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EstimatingfundamentalSharperatios:A

Kalmanfilterapproach

Research·May2015

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:

i

i

l

i

i

h

iek

ke

(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

-)² 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.