New Test Statistics for Market Timing with Applications to Emerging Markets Hedge Funds∗ Alessio Sancetta Department of Applied Economics, University of Cambridge Stephen E. Satchell Faculty of Economics and Politics, University of Cambridge April 2004

Abstract We provide a new framework for identifying market timing. Our analysis focuses on the local joint history of the hedge fund with the benchmark. The approach is fully nonparametric. Therefore, it has the advantage of avoiding the misspecification problems so common in this literature. The test statistic is some rank preserving function of a second order Uprocess. This empirical process allows us to define a set of statistics for market timing. We state the relevant asymptotic distribution. Some of these statistics are used to study the timing component of emerging markets funds using the dataset of Hwang and Satchell (1999). Key Words: Empirical Process; Kendall’s Tau; Nonparametric Estimation; Performance Measurement. ∗ Part

of this research has been supported by the ESRC Award 000-23-0400. We thank the referee for several

suggestions that improved the presentation of the paper.

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1

Introduction

This paper proposes new statistics for market timing which can be used for inference and testing purposes. The procedure advocated has broad applicability and it is not restricted to the class of linear market models such as the Henriksson and Merton model (1981). The purpose is to improve on parametric models which are frequently misspecified and allow no correct inference to be drawn: e.g. the parameters corresponding to the selectivity skills and the market timing skills are often confounded (Kothari and Warner, 1997). Furthermore, negative correlation between the above two measures has been reported by some authors, Kon (1983), Henriksson (1984) and Jagannathan and Korajczyk (1986). This has led some other researchers to define global measures of performance in order to circumvent some of these problems, e.g. avoid the problem of distinguishing between selectivity skills and timing skills (inter alia, Grinblatt and Titman, 1989, 1994, Cumby and Glen, 1990, Glosten and Jogannathan, 1994, et cetera). Also, problems can occur when the data considered exhibit evident non-linearity as in the case of emerging markets data. In this context, Hwang and Satchell (1999) use a three moment CAPM as proposed by Kraus and Litzenberg (1976). Linear models seem incompatible with the documented nonlinearities of financial data, especially hedge funds. Many Hedge funds returns are known to have implicit optionality embedded with total returns. This has been registred by Fung and Hseih (2000) and Agarwal and Naik (2001). Once we abandon the linear model, the choice is virtually infinite and any particular choice is hard to justify on a theoretical basis. For this reason, we propose a fully nonparametric testing procedure. The nonparametric approach is very common in econometrics, but the literature on performance measurement has not yet considered this more flexible approach. Another reason why nonparametric performance measurement is important is the perceived need to assess the track record of hedge funds combined with their reluctance to reveal their positions and strategies. As we shall see, it is possible to assess different strategies in broad terms relative to a given benchmark.

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Both in the parametric and nonparametric case, one would be expected to conduct a test based on the previously estimated conditional expectation function of the fund’s returns with the market or benchmark returns. Our approach is different from this. We use an empirical process that can capture the nonlinear dependence of the fund’s returns with the benchmark returns. However, the chosen empirical process is such that association implies association in the regression order, hence in terms of the conditional expectation function without the need of knowing its exact functional form. In fact, the process, which is a smoothed version of Kendall’s tau, has been proposed by Ghosal et al. (2000) to test for monotonicity of the unknown regression function. The process has also been studied in detail in Sancetta (2002) under strong mixing. Using this process as the main tool, our nonparametric approach is to construct statistics that allow us to identify market timing. In this respect the empirical process we use is such that we can speculate on the strategies of the fund manager only from observed market and fund’s returns. Inferring the strategies, we could also price the manager’s skills in terms of the financial instruments it replicates. While this issue is of great interest we do not, formally, address it. Nevertheless, we provide some informal discussion in our section on the empirical application. The plan for the paper is as follows. We review two of the most important market timing models. Abstracting from them, but considering their salient characteristics, we provide a general view of the statistical issues involved in performance measurement. This is done in Section 2. Then we consider the actual implementation of this discussion from the quantitative point of view: in Section 3, we introduce the empirical process which is the main technical tool for the market timing definition of many statistics and tests. The properties of this empirical process are briefly outlined, then the application to market timing is fully explored. In Section 4 we put our procedure to work. We choose to work with the emerging markets data used in Hwang and Satchell (1999). Conclusions are provided in Section 5. In the Appendix, we report general limit theorems for the asymptotic distribution for several of these statistics and limit theorems for the bootstrap. The asymptotic distribution of some of these statistics is unknown, so consistency of

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the bootstrap is fundamental when conducting inference.

2

Identifying Market Timing: Methodology

If the market goes up, and managers can forecast this, then they will strengthen their portfolio market exposure by buying high beta stocks and conversely, if the market goes down, they will decrease their market exposure. Therefore, market timing implies positive monotonicity of regression in bull markets as well as global convexity. We will make this clear below. First we will review the two mainstream parametric specifications for market timing. Then we present our general approach to the problem from a methodological point of view. In this case, the discussion tries to abstract from the actual quantitative statistical implementation.

2.1

Market Timing Definitions

Though Henriksson and Merton (1981), HM hereafter, provided two tests for market timing, one nonparametric and the other parametric, only the parametric one seems to be used in practical situations. In fact, the nonparametric test they propose requires knowledge of the timer’s forecast, which is rarely available. Let Zp,t and Zm,t be, respectively, the excess portfolio and market returns at time t. Then, the HM parametric model is given by Zp,t = α + βZm,t + γ max [0, −Zm,t ] + εt ,

(1)

where (εt )t∈Z is a mean zero iid innovation orthogonal to the other variables. A perfect market timer would have market coefficient β = 1 and timing coefficient γ = 1. Figure I plots the graph of the HM model when α = 0.5, β = 1 and γ = 1 together with the market line. On the other hand, Treynor and Mazuy (1996), TM hereafter, propose the following parametric specification Zp,t

= α + βZm,t θ (Zm,t ) + εt

θ (Zm,t ) = (1 + γZm,t ) , 4

(2)

which is a quadratic regression. θ (Zm,t ) can be interpreted as the first two terms in a Taylor expansion. While the β coefficient controls the market exposure, γ is the timing coefficient. Notice that α in both models is a measure of selectivity skills. Figure II shows the graph of the TM’s model when α = 0.5, β = 1 and γ = 0.1 together with the market line.

Figure I. Henriksson and Merton

Figure II. Treynor and Mazuy

(α = .5, β = 1, γ = 1)

(α = .5, β = 1, γ = .1)

As Figures I and II show, the common feature of the models is non-linearity and convexity for negative market returns. This convexity is fundamental in the definition of market timing because it captures the ability of the manager in choosing trading strategies that provide a protective put. This idea is explicitly modelled in the HM model by the γ term in (1). Generally, we would like to be able to distinguish between bull market timing, bear market timing, and global market timing. Given this information, we could select different managers for good and bad times. Both the HM and TM models provide this flexibility, but within a possibly misspecified parametric framework. Figures III and IV plot, respectively, the HM’s model with α = 0.5, β = 1 and γ = 1.5, and the TM’s model with α = 0.5, β = 1 and γ = 0.5. As before, the

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diagonal line is the market line.

