the four horsemen: heavy-tails, negative skew, volatility cl

Discussion Paper: 2014-004

1

Business School

Discipline of Finance

Discussion Paper

2014-004

“The Four Horsemen:

Heavy-tails, Negative Skew, Volatility

Clustering, Asymmetric Dependence”

David Allen Pembroke College

University of Cambridge

Stephen Satchell Trinity College Cambridge /

University of Sydney Business School


2

WORKING PAPER

December 4th

, 2013

ABSRACT

In the wake of the worst financial crisis since the Great Depression, there has been a

proliferation of new risk management and portfolio construction approaches. These

approaches endeavour to capture the “stylised facts” of financial asset returns: heavy tails,

negative skew, volatility clustering and asymmetric dependence. Many approaches

capture two or three characteristics, while capturing all four in a scalable framework

remains elusive. We propose a novel approach that captures all four stylised

characteristics using EGARCH, the skewed-t copula and extreme-value theory. Using

eight data sets we show the approach is superior to eight benchmark models in both a

VaR forecasting and a dynamic portfolio rebalancing framework. The approach generates

significant economic value relative to the 1/N rule and the Gaussian approach. We also

find that accounting for asymmetric dependence leads to a consistent improvement in

VaR prediction and out-of sample portfolio performance including lower drawdowns.

I. Introduction

Since 2008, 465 banks have failed in the United States alone, amounting to $687 bn. in total assets1.

The failures at systemically important institutions, in particular Bear Stearns, Lehman Brothers, AIG,

Fannie Mae and Freddie Mac, reverberated around the global economy, wiped 23.4% off the net

wealth of American households between 2007 and 2009 (Kennickell, 2011) and precipitated the Great

Recession. The Financial Crisis Inquiry Commission concluded that rather than being due to

exogenous unavoidable events, the “dramatic failures of corporate governance and risk

management at many systemically important financial institutions were a key cause of this crisis”. In

2012, JP Morgan suffered a $5.8 bn. loss on a single derivative trade amounting to 25% of the firm’s

annual profits. The potential for trading losses to erode shareholder wealth and to trigger systemic

events has galvanised renewed interest in approaches to risk management and portfolio construction.

In the current work we develop a multivariate model that captures the four stylised facts of financial

asset returns: heavy tails, negative skew, volatility clustering, and asymmetric tail dependence. We

find the approach produces superior VaR forecasts and adds significant economic value in out-of-

sample tests relative to eight benchmark methodologies. This finding is robust across eight data sets

including a range of asset classes. The majority of the uplift in performance derives from accounting

for non-stochastic volatility in line with the empirical work of Fleming, Kirby, and Ostdiek (2001,

2003) and Kirby, and Ostdiek (2012) and the analytical work of Allen, Lizieri and Satchell (2013).

From the viewpoint of the corporate entity it may appear uncontroversial that prudent risk

management increases the value of a firm and is worthwhile. However, as McNeil, Frey, and

Embrechts (2005) point out, from a corporate finance perspective it is by no means obvious that in a

world with perfect capital markets, including no informational asymmetries, no taxes, transaction

costs, or bankruptcy costs, that risk management should enhance shareholder value. If shareholders

have access to perfect capital markets then they can undertake their own risk management

transactions and will not pay a premium for the corporation to do so on their behalf. The potential

irrelevance of corporate risk management follows from the Modigliani-Miller (1958) theorem on

1 Federal Deposit Insurance Corporation, Failed Bank List

http://en.wikipedia.org/wiki/Corporate_governance

http://en.wikipedia.org/wiki/Risk_management

http://en.wikipedia.org/wiki/Risk_management


3

capital structure and the value of the firm. In practice informational asymmetries and inferior access

to capital markets limit the ability of shareholders to undertake risk-management positions. Further,

risk management makes bankruptcy less likely and less costly.

Indeed, sensible capital allocation and risk management are widely seen as the essential core

competencies of a financial institution (Allen and Santomero, 1998). If capital reserves are excessive,

then shareholder returns, business and consumer credit, and economic growth are inhibited. If capital

reserves are too low, there is an increased risk of bank failures, erosion of shareholder value and a

contraction in economic growth. The potential for trading losses to erode shareholder wealth and

imperil the very survival of financial entities has been a recurring theme over the last two decades. In

1995, Barings Bank, the oldest merchant bank in the United Kingdom was ruined by a $1.79 bn. loss

betting on equity futures. In 1998, Long Term Capital Management suffered a $5.85 bn. loss on

interest rate and equity derivatives prompting the Federal Reserve to coordinate a bailout. In 2006,

Amaranth Advisors collapsed after sustaining a $6.69 bn. loss on natural gas futures. In 2008,

Morgan-Stanley lost 8.67 bn. on credit default swaps relating to the subprime market. More recently,

in 2012, JP Morgan suffered a $5.8 bn. loss on a single derivative trade amounting to 25% of the

firm’s annual profits. The potential for trading losses to destroy shareholder wealth and to trigger

systemic events has galvanised renewed interest in approaches to risk management and portfolio

construction.

From a regulatory perspective, the recent transition to Basel 2.5 has magnified the importance of

quantifying risk accurately. The Basel 2.5 reforms are part of the Basel Committee on Banking

Supervision’s (BCBS) comprehensive response to the 2008 financial crisis. Under Basel 2, total

required market risk capital was given by the sum of a market value-at-risk (VaR) component and a

standardised specific risk measure. Under Basel 2.5, risk capital must also cover a stressed VaR

measure and two incremental risk charges relating to unsecuritised credit positions and correlation

trades. The BCBS quantitative impact study estimates that the adoption of Basel 2.5 will lead to a

three-fold increase in market risk regulatory capital2. The “traffic-light” system that operated under

Basel II is preserved under Basel 2.5 such that an excessive number of VaR model violations trigger

increased capital requirements and increased regulatory scrutiny.

It is being increasingly recognised that financial markets may be inherently unstable. For a long time

the neoclassical view of markets held sway. The efficient market hypothesis (EMH) of Fama (1965)

maintains that agents are rational and that prices reflect all available information instantaneously.

Under the EMH, the market is always in equilibrium and speculative asset bubbles are not possible.

The Great Moderation in the variability of GDP observed in the United States (Stock and Watson,

2002) and most OECD countries from the mid-1980s onwards was seen by many as evidence of the

efficiency and stability of markets and the success of monetarism (Bernanke, 2005, Summers, 2005).

Since the late 1970s behavioural evidence has been accumulating that suggests that humans are not

the hyper-rational computing machines (Kahneman and Tversky, 1979) of economic textbooks and

that markets may not be perfectly efficient (Shiller, 1981)3. The Great Recession has led to a

resurgence of the idea that free-market systems are inherently unstable. Alan Greenspan admitted that

he assumed that rational firms would not expose themselves to annihilation and that the “whole

intellectual edifice had collapsed”4.

2 Basel Committee on Banking Supervision, Analysis of the trading book quantitative impact study, October,

2009 3 See section 2 for an expanded list of references

4 House Committee on Oversight and Government Reform


4

The idea that disequilibrium may be the rule rather than the exception dates back to Schumpeter

(1928) and Fischer (1933). Fischer (1933) formally an apostle of the neoclassical school recanted his

beliefs during the Great Depression arguing that it is as absurd to assume that markets will stay in

equilibrium as it is “to assume that the Atlantic Ocean can ever be without a wave”. The work of

Minsky (1982), long dismissed by the neoclassical school is now being taken seriously. Minsky

(1982) argues that periods of consistent economic growth lead to upward revisions in expectations, a

greater willingness to borrow and lend, the extrapolation of growth rates and in time asset bubbles. It

is in this sense that Minsky (1982) argues that “stability is destabilising”. The long history of booms

and busts in free-market systems, including the South Sea Bubble, the Tulip mania, the 1970s Savings

and Loan crisis, the early 2000s Technology bubble to name but a few is certainly consistent with this

view. If markets are inherently unstable, and periods of tranquillity presage periods of turbulence, we

argue it is all the more important to understand the strengths and weaknesses of risk management and

portfolio construction approaches.

The remainder of this paper is organised as follows. In Section II, we provide a survey of the

literature. In section III we introduce the GSEV model. Section IV describes our eight benchmark

portfolio construction models and discusses our data and methodology for evaluating VaR forecasts,

and our dynamic rebalancing and performance evaluation frameworks. Section V discusses our

findings, and section VI concludes.

II. Literature Survey

Gaussian approaches to risk management have been roundly criticised in the literature. Li’s (2000)

Gaussian copula model for CDOs has been described as the “formula that felled Wall Street”5 and

“instrumental in causing the unfathomable losses that brought the world financial system to its

knees”6. The mainstay of modern portfolio theory, the mean-variance model has been criticised in

both the finance literature and the financial press. DeMiguel, Garlappi, and Uppal (2009) conclude

that there are “many miles to go” before the promised benefits of optimal portfolio choice can be

realised out of sample. Taleb (2009) describes mean-variance as “hot air” and a “quack remedy”.

Even the Babylonian Talmud has contributed to the debate with the sage advice that “one should

always divide his wealth into three parts: a third in land, a third in merchandise, and a third ready to

hand”7. The criticisms of the mean-variance approach centre on the alternative maintained hypotheses

that investors have mean-variance utility or that portfolio returns follow the normal distribution

(Cootner, 1964 and Lintner, 1972) and are independently and identically distributed. A careful

reading of Markowitz (1952) however reveals no mention of the Gaussian distribution. In fact

Markowitz and Usmen (1996a, 1996b), in Markowitz’s sole investigation of the return generating

process, conclude that the log returns of the S&P 500 are well described by a student-t distribution

with between four and five degrees of freedom. Nor did Markowitz assume that investors have mean-

variance utility. Rather, Levy and Markowitz (1979) find that the mean-variance approach serves as a

robust approximation to a wide range of utility functions, a finding that has been widely replicated

(Pulley, 1981, Kroll, Levy and Markowitz, 1984, and Simaan, 1993).

Mandelbrot (1963) finds that the commodity return distributions are heavy tailed, an observation that

has since been made in every major investment class including equities (Fama, 1963), fixed income

5 Jones, S. The formula that felled Wall St, Financial Times, April 24

th, 2009

6 Salmon, F. Recipe for Disaster: The formula that killed Wall street, Wired magazine, 2009

7 This quote appeared in DeMiguel, Garlappi, and Uppal (2009) and is attributed to Rabbi Issac bar Aha in the

Babylonian Talmud: Tractate Baba Mezi’a, folio 42a


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(Amin and Kat, 2003), currencies (Westerfiled, 1977), REITS (Lizieri, Satchell, Zhang, 2007), and

hedge funds (Agrawal and Naik, 2004). Further, it is well established that financial assets are often

negatively skewed where large declines are more common than large rallies (Kraus and Litzenberger,

1976, Beedles, 1979, Alles and Kling, 1994, Harvey and Siddique, 1999). Mandelbrot (1963) also

identifies volatility clustering, where “large changes tend to be followed by large changes, of either

sign, and small changes tend to be followed by small changes”. Again this finding is pervasive across

asset classes including equities (Fama, 1965), fixed income (Weiss, 1984) and foreign exchange

(Baillie and Bollerslev, 1989). It was later shown that financial assets exhibit a “leverage” effect

where negative innovations lead to larger upward revisions in conditional volatility than downward

revisions (Black, 1976, Christie, 1982, Glosten, Jaganathan and Runkel, 1993, Hansen and Lunde,

2005)8. It is generally acknowledged that the increase in financial leverage alone is insufficient to

account for the observed increase in volatility following market downturns (Bollerslev, 2009), and

that behavioural factors may be at work. We also capture this effect within our approach.

More recently, the asymmetric tail dependence between assets has been identified, aligning with the

market adage, that “when the market crashes all correlations go to one”. Erb, Harvey and Viskanta

(1994) show that the correlations between the G7 country indices are higher in up markets than in

down markets. Karolyi and Stulz (1996) find that the dependence between U.S. and Japanese stocks

increases during large shocks. Ang and Bekaert (2002) show that the correlations between

international equity indices tend to increase during volatile periods. The same pattern is evident within

countries. Longin and Solnik (2001) and Ang and Chen (2002) show that the dependence between

individual stocks and the aggregate market index is significantly higher for downside moves than for

upside moves9. Patton (2004) finds evidence of asymmetric dependence for indices of U.S. large and

small-cap portfolios. Hong, Tu and Zhou (2007) provide a model-free test for asymmetric correlation

and conclude that there is strong evidence for asymmetries for the “size” and “momentum” portfolios.

Beine, Cosma and Vermeulen (2010) argue that financial liberalization has significantly increased

left-tail comovement in international equities. Lower tail dependence reduces the ability of an agent to

protect against downside risk. Moreover the agent that does not account for tail dependence will

structurally underestimate downside risk. Despite the recent advances in the understanding of the

dependence structure between assets, linear correlation and dependence are often used

interchangeably. Correlation however is only appropriate for elliptical distributions. To take an absurd

example, consider the case of the unit circle around the origin. The Pearson’s correlation coefficient

will equal zero despite perfect dependence.

Collectively these distributional characteristics are often referred to as “stylised facts” of financial

markets 10

.

I. Leptokurtosis or “heavy tails”

II. Negative skew

III. Heteroskedasticity and “volatility clustering”

IV. Tail dependence

There is a growing body of evidence indicating that the accuracy of VaR estimates and the

performance of portfolio construction techniques can be improved by accounting for the respective

stylised facts. Lucas and Klaasen (1998) show that the failure to account for heavy tails leads to the

underestimation of the true risk by 25-30% and overly aggressive portfolio allocations at the 1% VaR

8 It is generally acknowledged that the asymmetry is not however present for currencies

9 In particular for value and small capitalisation stocks

10 See for example Cont (2001) or McNeil, Frey and Embrechts (2005)


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confidence level; at the 5% VaR level the Gaussian assumption leads to an overestimation of risk and

overly conservative allocations. McNeil, Frey and Embrechts (2005) show that the standard

unconditional Gaussian approach systematically results in too many VaR violations at the 1% level11

for a collection of equity indices. Harvey, Liechty, Liechty, and Muller (2010) show that

incorporating the effect of skew into the investor’s utility function leads to significant increases in

expected utility. This result mirrors the earlier results of Prakash, Chang, and Pactwa (2003) using a

Polynomial Goal Programming approach. Accounting for stochastic volatility has also been shown to

significantly improve outcomes for investors. Fleming, Kirby, and Ostdiek (2001) show that

“volatility timing” leads to a significant uplift in investor welfare that exceeds typical active

management fees. There is also evidence to suggest that accounting for the asymmetric dependence

between assets can improve investor welfare. Kole, Koedijk, and Verbeek (2007) show that the

Gaussian copula results in a significant underestimation of joint extreme downward realisations and

overestimates potential diversification. Alcock and Hatherley (2009) show that accounting for

asymmetric dependence leads to significant gains in economic value.

The deficiencies of the Gaussian approach have been well recognised for some time. Alan Greenspan

writes:

From the point of view of the risk manager, inappropriate use of the normal distribution

leads to an understatement of risk, which must be balanced against the significant

advantage of simplification. From the central bank’s corner, the consequences are even

more serious because we often need to concentrate on the left tail of the distribution in

formulating lender of last resort policies. Improving the characterisation of the

distribution of extreme values is of paramount importance.

Joint Central Banks Research Conference, 1995

It is also well established that investors like positive skew and dislike excess kurtosis. Harvey and

Siddique (2000) put forward an asset pricing model that incorporates skewness and find that an

investor may be willing to accept a negative return for high positive skewness. The popularity of

lottery tickets is an example of this preference. Dittmar (2002) employs nonlinear pricing kernels to

demonstrate that higher moments drive out the explanatory power of the Fama-French factors. Hwang

and Satchell (1999) find evidence that emerging market returns are better explained by incorporating

co-skewness and co-kurtosis. At the investor level, Mitton and Vorkink (2007) look at the allocations

of 60,000 accounts and find that investors systematically under-diversify to achieve positively skewed

portfolios.

At first blush it makes sense to abandon the mean-variance approach in favour of a more sophisticated

approach that accounts for departures from the i.i.d. Gaussian assumption and allows for investor

preferences for moments higher than order two. The mean-variance approach is of course deeply

embedded in the financial industry. Fabozzi, Focardi and Jonas (2007) survey 38 medium and large-

sized equity investment managers in North America and Europe totalling $4.3 trillion in assets under

management. The authors find that 97% of managers use variance to measure risk and that 83% of

managers employ mean-variance optimisation. The EDHEC European Practices Survey, 2008,

examines the investment behaviour of 229 investment managers including fund management firms,

pension funds, private banks, investment banks, family officers and consultants. The survey finds that,

57% of investment managers use the absolute return variance as the risk objective when performing

11

Approximately three times too many between 1996-2003 for a composite portfolio of the S&P 500, the FTSE

100, and Swiss Market indices.


