98% 9.92% 0.00000 87.30% 8.66% 0.00000 87.01% 7.54% 0.00000 88.36% 6.66% 0.00000 88.51% 5.89% 0.00000 88.55% 5.22% 0.00000 89.22% 4.66% 0.00000 90.34% 4.21% 0.00000 89.84% 3.78% 0.00000 89.29% 3.37% 0.00000 88.67% 2.99% 0.00000 88.72% 2.65% 0.00000 88.14% 2.34% 0.00000 89.42% 2.09% 0.00000 91.40% 1.91% 0.00000 90.59% 1.73% 0.00000 90.91% 1.57% 0.00000 91.43% 1.44% 0.00000 90.63% 1.30% 0.00000 89.66% 1.17% 0.00000 90.38% 1.06% 0.00000 93.62% 0.99% 0.00000 93.18% 0.92% 0.00000 92.68% 0.85% 0.00000 92.11% 0.79% 0.00000 91.43% 0.72% 0.00000 90.63% 0.65% 0.00000 89.66% 0.58% 0.00000 88.46% 0.52% 0.00000 86.96% 0.45% 0.00000 85.00% 0.38% 0.00000 88.24% 0.34% 0.00000 86.67% 0.29% 0.00000 84.62% 0.25% 0.00000 81.82% 0.20% 0.00000 77.78% 0.16% 0.00000 71.43% 0.11% 0.00000 80.00% 0.09% 0.00000 75.00% 0.07% 0.00000 66.67% 0.05% 0.00000 50.00% 0.02% 0.00000 0.00% 0.00% 0.00000
Survival Function of POSITIVE Market Momentum (standard resampling) Ordered KM Variance Significan failure estimator ce Test time j
t(j) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49
S(t) 0.830359 0.703226 0.597767 0.507284 0.428908 0.360757 0.301488 0.251413 0.209215 0.175185 0.14655 0.122767 0.102891 0.086338 0.072235 0.060631 0.05076 0.042446 0.035778 0.030322 0.025781 0.021876 0.018632 0.015824 0.013358 0.011191 0.009295 0.007766 0.006512 0.005393 0.00448 0.003658 0.002951 0.002359 0.00188 0.001468 0.001102 0.000827 0.00062 0.000413 0.000252 0.000184 0.000138 0.000115 9.19E-05 6.89E-05 0.000046 0.000023
*** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** ** ** ** * *
VAR(S(t)) 0.004816 0.007976 0.010314 0.012021 0.012928 0.013565 0.013977 0.014434 0.014863 0.014548 0.014372 0.013531 0.012718 0.012309 0.011623 0.010646 0.010056 0.009834 0.00938 0.008775 0.008089 0.007473 0.006732 0.006125 0.005568 0.005034 0.004603 0.00421 0.00384 0.003558 0.003258 0.002905 0.0026 0.002265 0.001934 0.001639 0.001339 0.001032 0.000852 0.0007 0.000578 0.000506 0.000436 0.000363 0.000291 0.000218 0.000145 7.27E-05
T-Stat 172.4284 88.1661 57.95755 42.19906 33.17545 26.5955 21.57043 17.41832 14.07602 12.04209 10.19654 9.072796 8.090354 7.013936 6.214947 5.695002 5.04795 4.316327 3.814482 3.455387 3.187164 2.9271 2.767654 2.583668 2.39928 2.223252 2.019462 1.844579 1.695652 1.515739 1.375252 1.259453 1.135229 1.041379 0.971857 0.896062 0.822616 0.801097 0.727677 0.590549 0.43658 0.36319 0.316228 0.316228 0.316228 0.316228 0.316228 0.316228
It is evident that the simulated benchmark survival curve is shorter than the empirical curve. The average survival time of the benchmark curve (as per equation (16), obtained by regular permutation, is 5.27 days. 15
It is the centre of our analysis to assess whether the differences between empirical and theoretical survivorship curves are of statistical significance and of structural nature. This is done by comparing average survival times and analysing realised differences between empirical and simulated curve. For this purpose, we use the log rank test, introduced in Section 3. The analysis is based on 500 iterations and carried out for 90 currency crosses for ten base currencies, using Dataset I. For each of the currency pairs the moving average combination of Short Run (SR: 1, 2, 3, 4, 5, 10, 15, 20, 25) and Long Run (LR: 5, 10, 15, 20, 25, 30) are tested. This equates to 3510 moving average combinations to be tested. Given the large number of tests, we present the results of the log rank test in the form of heat maps to allow more intuitive interpretation of results. 4.1.1.