Figure III. Henriksson and Merton

Figure IV. Treynor and Mazuy

(α = .5, β = 1, γ = 1.5)

(α = .5, β = 1, γ = .5)

As the above diagrams make clear, market timing involves issues of a very complex nature, not least the choice of benchmark. In all four diagrams we plot the market line as well; however, in practice, the market line need not exist. Moreover, even when the benchmark is identified, it remains to determine the nature of the timing strategy, i.e. global, bull market, or bear market. Clearly, these are only some of the almost infinite possible cases, e.g. we may have strategies that outperform the market only during extreme absolute movements of the market. This strategy could be achieved buying volatility, e.g. straddles via option strategies. Therefore, it is relevant to devise a set of statistics that are flexible enough for conducting inference over a broad range of possible strategies.

2.2

Timing Skills Versus Selectivity Skills

Before proceeding any further, it is important to distinguish the concepts of market timing and selectivity skills. Both are components of performance measurement, but with different implications. Our purpose in the sequel is to study market timing only, therefore finding ways to decompose the performance of a fund in a well defined fashion in order to avoid the problems mentioned in the introduction. Figure V below shows different graphs together with the bench-

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mark line.

Figure V. Same Timing Skills, Different Selectiviy Skills The reader should notice that while the graphs represent different performances, their market timing component is exactly the same, i.e. all differences are due to different selectivity components. An adequate market timing statistical procedure should be such that the above funds are indistinguishable. Then, statistics such as mean returns of the above funds would allow us to extrapolate the selectivity skills in a completely independent way. This point implies that the fund with the best mean return is not necessarily a market timer.

2.3

Identifying Market Timing

Here we provide a methodology to be followed when trying to identify the timing component in a hedge fund manager’s strategy. Clearly, we do not necessarily know his strategy, but only the outcome. In order to be more concrete, consider a researcher who wants to test for market timing in the classical framework as exemplified in Figures I and II, i.e. fully exposed to the market when it goes up, and down weighted when it is low. It this case, it may seem natural to proceed in three steps: (1.) identify the benchmark; (2.) extrapolate the degree of monotonicity of the funds’ excess returns with respect to the benchmark over some fixed positive region; (3.) verify the degree of protection that the strategy provides when the benchmark’s returns are negative. We explain each of the steps. Comments are deferred to Section 2.4.

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2.3.1

(1.) Identifying the benchmark

Market timing is a relative concept. Therefore, we need to define a benchmark. Let (Zm,t )t∈Z be the benchmark returns, and (Zp,t )t∈Z the fund returns. We believe that Zp,t = m (Zm,t ) + εt , where m (...) is a function to be defined, and εt is a martingale difference. Given that we have a common benchmark for these funds, we identify the general buy and hold position, i.e. buying the benchmark. In this case, Zp,t = Zm,t + εt ,

(3)

where the error εt is due to the impossibility of exactly buying the market. (For well developed markets like the S&P and the Nasdaq, it is actually possible to buy the market, i.e. Spiders and Cubes.) It follows that our performance measurement will be with respect to this buy and hold position. Clearly, we could also allow for Zp,t = βZm,t + εt , where β ∈ R++ is not necessarily equal to one. However, we use the word benchmark and not market in order to avoid this possibility, i.e. the benchmark may be some fraction of the market so that β is already incorporated in the definition of benchmark: this is merely for notational convenience. Then, we define the fund excess returns with respect to the benchmark, i.e. Z˜p,t = Zp,t − Zm,t .

(4)

³ ´ Therefore, we have the two dimensional series Z˜p,t , Zm,t . For conceptual simplicity, we divide

the series into two new series in order to mimic the two aspects of market timing in which our ³ ´ interest lies. Divide Z˜p,t , Zm,t into one series where Zm,t ≥ ω and one where Zm,t ≤ ω, say ³ ´ ³ ´ + − Z˜p,s , Zm,s and Z˜p,s , Zm,s , respectively, where ω is a real number which in many cases would

be equal to zero, but we want to allow for extra flexibility. Notice that when we are interested in

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timing components as in Figure I, ω is zero; this is the case considered here. We have changed the subscript from t to s in order to make clear that we are not necessarily considering the entire sequence (Zp,t )t∈Z , but only those values that satisfy the above criteria. In practice there is no need to impose such a strict dichotomy (see Section 5).

2.3.2

(2.) Degree of monotonicity over positive regions

We need to consider only the performance of the fund with respect to the positive returns of the benchmark. Conceptually, we would conduct some sort of positive monotonicity test. In particular, consider the following null © ª Ho+ = m+ (x2 ) − m+ (x1 ) ≥ 0, ∀ (x1 , x2 ) , x2 > x1 , where m+ is given by ¡ + ¢ Z˜p,s = m+ Zm,s + εt . The fund that violates the null of positive monotonicity cannot do as well as our chosen benchmark when this goes up. Clearly, a fund that claims to follow our benchmark must satisfy m+ ∈ Ho+ , otherwise it is not a market timer; i.e. conditional on its target, it does not achieve it. Those funds that hold low level beta stocks will underperform in this test even though they provide more downside protection. Clearly, these funds either do not exactly use our benchmark as reference, or they just underperform. An example where we would like the null to be rejected is given by the funds in Figure V.

2.3.3

(3.) Degree of downside protection

³ ´ − Following the same reasoning of step 2, we only consider Z˜p,s , Zm,s . Define the following set © ª Ho− = m− (x2 ) − m− (x1 ) ≤ 0, ∀ (x1 , x2 ) , x2 > x1 ,

where m− is given by ¡ − ¢ Z˜p,s = m− Zm,s + εt . 9

If this were a formal test, the null Ho− would be rejected when the fund loses more than the benchmark.

2.4

Comments on the Methodology

Some comments on the methodological aspect of the outlined steps are required. First of all we notice that there are some limitations derived by the use of the strict definition of monotonicity. In fact a function m (...) is positive monotonic if m (...) ≥ 0, which includes m (...) = 0. A more detailed discussion on this is given in Section 3 where the statistic involved in the testing procedure is discussed. In what follows, to aid implementation, we comment on each step.

2.4.1

(1.) Identifying the benchmark

We assume that the person who conducts the test conditions on his previous information and tests for a specific strategy which is defined before the test is conducted. This step involves the crucial assumption in the market timing testing procedure, i.e. the choice of benchmark. In the literature, this problematic issue has only been partially addressed. In general, it is assumed that the benchmark should be an efficient portfolio (e.g. Roll, 1978, in a mean variance, or quadratic utility world). A portfolio is efficient only relative to given axioms pertinent to decision making under uncertainty. Without being drawn into the controversial literature on choice under uncertainty, the chosen benchmark is preference dependent. This point is of particular relevance, and justifies the use of (3). Without restrictions on preferences, we cannot define a benchmark. In practice, it is convenient to choose well known market indices as reference. However this is restrictive and may fail in some instances, e.g. hedge funds. Hedge funds, which are often market neutral, are the epiphenomenon of the preference based benchmark choice. Consequently, market timing is not only a relative concept, but also highly dependent on preferences. These preferences should be incorporated into the choice of benchmark. For this reason we leave the choice of benchmark undefined and intentionally vague.

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2.4.2

(2.) Degree of monotonicity over positive regions

All the issues regarding preference are resolved in the first step. Conditionally on these preferences, we are just led by the simple idea that more is preferred to less, where the whole analysis is in relative terms, i.e. excess returns with respect to the benchmark. This simple idea has to be decomposed into two natural and different components, selectivity skills, and timing skills; see Section 2.2 and the references in the introduction for further details. In the linear framework, the timing skills are quantitatively defined in terms of the slope, i.e. β and γ in (1); in the more general nonlinear case, it is natural to translate it in terms of positive monotonicity of regression of the fund returns with the benchmark. Therefore, focusing attention on the fund excess returns with respect to the market allows us to consider just the timing skills, solving the frequent problem of confounding the timing and selectivity coefficients that occurs with other approaches.

