then this is the basis for a successful technical model. We can explore a similar processto that undertaken by the fundamental modeller above. Let’s set the scene.10.3.2.1 HypothesisLet us suppose that the hypothesis is that exchange rates exhibit trends. This sounds like arather simple statement, but establishing exactly what it means, and testing it, is far fromsimple. For example, the signals can be based on any price – spot is ﬁne if it works, but theassessment of whether signals can make money must be based on currency surprise. Trends inone direction do not last for ever, and so there will be decisions to make about the horizon ofany trend signal, and turning-point processes.10.3.2.2 InputsWe need a daily database with spot and forward rates. We may need higher frequency dataif intra-day strategies are to be modelled, and even higher frequency if any signal is to be‘instantaneously’ actioned. We also need to pick a currency pair to start with. We will useagain USD/GBP as an example, and data 1973–2002.10.3.2.3 CorrelationWe can run correlations between the currency returns and prior period currency returns, or inalgebraic form:Correlation (yt , yt−1 ), t = 1972–2002,
variable frequency
We can vary the horizons of the returns (i.e. one-day correlated with prior one-day, or ﬁve-daycorrelated with prior ﬁve-day) to see whether any autoregressive effect changes with horizon.Positive correlations imply trends (positive serial correlation), negative imply mean reversion(negative serial correlation). We are not constrained to the same horizons for the past returnsand the future returns, i.e. we could test the correlation between one-day returns, and the priorthree-day returns. We will explore this route of enquiry in more detail in Section 10.3.2.4. Forthis exercise, we can show the results from varying the horizon of the returns similarly forboth series. If we use about 30 years of daily data, we have about 7550 points in the series. Asthe horizons extend beyond one day, the number of independent observations falls – two-dayreturns have 3775 independent observations, etc.
220
Currency Overlay
Correlation coefﬁcients can be tested for statistical signiﬁcance. The formula for this is:√n−2t =r √1 − r2where t is a t-stat measure, r is the observed correlation coefﬁcient and n is the number of pairsof observations. If t > 1.96, then the observed correlation is positive and signiﬁcant at the 95%level (two-sided test). Similarly, if t < 1.96, then the correlation is signiﬁcantly negative. For7500 observations, the signiﬁcance level is 0.022 (or −0.022) (i.e. apparently very low). Asthe horizon period rises, the level required for signiﬁcance will also rise. We show the resultsfor the 1973–2002 period in Figure 10.1.0.160.14
95% Confidence limit
Correlation coefficient
0.120.100.080.060.040.020.000
5
10
15
20
25
30
35
40
Data frequency (days)
Figure 10.1 Correlation coefﬁcient curve (USD/GBP, 1973–2002, lagged dependent variable, variabledata frequency)
Only daily data appears to offer any real signiﬁcance over this long data period, despitethe marginal signiﬁcance of 25-day data (or roughly monthly). All the horizons show positivecorrelation, indicating that this relationship has trended over history. However, if we werebuilding a model to use in live trading, we would be building it over the ‘testing period’,and running it over a ‘live’ period, outside the testing period. This requires the relationshipwe establish in the testing period to persist into the live period. This ‘stability’ is core to themodelling process, and is discussed in Chapter 10 in more detail.For this example, we can look at the ‘split test’ of correlation: this is dividing the datainto three roughly equal-length periods – 73–82, 83–92, 93–02. Each of these can be testedindependently, as the graph in Figure 10.2 shows.Regrettably, this new information thoroughly undermines our previous hypothesis. It isnot so much that the USD/GBP does not trend, because it clearly did in the 20 years 1973–1992, but that it changed its behaviour in the last 10 years, and became a mean-reverting
Active Currency Overlay – Management Styles
221
0.373–82
Correlation coefficient
0.2
83–92
0.1
0.0
−0.1
93–02
−0.295% Confidence limits
− 0.30
5
10
15
20
25
30
35
40
Data frequency (days)
Figure 10.2 Correlation coefﬁcient curves (USD/GBP, 1973–2002, lagged dependent variable, splittests, variable frequency)
