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Sharpe Ratios and Their Fundamental Components: An Empirical Study

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  • IESEG School of Management, Paris, France

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In this article, we considered a risk-adjusted performance measure which benefits from a large success among the portfolio management community. Namely, Sharpe ratio considers the ratio of a given stock's excess return to its corresponding standard deviation. Excess return is commonly thought as a performance indicator whereas standard deviation is considered as a risk adjustment factor. However, such considerations are relevant in a stable setting such as a Gaussian world. Unfortunately, Gaussian features are scarce in the real world so that Sharpe performance measure suffers from various biases. Such biases arise from deviations from normality such as skewness and kurtosis patterns, which often exhibit the non-negligible weights of large and/or extreme return values. To bypass the potential biases embedded in Sharpe ratios, we propose a robust filtering method based on Kalman estimation technique so as to extract fundamental Sharpe ratios from their observed counterparts. Obtained fundamental Sharpe ratios are free of bias and exhibit a pure performance indicator. Results are interesting with regard to two findings. First, fundamental Sharpe ratios are obtained after removing directly the market trend impact whereas the kurtosis bias is removed at the volatility level. Second, fundamental Sharpe ratios exhibit a cross section dependency in the light of the well known size and book-to-market factors of Fama and French [1993]. Consequently, it is possible to extract pure performance and bias-free indicators, which are of primary importance for asset selection and performance ranking. Indeed, such concern is of huge significance given that the asset allocation policy, performance forecasts and cost of capital assessment, among others, are driven by performance indicators (Farinelli, Ferreira, Rossello, Thoeny, and Tibiletti [2008]; Lien [2002]; Christensen, and Platen [2007]).
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Electronic copy available at: http://ssrn.com/abstract=2838935
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EstimatingfundamentalSharperatios:A
Kalmanfilterapproach
Research·May2015
DOI:10.13140/RG.2.1.3528.5923
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Electronic copy available at: http://ssrn.com/abstract=2838935
1
Estimating fundamental Sharpe ratios: A Kalman filter approach
Hayette Gatfaoui
Associate Professor at NEOMA Business School,
Finance Department, 1 Rue du Maréchal Juin, BP 215, 76825 Mont-Saint-Aignan Cedex
France, Phone: 00 33 2 3282 5821, Fax: 00 33 2 3282 58 34, hayette.gatfaoui@neoma-bs.fr
First draft: July 2009, Revised draft: February 2015
Abstract: A wide community of practitioners still focuses on classic Sharpe ratio as a risk-
adjusted performance measure due to its simplicity and easiness of implementation.
Performance is computed as the excess return relative to the risk free rate whereas risk
adjustment is provided by the asset return’s volatility as a denominator. However, such risk-
return representation is only relevant under a Gaussian world. Moreover, Sharpe ratio
exhibits time variation and can also be biased by market trend and idiosyncratic risk. As an
implementation, we propose to filter out classic Sharpe ratios (SR) so as to extract their
fundamental component on a time series basis. Time-varying filtered Sharpe ratios are
obtained while employing the Kalman filter methodology. In this light, fundamental/filtered
Sharpe ratios (FSR) are free of previous reported biases, and reflect the pure performance of
assets. A brief analysis shows that SR is strongly correlated with other well-known
comparable risk-adjusted performance measures while FSR exhibits a low correlation.
Moreover, FSR is a more efficient performance estimator than previous comparable risk-
adjusted performance measures because it exhibits a lower standard deviation. Finally, a
comparative analysis combines GARCH modeling, extreme value theory, multivariate copula
representation and Monte Carlo simulations. Based on 10 000 trials and building equally-
weighted portfolios with the 30 best performing stocks according to each considered
performance measure, the top-30 FSR portfolio offers generally higher perspectives of
expected gains as well as reduced Value-at-Risk forecasts (i.e. worst loss scenario) over one-
week and one-month horizons as compared to other performing portfolios.
JEL Codes: C15, C16, G12.
Keywords: Extreme Value Copula, Kalman Filter, GARCH, Latent factor, Pure Performance,
Sharpe ratio, Value-at-Risk.
1. Introduction
A wide community of practitioners still focuses on classic Sharpe ratios as a tool to assess
assets’ performance (Bhargava et al. 2001; Elyasiani and Jia 2011; Ho et al. 2011; Robertson
2001; Scholz and Wilkens 2005b). Basically, portfolio managers use extensively such risk-
adjusted performance measure due to its simplicity and easiness of implementation. Sharpe
ratio is a performance measure whose assumptions come from the Capital Asset Pricing
Electronic copy available at: http://ssrn.com/abstract=2838935
2
Model (CAPM). The CAPM is an equilibrium relationship between security returns, which is
derived under a basic and restrictive setting (Lintner 1965a, 1965b; Mossin 1966; Sharpe
1963). In particular, performance is computed as the excess return relative to the risk free rate
(e.g. 1-month T-bill) whereas risk adjustment is provided by the asset return’s volatility, or
equivalently, the return’s standard deviation as a denominator (Sharpe 1964). Hence, Sharpe
ratio expresses the excess return, or equivalently, the investor’s reward per unit of (total) risk.
Such risk-return representation is only relevant under a Gaussian world while assuming the
total risk to result exclusively from market risk (e.g. diversified and efficient portfolios,
Sharpe 1964). However, Sharpe ratios can be biased because of existing idiosyncratic risk in
considered financial assets and/or due to existing portfolio underdiversification (Hwang et al.,
2012; Van Nieuwerburgh and Veldkamp, 2010). In particular, idiosyncratic risk can
contribute to increase volatility and exacerbate skewness and kurtosis effects in asset returns
(Angelidis and Tessaromatis, 2009; Yan, 2011). Therefore, deviations from normality as
materialized by skewness and kurtosis patterns (Black 2006; Eling and Schuhmacher 2007)
generate biases in asset performance valuation (Hodges 1998; Klemkosky 1973; Spurgin
2001; Zakamouline and Koekebakker 2009). For example, existing jumps in asset prices
generate skewness in corresponding returns so as to invalidate CAPM-based relationships in
their classic form, and therefore engender classic Sharpe ratio’s misestimation (Christensen
and Platen 2007; Platen 2006). One statistically significant outlier return suffices to bias
upward or downward Sharpe ratio1 (Gatfaoui 2012) due to the impact of the outlier on both
the average return level and corresponding standard deviation. Moreover, Goetzmann et al.
(2007) and Spurgin (2001) show that managers can manipulate Sharpe ratio. Indeed,
managers can bias upward Sharpe ratio estimates by taking well-chosen derivatives positions,
which artificially lower standard deviation without really lowering the investments’ risk.
Incidentally, Sortino (2004) shows that standard deviation underestimates risk during
upward market trends while it overestimates risk during downward market trends. Thus,
market trends distort Sharpe ratio, which should therefore account dynamically for market
trend bias (e.g. time variation in returns’ performance). Additionally, current research also
advocates time variation in Sharpe ratio (Tang and Whitelaw 2011; Woehrmann et al. 2005).
Specifically, the equity premium and Sharpe ratio vary over the business cycle (Kocherlakota
1996). Such time variation may reflect changes in agents’ risk aversion (Raunig and Scharler
2010) as well as cyclical/seasonal patterns (Tang and Whitelaw 2011). Therefore, if risk
assessment is biased, the reward-to-risk assessment becomes a biased performance indicator
(Sortino 2004). As a consequence, the stock or asset selection process resulting from such
performance measure yields a flawed asset picking because it is based on a misestimated
selection tool (Klemkosky 1973). As an improvement, we propose to filter out classic Sharpe
ratios (SR) so as to extract their fundamental components on a time series basis. The obtained
time-varying fundamental Sharpe ratios, or equivalently, filtered Sharpe ratios (FSR) are free
of previous reported biases, and reflect the pure performance of assets. On a practical
viewpoint, classic Sharpe ratios represent a noisy performance measure from which we
extract the pure performance component, namely the FSR. In particular, the performance
noise embedded in classic Sharpe ratios results from the market’s impact and idiosyncratic
risk among others. The employed Kalman filter model suggests that fundamental Sharpe
ratios are obtained after removing directly the market’s trend and volatility impact from
1 Gatfaoui (2012) assesses the impact of returns’ asymmetry, as represented by skewness and kurto sis patterns,
on Sharpe ratio through a simulation study. Introducing one outlier in normally distributed returns, the author
quantifies the bias generated by such an outlier return on Sharpe ratio estimates. Such a bias is driven by the
investment horizon, the frequency of the data, and the propensity of the outlier return to deviate from the average
return level.
3
observed Sharpe ratios. Additionally, a comparison with six other well-known risk-adjusted
performance measures highlights the clear discordances between the investment ranks
resulting from those performance measures and the ones inferred from FSR. Those
performance measures tend rather to track the performance classification, which is implied by
classic Sharpe ratios to a large extent. Furthermore, FSR is a more efficient performance
estimator as compared to such comparable risk-adjusted performance measures (RAPMs)
because it exhibits a lower standard deviation. Finally, a comparative risk analysis (i.e. market
risk exposure) accounts for the time-varying volatility and tail risk of stock returns as well as
correlation risk across stock returns among others. Portfolios composed of FSR-based
winning stocks offer higher expected gains and reduced Value-at-Risk levels over one-week
and one-month horizons as compared to other RAPM-based performing portfolios.
Our paper is organized as follows. In the second section, we present the Kalman filter
model. Filtering out observed Sharpe ratios, we extract unobserved fundamental Sharpe ratios
after removing the noise resulting from the financial market’s influence and existing
idiosyncratic risk. In the third section, we then introduce the stock returns under
consideration, namely 85 return time series so as to draw statistical inference and exhibit
stylized facts. Further, section 4 leads a back-testing analysis proving the soundness of the
model and related measurement’s robustness. As an extension, section 5 proposes a
comparative study relative to other risk-adjusted performance measures (RAPMs) such as
Sortino, Omega, Kappa, and Upside potential ratios. Most RAPMs are strongly correlated
with SR and low correlated with FSR, suggesting structural differences between SR-based
and FSR-based stock picking processes. Finally, section 6 proposes a comparative analysis
with respect to RAPMs’ efficiency and relevance. Firstly, FSR is a more efficient
performance estimator than other RAPMs because it exhibits a lower standard deviation.
Secondly, the market risk analysis combines GARCH modeling, extreme value theory,
multivariate copula representation and Monte Carlo simulations (i.e. GARCH-EVT-Copula
model). Based on 10 000 trials and building equally-weighted portfolios with the 30 best
performing stocks according to each considered performance measure, the top-30 FSR
portfolio offers generally higher perspectives of expected gains as well as reduced Value-at-
Risk forecasts (i.e. worst loss scenarios) over one-week and one-month horizons as compared
to other performing portfolios. Then, major findings are summarized in section 7, which also
introduces concluding remarks.
2. The model
Classic Sharpe ratios are biased/noisy performance indicators whose biases result from
market climate and idiosyncratic risk among others. In that way, they represent disturbed risk-
adjusted performance measures, which require to be cleaned.
2.1. Motivations
Sharpe ratios and then related performance assessment are subject to three types of bias,
namely non normality, market climate and time variation. The latter bias requires a dynamic
performance measurement.
The first bias results from deviations from normality as illustrated by stock returns’
skewness and kurtosis patterns. Such deviations invalidate the appropriateness of both the
mean return as a performance indicator and the standard deviation as a risk measure. Hence,
4
the Sharpe ratio of non-Gaussian stock returns provides an erroneous performance measure.
The second bias arises from the trend of the financial market, which impacts the reliability of
Sharpe performance measure (Krimm et al. 2012; Scholz and Wilkens 2005a; Scholz 2007;
Sortino 2004). As an example, Sharpe ratio overestimates the performance of poorly
diversified portfolios or funds during bear markets while it underestimates the performance of
such funds or portfolios during bull markets (Krimm et al. 2012; Scholz and Wilkens 2005a).
Market climate impacts then performance valuation and related investment rankings (Sortino
2004). As a result, economic variables represent key factors explaining the predictable time-
variation of investment returns. Incidentally, Ferson and Harvey (1991) show the significance
of the market risk premium for stock-specific investment returns. In particular, the risk of
change (i.e. time-variation) in investment returns exhibits a common component according to
Alexander (2005). Additionally, Sharpe (1963) exhibits the common systematic component in
stock excess returns through the CAPM. Therefore, the market risk premium represents a
systematic component of stock returns risk premium and captures the market climate bias
(e.g. time variation, market-based structural changes). Following Fama and French (1993),
Sharpe ratios should therefore be linearly linked with the market risk premium, which is an
explicit market bias indicator.
As regards the third bias, current research exhibits the cyclical pattern of Sharpe ratio. For
example, Lettau and Ludvigson (2010), Lustig and Verdelhan (2012), Whitelaw (1997) as
well as Woehrmann et al. (2005) exhibit countercyclical Sharpe ratios. Such countercyclical
pattern can result from changes in investors’ sentiment (Doran et al. 2009; Tang and
Whitelaw 2011) or in aggregate risk aversion over the business cycle (Kamstra et al. 2003;
Tang and Whitelaw 2011). Incidentally, stock returns’ seasonality also supports Sharpe ratio’s
cyclical feature (Fiore and Saha 2015). Moreover, the implied volatility index (VIX) appears
as a central factor for performance assessment since it represents a proxy of time varying
volatility (Brandt and Kang 2004) as well as investors’ fear gauge. In particular, Brandt and
Kang (2004) show that time variation in volatility explains well-known deviations from the
positive relationship between risk premium and volatility, which is advocated by the CAPM.
Indeed, the arrival of new information as well as changes in the economic outlook or in
investors’ risk aversion can generate jumps in stock returns (Raunig and Scharler 2010).
Hence, equity returns exhibit volatility regimes so that volatility is subject to shifts (i.e. jump
risk) over the business cycle (Cremers et al. 2015; Hamilton and Lin 1996; Santa-Clara and
Yan 2010; Schwert 1990). Incidentally, Whitelaw (2000) advocates time-varying probabilities
of regime switches while Santa-Clara and Yan (2010) exhibit time-varying jump and
volatility risks. Following previous findings, we consider the impact of stock market volatility
(as represented by VIX) on stock returns’ level.2 Hence, the impact of a change in market
volatility on stock returns (i.e. volatility feedback) is linearly taken into account through VIX
indicator in accordance with Maheu et al. (2013). Such feature is also supported by the fact
that volatility feedback generates return asymmetry and therefore skewness (Campbell and
Hentschel 1992; Gatfaoui 2013). In this light, the impacts of structural changes in both the
mean and variance of the stock market’s return require to be taken into account (i.e. time
variation at the return and volatility levels). Thus, biases resulting from the financial market’s
trend regimes and volatility regimes will be captured and quantified.
Consequently, classic Sharpe ratios need to be filtered out so as to correct them for their
hidden biases. For this prospect, we select the linear Kalman filter methodology. In particular,
classic Sharpe ratios are considered as noisy performance signals, which are disturbed/altered
2 Non-linearity in financial markets often arises from time-varying volatility patterns (i.e. volatility relationships,
see Nam et al. 2006).
5
by specific biases. The linear Kalman filter helps clean the noisy observed signals in order to
extract the true and unbiased corresponding signals, which are unobserved components
(Durbin and Koopman 2001; Harvey 1989; Kalman 1960; Kalman and Bussy 1961). Once
classic Sharpe ratios go through the Kalman filter, they yield their pure signal counterparts,
which are freed from existing biases (i.e. fundamental Sharpe ratios or FSRi). Moreover, the
linear Kalman filtering methodology is efficient in correcting observed Sharpe ratios because
monthly Sharpe ratios are stable, and can be considered as approximately linear (see Table 1).
2.2. Specification
Applying an unobserved component methodology (Harvey et al. 2004; Harvey and
Koopman 2009), we clean observed classic Sharpe ratios (i.e. the noisy signal) and extract the
denoised signal corresponding to unbiased/fundamental Sharpe ratios. Specifically, a linear
Kalman filter model extracts unobserved fundamental Sharpe ratios from observed stock-
specific Sharpe ratios along with relevant market indicators and stock returns’ stylized facts.
Given that monthly SRs exhibit no first order autocorrelation (see section 3) and that they
depend on the market climate, we specify the fundamental Sharpe ratios (FSRi) as a random
cycle component with deterministic frequency, amplitude and phase. Moreover, the market
premium and the natural logarithm of VIX are introduced in the dynamics of monthly Sharpe
ratios (i.e. linear link between SRi and factors).3 Hence, fundamental Sharpe ratios are
unobserved and free of market climate and time variation biases (i.e. unbiased). Previous
considerations yield the following specification for any time t and any stock i with i
{1,…,85} and t {1,…,170}:4
 