2.4.3

(3.) Degree of downside protection

This step has two main purposes. First, it allows us to avoid the problem of confounding a market timer with a speculator who holds leveraged positions, e.g. invests her capital in futures. Figure VI depicts this case, where the strategy represented by the line with a higher slope, relative to the market, could be approximately achieved by buying and holding futures on the benchmark. (Clearly, this argument has been simplified for illustrative purposes.)

Figure VI. Speculator 200% Exposure to the Benchmark The other important reason is that it allows us to assess if the manager’s strategies replicate a protective put as in the HM model, and more generally, a speculative put as is the case for 11

global market timing, e.g. see Figures III and IV above. The study of strategies that replicate a speculative put can be conducted by redefining the excess returns as Z˜p,t = Zp,t − [Zm,t + max (0, −Zm,t )] .

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(5)

The Empirical Process

In order to accommodate from a quantitative point of view the discussion of the previous section, we propose to use the empirical Kendall’s tau process considered in Ghosal et al. (2000) for the iid case, and studied in Sancetta (2002) under more general conditions. Then, a test statistic would usually be some norm of this process. We recall the properties of this process. Definition 1. For two series Y1 , ..., Yn and X1 , ..., Xn , the empirical Kendall’s tau process is a second order U-process given by Un (t) =

2 n (n − 1)

X

1≤i
sign (Yi − Yj ) sign (Xi − Xj ) khn (Xi − t) khn (Xj − t) ,

(6)

¡ −1 ¢ where khn (s) = h−1 n k hn s is a kernel smoother with argument s and bandwidth hn ; hn being a decreasing function of n.

Notice that a U-process is a U-statistic where the U-kernel is a class of functions. Let f be a fixed function with v dimensional argument. Then, a U-statistic of order v and U-kernel f is given by µ ¶−1 X n f (Xi1 , ..., Xiv ) , v (v,n)

where

P

(v,n)

stands for summation over all possible combinations of v indices out of n (see

Serfling, 1980, for details on U-statistics). Let f ∈ F where F is some arbitrary, but fixed class of functions; the U-process of order v is given by µ ¶−1 X n f (Xi1 , ..., Xiv ) , f ∈ F. v (v,n)

In our case, the class of functions is indexed by t. The reader should not confound the term Ukernel, with kernel smoother, which is just a density function used to convolve the observations,

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e.g. k (s) is a density function. The U-kernel of Un (t) is given by hi,j (t) ≡ sign (Yi − Yj ) sign (Xi − Xj ) khn (Xi − t) khn (Xj − t) .

(7)

Then Un (t) is just the sample average of hi,j (t); i.e. it is the sum over all the different ways of selecting two objects out of n regardless of their order divided by the number of elements. Thus, it is a smoothed version of Kendall’s tau statistic also known as the sign correlation coefficient. Unlike Pearson’s correlation coefficient, Kendall’s tau (as well as Spearman’s rho) allows us measurement of non-linear dependence (e.g. Joe, 1997). This is not the case for Pearson’s correlation coefficient which is just equal to the cosine of the angle between two vectors (i.e. two series of length n can be seen as two points in a n dimensional space). This process (notice it is a function of t) is such that it stays close to zero for any t if (Yi )i∈Z and (Xi )i∈Z are independent, while it spends most of the time above zero if they are positively associated, and below zero if negatively associated. Clearly, (Yi )i∈Z and (Xi )i∈Z may not display a monotonic dependence, in which case Un (t) fluctuates above and below zero, where the magnitude of the fluctuations are proportional to the degree of association between (Yi )i∈Z and (Xi )i∈Z . Notice that (6) can capture association in the regression order, namely, positive values of (6) imply (asymptotically) E (Y |X) is increasing in X, i.e. Yt = m (Xt ) + εt ,

(8)

where m is an order preserving mapping and (εt )t∈Z is identically distributed. To see this, consider, for simplicity, the sign correlation coefficient, i.e. Kendall’s tau: Un =

2 n (n − 1)

X

1≤i
sign (Yi − Yj ) sign (Xi − Xj ) ,

where Un → U in some mode of convergence and U = E [sign (Yi − Yj ) sign (Xi − Xj )] (see Hoeffding, 1948). Assuming iid observations for the moment, let F and G be the distribution

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functions of (Xi )i∈Z and (εi )i∈Z in (8). Then, when (8) holds, U

=

Z

···

Z

sign (m (Xi ) + εi − m (Xj ) − εj ) sign (Xi − Xj )

×F (dXi ) F (dXj ) G (dεi ) G (dεj ) ZZ h i ˜ (m (Xj ) − m (Xi )) sign (Xi − Xj ) F (dXi ) F (dXj ) , = 1 − 2G

(9)

˜ (A) ≡ Pr (εi − εj < A). Without loss of generality let Xi ≥ Xj . Since G ˜ (0) = 1 , i.e. the where G 2 random variable (εi − εj ) is symmetric by construction (i.e. (εi )i∈Z is an iid process), we have that U ≥ 0 if and only if m is a rank preserving transformation, i.e. (m (Xi ) − m (Xj )) ≥ 0 =⇒ ˜ (m (Xj ) − m (Xi )) ≤ 1/2. When kh (x) is everywhere non-negative the result is directly seen G n to apply to Un (t) as well. However, the difference is that we would be able to extrapolate local information on m, i.e. for particular values of X. The above implications are valid under weaker conditions than independence of (εi )i∈Z . If εi and εj are not independent, but identically distributed, symmetry would follow if and only if their copula function is symmetric. Recall that if H (εi , εj ) is the joint distribution of εi and εj , their copula is given by C (G (εi ) , G (εj )) = H (εi , εj ) , where G is the marginal distribution of εi and εj , i.e. the copula is the joint distribution of uniform [0, 1] mariginals. Both for practical and theoretical purposes, we cannot work with Un (t) directly, but we need a standardized version Un∗ (t) ≡

√ nhn Un (t) , cˆ (t)

(10)

where cˆ (t) ≡

4hn 3n (n − 1) (n − 2)

X

1≤i,j,k≤n i6=j6=k

2

sign (Xi − Xj ) sign (Xi − Xk )

×khn (Xi − t) khn (Xj − t) khn (Xk − t)

(11)

is a function of order one. This function is an asymptotic estimator for the variance adjustment (see Ghosal et al., 2000, and Sancetta, 2002, for details). Further, khn (s) is the same kernel ¡ −1 ¢ smoother used in (6). Notice that khn (s) ≡ h−1 n k hn s should satisfy some conditions (see 14

Condition 2 in Section 4), but apart from this, it is arbitrary. A particular choice of k (s) is given by (14), which is the one used for our empirical application in Section 6.

3.1

The Statistics

Without being drawn into technical details, we provide example of test statistics in order to be more concrete. To draw statistical inference on Un∗ (t) we can either look at some specific t or consider the process as a whole. In the first case, we only need to know its univariate finite dimensional distribution. This corresponds to the case in which we are particularly interested in the value at some t. Notice that t is the parameter in the kernel smoother, therefore this implies testing at some particular X. If we consider the process as a whole, which is of more relevance in our case, we need to compute either some norm of the process, or some integral of it with respect to t. In general, the test statistic is given by ψ (Un∗ (t)) ,

(12)

where ψ (...) is a positive monotonic transformation. From (9) and its following discussion, we can use the statistic defined by (12) to test the following hypothesis Ho = {m (x2 ) − m (x1 ) ≥ 0, ∀ (x1 , x2 ) , x2 > x1 } , where m (...) does not need to be known. The null would be rejected for small values of (12). Depending on the specific application, we may wish to consider ψ (−Un∗ (t)) instead, so that we reject for large values. We devote this subsection to the discussion of useful choices of ψ (...) for our purposes, i.e. the supremum, the time spent statistic, and the area above a certain threshold. The discussion abstracts from the process Un∗ (t) , so we will use ξ (t) to denote some arbitrary process. For further details on these statistics, the reader is referred to Cramér and Leadbetter (1967).