currency with negative autoregressive behaviour (although at longer horizons than the trendingbehaviour).10.3.2.4 RegressionWe can express a basic trends hypothesis in the following format:yt = f (yt−1 , yt−2 , yt−3 , . . . , yt−n , e)where yt = ct+n − ct , where ct = interest-rate-adjusted log currency index, and e is the error term (i.e. the unexplained variations in y). Expressed in words, this is a rather generalmodel which says that returns in period t are (partially) explained by returns in periods t − 1,t − 2, t − 3, . . . , t − n. This type of model is called a lagged dependent variable model. Thedependent variable is yt , called such because in the model context it is dependent on the explanatory variables. If we can get a dependent variable for a future period to be substantiallyexplained by variables whose values we already know (lagged), then we have the makings ofa forecasting model.We could also re-specify the model slightly into:yt = f (xt−1 , xt−2 , xt−3 , . . . , xt−n , e)where xt = pt+n − pt , and pt = log spot price. This might reﬂect more closely what drivesthe price – and despite appearances, this is still a lagged dependent variable model.It is usual to model y, the log currency surprise (sometimes called currency returns), ratherthan c, the log interest-rate-adjusted price. Returns are the rate of change of prices, and incontinuous maths would be called the ﬁrst differential. Modelling returns (rather than price)
222
Currency Overlay
is more useful because the random-walk-like nature of prices means that today’s price isalways 95%+ determined by yesterday’s price, so price models give overwhelming (but ratherindiscriminate) credence to lagged dependent variables. By contrast, even if there is serialcorrelation in currency returns, it is weak. So currency returns today are determined only to asmall extent by yesterday’s returns: this is the value we want to get as high as possible.I have chosen to model log currency surprise (yt ) with ﬁve lagged dependent variables, eachone period earlier than the other. I have used linear regression, the statistical details of whichare available in any good econometric modelling textbook.10 The regression model is:yt = a + b1 yt−1 + b2 yt−2 + b3 yt−3 + b4 yt−4 b5 yt−5 + ewhere b1 , b2 , b3 , b4 and b5 are constants (parameters) and e is the error term (i.e. picking upall the unexplained variations.Again, like the correlation model, we can vary the frequency of the data – I have tested1–30-day frequency. Just to be clear, a 30-day frequency model would have the dependentvariable as the 30-day currency return (i.e. log currency surprise), let us say starting today andending 30 days in the future. The ﬁrst lagged dependent variable would be the 30-day currencyreturn ending today, starting 30 days prior; the second lagged dependent variable would be the30-day currency return ending 30 days prior, and starting 60 days prior, and so on.To complete the testing, I examined the regression results using only one lagged dependentvariable – the lagged-one-period dependent variable. I then ranked the explanatory power ofall the alternative regressions by an F-ratio – essentially a statistical test of how conﬁdent wecan be that the amount of variation explained by the model is not zero. A high ratio meanswe can be very conﬁdent; conﬁdence levels are given by the F-distribution. The model withthe highest F-ratio, giving us the highest conﬁdence that the model has genuine explanatorypower, was the simplest – the single one-day lagged dependent variable on daily data.Summary results for this example regression model are shown in Table 10.2. To the uninitiated, it may be surprising that a very simple model gives us most conﬁdence. It does not giveus the best R 2 – a 5-lagged-dependent-variable 18-day frequency regression explains over ﬁvetimes as much of the variation. But as we add explanatory variables, we add signiﬁcantly tothe chance that their explanations are spurious.Table 10.2 Regression results (USD/GBP, 1973–2002, autoregressive model)Criteria for ‘best regression’F-ratioF-test conﬁdenceBest frequencyb1 coefﬁcient valuet-stat for b1 coefﬁcientBest model speciﬁcation
F-test21.5199.99964%1 day+0.053(+ = trends; − = mean reverting)4.4Independent variables: single one-period laggedZero intercept
Finally, we need to look at a split test. Having seen the results of the correlation split test,we can already guess the result: 1993–2002 will have a negative b1 coefﬁcient.10
For example Johnston and Dinardo, Econometric Methods, 4th edn, McGraw-Hill, 1996, ISBN 0-07-913121-2.
