Estimating the basis risk of index-linked hedging strategies using multivariate extreme value theory

Journal of Banking & Finance 37 (2013) 4353–4367

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Estimating the basis risk of index-linked hedging strategies using multivariate extreme value theory Ralf Kellner a,⇑,1, Nadine Gatzert b,2 a b

Schinkelstraße 43, 80805 München, Germany Friedrich-Alexander University Erlangen-Nuremberg, Lange Gasse 20, 90403 Nürnberg, Germany

a r t i c l e

i n f o

Article history: Received 5 June 2012 Accepted 27 July 2013 Available online 8 August 2013 JEL classification: C22 C51 G11 G32 Keywords: Extreme value theory Index-linked hedging instruments Copulas

a b s t r a c t This paper studies the empirical quantification of basis risk in the context of index-linked hedging strategies. Basis risk refers to the risk of non-payment of the index-linked instrument, given that the hedger’s loss exceeds some critical level. The quantification of such risk measures from empirical data can be done in various ways and requires special consideration of the dependence structure between the index and the company’s losses as well as the estimation of the tails of a distribution. In this context, previous literature shows that extreme value theory can be superior to traditional methods with respect to estimating quantile risk measures such as the value at risk. Thus, the aim of this paper is to conduct an empirical analysis of basis risk using multivariate extreme value theory and extreme value copulas to estimate the underlying risk processes and their dependence structure in order to obtain a more adequate picture of basis risk associated with index-linked hedging strategies. Our results emphasize that the application of extreme value theory leads to better fits of the tails of the marginal distributions in the considered stock price sample and that traditional methods in regard to estimating marginal distributions tend to overestimate basis risk, while basis risk can in contrast be higher when taking into account extreme value copulas. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction Index-linked hedging strategies are of high relevance in finance and insurance and comprise derivatives such as futures or options as well as alternative risk management instruments such as cat bonds or industry loss warranties. One central problem associated with these risk management instruments is basis risk, which arises if the risk process underlying the hedge, e.g., a stock price index or a catastrophic loss index, and the hedging firm’s position are not perfectly dependent. This implies a risk of non-payment if the index does not exceed the contractually defined (high) trigger level,

⇑ Corresponding author. Tel.: +49 15140022413. E-mail addresses: [email protected] (R. Kellner), [email protected] (N. Gatzert). 1 During the development of the paper, Ralf Kellner worked at the chair for insurance economics and risk management at the Friedrich-Alexander University in Erlangen-Nuremberg. Currently, he is working in a reinsurance company. 2 Nadine Gatzert is head of the chair for insurance economics and risk management at the Friedrich-Alexander University in Erlangen-Nuremberg. 0378-4266/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jbankfin.2013.07.043

even though the company suffers a critical loss.3 This is not only critical from the perspective of the hedging firm, but also for regulators with respect to acknowledging these instruments as a risk transfer. There are two central aspects associated with the estimation of basis risk that require special attention, namely an adequate estimation of the tails of the marginal probability distribution function of the two risk processes (index and company’s losses) and estimating the dependence structure between the two processes. In this context, previous literature has shown that extreme value theory (EVT) allows a better assessment of quantile risk measures as com-

3 As an example, consider a company that purchases a put option to hedge against high losses of its own firm value, which cannot be exercised (the put option is not triggered) since the put option’s underlying does not suffer a loss that is high enough, i.e. the underlying does not fall below the strike price, given that the company’s loss exceeds a critical loss level. Besides hedging financial market risk by means of financial derivatives, basis risk is also of high relevance in the context of risks related to insurance business operations, where risks of the insurer’s underwriting operations are transferred by means of index-linked products. In this case, the buyer of the contract receives a payout, if an index (usually an industry, parametric or modeled loss index, see, e.g. SwissRe, 2009) exceeds a certain threshold. If this threshold is not exceeded, i.e. the industry loss for, e.g., a certain region in the U.S. or Europe is low, the contract does not provide the risk management measure needed, even though the buyer experiences a high loss from insurance contracts sold in the contractually defined region.

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pared to traditional approaches. Hence, the aim of this paper is to combine multivariate extreme value theory and extreme value copulas to estimate basis risk in order to obtain a more adequate picture of the risk associated with index-linked hedging strategies. This is done based on an empirical analysis, where we compare this method to traditional approaches. In the literature, basis risk plays an important role for hedging strategies using financial derivatives such as futures or options, whose payments depend on changes in stock prices or indices (see, e.g. Figlewski, 1984; Moser and Helms, 1990; Castelino, 1992; Netz, 1996) as well as in the context of alternative risk transfer instruments in insurance markets (see, e.g. Major, 1999; Harrington and Niehaus, 1999; Cummins et al., 2004; Zeng, 2000; Gatzert and Kellner, 2011). Further important fields of interest with respect to basis risk are weather related risks (see Golden et al., 2007; Manfredo and Richards, 2009; Yang et al., 2009) and hedging strategies against price changes in the energy markets (see Haushalter, 2008). Basis risk is also of relevance when pricing hedging instruments (Wang and Wu, 2008; Lee and Yu, 2002). However, when aiming to quantify basis risk or downside risk measures in general based on an empirical data sample, one problem is that the true probability distribution of the underlying risk process is usually unknown. Thus, traditional methods either use the empirical distribution function or estimate the whole marginal distribution based on the data sample. Even if these methods might be sufficient for most of the sample’s observations, a drawback lies in the potential misestimation of the tails of the probability distribution and the non-consideration of extreme events such as catastrophic losses or a financial market crash. However, it is particularly the tails of the distribution, which are of relevance for most risk measures and in particular basis risk. An alternative to traditional approaches is the threshold exceedances method, which is based on extreme value theory and exclusively estimates the tail of a distribution, thus reducing potential misestimations in the tail. Moreover, the excess distribution in the tail converges to one of three possible distributions, whereas multiple distributions have to be estimated and compared if whole marginal distributions are considered. Furthermore, EVT and specifically the threshold exceedances method explicitly allow for the occurrence of extreme events, as heavy tails can be accounted for through the generalized Pareto distribution (GPD), which provides good estimates with respect to, e.g., the value at risk in the uni- and multivariate case and exhibits advantages as compared to traditional methods (see, e.g. Longin, 2000). Due to these advantages, several papers apply extreme value theory to different fields of interest, including wind storm losses (see Rootzén and Tajvidi, 1997), loss distributions (see McNeil, 1997) or operational losses (see, e.g. Gourier et al., 2009). Extreme value theory is further used to examine effects arising from the dependence structure in the tails of multivariate distributions. While Zhou (2010) studies the impact of tail risks on diversification effects in a portfolio using the multivariate extreme value approach, Longin and Solnik (2001) and Poon et al. (2004) focus on the dependence structure among stock market indices and detect that common dependence models such as multivariate normality tend to underestimate tail dependencies, which might lead to an underestimation of the actual risk situation. In the field of insurance, Cébrian et al. (2003) price an excess of loss reinsurance contract whose payment depends on indemnity payments and allocated loss adjusted expenses, thereby taking into account the dependence structure through extreme value copulas. Further applications of multivariate extreme value theory to actuarial problems such as joint- and last-survivor annuities, hurricane losses in different regions or insurance portfolio composition are conducted by Dupuis and Jones (2006) as well as Brodin and Rootzén (2009).