Full sample period results
Tables 4 and 5 show the log rank test results for positive and negative momentum signals. Each of the tables shows fifteen heat maps. The first ten heat maps show the outputs sorted for each of the ten base currencies tested. For instance, in Table 4, the USD heat map shows the median log rank test results from each of the ten USD currency crosses. However, it has to be noted that any positive moving average rule is associated with appreciating foreign currencies, hence a depreciating base currency and vice versa for any negative moving average rule. The remaining five heat maps in each table take all 90 currency-crosses together and show the 10th 25th, 50th, 75th and 90th percentile of each of the currency pairs. The numbers reported in Tables 4 and 5 are Z-values of a standard normal distribution. The darkest red shade represents a significance level of less than 5%, which means that the empirical observation is significantly shorter than what is suggested by the benchmark process. The colour shades then change incrementally up to the darkest blue shade, which indicates that the empirical momentum signals live longer than the benchmark ones, at a confidence level of 95% and more. The results show that, firstly, a high number of log rank tests exhibits statistically significant results, both for positive and negative momentum. This indicates that that there are structural inefficiencies within the currency market. Secondly, all of the heat maps follow similar pattern. Namely, the trading rules based on shorter moving average combinations have empirical momentum survival times that are longer than those of the benchmark simulation. Specifically, the combinations SR (1, 2, 3, 4, 5) and LR (5, 10) show the highest degree of
16
positive deviation from the benchmark simulations. Conversely, trading rules based longer term moving average combinations suggest empirical survival times that are significantly shorter than what can be expected. Thirdly, various currencies exhibit differing levels of significance. For USD currency pairs, for instance, the empirical curve does not outlive the simulated curve; it instead survives less long than what is suggested by the benchmark processes. Smaller currencies on the other hand, such as the NZD, AUD, NOK, SEK show zscores of the log rank test that are higher (specifically for shorter moving average combinations), implying that moving average signals in these currencies consistently outlive the generated benchmark process.
17
TABLE 4 Log rank test heat map output for positive moving average signals (Regular permutation) USD
GBP
LR 10 LR 15 LR 20 LR 25 LR 30
SR 1
0.50
-0.47
-1.09
-0.90
-1.14
-1.21
SR 1
0.90
0.63
-0.01
-0.20
-0.24
-0.29
SR 2
0.24
-1.00
-1.39
-1.26
-1.37
-1.39
SR 2
0.84
0.27
-0.09
-0.29
-0.23
-0.35
SR 3
0.26
-1.12
-1.49
-1.40
-1.42
-1.44
SR 3
1.18
0.29
-0.19
-0.31
-0.30
SR 4
0.69
-1.11
-1.46
-1.35
-1.46
-1.38
SR 4
1.41
0.28
-0.10
-0.30
-1.08
-1.42
-1.25
-1.43
-1.36
SR 5
0.34
-0.06
-1.34
-1.29
-1.11
-1.17
SR 10
0.13
-1.12
-1.07
-1.00
SR 15
-1.01
-0.96
SR 20
-0.89
SR 25
SR 5 SR 10 SR 15 SR 20 SR 25
LR 5
JPY
LR 5
LR 10 LR 15 LR 20 LR 25 LR 30
EUR LR 5
LR 5
LR 10 LR 15 LR 20 LR 25 LR 30
SR 1
0.02
-0.69
-1.01
-0.99
-1.08
-0.85
SR 2
-0.15
-1.07
-1.23
-1.32
-1.24
-0.98
-0.30
SR 3
-0.12
-1.23
-1.53
-1.40
-1.29
-1.13
-0.21
-0.20
SR 4
0.68
-1.38
-1.53
-1.35
-1.28
-1.19
-0.23
-0.18
-0.22
SR 5
-1.15
-1.47
-1.43
-1.31
-1.23
-0.17
-0.07
-0.18
SR 10
-1.36
-1.44
-1.27
-1.13
0.01
-0.13
-0.29
SR 15
-1.34
-1.16
-1.03
-0.08
-0.21
SR 20
-1.08
-0.99
-0.11
SR 25
-1.09
CHF LR 5
LR 10 LR 15 LR 20 LR 25 LR 30
SR 1
0.65
0.06
-0.42
-0.42
-0.53
-0.47
SR 1
0.34
-0.14
-0.60
-0.62
-0.46
-0.53
SR 1
1.97
1.17
0.48
0.24
0.07
-0.03
SR 2
0.33
-0.31
-0.68
-0.71
-0.63
-0.66
SR 2
-0.16
-0.57
-0.74
-0.99
-0.70
