Master’s Paper of the Department of Statistics, the University of Chicago(Internal document only, not for circulation)

Estimation of the Tail Behavior of Mutual FundReturns: An EVT-based Approach

Jin S. Choi

Advisor: Prof. Wei-Biao Wu

Approved

Date

February 17, 2011

Estimation of the Tail Behavior of Mutual Fund Returns: An

EVT-based Approach

Jin S. Choi

Department of StatisticsThe University of Chicago

February 17, 2011

Abstract

This study employs the conditional EVT method developed by McNeil and Frey (2000) to in-vestigate the tail behavior of distributions of returns on mutual funds that are assumed to exhibitvolatility clustering as usually observed in financial time series. After a Monte Carlo simulationconducted to find a sufficiently high threshold necessary to use the generalized Pareto distributionfor modeling extreme returns, the shape parameter ξ that quantifies the heaviness of the tails isestimated for each of actively-managed equity mutual funds based on the data obtained from theCRSP Survivor-Bias-Free U.S. Mutual Fund Database. With the quintile portfolios of mutual fundsformed on the magnitude of the shape parameter, annualized return and volatility are computed,and risk-adjusted excess returns are estimated as well by using well-known asset pricing models toexamine a relation between the tail shape and the return performance. Even though there appearsno significant linear correlation, an intriguing pattern between them is detected and later discussed.

1

Contents

1 Introduction 3

2 Extreme Value Theory 5

2.1 Fisher–Tippett–Gnedenko Theorem and the BM Method . . . . . . . . . . . . . . . . . . 5

2.1.1 Maximum Domain of Attraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.2 Issue of Time Dependence among the Maxima . . . . . . . . . . . . . . . . . . . . 6

2.2 Pickands–Balkema–de Haan Theorem and the POT Method . . . . . . . . . . . . . . . . . 7

2.2.1 Quantile Estimation of the Underlying Distribution . . . . . . . . . . . . . . . . . 8

3 Method for Analysis 8

3.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 Method and Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2.1 Stochastic Volatility Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2.2 GPD model for zt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2.3 Portfolios of Monthly Returns and Time Series Regressions . . . . . . . . . . . . . 19

4 Results of the Analysis 20

5 Conclusion 25

6 Appendix 26

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1 Introduction

While assessment and of stock or equity portfolio performance has long been a significant, if not the most,concern of investors, there has not been a general consensus on which measure would be the standardfor comparing a financial product with another. Traditional risk-adjusted performance measures includeTreynor index (1965), Sharpe ratio (1966) and Jensen’s alpha (1968) that are all based on the classicalmean-variance framework introduced by Markowitz (1952), namely use the variance of portfolio returnsthemselves or the covariance with market returns represented by the sigma or the beta of the capitalasset pricing model (CAPM), respectively. Even though there have been much efforts on developingmore sophisticated asset pricing models such as Fama-French three-factor model (1992) or Carhart’salpha (1997) over the last four decades since then, many investors and professionals still give muchcredibility to these classic indicators as tools for making financial decisions. From the risk managementperspective, meanwhile, Value-at-Risk (VaR), popularized along with the publication of RiskMetricsin 1994 by JPMorgan, is another way to measure risk defined as the maximum loss expected over agiven time period, given a specified probability. So far modified and extended versions of VaR havebeen introduced in effort to have actual aspects of return distributions incorporated, but widely-appliedtraditional VaR computations simply assume that returns follow a Normal distribution that can bedefined by only two parameters, mean and variance.

It is a well-known fact in the recent academic literature that there are evidences of skewness and heavytails in distributions of financial asset returns as numerous studies by scholars including Hols and de Vries(1991), Loretan and Phillips (1994) and Ghose and Kroner (1995) have already demonstrated since early1990s. The conventional measures mentioned above take into account only simple expected returns andhistorical volatilities of movements, which are the first and second moments of asset return distribution,while leaving higher order of moments out of consideration that rather come into play an important roleto explain the asymmetric and leptokurtic characteristics of asset return distributions. Consequentlycommon industry practices dealing with financial equity products based on such methodologies withclear limitations could lead to settling with lower overall returns and confronting extreme losses morefrequently than predicted.

Black Monday in 1987, the burst of Dotcom bubble at the dawn of the new century, and the failuresof so-called too big to fail Bear Stearns and Lehman Brothers a few years ago all exemplify seeminglyimpossible disasters in the U.S. financial market that actually happen with probabilities once perceivedas virtually non-existent. The aftermath of extreme losses is considered substantial to the performanceof stock market as a whole; Wu et al. (2009) state that for all the common stocks traded on NYSE,AMEX, and NASDAQ from 1963 to 2005, the average daily return would be more than doubled from0.09% to 0.23% if the bottom 1% returns for each stock within each year. Putting the fact aside thatthe probabilities associated with equal amounts of gains and losses are not identical, however, noneof indices under the normality assumption, or in which only means and variances embedded, do notcorrectly predict how an equity security in question would respond in such cases of market-wide eventof extreme movement at a given magnitude.

Denoted as Extreme Downside Risk by Wu et al. (2009), the potential loss from extreme undesirablereturns, realized by higher probabilities than implied according to a Normal distribution or even a

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Student’s t-distribution with heavier tails, should be another dimension of risk that rational risk-averseinvestors are concerned. It would be reasonable to consider that it does not only bring up an argumenthow a measure of risk would be rendered as a component of performance indices and ultimately ofportfolio optimization, but also becomes a significant factor in asset pricing models. In other words,investors require higher returns as additional premium in exchange for taking positions more likely toface severe downturns, namely beyond two to three times more of the estimated annualized volatilities,compared to other positions in the identical asset class, e.g. portfolios of only equity securities tradedin major U.S. exchanges.

In search of an appropriate proxy or a quantified representation of such risk measure of the extreme equityreturn movements, this study is largely motivated by the idea from the research done by Wu et al. (2009)that has been referred to above and other studies in empirical finance and statistics. Indeed, during therecent years, there have been active researches with regards to expanding the concept of financial riskfurther from the traditional Gaussian framework limited up to the second moment of returns viewedas random variables. Harvey and Siddique (2000) and Dittmar (2002) introduce coskewness (the thirdmoment) and cokurtosis (the fourth moment) of asset returns with systematic risk factors other thannon-market ones stated in Fama and French (1992) to asset pricing models, respectively. Later Chung etal. (2006) examine whether Fama-French factors lose their explanatory powers when a set of systematicco-moments of third order through tenth order are included in the model, which will be reviewed later inthis paper. In sum, all of the studies provide empirical evidences supporting the conclusion by Rubinstein(1973) that if returns are not normal, measuring risk requires more than just measuring covariance.

Following a different approach other than ones for the most of researches above that emphasize thedecomposition of returns into the distributional characteristics of the exogenous factors, this paperpresents the direct investigation on the tail behaviors of the asset return distribution as proposed by Wuet al. (2009). While the approach is fundamentally similar to their work in terms of using extreme valuetheory (EVT), over the course of the research the alternative method is considered despite the fact thatthey are all originated from the same theory, in an effort to maintain key assumptions that make themodel statistically valid. What makes this analysis diverged further from the previous endeavor is that amodel is constructed on samples of mutual fund returns rather than those of individual stocks themselvesin order to bring up another issue in question. For actively managed mutual funds, idiosyncratic riskof each stock is assumed to be already controlled in a way of optimizing portfolios, and managers areseeking to maximize the “alpha” to let them attractive and competitive in the market. In terms of riskmanagement, a certain levels of VaR and expected shortfall (ES), either traditional or more advanced,would have already been set as well. In this sense, this paper also examines if the EVT-based estimationof tail behavior can be meaningfully applied to “optimal” portfolios purposefully formed on such modelsand measures that are already in place.

Having been widely accepted and frequently used in the fields of engineering, actuarial sciences andenvironmental sciences, EVT has proven to be an effective statistical tool for analyzing probabilisticcharacteristics, i.e. the shape of the tails of a distribution, of extraordinary events such as naturaldisasters or large insurance claims. Along with the rapid growth of understanding the importance offinancial risk management, there is a recent interest for EVT in the finance literature as well; theapplication of EVT to financial data is first demonstrated in his seminal study by Longin (1996) who

4

identified the general extreme value (GEV) distribution as the asymptotic distribution of the extremedaily returns of the S&P500 index. Longin and Solnik (2001), Jondeau and Rockinger (2003) andPoon, Rockinger and Tawn (2004) thereafter follow with studies on the possible correlations betweenextreme returns in international markets. Danielsson and de Vries (1997), McNeil and Frey (2000) andBali (2003) suggest ways to improve the estimations of VaR and ES by deriving the tail distributionfunction by EVT. Wu et al. (2009) argue, however, that there have been only few researches, if notnone, incorporating EVT into asset pricing studies, and overall applications so far have been limited toconfirming the evidences for heavy tails of equity returns; the efforts to be presented in this paper is inline with theirs to explore another direction of application.

Characterized by volatility clustering (Mandelbrot, 1963), it would definitely be a strong assumptionthat returns, or financial time series in general, are independent and identically distributed (iid) randomprocesses. Even though many studies including Chavez-Demoulin and Embrechts (2008) and Wu etal. (2009) argue that implementing the block maxima method, one of parametric approaches to bediscussed later, can fulfill the assumption without further considerations, the classic EVT approach hassome drawbacks especially when a sample size is not large enough. McNeil and Frey (2000) establishesan alternative method as a remedy for non-iid samples exhibiting heteroscedasticity by incorporatinga “pre-whitening” process based on the GARCH-type of model into the relatively modern peak overthreshold (POT) method. The conditional EVT method, named by McNeil and Frey, is employed forthe analysis with a couple of modifications that suit the mutual fund data of interest.

