## 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.

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## Notes

Lo’s (2002) efforts have however, come under close scrutiny regarding the generalizability of his analysis. For example, Wolf (2003) highlights that the main distributional result for the Sharpe ratio in Lo (2002) critically assumes normality of returns. As such, the confidence intervals circumstances will be biased in the direction of being too narrow. In response, Lo (2003) clarifies the more general result for the more realistic setting of non-normal distributions.

In the recent times there have been a few new approaches to measure managed funds performance. For instance, Chang et al. (2003) measure hedging timing ability of fund managers along with their market timing and security selection skills.

Investment and Financial Services Association Limited (IFSA, Australia).

This section draws heavily on Kim and In (2005a).

For more details of the derivation of the adjusted Sharpe ratio and its algorithm, see Kazemi et al. (2003). To calculate the adjusted Sharpe ratio, it is important to match the distribution of a portfolio to that of benchmark. To do so, the first step is to estimate an approximation of function

*F*(1 +*R*_{ i }). Kazemi et al. (2003) show that after transforming the return of a portfolio, its distribution is almost same as that of benchmark. Based on the function*F*(1 +*R*_{ i }), we estimate the value of P. In our paper the calculated values of P are subject to sampling variation. For this reason, we calculate the statistical inference using bootstrap methods by generating 3,000 random returns from the risk neutral distribution. The average across replications is very similar to the original estimates of the value of P, which means that no correction for small-sample bias is needed (we do not present the results but they are available on request).More detailed explanation of wavelet analysis and the technical details of how we implement wavelet decomposition, is provided in the technical appendix.

The Australian 13-week Treasury notes were not used for the risk-free rate since they were discontinued in August 2000.

LB(15) is the Ljung-Box statistic for up to 15 lags, distributed as χ2 with 15 degrees of freedom.

Considering the balance between the sample size and the length of the wavelet filter, we settle with the Daubechies extremal phase wavelet filter of length 4 (D(4)). For the various wavelet filters, see Gençay et al. (2002, pp. 114–115).

According to Gençay et al. (2003), the spectrum of raw monthly return series contains all frequencies between zero and 1/2 cycles, equivalent to the 0 to 2-month period in our data frequency.

For statistical inference, we also calculate the confidence interval for six series. However, for the sake of a clear presentation we do not plot the confidence intervals.

These figures are calculated by the normalization of wavelet variance using the sample variance. For more details, see In and Kim (2006).

We choose to benchmark against the large wholesale funds because this achieves the twin objective of having a ‘market like’ benchmark, while allowing a direct examination of the impact of fund size.

The logic is as follows. Financial managed fund markets are heterogeneous and made of investors and traders with different investment time horizons. Consider the large number of investors who trade in the security market and make decisions over different time scales. One can visualize traders operating minute-by-minute, hour-by-hour, day-by-day, month-by-month, and year-by-year. In fact, due to the different decision-making time scales among traders, the true dynamic structure of the relationship between the performance of the managed funds and risk factors will

*vary*over different time scales associated with those different horizons. Therefore, it is imperative to know that the true performances of the different managed funds will only be revealed when the Sharpe ratio is decomposed by the*different time scales*or*different investment horizons*.Wavelet analysis is relatively new in economics and finance, although the literature on wavelets is growing rapidly. Applications in these fields include decomposition of economic relationships of expenditure and income (Ramsey and Lampart 1998a, b), systematic risk in the capital asset pricing model (Gençay et al. 2003), the multiscale relationship between stock returns and inflation (Kim and In 2005b), and a multiscale hedge ratio (In and Kim 2000).

According to Ramsey (2002), one can approach the analysis of the properties of wavelets either through wavelets or through the properties of the filter banks.

In this context, “first” is used because the Haar wavelet is firstly developed among the various wavelet filters.

Additional information regarding wavelet and scaling filters (including the Haar and longer compactly supported orthogonal wavelets) and their properties may be found in Gençay et al. (2002).

The series stop at 3. As can be seen above Eq. A.5 we define \(h^{\prime}=(h_0,0,h_1)=(1/\sqrt{2},0,-1/\sqrt{2}).\) If we fully expand the equation (A.5), it has the following form:

$$ \begin{aligned} &\hbox{When}\;i=0,\;h_{2,0}=g_0{h}_0^{\prime}+g_1{h}_{-1}^{\prime} =\frac{1}{\sqrt{2}}\times\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}\times0=\frac{1}{2},\\ &\hbox{When}\;i=1,\;h_{2,1}=g_0{h}_1^{\prime}+g_1{h}_0^{\prime} =\frac{1}{\sqrt{2}}\times0+\frac{1}{\sqrt{2}}\times\frac{1} {\sqrt{2}}=\frac{1}{2}\\ &\hbox{When}\;i=2,\;h_{2,2}=g_0{h}_2^{\prime}+g_1{h}_1^{\prime} =\frac{1}{\sqrt{2}}\times-\frac{1}{\sqrt{2}}+\frac{1} {\sqrt{2}}\times0=-\frac{1}{2}\\ &\hbox{When}\;i=3,\;h_{2,3}=g_0{h}_3^{\prime}+g_1{h}_2^{\prime} =\frac{1}{\sqrt{2}}\times0+\frac{1}{\sqrt{2}}\times -\frac{1}{\sqrt{2}}=-\frac{1}{2}\\ \end{aligned} $$In the above calculation,

*h*_{−1}and*h*_{3}are considered as 0 because it is not defined. This happens when we generalize notation using summation.For other filters such as the Daubechies least asymmetric wavelet filter of length 8 (LA(8)), the construction of the higher scale filters are the same as in this example. For instance, the scale 1 LA(8) scaling filter,

*g*, is (−0.0758, −0.0296, 0.4976, 0.8037, 0.2979, −0.0992, −0.0126, 0.0322). In this case,*g*′ is defined as (−0.0758, 0, −0.0296, 0, 0.4976, 0, 0.8037, 0, 0.2979, 0, −0.0992, 0, −0.0126, 0, 0.0322). For the various scaling filter coefficients, see Gencay et al. (2002, pp. 114–11).

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## Acknowledgments

The authors thank an anonymous referee for helpful comments and are pleased to acknowledge the support of the Australian Research Council, grant number DP0557172.

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## Technical appendix wavelet analysis

### 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.^{Footnote 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.^{Footnote 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 first^{Footnote 17} wavelet filter.^{Footnote 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:

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:

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:

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:

^{Footnote 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 2*d*
_{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:^{Footnote 20}

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 2*s*
_{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:

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:

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:

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In, F., Kim, S., Marisetty, V. *et al.* Analysing the performance of managed funds using the wavelet multiscaling method.
*Rev Quant Finan Acc* **31**, 55–70 (2008). https://doi.org/10.1007/s11156-007-0061-8

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DOI: https://doi.org/10.1007/s11156-007-0061-8

### Keywords

- Performance measure
- Wavelet analysis
- Sharpe ratio
- Australian managed funds

### JEL Classifications

- G23
- G21
- G10