-0.68
SR 2
1.41
0.61
-0.27
-0.21
-0.47
-0.26
SR 3
0.40
-0.41
-0.77
-0.77
-0.63
-0.83
SR 3
0.02
-0.76
-1.02
-0.96
-0.85
-0.82
SR 3
1.42
0.27
-0.46
-0.42
-0.52
-0.42
SR 4
0.74
-0.49
-0.75
-0.74
-0.80
-0.89
SR 4
0.79
-0.80
-0.93
-1.04
-0.79
-0.82
SR 4
2.12
0.14
-0.66
-0.48
-0.51
-0.53
-0.48
-0.70
-0.68
-0.80
-0.83
SR 5
-0.81
-0.97
-0.98
-0.87
-0.87
SR 5
0.22
-0.57
-0.43
-0.47
-0.54
-0.59
-0.70
-0.71
-0.69
SR 10
-0.77
-0.86
-0.80
-0.85
SR 10
-0.23
-0.54
-0.54
-0.60
-0.51
-0.73
-0.62
SR 15
-0.77
-0.81
-0.86
SR 15
-0.34
-0.43
-0.52
-0.57
-0.68
SR 20
-0.70
-0.79
SR 20
-0.34
-0.47
-0.59
SR 25
-0.78
SR 25
SR 5
LR 10 LR 15 LR 20 LR 25 LR 30
NOK
SR 10 SR 15 SR 20 SR 25
SEK LR 5
LR 5
LR 10 LR 15 LR 20 LR 25 LR 30
-0.27
CAD
LR 10 LR 15 LR 20 LR 25 LR 30
LR 5
AUD
LR 10 LR 15 LR 20 LR 25 LR 30
LR 5
LR 10 LR 15 LR 20 LR 25 LR 30
SR 1
1.42
0.88
0.47
0.45
0.35
0.26
SR 1
1.13
0.34
-0.05
-0.29
-0.53
-0.60
SR 1
1.94
1.37
0.67
0.33
0.04
-0.07
SR 2
0.93
0.25
-0.01
0.02
0.02
-0.02
SR 2
0.58
-0.16
-0.35
-0.67
-0.74
-0.68
SR 2
1.51
0.99
0.25
-0.01
-0.21
-0.35
SR 3
1.00
0.10
-0.25
-0.13
-0.04
-0.23
SR 3
0.86
-0.25
-0.53
-0.64
-0.94
-0.83
SR 3
1.64
0.75
0.02
-0.17
-0.44
-0.38
SR 4
1.58
0.11
-0.31
-0.18
-0.11
-0.36
SR 4
1.36
-0.28
-0.61
-0.72
-1.01
-0.81
SR 4
2.25
0.71
-0.06
-0.31
-0.49
-0.51
0.05
-0.19
-0.23
-0.12
-0.43
SR 5
-0.18
-0.64
-0.67
-0.99
-0.77
SR 5
0.65
-0.09
-0.39
-0.62
-0.50
-0.24
-0.30
-0.19
-0.43
SR 10
-0.38
-0.41
-0.69
-0.62
SR 10
0.08
-0.28
-0.46
-0.36
-0.10
-0.06
-0.26
SR 15
-0.30
-0.60
-0.59
SR 15
-0.03
-0.34
-0.27
-0.11
-0.23
SR 20
-0.36
-0.51
SR 20
-0.24
-0.25
0.01
SR 25
-0.43
SR 25
SR 5 SR 10 SR 15 SR 20 SR 25
NZD LR 5
-0.20
10 PCTL. LR 5
LR 10 LR 15 LR 20 LR 25 LR 30
LR 5
LR 10 LR 15 LR 20 LR 25 LR 30
SR 1
2.34
1.84
1.17
0.89
0.81
0.51
SR 1
-0.59
-0.73
-1.31
-1.15
-1.37
-1.17
SR 1
0.22
-0.40
-0.71
-0.80
-0.87
-0.92
SR 2
1.68
1.26
0.54
0.36
0.14
0.09
SR 2
-0.54
-1.13
-1.53
-1.52
-1.46
-1.37
SR 2
-0.04
-0.68
-0.86
-1.03
-0.97
-1.03
SR 3
1.78
0.95
0.18
0.33
0.06
-0.02
SR 3
-0.34
-1.37
-1.73
-1.53
-1.48
-1.40
SR 3
0.14
-0.80
-1.11
-1.13
-1.15
-1.15
SR 4
2.62
0.81
0.10
0.16
0.03
-0.16
SR 4
0.02
-1.31
-1.61
-1.55
-1.53
-1.45
SR 4
0.83
-0.86
-1.16
-1.17
-1.18
-1.20
0.75
0.10
0.06
-0.10
-0.25
SR 5
-1.38
-1.52
-1.54
-1.64
-1.50
SR 5
-0.76
-1.14
-1.05
-1.20
-1.12
0.51
0.06
-0.10
-0.20
SR 10
-1.50
-1.47
-1.48
-1.36
SR 10
-0.93
-1.07
-1.07
-1.04
0.39
-0.01
-0.33
SR 15
-1.44
-1.38
-1.35
SR 15
-0.90
-1.05
-1.02
0.10
-0.27
SR 20
-1.28
-1.22
SR 20
-0.93
-0.96
-0.01
SR 25
-1.20
SR 25
SR 5
LR 10 LR 15 LR 20 LR 25 LR 30
25PCTL.
SR 10 SR 15 SR 20 SR 25
MEDIAN LR 5
-0.86
75 PCTL.
LR 10 LR 15 LR 20 LR 25 LR 30
LR 5
90PCTL.
LR 10 LR 15 LR 20 LR 25 LR 30
LR 5
LR 10 LR 15 LR 20 LR 25 LR 30
SR 1
1.02
0.48
-0.22
-0.17
-0.38
-0.39
SR 1
2.12
1.41
0.61
0.38
0.31
0.20
SR 1
2.78
1.85
1.29
1.00
0.95
0.87
SR 2
0.65
0.00
-0.38
-0.57
-0.63
-0.58
SR 2
1.47
0.55
0.15
-0.04
-0.13
-0.09
SR 2
1.96
1.41
0.66
0.57
0.47
0.56
SR 3
0.78
-0.25
-0.63
-0.59
-0.69
-0.75
SR 3
1.53
0.56
-0.02
-0.22
-0.19
-0.05
SR 3
2.23
1.12
0.43
0.42
0.32
0.36
SR 4
1.39
-0.22
-0.57
-0.65
-0.66
-0.71
SR 4
2.04
0.42
-0.16
-0.21
-0.10
-0.25
SR 4
2.86
1.02
0.44
0.30
0.24
0.19
-0.23
-0.52
-0.68
-0.72
-0.72
SR 5
0.53
-0.07
-0.12
-0.18
-0.30
SR 5
1.14
0.40
0.31
0.31
0.11
-0.47
-0.62
-0.65
-0.66
SR 10
0.11
-0.09
-0.20
-0.24
SR 10
0.74
0.31
0.23
0.12
-0.43
-0.54
-0.62
SR 15
0.05
-0.08
-0.24
SR 15
0.59
0.41
0.17
-0.51
-0.51
SR 20
-0.02
-0.13
SR 20
0.48
0.19
-0.49
SR 25
-0.02
SR 25
SR 5 SR 10 SR 15 SR 20 SR 25
Significance levels
1.00
0.95
0.90
0.80
0.70
0.50
0.30
0.20
0.10
0.05
0.39
0.00
The darkest red shade represents a significance level of less than 5%, the colour shades then change incrementally, as indicated in the legend of each of the tables up to the darkest blue shade, which indicates a confidence level of 95% and more.