ittiiitit uVIXbaFSRSR ln
t
MktPremium
(1)
 
itiiiiit vtgftdcFSR sincos
(2)
where equations (1) and (2) represent the dynamic and state equations respectively;SRit
represents the Sharpe ratio of stock i over time t; MktPremiumt and VIXt represent the market
premium and VIX over time t; (uit) and (vit) are serially independent and correlated Gaussian
white noises with a zero mean (i.e. dynamic and state errors over time t);5 FSRit is the
unobserved/latent component in SRit over time t, the equation errors are assumed to follow a
two-dimensional Gaussian variable with a covariance matrix i defined as follows:
(3)
where a, b, c, d, f, g, h, k and l are constant parameters.6 Hence, we consider a stationary
representation, which is advocated by the stationary pattern of the data. The previous
3 Model selection is based on information criteria (Akaike 1974; Hannan and Quinn 1979; Schwarz 1978).
4 In unreported results, we tested for several specifications and concluded that either VIX or both SMB and HML
factors can be used additionally to the market premium. However, most relevant results are obtained with VIX,
which supports the findings of section 3.2. Hence, the market premium and VIX factors are incorporated to the
model while dropping SMB and HML explanatory factors. Moreover, we also considered an additional
unobserved trend component but it revealed also to be insignificant as compared to the cycle component in FSR.
5 They represent residual idiosyncrasies.
6 Gatfaoui (2010) proposes a simulation study, which evaluates the impact of return asymmetry on Sharpe ratio.
Applying asymmetric shocks to normally distributed returns, the author considers the distortion of the Sharpe
6
representation’s consistency is twofold. First, such specification is relevant since classic
Sharpe ratios are a special case of the previous representation under a Gaussian setting.
Indeed, Gaussian returns imply at least that h tends towards minus infinity, and even that a=0
and b=0 under a neutral market trend and volatility assumption. Thus, the variance of errors
(ui) is zero, and then SRi = FSRi (i.e. no dynamic error since the related mean and variance are
zero). In such case, the presumed noisy signal (SRi) coincides with its filtered and unobserved
signal counterpart (FSRi), which means that potential biases are not altering the observed
signal such as the observed Sharpe ratios (SRi). Second, the possible correlation between state
and dynamic errors accounts for possible remaining market-based commonalities (i.e. residual
correlation risk) such as liquidity commonality, or equivalently, systematic liquidity risk in
stock prices (Acharya and Pedersen 2005; Chordia et al. 2000; Hasbrouck & Seppi 2001;
Keene and Peterson 2007; Kempf and Mayston 2008), and potentially unaccounted market
volatility regime or jumps among others (Ammann and Verhofen 2009; Chang 2009; Chu,
Santoni, and Tung 1996; Kim, Morley, and Nelson 2004).
Discussion about interest and befits of FSR
Sharpe ratio assumes risk symmetry and penalizes the average performance, as measured by
the average excess return beyond the risk free rate, by the downside and upside variances
which are embedded in stock returns’ global variance. The upside and downside variances are
simply the variances of positive and negative excess returns respectively. SR does not
distinguish between the upside potential of stock returns and related downside potential.
However, rational investors favor stocks exhibiting highly variable gains, or equivalently,
high upside potential (Zakamouline 2011). In this light, the SR-based selection process favors
a stock return exhibiting a low downside variance (higher SR) as compared to a stock return
exhibiting a high upside variance (lower SR). Hence, inconsistencies about risk perception
appear, generating then a biased stock selection process. Stock picking is also biased by the
stock market’s impact among which its trend. Such biases are handled within the proposed
Kalman-based estimation. First, average computations on approximately one-month non-
overlapping windows yield Gaussian SRs, which conforms to model assumptions. Second, the
stock market’s bias is handled through the stock market trend and volatility factors (e.g. trend
and volatility regimes), releasing therefore FSR from the dependency on market climate and
market volatility regimes. Finally, the cyclicality of FSR is acknowledged in line with
financial markets’ oscillatory pattern (Dayri 2011) and related sensitivity to business cycle
(Lettau and Ludvigson 2010; Lustig and Verdelhan 2012; Whitelaw 1997; Woehrmann et al.
2005) among others. As a result, proposed performance measurement accounts for the
reported time variation, which results from market trend and volatility regimes as well as
cyclical/seasonal patterns. Obtained FSR is dynamically adjusted to structural changes
providing therefore a bias correction.
Moreover, corresponding model implications are threefold. First, estimating fundamental
Sharpe ratios requires removing directly the market trend and volatility biases from observed
monthly Sharpe ratios. Such representation follows the findings of Fama and French (1993) as
ratio, which is induced by the resulting return skewness and kurtosis. Such a distortion allows for quantifying the
bias in the Sharpe ratio, which results from return asymmetry. The author handles then a filtering process based
on Kalman methodology in order to remove the bias in the Sharpe ratio. However, firm-specific and market-
specific factors are not taken into account in the simulation study. Hence, our study improves and strengthens the
work of Gatfaoui (2010) while illustrating the reality of financial markets and stocks' stylized features. In this
light, we therefore propose a more sophisticated and more robust study.
7
well as stochastic volatility patterns (Maheu et al. 2013). Second, accounting for possibly
remaining market commonalities through equation errors’ covariance (i.e. correlation,
volatility linkages) avoids model misspecification. Third, fundamental Sharpe ratios consist of
a random cycle component, which conforms to the oscillatory (Mishchenko 2014), scaling
(Dayri et al. 2011) and cyclical (Lettau and Ludvigson 2010; Woehrmann et al. 2005) patterns
of financial markets. In particular, FSR encompasses a predictable cyclical trend and an
unpredictable white noise component. The predictable cyclical trend is of interest to investors
who target a market timing strategy and therefore bet on cycle reversals (i.e. active investment
strategy as opposed to static buy and hold strategy). However, such predictable cyclical trend
is balanced with an unpredictable shock, which illustrates prevailing uncertainty. Uncertainty
materializes as either reinforcing or counteracting deviations from predictions. Thus, we are
able to characterize the predictable variation in fundamental Sharpe ratios (Tang and
Whitelaw 2011) and its uncertainty. Predictability is preserved when the unpredictable shock
emphasizes the directional and cyclical trend so that a winning asset allocation is favored
(when investors attempt to time cycle reversals from one month to another).7 In the reverse
case, predictability is compromised and asset allocation performs poorly because related
market timing strategy fails. As a consequence, performing active investors consist of
portfolio/fund managers who are able to mitigate uncertainty at the portfolio level (e.g.
mitigating idiosyncratic shocks) so as to rely mainly on the predictable performance
component conditionally on market climate (e.g. living over trend and volatility regimes).
3. Data
The data under consideration deal with stock prices and corresponding relevant
fundamental factors on the U.S. financial market. Such factors bring in information about the
performance of stocks. We first introduce the data and their properties, and then arrange the
data in order to run the Kalman filter estimation on a time series basis.
3.1. Description and properties
3.1.1. Data description
The daily returns of 85 stocks are considered between 2000/01/04 and 2014/04/30, namely
3398 observations per series. Related risk premia versus the one-month T-Bill rate are
computed over this investment horizon. Namely, yield differences such as Rit-Rft are
computed on each day t {1,…, 3398} for any stock i {1,…,85} where (Rft) is the one-
month T-Bill rate on day t and (Rit) is stock i’s return on day t. The sample stocks are picked
randomly on the U.S. market and also belong to specific Standard & Poor’s indexes. Among
the 85 stocks, 1 return is the S&P500 index return, 63 stocks belong to the S&P500 index (i.e.
large-cap market), 9 stocks belong to the S&P MidCap 400 index (i.e. medium sized
companies’ equity market), and 12 stocks belong to the S&P SmallCap 600 index (i.e. small
sized equity market segment).8 Each stock is identified by a number i running from 1 to 85,
and each stock i exhibits a specific daily Sharpe ratio (SRi) over the investment horizon.
7 Following their predictions, aggressive investors focus on tactical asset allocation in order to maximize their
investment returns in the short run. Successful active investors will enter the market after a favorable cycle
reversal and surf on the upward performance trend. They will also be able to exit the market before the next
cycle reversal so as to avoid any downward performance trend.
8 The S&P SmallCap 600, S&P MidCap 400 and S&P500 indexes represent 3%, 7% and 75% of the U.S. equity
market respectively.
8
Those stock-specific Sharpe ratios are computed as the ratios of average excess returns to
corresponding excess returns' standard deviations over the sample horizon. Moreover, the
three Fama and French (1993) factors are considered, namely the market portfolio’s premium
(MktPremium), the return of the Small minus Big portfolio9 (SMB) and the return of the High
minus Low portfolio10 (HML). The daily implied volatility index level (VIX) is also
considered as a market volatility indicator. All stock data as well as VIX values are extracted
from Yahoo Finance website whereas Fama and French (1993) factors come from the authors’
website.11
3.1.2. Properties
Stock returns exhibit generally a non-Gaussian behavior.12 As a rough guide, Table 1
displays some descriptive statistics about the estimated stock-specific SR, mean, median,
standard deviation, skewness and kurtosis across stocks. The risk profile of each stock return
series results from the tradeoff between existing tails and their heaviness as represented by
skewness and kurtosis (but also by the divergence between mean and median values).
[Insert Table 1 about here]
Results are mitigated and exhibit heterogeneous stock returns’ behaviors. In particular,
stock returns exhibit non-negligible skewness and kurtosis patterns on a daily basis.
3.2. Data transformation
In order to run a time series analysis based on Sharpe ratios, we operate a simple data
transformation. Such transformation conforms to the financial practice while assessing
investment or fund performance. In this light, we compute the monthly Sharpe ratios as well
as corresponding average Fama and French (1993) factors and VIX levels. Monthly data are
computed from daily data based on non-overlapping periods of 20 working days (i.e.
approximately one month). Hence, our daily sample horizon comprises 170 monthly data, the
first month (January 2000) encompassing 18 working days. In unreported results, we find that
monthly SR series are Gaussian for all stock returns under consideration. Moreover, they are
stationary, behave globally like white noises, and exhibit at least no first order
autocorrelation.
As an extension, Table 2 displays the properties of Kendall tau b correlations between
monthly Sharpe ratios and the four explanatory factors, namely the three Fama and French
factors (market premium, SMB, HML) and the VIX.
[Insert Table 2 about here]
9 It comes from the difference between small-cap-specific portfolios and large-cap-specific portfolios (i.e. size
indicators).
10 It results from the difference between value stocks’ portfolios and growth stocks’ portfolios (i.e. book-to-
market indicators).
11 See http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.
12 Unreported Jarque Bera statistics confirm generally such a stylized fact (Jarque and Bera 1980, 1981, 1987).
Indeed, the Jarque Bera statistics of our 85 stocks range from 371.4619 to 1473438, which invalidate the
Gaussian distribution assumption at a 5% test level. Moreover, returns exhibit high excess kurtosis levels.
9
The market premium is non-negligibly linked with stocks’ monthly Sharpe ratios whereas the
linkages between remaining explanatory factors and monthly SRs are weaker. However, the
Kendall correlations of market premium and VIX with monthly SRs exhibit the lowest
coefficient of variations. Hence, former correlation coefficients display a low dispersion,
which may suggest a more homogeneous link between monthly SRs and both the market
premium and VIX factors. As a result, the market premium and VIX are the most relevant
factors to describe stocks’ monthly Sharpe Ratios as compared to the influence of SMB and
HML factors, the latter influence being mitigated across stocks on a monthly basis.
4. Econometric results and robustness check
4.1. Estimations
The model is estimated separately for each of the 85 stock return series under
consideration. Model parameters are estimated along with the log-likelihood maximization
principle under the Gaussian joint multivariate distribution of state and dynamic errors.
Hence, we obtain 85 series of FSRs, which are computed over 170 months. Moreover,
unreported Cramer-Von-Mises statistics confirm the Gaussian behavior of monthly SRs and
FSRs series for the 85 stock returns under consideration at a five percent test level (Cramer
1928; von Mises 1931).13
Given the Gaussian nature of obtained FSRs, we can consider the median FSR as a good
performance proxy since it is an appropriate location indicator. Moreover, the Cramer-von
Mises statistic of median FSR equals 0.0178, which validates its Gaussian behavior at a five
percent test level. Differences between Sharpe ratios and their median fundamental
counterparts result from market trend and volatility biases as well as idiosyncrasies. Such
discrepancies highlight overestimation or under-estimation issues while assessing stock
performance with classic Sharpe ratios. Conversely, any coincidence between SR and FSR
indicates the goodness of Sharpe ratio as a performance measure in the light of possible bias
factors. In such situation, either bias factors may not exist or the trade-off in between such
factors eliminates potential biases. Moreover, Kendall’s tau b correlation between SRs and
median FSRs equals 0.2443 and is significant at a five percent test level. Hence,
commonalities in SR- and FSR-based performance analyses arise in 24.43 percent of cases.
However, such degree of commonality is still low. Investors can thus question the impact of
SR and FSR differences on both related stock selection processes and resulting portfolios
performance.
Although results highlight the relevance of the employed Kalman filter model, the next step
consists of investigating model’s robustness. A robustness check is necessary so as to ensure a
non-spurious relationship (i.e. ensuring an appropriate description of the phenomenon under
consideration).
4.2. Robustness investigation
Checking for regression residuals’ behavior, Fig. 1 plots Cramer-Von-Mises statistics as
well as their five percent critical threshold. Cramer-Von-Mises normality test is validated as
long as corresponding statistics avoid being higher than the critical threshold. Hence,
13 The critical value of the Cramer-Von-Mises statistic is 0.2200 for a sample size of 100 and a 5% test level.
10
estimated Cramer-Von-Mises statistics show the Gaussian behavior of dynamic (ui) and state
(vi) errors at a 5% test level. Thus, dynamic and state errors conform to the theoretical
Gaussian assumption, which supports Gaussian SRs and FRSs.
[Insert Fig. 1 about here]
As a complementary diagnostic test, unreported Ljung-Box statistics (Ljung and Box 1978)
underline generally the white noise property of dynamic and state errors. For example, the
Ljung-Box statistics of order 1 range from 0.0013 to 3.6268, and from 0.0000 to 0.2301 for
the dynamic and state errors respectively, and are below the 3.8415 critical value at a five
percent test level. The same remark also holds for squared dynamic and state errors whose
first order Ljung-Box statistics range from 0.0041 to 3.8051 and from 0.0011 to 3.7269
respectively. Hence, model residuals exhibit no heteroskedastic pattern (i.e. time-varying
variance) as well as independency, which strengthen the robustness of our estimation method.
Moreover, an unreported Phillips-Perron unit root test without trend and without constant
term (Phillips and Perron 1988) supports stationary model residuals at a five percent level. As
a conclusion, the Gaussian white noise assumption about dynamic and state errors is accepted.
Our previous study yields thus interesting results while providing a possible description of
Sharpe ratio’s biases. Moreover, the proposed filtering methodology is robust and in line with
previous findings. The obtained fundamental Sharpe ratios can help building powerful
performance assessment tools in a very simple framework.
5. Comparative study
We look for comparing our FSR with SR and existing RAPMs in the light of current
literature review. We split the RAPMs into two distinct groups. The first group relies on
normality assumptions about stock returns such as Sortino ratio whereas the second group
relies on scale-independent RAPMs such as Omega, Kappa and Upside potential ratios.
5.1. Considering 6 other RAPMs
The first group of RAPMs is the Sortino ratio (Sortino 1991, 2004; Sortino and Price
1994; Sortino and Forsey 1996). Sortino ratio corresponds to the ratio of the excess return
(relative to the risk free rate) to the standard deviation of losses as represented by the
downside risk. The downside risk is defined as the lower partial moment of order one over the
investment horizon. In particular, Fishburn (1977) defines the lower (LPM) and upper (UPM)
partial moments of order n as follows:
UPM(i,r,n) = E[Max(Ri-r,0)n] (4)
LPM(i,r,n) = E[Max(r- Ri,0)n]= (-1)n E[Min(Ri-r,0)n] (5)
where E[.] is the expectation operator,14 Ri is the return on stock i, n is the order of the
moment, and r is the minimum targeted return of the investor. We’ll set this latter variable to
the risk free rate Rf for comparability reasons relative to SR and FSR. Hence, the Sortino ratio
writes:
14 In the simplest situation, it is simply the arithmetic mean of observed values over our investment period.
11
 
