Analysing the performance of managed funds using the wavelet multiscaling method

Abstract

We propose a new approach for investigating the performance of managed funds using wavelet analysis and apply it to an Australian dataset. This method, applied to a multihorizon Sharpe ratio, shows that the wavelet variance at the short scale is higher than that of the longer scale, implying that an investor with a short investment horizon has to respond to every fluctuation in the realized returns, while for an investor with a much longer horizon, the long-run risk associated with unknown expected returns is not as important as the short-run risk. Using multihorizon Sharpe ratios of six groups of managed funds, we find that none of the fund groups are dominant over all time scales.

Technical appendix wavelet analysis

Wavelet analysis refers to the representation of a signal in terms of a finite length or fast decaying oscillating waveform (known as the mother wavelet). Financial markets are comprised of investors and traders with different investment time horizons—there is considerable heterogeneity of investment time horizon. At the heart of most trading mechanisms are the market makers. At the next level up are the intraday investors who carry out trades only within a given trading day. Then there are day investors who may carry positions overnight, and then short-term traders and finally at the top of the tree are long-term traders. Overall, it is the aggregation of the activities of all investors for all different investment horizons that ultimately generates market prices. Therefore, market activity is heterogenous with each investment horizon dynamically providing feedback across the different time scales (Dacorogna et al. 2001). The implication of a heterogenous market is that the true dynamic relationship between the various aspects of market activity will only be revealed when the market prices are decomposed by the different time scales or different investment horizons. The main advantage of wavelet analysis is the ability to decompose a time series, measured at the highest possible frequency, into several time scales.¹⁵

A major innovation of our paper is the introduction of a new approach—the wavelet multiscaling method—for investigating the multihorizon Sharpe ratio. We achieve this through an application to the performance of a sample of Australian managed funds. In this section, we briefly review how to derive the wavelet coefficients and how to derive wavelet variance over various time scales.

One useful way to think about wavelets is to consider wavelets and scaling filters.¹⁶ The relationship between filter banks and wavelets is extensively discussed in Strang and Nguyen (1996) and Percival and Walden (2000). Like conventional Fourier analysis, wavelet analysis involves the projection of a signal onto an orthogonal set of components—sine and cosine functions in the case of Fourier analysis and wavelets in the case of wavelet analysis. Wavelet analysis enables us to decompose the data into several time scales. To show the multiscale decomposition of a portfolio return series by wavelet analysis briefly, we employ the simple Haar wavelet, which is the first¹⁷ wavelet filter.¹⁸ The Haar wavelet filter coefficient vector, of length L = 2, is given by $h=(h_0, h_1)=(1/\sqrt{2},-1/\sqrt{2}).$ The complementary filter to h is the Haar scaling filter $g=(g_0, g_1)=(1/\sqrt{2},1/\sqrt{2}).$ These filters possess the following attributes:

$$ \sum_l{h_l}=0, \quad \sum_l{h_l^2}=1,\quad\hbox{and}\quad\sum_l{h_l h_{l+2n}}=0\quad \hbox{for all integers}\quad n\neq0. $$

(A.1)

$$ \sum_l{g_l}=\sqrt{2},\quad\sum_l{g_l^2}=1,\quad \hbox{and}\quad\sum_l{g_l g_{l+2n}}=0\quad \hbox{for all integers}\quad n\neq0. $$

(A.2)

Equation A.1 indicates the basic three properties of the wavelet filter, namely, it: (1) sums to zero, (2) has unit energy, and (3) is orthogonal to its even shifts. The scaling filter (Eq A.2) follows the same orthonormality properties of the wavelet filter, namely, having unit energy and orthogonality to even shifts, but instead of differencing consecutive blocks of observations, the scaling filter averages them. Thus, g may be viewed as a local averaging operator.

Using the Haar wavelet filter coefficients h, when applied to a return series R _t , the wavelet coefficients can be obtained by:

$$ \sqrt{2}d_{1t}=h_0 R_t+h_1 R_{t-1}, t=0, 1, \ldots, T-1. $$

(A.3)

The factor of $\sqrt{2}$ is required to ensure that the squared norm of the wavelet coefficients is equivalent to the squared norm of the return series. From this equation, it is observed that the wavelet coefficient d _1t is a weighted difference between consecutive returns. In a similar manner, the scaling coefficients can be obtained using the Haar scaling filter coefficient vector g as follows:

$$ \sqrt{2}s_{1t}=g_0 R_t+g_1 R_{t-1}, t=0, 1, \ldots, T-1. $$

(A.4)