18
TABLE 5 Log rank test heat map output for negative moving average signals (Regular permutation) USD
GBP
LR 5
LR 10 LR 15 LR 20 LR 25 LR 30
SR 1
0.83
-0.08
-0.77
-0.82
-0.72
-0.77
SR 2
0.50
-0.47
-1.01
-0.93
-0.96
-0.86
SR 3
0.58
-0.70
-1.23
-0.90
-0.96
SR 4
1.07
-0.73
-1.18
-0.95
-0.69
-1.15 -0.91
SR 5 SR 10 SR 15
LR 10 LR 15 LR 20 LR 25 LR 30
SR 1
0.01
-0.13
-0.49
-0.52
-0.25
-0.44
SR 1
0.89
0.43
-0.19
-0.21
-0.46
-0.40
SR 2
-0.10
-0.41
-0.75
-0.75
-0.45
-0.63
SR 2
0.81
0.20
-0.38
-0.39
-0.50
-0.38
-1.03
SR 3
0.22
-0.47
-0.82
-0.66
-0.56
-0.62
SR 3
0.94
0.14
-0.45
-0.36
-0.53
-0.43
-1.00
-1.03
SR 4
0.68
-0.34
-0.75
-0.59
-0.46
-0.65
SR 4
1.47
0.11
-0.40
-0.31
-0.45
-0.34
-0.95
-0.99
-0.92
SR 5
-0.38
-0.67
-0.50
-0.54
-0.65
SR 5
0.33
-0.25
-0.36
-0.40
-0.38
-0.74
-0.72
-0.74
SR 10
-0.36
-0.49
-0.49
-0.61
SR 10
-0.26
-0.35
-0.24
-0.20
-0.81
-0.73
-0.63
SR 15
-0.31
-0.44
-0.63
SR 15
-0.27
-0.28
-0.20
-0.88
-0.67
SR 20
-0.39
-0.42
SR 20
-0.32
-0.34
-0.83
SR 25
-0.40
SR 25
SR 20 SR 25 EUR LR 5
JPY
LR 5
LR 5
LR 10 LR 15 LR 20 LR 25 LR 30
-0.29
CHF
LR 10 LR 15 LR 20 LR 25 LR 30
LR 5
NOK
LR 10 LR 15 LR 20 LR 25 LR 30
LR 5
LR 10 LR 15 LR 20 LR 25 LR 30
SR 1
0.88
0.31
-0.03
-0.25
-0.38
-0.30
SR 1
1.19
0.95
0.24
0.08
-0.03
-0.10
SR 1
1.55
0.78
0.07
0.16
-0.11
-0.19
SR 2
0.51
0.06
-0.33
-0.67
-0.46
-0.43
SR 2
0.89
0.49
-0.10
-0.16
-0.10
-0.13
SR 2
1.02
0.13
-0.38
-0.32
-0.46
-0.41
SR 3
0.78
-0.05
-0.46
-0.60
-0.49
-0.55
SR 3
0.97
0.43
-0.21
-0.22
-0.23
-0.17
SR 3
1.14
-0.31
-0.68
-0.54
-0.55
-0.57
SR 4
1.22
-0.15
-0.50
-0.68
-0.61
-0.58
SR 4
1.43
0.46
-0.12
-0.12
-0.19
-0.16
SR 4
1.94
-0.42
-0.74
-0.64
-0.65
-0.67
-0.03
-0.55
-0.54
-0.58
-0.59
SR 5
0.46
-0.18
-0.16
-0.25
-0.14
SR 5
-0.40
-0.82
-0.57
-0.67
-0.69
-0.37
-0.45
-0.40
-0.53
SR 10
-0.16
-0.16
-0.15
-0.12
SR 10
-0.46
-0.72
-0.66
-0.65
-0.32
-0.47
-0.48
SR 15
-0.06
-0.03
-0.09
SR 15
-0.50
-0.59
-0.67
-0.51
-0.39
SR 20
0.02
-0.03
SR 20
-0.46
-0.61
-0.34
SR 25
-0.05
SR 25
SR 5 SR 10 SR 15 SR 20 SR 25
SEK LR 5
-0.43
CAD LR 5
LR 10 LR 15 LR 20 LR 25 LR 30
SR 1
1.57
0.89
0.55
0.23
0.04
-0.19
SR 1
0.86
-0.10
-0.53
-0.44
-0.90
-0.86
SR 1
1.62
0.69
0.04
-0.33
-0.40
-0.42
SR 2
1.05
0.30
0.11
-0.27
-0.46
-0.47
SR 2
0.36
-0.49
-0.77
-0.89
-0.96
-0.97
SR 2
0.92
0.00
-0.61
-0.80
-0.94
-0.92
SR 3
1.13
0.20
-0.09
-0.32
-0.51
-0.68
SR 3
0.54
-0.74
-0.90
-0.95
-1.07
-1.00
SR 3
0.96
-0.23
-0.85
-1.09
-1.09
-1.05
SR 4
1.72
0.05
-0.19
-0.47
-0.64
-0.78
SR 4
1.01
-0.71
-0.90
-1.05
-1.05
-1.10
SR 4
1.76
-0.38
-0.99
-1.04
-1.19
-1.17
0.22
-0.18
-0.58
-0.73
-0.82
SR 5
-0.78
-0.98
-1.10
-1.13