Active Currency Overlay – Management Styles
223
The results to prove it are shown in Table 10.3. We see that b1 is negative, but not signiﬁcantlyso over this period, and the F-ratio is not high enough for us to be at all conﬁdent that themodel explains anything at all.Table 10.3 Regression results – split test (USD/GBP, 1993–2002,autoregressive model)Criteria for ‘best regression’F-ratioF-test conﬁdenceb1 coefﬁcient valuet-stat for b1 coefﬁcient
F-test1.7081%−0.028 (+ = trends; − = mean reverting)−1.42
Regression models are explanatory, not trading processes. To convert signiﬁcant regressionresults into a trading model, the modeller is required to develop a process. This might bea ‘momentum’ model, in which the parameters that are identiﬁed in a regression as havingsigniﬁcant explanatory power are then pressed into service to take an active currency position(proportionately ‘long’ for predicted price rises and proportionately ‘short’ for predicted pricefalls). I show results of a simple version of such a model below.10.3.2.5 Trading modelsThe rules of trading models can be inﬁnitely varied. Their purpose is to codify actions that acurrency manager needs to take at time t to ‘maximise’ the chance that the result, once theposition is closed, is a proﬁt.I have constructed and tested two models. The ﬁrst is a moving average model, the secondis a momentum model.Moving average model: A moving average model is one where a simple trading rule isestablished: buy the currency that is rising at the moment a shorter trailing moving average ofthe spot rate crosses a longer trailing moving average. Such a model will exploit positive serialcorrelation if it exists; the reverse rule would be applied if it is believed that markets are meanreverting. Note that it is perfectly possible for markets to be both trending and mean-revertingat the same time. The effects can apply at different horizons – say trending up to 3 months, butmean-reverting at 2 years. I will be testing shorter effects in this modelling. Moving averagesare expressed in days, and in modelling this kind of process, one can visualise the shortertrailing moving average (say 2 days) being a damped proxy for the current price, and thelonger trailing moving average (say 18 days) being a proxy for a ‘trend’ line. When the shortercrosses the longer, the model is implicitly saying that the trend has changed, and therefore theposition should too.I have constructed a moving average model which uses the spot rate as the trend indicator(i.e. as the constituent parts of the moving averages), and which takes a 1 or −1 position (longand short) depending on the relative values of the moving averages. However, the proﬁts orlosses from this position are recorded from the subsequent currency surprise, not the spot rate.It is in theory possible (although only with wide interest rate differentials) that the movingaverage model could correctly predict the spot rate, but that the position would be loss-makingbecause of the interest rate differential erosion. The model also builds in transaction costs
224
Currency Overlay
Box 10.7
Mean reversion
This is a common expression used to describe a particular type of market behaviour, butwhat exactly does it mean? It is a rather loose expression, but in ‘non-mathematical’ speakit means that (a) market movements are not random and (b) the market is more likely toreverse a previous move than continue it. The horizon over which mean reversion appliesis very important in determining its effect on market prices, and the kind of active processto exploit it.Very short horizon mean reversionThis is where the mean reversion applies over the shortest of horizons – say one day. Thismeans that if the market goes up one day, it is more likely to reverse that move tomorrowrather than continue it. This behaviour is characteristic of illiquid markets, and the ‘choppy’market prices this brings about are often the result of the absorption process of large ordersin the market.Mid-horizon (moving average) reversionUnder this model, the market has a ‘memory’ of a moving average of past prices, and thereis an ‘elastic band’ which is trying to move the market level back towards these historicallevels. The moving average can be of a wide range of maturities. It is supportable as aproposition that exchange rates are mean-reverting over 5-year horizons.Fixed meanUnder this model, the mean is not a moving average of historic prices, but a broadly ﬁxedvalue. The most obvious potential example of this is real (inﬂation-adjusted) interest rates;they vary day-by-day and year-by-year, but they tend to revert to a value centred around(say) 3%. This is perhaps the purest example of mean reversion.Mean reversion and trending can co-exist. Indeed there is strong evidence that exchangerates are trending over ‘market memory’ horizons – say 0–1 years, but mean-reverting over5-year horizons. This is not particularly surprising: market movements are the result of thecollective behaviour of the market’s participants, and the complex interactions of sentimentand underlying economic variables may perfectly well have different horizons.for changes in the position. This kind of detail is important in modelling, since one key andnecessary