7

portfolio optimisation. For managers that set relative risk objectives, 76% use the tracking error,

which is, the square-root of the variance of excess returns. Of the managers that perform optimisation

subject to value at risk (VaR) and expected shortfall constraints, the majority use the normal

distribution to derive their estimates, which of course yields the same set of portfolios as mean-

variance analysis. The mean-variance approach is also a well understood, tractable approach with

closed form expressions for optimal portfolios, expected utility and the decomposition of risk. The

mean-variance approach is also a key building block for other pillars of modern financial theory,

including the capital asset pricing model. Before we abandon the mean-variance approach in favour of

an ostensibly more appropriate alternative it makes sense to quantify the loss in investor welfare from

assuming returns are normally distributed.

A number of modelling approaches capture two or three of the stylised facts; however the holy grail

of all four in a scalable framework remains elusive. Nystrom and Skoglund (2002)12

capture heavy-

tails, negative skew and volatility clustering, but not asymmetric dependence13

. Xiong (2010)14

accounts for heavy tails and skew, but ignores volatility clustering and asymmetric dependence.

Sortino (2010)15

captures heavy tails, skew and asymmetric tail dependence, but does not account for

volatility clustering. Hu and Kercheval (2007)16

account for skew, asymmetric dependence, volatility

clustering and semi-heavy tails17

. Patton (2004)18

and Viebig and Poddig (2010)19

develop models that

account for all four stylised facts, however generalising the Archimedean copula approach they use to

higher dimensions ( ) requires restrictive assumptions. Filtered Historical Simulation (FHS)

proposed by Barone-Adesi, Bourgoin and Giannopouos (1998) captures all four stylised

characteristics, although the approach is non-parametric. The approach we propose captures all four

stylised facts and can be employed in high dimensions.

There is a vast literature on VaR estimation, and we do not discuss all of the findings here. There is a

consensus that models that account for stochastic volatility outperform static models (Pritsker, 2001,

Berkowitz and O’Brien, 2002, McAleer and da Veiga, 2008, Skoglund, Erdman, and Chen, 2010) and

that models that employ non-Gaussian innovations tend to outperform Gaussian models (McAleer and

da Veiga, 2008). Further, there is a general consensus that single index approaches tend to outperform

portfolio approaches (Berkowitz and O’Brien, 2002, Brooks and Persand, 2003, Bauwens, Laurent,

and Rombouts, 2006, Christoffersen, 2009, and McAleer, 2009)20

. Interestingly, the arguably more

intuitive historical simulation and its variant the filtered historical simulation approaches are currently

the most widely used methods at commercial banks (Christoffersen, 2006, Pérignon and Smith, 2010).

While there is an abundance of work showing the superiority of novel downside portfolio construction

techniques in-sample, there is a relative paucity of robust out-of-sample evidence. The techniques

proposed in the literature tend to have a large number of parameters, so it is of little surprise that in-

sample the techniques perform well. As Patton (2004) notes, a common finding in the point

forecasting literature is that complex models often provide inferior forecasts than simple misspecified

models (Swanson and White, 1997, Stock and Watson, 1999). It is critical to evaluate the out-of-

12

GARCH/EVT/t-copula approach 13

The t-copula is radially symmetric leading to the same degree of upper and lower tail dependence 14

Truncated Levy-flight distribution 15

Historical simulation 16

GARCH-multivariate skew-t 17

Misorek and Weron, 2010 18

Time varying copulas and moments up to the fourth order that are functions of exogenous variables 19

GARCH/EVT/Archimedean-copula approach 20

The results of McAleer and da Veiga (2008) however are mixed


8

sample evidence to ensure the benefits of a technique are robust to the increase in estimation that

tends to accompany an increase in parameters. Brandt, Santa-Clara, and Valkanov (2009) argue that

“extending the traditional approach beyond first and second moments, when the investor’s utility

function is not quadratic, is practically impossible because it requires modelling not only the

conditional skewness and kurtosis of each stock but also the numerous high-order cross-moments”.

The bulk of existing studies exclusively present in-sample portfolio performance evidence (Campbell,

Huisman, Koedijk, 2000, Consigli, 2002, Nystrom and Skoglund, 2002b, Prakash, Chang, and

Pactwa, 2003, Tokat, Rachev, and Schwartz, 2003, Morton, Popova and Popova, 2005, Jarrow and

Zhao, 2006, Jondeau and Rockinger, 2006 , Harvey, Liechty, Liechty, and Mulleer, 2010, Viebig and

Poddig, 2010, Sortino, 2010). In this sense, the literature concentrates on the demonstration of

techniques rather than providing hard evidence of efficacy.

The number of studies that examine the performance of downside portfolio construction approaches

out-of-sample is surprisingly low. Out-of-sample analyses typically involve the estimation of complex

multivariate distributions each period over the back-test window. The estimation of these models

often involves Markov-chain Monte Carlo or the Expectations-Maximisation algorithm and can be

both computationally demanding and time consuming. Guastaroba, Mansini, Speranza (2009)

evaluate the computational burden posed by several prominent scenario generation models. As an

example, the authors find that it takes 60 minutes to estimate and then simulate 10,000 times from a

multivariate GARCH model with student-t innovations21

. If we are re-estimating the models daily or

even weekly it is easy to see that meaningful out-of-sample tests can be prohibitively time consuming

even for a short back-test. We speculate that this may explain the small number of out-of-sample

studies in this important area.

Of the out-of-sample analyses, Patton (2004) finds that accounting for asymmetry and skewness using

skewed-t marginals and the rotated Gumbel copula leads to a small improvement in realised utility

relative to 1/N. Patton using a stationary boot-strap approach demonstrates that for all levels of risk

aversion the utility of the Gumbel copula approach is statistically significantly larger than the utility

of the Gaussian copula approach. Adler and Kritzman (2007) employ full-scale optimisation in

conjunction with S-shaped and bi-linear utility. Full-scale optimisation maximises utility directly and

makes no parametric assumptions, mitigating the effect of estimation error. The authors conclude that

the in-sample superiority of full-scale optimisation prevails out of sample. In a related approach

Brandt, Santa-Clara, and Valkanov (2009) address the portfolio construction problem by modelling

the weight of each asset as a function of its characteristics. The idea is to massively reduce the

dimensionality of the problem by maximising expected utility directly as a function of loadings on

common factors. The advantage of the Brandt et al. (2009) approach is that it circumvents the

estimation of complex multivariate distributions thereby reducing estimation error. Further, the

approach is also utility function agnostic. The disadvantage of the approach is that it does not lead to

downside risk estimates such as VaR and CVaR. Brandt et al. (2009) find that the characteristic-based

approach leads to substantial uplifts in utility that persists out of sample.

Alcock and Hatherley (2009) employ Gaussian marginals in combination with the Clayton copula.

The authors find that accounting for asymmetric dependence leads to a significant uplift in average

returns and statistically superior returns in down-markets. Guastaroba, Mansini, Speranza (2009)

compare the performance of the standard boot-strap, the block-boot strap, multivariate GARCH and

historical simulation. The authors conclude that the block-boot-strap consistently outperforms all the

other scenario generation techniques. Martellini and Ziemann (2010) use a Taylor Series approach to

21

For a FTSE 100 universe with a Pentium III processor and 1 GB of RAM


9

investigate the effect of incorporating investor preferences for moments higher than two. The authors

find that incorporating higher moments does not improve investor welfare unless sophisticated

shrinkage estimators of higher moments are employed. Xiong (2010) uses the truncated Lévy-flight

distribution to capture skew and heavy tails. The standard Levy-stable distribution was used by

Mandelbrot (1963) and Fama (1965) to model asset process. The Levy-stable distribution however

has an infinite variance, violating common sense and creating problems for the expected utility

framework. The truncated Levy-flight distribution obviates this shortcoming. Xiong (2010) shows the

approach outperforms mean-variance marginally during the 2008 financial crisis.

The out-of-sample evidence also lacks depth. It is notable that of the seven key out-of-sample studies

discussed above not a single one uses more than one data set. Because financial data are inherently

noisy and the extreme market conditions that provide stress tests of an approach are by definition rare,

we argue for employing as much high quality data as possible. By way of illustration, consider a top

quartile fund manager with an information ratio22

, defined as the annual excess return divided by the

annual tracking error23

, of 0.524

. Using the standard-error formula given in Lo (2002) it is

straightforward to show that no less than 18 years of data are required to reject the hypothesis that the

true information ratio is greater than zero. The information ratio is of course a Gaussian- based metric.

Intuitively, when returns and the performance metrics employed are non-Gaussian it will take even

longer to establish statistical significance, particularly when the gains of a given technique may be

marginal. Employing multiple data sets also limits the scope for the “cherry picking” of results.

Further, of the seven studies, five pertained to equities while only one included bonds, and another

included hedge fund indices. A further concern is the absence of tests to determine whether the

supposed uplifts in performance are statically significant. In order to assert that a given approach is

superior to an existing benchmark approach establishing statistical significance would appear

essential. The lack of tests for statistical significance is probably due to the lack of closed form

solutions for non-Gaussian performance metrics.

To establish statistical significance we follow Patton (2004) and use the stationary boot-strap

procedure of Politis and Romano (1994). A final criticism of the downside portfolio construction

literature is the lack of work comparing the prominent approaches on a like for like basis across the

same investment problems. The empirical research examining downside portfolio construction

techniques lacks the thoroughness of the literature analysing the mean-variance approach. For

example, DeMiguel, Uppal and Garlappi (2009) examine fourteen extensions of the mean-variance

approach assessing statistical significance across seven data sets. Kritzman, Page and Turkington

(2010) compare the performance of the mean-variance approach to 1/N out of sample across thirteen

data sets using seven asset classes.

In general, the literature suggests that incorporating non-Gaussian characteristics into portfolio

construction methods results in improvements in investor welfare. It appears that there are gains from

accounting for each of the stylised facts relative to the mean-variance approach. There is a lack of

consensus however, as to which portfolio construction approach should be preferred. It is also

difficult to infer from the literature the relative importance of accounting for each of the stylised facts.

We believe that we need to provide the same level of thoroughness described in the mean-variance

literature to help build consensus and move the discussion forward. This means comparing multiple

22

The annual excess return divided by the tracking error 23

The tracking error is defined as the standard deviation of returns in excess of a benchmark 24

Grinold and Kahn (2000)


10

approaches out-of-sample across a large number of investment problems, with multiple asset classes,

and testing for statistical significance.

III. The GSEV Model

In this section, we introduce the GSEV25

model comprised of ARMA/EGARCH, the skewed-t copula

and extreme-value theory. An approach that has gained popularity in the risk modelling literature is to

use an ARMA/GARCH filter to capture autocorrelation and stochastic volatility and non-Gaussian

marginal distributions to accommodate asymmetry and heavy tails26

. We break the multivariate

estimation problem down into three parts: the fitting of an ARMA/EGARCH process, the estimation

of the univariate marginals, and the modelling of the dependence structure. The advantages of this

approach are manifold. Filtering using ARMA/GARCH yields a series that is i.i.d., a prerequisite for

fitting a parametric distribution. The estimation procedure is also simplified and accelerated with a

minimal loss in efficiency (see for example the simulation studies of Joe, 2005 and Patton, 2006). By

fitting the dependence structure separately from the univariate marginals we also have the ability to

use a different distribution for the copula and the univariate marginals27

. We now discuss the three

25

We abbreviate the rather onerous ARMA/EGARCH-Skewed-t copula-Extreme-value theory model to the

GSEV model 26

Key examples include Barone-Adesi, Bourgoin and Giannopouos (1998) using non-parametric marginals, and

McNeil, and Frey (2000), Nystrom and Skoglund (2002), Kuester, Mittnik, and Palollea (2006), Hu and

Kercheval (2007), Viebig and Poddig (2010) and Hilal, Poon, and Tawn (2011) using parametric marginals 27

It is also possible to use different univariate marginals for each asset class

Heavy tails

Tail dependence

Volatility Clustering

Skew

Barone-Adesi, Bourgoin and

Giannopouos (1998)

Xiong (2010)

Harvey, Liechty, Liechty

and Muller (2004)

Nystrom and Skoglund (2002)

Bonato (2003)

Effron and Tibshirani

(1993)

Sortino (2010)

Alcock and Hatherley

(2009)

Hu and Kercheval (2007)

RiskMetrics™, JP Morgan

Figure 1 – Literature Venn diagram. Figure 1 shows a Venn diagram of the multivariate financial return literature


11

components: the type of ARMA/GARCH process, the non-Gaussian distribution, and the copula

function.

A. Modelling Heteroskedasticity: ARMA/EGARCH Process

As we show in section IV, several of our data-sets display weak first-order autocorrelation. We

therefore apply an AR(1) to each asset as follows.

(1)

We use exponentially-weighted GARCH (EGARCH) developed by Nelson (1991). EGARCH is an

asymmetric GARCH process that models the logarithm of the conditional volatility rather than the

conditional variance thereby obviating the need for parameter constraints. Alexander (2009) shows

that the EGARCH model provides a superior fit relative to competing asymmetric and symmetric

GARCH models28

.

Definition 3.1 EGARCH Process: An exponential GARCH(p,q) process where the innovation

distribution is Gaussian is given by

In addition to capturing heteroskedasticity, the EGARCH process generates heavy-tails in the

unconditional distribution of returns. McNeil, Frey and Embrechts (2005) provide the kurtosis of

as follows for a standard GARCH (1,1) process as follows

(2)

where refers to the ARCH coefficient, refers to the GARCH coefficient, the . Thus,

if , for example using Gaussian or scaled student-t innovations, the kurtosis of will be

strictly greater than the kurtosis of the innovations, .

B. Modelling the tails: Extreme-Value Theory

There are a number of distributions, with varying levels of theoretical support that can accommodate

heavy tails and asymmetry including the log-normal, the Generalised Hyperbolic, and Levy-alpha

stable distributions. Rather than imposing an arbitrary distribution on the data we draw on extreme

value theory (EVT). Several studies show the benefits of combining a conditional volatility model

with EVT including, McNeil and Frey (2000) and Nystrom and Skoglund (2002), Kuester, Mittnik,

and Palollea (2006), Viebig and Poddig (2010) and Hilal, Poon, and Tawn (2011). EVT provides the

theoretical basis for how the tails of all i.i.d. distributions behave asymptotically and is used to model

rare events in a variety of fields29

. In hydrology, EVT is used to help determine the size that damn

28

This finding however is not unanimous. Engle and Ng (1993) find that the GJR-GARCH model is superior to

EGARCH. In unreported results we find that our conclusions are robust to the choice of asymmetric GARCH

process. 29

For example, predicting the likelihood of forest fires, or core-melt in nuclear power plants.


12

walls need to be to withstand a 100-year flood. EVT can be thought of as the counterpart of the

central limit theorem (CLT).

Definition 3.2 Central Limit Theorem: If are n independently and identically

distributed random variables with mean and standard-deviation, , and , then

Whereas the CLT is concerned with the aggregation of fluctuations around a mean value, EVT deals

with the asymptotic behaviour of extreme departures from the mean30

. The Fisher-Tippett (1928)

theorem gives the limiting distribution of a sequence of block-maxima.

Definition 3.3 Fisher-Tippet Theorem: If there are two normalising constants, and , and a

non-degenerate distribution, , such that

then converges to the generalised extreme value distribution (GEV) given by31

where , , and are the location, scale and shape parameters respectively.

The generalised extreme-value distribution is generalised in that it subsumes the Weibull ,

Cauchy and Frechet distributions . The block-maxima of a set of data can be used to

calibrate the generalised extreme-value distribution. The approach however is wasteful of scarce data

and in practice has largely been superseded by methods based on threshold exceedances that use all

data that is designated as extreme.

Definition 3.4 Limiting Distribution of Threshold Exceedances: For a threshold, , and a

positive function , such that for all ,

The right-hand side of definition 3.4 is the familiar Generalised Pareto distribution.

Definition 3.5 Generalised Pareto Distribution: The Generalised Pareto distribution is defined

as follows

30

The CLT says that the mean of a sufficiently large number of independent random variables will be

approximately normally distributed. The CLT does not however describe the behaviour of the tails of a

distribution. 31

Using the von Mises representation


13

where and are the shape and scale parameters.

We follow McNeil and Frey (2000) and Nystrom and Skoglund (2002), and employ the GPD to

model the tails of the distribution, and a Gaussian smoothing kernel to model the body, defined as the

inner 90% of the density32

. Intuitively, this piecewise approach enables us to model extreme events,

the key area of concern to the risk manager, more accurately than the standard maximum likelihood

approach that does not place much weight on the tails. Further, modelling the lower and upper tail

independently allows for asymmetry.