 
2,,riLPM
rRE
Sortino i
i
(6)
Within the second group of RAPMs, the Omega ratio is the ratio of all gains to all losses over
the investment horizon. In other words, it is the ratio of positive performance to negative
performance in absolute terms (see Keating and Shadwick 2002). Considering the higher
moments of the return distribution, it then writes:
 
 
1,,
1,,
riLPM
riUPM
Omegai
(7)
In the same line, the Omega-Sharpe ratio corresponds to the Omega ratio minus 1, and
illustrates a modified risk measure (Bacon 2008; Bernardo and Ledoit 2000; Kazemi et al.
2004). The risk measure in use focuses on all the losses over the investment horizon so that
Omega-Sharpe ratio writes:
 
 
1,,
1Sharpe-Omega riLPM
rRE
Omega i
ii
(8)
Incidentally, Omega and Omega-Sharpe yield the same investment rankings. Therefore, their
respective performance classifications should coincide. As a modification of Sortino ratio, the
Kappa ratio accounts for higher moments (Kaplan and Knowles 2004) and focuses on losses
over the investment horizon. It is the ratio of the excess return to a given lower partial
moment and writes:
 
 
 
 
 
n
i
n
i
in nriLPM
rRE
nriLPM
rRE
Kappa /1
,,,
,,
(9)
When the targeted minimum return is the risk free rate, Omega-Sharpe and Sortino ratios are
particular cases of Kappa ratio as follows:
Kappa1,i = Omega-Sharpei (10)
Kappa2,i = Sortinoi (11)
Caring about order 3 and 4 moments, we consider additionally Kappa3 and Kappa4 ratios. As
a last performance measure, the Upside potential ratio (UPR) considers finally the ratio of all
gains to the downside risk over the investment horizon (Sortino et al. 1999) and writes:
 