condition of a model that proves successful in practice is one that fully representsall the factors encountered in the live trading process.The results for this model are shown in Table 10.4. I have used ‘information ratios’ (broadlyspeaking, annualised return divided by annualised standard deviation of returns – see Box10.8) as the ‘performance’ result, and this is calculated from the daily value-added calculatedfrom this model. From the table, the reader can see that the best result was from the 2- and18-day combination.The table also neatly illustrates the effect of transaction costs. The values for 10- and 12-daymoving averages are repeated in both row and column. The difference of 0.2 (i.e. 0.52 − 0.72)information ratio is twice the effect of the transaction costs, which is therefore 0.1 information
Active Currency Overlay – Management Styles
225
Table 10.4 Information ratios of moving average model, daily data (1973–2002)Longer moving average (days)Shorter moving average (days)
10
12
14
16
18
20
22
124681012
0.580.560.610.480.45
0.540.600.650.600.590.52
0.640.630.580.520.420.440.23
0.730.750.600.420.350.380.33
0.710.760.430.290.270.320.37
0.670.610.410.360.310.400.28
0.660.520.390.340.300.330.18
Box 10.8
−0.72
Information ratio
The expression ‘information ratio’ was ﬁrst coined by Nobel Laureate William Sharpe asa useful way of measuring the ‘skill’ of active managers. The principle on which it rests,‘risk-adjusted excess return’, is the same as his parallel measure for securities – the Sharperatio.The information ratio is the average added value of a portfolio (portfolio return lessbenchmark return) divided by the tracking error (see Box 6.1). This value (generally lyingin the −1 to +1 area) records how much risk a manager is taking to achieve the recordedvalue added.Information ratios are calculated on an annual basis, so if quarterly, monthly or higherfrequency data is used, it must be annualised ﬁrst. Information ratios greater than 0 indicatethat a manager might have demonstrated skill. But active managers have to deviate frombenchmark returns to add any value at all, and so there is, on the face of it, a roughly 50%chance that a manager with an information ratio greater than 0 will have achieved it bychance. We can test this by using the property of the very useful Central Limit Theorem. Insummary, if we multiply the information ratio by the square root of the number of years forwhich the averages are calculated, we will get a ‘t-stat’. Roughly speaking, if this numberis greater than 1.7, it tells us that there is a 95% chance that the manager has demonstratedskill not luck.There is a ‘rule of thumb’ in evaluating managers that an information ratio of 0.5 isgood, and 0.75 is excellent. It is highly unlikely that any active manager can demonstrate aprolonged information ratio of 1. An active manager with a (more typical) 0.4 informationratio would have to show a 16-year track record for this to prove skill, whereas a 0.75information ratio manager could prove this in 5 years.ratio. The effect of transaction costs is less with wider differences of moving average horizon –at 2 and 18 it is about 0.06 information ratio.This apparently stable state of affairs is, unfortunately, upset when we apply the same splittest as in the case of correlation above. If we take the best moving averages (2 and 18), we cancalculate the information ratios for each of the split periods (Table 10.5). So the last 10 yearshas shown negative returns for this strategy, while the previous 20 years gave a staggering
226
Currency OverlayTable 10.5 Moving average model – USD/GBP, 2- and18-day, information ratios for different periodsPeriodWhole period (1973–2002)1973–821983–921993–2002
Information ratio0.761.521.22−0.54
average of an information ratio of 1.37. This ‘change of market nature’ ﬁts in with the negativeserial correlation we detected.The temptation is to attempt to develop the next level of reﬁnement to provide a ﬁlter togive ‘early warning’ of the change from positive to negative serial correlation. This kind ofthinking is appealing, but has to be treated with caution. This is for the reasons set out earlierin the chapter. It encourages the idea that with enough explanatory variables, all the ‘movingparts’ of this series’ generator will be captured. A better response, in my opinion, is for themodeller to take the trouble to try to understand, if such a thing is possible, the reason(s)for positive serial correlation appearing in the data in the ﬁrst place. If these are accuratelyidentiﬁed, then if these factors change or disappear, independently of the data, the modellercan progress to predicting such market behaviour changes with a well-founded likelihood ofsuccess.Momentum model: A momentum model can take many forms. The moving average modelis a particular type of momentum model. In this section, we will look at another type of model,which in this example is our regression model above.I will illustrate one class of models in the large universe of possible autoregressive models.Interested readers can ﬁnd a great deal on univariate11 time series analysis, as it is called,in econometric and statistics textbooks. It is worth noting that this universe also covers autoregressive