C. Modelling the dependence structure: The Skewed-t Copula

Having characterised the univariate densities of each asset’s returns, we model the dependence

structure using copulas. The use of copula functions to model asset returns has increased dramatically

in recent years33

. Copulas have proven a valuable addition to the econometrician’s toolbox because

they enable the researcher to model the dependence structure separately from the univariate densities

using the inference function for margins (IFM) method of Joe and Xu (1996). Multivariate

distributions that were extremely difficult and time consuming to fit can now be estimated rapidly. Joe

and Xu’s (1996) method follows from Sklar’s (1959) theorem which says that if is a multivariate

distribution function, and denotes the marginal distribution, , then a copula, exists such that for

all in = .

Definition 3.6 Sklar’s (1959) Theorem: Let F be a joint distribution function with margins

. Then there exists a copula such that, for all in

A parallel of to the inference function for margins method can be seen in the estimation of CCC

(Bollerslev, 1990) and DCC (Engle, 2002) multivariate GARCH models. These approaches separate

the estimation of the variance and covariance matrix into two parts. In the same way, we estimate the

marginal distributions of each asset separately from the dependence structure. Before we introduce the

skewed-t copula we define the lower and upper tail dependence between assets and as follows.

Definition 3.7 Tail dependence: The lower tail dependence, , and upper tail dependence, ,

between two stochastic variables and with cumulative density functions of and are given

by

32

In this way we eliminate the stair-case pattern evident in empirical density functions 33

Mikosch (2005) notes that a Google search of the word “copula” in 2003 produced 10,000 hits. In 2013, the

same search yields 1.63 million hits


14

To capture the dependence patterns discussed in section II, we argue that the ideal copula should be

able to accommodate the following four characteristics:

I. Tail dependence: , for all and

II. Asymmetric tail dependence: for all and

III. Heterogeneous tail dependence: and for all assets , , and

IV. Scalable in high dimensions

The first requirement says that the copula should allow for asymptotic dependence. The Gaussian

copula in contrast is asymptotically independent (Sibuya, 1960). As returns become more extreme

they become uncorrelated, for all , a potentially hazardous characteristic when modelling

financial asset returns. Indeed Li’s (2000) Gaussian copula model for CDOs has been widely blamed

for the mispricing of derivative instruments and the massive losses taken by investment banks during

the 2008 financial crisis34

. The t-copula accommodates tail dependence as can be seen below.

Definition 3.8 Tail dependence of the t-copula: The tail dependence of the t-copula, where

represents the tail of the univariate t-distribution, is the degrees of freedom parameter, and is the

correlation coefficient is given by

The second requirement, asymmetric tail dependence, says that the upper and lower tail dependence

of a given asset pair should be able to differ. For example, in “bear” markets dependence tends to be

higher than in “bull” markets. As shown in definition 3.8, the t-copula is radially symmetric. Several

copulas provide radially asymmetry including the Clayton and Gumbel copulas of the Archimedean

family.

Our third requirement, heterogeneous tail dependence, stipulates that the ideal copula should allow for

different dependence structures across asset pairs. For example, empirically the structure of tail

dependence is very different between developed equities and bonds than it is between developed

equity and emerging market equity or between different pairs of hedge fund strategies (Viebig and

Poddig, 2010). Unfortunately, standard Archimedean copulas do not generalise readily to higher

dimensions and it is necessary to impose severe constraints on the dependence structure, tantamount

to forcing the off-diagonal terms in the correlation matrix to be equal. The multivariate Levy-Stable

distribution also allows for radial asymmetry; however it imposes a homogenous dependence

structure across asset pairs. The fourth requirement, to be useful to practitioners, a copula should be

scalable in dimensions . Again this is a problem with the Archimedean copulas.

The skewed-t copula is perhaps unique in that it fulfils all four requirements. Fitting the skewed-t

copula however is non-trivial. Smith, Gan and Kohn (2010) use a Bayesian Markov Chain Monte

Carlo approach, while Kollo and Pettere (2010) employ a GMM approach. We follow Sun, Rachev,

Stoyanov and Fabozzi (2008) and extract the copula from the multivariate skewed-t distribution. We

can extract the dependence structure from the multivariate distribution in conjunction with the

marginal distributions by Sklar’s theorem (1959).

We first need to discuss the skewed-t distribution. There is a bewildering array of variants of the

skewed-t distribution including Hansen (1994), Fernandez and Steel (1998), Branco and Dey (2001),

34

Salmon (2009)


15

Bauwens and Laurent (2002), Azzalini and Capitanio (2003), Jones and Faddy (2003), Sahu, Dey and

Branco (2003), and Patton (2004). Each of these distributions have polynomial upper and lower tails,

so while they can fit heavy-tailed data well, they cannot accommodate significant asymmetry (Aas

and Haff, 2006). We employ the skewed-t distribution of DeMarta and McNeil (2005) that derives

from the Generalised Hyperbolic (GH) distribution. Unique among skewed-t distributions, the GH

skewed-t distribution can accommodate significant asymmetry.

Definition 3.9 Generalised Multivariate Hyperbolic Distribution: The Generalised Multivariate

Hyperbolic distribution where is a vector, is a matrix, is a vector, and and are constants is

given by

where the normalising constant is

The GH family incorporates several important distributions. If , we have the

hyperbolic distribution, and if , we attain the normal-inverse Gaussian distribution. If

, we obtain the GH skewed-t distribution following McNeil, Frey and Embrechts (2005).

Definition 3.13 Multivariate Skewed-t Distribution: The Multivariate Skewed-t distribution

where is given by

where the normalising constant is

If , we recover the familiar multivariate student-t distribution.

In order to simulate from the skewed-t copula, we require the covariance matrix, Σ. We can derive this

from the stochastic equation of the multivariate skewed-t distribution as follows.

(3)

The covariance matrix is then comprised of a skew-based component and a heteroskedastic Gaussian

component. For the skewed-t distribution, . Employing the expected values of the


16

first and second moments of the inverse gamma distribution we can estimate the covariance term as

follows

(4)

We are now armed with the requisite components to extract the skewed-t copula from the multivariate

skewed-t distribution. We follow the algorithm of Sun, Rachev, Stoyanov and Fabozzi (2008) set out

below.

Algorithm 3.1 Skewed-student-t Copula Simulation

1. Estimate the parameters for the univariate skewed-t distributions using MLE35

2. Estimate the covariance matrix using equation (4)

3. Draw independent n-dimensional vectors from the multivariate skewed-t distribution using

the stochastic representation given by (2). The result is a N-by-d matrix of simulations, .

a. Draw independent d-dimensional vectors from the multivariate

Gaussian distribution defined by

b. Draw independent random numbers from the inverse gamma

distribution defined by

c. Substitute and into equation (3)

4. Generate the cumulative distribution functions for the univariate marginals in step 1 through

numerical integration36

5. Transform the simulations, , to uniformly distributed variables using the cumulative

distribution functions in step 4

An alternative procedure to MLE is to use the expectation-maximisation algorithm (EM)37

. In the case

of the univariate skewed-t distribution that we are using here, we have three free parameters, and

and a standard solving method such as quasi-Newton suffices38

.

In each panel of figure 2, we show 10,000 simulations for different permutations of and of the

bivariate skewed-t copula using steps 1-5. The upper left hand panel exhibits lower tail dependence,

while the lower right panel exhibits upper tail dependence. The copula in the centre is radially

symmetric and is identical to the standard-t copula. The remaining panels are not exchangeable and

show a range of tail behaviour indicative of the flexibility of the skewed-t copula.

35

Note that we do not lose any flexibility in fixing the degrees of freedom parameter, . 36

We are unaware of a tractable analytical solution for the c.d.f. of the skewed-t distribution 37

Liu (2012) 38

McNeil, Frey and Embrechts (2005)


17

In summary, we employ EGARCH to capture time varying volatility and the leverage effect, extreme

value theory (EVT) to capture negative skew and heavy tails, and the skewed-t copula to capture

heterogeneous asymmetric tail dependence.

E. Applying the GSEV model to the S&P 500

In this subsection we provide a worked example of the proposed GSEV model. We employ daily data

of the S&P 500 index from 1/1983-1/2013. To proxy for a small-cap index we use the value-weighted

portfolio of the smallest quintile of firms in the CRSP universe39,40

. The data displays strong

departures from normality failing the Jacque-Bera test at the 1% level. We apply the Jarque-Bera test

to annual non-overlapping sub-periods and find that in 100% of years, normality can be rejected for

both indices.

39

Courtesy of K.R. French 40

Commercial small-cap indices did not start providing daily indices until more recently, for example the MSCI

U.S.A. Small Cap Total Return index commenced on 1/2001.

-

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Figure 2 – Skewed-T Copula Simulation. Each panel of figure 2 shows 10,000 draws from the bivariate

skewed-t copula. and refer to the skew parameters of the two variables. Simulations were performed

with , and .


18

Table I

Summary Statistics: S&P 500

S&P 500

Index

Small Cap

Index

Annualised Return 10.7 9.7

Standard-deviation 18.3 17.2

Skewness -0.6 -0.7

Kurtosis 16.1 11.1

Jacque-Bera 0.001 0.001

% Jarque-Bera failures 100 100

% stat. sign skew 65 89

% stat. sign kurtosis 100 92

-0.0 0.1

0.1 0.4

In figure 3, we show the sample autocorrelation function of the squared daily returns for the S&P 500

(panel A) and U.S. Small Cap (panel B) indices. The horizontal red lines delineate statistical

significance at the 5% level. The squared returns for both indices follow an autoregressive structure

consistent with volatility clustering. We then apply an ARMA/EGARCH model to each return series.

The maximum-likelihood estimates are given in table II. The leverage effect is negative and

statistically significant for both indices indicating that negative returns lead to larger increases in

volatility than positive returns of the same magnitude. Moreover, the EGARCH model provides a

superior fit to the standard GARCH model for both indices.

Table II – ARMA/EGARCH Parameters

S&P 500: GARCH S&P 500: EGARCH

Coefficient Error t-stat

Coefficient Error p-value

C 0.07 0.01 7.25

0.04 0.01 0.00

0.02 0.01 1.21

0.02 0.01 0.07

0.01 0.00 11.71

0.00 0.00 0.01

0.91 0.00 316.91

0.98 0.00 0.00

0.08 0.00 36.82

0.13 0.01 0.00

-0.09 0.00 0.00

-10147.16

-10043.10

Small Cap Index: GARCH

Small Cap Index: EGARCH

Coefficient Error t-stat

Coefficient Error p-value

C 0.05 0.01 5.78

0.04 0.01 0.00

0.21 0.01 15.79

0.22 0.01 0.00

0.01 0.00 11.71

-0.00 0.00 0.00

0.84 0.00 292.51

0.97 0.00 0.00

0.15 0.00 70.47

0.25 0.01 0.00

-0.07 0.00 0.00

Table I provides the summary statistics for S&P 500 index and a U.S. Small Cap index comprised of the value-weighted

portfolio of the smallest quintile of firms listed on the NYSE, AMEX or NASDAQ exchanges for the period 1/1983-

12/2012. The annualised return is calculated geometrically. The % of Jarque-Bera failures refers to the proportion of 260

day sub-periods where normality is rejected at the 5% level. The % of statistically significant skewness and % of

statistically significant kurtosis refers to the proportion of sub-periods where the skewness and kurtosis is statistically

significant. refers to the autocorrelation at one lag. refers to the one-period autocorrelation

of the absolute value of returns.

Table II provides the ARMA/GARCH and ARMA/EGARCH(1,1) maximum-likelihood estimates for the S&P 500 and U.S.

Small Cap indices using daily returns between 1/1983-12/2012.


19

-8537.08

-8518.97

In figure 3, panels C and D we show the autocorrelation function of the squared standardised

ARMA/EGARCH residuals. The process has removed the autoregressive structure in the squared

returns, and we then fit a generalised Pareto-distribution to the lower 5% quantile of the standardised

residuals again using maximum-likelihood.

-0.20

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Panel A: Sample ACF of squared daily returns for the S&P 500 Index: 1/1983-12/2012

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0.00

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Panel C: Sample ACF of squared daily ARMA/EGARCH residuals for the S&P 500

Index: 1/1983-12/2012

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Panel B: Sample ACF of squared daily returns for the U.S. Small Cap. Index: 1/1983-12/2012

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0.00

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0.40

0.60

0.80

1.00

0 5 10 15 20

Lag

Panel D: Sample ACF of squared daily ARMA/EGARCH residuals for the U.S. Small

Cap Index: 1/1983-12/2012

Figure 3 – Autocorrelation Function of squared daily S&P 500 and Small Cap index returns. Figure 3 shows the

autocorrelations for the standardised residuals for the daily S&P 500 and Small Cap index returns between 1/1983 and

12/2012 for 0 to 20 lags. The horizontal blue lines delineate statistical significance at the 5% level. Standardised residuals

are obtained by dividing by the EGARCH residuals by the EGARCH conditional standard-deviation estimates.


20

The estimated cumulative distribution function maps on well to the empirical c.d.f. for both indices.

The Kolmogorov–Smirnov test fails to reject the null hypothesis of a GPD in both cases at the 1%

level. The estimated coefficients are shown in table III.

In order to forecast portfolio risk and construct portfolios, we now need to combine the univariate

GPDs into a multivariate distribution. As detailed in section III B, we do this using the skewed-t

copula. To simulate the copula function, we follow algorithm 1.1. (Sun, Rachev, Stoyanov and

Fabozzi, 2008). The first stage is maximum-likelihood to estimate the parameters of the univariate

0

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-8 -7 -6 -5 -4 -3 -2 -1 0

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Panel A: Lower 5% quantile of standardised residuals for S&P 500 Index: 1/1983-12/2012

0

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Panel B: Lower 5% quantile of standardised residuals for U.S.Small Cap Index: 1/1983-12/2012

KS-test p-value

S&P 500 index -1.66 0.218 0.494 0.88

US Small Cap index -1.72 0.193 0.569 0.14

Table III – MLE estimates for GPD distribution for 5% quantile of daily S&P 500 and US Small Cap index

ARMA/EGARCH residuals. Table 8 provides the maximum likelihood estimates for the GPD for the ARMA/EGARCH

residuals estimated for the period 1/1983-12/2012. We also provide the p-value for the Kolmogorov–Smirnov goodness of

fit test.

Figure 4 – Cumulative Density Function for the 5% quantile using the Fitted Generalised Pareto Distribution and the

Empirical distributions for the ARMA/EGARCH residuals for the S&P 500 and US Small Cap indices. Figure 9

shows the cumulative density functions for 5% quantile using the fitted GPD and empirical distributions for the

ARMA/EGARCH residuals for the period 1/1983-12/2012 for the S&P 500 and US Small Cap indices.


21

skewed-t distributions, and for a fixed degrees of freedom, . We constrain to zero which is a

common assumption for short forecast horizons.

Definition 3.14 Univariate Skewed-t Distribution: The Univariate Skewed-t distribution is given

by

We then estimate the covariance matrix of the multivariate skewed-t distribution using equation (4)

reproduced below.

We then simulate the multivariate skewed-t distribution using the stochastic representation given by

equation (3)

where , and where A is an upper triangular matrix.

To simulate the copula we transform the columns of the simulated return matrix to uniform

distributions by taking the inverse of the c.d.f. of the skewed-t distribution using the fitted parameters.

We then take the bivariate uniform distributions and draw from the fitted GPD to generate the

unconditional GSEV distribution. The final stage is to condition on volatility using the unconditional

GSEV distribution as innovations for the ARMA/GSEV process.

By way of illustration, at the onset of the global financial crisis in January 2008, the maximum

likelihood estimates for the asymmetry parameters, for the S&P 500 and US Small Cap indices were

=-0.4 and =-0.49. Figure 5 shows the simulated bivariate skewed-t copula using these

parameters. Note the increase in dependence in the left-hand tail.


22

IV. Data and Methodology

Capturing the four stylised facts in a coherent scalable framework is not straight-forward. In the case

of the GSEV model, a practitioner requires advanced knowledge of GARCH processes, copula theory,

the multivariate skewed-t distribution and extreme value theory. It is therefore important to quantify

the likely benefits of the approach relative to more parsimonious approaches. We compare the GSEV

approach to portfolio construction models commonly used by practitioners or that are prominent in

the literature. Our empirical approach is made up of two parts. In section V A we compare the

performance of the value-at-risk (VaR) forecasts of the GSEV model to seven benchmark models. In

section V B we compare the performance of the GSEV model to eight benchmark models in a

dynamic portfolio rebalancing framework. To test the generality of our conclusions we utilise eight

data sets including five asset classes.

A. Benchmark Models

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Figure 5 – Simulated Bivariate Skewed-t Copula and GSEV model returns for S&P 500 and US Small Cap indices.

Panel A shows 10,000 draws from the bivariate skewed-t copula with parameters and , and .

Panel B shows 10,000 draws from the conditional GSEV daily returns for the S&P 500 and the US Small Cap index

estimated at 1/2008.


23

The eight benchmark models are divided into unconditional and conditional methods. We discuss the

motivation for and the estimation of each of these models in detail in appendix A. Table IV provides

an overview. As well as serving as performance benchmarks, these particular models have been

selected to help elucidate the desirable characteristics of a risk modelling approach, for example

conditional or non-conditional, parametric or non-parametric, symmetric or asymmetric dependence,

Gaussian or non-Gaussian marginal distributions.