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3.1.1

Supremum

The supremum is the more common choice of directional deviation for stochastic processes, i.e. ψ (ξ (t)) = sup (ξ (t)) . t∈T

This choice of ψ (...) allows us to consider the maximal deviation of the process either for positive or negative values, i.e. just look at −ξ (t). The drawback of the supremum is that it only considers the maximal deviation, without any reference to the general behaviour of the process.

3.1.2

Time spent statistic

The time spent statistic is defined as 1 ψ (ξ (t) , u) = T

ZT

I{ξ(t)>u} dt,

0

where I{A} is the indicator of the set A, and u is some chosen real number. Looking at I{ξ(t)
3.1.3

The area above u

We can define several statistics by integrating some function of the empirical process. For example, the area above u is given by 1 ψ (ξ (t) , u) = T

ZT

ξ (t) I{ξ(t)>u} dt,

0

so that the time spent statistic is recognized as a special case. Reversing the inequality allows us to find the area below u. This statistic seems to be a compromise between the two above. It is sensible for both large deviations of the process and persistent violation of the threshold.

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3.2

The Application of Un (t) to Market Timing via Monotonicity

The empirical process in (6) can be used as a mean for testing association in the regression order ³ ´ for Z˜p,t , Zm,t , i.e. in place of (Yi )i∈Z and (Xi )i∈Z . The process can be plotted, hence we can

visualize the behaviour of different funds with respect to the benchmark. Visual inspection is always very important and complementary to the quantitative result; see Section 5. The market

timing test becomes a test of the behaviour of Un (t) , or some norm of it. Recall that Un (t) ³ ´ is a smoothed version of the sign correlation, in our case between Z˜p,t , Zm,t . Notice that a plot of the graph of Un (t) should be interpreted as follows: negative values imply negative sign

correlation, consequently m0 ≤ 0, where m0 is the first derivative of m when it exists; the sign is reversed for positive values. For example, if Un (t) is always above zero for t ∈ [c, d] (c, d ∈ R ³ ´ arbitrary, but fixed), we can infer that Z˜p,t , Zm,t , having positive sign correlation in [c, d], are positive monotonic to one another, but only in [c, d].

Let us go back to step (3.) in the procedure outlined in Section 2.3. We saw that using the idea of monotonicity, rejection of the null did not necessarily imply the use of a protective put, but only that the fund did not underperfom. Recall that this is due to the fact that m is positive monotonic if m ≥ 0, which includes m = 0, with a negative sign for negative monotonicity. When conducting a test, if we do not reject the null we cannot say anything about the degree of monotonicity. Fortunately, we derive our statistic directly from the empirical process in which we translated all the information regarding the conditional mean function of the fund. Once market timing is not rejected, we can compute other statistics directly from Un (t) in order to provide a ranking among the funds. For example, we can define the testing procedure as a preliminary stage in order to identify the market timers. Then we can compute similar statistics as the one used in the test in order to rank the funds. Within the framework of the Kendall’s tau process, the choice is quite flexible and should accommodate both the needs of the applied researcher and the practitioner. On the other hand we may modify a monotonicity test so that we can infer m (...) > 0. We

17

propose two possible solutions to the problem: (1.) sequential testing, and (2.) redefinition of the benchmark.

3.2.1

(1.) Sequential testing

Consider the null given by the following set {m (x2 ) − m (x1 ) ≥ 0, ∀ (x1 , x2 ) , x2 > x1 } . If the null is not rejected, rejection of {m (x2 ) − m (x1 ) ≤ 0, ∀ (x1 , x2 ) , x2 > x1 } would imply the desired result. This approach has the drawback of involving sequential testing. As an advantage it has a clear interpretation in terms of the behaviour of the empirical process. Figure VII shows the asymptotic 2 standard deviations confidence region for Un∗ (t) . For m = 0, under specific circumstances, Un∗ (t) could be bounded by these lines for most values of t ∈ T . (This is not correct in general as the distribution of the supremum of the process would depend on the covariance function and the size of the interval. Specific numbers in this example are used only for illustrative purposes.) On the other hand, for m > 0, the process would be likely to cross from below the top line, but never downcross the bottom one.

Figure VII. Example of Confidence Lines for U ∗ (t) In this particular case, one would work with the extrema of the process. However, the same idea works as well with the other statistics considered above.

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3.2.2

(2.) Redefinition of the benchmark

If we allow the benchmark to be defined as in (5), but with a slight modification, e.g. Z˜p,t = Zp,t − [Zm,t + γ max (0, −Zm,t )] , for γ > 0, then we can surely say m (...) > 0, and the magnitude of γ at which we reject the test provides the degree of market timing as expressed by the protective put.

4

Application to Emerging Markets Data

As an application of the above theoretical discussion, we study the performance of fifteen emerging markets funds using sixty-six monthly observations, i.e. from January 1992 to August 1997. The dataset is the same one used by Hwang and Satchell (1999) and details can be found there. The choice of benchmark, as discussed in Hwang and Satchell (1999) is a controversial issue in this context and our results show that different conclusions may be obtained for different benchmarks. Because of the problem related to emerging markets indices being inefficient portfolios (see Masters, 1998), we follow Hwang and Satchell (1999) in using the MSCI Emerging Markets Free (ESCI-EMF) and the IFC Investible (IFCI) as a benchmarks. Broader indices may not be adequate because of investment restrictions to foreign investors, e.g. the MSCI Emerging Markets Global and the IFC Global. We report summary statistics in Table I; in particular, the first four cumulants for all the time series. Whether it is appropriate to regard these funds as hedge funds is a moot point; the key issue, as can be seen in Table I, is that the data are highly non-normal.

19

TABLE I SUMMARY STATISTICS (Excess Returns) mean s. deviation MFCI-EMF 0.363 5.047 IFCI-C 0.191 5.369 CL Developing Markets 0.886 5.654 Genesis Emerging Markets 0.844 4.394 PFC Emerging Markets Ptfl 0.546 4.994 KB Emerging Markets 0.545 4.835 Templeton GS Emerging Mkts 0.529 4.421 INVESCO Pioneer Markets 0.429 4.997 INVESCO PS Glbl Emer Mkts 0.262 5.004 Royal Life/INVESCO PioneerMkt 0.256 6.053 Hansard\GT Emerging Markets 0.149 4.579 GT Emerging Markets B 0.095 4.785 Baring EMUF Global Emerg Mkts 0.083 5.619 Baring Chrysalis Undiluted 0.057 5.138 Groupe Indosuez Dev Mkts -0.003 5.171 Gartmore CSF Emerging Mkts -0.118 5.15 MBf Venture Portfolio -0.898 6.42

skewness excess kurtosis -0.322 1.236 -0.431 1.413 0.256 0.278 0.124 0.035 -0.081 0.813 -0.447 0.914 -0.223 0.092 -0.203 1.205 -0.272 1.461 -0.267 1.162 -0.318 1.397 -0.767 1.871 -0.173 1.216 -0.101 0.385 -0.747 1.594 -0.661 1.599 0.266 0.242

We used two simple linear transformations of the data. Using (4), we generated series of excess returns with respect to each benchmark, i.e. the MSCI-EMF and the IFCI. Then, to make results comparable and for computational reasons, we transformed the benchmark data as follows, Z˜m,j =

Zm,j − min (Zm,j ) j

max (Zm,j ) − min (Zm,j ) j

.