second and higher moments – like autoregressive volatility12 and so on. Finally,staying with univariate modelling, we could (although not here) move away from the strictlystatistical into pattern recognition and pattern matching. This moves us into the world of chartsand chartists, and more esoteric processes like fractals.13In an effort to keep the example manageably simple, for these illustrative purposes I havebuilt a momentum trading model that applies the ‘best model’ from the regression analysisabove, and turns it into a signal-generation process for position-taking. This is not the principaltechnique applied to building momentum models, which will be simulation techniques (seeBox 10.9). However, in this example, I have chosen to set the size of the position (long or shortof GBP) to the size of the predicted move from the autoregressive model. Since the ‘best’ modelis so simple, this is actually proportionate to the size of the move the previous day. The modelis reset daily, and gives daily returns, which makes it easy to calculate information ratios. Ihave chosen to show the results with and without transaction costs. This is because a very ‘hairtrigger’ model like this, which changes the position every day, and which tries to exploit a very11
Univariate means that there is one time series (the price), and all modelling is conducted on the basis of this information alone.For example GARCH (Generalised Auto-Regressive Conditional Heteroscedasticity), a univariate modelling process for nonstationary volatility.13Fractals are self-similar geometric shapes. ‘Self-similar’ means that they look the same however much you enlarge or reducethem. There is a lot on fractals in maths and art textbooks, and on the web.12
Active Currency Overlay – Management Styles
Box 10.9
227
Simulations
A signiﬁcant part of the research and development process of active currency overlaymanagers is spent building and testing trading models. These are routinely tested in historicalsimulations. In a simulation, a model which has been developed using a range of historicaldata (usually up to the present time) is ‘run’ over history to simulate the trades it would havegenerated had it been live at the time. These simulations can be used to reﬁne models toproduce very high value added and low tracking errors (leading to high information ratios)for the whole period tested.Well-designed simulations are a very valuable tool for managers in their active processdevelopment efforts. However, good simulation design requires:
r Full accounting for transactions costs;r Care over the exact historical timing of information arrival (to avoid trades being ‘executed’ prior to the arrival of the relevant information);
r High-quality price information;r Care taken to ensure that the simulated trading process is replicable in live trading (i.e.frequency of price monitoring, lags in executing real transactions).However, the most important aspect of simulations is that they will generally producehistorical results that are consistently better than those achievable in live trading. Why isthis?In the currency markets, it is not primarily due to lack of liquidity or optimistic pricing in the model. Rather, it is the conceptual problem of model selection. Models whichsurvive the ‘cuts’ imposed by the testing and simulation process will be ‘better’ (i.e. havebetter historical returns) than those that do not. Surviving models may have a proportionof their good performance explained by real and stable inefﬁciencies, but also a proportion explained by chance. The random element is likely to revert to the mean (i.e. zeroperformance) for the future (live) period; only the ‘real’ element will contribute any valueadded.The disappointment engendered by underperformance of live models has tarnished simulations as a tool. This is shooting the messenger – the problem is not the simulations butthe modelling process. The most effective modellers are those that understand the survivalproblem, and build models which may on the face of it produce inferior historical returns,but which continue to produce similar returns in live trading.weak effect over very short intervals of time (one day), has a very high cost/beneﬁt ratio. Thisis clearly shown in the results (Table 10.6).The results are not particularly appealing – transactions costs (even at the low level achievablein the currency markets, I have assumed 3 bp round-turn on outright forwards) have a verydebilitating effect on results. The 93–02 period stands out as being pretty disastrous. Clearlythere is some work to do on this model to make it remotely usable.10.3.2.6 Summary technical modelling resultsSummary results are provided in Table 10.7. As we have seen, behind these superﬁcially attractive results is a great deal of non-stationarity. 30 years of daily data is a huge amount
228
Currency OverlayTable 10.6 Momentum model – USD/GBP, ‘best ﬁt’ regression, information ratios fordifferent periodsPeriod
Information ratio(without transactions costs)
Information ratio(with transactions costs)
0.670.501.37−0.36
0.360.211.08−0.76
Whole period (1973–2002)1973–821983–921993–2002
Table 10.7 Technical model resultsType of model1 Correlation2 Regression3 Trading model –moving averages4 Trading model –momentum