Furthermore, the estimation of quantile risk measures by means of extreme value theory is subject to several analyses. Longin (2000) uses the block maxima method to estimate the value at risk of a long and short position in the S&P 500 and mixed portfolio positions, thereby using a linear model with respect to the dependence structure in the portfolio. This method is then compared to traditional approaches, i.e. the empirical and normal distribution as well as GARCH processes. The results suggest that the block maxima method might be better suited for the value at risk estimation, since the tails of the normal distribution may not be well fitted and GARCH processes might underestimate large unexpected market shocks. Hotta et al. (2008) compare the value at risk of a two asset portfolio using either the empirical distribution or a mixed distribution consisting of empirical distribution values and a GPD in the tail of the marginal distribution. They focus on two specific dependence models, logistic and asymmetric logistic dependence, and quantify the difference between these two approaches (with or without the use of the GPD) by means of backtesting without including the impact of different marginal distributions. A similar procedure is applied by Ghorbel and Trabelsi (2009) who estimate the value at risk of stock index portfolio positions and further include multivariate GARCH models in their comparison. With respect to their backtesting results, Hotta et al. (2008) and Ghorbel and Trabelsi (2009) both detect the extreme value approach to be superior to traditional methods. In this paper, we contribute to previous literature by combining the two strands of the literature described above, the one on basis risk and the one on the application of extreme value theory in finance and insurance. Our aim is to show how basis risk can be estimated empirically in order to better assess the risk associated with index-linked hedging strategies. This is done by using multivariate extreme value theory and extreme value copulas. Toward this end, we conduct an empirical analysis using the S&P 500 as the index underlying the hedge and several firms listed in the S&P 500 as the assumed hedgers. We provide a formula for basis risk based on the estimated marginal distributions and dependence structure for the risk processes. Furthermore, in contrast to, e.g., Longin (2000), Hotta et al. (2008), and Ghorbel and Trabelsi (2009), we conduct a comparison of different estimation methods that is intended to provide an isolated analysis of the impact of potentially misestimating the tails of the marginal distributions and the dependence structure between these tails. In particular, we first focus on marginal distributions and compare a traditional approach, which fits the marginal distributions of stock prices as a whole (based on the t and logistic distribution), with an EVT method that uses a GPD to estimate the probability values in the upper tail of the marginal distributions. Second, the impact of the dependence structure is taken into account with and without taking into account extreme value copulas. This procedure allows isolated insights regarding the effects of misestimating the marginal distribution and the dependence structure. Our results show that for all marginal distributions used in our analysis, the fit of the distribution can be improved if the data above the threshold is estimated through a GPD. Furthermore, for approximately half of our data, extreme value copulas capture the dependence structure in the joint tail distribution better than regular copulas that are estimated based on the whole distribution, and can thus be considered as a viable alternative. One major finding is that traditional methods regarding the marginal distributions tend to overestimate basis risk in the considered examples. In addition, the comparison between traditional and extreme value copula models emphasizes that the degree of asymptotic dependence is a key driver for basis risk. The remainder of this paper is structured as follows. Section 2 presents the theoretical background regarding extreme value theory and (extreme value) copulas as well as the

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R. Kellner, N. Gatzert / Journal of Banking & Finance 37 (2013) 4353–4367 Table 1 Relevant copula functions for the analysis in Section 3. Copula

Function

Coefficient of upper tail dependence

Archimedean and elliptical copulas (traditional and mixed-EVT method) n o ðexpðhu1 Þ1Þðexpðhu2 Þ1Þ Frank 1 C Fr ; h2R h ðu1 ; u2 Þ ¼  h ln 1 þ expðhÞ1

0

Gumbel

v ¼ 2  2h

h h 1=h g; C Gu h ðu1 ; u2 Þ ¼ expfðð lnðu1 ÞÞ þ ð lnðu2 ÞÞ Þ 1 1 ðu ; u Þ ¼ U ð U ðu Þ; U ðu ÞÞ C Ga P 1 2 1 2 P 1 C tv ;P ðu1 ; u2 Þ ¼ tv ;P ðt1 v ðu1 Þ; tv ðu1 ÞÞ

Gauss t

1

1
0



pffiffiffiffiffiffiffipffiffiffiffiffiffiffi

pffiffiffiffiffiffiffi1q v ¼ 2tmþ1  mþ1 1þq

Extreme value copulas (tail-EVT method) n o Gumbel h h 1=h C Gu ; 1
1

v ¼ 2  2h

1h

2  2  Að0:50Þ, AðwÞ ¼ 1  ðwh þ ð1  wÞh Þ 2  2  Að0:50Þ, AðwÞ ¼ h  w2  h  w þ 1

Notes: UP (tv ;P ) is the joint distribution function of the bivariate standard normal (t) distribution with linear correlation matrix P and U (t v ) denotes the standard univariate normal (t) distribution.