-1.09
SR 5
-0.42
-1.07
-1.05
-1.20
-1.21
-0.01
-0.52
-0.75
-0.68
SR 10
-0.95
-1.10
-1.15
-1.17
SR 10
-0.81
-1.04
-1.09
-1.12
-0.10
-0.53
-0.56
SR 15
-0.93
-1.15
-1.12
SR 15
-0.70
-0.94
-1.06
-0.25
-0.53
SR 20
-0.97
-1.01
SR 20
-0.81
-0.93
-0.35
SR 25
-0.88
SR 25
SR 5
LR 10 LR 15 LR 20 LR 25 LR 30
AUD
SR 10 SR 15 SR 20 SR 25
NZD LR 5
LR 5
LR 10 LR 15 LR 20 LR 25 LR 30
-0.81
10 PCTL. LR 5
LR 10 LR 15 LR 20 LR 25 LR 30
LR 5
LR 10 LR 15 LR 20 LR 25 LR 30
SR 1
1.74
1.15
0.65
0.57
0.49
0.26
SR 1
-0.49
-0.76
-1.31
-1.13
-1.40
-1.23
SR 1
0.30
-0.41
-0.67
-0.86
-0.91
-0.90
SR 2
1.23
0.54
0.07
0.05
-0.12
-0.07
SR 2
-0.61
-1.10
-1.55
-1.51
-1.47
-1.35
SR 2
-0.03
-0.71
-0.98
-1.01
-1.05
-1.02
SR 3
1.24
0.24
-0.18
-0.31
-0.33
-0.28
SR 3
-0.30
-1.32
-1.66
-1.57
-1.45
-1.39
SR 3
0.13
-0.87
-1.09
-1.14
-1.14
-1.15
SR 4
2.02
0.09
-0.39
-0.33
-0.43
-0.35
SR 4
0.06
-1.35
-1.59
-1.56
-1.57
-1.50
SR 4
0.76
-0.92
-1.13
-1.13
-1.20
-1.14
0.05
-0.29
-0.36
-0.45
-0.48
SR 5
-1.32
-1.66
-1.54
-1.63
-1.46
SR 5
-0.79
-1.13
-1.03
-1.20
-1.17
-0.16
-0.36
-0.19
-0.47
SR 10
-1.52
-1.45
-1.40
-1.31
SR 10
-0.92
-1.04
-1.03
-1.04
-0.11
-0.16
-0.42
SR 15
-1.52
-1.38
-1.31
SR 15
-0.87
-1.04
-1.00
0.05
-0.33
SR 20
-1.24
-1.24
SR 20
-0.98
-0.91
-0.05
SR 25
-1.23
SR 25
SR 5
LR 10 LR 15 LR 20 LR 25 LR 30
25PCTL.
SR 10 SR 15 SR 20 SR 25
MEDIAN LR 5
-0.87
75 PCTL.
LR 10 LR 15 LR 20 LR 25 LR 30
LR 5
90PCTL.
LR 10 LR 15 LR 20 LR 25 LR 30
LR 5
LR 10 LR 15 LR 20 LR 25 LR 30
SR 1
1.01
0.48
-0.20
-0.20
-0.33
-0.42
SR 1
2.03
1.31
0.56
0.41
0.33
0.15
SR 1
2.80
1.92
1.25
0.98
0.95
0.88
SR 2
0.65
-0.01
-0.42
-0.61
-0.56
-0.61
SR 2
1.50
0.60
0.15
-0.06
-0.10
-0.03
SR 2
2.00
1.49
0.69
0.56
0.49
0.55
SR 3
0.83
-0.19
-0.62
-0.61
-0.68
-0.68
SR 3
1.56
0.53
-0.04
-0.20
-0.17
-0.12
SR 3
2.13
1.07
0.42
0.37
0.36
0.37
SR 4
1.34
-0.21
-0.57
-0.63
-0.66
-0.72
SR 4
2.14
0.47
-0.21
-0.18
-0.16
-0.25
SR 4
2.93
1.04
0.33
0.36
0.19
0.25
-0.20
-0.62
-0.62
-0.76
-0.69
SR 5
0.50
-0.07
-0.11
-0.17
-0.23
SR 5
1.16
0.34
0.26
0.29
0.17
-0.49
-0.62
-0.66
-0.64
SR 10
0.06
-0.16
-0.17
-0.25
SR 10
0.63
0.31
0.28
0.15
-0.45
-0.52
-0.62
SR 15
0.14
-0.09
-0.23
SR 15
0.60
0.40
0.19
-0.54
-0.54
SR 20
-0.08
-0.12
SR 20
0.45
0.16
-0.46
SR 25
0.02
SR 25
SR 5 SR 10 SR 15 SR 20 SR 25
Significance levels
1.00
0.95
0.90
0.80
0.70
0.50
0.30
0.20
0.10
0.05
0.38
0.00
The darkest red shade represents a significance level of less than 5%, the colour shades then change incrementally, as indicated in the legend of each of the tables up to the darkest blue shade, which indicates a confidence level of 95% and more.