Table IV – Benchmark Methodologies

Methodology Type Components Key references

1/N Unconditional, Heuristic -Equal portfolio weights Bernatzi and Thaler (2001),

DeMiguel, Garlappi and

Uppal (2009)

Gaussian Unconditional,

Parametric

Gaussian distribution Bloomfield, Leftwitch and

Long (1977)

Jobson and Korkie (1981)

Student-t Unconditional,

Parametric

Student-t distribution Lauprete, Samarov and

Welsch (2001)

Hu and Kercheval (2010)

Historical Simulation Unconditional, non-

Parametric

Bootstrapped from historical

distribution

Efron and Tibshirani (1993),

Sortino (2010)

Gaussian Marginals/

Clayton-Copula

Unconditional

parametric

Gaussian marginals and the

Clayton copula

Alcock and Hatherley

(2009)

EWMA Conditional, Parametric -Exponentially-weighted

covariance matrix

RiskMetrics Technical

Document, 1996

Filtered Historical

Simulation

Conditional, non-

Parametric

-EGARCH simulation applied

to bootstrapped returns

Barone-Adesi, Bourgoin and

Giannopouos (1998)

GGEV Conditional, Parametric -EGARCH

-Generalised Pareto distribution

to fit tails

-Gaussian Copula

Nelson (1991)

Nystrom and Skoglund

(2002)

B. Risk Model Forecast Evaluation

The accurate quantification of tail-risk is essential to the efficient operation and indeed survival of

financial institutions. If capital reserves are excessive, then shareholder returns, business and

consumer credit, and economic growth are inhibited. If capital reserves are too low, there is an

increased risk of bank failures, erosion of shareholder value and a contraction in economic growth.

Further, under Basel III, internal VaR models that perform poorly trigger increased capital

requirements. To prepare the ground, we begin by defining the loss function.

Definition 4.1 Loss Function: The loss function is given by the change in value, , of the

portfolio between time and .

By convention, the loss function is usually expressed as a positive value and we are concerned with

the right-hand tail. VaR is defined as the loss that a portfolio will not exceed a given probability of the

time, . Mathematically, VaR refers to the -quantile of a distribution.

Definition 4.2 Value-at-Risk: The value-at-risk, for confidence level , where

is the cumulative distribution function for the loss associated with is given by

Table IV provides an overview of the eight benchmark methodologies and the GSEV model.


24

We re-estimate the respective VaR models daily and produce daily VaR forecasts for the equally

weighted portfolios for each asset class. Under Basel II a 99% confidence level is used to estimate the

10-day VaR. In addition to the 99% VaR level we evaluate the 99.5% VaR estimate over a one day

horizon as is common in the VaR validation literature (Skoglund, Erdman, and Chen 2010). The Basel

Committee on Banking Supervision is currently considering the standardisation of VaR model

estimation windows to between two and five years. For our estimation window, we follow McNeil,

Frey and Embrechts (2005) and use a 1000-day estimation window41

. In the Gaussian case, VaR can

be estimated analytically.

Definition 4.3 Value at Risk for the Gaussian Distribution: The conditional value-at-risk for the

Gaussian distribution, where is the confidence level, , is the standard Gaussian density function,

and is the standard-deviation of , is given by

Similarly if the return series is distributed as a student-t, we have the following relation.

Definition 4.4 Value at Risk for Student-t Distribution: The value-at-risk for the student-t

distribution, where is the confidence level, is the degrees of freedom, is the density function of

the student-t, and is the standard-deviation of , is given by

To estimate the VaR of the non-elliptical models we use Monte Carlo simulation to generate 100,000

daily returns for each asset for each model, each day. We then estimate the VaR as the quantile of

the simulated return series.

Definition 4.5 Value at Risk for Discrete Data: The value-at-risk for discrete data, where is the

cumulative simulated density function of the discrete data, and is the confidence level, is given by

The process for generating the respective VaR forecasts is summarised below.

Algorithm 4.1 VaR Estimation

1. At time, , estimate each risk model using data from -1000 to -1

2. Simulate 100,000 daily returns for each risk model

3. Estimate the VaR for each model as the quantile of the simulated data42

4. Step forward one business day and repeat

In order to evaluate the quality of our VaR models, we first define an indicator variable of violations,

also known as exceedances, where the VaR estimate is breached43

. If the model is well calibrated, the

proportion of violations should approximate one minus the confidence level.

41

The 1000 day estimation window also ensures that we have sufficient data to estimate each of our nine models

for all of our data sets 42

For the Gaussian models we use the closed form solution instead of relying on quantiles 43

See “The Analytics of Risk Model Validation” Christodoulakis and Satchell (2008) for a review of novel VaR

model evaluation techniques


25

Definition 4.6 Indicator Variable of Violations: The indicator variable of violations where

is the loss in period and is the VaR estimate with confidence level made at point, , is

given by

We evaluate the quality of the VaR forecast using the unconditional coverage (UC), serial

independence (SI) and conditional coverage (CC) tests proposed by Christoffersen (1998). The

unconditional coverage (UC) compares the theoretical proportion of violations, , with the

observed proportion of violations, . If the model produces too many violations, , and the

VaR model underestimates risk. If the model produces too few violations, , and the VaR

model overestimates risk. The likelihood ratio of the unconditional coverage ratio is defined as

follows.

Definition 4.7 Likelihood Ratio for Unconditional Coverage: The Likelihood Ratio for

Unconditional Coverage where is the confidence level and is the maximum

likelihood estimate of , and is the number of observations with value is given by

The correct unconditional coverage is not however a sufficient condition for a well-behaved VaR

model, the violations should also be serially independent (SI). If the violations cluster together, the

firm’s capital reserves may prove insufficient to ward off the threat of bankruptcy. The likelihood

ratio for serial independence is defined as follows.

Definition 4.8 Likelihood Ratio for Serial Independence: The likelihood ratio for serial

independence where is the number of observations with the value , followed by . is given by

and

and

We then use Christoffersen’s (1998) joint test of unconditional coverage and serial independence

known as the conditional coverage test. The likelihood ratio of the conditional coverage statistic is

given by the sum of the conditional coverage and serial independence likelihood ratios.

Definition 4.9 Likelihood Ratio Conditional Coverage

C. Dynamic Portfolio Rebalancing Framework

We quantify the economic benefits of the GSEV model in a dynamic portfolio rebalancing framework

(Solnik, 1993, Fletcher, 1997). As in the preceding section we re-estimate each VaR model each day

for each data set. We then produce an optimal portfolio each day for each model for each dataset

using a rolling 1000 day window.


26

C1. Expected Returns and Estimation Error

A key goal of this paper is to quantify the economic value of a range of risk modelling approaches. It

is therefore important that our conclusions are not influenced by the estimation error inherent in

expected returns that can easily contaminate the conclusions of dynamic portfolio rebalancing

analyses. Chopra and Ziemba (1993) show that errors in expected returns are over ten times more

important than errors in variance and over twenty times more important than errors in covariance in

determining portfolio weights. Three approaches to estimating expected returns are commonplace in

the literature: the use of historical sample averages (Bloomfield, Leftwitch and Long, 1977, Jorion,

1991, Jagannathan and Ma, 2003, Duchin and Levy, 2009, DeMiguel Garlappi and Uppal, 2009),

predictive regressions based on fundamental variables44

(Solnik, 1993, Fletcher, 1997), and

equilibrium based models (Black and Litterman, 1992, and Pastor and Stambaugh, 1999, 2000). All

approaches however entail large amounts of estimation error that can easily confound the empirical

results. For example an equilibrium model may perform poorly leading to poor performance of the

out-of-sample portfolios even if the underlying risk model performs well. In effect, we have a dual

hypothesis problem.

Bayesian methods, first suggested by Zellner and Chet (1965) and improved on by Jorion (1986), and

heuristic methods, including resampling (Michaud, 1998) and constraints (Frost and Savarino, 1998),

have been applied to mitigate the effect of estimation error. Again, if we were to use one of these

heuristic techniques, it becomes difficult to disentangle the benefits of the heuristic and the benefits of

the portfolio construction approach. We take a different approach and assume identical expected

returns for each asset. This is consistent with a null prior and may build a level of conservatism into

our conclusions in that we are implicitly assuming that agents have no forecasting skill45

.

C2. CVaR and Coherent Measures of Risk

We now motivate our choice of conditional value-at-risk (CVaR), also known as expected shortfall, as

our primary risk measure. Artzner, Delbaen, Eber and Heath, (1998) provide four axioms of a

coherent risk measure.

Definition 4.10 Axioms of a Coherent Risk Measure (Arzner, Delbaen, Eber and Heath, 1998)

A1 Translation Invariance: For all and all real numbers , we have

A2 Sub-additivity: for all and

A3 Positive Homogeneity: for all and all

A4 Monotonicity: For all and with , we have

Translation invariance means that the risk measure should be linear with respect to returns.

Subadditivity requires the risk measure of a portfolio of two assets to be less than the sum of the risk

measures for each asset and hence reward diversification. Positive homogeneity requires that the risk

measure is a linear function of the portfolio exposure. Monotonicity means that if a portfolio has a

higher return than a second portfolio in all states of the world, the risk of the second portfolio must be

higher. In the Artzner et al (1999) framework there is no requirement that risk pertain to negative

outcomes running counter to one’ s intuitive grasp of risk. Indeed, Pedersen and Satchell (1998) in a

set of axioms that precedes Artzner et al (1999) propose that a basic property of a risk measure should

44

Campbell and Thompson (2008) provide an excellent overview of the explanatory power of a wide range of

variables 45

With regard to expected returns


27

be a focus on the downside. The downside focus is consistent with Prospect Theory, the behavioural

studies of Kahneman, Daniel, and Amos Tversky (1979), and common-sense. We therefore require

that our risk measure has a downside focus in addition to the axioms of a coherent risk measure.

Under Basel II, the Committee on Banking Supervision advocates for the use of VaR. Despite the

ubiquity of VaR, the approach does not satisfy the second axiom of a coherent risk measure because it

is not subadditive. Instances can occur where a portfolio of two assets has a higher VaR than the sum

of the VaRs of the two portfolios running counter to the principle of diversification. VaR is also

invariant to the size of losses beyond the VaR threshold. McNeil, Frey and Embrechts (2005) show

how the optimisation procedure can exploit this conceptual weakness of VaR and result in portfolios

that are highly risky and undiversified. From a practical point of view, using VaR to construct

portfolios leads to a non-smooth, non-convex, and difficult to solve optimisation problem. Despite

significant toil, efficient algorithms to solve VaR based optimisations in all but low dimensions have

not been forthcoming (Kast, Luciano and Peccati 1998, Basak and Shapiro, 2001, Puelz, 1999,

Emmer, Kluppelberg and Korn 2000). The Basel Committee on Banking Supervision have recently

argued for the adoption of conditional value at risk (CVaR)46

, also known as expected shortfall, over

VaR, due to the latter’s “inability to capture tail risk”47

. We define as follows

Definition 4.11 Conditional Value at Risk: , where is the loss function and is

the underlying probability density function is given by

The advantages of CVaR are numerous. Firstly, unlike variance, CVaR is downside focussed. CVaR

is a coherent measure of risk and hence sub-additive (Acerbi and Tasche, 2002). CVaR is therefore

also a convex risk measure and amenable to optimisation. CVaR, unlike VaR accounts for the size of

losses beyond the VaR threshold. CVaR is also intuitive and readily interpretable, capturing a key

quantum interest to practitioners and regulators: if there is an extreme negative realisation, how large

is it likely to be? For these reasons and owing to the recent Basel Committee endorsement, we employ

CVaR as our principle risk measure.

C3. Optimisation Problem

The standard approach in analyses of portfolio optimisation techniques is to maximise an objective

function that is a positive function of expected returns and a negative function of expected risk.

(5)

where is a vector of asset weights, is the coefficient of risk aversion, and is a risk function.

We instead follow Leibowitz and Kegelman (1991), Lucas and Klaasen (1998), Campbell, Huisman

and Koedijk (2001), and Chekhlov, Uryasev, and Zabarankin (2005) and maximise the expected

return of the portfolio, subject to a constraint on risk, in our case CVaR.

46

The notion of expected shortfall has been familiar to insurance practitioners for several decades (Misorek and

Weron, 2010). 47

Fundamental Review of the Trading Book, Consultative document, Basel Committee on Banking Supervision,

May 2012


28

(6)

where is a vector of portfolio weights, is a vector of ones, and is a function that estimates

the conditional value at risk. The origin of this approach traces back to Roy (1952) and the safety-first

principle. This approach is consistent with second order stochastic dominance. We set the one-day

99% CVaR target to 2%, corresponding to the historical 99% one-day CVaR of the average pension

fund allocation invested 50% in equities and 50% in fixed-income and cash48,49,50

.

There are several benefits of maximising returns subject to a CVaR constraint. First, this approach

aligns with the risk management practice at banks and hedge funds. For example, a portfolio manager

will in general attempt to maximise the returns he or she can generate in order to attract inflows and

earn a performance fee, while limiting the potential for a severe negative outcome that will lead to

outflows or the loss of the mandate. Second, constraining the CVaR of the portfolios also enables us

to see if the proposed model produces portfolios with the targeted CVaR out-of-sample. Third, we are

freed from the requirement to estimate the level of risk aversion. The expected utility maximisation

framework is consistent with our constrained approach and the efficient frontiers of the two

approaches coincide.

CVaR defined in 4.11 is not a smooth convex function leading to difficulties converging to the global

optimum portfolio. Rockafellar and Uryasev (2001) solve this problem by re-expressing the

optimisation problem as follows.

Definition 4.12 Rockafellar and Uryasev (2001) Conditional Value at Risk: The -CVaR,

where is the threshold level, is the loss function and is the underlying probability

density function, and when and 0 when , is given by

The discretised version is given below.

Definition 4.13 Conditional Value at Risk for Discrete Data: The value-at-risk for discrete data,

where is the number of violations of is given by

Under this approach, VaR enters the optimisation as an additional variable and is determined

simultaneously with CVaR. In this way the dependence on VaR is eliminated and the problem is

smooth and convex. We use the Rockafellar and Uryasev (2001) approach to estimate the CVaR using

100,000 simulated daily returns for each benchmark model51

.

Algorithm 4.2 Dynamic Portfolio Rebalancing

48

Estimated based on 50% weight in the S&P 500 index and 50% weight in the Barclays Aggregate

Government Bond index for the period 5/1990 to 10/2013 49

Ibbotson and Kaplan (2000) 50

The performance measures we describe in section IV D are invariant to the level of the CVaR constraint. 51

For the two elliptical cases we instead use the closed form solution for CVaR given in definition 4.1 and 4.2


29

1. At time estimate each risk model using data from -1000 to -1

2. Simulate 100,000 daily returns for each asset for each risk model

3. Maximise the objective function in (6) where is defined in 4.12 for each model

4. Record portfolio returns and weights at

5. Step forward one business day and repeat

D. Performance Evaluation

Our primary performance measure is the ratio of the mean historical return in excess of the risk free

rate divided by the historical . The Mean/ ratio is consistent with our optimisation

problem in section IV C3, the axioms of a coherent risk measure, the downside requirement of

Pedersen and Satchell (1998) and Basel III.

Definition 4.14 Mean/ Ratio: The Mean/ ratio, where is the mean historical

return, is the risk-free rate, and refers to the conditional value-at-risk with a confidence

level of is given by

We now describe our approach for evaluating the robustness of our findings to alternative

performance measures. We define a performance measure as a ratio of reward to risk and that is valid

with respect to a given utility function. In practice, investor preferences are heterogeneous and there is

no single utility function that is valid for all investors. Hence there is no single performance measure

that is appropriate for all investors and it makes sense to evaluate performance through the prism of a

number of different measures. We restrict ourselves to measures that are consistent with the key

axioms of a coherent risk measure, are implied by expected utility theory, and have a decision

theoretic basis.

In section IV B2 we used the axioms of a coherent risk measure as provided by Artzner et al (1998) to

motivate the use of CVaR. The Artzner et al (1999) axioms do not require that a risk measure should

be non-linear with respect to losses. For example CVaR is invariant to the distribution of losses

beyond the VaR threshold and is determined purely by the mean exceedance. CVaR is thus a risk-

neutral risk measure. So while CVaR has several attractive properties, including coherence,

convexity, interpretability, and consistency with Basel III, it is not a panacea.

Generalised lower partial moments (LPM) proposed by Fishburn (1977) and (Bawa, 1977) provide

the basis for our two supplementary performance measures. LPM follow directly from the utility

function proposed in Fishburn (1977) and Bawa (1978).

Generalised LPM subsume several notable risk measures including, the probability of below target

return ( for all ), expected regret ( ), and the Sortino ratio ( ). LPM

are widely used in the investment industry suggesting the metrics reflect the utility functions of

market participants.