 
2,,
1,,
riLPM
riUPM
UPRi
(12)
We handle therefore a range of various risk-adjusted performance measures or RAPMs, which
will be compared to our FSR as well as its classic SR counterpart.
5.2. Empirical results
As regards Sharpe ratios, Fig. 2 ranks classic Sharpe ratios by ascending order and plots
ordered classic Sharpe ratios against their fundamental counterparts. As a rough guide, a
regression line is plotted so as to check for a one-to-one correspondence between classic and
12
fundamental Sharpe ratio-based rankings. Obviously, the reported couples of classic and
median fundamental Sharpe ratios do not follow the regression line (or at least gather
homogenously around it) but rather split widely around. Such a plot distribution highlights the
wide ranking heterogeneity arising from classic and median fundamental Sharpe ratios.
Switching from classic to median fundamental Sharpe ratios does not preserve performance
ranks. Such discrepancies highlight the impact of previously reported market climate, time
variation and idiosyncratic biases. As a result, accounting for structural biases modifies non-
negligibly performance assessment, which should impact related investment selection and
resulting portfolios’ performance to a large extent. For example, ignoring market trend or
volatility regime generates misestimation in SR performance measure and corresponding
ranking. In this light, FSR proposes a correction for previous structural biases and therefore a
bias-free performance ranking.
[Insert Fig. 2 about here]
As regards the 6 previous RAPMs, we rank corresponding stocks accordingly and propose
then a comparative study in two steps. First, we investigate graphically the rank
commonalities of those 6 RAPMs with our median FSR estimates as well as corresponding
SRs. Then, we compare such rankings while testing for rank similarity and stability.
As a preliminary analysis, the panel (a) of Fig. 3 plots our 6 other RAPMs' ranks
relative to SR ranks (i.e. excluding FSR ranks). A clear correlation appears for the 5 RAPMs
above-mentioned since their respective relationships with SR ranks are almost perfectly
linear. Conversely, a noticeable discordance between SR ranks and the ranks inferred from the
Upside potential ratio is confirmed through the less linear correspondence between the two
types of ranks. The significance of such discordance will be investigated in a forthcoming
step. Panel (b) plots Upside potential ratio-based ranks against FSR-based ranks to check for
rank commonalities. Unfortunately, no link does appear.
[Insert Fig. 3 about here]
As a conclusion, the obtained median FSR yields a performance classification, which is
totally different from the RAPMs under consideration. Moreover, the other RAPMs’
performance classification tracks well the performance ranking of SR to some extent. The
latter result is indeed confirmed by a signed rank Wilcoxon test, which is displayed in Table 3
(see Bennett 1965; McCornack 1965; Wilcoxon 1945, 1947, 1949). In order to check for the
latter result, we performed a test of rank stability in between RAPMs’ rankings and SR’s
benchmark ranking. The null hypothesis under consideration states that the median difference
between the benchmark ranking and each ranking under consideration is zero. If the null
assumption is confirmed, then SR and other RAPMs yield the same ranking on average. The
first part of Table 3 displays corresponding results for a two-tailed signed Wilcoxon test based
on paired samples (i.e. performance rankings are linked to some extent because RAPMs deal
with the same stock returns). Of course, reported results confirm the acceptance of the null
assumption at a five percent level for the two-tailed test. Apart from FSR, previous RAPMs
yield therefore strongly correlated stock picking and performance-based investment strategies.
[Insert Table 3 about here]
The second part of Table 3 reports Kendall correlation coefficients between SR ranks and
other RAPM ranks, on one side, and between FSR ranks and other RAPM ranks, on the other
13
side.15 The reported statistic consists of Kendall’s tau b, which is significant in the SR case
and generally insignificant in the median FSR case at a five percent test level. Observed high
levels of Kendall’s tau highlight the ranking consistency and similarity between SR and other
RAPMs. Conversely, Kendall’s tau metric clearly underlines discrepancies between FSR
ranking and other RAPM rankings. Moreover, the previous feature also applies to the
comparison between SR and median FSR rankings since the observed significant correlation
level between SR ranks and FSR ranks is 0.2443 only. However, such a metric mitigates the
relationship between the rankings of both SR and the Upside potential ratio. Previous
mitigation probably results from the presence of a few non-negligible “outliers” or extreme
returns (e.g. extreme gains and/or losses). Such returns have a significant impact on the
Upside potential ratio since it balances stock returns’ upper tail (i.e. gains) with corresponding
lower tail (i.e. losses) on an absolute value basis (e.g. magnitude and significance of
distribution tails). Under UPR setting, gains are simply penalized by losses. Hence, stock
returns exhibiting a strong risk asymmetry will exhibit higher UPRs when, for example,
returns are more often over-performing than underperforming the benchmark return (i.e. right-
skewed returns with a fatter right tail). Conversely, stock returns will exhibit lower UPRs
when they are more often underperforming than over-performing the benchmark return (i.e.
left-skewed returns with a fatter left tail, or equivalently, positive excess kurtosis). As a result,
investors favor stock returns with a high upside potential and a low downside risk so that they
exhibit a stronger risk-aversion under the UPR-driven selection process (as compared to a SR-
driven selection process, Bacon 2008). Finally, UPR and SR differ because UPR handles risk
asymmetry in stock returns whereas SR assumes risk symmetry, yielding then erroneous stock
picking in the presence of highly skewed and fat tailed stock returns. Moreover, previous
results confirm the findings of Eling and Schuhmacher (2007) according to which the choice
of a performance measure has no impact on the performance ranking (of hedge funds).
Apparently, this result applies to risk-measures, which are founded on excess risk premia
relative to the risk free rate of interest (i.e. nature or structure of risk measure based on
normalized excess returns).
6. Efficiency and stock picking ability of FSR
We assess the efficiency of median fundamental Sharpe ratio (FSR) as compared to other
RAPMs such as Sharpe ratio. Incidentally, we investigate the stock picking and portfolio
performance implications of FSR and remaining RAPMs. In this light, a value-at-risk analysis
is proposed under various risk scenarios as a comparative study. Such analysis confirms the
relevance of FSR while backtesting FSR-based and competing performance-based investment
strategies.
6.1. Efficiency of performance measures
As a final investigation, we focus on the efficiency of FSR performance measure relative
to the other risk-adjusted performance measures under consideration. In this light, the
performance measures under consideration are envisioned as performance estimators. In
statistics, the quality of an estimator and its efficiency in particular, is assessed through its
variance. The more efficient the estimator is, the lower its variance should be. When the
15 Results remain the same when we compute the Kendall correlations of performance measures instead of their
respective rankings.
14
variance of the estimator is low, the accuracy of such an estimator is therefore high (Kennedy
1998). In particular, accuracy refers to a reduced estimation bias, or equivalently, a lowered
valuation error. For this purpose, Table 4 proposes the descriptive statistics relative to all the
performance measures under consideration, namely the SR, median FSR, Sortino, Omega,
Omega-Sharpe, Kappa 3, Kappa 4 and Upside potential ratios. Strikingly, the median FSR
exhibits the lowest standard deviation as compared to SR and the 6 other RAPM performance
measures. Hence, FSR is a more accurate performance measure as compared to SR and other
RAPMs.
[Insert Table 4 about here]
6.2. Scenario analysis and value-at-Risk
For risk analysis prospects, we consider equally weighted portfolios composed of the 30
top-stock group and the 30 bottom-stock group. The portfolios under consideration are
composed of the 30 best performing and the 30 worst performing stocks (i.e. over-
performing/winning stocks versus underperforming stocks) in accordance with RAPMs such
as the FSR, Sharpe ratio (SR), Omega ratio and Upside potential ratio (UPR).16 The 30
top/bottom-stock portfolios differ from each other due to the benchmark performance
measure, which is employed to select the 30 best/worst performing stocks. We label such
portfolios the top/bottom (median) FSR, top/bottom SR, top/bottom Omega and top/bottom
UPR portfolios respectively.
The risk analysis focuses on the market risk exposure of previous stock portfolios, which
is measured with the Value-at-Risk (VaR) (Alexander, 2009; Dowd and Blake, 2006;
Gourieroux and Jasiak, 2010). The VaR measures downside risk while providing investors
with the worst possible loss at a given confidence level (i.e. under a specified risk scenario).
In this light, we apply a four-step methodology in line with Jondeau and Rockinger (2006),
Kuester and Mittnik (2006), McNeil and Frey (2000), McNeil et al. (2005), Nyström and
Skoglund (2005), Rockafellar and Uryasev (2002), Poon et al. (2004). More specifically,
previous portfolios’ VaRs are computed while combining Generalized Autoregressive
Conditional Heteroskedastic (GARCH) modeling, copula models, extreme value theory
(EVT) and Monte Carlo simulations (i.e. GARCH-EVT-Copula model, see Appendix). Based
on GARCH modeling, the first step captures the time-varying volatility of stock returns and
therefore portfolio returns. In accordance with EVT, the second step then handles stock
returns’ distributional asymmetries through the Generalized Pareto Distribution (GPD). As an
extension and third step, the copula approach describes the joint dependence structure of
constituting stocks within each portfolio under consideration. Finally, the fourth step use
previous results, namely historical daily data’s properties, to simulate stock returns and
therefore portfolios’ returns over a specified forecast horizon (e.g. one week, one month).
Such simulation framework captures the time-varying nature of returns’ volatility as well as
the existing correlation between stock returns (i.e. correlation between underlying market risk
16 Given that SR is strongly correlated with Sortino, Omega, Omega-Sharpe, Kappa 3 and Kappa 4 ratios, we
assimilate those five latter performance measures to SR performance measure. Hence, we disentangle three
performance metrics corresponding to SR, median FSR and UPR.
15
factors). Then, VaR is inferred as the appropriate quantile of the probability distribution of
portfolios’ returns, which are rebuilt over the target horizon.
Based on 10 000 trials, Table 5 displays the maximum gain and loss while Table 6 displays
the one-week and one-month VaR forecasts for various risk levels.
[Insert Table 5 about here]
Comparing the best performing portfolios selected in accordance with FSR, SR, Omega and
UPR RAPMs, the top FSR portfolio exhibits the lowest loss potential over one-week and one-
month horizons and the highest gain potential over one-month horizon (see Table 5).
Comparing the worst performing portfolios selected in accordance with FSR, SR, Omega and
UPR RAPMs, the bottom FSR portfolio exhibits the highest gain potential over one month
and lowest loss potential over one week. Over the one-month horizon, bottom portfolios’
results are mitigated. However, the FSR yields consistent results while building performing
portfolios composed of winning stocks. Such performing portfolios should exhibit the lowest
loss and highest gain potentials over the chosen target horizons.
[Insert Table 6 about here]
With respect to VaR forecasts in Table 6, the top FSR portfolio exhibits the highest one-
month VaR forecasts (i.e. highest negative quantile values, or equivalently, lowest loss