19
Finally, currencies such as the JPY or the CHF show opposing results of the log rank test for positive and negative moving average signals. This might be explained by the significant appreciation of both currencies over the observation time period. 4.1.2.
Sub-sample periods results
As it was pointed out, the moving average combinations SR (1, 2, 3, 4, 5) and LR (5, 10) show the highest positive deviation from the benchmark simulation. The question now arises as to whether this positive deviation is persistent over time across sub-samples. Figures 1 and 2 aim to answer this question for positive and negative momentum respectively, for all nine sub-sample periods (SS1-SS9). Figure 1 shows the median log rank test result across all moving average pairs that have been generated from positive momentum signals (dark bars). It also shows the difference between the median log rank test result of all moving average pairs and the median log rank test result of the short term moving average pairs (light bars).
Figure 1: Positive Momentum Signals Median Log rank test result from all moving average pairs vs. Difference between median of all moving average pairs and median of short term moving average pairs Median for all Currency Pairs 1
0.5
0
-0.5
-1 SS1
SS2
SS3
Median across all moving averages
SS4
SS5
SS6
SS7
SS8
SS9
difference in median between long term and short term moving averages
Figure 1 illustrates that the median log rank test result across all moving average pairs has increased from -0.5 in sub sample one to 0.2 in sub sample nine. This implies that over the 25 years observation period the empirical survival time has increased. In the first sub sample the empirical survival time is shorter than what is suggested by the benchmark simulation. In the 20
last sub sample it has become marginally longer than what the benchmark simulation indicates. In addition, Figure 1 shows that the difference between short term and long term moving average results has been decreasing over time. Similar conclusions can be drawn from Figure 2 which presents the same set of results for negative moving average combinations. Overall, we find that the log rank test results become less strong in the more recent time period, i.e. deviation from normality diminishes over time for both positive and negative momentum. An exception is notable in the last sub sample, suggesting a pickup in non-normality. Figure 2: Negative Momentum Signals Median Log rank test result from all moving average pairs vs. Difference between median of all moving average pairs and median of short term moving average pairs Median for all Currency Pairs 1
0.5
0
-0.5
-1 SS1
SS2
SS3
Median across all moving averages
SS4
SS5
SS6
SS7
SS8
SS9
difference in median between long term and short term moving averages
To assess whether the results outlined in this section are merely academic, or whether the survivorship analysis provides information that allows for the construction of profitable trading strategies that outperform generic ones, we devise an easily applicable trading rule, as per following section. 4.2.
Trading Rule Implementation
From the survivorship analysis it has become evident that short term moving average combinations exhibit survival rates that outlive the theoretical benchmark model, whilst moving average combinations that use longer moving averages indicate a shorter life expectancy than what is suggested by the theoretical benchmark model. Hence, going 21
forward, we compare the performance of a „benchmark‟ trading rule that uses all moving average signals against the „enhanced‟ trading rule based only on a subset of moving average signals suggested by survival analysis. 4.2.1. ‘Benchmark’ vs. ‘Enhanced’ Trading Rule To create a real-life „benchmark‟ trading rule we combine all individual moving average trading signals and create a composite moving average signal for each currency pair. Generating trades based on single trading signals might lead to frequent trading and a high degree of transaction cost which is likely to make any trading strategy unprofitable. In addition to that, a scenario may occur in which, for instance, longer term moving average combinations might point towards a long position, while shorter term moving averages might indicate an increasing short bias. Therefore, we view a composite trading rule as the appropriate benchmark for the enhanced trading rule strategy. The „benchmark‟ moving average trading rule is constructed by taking all 39 moving average combinations (as per Table 1), with each giving either a positive (buy) signal, that we denote as +1 or a negative (sell) signal, marked as -1 in our analysis. To generalise, by summing up the number of positive signals and then deducting the number of negative signals from it we obtain the composite trading signal. A number of +39 would indicate maximum positive momentum, because all moving average rules give positive trading signals, whilst a -39 would signal maximum negative momentum. By dividing the composite signal by the total number of moving averages tested, the „benchmark‟ trading rule is then standardised in the range between +1 and -1. To exemplify, for maximum positive momentum its value is: +39/39 = +1.
To assess whether results from survivorship analysis can be used to improve performance of a trading strategy, we construct the „enhanced‟ trading signal. The „enhanced‟ rule will compile aggregate trading signals for each of the currency pair based on the following moving average combinations: SR (1, 2, 3, 4, 5) and LR (5, 10, 15).The total of 14 moving average combinations is therefore used to obtain the „enhanced‟ signal following identical method as in generation of the „benchmark‟ signal. Both the „benchmark‟ and „enhanced‟ trading rules result in a series of buy and sell signals which enable us to construct long/short strategies for each rule. The performance of each rule 22
is evaluated using breakeven transaction costs explained in the following section. Both Dataset I and Dataset II that accounts for trading costs are used in the trading rules analysis.