Table 2 Procedure to determine basis risk under the traditional, mixed-EVT and tail-EVT method. Marginals

Traditional

Copula model

Regular copula

Step 1 Step 2

Determine the ARMAðp1 ; q1 Þ - GARCHðp2 ; q2 Þ model

Determine the ARMAðp1 ; q1 Þ - GARCHðp2 ; q2 Þ model ^ n;b ðy Þ for y ¼ x  t ; i ¼ 1; 2 ^ n;b ðy Þ; G Estimate the GPDs G

Estimate the marginal distributions b F ðx1 Þ: b F ðx2 Þ b and C b 0 among all realizations b Estimate the copula C F ðx1 Þ: b F ðx2 Þ

Estimate the marginal distributions b F ðx1 Þ: b F ðx2 Þ b mixed among all realizations b Estimate the copula C F ðx1 Þ: b F ðx2 Þ, xi 6 ti ; i ¼ 1; 2, F GPD ðx2 Þ, xi > ti ; i ¼ 1; 2 and b F GPD ðx1 Þ: b

) Choose best fit b using C

) Choose best fit b mixed using C

Step 3

Step 4

EVT Extreme value copula

Regular copula

Extreme value copula

1

b 0 among Additionally estimate the extreme value copula C F ðx1 Þ: b F ðx2 Þ=> the realizations for xi > ti ; i ¼ 1; 2 using b b and C b0 Choose best fit among C

Calculate basis risk according to Eq. (5)

definition and analytical calculation of basis risk. Section 3 contains results of the empirical analysis and Section 4 concludes. 2. Multivariate extreme value theory This section presents the theoretical background for assessing basis risk using the concepts of extreme value theory and copulas. First, extreme value theory and its application to marginal distributions are illustrated. Second, the marginal distributions are linked through the concept of copulas and, third, we show how basis risk can be calculated using the concept of copulas. In the last part of this section, the estimation methodology applied in the empirical analysis in Section 3 is laid out. 2.1. Extreme value theory – theoretical background Extreme value theory is concerned with distribution functions for extreme values of i.i.d. random observations X 1 ; . . . ; X n . The most frequent applications of EVT are the block maxima method to determine the distribution for a maximum or minimum, the threshold exceedances method, which focuses on the distribution for values of X1, . . . , Xn exceeding a predefined threshold and can be used to determine risk measures in the upper tail (e.g. value at risk, expected shortfall), and the point process method to model a series of events in time whose observations are determined with a generalized Pareto distribution from the threshold exceedance method (see McNeil et al., 2005, p. 275).4 Through the application of the block maxima method, the data available for estimation is 4 The threshold exceedances method is also known as the peaks-over-threshold method.

2

i

i

i

b mixed among the Additionally estimate the extreme value copula C 0 b GPD ðx1 Þ: F b GPD ðx2 Þ) Choose realizations for xi > ti ; i ¼ 1; 2 using F

b mixed b mixed and C best fit among C 0 Calculate basis risk according to Eq. (5)

heavily reduced by solely retaining the maximum from a sample block, which might lead to high standard errors for the estimates of the GEV. We therefore use the threshold exceedances method, which avoids this potential drawback by means of using all data y = x  t that exceed a certain threshold t. Let

F t ðyÞ ¼ PðX  t 6 yjX > tÞ ¼ < xF  t;

Fðy þ tÞ  FðtÞ ; 1  FðtÞ

xF 6 1;

06y ð1Þ

be the excess distribution function over the threshold t with xF ¼ supfx 2 R : FðxÞ < 1g as the right endpoint of F, then Ft(y) can be approximated through a GPD for t ? xF. According to the theorem by Pickands–Balkema–de Haan (see Balkema and de Haan, 1974; Pickands, 1975), analogously to deriving the distribution of maxima, one can find a positive-measurable function b(t) such that lim sup jF t ðbðtÞyÞ  Gn;b ðyÞj ¼ 0, if F lies in the maximum domain t!xF 06y6x t F of attraction (MDA) of Hn, F 2 MDAðHn Þ; n 2 R, with Hn being the generalized extreme value distribution. Gn,b(y) is a generalized Pareto distribution (GPD) given by

( Gn;b ðyÞ ¼

1  ð1 þ ny=bÞ1=n ; n–0 1  expðy=bÞ;

n¼0

;

where b > 0; y P 0 if n P 0 and 0 6 y 6 b=n if n < 0 (see, e.g. McNeil, 1999; Longin, 2005). For n > 0ðn ¼; n < 0Þ the GPD is a Pareto (exponential, Pareto type II) distribution. An important implication of the threshold exceedances method is that the tail of the unknown distribution function F for i.i.d. random observations X1, . . . , Xn can be estimated by means of the GPD, if the observation lies in the area of the threshold exceedances. Based on the theorem by Pickands–Balkema–de Haan, it is reasonable to assume that for a

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Table 3 Improvement for marginal distributions’ fit if upper tail is estimated through GPD. Company

Traditional distribution

p-Value marginal distribution

p-Value mixed marginal distribution

Improvement through GPD (%)

p-Value GPD

American Express Co. Aon Corp. Berkshire Hathaway Chevron Corp. Cisco Systems Coca Cola Co. ConocoPhillips Devon Energy Corp. eBay Inc. Exxon Mobile Corp. FedEx Corp. Hewlett–Packard Johnson & Johnson Kellog Co. March & McLennan Procter & Gamble SLM Corporation Southwest Airlines S&P 500 Wal-Mart Walt Disney Co. Yahoo Inc.

t

0.95

1.00

5.49

0.80

t t t t t t Logistic t t t t t Logistic t t t t Logistic t Logistic t

0.92 0.71 0.80 0.65 0.86 0.76 0.90 0.62 0.80 0.32 0.82 0.59 0.93 0.84 0.38 0.24 0.46 0.14 0.96 0.97 0.62