(13)

j

Therefore, the benchmark data for both the MSCI-EMF and the IFCI have their range in [0, 1] . We sorted the data in descending order once with respect to the MSCI-EMF and once with respect to the IFCI. With this data, we computed Un∗ (t) in (10) for each fund’s benchmark excess returns against the benchmark itself. In particular, we chose ¡ ¢ k (s) = 0.75 1 − s2 , s ∈ [0, 1]

(14)

= 0 elsewhere,

and bandwidth hn = 0.3, where we recall that khn (Xi − t) =

1 k hn

µ

Xi − t hn

¶

.

Figures VIII-XII shows a sample of the 30 processes calculated (15 funds for each benchmark). Before, commenting on these sample figures, we highlight the statistics we chose to compute. Notice that we used the whole set of sample observations without distinguishing between of 20

positive and negative benchmark returns. We did this so we could use all the sample observations. However, we chose to define two regions of interest for calculating our statistics, t ∈ [.05, .5], t ∈ [.5, .95] , and the subset [.05, .95] of [0, 1] for technical reasons related to smoothing over compact intervals. Recall the definition of t from (6); this is the parameter of the empirical process and corresponds to Z˜m,j in our case. Then we can interpret the values of the process for t ∈ [.05, .5] as the process being on the left of the median of the benchmark. We chose this subdivision because the median is quite a robust statistic and this allows easy comparison of results for different underlying benchmarks. Notice that Z˜m,j = .5 corresponds very closely to Zm,j = 0 for both benchmarks. Therefore, the two regions of interest are directly related to the process being defined for negative benchmark’s returns when t ∈ [.05, .5] and positive when t ∈ [.5, .95] . In each of these regions we computed the extreme points (supremum and infimum), and the time spent above and below zero as a fraction of the total time T = .95 − .05. The results are in Tables II-IV. (In Tables II and III, the time t on the left of inf is the time at which the infimum occurred, similarly for t on the left of sup, i.e. the supremum.) EXTREMA OF THE PROCESS Benchmark: MFCI-EMF CL Developing Markets Genesis Emerging Markets PFC Emerging Markets Ptfl KB Emerging Markets Templeton GS Emerging Mkts INVESCO Pioneer Markets INVESCO PS Glbl Emer Mkts Royal Life/INVESCO PioneerMkt Hansard\GT Emerging Markets GT Emerging Markets B Baring EMUF Global Emerg Mkts Baring Chrysalis Undiluted Groupe Indosuez Dev Mkts Gartmore CSF Emerging Mkts MBf Venture Portfolio

t 0.13 0.1 0.16 0.13 0.2 0.08 0.08 0.4 0.05 0.23 0.05 0.11 0.05 0.05 0.05

TABLE II t<.5 inf t -1.94 0.5 -2.69 0.38 -1.17 0.32 -1.69 0.44 -1.62 0.41 -3.41 0.45 -2.35 0.31 -0.82 0.22 -4.08 0.39 -0.53 0.07 -1.69 0.38 -2.42 0.5 -1.57 0.47 -1.75 0.3 -1.82 0.35

21

t>.5 sup 0.06 0.38 -0.17 1.57 -1.62 1.43 0.91 1.56 2.59 1.44 0.49 0.92 0.98 1.23 1.91

t 0.93 0.95 0.5 0.83 0.77 0.86 0.75 0.89 0.89 0.87 0.95 0.95 0.92 0.92 0.71

inf -0.77 -2.41 -0.24 -0.77 -0.70 -0.97 -2.00 -1.33 0.60 -1.15 -1.60 -1.99 -0.36 0.88 -0.04

t 0.83 0.50 0.89 0.50 0.89 0.95 0.92 0.78 0.50 0.57 0.50 0.81 0.50 0.81 0.95

sup 0.32 0.23 2.01 1.41 0.58 1.57 -0.10 1.39 2.39 1.56 0.34 1.23 0.93 0.88 3.97

EXTREMA OF THE PROCESS Benchmark: IFCI-C CL Developing Markets Genesis Emerging Markets PFC Emerging Markets Ptfl KB Emerging Markets Templeton GS Emerging Mkts INVESCO Pioneer Markets INVESCO PS Glbl Emer Mkts Royal Life/INVESCO PioneerMkt Hansard\GT Emerging Markets GT Emerging Markets B Baring EMUF Global Emerg Mkts Baring Chrysalis Undiluted Groupe Indosuez Dev Mkts Gartmore CSF Emerging Mkts MBf Venture Portfolio

t 0.13 0.26 0.16 0.15 0.11 0.09 0.09 0.17 0.07 0.14 0.19 0.27 0.08 0.05 0.05

TABLE III t<.5 inf t -2.19 0.41 -2.55 0.15 -1.99 0.29 -0.80 0.44 -2.39 0.43 -2.54 0.4 -1.75 0.41 -0.16 0.05 -1.06 0.42 -1.90 0.5 -0.68 0.05 -0.94 0.08 -1.99 0.5 -1.63 0.22 -3.56 0.5

t>.5 sup 0.13 -0.42 0.07 0.84 1.65 0.45 2.31 2.32 1.05 0.77 0.90 0.33 1.76 2.42 1.52

t 0.87 0.79 0.69 0.88 0.76 0.86 0.88 0.93 0.76 0.7 0.88 0.91 0.91 0.92 0.86

inf -0.48 -2.32 -1.11 -1.87 0.11 -2.13 -0.59 -2.17 0.25 0.66 -0.58 -3.00 0.33 -0.70 0.07

t 0.73 0.69 0.95 0.95 0.5 0.5 0.5 0.79 0.5 0.84 0.55 0.58 0.66 0.75 0.61

sup 0.90 -0.92 0.91 0.05 1.53 -0.03 2.16 0.87 0.94 1.52 0.83 -0.13 2.18 1.04 1.86

TABLE IV TIME SPENT BY THE PROCESS Below 0 Above 0 Below 0 Above 0 Below 0 Above 0 Below 0 Above 0 MFCI-EMF IFCI-C Benchmark: t<.5 t>.5 t<.5 t>.5 CL Developing Markets 46% 4% 22% 28% 37% 13% 39% 11% Genesis Emerging Markets 31% 19% 41% 9% 50% 0% 50% 0% PFC Emerging Markets Ptfl 50% 0% 21% 29% 45% 5% 29% 21% KB Emerging Markets 27% 23% 19% 31% 17% 33% 35% 15% Templeton GS Emerging Mkts 34% 16% 16% 34% 26% 24% 0% 50% INVESCO Pioneer Markets 23% 27% 16% 34% 27% 23% 50% 0% INVESCO PS Glbl Emer Mkts 27% 23% 50% 0% 13% 15% 37% 35% Royal Life/INVESCO PioneerMkt 22% 28% 15% 35% 4% 46% 12% 38% Hansard\GT Emerging Markets 18% 32% 0% 50% 22% 28% 0% 50% GT Emerging Markets B 14% 36% 21% 29% 30% 20% 0% 50% Baring EMUF Global Emerg Mkts 26% 24% 27% 23% 14% 36% 30% 20% Baring Chrysalis Undiluted 21% 29% 8% 42% 39% 11% 0% 50% Groupe Indosuez Dev Mkts 26% 24% 0% 50% 26% 24% 0% 50% Gartmore CSF Emerging Mkts 18% 32% 4% 46% 7% 43% 12% 38% MBf Venture Portfolio 15% 35% 3% 47% 31% 19% 0% 50%