30

Definition 4.15 Generalised Lower Partial Moments: The Generalised Lower Partial Moment,

, where is the threshold return, is the portfolio return, and is the moment order is defined

as follows.

Generalised Lower Partial Moments (LPM) are not translation invariant. Barbosa and Ferreira (2004)

argue that non-observance of the translation invariance axiom is not a serious drawback because

increasing the weight of the risk-free asset in a risky portfolio will still lead to a decrease in risk52

.

LPM satisfy the homogeneity axiom for , and the subadditivity axioms for . We therefore

restrict ourselves to LPM with and . Setting the target return to zero also ensures that the

LPM-based metrics provide a complimentary perspective of risk to Mean-CVaR where the threshold

return is in the far left-hand tail. The first supplementary second measure we use is the Mean/

ratio and is defined as follows.

Definition 4.16 Mean/ Ratio: The Mean/ ratio, where is the mean historical return,

is the risk-free rate, and refers to the second order lower-partial moment of returns with

respect to the origin, is given by

It can easily be shown that the Mean/ ratio is equivalent to Keating and Shadwick’s (2002)

Omega ratio minus one53

. The Omega ratio is equal to the sum of the probability above the target

return divided by the probability below the targeted return. Keating and Shadwick (2002) argue that in

this sense the Omega ratio summarises the entire probability density in a single number. Mean/ ,

like Mean/ is a risk-neutral measure of risk.

The second performance measure we use is the Mean/ ratio, and is equivalent to the Sortino

ratio with a target return of on the numerator and zero on the denominator. The is convex in

losses and is therefore a risk averse measure and the Mean/ ratio provides a complimentary view

of risk to the Mean/ and Mean/ measures.

Definition 4.17 Mean/ Ratio: The Mean/ ratio, where is the mean historical return,

is the risk-free rate, and refers to the second order lower-partial moment of returns with

respect to the origin, is given by

E. Establishing Statistical Significance

In the extant literature, there is almost a complete absence of studies that evaluate the statistical

significance of the difference in performance between non-Gaussian and benchmark portfolio

construction techniques. We attempt to fill this gap. For Gaussian performance ratios such as the

Sharpe ratio, it is trivial to evaluate the statistical significance of the difference between two strategies

52

Note however, that LPM satisfy the “shift invariance” property of Pedersen and Satchell (1998) 53

where the target return on the numerator and denominator is given by


31

using the Jobson and Korkie (1981) t-statistic with Memmel’s (2003) correction and we do so in the

current work.

There is not however a similarly developed literature to evaluate the statistical significance of non-

Gaussian measures, such as those identified in section IV D. We employ a block bootstrap technique

to estimate the standard-errors of our performance measures. It is necessary to use a block bootstrap to

account for the observed autocorrelation in portfolio returns. We select the block size, , to ensure the

fastest possible rate of convergence given the observed level of dependence using the algorithm of

Politis and White (2004).

where

and

and

We bootstrap from the observed strategy returns 10,000 times and compute 10,000 estimates for each

performance metric.

F. Quantifying Economic Value Added

In this subsection we discuss our methodology for quantifying the economic value added of a given

approach. Performance ratios allow for many insights. They do not however allow us to readily gauge

the uplift in investor welfare of a portfolio construction technique. We follow West, Edison, and Cho

(1993) and Fleming, Kirby and Ostdiek (2001) and estimate the value added, , as the fee that equates

the expected utility of the proposed approach with a benchmark approach, in our case the

unconstrained Gaussian model54

. To help ensure our results are robust to the choice of utility function,

we employ the mean-variance utility function and the power utility function. While the mean-variance

utility function is more widely used by practitioners, it does not of course account for higher

moments. Nevertheless, we include it for completeness. We use three levels of risk aversion

54

Fleming, Kirby and Ostdiek (2001) among others refer to this metric as the performance fee. We use the term

“value added” to avoid confusion with the terminology used in the mutual fund industry where a performance

fee is charged as a percentage of the return


32

representing a conservative ( =0.028), a moderate ( =0.05), and an aggressive investor ( =0.089).

These values are derived from the first order conditions of the mean-variance investors using the

weights of an aggressive, moderate and conservative investor.55

Definition 4.18 Value Added – Mean-Variance Utility: The value-added refers to the management

fee, , that solves the following equation where refers to the coefficient of risk aversion,

and refers to the return of the model and benchmark portfolios in period .

Conversely, power utility does account for higher moments and is generally considered a more

plausible model of agent behaviour, exhibiting constant relative risk aversion. Again we use three

levels of risk aversion, =5, =10, and =15, consistent with estimates given in the literature.56

Definition 4.19 Value Added – Power Utility: The value-added refers to the management fee, ,

that solves the following equation where refers to the coefficient of risk aversion, and

refers to the return of the model and benchmark portfolios in period .

F. Data

We employ eight data sets across all of the major asset classes as summarised in table V. In the vast

majority of the related literature, one or perhaps two data sets are used per study to validate a given

portfolio construction technique. Because financial data are inherently noisy and the extreme market

conditions that provide stress tests of an approach are by definition rare, we argue for employing as

much high quality data as possible. By way of illustration, consider a top quartile fund manager with

an information ratio57

of 0.558

. Using the Jobson and Korkie (1981) t-statistic it can be shown that we

requires no less than 26 years of data to reject the hypothesis that the true information ratio equals

zero. The information ratio is of course a Gaussian- based metric. Intuitively, when returns and the

performance metrics employed are non-Gaussian it will take even longer to establish statistical

significance. Employing multiple data sets also limits the scope for “cherry picking”.

Our data sets span multiple periods of market turbulence, including the 1987 stock-market crash, the

1997 Asian financial crisis, the Russian debt default and the demise of Long-Term Capital

Management in 1998, the dot-com crash in 2000, the 2001 terrorist attacks, and the 2007-8 financial

crisis. We limit ourselves to value-weighted portfolios and indices to help ensure that the strategies

are robust to trading costs. The first data set, replicates the asset allocation problem of the institutional

investor, and is comprised of U.S. equities (S&P 500), international equities (MSCI EAFE), U.S.

corporate bonds (Barclays Corporate Bond index), a broad-based commodities index (GSCI), and

U.S. REITs (FTSE/EPRA U.S. REITs). The second, third, and fourth data sets include U.S. sectors at

three levels of granularity and have been selected to replicate the investment universe of an equity

55

Allen, Lizieri and Satchell (2012) 56

For example, Jondeau and Rockinger (2012) 57

The annual excess return divided by the tracking error 58

Grinold and Kahn (2000)


33

portfolio manager; a typical structure for a fund management firm is for the portfolio manager to

oversee sector analysts and to allocate capital to each sector. The fifth data-set includes the ten largest

U.S. equities that have been continuously listed since 1985, and has been selected to represent the

stock picker’s problem. The sixth and seventh data sets have been chosen to replicate the investment

universe of a fund of funds manager that allocates between large cap and small-capitalisation funds,

and value and growth funds. Our eighth data set includes the three Fama-French factors (Fama and

French, 1993), the momentum factor used in the four factor market of Carhart (1997) and the short-

term reversal factor (Jegadeesh, 1990) utilised by statistical arbitrage funds. This data set is analogous

with the opportunity set of a fund of hedge fund manager allocating between different hedge fund

styles. All our data series are daily including dividends where relevant.

Table V – Data Sets

Data set Description Source Period

1. 5 Asset Classes US Equities (S&P 500), EAFE Equities

(MSCI), Corporate Bonds Index (Barclays) ,

Commodities (GSCI), US REITs

(FTSE/EPRA)

Datastream,

Bloomberg

12/89-5/13

2. 5 US Industries Value weighted K.R. French 12/89-5/13



5. 10 Equities Largest 10 U.S. Equities59 continuously listed

for 31/12/1985 to 31/5/2013

Factset, Ex-share

database

12/84-5/13

6. 5 Market Capitalisation Portfolios Value-weighted quintiled by market

capitalisation

K.R. French

7. 5 Book-to-Market Portfolios Value-weighted quintiled by Book-to-Market

ratio

K.R. French

8. Fama-French Factors Market, Size, Value, Momentum, Short-term

Reversal

K.R. French 12/89-5/13

In table VI, we show the average summary statistics for each of the eight data sets60

. We divide the

data sets into overlapping 1000-day sub-periods and conduct Jarque-Bera tests for normality. All data

sets show marked departures from normality. In roughly 80% of sub-periods normality is rejected for

each asset class. Statistically significant excess kurtosis is more common than statistically significant

skew. In general, our data sets exhibit insignificant levels of first-order autocorrelation. There is

however pronounced autocorrelation in the absolute value of returns consistent with

heteroskedasticity.

59

at 31/12/1985 60

The full summary statistics for each asset class are available from the author on request

Table V summarised our eight datasets used to validate the respective approaches to VaR estimation and portfolio

construction.


34

Table VI

Average Summary Statistics: All Asset Classes

V. The Benchmark Models

Asset

Allocation

Problem

5

Industries

10

Industries

30

Industries

10

Stocks

Size

Portfolios

Value

Portfolios

Fama-

French

Factors

Annualised Return 5.9 11.6 11.5 11.0 10.8 11.1 11.9 9.0

Standard-deviation 20.5 19.5 20.3 22.9 30.6 18.2 18.1 12.4

Skewness -0.6 -0.5 -0.4 -0.3 -0.1 -0.6 -0.7 -0.2

Kurtosis 16.3 15.5 14.4 12.5 20.3 12.1 17.6 18.0

Maximum 9.6 12.2 12.8 13.8 22.0 9.8 11.1 8.0

Minimum -10.0 -17.5 -17.4 -17.7 -25.6 -13.9 -17.4 -9.8

Jacque-Bera 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001

% of Jacque-Bera failures 80 79 81 80 85 82 80 79

% of stat. sig. skew 43 46 46 46 50 57 46 57

% of stat. sig. kurtosis 82 79 81 80 82 82 79 78

0.04 0.01 0.02 0.03 -0.04 0.04 -0.00 0.10

0.21 0.17 0.18 0.20 0.23 0.28 0.18 0.25

We quantify the degree of asymmetric dependence (AD) using the concept of exceedance

correlations.

Definition 4.20 Exceedance Correlations: The positive and negative exceedance correlations

where and are random variables, and is the exceedance standard-deviation are defined as

To determine whether or not the observed asymmetry is statistically significant, Hong, Tu and Zhou

(2007) propose the model-free J-test. The J-test is distributed as a where is the number of

exceedance thresholds.

Definition 4.21 J-test for Asymmetric Correlation (Hong, Tu, and Zhou, 2007): The J-test

statistic where is the number of data points, and are the upper and lower exceedance

correlation coefficients, and is an estimate of the covariance matrix of is given by

We use the signed J-test of Alcock and Hatherley (2009) that also allows us to identify whether the

data displays net upper or lower tail dependence with exceedance thresholds of

Table VI provides the median summary statistics for the nine data sets listed in table I. The time period is 1/1983-12/2012 for

each all data sets except for the asset allocation problem (121989-5/2013) and the stock selection data set (12/1983-3/2013).

The annualised returns are calculated geometrically. The % of Jarque-Bera failures refers to the proportion of 1000 day sub-

periods where normality is rejected at the 5% level. The % of statistically significant skewness and % of statistically

significant kurtosis refers to the proportion of sub-periods where the skewness and kurtosis is statistically significant at the

5% level. The mean positive (negative) exceedance ρ refers to the average exceedance correlation above 1.5 standard-

deviations. The % stat. sign. net +ve exceedance ρ refers to the proportion of sub-periods where the net exceedances

correlation is statistically significant using the signed J-test. refers to the autocorrelation at one lag.

refers to the one-period autocorrelation of the absolute value of returns.


35

.61 We calculate the signed J-statistic for non-overlapping 260 day sub-

periods. The mean “bear” (downside) and “bull” (upside) correlations are shown in table VII for

absolute returns in excess of one-standard-deviation62

. In six out of eight of the data sets the mean

bear correlation exceeds the mean bull correlation. In seven out of the eight data sets, statistically

significant downside asymmetric dependence is more common than upside asymmetric dependence.

In general statistically significant asymmetric dependence however is the exception rather than the

rule. The Fama-French data exhibits asymmetric dependence most frequently, approximately twice as

often as would be expected by chance alone. The finding that asymmetric dependence is not universal

is perhaps under appreciated. The fact that asymmetric dependence appears to be episodic does not

however mean that it is unimportant. It is well established that dependence tends to increase during

times of market stress coinciding with periods when investors value diversification the most. It is

plausible therefore that although asymmetric dependence is not an ever-present feature of financial

markets, investors may still benefit from accounting for it.

Table VII

Asymmetric Dependence: All Asset Classes

Asset

Allocation

Problem

5

Industries

10

Industries

30

Industries 10 Stocks

Size

Portfolios

Value

Portfolios

Fama-

French

Factors

Mean "bear" ρ 0.13 0.67 0.57 0.48 0.30 0.81 0.79 0.16

Mean "bull" ρ 0.17 0.55 0.44 0.36 0.18 0.70 0.72 0.21

% sign AD 8.33 1.72 2.15 4.30 7.33 0.34 0.34 11.03

% sign + AD 5.00 0.34 1.38 2.43 4.86 0.0 0.0 12.07

V. Results

We proceed as follows. In section V A we evaluate the model fit of the GSEV approach. In section V

B we present the VaR forecasting results for the eight models. In section V C we discuss the out-of-

sample performance of the respective models and quantify the uplift in investor welfare.

A. Evaluating Model Goodness of Fit

It is important to ensure that the marginal density functions are consistent with the data as an

inappropriate specification of the marginal distributions will lean to a misspecification of the copula

function. We evaluate the fit of the Gaussian, student-t, and GSEV distributions to the univariate

ARMA/EGARCH residuals using the two-sample Kolmogorov-Smirnov and Cramer von-Mises

tests63

. We perform the tests each month on overlapping sub-periods of 1000 days each for each asset.

Table VIII shows the proportion of sub-periods that each distribution is rejected for each of the data

sets. Both the two-sample Kolmogorov–Smirnov and Cramer von-Mises tests indicate that normality

can be rejected approximately half the time. This is a reduction from what we saw in table VI

61

Hong, Tu and Zhou (2007) find that has reasonable finite sampling properties. We find that

this choice leads to unstable estimates owing to our shorter estimation window. We therefore use

consistent with Alcock and Hatherley, (2009) 62

Theoretically we would prefer to use a higher standard-deviation threshold; however a higher threshold would

lead to sampling issues 63

Evaluating the relative model fit using, for example, Akaike’s information criterion is inappropriate as we are

using a smoothing kernel for the body of the distribution for the GSEV model.

Table VII provides mean exceedance correlations for each asset class. The upside and downside exceedance correlations are

estimated using returns in excess of one standard deviation from the mean using non-overlapping 260 day sub-periods.

Statistical significance is determined using the singed J-test at the 5% level.


36

indicating that the ARMA/EGARCH process is accounting for a portion of the non-normality of the

data consistent with the discussion in section III64

. The student-t distribution while an improvement

over the Gaussian distribution is still rejected in 17% of cases. The GSEV approach however is

rejected less than 1% of the time. The low level of rejections is not a surprise given the use of a

smoothing kernel for the body of the data.

Table VIII Goodness of Fit and Model Comparison: All Data Sets

Kolmogorov–Smirnov Cramer von-Mises

Gaussian Student-t GSEV Gaussian Student-t GSEV

Asset Allocation Problem 48% 22% 0% 42% 17% 0%

5 Industries 40% 8% 0% 37% 5% 0%

10 Industries 34% 7% 0% 35% 3% 0%

30 Industries 86% 31% 1% 89% 26% 1%

10 Stocks 34% 7% 0% 35% 3% 0%

Size Portfolios 57% 36% 0% 55% 31% 0%

Value Portfolios 46% 16% 0% 44% 8% 0%

Fama-French Factors 42% 9% 0% 43% 5% 0%

Mean 48% 17% 0% 47% 12% 0%

Given our focus on downside risk, we also perform goodness-of fit tests on the lower 5% quantile of

the univariate residuals. This can be seen as viewing the lower 5% quantile as a complete distribution.

In the case of the GSEV model, the lower 5% quantile is identical to the generalised Pareto

distribution. The results are shown in table IX. In general the rate of rejection for the tail is higher

than for the entire distribution. The student-t improves on the Gaussian distribution, while the GSEV

model has the lowest rate of rejection for each data set, in the region of 2-10%. The GSEV model

therefore appears to fit the body and the lower tail of the data well in sample.