1.00 0.97 0.99 0.95 0.99 0.97 1.00 0.94 0.99 0.77 0.96 0.94 1.00 0.99 0.80 0.64 0.87 0.54 1.00 1.00 0.95

8.76 36.57 24.11 46.23 14.97 27.68 10.98 50.03 23.67 143.29 20.69 57.80 7.59 17.73 109.15 167.55 87.62 274.32 3.87 3.06 52.70

0.98 0.84 0.55 0.37 0.76 0.34 0.19 0.47 0.82 0.41 0.89 0.54 0.91 0.62 0.78 0.56 0.14 0.99 0.66 0.78 0.47

Table 4 Results for the chosen copula models and corresponding basis risk values in case of the traditional and the EVT method (in case of an extreme value copula in bold). Company

American Express Co. Berkshire Hathaway Chevron Corp. Coca Cola Co. (regular copula only) Devon Energy Corp. Exxon Mobile Corp. (regular copula only) FedEx Corp. (regular copula only) Hewlett–Packard Johnson & Johnson Wal-Mart (regular copula only) Walt Disney Co. (regular copula only)

Traditional method – regular and extreme value copula

EVT method – regular and extreme value copula BR

Copula

Copula parameter (standard error)

Least sum of squared values

BREVT

2.65E04 2.53E04 1.49E04 3.83E05

0.50 0.84 0.55 0.60

Gauss t Gauss t

0.64 0.20 0.44 0.43

(0.01) (0.02) (0.02) (0.03)

2.95E04 1.07E04 2.54E04 5.01E05

0.47 0.70 0.68 0.54

0.42 (0.02) 0.33 (0.02) 0.51 (0.03)

4.13E05 4.42E04 1.69E04

0.59 0.79 0.58

– Gauss Gauss

– 0.33 (0.02) 0.50 (0.01)

– 3.92E04 3.32E03

– 0.76 0.63

t t

0.51 (0.02) 0.95 (0.02)

2.06E04 1.35E03

0.50 0.02

Galambos

1.70 (0.02)

1.03E04

0.08

Gauss t Galambos

0.56 (0.01) 0.39 (0.02) 0.56 (0.02)

9.65E05 5.62E05 6.69E05

0.62 0.59 0.55

t Gauss Gauss Galambos

0.96 0.56 0.40 0.56

1.53E03 1.40E04 2.20E03 1.08E04

0.02 0.57 0.71 0.50

t Galambos

0.54 (0.02) 0.57 (0.03)

3.60E04 5.01E04

0.50 0.68

t Tawn

0.54 (0.02) 0.65 (0.02)

5.32E04 7.63E04

0.45 0.47

t

0.57 (0.01)

1.37E03

0.62

t

0.57 (0.01)

1.50E03

0.42

Copula

Copula parameter (standard error)

Least sum of squared values

Gauss Gauss t Tawn

0.63 0.23 0.45 0.51

(0.01) (0.02) (0.02) (0.02)

t Gauss Tawn

certain high threshold t, Ft(y) = Gn,b(y) for some n and b (see McNeil, 1999). Setting x = y + t and combining this with Eq. (1), it follows that F(x) = (1  F(t))Gn,b(x  t) + F(t), x > t. Using (n  Nt)/n as an empirical estimate for F(t), where n is the number of all observations and Nt is the number of observations exceeding the threshold, and estimating the parameters for the GPD Gn,b(y), the tail estimate F GPD ðxÞ, can be determined through of F; b

8  1=n > < 1  Nt 1 þ ^n xt ; ^n–0 ^ n b b F GPD ðxÞ ¼    > : 1  Nt exp xþt ; n¼0 ^ n b

trad

(0.002) (0.01) (0.02) (0.02)

due to the scarceness of data for values in the tail (see McNeil, 1999).5 The methods presented in this section are generally suited for i.i.d. data. This assumption might be reasonable for insurance loss data, e.g. storm or fire damages. However, if the analysis is focused on data for which the i.i.d. assumption might not be satisfied, e.g. time series of financial data, the dependence structure within the time series can be modeled using ARMA(p1, q1) models with

ð2Þ

if x > t. Note that F(t) is estimated through the empirical distribution function, which should not be applied to the tails of a distribution

5 For instance, imagine two random samples with 1000 observations from the same theoretical distribution. While the estimate for the 50% quantile of the data might be very similar for both samples, the estimate for the 99% quantile might strongly differ as it only depends on the ten highest values of the sample.

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S&P 500, Tailfit - GPD

2.5 1.5

2.0

Sample quantiles

2.5 2.0 1.0

1.0

1.5

Sample quantiles

3.0

3.0

S&P 500, Tailfit - Logistic

1.0

1.5

2.0

2.5

3.0

1.0

1.5

Theoretical quantiles

2.5

3.0

American Express Co., Tailfit - GPD

1.0

2.5 2.0 1.0

1.5

1.5

2.0

Sample quantiles

2.5

3.0

3.0

American Express Co., Tailfit - t

Sample quantiles

2.0

Theoretical quantiles

1.0

1.5

2.0

2.5

3.0

Theoretical quantiles

1.0

1.5

2.0

2.5

3.0

Theoretical quantiles

Fig. 1. QQ Plots for marginal distributions of the S&P 500 and American Express Co. in case of traditional (left) and GPD (right) distributions for values above the threshold. Dashed lines represent the empirical estimate for the trigger and the critical loss level, respectively.