4.2.2. Breakeven Transaction Costs Due to different time zones between countries, the currency market is effectively a 24 hour market where it is possible to initiate a trade at any point during the day. However, at some hours of the day trading volumes could be rather thin. Hence, once the trading rule indicates a buy (sell) signal, one can only gain exposure (exit the position) gradually in our strategy. This comes at an opportunity cost known as implementation shortfall, representing a form of „implicit‟ transaction cost. To account for the implementation shortfall we assume that it takes one day12 to get fully exposed to (exit) the currency position. Specifically, we deduct the following 24 hour return from any buy signal (add the following 24 hour return to any sell signal). Only after 24 hours the performance of the trading rule is evaluated. In addition to this, the „explicit‟ transaction costs such as the bid-ask spread are also taken into account by applying trading strategies to Dataset II and buying currencies at the ask price and selling them at the bid. To evaluate performance of long/short strategies, we resort to breakeven transaction costs analysis, which calculates the level of transaction cost per trade incurred when the trading rule yields a risk adjusted return13 of zero. This ensures that breakeven costs are comparable across moving average combinations and base currencies. Hence, if the breakeven transaction cost level is higher than the actual trading cost, then the strategy is profitable. Breakeven costs analysis for the composite trading signal incorporates the element of turnover. Hence, if the composite trading signal indicates to turn over the position by 60% then the transaction costs are only applied to 60% of the portfolio. This ensures the comparability of different composite trading rules and gives a representative indication of returns obtainable from a real life trading strategy.
12
24 hour lag can be regarded as a reasonably conservative symmetric measure for the implementation shortfall, incurring profits or losses, depending on the currency movement 13 Measured by Coefficient of Variation, which represents a ratio of standard deviation over expected return 23
4.2.3. ‘Enhanced’ Trading Rule Performance 4.2.3.1.
Accounting for implicit transaction costs
Figure 3, based on Dataset I, suggests that over the 25 years period, the „enhanced‟ trading strategy delivers considerably higher breakeven transaction cost levels than the „benchmark‟ trading strategy across currencies. Figure 3 Breakeven transaction cost levels for ‘enhanced’ and ‘benchmark’ long/short moving average trading rules (across base currencies) in Bps. Full Sample 20 16 12 8 4 0 MEDIAN
USD
GBP
JPY
EUR
Benchmark Long Short trading strategy Difference between E and BM strategy
CHF
NOK
SEK
CAD
AUD
NZD
Enhanced Long Short trading strategy
Figure 3 illustrates that the median breakeven transaction cost level for the „benchmark‟ strategy is 7.8 basis points (bp), whilst that for the „enhanced‟ strategy is 14.5 bp, amounting to a performance difference of 6.7 bp per trade. Figure 4 shows the evolution of the median breakeven transaction cost levels over time. It can be observed that as the time passes, the benchmark strategy sees an erosion of profitability (exception are the last two sub-samples), while the enhanced strategy remains profitable. The breakeven transaction costs of the benchmark strategy range from 1.5bp to 15.4bp, whilst those of the enhanced trading strategy span from 11.2bp to 18.3bp, corroborating superior performance of enhanced strategy over time.
24
Figure 4 Median breakeven transaction cost levels for Enhanced and Benchmark long short moving average trading rules (across sub samples) in Bps.
Median 20 16 12
8 4
0 SS1
SS2
SS3
SS4
SS5
Benchmark Long Short trading strategy Difference between E and BM strategy
4.2.3.2.
SS6
SS7
SS8
SS9
Enhanced Long Short trading strategy
Accounting for explicit transaction costs
The analysis so far does not consider the impact of bid/ask spreads. Dataset II, which appropriately reflects the dynamics of the bid/ask spread will facilitate incorporation of that explicit element of transaction cost. Figure 5 shows the median transaction cost breakeven levels for the benchmark and the enhanced trading rule. The analysis has been carried out for sub-sample 8 and sub-sample 9, given the restricted time span of Dataset II. It is evident that applying the benchmark composite trading rule either destroys value (sub-sample 8) or fails to add value (sub-sample 9). Enhanced trading strategy, on the other hand, will create value even in periods where a generic trading rule fails to perform. Another aspect that comes out of this analysis is the magnitude of difference between breakeven transaction cost before and after the incorporation of bid-ask spreads.
25
Figure 5 Dataset 2: Breakeven transaction cost levels for Enhanced and Benchmark long short moving average trading rules including Bid/Ask spread(across sub samples) Median
15
10
5
0
-5
-10
-15 SS1
SS2
SS3
SS4
Benchmark Long Short trading strategy
SS5
SS6
SS7
SS8
SS9
Enhanced Long Short trading strategy
In the case of the „enhanced‟ trading strategy, adding the bid-ask spreads leads to a small decrease of breakeven transaction cost level by 0.3bps in sub-sample 8 and 4bps in subsample 9. The differences in the „benchmark‟ trading rule are much more prominent (18.5 bps for SS8 and 7.3bps for SS9). The most likely explanation for this are the dynamics of the bid-ask spreads in periods of stress in financial markets (SS8 and SS9) and the element of continuous compounding. Specifically, whenever there is stress in markets, then bid/ask spreads tend to widen, hence, any trade that is undertaken will be more expensive than under normal circumstances. If a trading strategy is in loss during stressed market environments these losses will be aggregated by wide bid/ask spreads. Furthermore it is not difficult to see that the benchmark trading strategy is more likely to be negatively exposed to market shock, whilst the enhanced strategy most likely benefits form market shock, given its short term nature. Hence, by the effect of continuous compounding the results of both trading strategies might well drift apart, as in SS8 and SS9. 4.3.