Embrechts et al. (1997) and McNeil et al. (2005) deliver the comprehensive coverage of the theoreticalparts of EVT and its applications, and in the next section I briefly present an overview of two parametricEVT approaches mentioned previously. In Section 3 the data and the analysis method modified from theproposed model by McNeil and Frey (2000) are described in detail, and its result is discussed in Section4. Section 5 concludes and provides several suggestions for the further possible use of the method.

2 Extreme Value Theory

2.1 Fisher–Tippett–Gnedenko Theorem and the BM Method

The central limit theorem states that a normal distribution is the limiting distribution for sums andaverages of iid random variables regardless of the underlying distribution as long as the size of samplesis sufficiently large. Likewise, the extreme value theorem, also known as Fisher-Tippett (1928) theorem,deals with the convergence of sample maxima to the GEV distribution with the distribution function asfollows

Hξ,µ,σ (x) =

exp(−(1 + ξ x−µσ

)− 1ξ

), ξ 6= 0

exp(− exp

(x−µσ

)), ξ = 0

The block maxima (BM) method uses the GEV distribution to approximate the true distribution of a setof maxima. It is generated by identifying the maximum value denoted asMn,i = max (Xn·i+1, . . . , Xn·i+n)

for i = 0, . . . ,m− 1 of each of m blocks with a homogeneous size of n that is assumed to be sufficiently

5

large, into which a sequence of iid random variables Xj for j = 1, . . . ,m · n from an unknown underly-ing distribution F is divided. As the GEV distribution is fitted to the maxima using various methodssuch as maximum likelihood (ML), the method of moment (MoM), or probability weighted moments(PWM), three parameters are subsequently estimated. µ ∈ R and σ > 0 are a location and a scaleparameters that properly normalize the block maxima so that (Mn − µ) /σ has the GEV distributionHξ,µ,σ (x) := Hξ ((x− µ) /σ) as the limit distribution.

The other parameter ξ is the shape parameter and determines what type of the limit distribution Hξ itis. When ξ > 0, the distribution is called the Fréchet distribution with heavy tails; the larger the shapeparameter is, the more fat tailed the distribution have. If ξ < 0, the distribution is called the Weibulldistribution, and at last ξ = 0 identifies that it is the Gumbel distribution. The Fisher-Tippett theoremsuggests that the asymptotic distribution of the maxima belongs to one of the three distributions above,regardless of F , the distribution of the observation samples.

Gnedenko (1943) later shows that if the tail of F decays like a power function, then it is in the domainof attraction for the Fréchet distribution. The class of distributions with tails decaying like a powerfunction include the Pareto, Cauchy and Student’s t distributions, which are the well-known heavy taileddistributions. The distributions in the domain of attraction of the Weibull distribution, i.e. ξ < 0, are theshort tailed distributions such as a uniform distribution. The distributions in the domain of attractionof the Gumbel distribution, i.e. ξ = 0 include the normal, exponential and gamma distributions whosetails are decaying exponentially.

2.1.1 Maximum Domain of Attraction

An important concept for the application of EVT is Maximum Domain of Attraction, and it establishesthe relation between the GEV distribution and the generalized Pareto distribution (GPD) as shown inthe next section. Theoretically speaking, a distribution F that a sequence of random variables Xi followsis said to belong to the maximum domain of attraction of the extreme value distribution H, denoted asF ∈ MDA(H), if and only if the Fisher–Tippett theorem Mn−bn

an

d−→ H, where an > 0 and bn ∈ R ∀nas n →∞, and Mn are sample maxima among Xi for i = 1, . . . , n as already introduced. For empricalstudies, it is sometimes a difficult task to recognize which distribution observations truly follow, and aslong as modeling the extreme value is the interest of studies, the exact form of the underlying distributionbecomes a trivial issue by the theorem and the concept of MDA. Further, McNeil et al. (2005, pp. 267,278) assure that it is sufficient to say that essentially all the common continuous distributions are inMDA(Hξ), for some value of ξ, and per Gnedenko their maxima limiting distributions is one of threeclasses of the GEV distribution.

2.1.2 Issue of Time Dependence among the Maxima

Followed by the rationale for employing the block maxima method in Wu et al. (2009), Dowd (2002)asserts that selected maxima are less clustered than the underlying data from which they are drawn,and become even less clustered as the period of time from which they are drawn, i.e. the size of block,

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become longer. Hence it is possible to eleminate time dependence completely if block periods are chosenlong enough. However, it may be a strong assertion that the time dependence would be completelyremoved as there might exist a long term seasonality such as January effect. (Keim, 1983; Kramer,1994) For instance, the maxima indentified from blocks that contain a year-long range of monthly ordaily return sequence with a size n = 12 or 252, respectively, would be mostly ones of returns realizedin January. Even if they appear not to be serially correlated, i.e. cov (Mn,i = Xk,Mn,j = Xl) = 0 fori < j, k+ 252 ≤ l, as rather the observations distant from each other in a random process, they could beconsidered implicitly time dependent. Meanwhile, McNeil and Frey (2000) implies cov

(M2n,i,M

2n,j

)> 0

by pointing out that Longin (1996) ignores the stochastic volatility exhibited by most financial returnseries and simply applies estimators for the iid case.

2.2 Pickands–Balkema–de Haan Theorem and the POT Method

As an alternative to studying the maxima of observations, the behavior of number of observations thatexceed a certain level of threshold can be considered from the perspective of order statistics. Supposethat iid random variables X1, X2, . . . , Xn are from an unknown distribution function F ∈ MDA (Hξ)

for some ξ ∈ R. Then the corresponding distribution function of exceedances over a high threshold u,denoted as Y1, Y2, . . . , YNu , Nu = card {i : i = 1, . . . , n, Xi > u}, is given by

Fu (y) = Pr {Y = X − u ≤ y|X > u}

=Pr {u < X ≤ y + u}

Pr {X > u}

=F (y + u)− F (u)

1− F (u)

for 0 ≤ y < xF − u, where xF is either the finite or infinte right endpoint of the underlying distributionF . Fu describes the distribution of the excess loss over the threshold u, given that u is exceeded. Bythe second theorem in extreme value theory (Balkema and de Haan, 1974; Pickands, 1975)

limu→xF

sup0<y<xF−u

∣∣Fu (y)−Gξ,β(u) (y)∣∣ = 0 ⇐⇒ F ∈ MDA (Hξ) , ξ ∈ R

, it is shown that for the sufficiently large u, the excess distribution Fu converges to the GPD, which isdefined as

Gξ,β (y) =

1−(

1 + ξ yβ

)− 1ξ

, ξ 6= 0

1− exp(− yβ

), ξ = 0

where β > 0, and y ≥ 0 when ξ ≥ 0 and 0 ≤ y ≤ −β/ξ when ξ < 0. Embrechts et al. (1997, pp. 165)provide the detailed proof of the theorem.

Similarly to the GEV distribution, the GPD transforms into a number of other distributions dependingon the value of ξ. When ξ > 0, it takes the form of the ordinary Pareto distribution. This case wouldbe the most relevant for financial time series data as it has a heavy tail. If ξ = 0, the GPD correspondsto exponential distribution, and it is called a short-tailed, Pareto II type distribution for ξ < 0. In theheavy-tailed case, ξ > 0, E

(Xk)

= ∞ for k ≥ 1ξ , e.g. the GPD has an infinite variance for ξ = 0.5 and

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has an infinite fourth moment when ξ = 0.25. For asset returns, the estimates of ξ are expected to beusually less than 0.5, implying that the returns have finite variance (McNeil, 1998).

The importance of the Pickand–Balkema–de Haan theorem lies on that the distribution of exceedancesmay be approximated by the GPD by choosing ξ and β and setting a high threshold u. MLE iscommonly and easily implemented to fit the GPD if the exceedances can be assumed to be realizationsof independent random variables, while there are other various ways including PWM. Especially forξ > − 1

2 , Hosking and Wallis (1987) presents evidence that maximum likelihood regularity conditionsare satisfied, and the ML estimators are asymptotically normally distributed. Therefore, the standarderrors for the ML estimators ξ and β can be obtained from the inverse of Fisher’s information matrix asdone in the usual estimation procedure.