GARCH(p2, q2) errors, such that the time series Xt is assumed to follow a stochastic process

X t ¼ lt þ rt Z t ;

lt ¼ l þ

with

p1 q1 X X /i ðX ti  lÞ þ hj ðX tj  ltj Þ; i¼1

and

ð3Þ

2 t

r ¼ a0 þ

j¼1 p2 X

2

ai ðX ti  lti Þ þ

i¼1

q2 X

2.2. Estimating the marginal distributions and the dependence structure using copulas If focus is laid on determining a risk measure, which depends on multivariate risk processes and is thus calculated based on a joint distribution such as basis risk, the dependence structure including the degree and type of dependence among risk processes plays an important role (see, e.g. Zhou, 2010). The benefit of copulas for our analysis can be seen from Sklar’s theorem

Fðx1 ; . . . ; xd Þ ¼ CðFðx1 Þ; . . . ; Fðxd ÞÞ; 2 tj ;

cj r

j¼1

with p1, q1 and p2, q2 representing the order, l and a0 the intercept and ui, hj, ai, cj the coefficients of the ARMA(p1, q1) and GARCH(p2, q2) processes, respectively, and Zt representing the i.i.d. residuals from the stochastic process following a certain distribution function. To determine risk measures for the time series data, first the ARMA(p1, q1)  GARCH(p2, q2) model must be estimated. The GPD tail estimation can then be applied to the estimated i.i.d. residuals Zt of the process (see, e.g. McNeil et al., 2005, p. 292; Hotta et al., 2008).

which illustrates how the joint distribution function for d processes F(x1, . . . , xd) can be modeled by coupling the marginal distributions F(x1), . . . , F(xd) with copula C (see, e.g. Nelsen, 2006). This method provides a way to assess basis risk by means of determining the marginal distributions for the risk processes F(x1), F(x2) and the corresponding copula C. However, when this method is applied to empirical data, the marginal distributions as well as the copula are usually unknown and must be estimated to derive the estimated joint distribution b F ðx1 ; . . . ; xd Þ. In the case of two relevant processes X1 and X2, for instance, the usual procedure which we refer to as the ‘‘traditional’’ F ðx1 Þ; b F ðx2 Þ method, is to estimate the marginal distributions b b (see, e.g. Cébrian et al., 2003) and the corresponding copula C

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based on all available data. Potential drawbacks of this method are that (1) estimating (complete) marginal distributions can be difficult, e.g. normal distributions for log-returns of stocks are often rejected (see, e.g. Rufino and de Guia, 2011) and (2), even if the marginal distribution can be fitted, the risk in the tails is often underestimated (see, e.g. Longin, 2000), which is of special relevance when determining a downside risk measure. Therefore, in the following, we compare this traditional method to the use of extreme value theory, referred to as the ‘‘EVT method’’, to quantify downside and basis risk. With respect to the quantification of marginal probability distributions, we fit the marginal GPDs in the tails of the distribution and calculate b F GPD ðx1 Þ; b F GPD ðx2 Þ for x1 > t1 ; x2 > t 2 using Eq. (2). Regarding the estimation of a copula and thus the dependence structure among the marginal distributions, the traditionally estimated marginal distributions b F ðx1 Þ; b F ðx2 Þ are used to determine marginal probability values below the relevant thresholds, while b F GPD ðx1 Þ; b F GPD ðx2 Þ are used for the estimation of probability values above the thresholds t1 and t2 (see, e.g. Hotta et al., 2008). Based on these probability values b for the risk processes ðb F ðx1 Þ; b F ðx2 Þ; b F GPD ðx1 Þ; b F GPD ðx2 ÞÞ, the copula C X1 and X2, referred to as the ‘‘regular’’ copula, can be estimated and used to asses basis risk (see, e.g. Dupuis and Jones, 2006). However, analogously to the univariate case, a potential drawback might exist when applying this method. Even though the tail estimation for marginal distributions might be improved by using an estimate according to the GPD (see Eq. (2)) for probability values above the thresholds t1 and t2, the ‘‘regular’’ copula is estimated based on the (complete) marginal distributions and, thus, the dependence structure in the tail might be misestimated. Hence, as an alternative to estimate the dependence structure on the basis of all marginal data, we focus on the tail of the bivariate joint disb from tribution. In the tail, we then substitute the regular copula C the traditional approach with its extreme value limit, i.e. the ‘‘exb 0 (see, e.g. McNeil et al., 2005, p. 320; Ghortreme value copula’’ C bel and Trabelsi, 2009).6 In general, extreme value copulas are used to capture the joint distribution among block maxima (see Section 2.1). Let X 1;i ; X 2;i ; i ¼ 1; . . . ; n be two risk processes with n independent observations with marginal distribution functions F(x1), F(x2) and M1,n, M2,n denoting the corresponding maxima for X 1;i ; X 2;i ; i ¼ 1; . . . ; n, then PðM 1;n 6 x1 Þ ¼ F n ðx1 Þ and PðM2;n 6 x2 Þ ¼ F n ðx2 Þ (see, e.g. Longin, 2000; Cotter, 2001). To model the dependence structure among the two risk processes X1 and X2, Sklar’s theorem can be used to separate the joint distribution of the componentwise maxima Fn(x1, x2) according to Fn(x1, x2) = Cn((Fn(x1))1/n, (Fn(n (x1, x2) = Cn((Fn(x1))1/n, (Fn(x2))1/n), (see Gudendorf and Segers, 2010). Extreme value copulas can be seen as the limiting copulas for maxima and, according to Nelsen (2006), are defined such that C0 is an extreme value copula if there exists a copula C such that

  1=n lim C n u1=n ¼ C 0 ðu1 ; . . . ; ud Þ; 1 ; . . . ; ud

n!1

ðu1 ; . . . ; ud Þ 2 ½0; 1d ;

ð4Þ

where (u1, . . . , ud) represent the uniform transformed margins of (M1,n, . . . , Md,n). Furthermore, C is in the domain of attraction of C0. With respect to the bivariate case, according to Pickands representation, a bivariate copula is an extreme value n  copula, o if and only lnðu1 Þ if it takes the form C 0 ðu1 ; u2 Þ ¼ exp lnðu1 u2 ÞA lnðu , where A is u Þ 1 2 a convex dependence function, which can be used to construct the 6 Especially when using the mixed marginal distributions and the extreme value copulas, the complete marginal distributions as well as the corresponding copula do not need to be derived and remain unknown, as the extreme value copula can be estimated by means of censored data, such that only the probability distribution values F^GPD ðx1 Þ and b F GPD ðx2 Þ above the thresholds t1 and t2 are necessary for the b 0 in the tail. Furthermore, by estimating the marginal GPDs and estimation of C b 0 in the tail, the fitting procedure is focused on the area of the extreme value copula C joint distribution that is relevant for downside risk and basis risk in particular.