Linkage between survival analysis and the trading rule results
Comparison of the results from the full sample log rank tests and the trading rule results reveals that survivorship analysis captures the differences in trading rule profitability. The survival analysis shows that shorter term moving averages outlive the benchmark
26
simulations, whilst longer term moving average combinations tend to have a shorter life expectancy than what is suggested by the benchmark model. This is also true when it comes capturing the profitability of all trading rules over time. Figure 6 proves this. The left hand side axis denotes the median breakeven transaction cost levels for benchmark trading strategy for all currency pairs and the right hand side axis is the absolute z-score of the average median log rank test across positive and negative momentum signals (RHS).The reason for taking the absolute value of the average across median log rank test values for positive and negative momentum signals is the fact that the trading strategy is a long/short strategy. Hence it should capture deviation from market normality either way regardless of whether the sign is positive or negative. Figure 6 LHS: Breakeven transaction cost levels for benchmark trading rules in bp and RHS: absolute z-score of median log rank test results (positive and negative combined) across sub samples
Median 20
0.6
0.5
16
0.4 12 0.3 8 0.2 4
0.1
0
0 SS1
SS2
SS3
SS4
SS5
SS6
SS7
SS8
SS9
Benchmark Trading rule transaction cost breakeven points in BPS Absolute level of median log rank test results (pos. neg. combined)
Figure 6 suggests that over 9 sub-samples, the findings from survivorship analysis are reflected in those of the benchmark trading strategy. Deviation from normality implied by survival analysis as well as breakeven transaction cost levels diminish over time, with a small pickup in the most recent period. The correlation between the two sets of findings in Figure 6 is 0.786.
27
On the contrary, comparing the breakeven transaction cost levels of the enhanced trading strategy with the log rank test results from short term moving averages; such a positive relationship cannot be established. Figure 7 corroborates this. The correlation between the two sets of results is -0.106. This result suggests that although the survivorship analysis does have the power to explain the change in overall trading rule profitability over time, it fails to capture the dynamics of the shorter term moving average rules. This allows for two further observations. Firstly, some aspects of the trading rule profitability are driven by deviations from market normality. The survivorship analysis is a statistical tool that allows researchers to model life expectancy. In the context of this paper the survivorship model aims to assess whether the life expectancies of various momentum trading signals are in line with what can be expected assuming market efficiency. The overall outcome of this analysis suggests that shorter term moving average trading signals are prone to exhibit higher deviation from market normality than signals created from other moving average combinations. The sub sample analysis suggests that these deviations diminish over time. As these deviations from market normality diminish, overall trading rule profitability diminishes as well. This suggests that the diminishing part of the trading rule profits can be attributed to diminishing market inefficiency.
28
Figure 7 LHS: Breakeven transaction cost levels for enhanced trading rules in bp and RHS: Absolute z-score of median log rank test results (positive and negative combined) for short moving average combination across sub samples
Median
20
0.6 0.5
16
0.4 12 0.3 8 0.2 4
0.1
0
0 SS1
SS2
SS3
SS4
SS5
SS6
SS7
SS8
SS9
Enhanced Trading rule transaction cost breakeven points in BPS Absolute level of median log rank test results (pos. neg. combined) for short moving average combinations
Secondly, there is a set of trading rules that maintains its level of profitability despite the fact that the survivorship analysis points towards diminishing deviation from market efficiency. This implies some of the trading rules have a return driver other than market inefficiency. Given the fact that the strategy has been implemented on a long/short basis makes an argument of systematic risk taking rather difficult.
5. Conclusions This paper introduces an alternative methodology of detecting currency market inefficiency, based on life time statistics and, in particular, survivorship analysis. The intuition behind the selected methodology finds its roots in the concept of runs test. However, the runs test only allows for a benchmark specification that follows a Bernoulli-type process, hence mere independence between returns. The survivorship analysis on the other hand has the flexibility of using different benchmark processes. Furthermore, a runs test can only ever be applied to a return stream. When it comes to assessing signals that are generated from complicated trading rules, the runs test specification breaks down, unlike survivorship analysis. Finally, when assessing market inefficiency by the specification of runs test the aspect of hypothesis 29
testing becomes particularly problematic. This is not the case when applying the survivorship analysis. The log rank test, earlier introduced in this paper represents a reliable tool, to assess the statistical significance of results. Further to the methodological advances, this paper provides evidence of inefficiencies in the currency market. We analyse 90 G10 currency pairs over 9025 trading days from 04/01/1974 to the 31/12/2009. The survivorship analysis for the 25 year period suggests that 1) empirical momentum signals arising from crossovers of very short term moving average combinations outlive theoretical benchmark signals and 2) empirical momentum signals created from crossovers of longer term moving averages have lower lifetime expectancy than the theory would suggest. However, looking at 9 sub-samples we find that most of those deviations from market efficiency have been deteriorating over time, to the point where all of the momentum signals exhibit survival times that are statistically equivalent to what is suggested by benchmark processes. Implementation of trading rules on the same set of moving average crossover signals as tested in the survivorship analysis reinforce the validity of the survivorship methodology as a tool to detect market inefficiencies. The profitability of a benchmark trading rule that incorporates all moving average signals deteriorates over time (as is suggested by the survivorship analysis) to a point where the trading rule becomes unprofitable. Additionally, an enhanced trading strategy based on a sub-set of moving average signals persistently outperforms the benchmark trading strategy over time. This result counters the findings of survivorship analysis and implies that the source of these returns is something other than market inefficiency. Furthermore, given the fact that the trading strategy is implemented on a long/short basis makes the argument of systematic risk taking unlikely. Assessing the source of these returns is the subject of further analysis. Finally, it was not our aim to identify the most profitable trading rule; however, further advances of designing optimal trading rules based on survival analysis will be addressed in future research.
30
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