2.2.1 Quantile Estimation of the Underlying Distribution

In order to obtain a superior measure of tail risk based on EVT compared to the traditional VaRassociated with the normality assumtipon, the underlying distribution F is first decomposed as follows.Given that x > u,

F (x) = Pr {X ≤ x} = Pr {X ≤ u}+ Pr {u ≤ X ≤ x}

= Pr {X ≤ u}+ Fu (y) Pr {X > u} ∵ y = x− u from the eq. above

= F (u) + Fu (y)F (u)

or it can be simply derived from the distribution function of exceedances. Then F (u) that is theproportion of samples below the threshold and Fu (x− u = y) can be approximated by 1 − Nu

n andGξ,β (y), respectively. The tail estimator is then approximated as

F (x) ≈ 1− Nun

(1 + ξ

x− uβ

)− 1

ξ

, x > u

then the quantile estimation, or VaR in the risk management literature, for a given probability q > F (x)

is calculated as

q = 1− Nun

(1 + ξ

xq − uβ

)− 1

ξ

⇒ xq = u+β

ξ

((n

Nu(1− q)

)−ξ− 1

)

3 Method for Analysis

As stated in the introduction, the analysis is conducted using the conditional EVT approach proposed byMcNeil and Frey (2000). The POT approach, or often referred to as threshold exceedances approach, ischosen to be integrated in their method as it is favored over the alternative modeling schemes – the block

8

maxima method that is covered in the previous section and semi-parametric Hill’s estimator method.By setting a threshold, it is allowed to salvage apparently more data for actual distribution fitting whileconstructing large blocks is wasteful of data as only the maximum losses are retained. McNeil and Freyclaim that a distribution of asset returns sometimes appear to have rather light tails than heavy tails,but Hill’s estimator is specifically designed for the heavy-tailed case with the shape parameter ξ > 0.Furthermore, the qunatile estimator computed based on the GPD method is said to be more stable interms of mean squared error (MSE) with respect to the choice of the level of threshold u. The results ofcomparison is described in detail in their paper.

To put it simply, the conditional EVT method consists of two stages. The process starts with fittinga GARCH-type of model to the raw daily asset return data by making no specific assumptions on theunderlying distribution F , or more specifically the distribution of residuals Fz, therefore the parametersare estimated with pseudo- or quasi-Gaussian maximum likelihood (PML or QML) techniques. Afterthe raw return data are “whitened”, the iid residual sequence zi is fitted to the GPD to model the tail ofFz. to estimate the shape parameter ξ and the quantile estimator zq for q > 0.95.

Adapting the idea from Wu et al. (2009), one more stage of anlaysis is carried out. Based on themagnitude of sorted ξ or z0.99 that represent the likelihood of extreme realizations of asset returns, tenequally weighted portfolios of mutual funds are created by using their monthly return series, and alphasare estimated with the CAPM, Fama-French (1992) Three Factor model and Carhart (1997) Four Factormodel by conducting time series regressions.

These stages are described in detail in Section 3.2 and demonstrated with an example with using thedaily returns of the S&P 500 index from Sep 1, 1998 to Sep 30, 2010.

3.1 Data

The main data source used in this study is the CRSP Survivor-Bias-Free US Mutual Fund Database(MFDB) that was initially developed by Carhart (1997) for his dissertation at the University of ChicagoGSB. The CRSP MFDB provides information on daily and monthly fund returns at the share-classlevel, total assets under management (TNA), net asset value (NAV), investment objectives, and otherfund characteristics for each fund share class. Being updated quarterly, the MFDB currently offers dailyreturn data of all funds available from Sep 1, 1998 to Sep 30, 2010.

As mentioned repeatedly, among many information offered the variables actually used for the statisticalanalyses are daily and monthly return series of each fund. The main disadvantage of using the CRSPMFDB as a sole data source is that the daily returns and NAVs data became available since the late1998. Hence one should note as of 2010 this only allows approximately 3,000 observations at most, orequivalently, 144 monthly observations that unfortunately do not cover one of the major market crashesin the modern era, the 1987 crash.

While fee and expense structures of mutual funds, e.g. frond-end or back-end load, which determine ashare class of a fund portfolio, e.g. Class A, B, or C, is one of special characteristics what distinguish themfrom other standard financial securities such as stocks, the empirical tests in this paper are conducted

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Active-Equity funds selected (a) 15,484Total obs + missing from (a) (b) 22,339,154

Funds with >315 obs (c) 14,209Total obs + missing from (c) (d) 22,108,237Funds with >1 missing obs (e) 1,758

Total missing from (e) (f) 135,958Funds with >1% missing obs (g) 878

Total missing from (g) (h) 133,255Available funds for analysis (i) (c) - (g) 13,331

Total obs + missing from (i) (j) (d) - (h) 21,974,982Funds with missing obs included (k) (e) - (g) 880

Total missing from (k) (l) (f) - (h) 2,703

Table 1: Filtering actively-managed equity funds based on the criteria of 1. minimum sample size and2. maximum missing observation size. Note that the funds selected for analysis still include some returnsequences with missing value as shown in (k) and (l).

based on individual fund share classes instead of individual fund portfolio. In other words, despite thefact that “Fund Class A” and “Fund Class B” share a same investment strategy and portfolio, they areconsidered as different securities, if there are any, in this study as done in Hortaçsu and Syverson (2004).Even if this could lead to the double counting problem, funds with multiple share classes are observed tohave spreads in returns between share classes that become widened up as the time span of investment,because from the perspective of an investor who hold Class A shares the sales commissions paid at thetime of purchase should be compensated with higher returns than ones from Class B shares that haveback-end loads and are eventually converted to Class A shares in long term.

As the study is focused on actively-managed U.S. domestic equity funds, fund information on investmentobjectives from the CRSP MFDB is used to identify such funds from the whole funds data available. Inorder to filter out funds holding non-equity positions such as treasury, state or municipal debt securities,strategic insights (SI) objective codes and Lipper objective codes are used as the first step of the sortingprocedure. The investment objective variables cover different time periods; the SI objective covers forthe period up to 1999 and Lipper objective code is for the period after 2000. Equity funds are identifiedby taking the following values of the strategic insights objective code ‘AGG’, ‘GMC’, ‘GRI’, ‘GRO’,‘ING’ and ‘SCG’, and of Lipper objective codes ‘CA’, ‘EI’, ‘G’, ‘GI’, ‘I’, ‘MC’, ‘MR’ and ‘SG’. Thedetailed explanations on what each code refers to are available on the CRSP MFDB guide (2000).

As for criteria that select funds that are under active management, ‘index fund flag’ variable from thedatabase as well as full names of funds are utilized. The value ‘D’ for ‘index fund flag’ suggest that a fundis a pure index fund, while the ‘B’ or ‘E’ show that funds are formed based on indices as benchmarks, butfurther modifications are made to yield better returns than indices. Since the variable only covers theshorter range of time period than the actual target range, full names of funds are also used to identifyindex funds. If a fund name includes the case-insensitive word ‘index’ without the word ‘enhanced’combined with, the fund is classified as an index fund.

Since the sequence of daily returns of each fund is to be ultimately fitted to the GPD later in the secondstage of analysis, a certain level for the minimum length of each return series is required to ensurethe model has a good fit with the minimum MSE. At least 15 months or 315 days worth of return

10

observations for each fund appear to be required based on the result from a Monte Carlo simulationsimilar to the one conducted in McNeil and Frey (2000) and is presented in Section 3.2.2. Setting suchrequirement of minimum number of observations is also expected to eliminate distinctive upward biasesto fund returns introduced by incubation or back-filling documented by Evans (2010).

Missing observations in daily return series are the last concern in the sorting procedure. Among thetotal of 14,209 funds chosen after the processes described above, 1,758 funds appear to have at least oneobservation missing in their daily return sequences, and the total value of missing value is near 136,000.As there is no gold standard to handle missing data in the statistics or other academic literatures, asimple rule of thumb is used to disregard funds if the proportion of missing observations is greater thanone percent. This entails having 878 funds sorted out along with 133,255 missing points. It suggeststhat a relatively few funds account for the most of missing data and the database is somehow incompleteto include almost the whole sequences for several funds.

Summarized in Table 1, the resulting sample data include 13,331 funds with 21,972,279 daily returnobservation. 2,703 missing daily log return data from 880 funds out of the total are filled by using thesimple linear interpoation method. Despite the lack of accuracy, it appears to be better to have theresults of analysis regarding extreme values not influenced by large approximations possibly by othermethods for missing values. the Each fund is given a unique value of the identifier variable ‘fund number’,and it is extensively used throughout the analysis as a mapper that connects shape parameters ξ andquantile estimators zq estimated by daily returns, and monthly returns used to construct portfoliosaccordingly to the ranking of the estimators.

3.2 Method and Procedure

Per the CRSP MFDB Guide (2000), a simple daily return for a mutual fund is computed as follows,

Rt =NAVt × cumfact

NAVt−1− 1

and cumfact is an adjustment factor for the net asset value (NAV) that reflects an impact on the fundportfolio due to share splits or distributions (or dividends) of equity security holdings. Since its valueis not explicitly available, simple daily and monthly return sequences are directly used as stated in theprevious section, rather than computing them from NAVs, which can simply be considered as prices ofan asset.

3.2.1 Stochastic Volatility Model

What makes the method proposed by McNeil and Frey (2000) unique among studies in attempt to applyEVT to financial data is its effort to remove volatility clustering embedded in financial time series tosecure much more stable results than when applied to non-iid data. However, they also consider that thenon-iid nature of financial data is due to time-varying conditional mean returns and present the example

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30−Aug−1998 10−Sep−2002 21−Sep−2006 02−Oct−2010

−0.1

−0.05

0

0.05

0.1

30−Aug−1998 10−Sep−2002 21−Sep−2006 02−Oct−20100

0.02

0.04

0.06

Figure 1: Top: the series of log returns on S&P 500 index ranged from Sep. 1, 1998 to Sep. 30, 2010including the ’08-’09 market turmoil. Bottom: the series of fitted values of the conditional standarddeviation from the GARCH(3,4) model fitted based on the method of QML.

by Embrechts et al. (1997, pp 270) that illustrates inaccurate Hill’s estimators of the distributionssimulated from a heavy-tailed and symmetric AR(1) process with iid innovation.