extreme value copula (see Genest and Segers, 2009), for goodnessof-fit tests for bivariate extreme value copulas or to calculate the coefficient of upper tail dependence (see Coles et al., 2000). Thus, in regard to the EVT method, for any uniform margins (v1, v2) e (0, 1)2, Eq. (4) can be used to approximate Cððv 1 Þ1=n ; ðv 2 Þ1=n Þ  C 1=n for n sufficiently large. Substituting 0 ðv 1 ; v 2 Þ n ((v1)1/n, (v2)1/n) with (u1, u2) results in Cðu1 ; u2 Þ  C 1=n 0 ððu1 Þ ; ðu2 Þn Þ ¼ C 0 ðu1 ; u2 Þ, which leads to the closest approximations for large values of (u1, u2) as ((v1)1/n, (v2)1/n) ? (1, 1) for n ? 1 (see McNeil et al., 2005, p. 320). Hence, for the quantification of downside risk using extreme value copulas,

b 0ðb b F ðx1 ; x2 Þ ¼ C F ðx1 Þ; b F ðx2 ÞÞ and b F ðx1 ; x2 Þ b 0ðb F GPD ðx1 Þ; b F GPD ðx2 ÞÞ; ¼C

xi > t i ;

i ¼ 1; 2;

can be used to determine joint distribution values in the tail of the bivariate distribution, where xi > t i ; i ¼ 1; 2 is a necessary condition as the extreme value copulas can only be used for regular (and not maxima) data if the value of interest xi lies above the threshold ti .7 Thus, summing up, in order to estimate the bivariate distribution function, we can distinguish two cases for estimating the marginal distribution functions ((1) the traditional method and (2) the EVT-method with a mixed distribution with traditional below the threshold and GPD above the threshold) and two cases for the dependence structure ((1) estimating a regular copula using all marginal data or (2) estimating an extreme value copula based on tail data above the threshold). Starting with the traditional method, we thus first derive the estimate for the bivariate distribution function through using all traditional marginal functions. For these marginals, we first study the ‘‘regular’’ copula model  b b b F ðx1 ; x2 Þ ¼ C F ðx1 Þ; b F ðx2 Þ and then additionally  include the exb0 b treme value copula b F ðx1 ; x2 Þ ¼ C F ðx1 Þ; b F ðx2 Þ , x1 > ti ; i ¼ 1; 2, as an alternative to the regular copula model, which may provide a better fit to the tail data (of relevance for basis risk). Second, including extreme value theory for the marginal distribution functions, we use all mixed marginal distribution values (traditional below and GPD above threshold) and a ‘‘regular’’ copula model b mixed C

  8 b mixed b > F ðx1 Þ; b F ðx2 Þ ; > >C > >   > > >C b mixed b < F GPD ðx1 Þ; b F ðx2 Þ ; b   F ðx1 ; x2 Þ ¼ > b mixed b > C F ðx1 Þ; b F GPD ðx2 Þ ; > > > >   > > b mixed b :C F GPD ðx1 Þ; b F GPD ðx2 Þ ;

x1 6 t 1 ;

x2 6 t 2

x1 > t 1 ;

x2 6 t 2

x1 6 t 1 ;

x2 > t 2

x1 > t 1 ;

x2 > t 2

or additionally allow for marginal distributions values derived with the GPDs and approximate the dependence structure in the tail b mixed through an extreme value copula C 0 GPD GPD b mixed ð b b b ðx Þ; ðx ÞÞ, x > t ; i ¼ 1; 2, which uses Eq. F ðx1 ; x2 Þ ¼ C F F 1 2 i i 0 (4) to calculate the estimate. Table 1 illustrates the copula functions which are taken into account for the analysis in Section 3. We choose the Frank and Gumbel copula, which belong to the family of Archimedean copulas, and the Gauss and t copula, which belong to the family of elliptical copulas (see Embrechts et al., 2003), while the Gumbel, Galambos and Tawn copulas (see Ghorbel and Trabelsi, 2009) represent three possible extreme value copulas. In the context of downside risk measures, an important characteristic that can be explicitly captured through copula models is tail dependence, which measures 7 A further approach to model the dependence structure in the upper tail of the bivariate distribution is the application of so-called threshold copulas, which exclusively model the dependence structure among risk processes for values above the threshold (see Juri and Wüthrich, 2002). However, this approach leads to conditional probability values for the joint distribution, which is not in line with the basis risk definition later defined in Eq. (5).

R. Kellner, N. Gatzert / Journal of Banking & Finance 37 (2013) 4353–4367

the dependence in the tail of the bivariate distribution. The coefficient for upper tail dependence8 v for the two uniform random variables U1 and U2 is given through v ¼ limu!1 PðU 2 > ujU 1 > uÞ, with v ¼ limu!1 vðuÞ. Before the estimation of the copula model is conducted, the existence of upper tail dependence needs to be tested, i.e. if v > 0. However, exclusively taking v into account to draw conclusions about the type of dependence might not be sufficient as v = 0 indicates asymptotic independence, but not necessarily indepen in dence (see Poon et al., 2004). Thus, an additional measure v the case of asymptotic independent models is taken into account.  can be derived through its limit Analogously to v, v v ¼ limu!1 v ðuÞ with v ðuÞ ¼ 2loglogð1uÞ  1, for 0 6 u 6 1 (see Coles  Cðu;uÞ  represents the survival cop 6 1, while C et al., 2000), and 1 < v  ula of C with Cðu; uÞ ¼ 1  u  u  Cðu; uÞ (see, e.g. Embrechts et al.,  Þ should thereby be used 2003; Chollete et al., 2012). The pair ðv; v  ¼ 1Þ for the derivation of asymptotic dependence, where ðv > 0; v leads to the conclusion of asymptotic dependence, while  < 1Þ indicates asymptotic independence. In our analysis, ðv ¼ 0; v  < 1 against v ¼1 v = 0 versus v > 0 is tested through testing v using a generalized likelihood ratio test as described in Coles et al. (2000).