Table IX Goodness of Fit and Model Comparison: All Data Sets

Kolmogorov–Smirnov Cramer von-Mises

Gaussian Student-t GSEV Gaussian Student-t GSEV

Asset Allocation Problem 62% 46% 7% 70% 50% 5%

5 Industries 51% 30% 9% 55% 32% 6%

10 Industries 52% 33% 8% 53% 32% 6%

30 Industries 48% 33% 8% 50% 35% 7%

10 Stocks 63% 46% 10% 61% 46% 10%

Size Portfolios 69% 65% 3% 74% 66% 2%

Value Portfolios 55% 44% 3% 60% 46% 2%

Fama-French Factors 53% 32% 6% 55% 34% 4%

Mean 57% 41% 7% 60% 42% 5%

B1. VaR Prediction – 99% confidence level

64

This is also consistent with Cont (2001)

Table VIII shows the average proportion of 1000 day sub-periods where the GARCH residuals fail the two-sample

Kolmogorov–Smirnov and two-sample Cramer von-Mises tests for the Gaussian, student-t, and GSEV distributions for each

of the data sets.

Table IX shows the average proportion of 1000 day sub-periods where the lower 5% quantile of the GARCH residuals fail the

two-sample Kolmogorov–Smirnov and two-sample Cramer von-Mises tests for the Gaussian, student-t, and GSEV

distributions for each of the data sets.


37

We now turn to the evaluation of the VaR predictions. Fitting the data well in-sample is of little

consequence if we fail to forecast risk accurately out-of-sample. Table X shows the proportion of VaR

violations, the proportion of consecutive violations, and the p-values for the unconditional coverage,

serial independence and conditional coverage tests at the 99% confidence level. If the models perform

perfectly, there should be 1% of violations or one every 100 days. Across all data sets for the

Gaussian model there are approximately 130% more violations than there should be. The student-t

model and the exponentially weighted moving average (EWMA) improve on this; however violations

occur approximately twice as often as they should. The historical simulation (HS) and Clayton copula

models (CC) provide further improvement reducing the proportion of violations to 1.64% and 1.65%.

This makes sense given the arguably more appropriate dependence structure that these models

facilitate. Note that the Clayton copula model is only applicable when there is positive dependence

and it is hence omitted when this condition is not continuously met. On average, the final three

models, filtered historical simulation (FHS), the EGARCH/Gaussian copula/EVT model (GGEV), and

the GSEV model yield 1.3%, 1.38%, and 1.24% violations respectively. We have included the GGEV

model in order to gauge the benefits of accounting for asymmetric tail dependence through the

skewed-t-copula instead of the Gaussian copula which is asymptotically independent. The GSEV

model has the highest average p-value and the least number of significant unconditional coverage test-

statistics. As discussed in section IV, the correct number of violations is not a sufficient condition for

a well-behaved VaR model. It is also desirable for the violations to be independent. For the

unconditional approaches, the Gaussian, student-t, and historical simulation, we see a high proportion

of consecutive violations relative to the conditional approaches, EWMA, FHS, GGEV and GSEV.

From a practical perspective consecutive violations translate to successive demands on capital that

may be difficult to meet. For seven out of eight of the data sets the FHS, GGEV and GSEV models

pass the serial independence test. The final test of conditional coverage incorporates both

unconditional coverage and serial independence. We see that the FHS, GGEV and GSEV have the

highest p-values, with the GSEV approach having the highest p-value on average and for six out of

the eight data sets.


38

Table X VaR Forecast Evaluation: 1/N Portfolios, All Data Sets

β=99% Gaussian Student-t

Historical

Simulation

Clayton

Copula/ Gaussian

Marginals

Exp.

Weighted Moving

Average

Filtered Historical

Simulation GGEV GSEV

Asset Allocation

Violations 3.03% 1.97% 2.06% 2.24% 1.54% 1.62% 1.40%

UC: p-value 0.00 0.00 0.00 0.00 0.02 0.01 0.07

Cons. viol 13.0% 15.6% 12.8% 1.9% 2.8% 2.6% 3.0%

SI: p-value 0.00 0.00 0.00 0.86 0.60 0.67 0.50

CC: p-value 0.00 0.00 0.00 0.00 0.05 0.02 0.15

5 Industries

Violations 2.15% 1.90% 1.67% 1.86% 1.99% 1.26% 1.27% 1.13%

UC: p-value 0.00 0.00 0.00 0.00 0.00 0.04 0.03 0.28

Cons. viol 9.3% 9.7% 9.2% 9.9% 4.6% 1.2% 2.4% 1.4%

SI: p-value 0.00 0.00 0.00 0.00 0.06 0.97 0.41 0.86

CC: p-value 0.00 0.00 0.00 0.00 0.00 0.13 0.07 0.55

10 Industries

Violations 2.16% 2.04% 1.72% 1.52% 2.02% 1.26% 1.32% 1.10%

UC: p-value 0.00 0.00 0.00 0.00 0.00 0.04 0.01 0.41

Cons. viol 9.9% 9.8% 9.8% 11.1% 5.3% 1.2% 1.2% 1.4%

SI: p-value 0.00 0.00 0.00 0.00 0.02 0.97 0.90 0.82

CC: p-value 0.00 0.00 0.00 0.00 0.00 0.13 0.05 0.69

30 Industries

Violations 2.35% 2.38% 1.69% 1.41% 2.22% 1.38% 1.49% 1.30%

UC: p-value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.02

Cons. viol 9.8% 9.0% 8.2% 8.7% 4.1% 2.2% 1.0% 2.4%

SI: p-value 0.00 0.00 0.00 0.00 0.16 0.53 0.69 0.44

CC: p-value 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.05

10 Stocks

Violations 1.86% 1.55% 1.41% 0.57% 1.54% 1.23% 1.26% 1.27%

UC: p-value 0.00 0.00 0.00 0.00 0.00 0.09 0.05 0.04

Cons. viol 6.1% 7.4% 7.0% 14.3% 1.1% 1.3% 1.3% 1.3%

SI: p-value 0.01 0.00 0.00 0.00 0.67 0.95 1.00 1.00

CC: p-value 0.00 0.00 0.00 0.00 0.00 0.23 0.15 0.12

Size Portfolios

Violations 2.39% 2.29% 1.47% 2.32% 2.25% 1.23% 1.07% 1.03%

UC: p-value 0.00 0.00 0.00 0.00 0.00 0.08 0.55 0.82

Cons. viol 11.5% 11.4% 9.4% 11.3% 5.4% 2.5% 1.4% 1.5%

SI: p-value 0.00 0.00 0.00 0.00 0.02 0.36 0.78 0.72

CC: p-value 0.00 0.00 0.00 0.00 0.00 0.14 0.81 0.92

Value Portfolios

Violations 2.26% 2.14% 1.66% 2.20% 1.98% 1.34% 1.32% 1.25%

UC: p-value 0.00 0.00 0.00 0.00 0.00 0.01 0.01 0.05

Cons. viol 10.2% 9.4% 10.2% 9.8% 5.5% 2.3% 3.5% 3.7%

SI: p-value 0.00 0.00 0.00 0.00 0.02 0.48 0.14 0.10

CC: p-value 0.00 0.00 0.00 0.00 0.00 0.03 0.02 0.04

Fama-French Factors

Violations 1.95% 1.63% 1.43% 1.72% 1.18% 1.66% 1.40%

UC: p-value 0.00 0.00 0.00 0.00 0.15 0.00 0.00

Cons. viol 17.3% 21.7% 18.3% 7.1% 5.2% 5.6% 6.6%

SI: p-value 0.00 0.00 0.00 0.00 0.02 0.01 0.00

CC: p-value 0.00 0.00 0.00 0.00 0.02 0.00 0.00

Averages

Violations 2.27% 1.99% 1.64% 1.65% 2.00% 1.30% 1.38% 1.24%

UC: p-value - 0.00 0.00 0.00 0.00 0.05 0.08 0.21

Cons. viol 10.91% 11.73% 10.60% 10.84% 4.39% 2.34% 2.37% 2.65%

SI: p-value 0.00 0.00 0.00 0.00 0.23 0.61 0.57 0.56

CC: p-value 0.00 0.00 0.00 0.00 0.00 0.09 0.14 0.31

Table X shows the summary statistics for the VaR forecasting models for the equally weighted portfolio for all eight data sets for

the period 1983-2012. From the 1000th day onwards, each model is re-estimated each day using a 1000-day estimation window.

Table X provides the proportion of violations using a 99% confidence level, the proportion of consecutive violations, and the p-

values of the unconditional, serial-independence, and conditional coverage tests.


39

B2. VaR Prediction – 99.5% confidence level

The results for the β=99.5% are shown in table XI. If the models perform perfectly we expect 0.5% of

violations or one every 200 days. It is again apparent that the elliptical models in general

underestimate tail risk. For the Gaussian model there are on average 260% too many violations across

all data sets. Similarly the student-t and EWMA models yield 160% and 180% too many violations.

The FHS and GGEV models produce 50% and 70% too many violations whereas the GSEV model

produces only 20% too many. The unconditional coverage p-value of the GSEV model is the highest

out of all the models for six out of eight of the data sets. We can reject the hypothesis of serial

independence for all of the unconditional models for all of the datasets and only for one of the data

sets for the FHS, GGEV and GSEV models. The FHS, GGEV and GSEV models have the highest

conditional coverage p-values on average, with the GSEV approach having the highest p-value on

average and for six out of the eight data sets. While the performance of the GSEV and GGEV models

is similar at the 99% confidence level, the GSEV model outperforms the GGEV model consistently at

the 99.5% level. The GGEV model produces 66% too many violations, the GSEV model only

produces 20%. Further, the conditional coverage p-values of the GGEV model exceed 5% in six out

of eight of the data sets and only in one data set for the GSEV model. The only difference between the

GGEV model and the GSEV model is the use of the Gaussian copula that is asymptotically

independent instead of the skewed-t copula that accommodates heterogeneous asymmetric tail

dependence. The superior performance of the GSEV model is indicative of the importance of

accounting for asymptotic dependence patterns as we look further into the tails.


40

Table XI VaR Forecast Evaluation: 1/N Portfolios, All Data Sets

β=99.5% Gaussian Student-t Historical Simulation

Clayton

Copula/

Gaussian Marginals

Exp.

Weighted

Moving Average

Filtered

Historical Simulation GGEV GSEV

Asset Allocation

Violations 2.11% 1.32% 1.18% 1.45% 0.92% 1.01% 0.48%

UC: p-value 0.00 0.00 0.00 0.00 0.01 0.00 0.90

Consecutive viol 14.6% 13.3% 7.4% 2.9% 4.5% 4.2% 8.3%

SI: p-value 0.00 0.00 0.04 0.53 0.21 0.25 0.05

CC: p-value 0.00 0.00 0.00 0.00 0.02 0.01 0.15

5 Industries

Violations 1.79% 1.20% 0.95% 1.43% 1.43% 0.81% 0.75% 0.61%

UC: p-value 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.21

Consecutive viol 9.4% 10.3% 9.7% 11.8% 4.3% 1.9% 2.0% 2.5%

SI: p-value 0.00 0.00 0.00 0.00 0.06 0.46 0.39 0.25

CC: p-value 0.00 0.00 0.00 0.00 0.00 0.00 0.02 0.23

10 Industries

Violations 1.81% 1.46% 0.98% 1.17% 1.49% 0.81% 0.81% 0.64%

UC: p-value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.11


SI: p-value 0.00 0.00 0.00 0.00 0.07 0.46 0.46 0.28

CC: p-value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.16

30 Industries

Violations 1.86% 1.69% 0.97% 0.98% 1.46% 0.83% 0.86% 0.58%

UC: p-value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.36


SI: p-value 0.00 0.00 0.00 0.00 0.23 0.47 0.51 0.22

CC: p-value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.31

10 Stocks

Violations 1.47% 1.05% 0.77% 0.36% 1.05% 0.54% 0.75% 0.59%

UC: p-value 0.00 0.00 0.01 0.10 0.00 0.67 0.01 0.34


SI: p-value 0.00 0.00 0.05 0.00 0.72 0.19 0.38 0.22

CC: p-value 0.00 0.00 0.00 0.00 0.00 0.38 0.02 0.30

Size Portfolios

Violations 1.89% 1.37% 0.86% 1.83% 1.52% 0.69% 0.66% 0.58%

UC: p-value 0.00 0.00 0.00 0.00 0.00 0.04 0.08 0.36


SI: p-value 0.00 0.00 0.00 0.00 0.09 0.32 0.29 0.22

CC: p-value 0.00 0.00 0.00 0.00 0.00 0.07 0.13 0.31

Value Portfolios

Violations 1.86% 1.45% 0.92% 1.85% 1.43% 0.80% 0.65% 0.63%

UC: p-value 0.00 0.00 0.00 0.00 0.00 0.00 0.11 0.15


SI: p-value 0.00 0.00 0.00 0.00 0.21 0.44 0.28 0.26

CC: p-value 0.00 0.00 0.00 0.00 0.00 0.01 0.15 0.19

Fama-French Factors

Violations 1.60% 0.92% 0.72% 1.22% 0.66% 1.18% 0.69%

UC: p-value 0.00 0.00 0.02 0.00 0.08 0.00 0.04

Consecutive viol 17.3% 21.7% 23.4% 7.6% 2.3% 7.8% 2.2%

SI: p-value 0.00 0.00 0.00 0.00 0.29 0.00 0.32

CC: p-value 0.00 0.00 0.00 0.00 0.12 0.00 0.07

Averages

Violations 1.80% 1.30% 0.92% 1.27% 1.38% 0.76% 0.83% 0.60%

UC: p-value 0.00 0.00 0.00 0.02 0.00 0.10 0.03 0.31

Cons. viol 11.07% 11.29% 9.09% 10.65% 3.87% 2.45% 3.06% 3.23%

SI: p-value 0.00 0.00 0.01 0.00 0.24 0.35 0.32 0.23

CC: p-value - 0.00 0.00 0.00 0.00 0.08 0.04 0.22

Table XI shows the summary statistics for the eight VaR forecasting models for the equally weighted portfolio for all eight

data sets for the period 1983-2012. From the 1000th day onwards, each model is re-estimated each day using a 1000-day

estimation window. Table XI provides the proportion of violations using a 99.5% confidence level, the proportion of

consecutive violations, and the p-values of the unconditional, serial-independence, and conditional coverage tests.


41

Consistent with prior literature we have shown that conditional approaches outperform unconditional

approaches. We have also demonstrated the benefits of using extreme-value theory to produce more

accurate VaR predictions. Finally, we have highlighted the importance of accounting for asymmetric

dependence patterns between assets where the skewed-t copula outperforms the Gaussian copula. The

GSEV approach incorporates all of these elements to yield reliable VaR forecasts at the 99% and

99.5% confidence levels. Out of the models surveyed here, the GSEV model appears to be the most

promising.

C. Dynamic Portfolio Rebalancing

We now present the results for the nine portfolio construction models for the eight data sets. Tables

XII, and XIII summarise the performance of the models for each data set. Table XIV shows the

average summary statistics across all eight data sets.

C1. Risk Characteristics

For every data set, the out-of sample CVaRs of the Gaussian, student-t, HS, EWMA models deviate

significantly from the targeted 2% level. In contrast the FHS, GGEV and GSEV models generate

CVaRs that are quite close to the target level with the GSEV model producing the smallest CVaR

error65

on average. The GSEV approach produces the smallest CVaR error for seven out of eight of

the data sets. This mirrors findings in section V B. The kurtosis of the conditional models is also

significantly lower across all of the data sets. The FHS model produces the smallest kurtosis level on

average. The conditional models, the EWMA, FHS, GGEV and GSEV also provide a reliably lower

maximum drawdown, defined as the peak to trough return, a metric commonly used by practitioners.

The GSEV model produces the maximum drawdown with the smallest absolute value for all eight

investment problems66

.

C2. Sharpe Ratios

The unconditional approaches, 1/N, the Gaussian, student-t, historical simulation, and the Clayton

copula approach generate similar Sharpe ratios to each other. The average Sharpe ratio of the 1/N

strategy and the Gaussian strategy are almost identical at 0.51 and 0.52 as shown in table XIV. This

mirrors the results of DeMiguel, Garlappi and Uppal (2009) for constrained minimum-variance. The

relatively unimpressive performance of the Clayton copula model runs counter to Alcock and

Hatherley (2009) who show a substantial performance uplift relative to the Gaussian case. The Alcock

and Hatherley (2009) work considers triplets of industry indices. Given, that the standard Clayton

copula imposes an identical dependence structure across asset pairs and has a single dependence

parameter, it is conceivable that the performance of the approach degrades as the number of assets and

complexity increases. The GSEV approach generates the highest average Sharpe ratio of 1.13 across

all data sets. Using the Jobson and Korkie (1981) t-statistic, the difference in the Sharpe ratio of the

GSEV model is statistically different from the Gaussian model for each data set at the 1% level. On

average the uplift in the Sharpe ratio is 125%. The finding that accounting for non-Gaussian

characteristics is beneficial even for an investor that is only concerned with the first two moments is

consistent with the analytical findings of Allen, Lizieri, and Satchell (2013).