Target to be modelled
External variables
Results
Change in futurecurrency surpriseChange in futurecurrency surpriseCumulative proﬁts
Lagged dependentvariableLagged dependentvariable(s)Lagged dependentvariableLagged dependentvariable
t-stat = 4.65 at 1-dayautoregressiveBest F-ratio = 21.5 at1-day lagIR peaks at 0.76 at 2- &18-day MAsIR peaks at 0.36 at 1-dayfrequency
Cumulative proﬁts
(c. 7550 days), and the models have no difﬁculty whatever in ﬁnding highly signiﬁcant explanatory models in all of the techniques. However, none of these techniques, as they stand,would have allowed their users to make money in the last 10 years – in fact they would havelost money.In this exercise, I have looked at one exchange rate series over a particular period of time.Of the major currency pairs, USD/GBP happens to exhibit the most extreme changes to thenature of its distribution. Other pairs have changed less, or not at all, or have become moretrending with the passage of time. What this exercise will have illustrated, I hope, is thattechnical modelling can demonstrate persistent inefﬁciencies, but that exploiting these inefﬁciencies requires a well-founded belief in the underlying causes of the inefﬁciency as wellas technical modelling expertise. So successful modellers under this reasoning should be experts in the foreign exchange markets ﬁrst, and mathematicians second, not the other wayround.I have concentrated on one currency pair, but of course there are many. One of the advantagesof technical modelling is that it can choose, on the basis of recent currency behaviour, whichpairs are conforming to the core model. This is a further powerful tool, which, in the righthands, can make the technical style one of the most powerful.Finally, technical model building requires an understanding of the needs and constraintsof the client. Unconstrained technical models can undoubtedly make money over time, butif they require a US-based client to take positions, long or short, in, say, JPY/GBP, is thisacceptable? Certainly a conventional overlay mandate, as we have seen above, imposes certainconstraints on the manager: 0–100% hedge ratios, all positions being part of a pair with thebase currency, and the size of the position being determined by the underlying asset holdingsin those currencies.
Active Currency Overlay – Management Styles
229
10.3.3 Option-basedThe currency option market developed in the mid-1980s, and is now large and very liquid.It certainly qualiﬁes on these grounds for inclusion as a suitable instrument for the currencyoverlay manager. Options are instruments which, as we have seen in Chapter 3, give the optionholder rights but no obligations in return for the up-front payment of a fee (the premium).Conversely, options confer on the writer (seller) obligations but no rights in return for receiptof the fee. Given the complexity of the concepts, and the maths surrounding options, the readershould have familiarised himself with the contents of Chapter 3 before embarking on thissection.This section looks at the additional dimensions of decision-making that options give theoverlay manager, and the routes that might allow systematic exploitation for value added.Underlying these descriptions is the presumption, given very strong credence in the earliersections of this chapter and in Chapter 9, that currency markets are not efﬁcient, and thereforethat results from analysis and active management can be positive on average over time.10.3.3.1 Passive option buying in serially correlated marketsPassive option buying is not a popular currency management style. It looks too ‘passive’ formost active currency managers, and perhaps, more pertinently, their clients. However, there isa perfectly respectable and mathematically sound case to be made that passive option buyinghas positive expected value added.We have established from a variety of evidence offered in previous chapters that manycurrency pairs exhibit trends. I have argued that this trending behaviour has its foundations inthe structure of the foreign exchange market. It has shown persistence, and the structure of theforeign exchange market has not materially changed over the period – it is still dominated byproﬁt-insensitive players. Let us take an example of how such an option-buying process mightwork.We will make the working assumption that implied option volatilities are equal historicalvolatilities calculated from the daily vols, and we will test this assumption later. We will lookat 12-month options, since the evidence of trending indicates that we should extend the optionmaturities as far as possible. What are we trying to exploit? Figure 10.3, reproduced fromFigure 9.1, shows the volatility differentials for the major currency pairs.We will be trying to establish whether we can buy options at the daily volatility generatedBlack–Scholes model premium, and hold them to receive an average payoff that reﬂects theannual volatility. Let’s start with the theory, which I will summarise in a mathematicallysimpliﬁed way.The Black–Scholes model formula for an at-the-money forward option (put or call) boilsdown to:FV 1-year option premiums = (0.796/2) × volatility
(10.1)
Taking each of the terms in turn