As opposed to the AR(1)-GARCH(1,1) model that McNeil and Frey suggest to whiten raw return data,this study employs the concept of the weak-form of Efficient Market Hypothesis (EMH) by Fama (1965,1970) with the assertion that stock returns themselves do not have a predictive power for their futurereturns and suggests conditional mean dynamics become negligible. Consequently, the ‘parsimonious’linear AR(1) model as the authors call is dropped, and the model is modified to be GARCH(p,q) sincethe higher order terms turn out to be necessary for correct specification.

Suppose that rt = log (1 +Rt) , t ∈ Z is a strictly stationary time series of daily log return observationof a fund. As proposed, the GARCH(p,q) model is given by

rt = µ+ εt

εt = σtzt

σ2t = ω +

q∑i=1

αiε2t−i +

p∑j=1

βjσ2t−i

while a Gaussian GARCH model assumes that zt ∼ N (0, 1), in this paper the true distribution of eachfund return Fz is assumed to be unknown and rather have heavier tails than a normal distribution. Hencethe model is fitted using the QML method to obtain parameters θ =

(µ, ω, α1, . . . , αq, β1, . . . , βp

)′that

maximize the log-likelihood for the GARCH(p,q) with normal innovations. McNeil and Frey assures thatthe QML method yields a consistent and asymptotically normal estimator with the proof by Gouriérox(1997).

Note that even though a conditional mean model is omitted from the model above, the residual series

12

0 10 20 30−0.5

0

0.5

1

Lag

Sam

ple

Aut

ocor

rela

tion

ACF of rt

0 10 20 30−0.5

0

0.5

1

Lag

Sam

ple

Aut

ocor

rela

tion

ACF of rt2

0 10 20 30−0.5

0

0.5

1

Lag

Sam

ple

Aut

ocor

rela

tion

ACF of zt

0 10 20 30−0.5

0

0.5

1

Lag

Sam

ple

Aut

ocor

rela

tion

ACF of zt2

Figure 2: Top correlograms for the raw return series and its squared series showing that the data isapparently not iid, while the bottom correlograms for the standardized residual series confirms thatautocorrelations are disappeared.

εt is still obtained by subtracting the unconditonal mean µ from the return series. Despite EMH as thebasic rationale, demeaning is to some extent important for correct specification of a GARCH model.This can be explained by writing the true mean dynamics as

rt = µ+ εt

and the model for estimation without an unconditional mean is

rt = ut

then the estimated residuals areut = µ+ εt

If the mean dynamics is not correctly specified in the way that µ p9 0, and it follows that

u2t =

(µ2 + 2µεt

)+ ε2

t

⇒ σ2t = E

[µ2 + 2µεt|Ft−1

]+ E

[ε2t |Ft−1

]which does not necessarily converge to E

[ε2t |Ft−1

]if the mean dynamics is misspecified without the

unconditional mean, and the GARCH estimates are not consistent with maximum likelihood estimation.

13

The fitted conditional volatility σt and the unconditional mean estimate µ are used to compute stan-dardized innovation (or residual) series zt = rt−µ

σtwith zero mean and unit variance. In order to verify

that the GARCH(p,q) model adequately captures the time varying volatility and fits the return serieswell, the Ljung-box test is performed on the series of zt as well as the series of z2

t . If σ2t is the correct

conditional variance, then dividing each of demeaned return by its conditional standard deviation σt

should make result in a series with constant unit variance, and only if divided by the correct condi-tional standard deviation will this be true. Upon satisfactory results of the test showing no significantautocorrelations of the first and second moments, i.e. cov (zt, zt−i) = 0 and cov

(z2t , z

2t−i)

= 0 for all1 < i < t− 1, it is assumed (even though not robust) that the stadardized returns are iid processes andare ready to be fitted to the GPD in the next stage for the tail analysis.

Algorithm of Fitting the Model It is practically infeasible to fit 13,331 series to the modelandexamine their validity one by one manually. In the fitting procedure of the actual analysis, therefore,once after a series is confirmed to be strictrly stationary with an augmented Dickey-Fuller (ADF) test,iterative attempts to find a correct specification for a GARCH(p,q) model are initiated. Starting withp = 1 and q = 1, each residual and squared residual series from an iteration is tested by the Ljung-boxtest statistics at lag 10 and 20, i.e. Q(10) and Q(20), for autocorrelations and iid assumption. If theautocorrelations persist, then the ARCH order is first increased to q = 2 in the next iteration, and thenthe GARCH order is secondly increased to p = 2 while the ARCH order is brought back to 1. Thisstep-wise increasing order of ARCH and GARCH terms continues until the iteration when there appearno siginificant autocorrelation throughtout lag 20, or iteration is halted without a valid standardizedresidual series, i.e. the fund sample is discarded, if a model cannot be fitted to a series even withhigher orders of two terms beyond p = 10 and q = 10. Increasing orders step-wise one by one wouldnot be completely sound for modeling all data in interest as one would fit well to a GARCH(1,3) orGARCH(4,2) models for instance, yet the portion of samples are compromised in exchange for keepingthe fitting algorithm simple and time efficient.

3.2.2 GPD model for zt

As seen in Figure 3, illustrating the Q-Q plot and the histogram, the distribution of standardized residualsappear to retain heavy tails with kurtosis greater than 3. and the left tail, in this case which can beconsidered to represent likelihoods of losses from a long position of a mutual fund benchmarking S&P500 index, is fitted to the GPD model to estimate the shape parameter ξ and the quantile value zq.

Given the threshold u chosen in the manner that is discussed later in this section, the data of exceedancesis obtained by first building a set

{Z ∈ Z : (−Zt) > u ∀t

}. For each of these exceedances we calculate

the amount Yi =(−Zi

)−u, 1 < i < Nu < T of the excessive losses. Note that negative values of Zi are

taken since the behavior of extreme losses the interest of the study. From the cumulative density functionof the generalized Pareto distibution given in Section 2.2, with the the probability density function gξ,βgiven by

gξ,β (y) =1

β

(1 + ξ

y

β

)− 1ξ−1

14

−4 −2 0 2 4−8

−6

−4

−2

0

2

4

Standard Normal Quantiles

Qua

ntile

s of

Inpu

t Sam

ple

QQ Plot of Sample Data versus Standard Normal

−8 −6 −4 −2 0 2 40

50

100

150

Figure 3: QQ-plot of the standardized residuals zt of the returns on S&P 500 index against the normaldistribution and the histogram suggest that Fz is leptokurtic.

S&P 500 Daily Return u := −z(k+1) k T ξ s.e. β s.e. z0.99 Z0.99

Loss Tail 1.2792 304 3040 0.0338 0.0518 0.5618 0.0434 2.6243 2.3263

Table 2: The estimated parameters for the 90th percentile of negative log returns on S&P 500. Thequantile estimator is larger than Z-score.

the log-likelihood function written as follows

logL (ξ, β;Y1, . . . , YNu) =

Nu∑j=1

log gξ,β (Yj)

= −Nu log β −(

1

ξ+ 1

) Nu∑j=1

log

(1 + ξ

Yjβ

)

should be maximized subject to the constraints that β > 0 and 1 + ξYjβ > 0 for all j numerically, since

analytical maximization by taking first order condition of the function with respect to each parameteris not basically possible.

The authors of the other studies on the similar subject often consider dividing a sequence of returns intomultiple periods and fit each of them to the GPD model to see if shape parameters are time varying, orthe kurtosis of a distribution is ultimately time-dependent as well as the mean and the variance. Eventhough the time ranges are different for each of data sequence available for a fund, however, this analysissimply fits the whole data to the model at once, i.e. one shape parameter and one quantile estimatorfor a fund, based on the result of Longin (1996) that concludes the tails of the distribution are stableover various time periods.

Threshold Selection Since manually investigating on what value of threshold would be appropriatefor each tail distribution of standardized returns on funds is another impracticable task through theanalysis, the procedure modified and suggested by McNeil and Frey (2000) is considered as an alternativeremedy. Instead of specifying a threshold value u for each series, the authors fix the number of observation

15

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Exceedance

Pro

babi

lity

Den

sity

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fitted Generalized Pareto CDFEmpirical CDF

Figure 4: The fitted density curve seems following the shape of the data well in the histogram (left) giventhe estimated parameters reported in Table 2. The comparison between the fitted and the empiricalcumulative distribution functions (right)

in the tail to be Nu = k where k � T for all cases. This then allows a random threshold at the k + 1thorder statistic in the ordered standardized residuals −z(1) ≥ −z(2) ≥ . . . ≥ −z(k) ≥ −z(k+1) ≥ . . . ≥−z(T ). Hence u := −z(k+1) and the excess data

{−z(1) + z(k+1),−z(2) + z(k+1), . . . ,−z(k) + z(k+1)

}is

fitted to the GPD. Then the quantile estimator for Fz discussed in Section 2.2.1 becomes as

zq,k = −z(k+1) +β

ξ

((nk

(1− q))−ξ− 1

)

and the notation zq,k has k as a subscript since it is also a function of k. For a simpler method, in thisstudy a threshold u is chosen to be the

(1− k

T

)quantile value of a given sample distribution, and the

resulting excess data has the exact same number of exceedances with similar values.