forecast or lowest absolute VaR) at the 5% and 1% risk levels (i.e. at a 95% and 99%
confidence levels). Moreover, the bottom FSR portfolio exhibits the smallest one-week VaR
(i.e. largest/strongest loss forecast) at all risk levels while it exhibits the smallest one-month
VaR at 5% and 1% risk levels. Again, FSR yields generally consistent results for performing
portfolios, which are composed of 30 winning stocks as compared to SR-, Omega- and UPR-
based performing portfolios. Indeed, the top FSR portfolio, or equivalently the FSR-based
performing portfolio exhibits higher VaR levels, which translate into lower loss forecasts.
Previous features ensure the appropriateness of FSR as a sound performance measure all the
more that obtained FSR conforms to model assumptions. At the level of performing/top stock
portfolios, simulation results support FSR as a stock selection and risk management tool.
According to the profit and loss analysis, investors who rely on FSRs face a lesser degree of
downside risk and higher upside potential. Such feature supports the preference of rational
investors. Furthermore, the VaR study shows that FSR-based top portfolios face a lower risk
exposure/a lower loss risk than SR-based top portfolios. Given that investors target their
gains’ maximization, they have interest in choosing FRS (rather than SR) for selecting stocks
so that they access better gain possibilities with reduced loss risk and risk exposure.
7. Summary and conclusion
In this article, we considered a risk-adjusted performance measure, which benefits from a
large success among the portfolio management community. Namely, Sharpe ratio considers
the ratio of a given stock’s excess return to its corresponding standard deviation. However,
such metric is relevant in a stable setting such as a Gaussian world. Unfortunately, Gaussian
features are scarce in the real world, and Sharpe performance measure suffers from various
biases. Such biases arise from returns’ departure from normality, which often illustrates the
non-negligible weights of large and/or extreme return values. Moreover, Sharpe ratios exhibit
an upward bias during downward market trends and a downward bias during upward market
16
trends with respect to poorly diversified portfolios. Finally, Sharpe ratios also exhibit time-
variation resulting from business cycle and volatility regimes among others.
To correct for potential biases, we apply a robust filtering method based on Kalman
estimation technique. The Kalman approach helps extract fundamental Sharpe ratios from
observed classic Sharpe ratios. Obtained fundamental Sharpe ratios are free of bias and
exhibit a pure performance indicator. Indeed, removing market/systematic biases yields free-
of-bias performance ratios, which are immediately comparable. Such ratios help therefore
rank fairly investments on a pure performance basis because they belong to the same
measurement scale. Corresponding results are interesting with regard to two findings. First,
fundamental Sharpe ratios are obtained after removing directly the market trend and volatility
impact. Second, fundamental Sharpe ratios exhibit a cyclical pattern in line with listed
cyclical and oscillatory patterns of financial markets (Mishchenko 2014; Tang and Whitelaw
2011; Woehrmann et al. 2005).
Our comparative study exhibited an obvious discordance between FSR performance
classification and a set of well-known RAPMs’ performance rankings. Conversely, SR
performance classification on one side, and Sortino, Omega-type, Kappa-type, and Upside
potential ratios’ performance rankings on the other side exhibited a non-negligible correlation.
Hence, the question about the impact of a RAPM choice on its corresponding performance
ranking is still pending. The answer to such question probably depends on both the nature of
the applied measure and the significance of reported biases. However, a simple robustness
check highlights the consistency, effectiveness and efficiency of FSR in performance
assessment. FSR is indeed a more accurate performance estimator than other RAPMs.
Moreover, FSR-based winning portfolios offer lowest expected losses and reduced worst-case
losses (i.e. VaR) over one-week and one-month forecast horizons as compared to other
RAPM-based winning portfolios. The former portfolios offer rather highest expected gains
over a one-month forecast horizon. Consequently, it is possible to extract reliable
performance indicators, which are of primary importance for asset selection and performance
ranking. Such concern is of huge significance to asset allocation policy, performance forecasts
and cost of capital assessment, which are driven by performance indicators among others
(Farinelli et al. 2008; Lien 2002; Christensen and Platen 2007).
Acknowledgements
We thank participants at the AFBC conference (Sydney, Australia, December 2009), ISCEF
conference (Sousse, Tunisia, February 2010), and SWFA annual conference (Houston,
U.S.A., March 2011) whose questions helped improve the quality of this paper. We are also
grateful to two anonymous referees. The usual disclaimer applies.
Appendix A. Describing the GARCH-EVT-Copula approach
We describe the four steps building such simulation analysis, which yields VaR
computations.
Step 1: Data filtering
17
The first step consists of capturing the time-varying volatility of stock returns and therefore
portfolio returns. For comparability prospects and complying with statistical assumptions, we
filter the constitutive stock returns with a Threshold Generalized Autoregressive Conditional
Heteroskedastic (TGARCH) model (Glosten et al., 1993). In line with Hansen and Lunde
(2005), Nyström and Skoglund (2005) as well as Ashley and Patterson (2010), the applied
representation combines GARCH and ARCH effects of order 1 with a threshold effect of
order 1 so that a TGARCH(1,1,1) model is employed for the conditional variance while the
mean equation satisfies an autoregressive dynamic of order 1 or AR(1). Hence, we consider
an AR(1)-TGARCH(1,1,1) representation (see Appendix for explanatory details and
justifications; Ling and McAleer, 2003). The standardized residuals in the mean equation are
assumed to follow a Student t probability distribution, which conforms to the fat-tailed profile
of stock returns. Such asymmetric behavior underlines the existence of frequent extreme
return levels (i.e. fat tails in the probability distribution), which contradicts a Gaussian return
behavior. Indeed, a Gaussian behavior assumes extreme return levels to be rare (i.e. thin tails)
so that most of observed return values lie around the distribution’s average level. Such
GARCH representation is estimated for each of the 30 constitutive stocks within a given
portfolio while applying the Maximum Likelihood Estimation (MLE) method (i.e. 240
GARCH-type models are estimated for the 240 stocks constituting the 8 portfolios under
consideration). Moreover, the GARCH analysis disentangles a pre-, during- and post-crisis
period over the sample horizon. Hence, estimations are obtained over three different volatility
regimes of the stock market.
Step 2: Estimating the distribution of stock returns
Given returns’ distributional asymmetries and in line with EVT, we combine a Gaussian
kernel estimation method with a Generalized Pareto Distribution (GPD) to estimate the
cumulative distribution function of standardized residual series. The Gaussian kernel
methodology captures the most frequent behavior of stock returns while the GPD focuses on
their lower and upper tail behaviors. In particular, the GPD is calibrated to focus on the 10%
extreme residuals belonging to the tails while the Gaussian kernel describes the empirical
distribution of the remaining 90% of sample residuals. Namely, we consider extreme quantile
levels so that 10% of the residuals lie beyond those extreme thresholds. Hence, the GPD
describes that part of the residuals, which lie beyond the quantile thresholds (i.e. it describes
the distribution of exceedances/peaks over thresholds; Davison and Smith, 1990; Embrechts
et al., 1997; Smith, 1984). The GPD estimation process relies on MLE methodology. Thus,
the frequent behavior and possible extreme quantiles of standardized residuals are both
handled (see Appendix). Particularly, the interest of the GPD with respect to risk management
relies on risk extrapolation perspectives (e.g. scenario analysis) because quantiles can be
extrapolated to higher confidence level (i.e. stronger risk scenarios).
Step 3: Capturing the joint dependence structure of stock returns within a portfolio
18
We estimate the joint dependence structure of the 30 stock components within each best/worst
performing portfolio. Given stock returns’ tail fatness and corresponding Student t
distribution of standardized residuals, a multivariate Student t copula is selected and estimated
with the canonical MLE (CMLE) method to describe joint dependencies within stock
portfolios (Cherubini et al., 2004; Nelsen, 1999). In particular, the Student copula is calibrated
after transforming the standardized residuals with their respective empirical cumulative
distributions function (CDF). In other words, standardized residuals are transformed while
applying the CDF to them so that we obtain corresponding values, which lie between 0 and 1.
Then, the multivariate Student copula is calibrated to the 30 transformed standardized
residuals series with the MLE method for each portfolio under consideration. The estimation
process yields estimates of the degree of freedom and the correlation matrix, which are the
parameters of the Student copula. In this light, the Student copula captures the correlation
structure of standardized residuals (i.e. assessing risk dependencies within stock portfolios).
Step 4: Simulating portfolio returns and forecasting VaR
We simulate equally weighted portfolios composed of the 30 best/worst performing stocks
according to FSR, SR and Omega RAPMs (i.e. three performance measures and then 8 stock
portfolios among which 3 best and 3 worst performing ones). In this light, we propose three
stages. As a first stage, we simulate multivariate Student copula values based on previous
parameter estimates. Hence, we obtain random variates from the Student copula, which
correspond to the transformation of standardized residual series. Then, we invert the empirical
CDF of standardized residuals from previous random variates (i.e. from simulated copula
values) so as to obtain corresponding estimates of the standardized residual series, which
describe the 30 constitutive stocks within each portfolio. The 30 standardized residual series,
which are obtained, are independent and identically distributed (iid) processes, which also
exhibit joint dependencies (i.e. correlated time series). The simulation procedure runs over a
time horizon of 5 (i.e. one week of forecasts) and 22 (i.e. one month of forecasts) working
days from the end of the sample horizon. Over such time windows, each time series under
consideration is simulated 10 000 times, or equivalently, Monte Carlo simulations rely on a
number of trials equal to 10 000. Hence, each standardized residual series is
simulated/forecasted 10 000 times within a given portfolio over one week and over one month
respectively (i.e. simulating 10 000 times a set of 180 residual series). As a second stage, the
corresponding simulated returns are obtained while applying the GARCH representation,
which was initially estimated and calibrated to empirical data. Incidentally, such GARCH
simulation employs the last available value of previously estimated conditional variance as
well as simulated residual series. As a third and final stage, we build equally weighted
portfolios of 30 best and worst performing stocks and compute the corresponding logarithmic
as well as cumulative logarithmic returns for each trial (i.e. the return series of each portfolio
is simulated 10 000 times over a given forecast window). Recall that any portfolio returns
result from the aggregation of the simulated returns of its stock components. Thus, the 10 000
trials describe the distribution of each portfolio’s returns and cumulative returns from which
relevant statistics can be inferred for risk management prospects. Recall that the Value-at-
Risk is computed as follows for a given portfolio’s cumulative return Rp:
19
 