4.1

Comments on Results

To help the reader gain a better understanding of our results, we guide her over Figure VIII and related statistics as an example. Figure VIII is the plot of Un∗ (t) against t for CL Developing Markets and the MFCI-EMF as benchmark under the transformations in (4) and (13). In this case, we see that Un∗ (t) is largely negative for t < .5. This is reflected in the time spent statistic in Table IV, which gives a value of 46% over the total interval [.05, .95], i.e. 92% of the time over [.05, .5] . In this respect, Table V shows that this fund ranks as the second best. For illustrative 22

purposes, we take three points in Figure VIII, A, B and C. These points correspond to the infimum of the process when t < .5 and the two absolute extrema for t > .5 : t = .13, .83, and .93; see Table II. We see that Un∗ (.13) = −1.94. Since Un∗ (t) is asymptotically Gaussian with mean zero and variance 1 under Condition 3 above, the sign correlation is almost significantly negative. (This claim has to be understood in terms of the finite dimensional distribution at t = .13. Apart from very few exceptions, none of the statistics computed in terms of the supremum and infimum are significant at the 90% level.) Negative sign correlation implies good downside protection. Turning to t = .83, Un∗ (.83) = .32. This is not significant, but may provide weak evidence of an increasing β at this return level, i.e. positive sign correlation that means same direction. When t = .93, Un∗ (.93) = −.77. At this point, we see a drop in ”market exposure”, i.e. the β of the fund with the market may have decreased below one. This can be interpreted as either (1) failure to have high exposure with a market high, (2) willingness to sell market exposure before a peak and its consequent downturn. However, this is pure speculation on our part simply for illustrative purposes, as the behaviour of Un∗ (t) , in Figure VIII, would hardly be found to be significantly different from zero under any statistic when t > 0.5.

B

U* 0.2

0.4

0.6

0.8

-0.5 C -1 -1.5 -2

A

Figure VIII. Un∗ (t) , t ∈ [.05, .95] : CL Developing Markets (MFCI-EMF Benchmark) We now turn to an overview and discussion of the results for the funds. The funds in Table I-IV are ordered in descending order with respect to mean return. In this respect, CL Developing Markets and Genesis Emerging Markets outperform the MFCI and IFCI benchmarks by more 23

than 10% a year, and are the best performers in terms of mean return. Figures VIII and IX show the empirical process for these two funds: the diversity is apparent. As we concluded in the last paragraph, CL Developing Markets seems to follow a simple buy and hold strategy, but holds a very low beta portfolio when the benchmark goes down. The case of Genesis Emerging Markets seems completely different. The fund seems to hold a portfolio with a very low beta with respect to both benchmarks, thus providing little evidence of any timing ability. In this case the time spent statistic is again a clear indicator. Further, the extrema of the process both on the left and the right of the median are a clear indication of the degree of deviation from zero, i.e. a simple buy and hold position. For the case of Genesis Emerging Markets, one may argue that the fund strategy could be roughly synthesized by buying the benchmark together with a put option out of the money which is financed by selling a call out of the money. Similar kinds of strategies, but only with respect to the MFCI benchmark, seem to be followed by Baring EMUF Global Emerg Mkts and Baring Chrysalis Undiluted. The difference is that these two funds seem to provide some protection only on the far left of the median. In general, it is fair to say that those funds that have spent the highest proportion of time on the left of the median below zero (i.e. their excess returns with respect to the benchmarks are negatively associated to the benchmarks when the latter goes down) are the ones that had the best average returns. These funds are the ones that used the most conservative strategies, though possibly different strategies from each other, e.g. see Figures VIII and IX. Nevertheless, consider a fund like Baring Chrysalis Undiluted when compared with IFCI. This fund spends most of its time on the left of the median below zero, but its infimum at the right of the median shows clear bad timing; see Figure X. Tables V and VI shows a ranking of these funds based on two features that we believe to be relevant. One is the time spent below zero when the process is at the left of the median, the other is the supremum of the process when it is on the right of the median. In most cases the funds that rank highest in one case do not in the other. This shows that no fund both outperforms the

24

market in good times and provides downside protection during a bear market.

0.2

0.4

0.6

0.2

0.8

0.4

0.6

0.8

-0.5

-0.5

-1

-1

-1.5

-1.5

-2 -2 -2.5 -2.5

-3

Figure IX.

Figure X.

Genesis Emerging Markets

Baring Chrysalis Undiluted

(MFCI-EMF Benchmark)

(IFCI Benchmark)

TABLE V HIGHEST DOWSIDE PROTECTION (Time Spent Below Zero, t<.5) Ranking Benchmark: MFCI-EMF Benchmark: IFCI-C 1 PFC Emerging Markets Ptfl Genesis Emerging Markets 2 CL Developing Markets PFC Emerging Markets Ptfl 3 Templeton GS Emerging Mkts Baring Chrysalis Undiluted 4 Genesis Emerging Markets CL Developing Markets 5 INVESCO PS Glbl Emer Mkts MBf Venture Portfolio 6 KB Emerging Markets GT Emerging Markets B 7 Baring EMUF Global Emerg Mkts INVESCO Pioneer Markets 8 Groupe Indosuez Dev Mkts Templeton GS Emerging Mkts 9 INVESCO Pioneer Markets Groupe Indosuez Dev Mkts 10 Royal Life/INVESCO PioneerMkt Hansard\GT Emerging Markets 11 Baring Chrysalis Undiluted KB Emerging Markets 12 Gartmore CSF Emerging Mkts Baring EMUF Global Emerg Mkts 13 Hansard\GT Emerging Markets INVESCO PS Glbl Emer Mkts 14 MBf Venture Portfolio Gartmore CSF Emerging Mkts 15 GT Emerging Markets B Royal Life/INVESCO PioneerMkt

TABLE VI MOST SPECULATIVE (Highest Sup Norm, t>.5) Ranking Benchmark: MFCI-EMF Benchmark: IFCI-C 1 MBf Venture Portfolio Groupe Indosuez Dev Mkts 2 Hansard\GT Emerging Markets INVESCO PS Glbl Emer Mkts 3 PFC Emerging Markets Ptfl MBf Venture Portfolio 4 INVESCO Pioneer Markets Templeton GS Emerging Mkts 5 GT Emerging Markets B GT Emerging Markets B 6 KB Emerging Markets Gartmore CSF Emerging Mkts 7 Royal Life/INVESCO PioneerMkt Hansard\GT Emerging Markets 8 Baring Chrysalis Undiluted PFC Emerging Markets Ptfl 9 Groupe Indosuez Dev Mkts CL Developing Markets 10 Gartmore CSF Emerging Mkts Royal Life/INVESCO PioneerMkt 11 Templeton GS Emerging Mkts Baring EMUF Global Emerg Mkts 12 Baring EMUF Global Emerg Mkts KB Emerging Markets 13 CL Developing Markets INVESCO Pioneer Markets 14 Genesis Emerging Markets Baring Chrysalis Undiluted 15 INVESCO PS Glbl Emer Mkts Genesis Emerging Markets

25

We note that comparisons of funds is difficult because some funds seem to use strategies that a priori may be of great value despite the fact that ex post they were not. This reinforces the importance of being able to distinguish between timing and selectivity skills. The nicest example is MBf Venture Portfolio which has the lowest mean return performance. Figures XI and XII depict the process of this fund with respect to both benchmarks. Using the ranking in Tables V and VI, this fund is among the five best when the IFCI is used. This shows how dramatically the statistics are dependent on the benchmark. On the other hand, the performance of CL Developing Markets seems to be almost independent of the two benchmarks. This fund adopts the classical (and apparently successful) HM market timing strategy: hold the market in bull markets, but hold a low level beta with respect to the market in bear markets. As a consequence, it ranks low among the speculative funds, but quite high among the ones with highest downside protection. A clear analysis of all the funds reveal that this is the only one whose pattern of returns is consistent with an HM type of strategy. 4

2

3

1

2 0.2 1

0.4

0.6

0.8

-1 0.2

0.4

0.6

-2

0.8

-1

-3

Figure XI.