Similarly to the simulation study presented by McNeil and Frey to compare the efficiencies of theHill’s estimator and the GPD model with respect to choosing k and ultimately u, another Monte Carlosimulation is conducted in this study to investigate slightly different issues; returns on mutual fundsto be compared each other based on the magnitudes of the estimators mostly have different samplesizes, and fixing k for the all cases would yield unreliable estimators for some. As proposed earlier,therefore, rather the proportion of samples considered to be in the tail, i.e. k

T , is determined to be fixed.Assuming that the degree of tail heaviness varies across fund returns, the simulation is constructed withtwo Student t-distributions with different degrees of freedom, 4 and 6.

For given sample size n from 250 to 2500 with increment of 10 (approximately from 1 year to 10 yearslong worth or daily return observations), 1000 sequences of random variables are generated each fromF1 ∼ t4 and F2 ∼ t6, and ξ

(i)v,p and z(i)

0.99,v,p where p = 1 − k1000 , k = 30, 40, . . . , 400 and v = 4, 6. The

true values of ξv = 1v and z0.99,v are easily computed, and variances, biases and MSEs of estimators are

16

0500

10001500

20002500

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

percentile = (1−k/n)

Zq estimate for Student−t w/ df = 6

n

MS

E

0500

10001500

20002500

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

percentile = (1−k/n)

Zq estimate for Student−t w/ df = 4

n

MS

E

0500

10001500

20002500

0.6

0.7

0.8

0.9

1

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

percentile

n

Bia

s

0500

10001500

20002500

0.6

0.7

0.8

0.9

1

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

percentile

n

Bia

s

0500

10001500

20002500 0.6

0.70.8

0.910

0.1

0.2

0.3

0.4

0.5

0.6

0.7

percentilen

Var

0500

10001500

20002500 0.6

0.70.8

0.910

0.1

0.2

0.3

0.4

0.5

0.6

0.7

percentilen

Var

Figure 5: Estimated MSE, bias and variance of z0.99 against percentile of n iid random samples drawnfrom Student t-distribution with 4 (left) and 6 (right) degrees of freedom used for tail analysis.

17

0

1000

2000

3000

0.60.7

0.80.9

110

0.2

0.4

0.6

0.8

1

1.2

1.4

n

ξ estimate for Student−t w/ df = 4

percentile = (1−k/n)

MS

E

0

1000

2000

3000

0.60.7

0.80.9

110

0.2

0.4

0.6

0.8

1

1.2

1.4

n

ξ estimate for Student−t w/ df = 6

percentile = (1−k/n)

MS

E

0500

10001500

20002500

0.6

0.7

0.8

0.9

1

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

npercentile

Bia

s

0500

10001500

20002500 0.6

0.70.8

0.91

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

npercentile

Bia

s

0

1000

2000

3000

0.60.7

0.80.9

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

npercentile

Var

0

1000

2000

3000

0.60.7

0.80.9

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

npercentile

Var

Figure 6: Estimated MSE, bias and variance of ξ against percentile of n iid random samples drawn fromStudent t-distribution with 4 (left) and 6 (right) degrees of freedom used for tail analysis.

18

obtained in the following manner,

MSE(ξv,p

)=

1000∑i=1

(ξ(i)v,p − ξv

)2

= Var(ξv,p

)+(Bias

(ξv,p

))2

and visualized in Figure 5 and 6.

An important issue to note is that the 0.99 quantile estimator zq,k would not be a reliable measure forthe purpose of comparison of likelihood of extreme loss or risk managing performance among funds,since it appears that the mean squared error depends too much on the degrees of tail heaviness v as wellas the given sample sizes. On the other hand, the MSE of ξ is at the minimum level across the rangeof sample sizes no matter how heavy the tail is, as long as the proportion of samples to be included ismaintained above approximately 10%, or below the 90th percentile.

While the key feature of threshold selection is the variance-bias tradeoff, as long as ξ is considered tobe the sole estimator used for comparison, having the 90th percentile of ordered standardized residualsfitted to the GPD model does not appear to incur any serious problems from such tradeoff since asequence of returns at least has 315 observations. The major areas of rapid increasing and decreasingvariances and biases, respectively, shown in Figure 6 are avoided by doing so.

3.2.3 Portfolios of Monthly Returns and Time Series Regressions

After the parameters are all estimated in the same manner as described so far, Along with the ‘fundnumber’ identifier is assigned to each of ξi, all mutual funds are sorted into five quintile portfolios basedon their corresponding value of ξi, and series of monthly simple returns are retrieved from the databaseagain. Each portfolio is equally-weighted and rebalanced monthly, meaning that whenever a mutualfund selected to be in a certain portfolio becomes no longer available or newly incepted in the marketover the course of time range through which a portfolio is maintained, the weights of mutual funds in aportfolio are instantly adjusted monthly basis to reflect a new collection of equal weights for the fundsavailable at the moment.

In order to estimate return performances of portfolios while risk factors are controlled other than therisk of extreme events due to a heavy loss tail, three asset pricing models, the capital asset pricing model(CAPM), the Fama-French Three Factor as well as Carhart Four Factor OLS time-series regressions areimplemented as follows

Reit = αi + βiRMRFt + εi,t

Reit = αi + βiRMRFt + hiHMLt + siSMBt + εi,t

Reit = αi + βiRMRFt + hiHMLt + siSMBt +miMOMt + εi,t, t = 1, . . . , T

where Reit refers to the simple return on a portfolio i in excess of the one-month U.S. Treasury billreturn at time t. RMRF is the excess return on a value-weighted aggregate market proxy, and SMB,HML and MOM are returns on value-weighted, zero-investment, factor-mimicking portfolios for size,book-to-market equity, and one-year momentum in stock returns. The data of the risk factor loadings

19

are all obtained from the database Ken French provides. The Gibbons, Ross and Shanken’s (1989) GRSF -test is conducted to verify whether the intercepts of all time series regressions are simultaneously zeroas well as the t-test for checking individual significance. That is, it is testing the hypothesisH0 : αi = 0 ∀iwith the test statistics computed as

J =T −N −K

N

(1 + f ′Σ−1

f f)−1

α′Σ−1α ∼ FN,T−N−K

where f = (E [RMRF] , . . .)′ is a vector of the factor means and Σf and Σ are unbiased estimates of

the factor’s covariance (f−f)′(f−f)

T−1 and the residual covariance ε′εT−K−1matrices, respectively, while ε is

a T × N residual matrix. N is the number of portfolios, T is the time range, and K is the number offactors.

4 Results of the Analysis

Total 11,234 series of fund returns are fitted to a GARCH(p,q) model with 1 ≤ p, q ≤ 10, and asdiscussecd above, 13, 331 − 11, 234 = 2, 097 sample series of fund returns end up being discarded dueto the fact that they are not appropriately fitted with the numbers of ARCH and GARCH order lessthan 10. Since the GARCH model orders p and q necessary in order for a proper fit of fund returns tothe volatility model are not the best interest in this study, and the computation is integrated into theautomated iterative process, which does not require manual review, they are not separately reported inthis section.

Figure 7 visually reports the shape parameters estimated with the stadardized residual distributions ofall available sample funds in the sorted manner, from the lowest to the highest. For reference purpose,the ξ of the standardized reisudals of S&P 500 index returns, which have been used to illustrate theprocedures throughout the previous section, is also included indicated as the blue dash line.

It is worth noting that comparing the red line of sorted ξ to the grey plot illustrating the correspondingξ’s estimated with the same fund data, but instead by using raw returns and ignoring iid assumption,suggests how volatility clusturing contributes to heavy tails of return distributions. It is observablethat for the most of time the red line is placed beneath the grey plot, which also shows a subtle trendof increasing value as it goes to the right side. In a way, this reversely suggests to some extent thata stationary GARCH process with iid innovations zt belongs to the maximun domain of attraction ofthe Fréchet distribution, and further extracting stochastic volatilities has an equivalent effect of tailtrimming.

With the quintile portfolios formed with each of 2,246 funds based on the sorted shape estimators asdesigned and explained in the previous section, Table 3 reports the means and the medians of the shapeparameter of funds included in each portfolio as well as the Sharpe ratios calculated with the annualized

20

0 2000 4000 6000 8000 10000−2

−1

0

1

2

3

funds sorted by ξ est of zt

estim

atio

n of

ξ

ξ est (r

t) ξ est (z

t) ξ est SPX z

tξ est SPX r

t

0 2000 4000 6000 8000 100000

0.2

0.4

Sta

ndar

d E

rror

ξ e

st

0 2000 4000 6000 8000 100000

200

400

funds sorted by ξ est of zt

sam

ple

size

T

Figure 7: The plot on the top compiles the results from four analyses. ξ estimated with iid zt series arethe sorted variable and hence are lined up smoothly. Corresponding ξ estimated with non-iid rt seriesare included for the reference purpose. ξSP500 = 0.0338 is also shown. For the portion on the left sidestandard errors of ξ < − 1

2 are not computed as iterative numerical MLE becomes problematic. Thebottom plot shows the corresponding sizes of exceedances Nu = k.

ξ Ritiid∼ N

(µ, σ2

)assumed

Portfolio Mean Median Annualized Excess Return Annualized Volatility Sharpe Ratio1 Low -0.4834 -0.4402 -0.0263 0.1839 -0.14132 -0.1826 -0.1664 0.0057 0.1798 0.03173 -0.0599 -0.0593 0.0282 0.1713 0.16484 -0.0140 -0.0141 0.0169 0.1666 0.10205 High 0.0676 0.0419 0.0061 0.1631 0.0374

Table 3: Only 23.07% of the total funds appears to have heavy-tailed standardized loss distribution aftervolatility clustering is removed. Sharpe ratios are computed for each of quintile portfolios formed basedon the sorted ξ.