 
 
 
1
min Pr
min
P
P
P P R P
R
VaR R Q R R F R R R
R R F

 
 
(A. 1)
where
 
.
P
R
F
is the CDF of the portfolio’s return Rp and
 
1.
P
R
F
is its inverse function, Pr(.) is
the probability operator,
is the risk level (so that 1-
is the confidence level) and
 
P
QR
is
simply the cumulative return’s quantile for risk level
over the chosen investment horizon.
As a result, we have:
 
 
Pr PP
R VaR R

(A. 2)
Hence, the VaR informs the investor about possible thresholds of extreme loss risk over a
forthcoming investment horizon. In other words, the worst possible portfolio’s loss is
 
P
VaR R
in (1 -
) percent of cases. Equivalently, there is a
percent probability (i.e. risk
level, or equivalently, risk scenario) that the portfolio’s loss exceeds the
 
P
VaR R
level (i.e.
extreme negative return scenarios, or risk of VaR violations) over the target horizon.
Appendix B. Diagnosing stock returns
In present appendixes, we provide insightful explanations and details about the model and
methodology (i.e. the GARCH-EVT-Copula model) employed to compute the VaR of
portfolios’ cumulative returns over one week and one month forecast horizons. As an
example, we first plot the autocorrelation function of both the returns and squared returns of a
given sample stock.
[Insert Fig. B.1 about here]
Fig. B.1 exhibits a significant first order autocorrelation in stock returns while squared returns
exhibit stronger dependency over time (i.e. higher order autocorrelation). Hence, the first
order autocorrelation of stock returns (i.e. the AR(1) dynamic) is captured in the GARCH
model while adding a first order autoregressive or AR(1) component in the mean equation.
Moreover, the dependency of squared residuals illustrates the time-dependency in stock
returns’ variance (i.e. heteroskedasticity, which means that returns’ variance depends on
time), which supports the use of a GARCH representation. Most of the 180 stocks composing
the 3 top and 3 bottom performing portfolios under consideration exhibit such standard and
well-known behavior.
Appendix C. GARCH representation
The Generalized Autoregressive Conditional Heteroskedastic (GARCH) methodology
specifies simultaneously one mean and one variance equations, which describe the conditional
mean and the conditional variance of stock returns. Assume that Rt is the return of a given
stock at time t [1,1154] where T=1154 is the end of the investment horizon (i.e. sample
size). Then, the autoregressive asymmetric GARCH representation AR(1)-TGARCH(1,1,1)
writes:
20
1t t t
RR
 
 
(C. 3)
 
2
2 2 2
1 1 1t t t t
   
 
 
(C. 4)
where
t is the regression error/residual,
t
- =
t when residual
t < 0 and zero else,
t² is the
conditional variance of regression residuals, and the other parameters (
,
) are simply a
constant and a factor loading (i.e. AR(1) term) in the mean equation. To account for tail
heaviness in stock market returns, the standardized residuals (zt), so that
t =
t² zt for t
between 1 and T, are assumed to follow a Student t distribution and are independent and
identically distributed by assumption. With respect to the variance equation,
illustrates the
long-term average variance while (
t-1²), (
t-1²) and (
t-1
- are respectively called the
ARCH(1), GARCH(1) and Threshold(1) terms with their respective factor loadings
,
and
. In particular, (
t-1²) emphasizes the greater impact of negative residuals (i.e. bad news) on
stock returns as compared to positive residuals’ impact (i.e. good news). Stock returns’
volatility simply correspond to the square root of their respective variance, namely
t.
Moreover, the GARCH representation is estimated with the MLE methodology. As an
example, Fig. C.1 plots the filtered residuals (
t) as well as their corresponding conditional
volatility (
t) for a given stock over time (i.e. by observation number).
[Insert Fig. C.1 about here]
However, the time window under consideration encompasses the global financial crisis
following the subprime mortgage market crash in August 2007. In this light, we split the
sample into three sub-periods for the GARCH analysis according to the structural break dates
introduced by Breitenfellner and Wagner (2012). Hence, we consider a pre-crisis, a crisis and
a post-crisis period, which span from 2000/01/04 to 2007/07/02, from 2007/07/03 to
2009/05/01, and from 2009/05/02 to 2014/04/30 respectively. As a result, we consider three
volatility regimes over the sample period, which are delimited by the stock market’s structural
changes (see Fig. C.2).
[Insert Fig. C.2 about here]
As a diagnostic of standardized residuals, Fig. C.3 displays the autocorrelation function of
both standardized residuals (zt) and squared standardized residuals (zt²) to check for their
required white noise property. As expected, standardized residuals are independent and
exhibit homoskedasticity (i.e. constant variance over time).
[Insert Fig. C.3 about here]
Accounting for the stock market’s breaks and their corresponding volatility regimes reduces
the conditional volatility, and strengthens the residuals’ robustness.
Appendix D. Extreme value theory and GPD
The Generalized Pareto Distribution (GPD) is used to describe return values, which
exceed a given threshold such as a specified quantile level for example. It is designed to
characterize the behavior of returns’ distribution tails while modeling stock returns’
exceedances (e.g. the difference between stock returns and a specified threshold such as the
21
corresponding 10%, 5% or 1% quantiles). Under such setting, the cumulative distribution
function (CDF) of the GPD writes as follows for any exceedance y > 0:
 
1
1 1 0
1 exp 0
yif
Fy
yif

 



 


(D. 1)
where
> 0 is the scale parameter, and the shape parameter
defines the tail fatness. When
is negative, the distribution exhibits thin tails while the distribution exhibits fat tails when
is
positive. Distribution parameters are estimate with the MLE method.
Hence, the GPD is mixed with a Gaussian kernel method to estimate the empirical probability
distribution of stock returns’ standardized residuals. The Gaussian kernel illustrates 90% of
frequently observed standardized residuals while the GPD describes the remaining 10% of
observed standardized residuals (i.e. standardized residuals’ lower and upper tail behaviors).
As an example, Fig. D.1 plots the empirical CDF of a given stock return’s standardized
residuals while Fig. D.2 proposes a graphical assessment of the quality of the GPD fit for
upper tail exceedances of standardized residuals.
[Insert Fig. D.1 about here]
[Insert Fig. D.2 about here]
Appendix E. The Student t copula
The Student t copula illustrates the dependence structure of the standardized residuals
peculiar to the stock components of each portfolio under consideration. Given that each
portfolio encompasses 30 stocks, the copula under consideration has a dimension of 30 (i.e.
multivariate case) and captures the tail fatness of stock returns.
Let
be a correlation matrix,
a degree of freedom and u1,, u30 in [0,1], the Student t
copula density writes:
22
 
30
30 2
1
13 11
30
22
30 2
1
30 1
1
22
1
, , ; ,
11
22
t
i
i
c u u
  


 
 
 

 
 


 
 
 
 
 
 