C3. Downside Performance Metrics

65

defined as the realised CVaR divided by the target CVaR 66

In two cases the difference in the maximum drawdown is negligible


42

The Sharpe ratio is of course a Gaussian-based metric and does not account for higher moments. We

have defined the Mean/CVaR ratio as the average excess return divided by the 99% expected

shortfall. The average Mean/CVaR ratio across our eight data sets is 5.02 versus 1.72 for the Gaussian

model and the GSEV model outperforms the Gaussian model for every data set. In addition the

improvement is statistically significant at the 1% level in all cases. Moreover the 182% average

increase in Mean/CVaR is even larger than the increase in the Sharpe ratio. The GSEV approach also

dominates the other benchmark models. In six out of the eight data sets the GSEV approach produces

the highest Mean/CVaR ratio. The GGEV model that uses the asymptotically independent Gaussian

copula produces the highest Mean/CVaR ratio in only two of the data sets. This provides further

evidence of the importance of accounting of asymmetric dependence. As we noted in table VII, the

two investment problems with the most frequent statistically significant asymmetric dependence are

the Asset Allocation and the Fama-French problems. It is perhaps of no surprise that these are the two

data sets where the GSEV approach adds the most value relative to the GGEV approach. For the

Fama-French data set, the increase in the Mean/CVaR ratio relative to the GGEV model is 13% and

statistically significant.

In general, the results for our two alternative downside performance metrics, the Mean/LPM1 and the

Mean/LPM2 tell the same story. The GSEV model generates higher downside-risk adjusted returns

than the Gaussian, student-t, historical simulation, Clayton Copula, and EWMA models. The average

Mean/LPM1 and Mean/LPM2 ratios of the Gaussian and GSEV models are 24.51 and 55.05, and

10.08 and 25.64 respectively. The results of the lower-partial moment measures mirror the Mean-

CVaR results giving us confidence that our findings are robust to the chosen performance metric.


43

Table XII – Dynamic Portfolio Simulation Results

1/N Gaussian Student-t

Historical

Simulation

Clayton

Copula/

Gaussian

Marginals

Exp.

Weighted

Moving

Average

Filtered

Historical


Asset Allocation

Annualised return 4.8% 8.9% 2.9% 4.7% 17.0% 25.1% 24.9% 23.9%

Standard-dev. 16.5% 14.8% 11.7% 9.7% 14.3% 12.2% 12.3% 11.2%

Kurtosis 9.2 23.0 32.4 13.3 3.0 2.2 2.2 2.6

CVaR 99% -4.7% -4.8% -4.0% -2.9% -3.3% -2.5% -2.5% -2.3%

Risk Error (%) 133% 141% 98% 45% 67% 23% 26% 16%

Max. DD -52% -72% -64% -53% -50% -39% -39% -37%

Sharpe ratio 0.36 0.65 0.30** 0.52 1.16* 1.89** 1.86** 1.96**

Mean/CVaR 99% 1.30 2.00 0.89 1.75 5.00** 9.38** 9.12** 9.53**

Mean/LPM1 19.45 38.97 18.87 30.65 57.57** 98.83** 97.28** 103.9**

Mean/LPM2 8.21 13.51 6.14 11.25 26.20** 45.89** 45.23** 47.87**

5 Industries

Annualised return 6.6% 5.4% 4.6% 4.2% 5.4% 7.1% 8.8% 9.1% 8.8%

Standard-dev. 18.3% 13.6% 11.2% 10.2% 14.3% 13.6% 11.2% 11.3% 11.0%

Kurtosis 16.7 45.7 65.1 30.6 60.8 5.2 6.9 7.7 8.3

CVaR 99% -4.7% -3.5% -2.9% -2.6% -3.6% -3.1% -2.5% -2.6% -2.5%

Risk Error (%) 136% 74% 45% 30% 82% 57% 27% 29% 25%

Max. DD -51% -34% -29% -28% -38% -33% -27% -26% -26%

Sharpe ratio 0.44 0.45 0.45 0.45 0.44 0.57 0.80** 0.82** 0.81**

Mean/CVaR 99% 1.71 1.78 1.76 1.77 1.73 2.48** 3.56** 3.63** 3.60**

Mean/LPM1 22.79 23.38 24.15** 23.11 23.01 26.67** 38.39** 39.76** 39.22**

Mean/LPM2 9.97 10.02 9.95 10.00 9.72 12.81** 18.27** 18.80** 18.62**

10 Industries


Standard-dev. 17.8% 13.7% 11.8% 10.8% 14.1% 14.1% 11.8% 11.9% 11.4%

Kurtosis 18.8 31.1 33.4 26.9 29.2 4.3 5.3 6.6 7.2

CVaR 99% -4.7% -3.4% -3.0% -2.7% -3.5% -3.1% -2.6% -2.6% -2.5%

Risk Error (%) 135% 70% 50% 37% 77% 57% 29% 31% 24%

Max. DD -52% -35% -32% -28% -36% -33% -27% -28% -23%

Sharpe ratio 0.45 0.48 0.46 0.49 0.47 0.69 0.89** 0.92** 0.91**

Mean/CVaR 99% 1.71 1.96 1.84 1.94 1.88 3.13** 4.11** 4.16** 4.19**

Mean/LPM1 23.47 24.95 24.52 25.42* 24.49 32.88** 43.25** 44.70** 44.41**

Mean/LPM2 10.14 10.88 10.50 10.97 10.69 15.88** 20.73** 21.18** 21.16**

30 Industries


Standard-dev. 18.7% 13.8% 12.7% 11.3% 14.0% 15.9% 12.5% 12.8% 12.3%

Kurtosis 16.9 36.7 49.1 30.8 29.4 4.4 4.0 6.3 7.0

CVaR 99% -5.1% -3.5% -3.3% -2.9% -3.5% -3.6% -2.8% -2.9% -2.8%

Risk Error (%) 156% 75% 63% 44% 77% 80% 38% 45% 41%

Max. DD -60% -41% -36% -36% -34% -40% -30% -31% -30%

Sharpe ratio 0.33 0.34 0.34 0.36 0.35 0.54 0.71** 0.74** 0.70**

Mean/CVaR 99% 1.22 1.37 1.33 1.41 1.40 2.40** 3.25** 3.27** 3.07**

Mean/LPM1 17.49 17.80 18.20* 18.37** 18.36* 25.58** 34.32** 35.80** 33.92**

Mean/LPM2 7.46 7.65 7.61 7.93** 7.99* 12.33** 16.47** 16.90** 15.99**

Table XII shows the summary statistics for the eight portfolio construction methodologies for the Asset Allocation, 5

Industries, 10 industries, and 30 Industries data sets for the period 12/1986-12/2012. The annualised return is calculated

geometrically. The standard deviation is annualised. Kurtosis refers to excess kurtosis. CVaR 99% refers to the

conditional value at risk at the 99% confidence level. Risk error (%) refers to the percentage difference in the realised

CVaR 99% and the CVaR 99% target of 2%. The Sharpe ratio is calculated arithmetically and is annualised. Mean/CVaR

99% refers to the ratio of the annualised arithmetic excess return divided by the realised CVaR 99%. Mean/LPM1 refers

to the ratio of the annualised arithmetic excess return divided by the first lower partial moment with a return target of

zero. Mean/LPM2 refers to the ratio of the annualised arithmetic excess return divided by the second lower partial

moment with a return target of zero.


44

Table XIII – Dynamic Portfolio Simulation Results


Historical

Simulation

Clayton

Copula/

Gaussian

Marginals

Exp.

Weighted

Moving

Average

Filtered

Historical


10 Stocks


Standard-dev. 18.9% 12.6% 10.6% 11.3% 11.9% 15.6% 12.4% 12.0% 11.6%

Kurtosis 7.7 7.0 7.6 4.1 6.8 3.8 2.1 2.4 2.4

CVaR 99% -4.6% -2.9% -2.5% -2.6% -2.7% -3.3% -2.6% -2.5% -2.4%

Risk Error (%) 130% 45% 26% 30% 34% 63% 29% 24% 19%

Max. DD -56% -32% -26% -34% -27% -32% -32% -27% -26%

Sharpe ratio 0.62 0.78 0.72 0.73 0.56 0.86 0.75 0.84 0.85

Mean/CVaR 99% 0.98 1.30 1.18 1.23 0.96 1.59* 1.39 1.58** 1.60**

Mean/LPM1 11.88 14.84 14.03 13.72 10.60 15.51 13.45 15.14 15.23

Mean/LPM2 5.54 7.02 6.50 6.58 5.13 7.74* 6.81 7.63** 7.68**

Size Portfolios


Standard-dev. 18.4% 14.2% 12.9% 10.0% 15.1% 13.8% 10.8% 10.7% 10.6%

Kurtosis 10.8 33.9 45.5 17.5 49.4 7.3 6.1 6.9 7.2

CVaR 99% -4.8% -4.1% -3.8% -2.7% -4.5% -3.5% -2.5% -2.5% -2.4%

Risk Error (%) 141% 104% 91% 35% 124% 76% 25% 24% 22%

Max. DD -56% -47% -49% -37% -54% -43% -30% -29% -29%

Sharpe ratio 0.40 0.38 0.35 0.36 0.31 0.63* 1.10** 1.12** 1.12**

Mean/CVaR 99% 1.53 1.32 1.18 1.37 1.06 2.50** 4.80** 4.85** 4.86**

Mean/LPM1 20.66 20.28 19.26 18.71 17.30 30.81** 54.58** 55.55** 55.97**

Mean/LPM2 8.95 8.03 7.33 7.95 6.53 13.84** 25.07** 25.37** 25.49**

Value Portfolios


Standard-dev. 18.2% 13.8% 11.9% 10.1% 14.4% 13.5% 11.1% 11.1% 11.0%

Kurtosis 18.1 41.4 47.9 39.8 47.3 5.7 5.7 6.3 5.9

CVaR 99% -4.9% -3.6% -3.1% -2.7% -3.8% -3.1% -2.6% -2.6% -2.5%

Risk Error (%) 144% 79% 56% 33% 88% 54% 29% 28% 26%

Max. DD -56% -40% -30% -33% -41% -28% -28% -28% -28%

Sharpe ratio 0.43 0.53 0.53 0.48 0.51 0.67 0.68 0.68 0.69

Mean/CVaR 99% 1.63 2.05 2.03 1.83 1.97 2.98** 2.94** 2.97** 3.00**

Mean/LPM1 23.07 27.83 28.32** 25.30 27.39 32.17** 32.40** 32.65** 32.93**

Mean/LPM2 9.81 11.70 11.65 10.54 11.38 15.24** 15.27** 15.39** 15.55**

Fama-French Factors

Annualised return 5.5% 6.5% 5.3% 12.1% 21.5% 25.3% 27.1% 26.2%

Standard-dev. 5.4% 14.7% 10.7% 13.1% 14.9% 13.3% 13.8% 12.1%

Kurtosis 18.2 25.6 74.3 8.5 4.4 2.6 4.3 3.5

CVaR 99% -1.5% -3.6% -2.7% -3.0% -3.0% -2.5% -2.7% -2.3%

Risk Error (%) 25% 80% 37% 49% 48% 25% 36% 16%

Max. DD -14% -41% -35% -39% -33% -38% -38% -36%

Sharpe ratio 1.00** 0.50 0.53 0.93** 1.38** 1.76** 1.80** 1.98**

Mean/CVaR 99% 3.65** 2.05 2.07 4.13** 6.98** 9.40** 9.16** 10.3**

Mean/LPM1 62.82** 27.96 33.80** 51.86** 77.27** 99.59** 103.4** 114.7**

Mean/LPM2 23.85** 11.78 12.64 23.22** 36.01** 47.15** 47.40** 52.77**

Table XIII shows the summary statistics for the eight portfolio construction methodologies for the 10 Stocks, 5 Size Portfolios,

5 Value Portfolios, and Fama-French factors for the period 12/1986-12/2012. The annualised return is calculated

geometrically. The standard deviation is annualised. Kurtosis refers to excess kurtosis. CVaR 99% refers to the conditional

value at risk at the 99% confidence level. Risk error (%) refers to the percentage difference in the realised CVaR 99% and the

CVaR 99% target of 2%. The Sharpe ratio is calculated arithmetically and is annualised. Mean/CVaR 99% refers to the ratio

of the annualised arithmetic excess return divided by the realised CVaR 99%. Mean/LPM1 refers to the ratio of the annualised

arithmetic excess return divided by the first lower partial moment with a return target of zero. Mean/LPM2 refers to the ratio

of the annualised arithmetic excess return divided by the second lower partial moment with a return target of zero.


45

Table XIV – Dynamic Portfolio Simulation Results


Historical

Simulation

Clayton

Copula/ Gaussian

Marginals

Exp.

Weighted Moving

Average

Filtered Historical



Standard-dev. 16.5% 13.9% 11.7% 10.8% 14.0% 14.5% 11.9% 12.0% 11.4%

Kurtosis 14.5 30.6 44.4 21.4 37.2 4.8 4.4 5.3 5.5

CVaR 99% -4.4% -3.7% -3.2% -2.8% -3.6% -3.3% -2.6% -2.6% -2.5%

Risk Error (%) 125% 83% 58% 38% 80% 63% 28% 30% 24%

Max. DD -50% -43% -38% -36% -38% -36% -31% -31% -29%

Sharpe ratio 0.51 0.52 0.46 0.54 0.44 0.82 1.08 1.10 1.13

Mean/CVaR 99% 1.72 1.73 1.54 1.93 1.51 3.39 4.86 4.85 5.02

Mean/LPM1 25.21 24.51 22.65 25.90 20.20 37.31 51.86 53.05 55.05

Mean/LPM2 10.50 10.08 9.05 11.06 8.58 17.51 24.46 24.74 25.64

D. Economic Value Added

In the previous section we have shown that the GSEV approach produces portfolios with lower risk

errors, lower drawdowns and statistically significant uplifts in the Sharpe ratio and the downside-risk

performance metrics. We now quantify the economic significance of the uplift. Table XV provides the

value added for all the models relative to the Gaussian model based on the mean-variance utility

assumption. The three levels of risk aversion represent aggressive, moderate and conservative

investors and are derived by solving the first order conditions given observed portfolio weights67

. The

1/N strategy and the Clayton copula model result in a loss in economic value relative to the Gaussian

case for all three investors. On average the student-t model does not add value, while the historical

simulation adds value for the moderate and conservative investors. The exponentially weighted

moving average model that underlies the popular RiskMetrics™ system adds in excess of 4% p.a. for

all levels of risk aversion. Filtered historical simulation adds an average of 7.4% across all three

levels. It is the GGEV and GSEV models however that add the most value. Interestingly, on average,

the GGEV model generates a marginally higher uplift of 7.8% relative to 7.7% for the GSEV model.

Given that typical mutual fund fees are 1-2%, this is highly significant indicating that managers that

implement either model may be able to charge substantially higher management fees.

In table XVI, we show the economic value added for the power utility investor. Power utility is seen

as a more plausible model of investor preferences than mean-variance utility. Unlike the mean-

variance investor, the power utility investor displays constant relative risk aversion and rewards

positive skew and penalises excess kurtosis. The 1/N rule and the Clayton copula model again

subtract value. The average value added of the student-t and historical simulation models increase

from the mean-variance case from -0.1% to 1.4% and 1.1% to 3.5%. This makes sense given that

these are heavy-tailed models and the power utility function rewards distributions with less tail risk.

The exponentially weighted moving average model adds a similar amount of value on average under

67

Allen, Lizieri and Satchell (2013)

Table XIII shows the average summary statistics for the eight portfolio construction methodologies for across all eight data

sets for the period 12/1986-12/2012. The annualised return is calculated geometrically. The standard deviation is annualised.

Kurtosis refers to excess kurtosis. CVaR 99% refers to the conditional value at risk at the 99% confidence level. Risk error (%)

refers to the percentage difference in the realised CVaR 99% and the CVaR 99% target of 2%. The Sharpe ratio is calculated

arithmetically and is annualised. Mean/CVaR 99% refers to the ratio of the annualised arithmetic excess return divided by the

realised CVaR 99%. Mean/LPM1 refers to the ratio of the annualised arithmetic excess return divided by the first lower partial

moment with a return target of zero. Mean/LPM2 refers to the ratio of the annualised arithmetic excess return divided by the

second lower partial moment with a return target of zero.


46

mean-variance and power utility. The filtered historical simulation, GGEV and GSEV models

however add significantly more value, on average 2%, under power utility indicating that these

models are being rewarded for accounting for higher moments. In contrast to the value-added under

mean-variance utility, under power utility the GSEV model generates more economic value than the

GGEV model. This should not be surprising given the power utility function’s sensitivity to higher

moments. The average economic value added by the GGEV approach across the three levels of risk

aversion is 10% p.a. This is highly significant suggesting that the model deserves further attention.

Table XV – Economic Value Added: Mean-Variance Utility

1/N Student-t

Historical

Simulation

Clayton

Copula/

Gaussian

Marginals

Exp.