21

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 20090

500

1000

1500

2000

2500Year of Inception − Mutual funds with first 3000 lowest shape parameter

Figure 8: Histogram of year of inception of the funds with the first 3000 lowest ξ, indicating that theirloss distributions have light tail.

excess returns and volatilities computed in the following manner with the monthly return data,

Annualized Excess Return = exp

(12

T

T∑t=1

log(Rit −R

ft + 1

))

Annualized Volatility =√

12

√√√√ 1

T − 1

T∑t=1

(rt − r)2

Sharpe ratio =Excess Return

Volatility

where rt = log (1 +Rt)and r is the mean of monthly log returns.

It appears that results presented in Table 3 are intriguing as they are showing some divergences from theoriginal premises made before the study is conducted. Those funds with heavy-tailed standardized returndistributions only account for approximately 23 percent of the total equity mutual funds that have beenavailable from the late 90’s to date, and even when considering the clustered volatilities removed theportion is much smaller than ones of the plain-vanilla stocks documented in Wu et al. (2009) that exceeds60 percent. It is also interesting to find that the funds with rather light-tailed return distributions thelowest 3000 shape parameters estimated are mostly founded at the beginning of or amid the turmoil ofthe 2008-2009 financial crisis as shown in Figure 8, suggesting that they are managed to limit an extremeloss while it is most likely in the time range of data.

Most importantly, it seems that having a heavy tail, represented only by the whole funds in one portfolio(fifth quintile) and the slight portion of funds in another (fourth quintile), does not necessarily implya higher average return as a reward for taking a risky position more vulnerable to an extreme event.Rather, a negative relation between the shape parameter and the annualized return is observed in theupper quintile portfolios. It is however notable that the results are neither consistent with nor opposing

22

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

p1

p2

p3

p4

p5

Predicted E(Rei)=Σpβ

i,p*E(f

i,p)

Act

ual E

(Rei

)

Model Comparison: CAPM, FF3F and Carhart4F

p2

p3

p4

Figure 9: Average Returns E[Rei]vs. Predicted Return Values from each model; the CAPM (green),

the FF3F (red) and the Carhart 4F (blue) do not appear to deviate much from each other from across-sectional perspective in terms of model performance, assuming that αi is zero.

against the pattern of the traditional measures computed; the median (third quintile) portfolio performsbest among others and the excess return decreases as the shape parameter either increases or decreases.This certain pattern is also exhibited in Table 4, which shows the estimates of αfrom three differentmodels.

As shown, the individual αi’s of the both ends appear to be significantly different from zero, rathernegative, and the GRS tests also suggest they are simultaneously departing from zero level. To someextent, the result is in line with previous studies regarding mutual fund performances; Malkiel (1995)and Gruber (1996) argue that mutual funds underperformed the S&P 500 index even before expenses,which are in part considered in this study by containing all share classes of mutual funds for as thesample data. However the unexpected ‘moving up, then down’ pattern of excess returns and α’s shouldbe concerned as well. It is previously discussed that the distribution of returns on an actively-managedmutual fund could be perceived differently from the one of stock returns. It is becuase a fund portfoliodoes not only take long positions in stocks but also leveraged short positions as well as hedging positionswith derivatives to maximize returns while reducing risks. The weights of positions in fund portfoliosare changed from time to time at the discretion of fund managers as long as fund objectives are notviolated. In this sense, the distribution of a fund return can be said to be artificial to some degree.One possible interpretation is, assuming that the shape parameter of the median porfolio represents theoutcome of adequate efforts practiced for risk management, that higher or lower parameters could signaleven poorer performances of funds as results of failure to keep up with industry standards.

From the perspective of asset pricing, none of three models does not fully explain the persistence of“negative” alphas, which is interpreted as an “anomaly”, even though they are exhibiting the characteristicof arbitrage pricing theory (APT) style of model by having the R2 exceptionally high. Figre 9 showscorss-sectionally that the most of porfolios formed on the tail estimator of loss distribution are not linedup on the red 45-degree theoretical line, on which fitted values should be if a model correctly extracts

23

Portfolio 1 Low 2 3 4 5 High 3-1 Spread 3-5 SpreadMean Excess Return -0.08 0.18 0.36 0.25 0.16 0.44 0.2

Std Error 0.43 0.43 0.40 0.39 0.39 0.15 0.07

a. CAPMAlpha -0.36 -0.10 0.09 -0.01 -0.10 0.44 0.18

-2.67 -0.90 1.18 -0.19 -1.84 3.05 2.72RMRF 0.99 0.99 0.96 0.94 0.92 -0.03 0.04

36.76 45.52 65.84 89.01 85.28 -0.96 2.86Adjusted R2 0.90 0.94 0.97 0.98 0.98 0.01 0.06

GRS F -Statistic (p-value) 8.48 (0)

b. Fama-French Three Factor ModelAlpha -0.36 -0.20 -0.01 -0.04 -0.11 0.36 0.1

-2.68 -2.52 -0.15 -0.82 -2.44 2.51 2.6RMRF 0.99 0.94 0.95 0.96 0.95 -0.04 0

35.18 57.08 79.14 101.93 105.67 -1.39 0.37HML 0.00 -0.01 0.09 0.09 0.08 0.09 0.01

0.08 -0.48 5.85 7.05 6.98 2.31 0.99SMB 0.01 0.23 0.14 0.00 -0.04 0.13 0.18

0.30 10.59 8.99 -0.01 -3.48 3.4 18Adjusted R2 0.90 0.97 0.98 0.99 0.99 0.1 0.73

GRS F -Statistic (p-value) 7.69 (0)

c. Carhart Four Factor ModelAlpha -0.36 -0.20 -0.01 -0.04 -0.10 0.35 0.08

-2.65 -2.49 -0.21 -0.79 -2.30 2.46 2.48RMRF 0.98 0.94 0.95 0.96 0.93 -0.03 0.02

32.04 52.10 72.92 93.12 98.95 -0.98 2.63HML 0.00 -0.01 0.10 0.09 0.08 0.1 0.02

0.01 -0.53 5.92 6.85 6.47 2.41 2.14SMB 0.01 0.23 0.14 0.00 -0.03 0.13 0.17

0.36 10.48 8.72 0.06 -2.97 3.22 18.55MOM -0.01 0.00 0.01 0.00 -0.02 0.02 0.03

-0.39 -0.33 0.86 -0.43 -3.34 0.73 5.58Adjusted R2 0.90 0.97 0.98 0.99 0.99 0.1 0.77

GRS F -Statistic (p-value) 8.88 (0)

Table 4: The unit of mean excess returns, standard deviation and alphas is percent (%). The top rowof each panel report ξ-sorted quintile portfolio alphas measured as intercept in time series regressionsimplemented for the CAPM, the F-F three factor model and the Cahart four factor model. The italicizednumbers are t-statistics for the corresponding coefficients above of them. Spreads between the medianportfolio 3 with high alphas and other two portfolios 1 and 5 with low alphas are also fitted in the timeseries regressions.

24

risk premium contributed by the factors specified in the model from actual mutual fund excess returns.

5 Conclusion

This study is primarily concerned with tail estimation for financial asset return distributions, and par-ticularly mutual fund returns. Following the method proposed by McNeil and Frey (2000), by usingquasi-MLE method GARCH-type of models are fitted to return series with different orders specifiedfor each case, and standardized innovations computed with fitted conditional volatilities are verified tobe independently and identically distributed by the Ljung-box test before the EVT-based Peak OverThreshold method accompanied with the GPD model is applied to estimate the tail shape parameter.Since each return series included in the data set used in this study has a different time range, a MonteCarlo simulation is conducted in the manner similar to that of McNeil and Frey in order to find a univer-sally appropriate proportion of exceedances among the total observations fixed at k

T , which automaticallydetermines the level of threshold u in each case. At last, portfolios of mutual funds are formed basedon sorted shape parameters by their magnitudes to examine if there is a tangible connection betweenthe likelihood of extreme loss represented by the tail shape parameter and the expected return. Excessreturns of the portfolios are also decomposed into several well-known common risk factors by employingthe CAPM, the FF3F and the Carhart4F models, and alphas, the portion of returns not captured bythe factors, are obtained.

During the simulation study, it is found that the quantile estimator zq would not be a good basis forcomparisons among tail distibutions as the mean squared error drastically changes as a function ofT , the number of total observations, and v, the degrees of freedom of t-distribution that representsthe magnitude of tail heaviness in the simulation. The mean squared error of the shape parameterestimate remains stable no matter what v is, as long as a certain number of exceedances in terms ofthe proportion > 10% are secured. Beyond that step, it is the matter of bias-variance trade-off. Forthe analysis having 10% of total observations as exceedances is considered proper and the threshold u ischosen correspondingly.

It is one of the findings from this study that volatility clusters embedded in return series much contributeto the tail heaviness of return distributions as the estimated parameters from stadardized innovationdistributions appear to be usually smaller than the corresponding parameters from raw return distribu-tions despite being aware of that they are not reliable estimates as correlations between observations areignored when MLE is impletemented. For estimation purpose, Statistics Toolbox offered in MATLABis extensively utilized as maximizing log-likelihood of the GPD models can only be done by numericalmethods.