 
(E. 1)
where
and
-1 are a thirty-dimension matrix and its inverse respectively, |
| is the
determinant of the correlation matrix,

is the Gamma function,
is the vector (
1, ,
30) of
the inverse univariate Student17 cumulative distribution function, which applies to each
element u1,, u30, and finally
t is the transposed vector of
.
As an example based on the top FSR portfolio, we have = 25.3973 and the estimated
correlation matrix
for the 30 stock components is:
1.00 0.36 0.25 0.33 0.35 0.36 0.21 0.22 0.22 0.30 0.20 0.30 0.28 0.25 0.30 0.26 0.39 0.19 0.35 0.30 0.27 0.20 0.24 0.31 0.30 0.33 0.22 0.26 0.24 0.23
0.36 1.00 0.37 0.39 0.45 0.33 0.26 0.36 0.33 0.33 0.25 0.40 0.36 0.28 0.43 0.41 0.50 0.27 0.40 0.36 0.35 0.21 0.32 0.42 0.41 0.42 0.27 0.42 0.39 0.26
0.25 0.37 1.00 0.30 0.33 0.24 0.18 0.34 0.33 0.24 0.20 0.27 0.26 0.22 0.33 0.38 0.35 0.32 0.33 0.26 0.22 0.18 0.24 0.31 0.29 0.28 0.23 0.40 0.34 0.22
0.33 0.39 0.30 1.00 0.38 0.37 0.24 0.27 0.28 0.29 0.21 0.31 0.31 0.25 0.34 0.30 0.42 0.24 0.37 0.33 0.29 0.19 0.27 0.31 0.31 0.35 0.23 0.30 0.27 0.22
0.35 0.45 0.33 0.38 1.00 0.37 0.24 0.28 0.28 0.37 0.23 0.34 0.34 0.29 0.38 0.33 0.48 0.28 0.40 0.39 0.35 0.21 0.29 0.36 0.34 0.37 0.26 0.34 0.31 0.29
0.36 0.33 0.24 0.37 0.37 1.00 0.24 0.24 0.23 0.29 0.24 0.32 0.26 0.24 0.31 0.26 0.43 0.22 0.43 0.31 0.32 0.20 0.27 0.29 0.28 0.33 0.22 0.27 0.25 0.23
0.21 0.26 0.18 0.24 0.24 0.24 1.00 0.16 0.17 0.20 0.20 0.25 0.21 0.20 0.23 0.20 0.29 0.17 0.27 0.22 0.22 0.19 0.22 0.25 0.25 0.26 0.16 0.23 0.22 0.22
0.22 0.36 0.34 0.27 0.28 0.24 0.16 1.00 0.58 0.22 0.25 0.25 0.29 0.22 0.32 0.37 0.33 0.31 0.29 0.26 0.20 0.24 0.30 0.34 0.34 0.28 0.21 0.41 0.38 0.21
0.22 0.33 0.33 0.28 0.28 0.23 0.17 0.58 1.00 0.23 0.27 0.25 0.26 0.21 0.32 0.33 0.30 0.29 0.28 0.26 0.21 0.23 0.31 0.35 0.35 0.29 0.19 0.40 0.38 0.22
0.30 0.33 0.24 0.29 0.37 0.29 0.20 0.22 0.23 1.00 0.26 0.32 0.26 0.27 0.31 0.29 0.37 0.20 0.33 0.37 0.38 0.22 0.30 0.28 0.28 0.33 0.15 0.30 0.29 0.24
0.20 0.25 0.20 0.21 0.23 0.24 0.20 0.25 0.27 0.26 1.00 0.26 0.25 0.22 0.28 0.23 0.29 0.20 0.26 0.24 0.26 0.50 0.73 0.24 0.23 0.29 0.22 0.57 0.60 0.20
0.30 0.40 0.27 0.31 0.34 0.32 0.25 0.25 0.25 0.32 0.26 1.00 0.29 0.24 0.36 0.33 0.44 0.24 0.34 0.35 0.35 0.20 0.30 0.32 0.31 0.46 0.25 0.32 0.32 0.24
0.28 0.36 0.26 0.31 0.34 0.26 0.21 0.29 0.26 0.26 0.25 0.29 1.00 0.23 0.33 0.29 0.36 0.24 0.33 0.32 0.30 0.21 0.27 0.29 0.27 0.33 0.23 0.29 0.28 0.23
0.25 0.28 0.22 0.25 0.29 0.24 0.20 0.22 0.21 0.27 0.22 0.24 0.23 1.00 0.26 0.24 0.31 0.19 0.27 0.31 0.27 0.21 0.28 0.26 0.24 0.26 0.20 0.26 0.25 0.20
0.30 0.43 0.33 0.34 0.38 0.31 0.23 0.32 0.32 0.31 0.28 0.36 0.33 0.26 1.00 0.34 0.45 0.28 0.38 0.35 0.32 0.25 0.34 0.33 0.33 0.40 0.26 0.40 0.38 0.23
0.26 0.41 0.38 0.30 0.33 0.26 0.20 0.37 0.33 0.29 0.23 0.33 0.29 0.24 0.34 1.00 0.37 0.30 0.32 0.31 0.26 0.19 0.28 0.32 0.31 0.32 0.20 0.40 0.35 0.21
0.39 0.50 0.35 0.42 0.48 0.43 0.29 0.33 0.30 0.37 0.29 0.44 0.36 0.31 0.45 0.37 1.00 0.27 0.45 0.41 0.36 0.25 0.34 0.42 0.40 0.47 0.30 0.38 0.36 0.29
0.19 0.27 0.32 0.24 0.28 0.22 0.17 0.31 0.29 0.20 0.20 0.24 0.24 0.19 0.28 0.30 0.27 1.00 0.24 0.22 0.19 0.17 0.24 0.25 0.26 0.24 0.21 0.32 0.30 0.19
0.35 0.40 0.33 0.37 0.40 0.43 0.27 0.29 0.28 0.33 0.26 0.34 0.33 0.27 0.38 0.32 0.45 0.24 1.00 0.35 0.30 0.21 0.31 0.33 0.34 0.38 0.24 0.36 0.33 0.25
0.30 0.36 0.26 0.33 0.39 0.31 0.22 0.26 0.26 0.37 0.24 0.35 0.32 0.31 0.35 0.31 0.41 0.22 0.35 1.00 0.38 0.21 0.30 0.30 0.30 0.38 0.21 0.29 0.28 0.24
0.27 0.35 0.22 0.29 0.35 0.32 0.22 0.20 0.21 0.38 0.26 0.35 0.30 0.27 0.32 0.26 0.36 0.19 0.30 0.38 1.00 0.23 0.30 0.28 0.26 0.37 0.21 0.27 0.27 0.22
0.20 0.21 0.18 0.19 0.21 0.20 0.19 0.24 0.23 0.22 0.50 0.20 0.21 0.21 0.25 0.19 0.25 0.17 0.21 0.21 0.23 1.00 0.51 0.23 0.21 0.25 0.18 0.46 0.48 0.20
0.24 0.32 0.24 0.27 0.29 0.27 0.22 0.30 0.31 0.30 0.73 0.30 0.27 0.28 0.34 0.28 0.34 0.24 0.31 0.30 0.30 0.51 1.00 0.27 0.27 0.34 0.23 0.64 0.65 0.24
0.31 0.42 0.31 0.31 0.36 0.29 0.25 0.34 0.35 0.28 0.24 0.32 0.29 0.26 0.33 0.32 0.42 0.25 0.33 0.30 0.28 0.23 0.27 1.00 0.60 0.34 0.26 0.34 0.32 0.27
0.30 0.41 0.29 0.31 0.34 0.28 0.25 0.34 0.35 0.28 0.23 0.31 0.27 0.24 0.33 0.31 0.40 0.26 0.34 0.30 0.26 0.21 0.27 0.60 1.00 0.33 0.23 0.32 0.29 0.25
0.33 0.42 0.28 0.35 0.37 0.33 0.26 0.28 0.29 0.33 0.29 0.46 0.33 0.26 0.40 0.32 0.47 0.24 0.38 0.38 0.37 0.25 0.34 0.34 0.33 1.00 0.24 0.35 0.34 0.26
0.22 0.27 0.23 0.23 0.26 0.22 0.16 0.21 0.19 0.15 0.22 0.25 0.23 0.20 0.26 0.20 0.30 0.21 0.24 0.21 0.21 0.18 0.23 0.26 0.23 0.24 1.00 0.25 0.25 0.19
0.26 0.42 0.40 0.30 0.34 0.27 0.23 0.41 0.40 0.30 0.57 0.32 0.29 0.26 0.40 0.40 0.38 0.32 0.36 0.29 0.27 0.46 0.64 0.34 0.32 0.35 0.25 1.00 0.80 0.23
0.24 0.39 0.34 0.27 0.31 0.25 0.22 0.38 0.38 0.29 0.60 0.32 0.28 0.25 0.38 0.35 0.36 0.30 0.33 0.28 0.27 0.48 0.65 0.32 0.29 0.34 0.25 0.80 1.00 0.22
0.23 0.26 0.22 0.22 0.29 0.23 0.22 0.21 0.22 0.24 0.20 0.24 0.23 0.20 0.23 0.21 0.29 0.19 0.25 0.24 0.22 0.20 0.24 0.27 0.25 0.26 0.19 0.23 0.22 1.00
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30
TABLES
Table 1 Descriptive statistics of cross-section statistics
Statistics
SR*
Mean
Median
Std. Dev.
Skewness
Kurtosis
Mean
0.0195
0.0608
0.0200
2.7039
0.2855
16.5332
Median
0.0197
0.0522
0.0146
2.6094
0.2334
12.0354
Max
0.0497
0.1782
0.1114
4.5768
4.1608
104.9361
Min
-0.0024
0.0017
-0.0830
1.3352
-2.5719
6.0898
Std. Dev.
0.0120
0.0374
0.0369
0.7715
0.7195
14.8127
Skewness
0.2474
0.8269
-0.0881
0.3635
1.1214
3.7393
Kurtosis
2.6772
3.6313
3.4368
2.4952
14.3464
19.5086
JB Test**
YES
NO
YES
YES
NO
NO
* Stock-specific Sharpe ratio series computed over the whole sample horizon.
** Jarque Bera normality test at a 5% level of significance.
Note: Sharpe ratio (SR) as well as stock-specific mean, median, standard deviation, skewness
and kurtosis are computed for the 85 series of stock returns (i.e. 85 stocks are considered). We
obtain then six series composed of 85 observations and for which we display corresponding
descriptive statistics. For example, we have one series of stock-specific SRs and one series of
stock-specific means. The abbreviation “Std. Dev.” stands for standard deviation.
Table 2 Properties of Kendall tau b correlations between monthly Sharpe ratios and explanatory factors
Statistics
MktPremium
SMB
HML
VIX**
Mean
0.3306
0.0814
0.0175
-0.0732
Median
0.3264
0.0887
0.0328
-0.0669
Std. Dev.
0.0927
0.0805
0.0792
0.0448
Skewness
1.1751
-0.4008
-0.3524
-0.1303
Kurtosis
4.6585
-0.6787
0.0168
-0.6084
Coefficient of variation *
0.2806
0.9888
4.5272
-0.6127
# significant correlations***
85 (100%)
41 (48%)
21 (25%)
20 (24%)
* It is computed as the ratio of standard deviation to the mean.
** VIX and ln(VIX) yield the same results.
*** Number of significant correlation coefficients at a five percent bilateral test level.
Note: Monthly Sharpe ratios (SRs) are computed for the 85 stocks under consideration so as to obtain 85 series of
stock-specific monthly SRs. Corresponding average monthly Fama and French (1993) factors (e.g. market
premium [MktPremium], SMB and HML) as well as average implied volatility index (VIX) are also computed.
Then, we compute Kendall correlation between each SR series and the four previous explanatory factors (i.e. four
correlation coefficients per stock). Thus, we obtain four series of stock-specific correlations with the four
explanatory factors, each series encompassing 85 observations. We report finally the statistical properties of those
four correlation series.