2.3. Quantifying basis risk using multivariate extreme value theory As described before, basis risk generally occurs if the hedging instrument’s underlying is not perfectly dependent with the position that should be hedged. Basis risk BR can thus be defined as the conditional probability that the hedging instrument’s underlying I does not exceed its trigger level YT (i.e. Y 6 Y T ), given the hedger’s losses LH exceed a predefined critical loss level LH;critical , i.e.

BR ¼ PðY H 6 Y T jLH > LH;critcal Þ ¼

PðY H 6 Y T \ LH > LH;critcal Þ PðLH > LH;critcal Þ

;

(see, e.g. Zeng, 2000). Using the concept of copulas, one obtains

BR ¼

PðI 6 IT \ L 6 1Þ  PðI 6 IT \ L 6 Lcritical Þ 1  PðL 6 Lcritical Þ T

¼

F I;L ðI ; 1Þ  F I;L ðIT ; Lcritical Þ 1  F L ðLcritical Þ T

¼ ¼ ¼

CðF I ðI Þ; F L ð1ÞÞ  CðF I ðIT Þ; F L ðLcritical ÞÞ 1  F L ðLcritical Þ CðF I ðIT Þ; 1Þ  CðF I ðIT Þ; F L ðLcritical ÞÞ 1  F L ðLcritical Þ F I ðIT Þ  CðF I ðIT Þ; F L ðLcritical ÞÞ 1  F L ðLcritical Þ

;

ð5Þ

4359

2.4. Estimation methodology Table 2 provides an overview regarding the traditional and the EVT method along with the use of the regular copula models and the extreme value copula for estimating basis risk that will be compared in the numerical analysis. Each method starts with the determination of the ARMA(p1, q1)  GARCH(p2, q2) process to extract dependencies within each time series as laid out in Section 2.1 in order to apply the estimation methods in Steps 2–4 to the i.i.d. residuals of the process defined in Eq. (3).10 To determine the marginal distribution functions in Step 2, maximum likelihood estimation is applied to estimate b n;b ðy Þ; G b n;b ðy Þ; y ¼ xi  ti ; i ¼ 1; 2. After having b F ðx2 Þ and G F ðx1 Þ; b 1 2 i estimated the marginal distributions and GPDs, the null hypothesis that the sample data stem from the estimated marginal distribution is tested using the Kolmogorov–Smirnov test, whose test statistic is based on the supremum of the difference between the values from the empirical and the stated probability distribution, while p-values for the test statistic are determined according to Marsaglia et al. (2003). Copulas are estimated in Step 3 with a similar procedure as in Step 2. However, before the parameters of the copula are estimated using maximum likelihood estimation, first an independence test is conducted, whose null hypothesis states that the dependence structure among relevant risk processes is captured through the independence copula Pðu1 ; . . . ; ud Þ ¼ Pdi¼1 ui . The test is based on Genest and Rémillard (2004) and compares the empirical distribution function with the theoretical values that would occur in case of independence (see also Genest et al., 2009)). Second, a test for asymptotic dependence as laid out in Section 2.2 is conducted. According to the these test results, copula models with or without asymptotic dependence, respectively, are taken into account for the maximum likelihood estimation of the copula models as presented in Table 1 (see Kojadinovic and Yan, 2010). To derive the likelihood function for the extreme value copula, censored data is used, in that values lying below their threshold only contribute to the likelihood with the information that they lie below the threshold (see McNeil et al., 2005, p. 320)). Thus, by virtue of the different copula estimation methods (non-censored; censored), copula models are not compared according to the highest maximum likelihood. Instead, we calculate the sum of squared values between the empirical and the stated copula (see Genest et al., 2009), given both values X1 and X2 lie above their threshold, C(F(x1), F(x2)|x1 > t1, x2 > t2), as the upper tail of the joint distribution is relevant for basis risk. Among all copula models, the model with the least sum of squared values between the empirical and the stated copula is chosen. The accuracy of the chosen copula model is further examined through conditional PP plots, which plot the observed (empirical) cumulative probability values against the expected (theoretical) cumulative probability values, for probability values PðX 1 6 x1 ; X 2 6 x2 jX 1 > t 1 ; X 2 > t 2 Þ.

3. Empirical analysis where FI,L denotes the joint and FI and FL the marginal distributions of the hedging instrument’s underlying I and the hedger’s losses L. Dependent on the method, basis risk can be quantified through inserting b ðIT Þ, b ðLcritical Þ for the marginal F I ðIT Þ, b F L ðLcritical Þ and b F GPD F GPD I L mixed mixed b b b b distribution values and C , C 0 , C and C 0 for the copula function, respectively.9 8

In the context of basis risk upper tail dependence plays an important role, as basis risk in general decreases in the presence of upper tail dependence as under this type of dependence structure the payment of an index-linked hedging instrument gets more likely. 9 The superscript ‘‘mixed’’ refers to the mixed marginal distributions that are used for copula estimation.

Our data consists of negative daily changes in stock prices for 21 U.S. companies listed in the S&P 500 as displayed in Table 3. Basis risk of these companies is assessed by assuming that each company purchases a risk management instrument whose payoff is 10 Stationarity and independence of the time series is examined through conducting the Augmented Dickey–Fuller and the Box-Pierce test. If the time series is stationary and not independent, an ARMAðp1 ; q1 Þ model is fitted on the basis of time series analyses techniques, e.g. information criteria, plots for autocorrelation, and partial autocorrelation, such that the residuals of the ARMAðp1 ; q1 Þ model are independent. Subsequent to the determination of the ARMAðp1 ; q1 Þ model, time series analysis techniques are applied to fit the GARCHðp2 ; q2 Þ model.