About the three fourth of all mutual funds examined in this study appear to have rather light negativetails in their standardized innovation distribution while the one of S&P 500 still retains the leptokurticcharacteristic, and this finding somehow opposes conclusions of many studies in the finance literaturethat claim financial return series are mostly heavy-tailed. Hence McNeil and Frey’s assertion that theHill’s estimator is inappropriate for analysis sounds correct. The distribution of mutual fund returns

25

could be considered retrospectively as a result from artificial efforts on optimizing fund portfolios, ratherthan simple random realizations of stock price appreciations or depreciations caused by idiosyncraticshocks. This is then connected to another finding that the famous asset pricing models are not capableof fully explaining the anomaly, denoted as α, with the systematic risk factors and the style factor‘momentum’ in Carhart’s model. It is also important to conclude that there does not appear to be afavorable, positive correlation between the shape parameters and the returns. However, with the medianporfolio showing the best performance among all, it is reasonable to see it as the representation of thebest industry practice. In other words, even though there is not an absolute value of the shape parameterimplying better return performance at each moment, the median value would always be the standard.Further studies with fund samples of a longer time span, correctly identified to have equity positionsonly, would be desirable to confirm this conclusion.

6 Appendix

MATLAB codes created for this research are available for review in the next pages after the bibliography.

26

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2/20/11 5:15 PM C:\Users\Administrator\D...\CRSPdata2mat.m 1 of 1

% Raw Data transformation into a Matrix clear all; close all; data = load('c:\sort4_5.csv'); % monthly dataindex = load('c:\a5.csv');datedata = load('c:\date.csv');data(data(:,3)==-99,3) = NaN;tsdatafull = fints(datenum(num2str(datedata),'yyyymmdd'),... zeros(length(datedata),1), 'dummy','M'); % compiling time series of each fund provided as a vector into a matrix% with the size fundnumber*total time spancount = length(index);for ii = 1: count % examine each fund by looping fundid = index(ii); % extract a subset of one fund data corresponding to its id number fund = data(data(:,1)==fundid,:); fts = fints(datenum(num2str(fund(:,2)),'yyyymmdd'),... fund(:,3), ['s' num2str(fundid)],'M'); tsdatafull = merge(tsdatafull,fts);end tsdatafull = rmfield(tsdatafull, 'dummy');R = fts2mat(tsdatafull); % matrix formed.

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% Monte Carlo Simulation for Threshold Selection df = 4;q = .99;Zq = tinv(q,df);Xi = 1/df; VarXi4 = zeros(numel(.97:-0.01:.60),numel(250:10:2500));BiasXi4 = zeros(numel(.97:-0.01:.60),numel(250:10:2500));MSEXi4 = zeros(numel(.97:-0.01:.60),numel(250:10:2500));VarZq4 = zeros(numel(.97:-0.01:.60),numel(250:10:2500));BiasZq4 = zeros(numel(.97:-0.01:.60),numel(250:10:2500));MSEZq4 = zeros(numel(.97:-0.01:.60),numel(250:10:2500)); nn = 1; for n = 250:10:2500; XiHatKJ = zeros(1000,numel(.97:-0.01:.60)); ZqHatKJ = zeros(1000,numel(.97:-0.01:.60)); for j = 1:1000; Series = trnd(df,n,1); XiHatK = zeros(numel(.97:-0.01:.60),1); ZqHatK = zeros(numel(.97:-0.01:.60),1); ThresK = zeros(numel(.97:-0.01:.60),1); kk = 1; for percentile = .97:-0.01:.60; u = quantile(Series,percentile); Y = Series(Series>u)-u; k = numel(Y); [ParamHat] = gpfit(Y); XiHatK(kk) = ParamHat(1); BetaHat = ParamHat(2); ZqHatK(kk) = u + (BetaHat/XiHatK(kk))... *(((1-q)/(k/n))^(-XiHatK(kk))-1); kk = kk+1; end XiHatKJ(j,:) = XiHatK'; ZqHatKJ(j,:) = ZqHatK'; end VarXi4(:,nn) = var(XiHatKJ)'; BiasXi4(:,nn) = mean(XiHatKJ)'-Xi; MSEXi4(:,nn) = VarXi4(:,nn)+BiasXi4(:,nn).^2; VarZq4(:,nn) = var(ZqHatKJ)';

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BiasZq4(:,nn) = mean(ZqHatKJ)'-Zq; MSEZq4(:,nn) = VarZq4(:,nn)+BiasZq4(:,nn).^2; nn = nn+1;end df = 2;q = .99;Zq = tinv(q,df);Xi = 1/df; VarXi2 = zeros(numel(.97:-0.01:.60),numel(250:10:2500));BiasXi2 = zeros(numel(.97:-0.01:.60),numel(250:10:2500));MSEXi2 = zeros(numel(.97:-0.01:.60),numel(250:10:2500));VarZq2 = zeros(numel(.97:-0.01:.60),numel(250:10:2500));BiasZq2 = zeros(numel(.97:-0.01:.60),numel(250:10:2500));MSEZq2 = zeros(numel(.97:-0.01:.60),numel(250:10:2500)); nn = 1; for n = 250:10:2500; XiHatKJ = zeros(1000,numel(.97:-0.01:.60)); ZqHatKJ = zeros(1000,numel(.97:-0.01:.60)); for j = 1:1000; Series = trnd(df,n,1); XiHatK = zeros(numel(.97:-0.01:.60),1); ZqHatK = zeros(numel(.97:-0.01:.60),1); ThresK = zeros(numel(.97:-0.01:.60),1); kk = 1; for percentile = .97:-0.01:.60; u = quantile(Series,percentile); Y = Series(Series>u)-u; k = numel(Y); [ParamHat] = gpfit(Y); XiHatK(kk) = ParamHat(1); BetaHat = ParamHat(2); ZqHatK(kk) = u + (BetaHat/XiHatK(kk))... *(((1-q)/(k/n))^(-XiHatK(kk))-1); kk = kk+1; end XiHatKJ(j,:) = XiHatK'; ZqHatKJ(j,:) = ZqHatK'; end

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VarXi2(:,nn) = var(XiHatKJ)'; BiasXi2(:,nn) = mean(XiHatKJ)'-Xi; MSEXi2(:,nn) = VarXi2(:,nn)+BiasXi2(:,nn).^2; VarZq2(:,nn) = var(ZqHatKJ)'; BiasZq2(:,nn) = mean(ZqHatKJ)'-Zq; MSEZq2(:,nn) = VarZq2(:,nn)+BiasZq2(:,nn).^2; nn = nn+1;end df = 6;q = .99;Zq = tinv(q,df);Xi = 1/df; VarXi6 = zeros(numel(.97:-0.01:.60),numel(250:10:2500));BiasXi6 = zeros(numel(.97:-0.01:.60),numel(250:10:2500));MSEXi6 = zeros(numel(.97:-0.01:.60),numel(250:10:2500));VarZq6 = zeros(numel(.97:-0.01:.60),numel(250:10:2500));BiasZq6 = zeros(numel(.97:-0.01:.60),numel(250:10:2500));MSEZq6 = zeros(numel(.97:-0.01:.60),numel(250:10:2500)); nn = 1; for n = 250:10:2500; XiHatKJ = zeros(1000,numel(.97:-0.01:.60)); ZqHatKJ = zeros(1000,numel(.97:-0.01:.60)); for j = 1:1000; Series = trnd(df,n,1); XiHatK = zeros(numel(.97:-0.01:.60),1); ZqHatK = zeros(numel(.97:-0.01:.60),1); ThresK = zeros(numel(.97:-0.01:.60),1); kk = 1; for percentile = .97:-0.01:.60; u = quantile(Series,percentile); Y = Series(Series>u)-u; k = numel(Y); [ParamHat] = gpfit(Y); XiHatK(kk) = ParamHat(1); BetaHat = ParamHat(2); ZqHatK(kk) = u + (BetaHat/XiHatK(kk))... *(((1-q)/(k/n))^(-XiHatK(kk))-1); kk = kk+1; end

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XiHatKJ(j,:) = XiHatK'; ZqHatKJ(j,:) = ZqHatK'; end VarXi6(:,nn) = var(XiHatKJ)'; BiasXi6(:,nn) = mean(XiHatKJ)'-Xi; MSEXi6(:,nn) = VarXi6(:,nn)+BiasXi6(:,nn).^2; VarZq6(:,nn) = var(ZqHatKJ)'; BiasZq6(:,nn) = mean(ZqHatKJ)'-Zq; MSEZq6(:,nn) = VarZq6(:,nn)+BiasZq6(:,nn).^2; nn = nn+1end subplot(1,2,1);surfc([250:10:2500]',[.97:-0.01:.60]',MSEZq4)subplot(1,2,2);surfc([250:10:2500]',[.97:-0.01:.60]',MSEZq2)subplot(1,2,1);surfc([250:10:2500]',[.97:-0.01:.60]',MSEXi4)subplot(1,2,2);surfc([250:10:2500]',[.97:-0.01:.60]',MSEXi2) surfc([250:10:2500]',[.97:-0.01:.60]',VarXi2)hold on;surfc([250:10:2500]',[.97:-0.01:.60]',BiasXi2)hold off;shading interp; surfc([250:10:2500]',[.97:-0.01:.60]',VarXi4)hold on;surfc([250:10:2500]',[.97:-0.01:.60]',BiasXi4)hold off;shading interp;