Table 3 Statistical measures for paired RAPM rankings
RAPM
Signed Wilcoxon test
Kendall’s rank correlation¤
Wilcoxon statistic
p-value
SR
Median FSR
Sortino
-0.0022
0.9983
-0.9759*
-0.2370*
Omega
-0.0223
0.9822
-0.9401*
-0.2336
Omega-Sharpe
-0.0223
0.9822
-0.9401*
-0.2336
Kappa3
-0.0112
0.9911
-0.9266*
-0.2269
Kappa4
-0.0022
0.9983
-0.8689*
-0.2196
31
Upside potential ratio
-0.0131
0.9895
-0.4426*
-0.0812
¤ Kendall’s tau b statistics are displayed.
* Significant at the five percent level of a bilateral test.
Note: The table proposes diagnostic statistics to test for the similarity between the rankings obtained from the 7
risk-adjusted performance measures (RAPMs) on one side, and the Sharpe ratio (SR) and the median
fundamental Sharpe ratio (FSR) on the other side. Basically, we rank SR, median FSR and the six other RAPMs
by ascending order, which yields a performance ranking for the 85 considered stocks. Then, we compare the
resulting SR-, median FSR- and other RAPMs-based performance rankings across stocks. In particular, the
Wilcoxon test checks for ranking stability across performance measures. Differently, Kendall rank correlation
tests for ranking commonalities between SR/median FSR and the RAPMs.
Table 4 Descriptive statistics of cross-section risk-adjusted performance measures
SR
Median
FSR
Sortino
Omega
Omega-Sharpe
Kappa 3
Kappa 4
Upside
potential ratio
Mean
0.0195
0.0011
0.0287
1.0605
0.0605
0.0182
0.0130
0.5032
Median
0.0197
0.0005
0.0289
1.0600
0.0600
0.0190
0.0136
0.5062
Maximum
-0.0024
-0.0202
-0.0036
0.9926
-0.0074
-0.0023
-0.0017
0.4013
Minimum
0.0497
0.0277
0.0776
1.1663
0.1663
0.0491
0.0351
0.5603
Std. Dev.
0.0120
0.0095
0.0178
0.0380
0.0380
0.0113
0.0099
0.0320
Skewness
0.2519
0.1709
0.3051
0.3490
0.3490
0.2839
0.3068
-0.8250
Kurtosis
2.8416
3.0624
2.9600
3.0196
3.0196
2.9542
3.0144
3.9746
Note: We consider SR, median FSR, Sortino, Omega, Omega-Sharpe, Kappa 3, Kappa 4 and Upside potential
ratio (i.e. 8 performance metrics), which are computed for the 85 stocks under consideration. Thus, we get 8
series of performance measures, each series encompassing therefore 85 observations. This table displays the
descriptive statistics of the 8 resulting risk-adjusted performance series.
The standard deviation is labeled “Std. Dev.”.
Table 5 Maximum profit and loss (P&L) of simulated portfolio cumulative returns
One week maximum P&L
One month maximum P&L
Portfolios
Loss
Gain
Loss
Gain
Top Portfolios
Top FSR
10.6368%
9.7948%
18.6534%
25.7380%
Top SR
11.0550%
11.9937%
22.8710%
21.1377%
Top Omega
11.4982%
11.2303%
21.2303%
24.8223%
Top UPR
11.0695%
12.6595%
18.9490%
22.5885%
Bottom
portfolios
Bottom FSR
11.0722%
12.7898%
21.1547%
25.8513%
Bottom SR
12.9585%
12.2338%
23.9003%
23.7436%
Bottom Omega
12.1185%
13.0040%
24.5295%
24.8714%
Bottom UPR
11.2627%
14.7606%
17.5297%
24.1297%
Note: Estimated profits and losses result from Monte Carlo simulations with 10 000 trials. The
top/bottom portfolios are composed of the 30 best/worst performing stocks in accordance with
the selected performance measures. P&L are computed at the portfolio return’s level and
displayed in absolute value.
32
Table 6 One week and one month VaR forecasts for portfolios’ cumulative returns at various risk levels
One week VaR
One month VaR
Portfolios
10%
5%
1%
10%
5%
1%
Top Portfolios
Top FSR
-2.3271%
-3.2457%
-5.0585%
-3.9646%
-5.8087%
-9.6616%
Top SR
-2.5486%
-3.4805%
-5.6394%
-3.8283%
-5.9828%
-10.0816%
Top Omega
-2.4894%
-3.4739%
-5.3720%
-3.9259%
-6.1925%
-10.5474%
Top UPR
-2.5531%
-3.4690%
-5.4950%
-4.4003%
-6.5233%
-10.5849%
Bottom
portfolios
Bottom FSR
-2.5489%
-3.4821%
-5.4015%
-4.4874%
-6.3872%
-10.5160%
Bottom SR
-2.9006%
-3.8632%
-6.2153%
-5.2734%
-7.3936%
-12.0561%
Bottom Omega
-3.0204%
-4.0562%
-6.2925%
-5.1489%
-7.5758%
-12.5991%
Bottom UPR
-2.7269%
-3.7217%
-6.0780%
-4.4728%
-6.7970%
-11.2336%
Note: Simulated results are based on a GARCH-EVT-Copula methodology. The top/bottom portfolios are
composed of the 30 best/worst performing stocks in accordance with the selected performance measures. The
VaR is computed at the portfolio return’s level.
33
FIGURES
0
0.05
0.1
0.15
0.2
0.25
010 20 30 40 50 60 70 80 90
Cramer-von Mises statistics
Stock number
v
u
Critical value (5%)
Fig. 1. Normality test of dynamic and state errors. This figure draws the Cramer-von Mises statistics of dynamic
(ui) and state (vi) errors for each stock i.
y = 0,2432 x- 0,0036
R² = 0,0947
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
0.04
-0.01 0 0.01 0.0 2 0.03 0.04 0.05 0.06
FSR
SR
Fig. 2. Plotting classic Sharpe ratios against fundamental ones. The figure draws classic Sharpe ratios (SR)
against corresponding fundamental Sharpe ratios (Median FSR). A regression adjustment is proposed where x
represents ranked Sharpe ratios while y represents their fundamental counterparts. Only around 10 percent of SR
estimates coincide approximately with FSR estimates.
34
0
10
20
30
40
50
60
70
80
90
010 20 30 40 50 60 70 80 90
RAPM ranks
SR ranks
Sortino
Omega
Omega-Sharpe
Kappa 3
Kappa 4
Upside potential ratio
(a)
(b)
Fig. 3. RAPM ranks versus SR ranks. This figure plots the ranks of stocks according to the performance measures
under consideration, namely the classic Sharpe ratio (SR), the fundamental Sharpe ratio (Median FSR) and the 6
other risk-adjusted performance measures (RAPMs). The panel (a) plots the SR-based ranks against the ranks
induced by the other RAPMs. The panel (b) plots FSR-based ranks against the ranks induced by the Upside
potential ratio.
0 2 4 6 8 10 12 14 16 18 20
-0.2
0
0.2
0.4
0.6
0.8
Observation number
ACF
Sample ACF of Stock Returns
0 2 4 6 8 10 12 14 16 18 20
-0.2
0
0.2
0.4
0.6
0.8
Observation number
ACF
Sample ACF of Stock Squared Returns
Fig. B.1. Autocorrelation function (ACF) of stock returns (upper panel) and squared stock returns (lower panel).
35
-0.4
-0.2
0
0.2
0.4
0.6
Residuals
0500 1000 1500 2000 2500 3000
0
0.05
0.1
0.15
0.2
Observation number
Volatility
Fig. C.1. Residuals and conditional volatility of residuals in the AR(1)-TGARCH(1,1,1) without adjustment to
volatility regimes (estimation over the sample horizon).
-0.4
-0.2
0
0.2
0.4
Residuals
0500 1000 1500 2000 2500 3000
0
0.02
0.04
0.06
0.08
0.1
0.12
Observation number
Volatility
Fig. C.2. Residuals and conditional volatility of residuals in the AR(1)-TGARCH(1,1,1) after adjustment to
volatility regimes (estimation after splitting the sample horizon into a pre-, during- and post-crisis period).
36
0 2 4 6 8 10 12 14 16 18 20
-0.2
0
0.2
0.4
0.6
0.8
Lag
Sample Autocorrelation
Sample ACF of Standardized Residuals for Stock Returns
0 2 4 6 8 10 12 14 16 18 20
-0.2
0
0.2
0.4
0.6
0.8
Lag
Sample Autocorrelation
Sample ACF of Squared Standardized Residuals for Stock Returns
Fig. C.3. Autocorrelation function (ACF) of standardized residuals (upper panel) and squared standardized
residuals (lower panel).
-8 -6 -4 -2 0 2 4 6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Standardized Residuals of Stock Returns
Probability
Pareto Lower Tail
Kernel Smoothed Interior
Pareto Upper Tail
Fig. D.1. Empirical CDF of standardized residuals as a mixture of a Gaussian kernel estimate and a GPD.
37
0 0.5 1 1.5 2 2.5 3 3.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Exceedance
Probability
Upper Tail of Standardized Residuals of Stock Returns
Fitted Generalized Pareto CDF
Empirical CDF
Fig. D.2. Quality of the GPD’s fit for upper tail exceedances of standardized residuals.