Figure XII.

MBf Venture Portfolio

MBf Venture Portfolio

(MFCI-EMF Benchmark)

(IFCI Benchmark)

Hwang and Satchell (1999) find that PFC Emerging Markets Ptfl and GT Emerging Markets B show some timing components. Our findings show some evidence of market timing for PFC Emerging Markets Ptfl in terms of time spent below zero; see Table V. On the other side, the performance of GT Emerging Markets B seems to depend on the benchmark. Hwang and Satchell (1999) seem to support conclusions similar to ours in some cases. However, the major difference is that they find little evidence of market timing besides the two cases mentioned above. Depending 26

on the statistic to be used, this is not the case here. The time spent statistic gives a strong signal of downside protection for several of the funds. The issue in this case is to identify those funds that at the same time hold the market during bull markets. In this case, we agree with Hwang and Satchell (1999): the funds with the best mean returns performance are not necessarily market timers. This does seem to be the case for Genesis Emerging Markets despite its second best performance in terms of mean returns. Since the process we use allows us to avoid the issue of confounding timing skills with selectivity ones, we can infer that the high performance of Genesis Emerging Markets should be the result of selectivity skills. Another point we find in common with Hwang and Satchell (1999) is that the choice of benchmark seem to be particularly relevant; see the ranking in Tables V and VI. As a final remark, notice that one should be careful in interpreting the results as we get close to either .05 or .95. As shown in table VII, most of the observations are concentrated in the interval [.4, .6] . Therefore, this is the region of particular interest whereas outside this region sparsity of data may lead to less reliable statements. Nevertheless, recall that we worked with 1

Un∗ (t) ≡ (nhn ) 2 Un (t) /ˆ c (t) also to partially accommodate problems of this nature. TABLE VII Summary for [0,1] Linear Transformation of Benchmark MFCI-EMF Min. 1st Qu. Median Mean 3rd Qu. Max. 0 0.4086 0.5045 0.4963 0.5869 1 IFCI-C Min. 1st Qu. Median Mean 3rd Qu. Max. 0 0.4085 0.5038 0.4951 0.5893 1

5

Final Remarks

In this paper we defined many statistics for identifying the timing component in performance measurement. All these statistics are in terms of norms or integrals of a smoothed version of Kendall’s tau. This process allows us to capture the nonlinearity in the conditional expectation function of the fund with respect to the benchmark. The attractive feature is that the procedure

27

is nonparametric so that problems related to misspecification are avoided. Moreover, we recalled several results for the distribution of the process and the bootstrap approximation. The results on the asymptotic and approximate distribution of the process are necessary for inference. In this respect, we proposed a few approaches to inference that should encompass the most used testing procedure in the literature. Further, we put our procedure to work using the emerging markets data of Hwang and Satchell (1999). Our results are in some cases different from those reported in Hwang and Satchell (1999) where the most common performance measures are used. It seems that the traditional performance measures tend to prefer funds with high returns. While this is not in contradiction with the definition, it does not necessarily imply market timing. In fact any leveraged fund which tracks the market would outperform the market when this is positive. Some similarities are found with the three moment CAPM used by Hwang and Satchell (1999). While our procedure is non-parametric, simulation results from Ghosal et al. (2000) show that a monotonicity test under the sup norm has reasonable power under several alternatives for samples as small as n = 100, and very good power for n ≥ 200. The reader should refer to these authors for more details. In relation to this issue, we realize that our sample size of n = 66 might appear a bit small. This does not invalidate our analysis, but may reduce our beliefs on confidence levels based on the asymptotic results of Section 4. For this reason, we restricted the analysis to a pure relative comparison among the funds and did not provide, intentionally, any p-value. A larger dataset would be required for accurate statistical inference. The broad conclusion is as follows. To understand a hedge fund’s performance against a benchmark, we need to examine not just summary statistics such as alpha, but the joint history of the fund and the benchmark covariation (and directly related statistics) if we wish to interpret strategies. To do this with any degree of reliability, smoothing is essential.

28

A

Appendix: Limit Resuls for the Empirical Process

We have defined many useful statistics, yet we have to admit that for some of them, the asymptotic distribution is, as yet, unknown. We recall a few results from Sancetta (2004). In particular, we state the limiting distribution of the process under weak dependence. Further, we state the limiting distribution of its supremum and an approximation result for this distribution. Once we have the asymptotic distribution of the process, we can in principle compute the distribution of some function of it. Our process converges to a stationary Gaussian process. Therefore, the distribution for its supremum and the sojourn time are well known (e.g. Leadbetter et al., 1983, and Berman, 1992). Since the rates of convergence in the weak invariance principle are not sharp, we also state results in terms of the bootstrap, but only in distribution. The bootstrap can also be used to draw confidence intervals for statistics whose asymptotic distribution is unknown. All the results allow for some form of weak dependence, but for strictly stationary sequences. This is an ideal condition. The extension to the wealky stationary case could be achieved with little extra effort. However, the extension to the nonstationary case would require several changes including the bootstrap resampling scheme used here (see Sancetta, 2004, for a further discussion and details).

A.1

Conditions

Since the very notion of market timing implies some sort of dependence in the underlying observations, we allow for weak dependence as formalized by α mixing, whose definition we recall next. We recall the definition of strong mixing. Let (Xi )i∈Z be a sequence of random variables with values in (Ω, F). Let Fk := σ (Xi , i ≤ k) and F k := σ (Xi , i ≥ k) be, respectively, the sub σ-algebras generated by (Xi )i≤k∈Z and (Xi )i≥k∈Z (⊂ F) . We say that (Xi )i∈Z is α mixing if

29

¡ ¢ limn supk α Fk , F k+n → 0, where ¡ ¢ α Fk , F k+n

:

= sup |Pr (A ∩ B) − Pr (A) Pr (B)|

(15)

A,B

= sup |cov (IA , IB )| , A,B

¡ ¢ and A ∈ Fk , B ∈ F k+n . We call α (n) := supk α Fk , F k+n the mixing coefficient.