In contrast to d ₁, the scaling coefficients s ₁ are based on local averages (of length two) of the original returns. Collecting both sets of coefficients into w = (d ₁, s ₁) yields the high-frequency and low-frequency content from the original returns. To derive the higher scale wavelet and scaling coefficients, it is necessary to calculate the higher scale wavelet and scaling filter coefficients using the first scale wavelet and scaling filters, h and g. To derive the scale 2 wavelet and scaling filters, let us define a filter ${h}^{\prime}=(h_0, 0, h_1)=(1/\sqrt{2},0,-1/\sqrt{2})$ to be the Haar wavelet filter with a zero between the two coefficients. The scale 2 Haar wavelet filter is calculated by:

$$ h_{2,i}=\{g\,{\ast}\,{h}_i^{\prime}\}=\sum_{j=0}^{L-1}{g_j {h}_{i-j}^{\prime}},\quad \hbox{for}\;i=0, \ldots, 3, $$

(A.5)

¹⁹

From this equation, we can obtain the scale 2 wavelet filter, h ₂ = (1/2,1/2,−1/2,−1/2) with length L ₂ = 4. The wavelet coefficient d ₂ may be obtained using h ₂ via 2d ₂ = {h ₂*R _t } where the factor of 2 is a normalization constant. The scale 2 Haar wavelet filter first averages two pairs of returns and then proceeds to difference them. Thus, the wavelet coefficients d ₂ are associated with changes on a scale of two. By defining ${g}^{\prime}=(g_0,0,g_1)=(1/\sqrt{2},0,1/\sqrt{2}),$ the scale 2 Haar scaling filter is obtained using:²⁰

$$ g_{2,i}=\{g\,{\ast}\,{g}_i^{\prime}\}=\sum_{j=0}^{L-1}{g_j{g}_{i-j} ^{\prime}},\quad \hbox{for}\quad i=0, \ldots, 3, $$

(A.6)

so that g ₂ = (1/2, 1/2, 1/2, 1/2). It is observed that g ₂ is a simple average of four consecutive returns. The scaling coefficients s ₂ may be obtained directly using g ₂ via 2s ₂ = {g ₂*R _t }.

Analogously, the scale 3 wavelet and scaling coefficients may be obtained by increasing numbers of zeros inserted between the wavelet and scaling filter coefficients such as h and g with length L = 2. For example, ${h}^{\prime\prime}=(h_0,0,0,h_1)=(1/\sqrt{2},0,0,-1/\sqrt{2})$ and define the scale Haar wavelet filter to be $h_3=\{\{g\,{\ast}\,{g}_i^{\prime}\}\,{\ast}\,{h}_i^{\prime\prime}\}=\{g_2 {\,{\ast}\,}{h}_i^{\prime\prime}\}$with the wavelet coefficients:

$$ h_3=\left({\frac{1}{\sqrt{8}},\frac{1}{\sqrt{8}},\frac{1} {\sqrt{8}},\frac{1}{\sqrt{8}},\frac{-1}{\sqrt{8}},\frac{-1} {\sqrt{8}},\frac{-1}{\sqrt{8}},\frac{-1}{\sqrt{8}}}\right) $$

with length L ₃ = 8. Generally, the length of a scale j wavelet filter can be calculated by L _j  = (2^j−1)(L−1) + 1. Similarly, the scale 3 scaling filter coefficients are calculated as $g_3=\{\{g\,{\ast}\,{g}_i^{\prime}\}\,{\ast}\,{g}_i^{\prime\prime}\}=\{g_2 \,{\ast}\,{g}_i^{\prime\prime}\}$where ${g}^{\prime\prime}=(g_{0},0,0,g_1)=(1/\sqrt{2},0,0,1/\sqrt{2}),$ with coefficients:

$$ g_3=\left({\frac{1}{\sqrt{8}},\frac{1}{\sqrt{8}}, \frac{1}{\sqrt{8}},\frac{1}{\sqrt{8}},\frac{1}{\sqrt{8}}, \frac{1}{\sqrt{8}},\frac{1}{\sqrt{8}},\frac{1}{\sqrt{8}}} \right) $$

Similar to the calculation of the scale 2 wavelet and scaling coefficients, the scale 3 wavelet and scaling coefficients are calculated via $\sqrt{8}d_3=\{h_3\,{\ast}\,R_t\}$ and $\sqrt{8}s_3 =\{g_3\,{\ast}\,R_t\},$ respectively. This procedure may be repeated up to the scale J = log2^T with the resulting wavelet and scaling coefficients organized into the vector $w=(d_1,d_2,\ldots,d_J,s_J).$

The wavelet coefficients from scale j = 1, 2, ..., J are associated with the frequency interval [1/2^j+1,1/2^j] while the remaining scaling coefficients s _J are associated with the remaining frequencies [0,1/2^j+1]. In this specification, the wavelet coefficients d _J , ..., d ₁, which can capture the higher frequency oscillations, represent increasingly fine scale deviations from the smooth trend. s _J represents the smooth coefficients that capture the trend.

Finally, using the wavelet coefficients, the wavelet variance for scale λ _j can be estimated by:

$$ \sigma^{2}(\lambda_j)\equiv Var(d_{jt}) $$

(A.7)