Weighted

Moving

Average

Filtered

Historical


Mean-Variance

Utility: λ=.0278

Asset

Allocation -4.3% -4.9% -2.7% 7.3% 14.5% 14.3% 13.8%

5 Industries -0.4% -0.2% -0.4% -0.3% 1.6% 3.6% 4.0% 3.7%

10 Industries -0.5% -0.4% -0.3% -0.2% 3.0% 4.6% 4.9% 4.6%

30 Industries -0.8% 0.0% 0.2% 0.1% 3.0% 4.7% 5.1% 4.5%

10 Stocks -6.2% -1.5% -1.1% -2.9% 2.4% -0.4% 0.5% 0.3%

Size Portfolios 0.0% -0.4% -0.3% -1.0% 3.6% 7.8% 7.9% 7.8%

Value Portfolios -1.4% -0.3% -1.2% -0.1% 1.9% 1.2% 1.2% 1.2%

Fama-French 0.8% -0.2% 5.6% 13.2% 16.7% 17.9% 17.6%

Average -1.6% -1.0% 0.0% -0.7% 4.5% 6.6% 7.0% 6.7%

Mean-Variance

Utility: λ=.0278

Asset

Allocation -5.0% -3.9% -1.2% 7.4% 15.3% 15.1% 14.9%

5 Industries -2.1% 0.6% 0.6% -0.7% 1.6% 4.4% 4.7% 4.5%

10 Industries -2.0% 0.2% 0.6% -0.3% 2.8% 5.2% 5.5% 5.3%

30 Industries -2.7% 0.4% 1.0% 0.1% 2.2% 5.1% 5.4% 4.9%

10 Stocks -3.4% -0.9% -0.8% -2.7% 1.4% -0.3% 0.7% 0.6%

Size Portfolios -1.6% 0.0% 0.9% -1.4% 3.7% 8.8% 8.9% 8.9%

Value Portfolios -3.1% 0.3% -0.1% -0.3% 2.0% 2.0% 2.0% 2.1%

Fama-French 3.0% 1.0% 6.1% 13.2% 17.2% 18.3% 18.5%

Average -2.1% -0.3% 0.9% -0.9% 4.3% 7.2% 7.6% 7.5%

Mean-Variance

Utility: λ=.0278

Asset

Allocation -6.1% -2.1% 1.6% 7.8% 16.9% 16.7% 17.0%

5 Industries -5.4% 1.9% 2.4% -1.4% 1.6% 5.7% 6.0% 5.9%

10 Industries -4.9% 1.2% 2.2% -0.5% 2.6% 6.3% 6.6% 6.6%

30 Industries -6.2% 1.1% 2.4% 0.0% 0.9% 5.9% 6.1% 5.8%

10 Stocks -7.8% 0.1% -0.1% -2.3% -0.5% -0.2% 1.0% 1.2%

Size Portfolios -4.7% 0.8% 3.2% -2.0% 4.0% 10.7% 10.8% 10.8%

Value Portfolios -6.2% 1.3% 1.8% -0.7% 2.2% 3.4% 3.5% 3.6%

Fama-French 7.1% 3.3% 7.1% 13.1% 18.1% 18.9% 20.1%

Average -4.3% 1.0% 2.6% -1.2% 4.0% 8.3% 8.7% 8.9%

Grand Average -2.7% -0.1% 1.1% -0.9% 4.2% 7.4% 7.8% 7.7%

Table XV shows the value added of the given model relative to the Gaussian model for a mean-variance investor.

The value added is defined as the annual fee that equates the expected mean-variance utility of the given model

and the Gaussian model.


47

Table XVI – Economic Value Added: Power Utility

1/N Student-t

Historical

Simulation

Clayton

Copula/

Gaussian

Marginals

Exp.

Weighted

Moving

Average

Filtered

Historical


Power Utility:

γ=5

Asset

Allocation 8.6% -3.9% -1.2% 7.6% 15.5% 15.3% 15.0%

5 Industries -1.4% 0.5% 0.6% -0.4% 1.8% 4.5% 4.8% 4.6%

10 Industries -2.3% 0.1% 0.5% -0.3% 2.9% 5.3% 5.6% 5.3%

30 Industries -2.5% 0.4% 1.0% 0.1% 2.4% 5.2% 5.5% 5.0%

10 Stocks -3.1% -1.0% -0.8% -2.7% 1.5% -0.3% 0.7% 0.6%

Size Portfolios -1.4% 0.0% 1.0% -1.5% 3.8% 8.9% 9.0% 9.0%

Value Portfolios -2.9% 0.3% -0.1% -0.4% 2.2% 2.1% 2.1% 2.2%

Fama-French 2.9% 0.9% 6.1% 13.3% 17.2% 18.3% 18.5%

Average -0.3% -0.3% 0.9% -0.9% 4.4% 7.3% 7.7% 7.5%

Power Utility:

γ=10

Asset

Allocation -5.4% -1.5% 2.7% 8.6% 18.0% 17.8% 18.2%

5 Industries -5.4% 2.3% 3.3% -1.4% 2.5% 6.8% 7.1% 7.0%

10 Industries -6.2% 1.6% 2.7% -0.4% 3.1% 7.0% 7.2% 7.3%

30 Industries -3.2% 1.2% 3.0% 0.2% 1.3% 6.7% 6.8% 6.6%

10 Stocks -8.3% 0.2% 0.0% -2.2% -0.7% -0.1% 1.1% 1.2%

Size Portfolios -4.5% 1.0% 4.3% -2.8% 4.7% 11.8% 11.9% 12.0%


Fama-French 8.0% 3.7% 7.6% 13.7% 18.7% 19.5% 20.8%

Average -3.9% 1.3% 3.3% -1.3% 4.5% 9.2% 9.5% 9.7%

Power Utility:

γ=15

Asset

Allocation -5.6% 1.3% 7.4% 10.3% 21.4% 21.1% 22.2%

5 Industries -8.8% 4.8% 7.3% -3.6% 4.5% 10.5% 10.7% 10.9%

10 Industries -9.2% 3.5% 5.4% -0.6% 3.9% 9.5% 9.6% 10.0%

30 Industries -11.1% 2.3% 5.6% 0.7% 1.1% 9.2% 9.0% 9.1%

10 Stocks -13.7% 1.4% 0.8% -1.7% -3.1% 0.1% 1.5% 1.9%

Size Portfolios -7.2% 2.1% 8.5% -5.0% 6.4% 15.7% 15.9% 16.0%


Fama-French 13.9% 6.8% 9.7% 14.7% 20.9% 21.2% 23.7%

Average -6.4% 3.2% 6.4% -2.1% 5.4% 11.9% 12.2% 12.8%

Grand Average -3.5% 1.4% 3.5% -1.4% 4.8% 9.5% 9.8% 10.0%

Table XVI shows the value added of the given model relative to the Gaussian model for a power utility investor.

The value added is defined as the annual fee that equates the expected power utility of the given model and the

Gaussian model.


48

VI. Conclusions.

The massive losses at systemically important institutions during the 2008 global financial crisis and

the economic sequela have reaffirmed the importance of robust risk management and portfolio

construction approaches. Financial assets deviate markedly from the idealised Gaussian distribution,

exhibiting heavy tails, volatility clustering, asymmetry and complex patterns of dependence. Up until

this point, there has been almost a complete absence of scalable techniques that can deal with all four

phenomena. We have proposed the GSEV model that incorporates ARMA/EGARCH, extreme-value

theory and the skewed-t copula. EGARCH captures heteroskedasticity and the leverage effect, and

extreme-value theory that provides a robust theoretical framework for the asymptotic behaviour of the

tails. The skewed-t copula is perhaps unique among copulas in that it can accommodate

heterogeneous asymmetric dependence in high dimensions. The GSEV model produces superior VaR

risk forecasts to a range of benchmark methodologies commonly cited in the literature. The GSEV

model also outperforms the benchmark methodologies in an out-of-sample dynamic rebalancing

framework. The approach generates higher risk adjusted returns, lower drawdowns and more accurate

out-of-sample risk levels. The approach also outperforms the GGEV approach that employs the

Gaussian copula lending support to the skewed-t copula and the importance of accounting for

asymmetric dependence. We also show that the GSEV approach generates significant economic value

for the investor, far exceeding typical active management fees.

Appendix A – Benchmark Models

A. Unconditional Models

A1. 1/N

The first benchmark model we use is the so-called naïve, 1/N portfolio that weights each asset

equally. The 1/N benchmark is relevant for four reasons. Humans have an innate behavioural

tendency to equal weight the options presented to them. Bernatzi and Thaler (2001) find that many

investors equally weight the investment choices they are presented with68

. It appears few are immune

from this bias towards equal weighting. Markowitz himself, when probed about how he allocated his

retirement investments in his TIAA-CREF account, confessed: “I should have computed the historic

covariances of the asset classes and drawn an efficient frontier. Instead…I split my contributions fifty-

fifty between bonds and equities”69

. The 1/N rule is not dependent on return or risk expectations and

is therefore devoid of the estimation risk that potentially contaminates the more complex approaches.

The 1/N rule is also easy for investors to apply, and thus a viable alternative in practice. Finally, the

1/N approach is pervasive in the literature.

A2. Gaussian

The second benchmark model we use is a portfolio constructed with the Gaussian distribution. This

serves as the key benchmark against which the alternative approaches are evaluated. Because we

68

See Huberman and Jiang (2006) for a more recent example. 69

Zweig (1998)


49

construct all portfolios, except for the 1/N rule, to have a CVaR of 2%, and because we use equal

return forecasts across assets, this is equivalent to the minimum variance portfolio scaled to the

targeted risk level. Despite, or perhaps because the minimum-variance portfolio does not attempt to

account for expected returns, the minimum-variance portfolio has been shown to perform well in out-

of-sample simulations. Clarke, de Silva and Thorley (2011) show for the period 1967-2009, the

minimum variance portfolio has generated higher returns than the cap-weighted US equity market

with a 50% higher Sharpe ratio. The strong performance of the minimum-variance strategy prompted

MSCI to launch the Global Minimum Volatility Indices in 2008.

The multivariate-Gaussian approach is heavily used by leading risk-model vendors and practitioners.

Portfolio variance is estimated using the standard formulae

VaR is then estimated analytically using the following standard relation.

Definition A1 Conditional Value at Risk for the Gaussian Distribution: The conditional value-

at-risk for the Gaussian distribution, where is the confidence level, is the standard Gaussian

density function, and is the standard-deviation of portfolio returns, , is given by

A3. Multivariate Student-t

The student-t distribution can accommodate heavy tails and has been widely used to capture the tails

of financial data. Indeed the use of the distribution was supported by Markowitz and Usmen (1996a,

1996b). There are several formulations of the multivariate student-t distribution70

. The student-t

distribution, unlike the Gaussian distribution allows for tail dependence; however the dependence

structure is symmetrical between upper and lower tails, which in general is unrealistic for financial

assets. We employ the Generalised Hyperbolic multivariate Student-t distribution in keeping with our

use of the Generalised Hyperbolic skewed-t copula in section III.

Definition A2 Multivariate Student-t Distribution: The density of the multivariate student-t

distribution where is the degrees of freedom, is the number of dimensions, is a vector of means,

and is a covariance matrix is given by

We use the expectation conditional maximisation (ECM) algorithm of Lui and Rubin (1998) to

estimate the parameters of the model. As in the Gaussian case, the CVaR can be estimated using

standard formulae.

Definition A3 Condition Value at Risk for Student-t Distribution: The conditional value-at-risk

for the Student-t distribution, where is the confidence level, is the degrees of freedom, and is

the density function and the density of the student-t, is given by

70

see Kotz and Nadarajah (2004) for an extensive survey


50

A4. Historical Simulation

Historical simulation and its variant filtered historical simulation are currently the most widely used

methods for estimating VaR at commercial banks (Pritsker, 2006, Berkowitz, Christoffersen, and

Pelletier, 2008, Pérignon and Smith, 2010). Under standard historical simulation, the VaR and CVaR

estimates are generated by bootstrapping from the historical portfolio return generated using the

current portfolio weights. There is no need to specify the univariate marginals or the multivariate

dependence structure, and the potential for model risk is attenuated. Further, the approach is fast, and

intuitive. The approach is unable to adapt to shifts in the level of volatility and produce single period

returns that are outside of what has been observed historically. It is also well known that historical

simulation tends to perform poorly as we move further out into the tails and will perform poorly if

data is scarce. Historical simulation is not assumption free as is sometimes suggested. The implicit

assumptions are that the standardised distribution is stationary and the dependence structure is

persistent.

A5. Gaussian/Clayton Copula Model - Alcock and Hatherley (2009)

Alcock and Hatherley (2009) among others71

use the Clayton copula to capture lower-tail dependence,

and demonstrate economically significant gains from accounting for asymmetric dependence. The

Clayton copula was developed to model the incidence of disease within families (Clayton, 1975). The

Archimedean family of copulas to which the Clayton copula belongs have been formulated to have

convenient mathematical properties and are amenable to maximum likelihood estimation and

simulation. The key downside of Archimendean copulas in general is the lack of generalisability to

dimensions greater than two. Nested and Hierarchical approaches have been proposed to remedy this

shortcoming; however it is not obvious that these techniques provide definitive advantages. We

follow Alcock and Hatherley (2009), use Gaussian marginals and impose a symmetrical dependence

structure across assets. We use the inference function for margins approach, estimating the mean and

variance of the marginal distributions via maximum likelihood estimation, and then deriving the

copula parameter, .

The n-variate Clayton copula is given by

(7)

where

The Clayton copula density is given by

(8)

where is the Euler function.

71

Rodrigues (2003), Hurlimann (2004), Viebig and Poddig (2010)


51

We maximise (8) to yield an estimate of at each rebalance point for each data set72. To simulate the

Clayton copula, we use the Marshall-Okin (1998) method. The approach is fast and scalable.

Algorithm A1 Simulation of Clayton Copula: Marshall-Olkin (1988) method

1. Draw an independent n-dimensional vector from the gamma distribution:

2. Draw by i.i.d. realizations from the uniform distribution:

3. Generate Clayton copula uniforms for using

In each panel of figure 9 we show 10,000 draws from the Gaussian bivariate copula with

(left-hand panel), and the Clayton copula with (middle panel) and (right-hand panel).

B. Conditional Models

B1. Exponentially Weighted Moving Average (EWMA)

In 1989, JP Morgan developed the RiskMetrics™ model in order to quantify the firm’s market risk.

The approach uses an exponentially weighted covariance matrix to give more weight to recent

observations and to dampen the effect of observations falling out of the estimation window. The

exponentially weighed covariance matrix is given by

(9)

The EWMA approach can be seen as a special case of a GARCH process without a mean-reversion

term as follows.

(10)

Now if and , equation (11) reduces to

72

The standard Clayton copula can only accommodate positive dependence. We therefore ensure positive

dependence before we use MLE by estimating Kendaull’s tau.

0

0.5

1

0 0.5 1

Gaussian Copula, ρ=0.75

0

0.5

1

0 0.5 1

Clayton Copula, α=2

0

0.5

1

0 0.5 1

Clayton Copula, α=6


52

which is equivalent to the recursive form of (10).

The EWMA approach assumes that returns are conditionally Gaussian. The simplicity of the EWMA

approach has contributed to the widespread employment of the technique through the RiskMetrics

software and by other risk model vendors. We employ a decay factor of for daily data, in line

the RiskMetrics Technical Document and Fleming, Kirby and Ostdiek (2001).

B2. Filtered Historical Simulation

Filtered historical simulation (FHS) as proposed by Barone-Adesi, Bourgoin and Giannopouos (1998)

bootstraps from the historical standardised residuals, and scales by conditional volatility using a

GARCH model. FHS is a rare example of an approach that can produce stochastic volatility, skew,

heavy tails and tail dependence in high dimensions. The approach does not impose a parametric

distribution on the return innovations. Nor is it necessary to model the multivariate dependence

structure. Unlike standard historical simulation the approach is conditional and adapts to changes in

volatility. Again, the approach is not assumption free, implicitly assuming that the standardised

distribution is stationary and the dependence structure is persistent.

Algorithm 3.4 - Filtered Historical Simulation

1. Fit a GARCH process yielding residuals, and conditional volatility estimates

2. Standardise the residuals as follows

3. Bootstrap 10,000 times from the residuals to yield a simulated vector

where days

4. Scale the bootstrapped residuals using the GARCH volatility estimates

where is the simulated return in period , and

is the simulated volatility

estimate in period using the updated GARCH process73

B3 GGEV Model

Finally, we evaluate the performance of a variant of the GSEV model that employs the Gaussian

copula instead of the skewed-t copula. The motivation is to shed light on the importance of accounting

for tail dependence through the skewed-t copula in the GSEV model.

73

Note that in the first simulated period we use

where is estimated from the

final estimated residual in the GARCH process


53

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Financial Econometrics 4, 275-309.

Acerbi, C., and D. Tasche, 2002, On the coherence of expected shortfall, Journal of Banking

& Finance 26, 1487-1503.

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