Our results are derived over the following domain of interest: T := [a, b] which is some compact

subset of the observed range of (Xi )i∈[1,n] in the sample of interest. We introduce the following conditions. Condition 1(p). (Xi )i∈N , (Yi )i∈N and (εi )i∈N are stationary α mixing sequences of random variables with mixing coefficient bounded above by a non-increasing sequence α (1) , α (2) , ..., such that α (n) ≤ |C| /np+ , for some

> 0 and |C| < ∞. Moreover, the sigma algebras generated by

(Xi )i∈N and (εi )i∈N are independent. Notice that if α (n) satisfies Condition 1(p), p refers to the exponent in n−p− . Condition 2. k (...) is a density function with support in [−1, 1] , with two bounded derivatives. Clearly, this condition implies that ∃M < ∞ such that k (...) ≤ M . Condition 3. Eq. (8) holds with m (x) := 0 ∀x ∈ R. Condition 4. (Xi )i∈N has marginal distribution functions FX and density fX with compact support. We implicitly assume the probability space to be rich enough so that we can define uniform random variables which are independent of the original observations. Remark. The limits of integration, unless specified otherwise, are taken to be over the actual support of the variables concerned. We will use . to indicate dominance in asymptotic order, i.e. a . b implies ∃C ∈ (0, ∞) such that a ≤ Cb.

30

A.2

The Limiting Distribution of Un∗ (t)

Let (b − a) be the range of the observed realization of the process (Xi )i≤n , i.e. |T | = b − a where |T | is the Lebesgue measure of T . Define the following

R

ρ (s) = where

£ ¤ Tn = 0, (b − a) h−1 ; n

q (z) q (z − s) k (z) k (z − s) dz , R [q (z) k (z)]2 dz

q (ω) ≡

µZ

sign (ω − z) k (z) dz

¶

Then we have the following limit theorem for Un∗ (t) . ¡ ¢ Theorem 1. Let Conditions 1 (3) , 2, 3 and 4 be satisfied and hn = O n−β , β ∈ (0, 1).

Then there exists a sequence of stationary Gaussian processes ξ n defined on the same sample space and indexed by Tn with continuous sample paths, such that Eξ n (t) = 0, Eξ n (t) ξ n (t0 ) = ρ (t − t0 ) , t, t0 ∈ Tn , and ¶ µ ¯ ∗ ¢¯ ¡ −1 −18+167β−15β 2 √ ¯ ¯ 24(11−β) sup Un (t) − ξ n hn (t − a) = Op n ln n . t∈T

Remark. We can replace n

−18+167β−15β 2 24(11−β)

−5

1

with the worse but cleaner bound hn 8 n− 15 . Hence,

−18+167β−15β 2 √ 1 √ −5 in the above theorem, the error becomes hn 8 n− 15 ln n > n 24(11−β) ln n. This remark

applies to Theorem 2 and 3 as well.

A.3

The Limiting Distribution of SupUn∗ (t)

Set ϕn = χn =

r

r

2 log

b−a , hn √

λ2 log 2π b−a 2 log −q , hn 2 log b−a hn

31

where

R R 00 6 [2K (x) − 1] k 2 (x) k 0 (x) dx + [2K (x) − 1]2 k (x) k (x) dx λ2 = − , R 2 [2K (x) − 1] k 2 (x) dx

K (x) =

Z

k (x) dx Z = hn khn (hn x) dx,

i.e. khn (s) =

h−1 n k

µ

s hn

¶

,

and the integral is taken over the [0, 1] support of k (x) . Theorem 2. Define Mn := sup Un∗ (t) . Under the same conditions as in Theorem 1, if n

−18+167β−15β 2 24(11−β)

ln n → 0,

then, © ª lim Pr {ϕn (Mn − χn ) ≤ x} = exp −e−x ,

n→∞

so for any 0 < ϑ < 1, the test based on the critical region Mn

has asymptotic level ϑ.

1 ≥ χn + log ϕn

Ã

1 log (1 − ϑ)

−1

!

¡√ ¢ p log log (1 − ϑ)−1 − log μ2 /2π p = 2 log (|Tn |) − 2 log (|Tn |)

We can also use an alternative formula for the sup norm of the process. The next result is in the spirit of Aldous (1989). Before stating this result it is convenient to recall the definition of level upcrossings (see Cramér and Leadbetter, 1967, for details). The number of upcrossings of level x, say CU+ (x) , is defined to be the number of times the process U (t) crosses the line x from below during a unit interval, e.g. a ≤ t ≤ a + 1, a ≥ 0. Introduce the following notation p (s) :=

(1 − ρ (s))2 2

1 − ρ (s) + ρ0 (s) |ρ0 (s)| µ ¶ ν := min 1/2, inf p (s) . s≥0

32

Using the weak invariance principle for Un∗ (t) , we have the following result. Theorem 3. Under the same conditions of Theorem 1, if n

−18+167β−15β 2 24(11−β)

then ECU+∗ and if for fixed , K > 0,

√ ln n → 0,

½ 2¾ 1 12 x (x) = μ2 exp − , 2π 2

≤ |Tn | ECU+∗ (x) ≤ K,

© ª Pr {Mn ≤ x} = exp − |Tn | ECU+∗ (x) + O

A.4

µ

ln |Tn | |Tn |ν

¶

.

The Bootstrap Approximation

Since the convergence rates in the above theorems are not sharp, one may look at bootstrap confidence intervals. Moreover, some of the statistics discussed do not appear to have a known limiting distribution. In these cases the bootstrap can be used. When the empirical Kendall’s tau process is used for hypothesis testing, the bootstrap can also be used to draw confidence intervals. To do so, we use a model based bootstrap, i.e. the observations are drawn from the model under Condition 3. Despite Condition 1, we do not need to preserve the dependence structure of the process, as the variance of the process is asymptotically equivalent to the one in the case of iid observations. Hence we use a uniform bootstrap resampling scheme. Let (mi,n )i∈{1,...,n} be a sequence of iid uniform random variables in {1, ..., n} . Define Pn to be the empirical measure, i.e. suppose Z1 , ..., Zn is a sample from a random variable Z, then Pn Z := n−1

n X

δ Zi Z = n−1

i=1

n X

Zi ,

i=1

where δ Zi is the Dirac distribution at Z = Zi . In this case, the bootstrap empirical process is given by

r n n p ¡ ¢ hn X X nhn Un,qn (t) = 2 I {mj,n = i} δ xi ,εi Pn f − P2n f . n i=1 j=1 33

(16)

We are not bootstrapping the whole U-process, but only the first term in its Hoeffding’s decomposition (see Sancetta 2004). Notice that Un,qn (t) is centered conditioning on the sample observations. The variance standardized bootstrap empirical process is given by ∗ Un,q n

where cˆ (t) was given in (11).

√ nhn Un,qn (t) p , (t) = cˆ (t)

(17)

For statistical applications of the bootstrap, what matters is that the process is asymptotically ∗ pivotal, in practice that Un∗ (t) and Un,q (t) have same second moment independent of any estin

mated parameter. The bootstrap must allow us to derive confidence intervals for the theoretical a.s.

model given by Condition 3, i.e. (Yj )j∈N = (εj )j∈N even when this condition is not supported by the sample. In this case, the bootstrap is consistent. Theorem 4. Let

¡ 7 ¢− 12 hn n → 0, and hn → 0. Suppose Un∗ (t) is the empirical Kendall’s

tau process that satisfies Condition 3, then, for any possible sample sequence, not necessarily satisfying Condition 3, under Conditions 1(2), 2 and 4, the empirical bootstrap process (17) converges weakly (a.s.) to a Gaussian process with same mean and covariance function as the one in Theorem 1. By the continuous mapping theorem, Theorem 4 allows us to approximate the distribution of the supremum of the bootstrap empirical process by the distribution of the supremum of the Gaussian process in Theorem 1. Hence, Theorem 2 and 3 directly apply to the Bootstrap empirical process, but with a better rate of approximation.

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