2/20/11 5:14 PM C:\Users\Administrator\Doc...\GPDcompute.m 1 of 3

% GPD Fitting Code% clear all; close all; data = load('c:\sort2_3.csv');index = load('c:\fundindex3.csv');data(data(:,3)==-99,3) = NaN; count = length(index);output = zeros(count,16);for ii = 1: count % examine each fund by looping fundid = index(ii); % extract a subset of one fund data corresponding to its id number fund = data(data(:,1)==fundid,:); % identify first/last available (non-NaN) data points fundbeg = find(~isnan(fund(:,3)),1,'first'); fundend = find(~isnan(fund(:,3)),1,'last'); fund = fund(fundbeg:fundend,:); % save basic fund life data [start end no.days] life = [fund(1,2) fund(end,2) size(fund,1)]; % replace NaN with approximated and construct log return series if sum(isnan(fund(:,3)))>0 % create financial time series object for using fillts fts = fints(datenum(num2str(fund(:,2)),'yyyymmdd'),... log(fund(:,3)+1), [],'D'); ftsfill = fillts(fts,'l'); rt = fts2mat(ftsfill); else rt = log(fund(:,3)+1); end % FIRST FLAG % Augmented Dickey-Fuller test to see if the series is stationary if ~adftest(rt); output(ii,:) = [fundid NaN(1,14) 88]; continue end % By the weak form of EMH, conditional mean dynamics of equity security % returns are assumed to be negligible. empirical tests with several % stock returns suggest that the AR coefficients appear to be % insignificant when jointly estimated with GARCH models. % Mean Model: r(t) = mu + e(t), e(t) = s(t)*z(t)

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% Volatility Model: s(t) = omega + sum(alpha(q)*e^2(t-q)) + % ... sum(beta(p)*s^2(t-p)) : GARCH(p,q) model mu = mean(rt); % unconditional mean of return series et = rt-mu; % demeaned return (innovation) series pp = 1; % GARCH order -- start with GARCH(1,1) model qq = 1; % ARCH order Spec = garchset('C',0,'P',pp,'Q',qq,'variancemodel',... 'GARCH','display','off','FixC',1); indep = 1; % preparing for while loop while indep~=0 [paramVOL,seVOL,logLVOL,e,s] = garchfit(Spec,et); % fitted to the residual series z = e./s; % standardized innovation zcorr = sum(lbqtest(z,'lags',10:10:20)); zsqcorr = sum(lbqtest(z.^2,'lags',10:10:20)); indep = zcorr+zsqcorr; % independence of z(t) check upto second moment % increase ARCH order first then GARCH order one by one if pp<qq qq = qq-1; % switch order pp = pp+1; else qq = qq+1; end % stop loop when both orders P and Q are above 6. if pp+qq>=20 break end Spec = garchset(Spec,'P',pp,'Q',qq); end % SECOND FLAG % if GARCH ARCH order cannot be limited to (10,10) then stop anlaysis if pp+qq>=20 output(ii,:) = [fundid NaN(1,14) 99]; continue end [JB, JBp, JBstat] = jbtest(z); % normality check % fitting loss tail to GPD Distribution z = -z; % convert to the series of NEGATIVE standardized residual percentile = .90; % set top 80 percentile of -z for the Loss tail % threshold u is chosen such that prop of z>u are 20 percent u = quantile(z,percentile); excess = z(z>u)-u; k = numel(excess);

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[paramTAIL, ciTAIL] = gpfit(excess); [NlogLTAIL, covTAIL] = gplike(paramTAIL,excess); XiHat = paramTAIL(1); XiSE = sqrt(covTAIL(1,1)); BetaHat = paramTAIL(2); BetaSE = sqrt(covTAIL(2,2)); Z99Hat = u + (BetaHat/XiHat)*((((1-.99)/(k/life(3)))^(-XiHat))-1); % Output table for Prelim Anal-- % [1-FundID 2-BeginDate 3-EndingDate 4-ObsDays 5-GARCHorder % 6-ARCHorder 7-jacberaPval 8-ThresholdU 9-NumExcess 10-ShapeEST % 11-ShapeSE 12,13-95CIShape 14-ScaleEST 15-ScaleSE 16-Z99val] % (IF filled with NaN and only identified with fund id, then its % return series does not considered to be a stationary process % by Augmented Dickey-Fuller test (Err code: 88) or cannot achieve % iid Z sequence by fitting upto GARCH(6,6)) output(ii,:) = [fundid life paramVOL.P paramVOL.Q JBp u k XiHat XiSE ... ciTAIL(:,1)' BetaHat BetaSE Z99Hat]; end

2/20/11 5:15 PM C:\Users\Administrator\Doc...\regression.m 1 of 2

% Asset Pricing Model Application load('output_portfolios_formed_sorted.mat') r1 = nanmean(port1')';r2 = nanmean(port2')';r3 = nanmean(port3')';r4 = nanmean(port4')';r5 = nanmean(port5')'; % equally-weighted portfolio return x = load('c:/factors.csv'); rmrf = x(:,2); smb = x(:,3); hml = x(:,4);umd = x(:,5);rf = x(:,6);labels = { 'p1' 'p2' 'p3' 'p4' 'p5'}; r = [r1 r2 r3 r4 r5]*100;rx = r-rf*ones(1,5);arx = mean(log(rx/100+1))*12; % annualized excess returnav = std(log(r/100+1))*sqrt(12); % annualized volatilitysr = arx./av;T = size(rx,1);N = size(rx,2);f = [rmrf];[alpha, beta, R2, siga, sigb, chi2stat, chi2vals, chi2pv, N, Fstat, Fpvals, Fpv, TNK] = tsregress2(rx,f); disp('CAPM'); fprintf(' Mean xs return %7.2f %7.2f %7.2f %7.2f %7.2f \n', mean(rx)); fprintf(' std err Mean %7.2f %7.2f %7.2f %7.2f %7.2f \n', std(rx)./T^0.5); fprintf(' CAPM alpha %7.2f %7.2f %7.2f %7.2f %7.2f \n', alpha'); fprintf(' t(alpha) %7.2f %7.2f %7.2f %7.2f %7.2f \n', alpha'./siga'); fprintf(' CAPM beta %7.2f %7.2f %7.2f %7.2f %7.2f \n', beta(:,1)'); fprintf(' CAPM R2%7.2f %7.2f %7.2f %7.2f %7.2f \n', R2'); fprintf('chi2 statistic, prob value %7.2f %7.2f \n' , [chi2stat 100*chi2pv]);fprintf('10, 5, 1%% p values of chi2(N) %7.2f %7.2f %7.2f \n', chi2vals);fprintf('F statistic, prob value %7.2f %7.2f \n', [Fstat 100*Fpv]);fprintf('10, 5, 1%% p values of F %7.2f %7.2f %7.2f \n', Fpvals); f = [rmrf hml smb];[alpha, beta, R2, siga, sigb, chi2stat, chi2vals, chi2pv, N, Fstat, Fpvals, Fpv, TNK] = tsregress2(rx,f); disp('FF3F'); fprintf(' 3F alpha %7.2f %7.2f %7.2f %7.2f %7.2f \n', alpha'); fprintf(' t on 3F alpha %7.2f %7.2f %7.2f %7.2f %7.2f \n', alpha'./siga'); fprintf(' 3F b %7.2f %7.2f %7.2f %7.2f %7.2f \n', beta(:,1)'); fprintf(' 3F h %7.2f %7.2f %7.2f %7.2f %7.2f \n', beta(:,2)');

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fprintf(' 3F s %7.2f %7.2f %7.2f %7.2f %7.2f \n', beta(:,3)'); fprintf(' 3F R2 %7.2f %7.2f %7.2f %7.2f %7.2f \n', R2'); fprintf('chi2 statistic, prob value %7.2f %7.2f \n' , [chi2stat 100*chi2pv]);fprintf('10, 5, 1%% p values of chi2(N) %7.2f %7.2f %7.2f \n', chi2vals);fprintf('F statistic, prob value %7.2f %7.2f \n', [Fstat 100*Fpv]);fprintf('10, 5, 1%% p values of F %7.2f %7.2f %7.2f \n', Fpvals); f = [rmrf hml smb umd];[alpha, beta, R2, siga, sigb, chi2stat, chi2vals, chi2pv, N, Fstat, Fpvals, Fpv, TNK] = tsregress2(rx,f); disp('carhart portfolios, 4 factor model'); fprintf(' 4F alpha %7.2f %7.2f %7.2f %7.2f %7.2f \n', alpha'); fprintf(' t on 4F alpha %7.2f %7.2f %7.2f %7.2f %7.2f \n', alpha'./siga'); fprintf(' 4F b %7.2f %7.2f %7.2f %7.2f %7.2f \n', beta(:,1)'); fprintf(' 4F h %7.2f %7.2f %7.2f %7.2f %7.2f \n', beta(:,2)'); fprintf(' 4F s %7.2f %7.2f %7.2f %7.2f %7.2f \n', beta(:,3)'); fprintf(' 4F u %7.2f %7.2f %7.2f %7.2f %7.2f \n', beta(:,4)'); fprintf(' 4F R2 %7.2f %7.2f %7.2f %7.2f %7.2f \n', R2'); fprintf('chi2 statistic, prob value %7.2f %7.2f \n' , [chi2stat 100*chi2pv]);fprintf('10, 5, 1%% p values of chi2(N) %7.2f %7.2f %7.2f \n', chi2vals);fprintf('F statistic, prob value %7.2f %7.2f \n', [Fstat 100*Fpv]);fprintf('10, 5, 1%% p values of F %7.2f %7.2f %7.2f \n', Fpvals);