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linked to the development of negative daily changes in the S&P 500 index values. An example for such a hedging strategy could be the purchase of a put option on the S&P 500, which is triggered if the index falls below a predefined strike price, i.e. the loss in the index is high enough. Negative values for daily changes are used, as the application of extreme value theory is usually dedicated to the upper tail of probability distributions. Our sample period spans from January 2000 until December 2010. Data is taken from the Thomson Datastream advance database.

3.1. Determination of marginal probability distributions In a first step, the ARMA(p1, q1)  GARCH(p2, q2) models are estimated to isolate time dependencies among relevant risk processes (see Section 2.1).11 For the estimation of the marginal distributions, we take into account the normal, logistic and non-central t distribution (see, e.g. Longin, 2000; Ghorbel and Trabelsi, 2009), since the first is often assumed in financial analysis and the latter allow higher probability masses in the tails of the density distributions as compared to the normal distribution. After the parameters of the normal, logistic, and non-central t distribution are estimated using the maximum likelihood method as described in Section 2.4, a Kolmogorov– Smirnov test is applied for each probability distribution. With a p-value of less than 5%, the null hypothesis that the sample data follows the stated distribution is rejected. If the null hypothesis is not rejected for more than one stated distribution, we choose the distribution with the highest p-value, i.e. the highest probability that the sample data follow the stated distribution. We thus choose the most likely distribution among the three tested. The chosen distributions along with their p-values are displayed in Table 3. The normal distribution is rejected for all companies except for the residual distribution of the Kellog Co. In 4 out of 22 cases, the logistic, and in 18 out of 22 cases, the non-central t distribution provides the best fit.12 After the marginal distributions of each company and the S&P 500 are fitted, we next focus on the upper tail of each distribution and fit a GPD (Step 2, see Table 2). With respect for the threshold of the GPD estimation we use the 90%-quantile for all cases. This decision is based on sample mean excess plots (see McNeil et al. (2005, p. 279)) and the tradeoff between lower thresholds resulting in lower standard errors (due to more data that is available) and higher thresholds improving the approximation of the excess distribution function through the GPD. From Table A.2 in the Appendix it can be seen that except for Wal-Mart, heavy tailed distributions can be observed as n > 0. Table 3 displays p-values for the traditional approach (whole marginal distribution is fitted) and the mixed marginal distributions, where the tail is fitted using a GPD. The results show that all fitted GPDs are not rejected through the Kolmogorov–Smirnov test and exhibit high p-values (see right column, Table 3), even though these values cannot directly be compared with p-values from the marginal distributions (logistic and t) due to the difference in distributions (unconditional in case of the traditional approach and conditional in case of the GPD).13 Even though standard errors for the estimates according to the GPDs are higher than for the marginal distributions (see Table A.2), the use of the GPDs in the upper tail of the marginal distributions can improve the accuracy for estimating basis risk,14 as the GPD increases the overall fit of the marginal dis11 For a detailed illustration including all parameter values for the ARMAðp1 ; q1 ÞGARCHðp2 ; q2 Þ models, see Table A.1 in the Appendix. 12 A detailed overview of all estimated parameters for the marginal distributions is provided in Table A.2 in the Appendix. 13 A detailed overview of all estimated parameters for the GPDs is given in Table A.2 in the Appendix. 14 Higher standard errors stem from the lower amount of data that is given for the estimation of the GPD in comparison to the traditional marginal distributions.

tribution through replacing the values above the threshold with the corresponding estimators according to Eq. (2). This can be observed when looking at the p-values of the Kolmogorov–Smirnov test for this distribution, which is denoted as the ‘‘mixed’’ marginal distribution. When comparing p-values in the third and fourth column of Table 3, it can be seen that each distribution’s overall fit is improved if the values above the threshold are replaced through the GPD estimators. Furthermore, these results indicate that the highest risk of misestimation of the traditional approach lies in the upper tail, as the Kolmogorv–Smirnov test statistic is based on the supremum of the difference values between the empirical and the theoretical probability distribution function. The misestimation in the tail can be considerably decreased through the GPD estimators as the p-values of the mixed marginal distributions increase. 3.2. Determination of the dependence structure through copulas In Step 3 (see Table 2), first an independence test as laid out in Section 2.4 is conducted to identify whether dependencies among the S&P 500 and each of the companies exist. Second, the degree of dependence is determined through Kendall’s rank correlation (Kendall’s tau). Based on the results, which are illustrated in Table A.3 in the Appendix, 10 companies are excluded from the quantification of basis risk as under independence or a low (in case of March & McLennan) or negative value (in case of Kellog Co.) of Kendall’s tau, the purchase of an index-linked risk management instrument would not be reasonable. For the remaining eleven companies, we conduct a test for asymptotic dependence as laid out in Section 2.2 with a significance level of 5%. When the test is performed, marginal values are transformed to standard Fréchet margins, such that two possible marginal distributions can be taken into account for the transformation (traditional and mixed marginal distribution). As a consequence, two test results exist for each company (Table A.3) and are used to decide which copula model should be chosen among the copula models displayed in Table 1. In case of asymptotic dependence, we choose among copula models exhibiting tail dependence, while the Gauss and Frank copulas are taken into account for the remaining cases. In a next step, for each method – traditional and EVT – a copula model (see Table 1) is chosen according to the least sum of squared values between the empirical and the stated copula, given that both values X1 and X2 lie above their threshold. Table 4 exhibits the copula models that fit best for the tail data, thereby allowing for both regular and extreme value copulas. In addition, the corresponding least sums of squared values between the empirical and the stated copula in the tail are provided in Table 4 along with the resulting basis risk calculated based on the chosen copulas. For the traditional (EVT) method, the Gauss copula and thus an asymptotically independent model is chosen in four (six) cases, while among the remaining seven (five) cases four (three) extreme value copula models and (one) three t copula model(s) fit best in the tail. This shows that approximating the dependence structure through an extreme value copula can be beneficial for the estimation’s fit of the joint tail in case of asymptotic dependence as it focuses on the tail only and thus on the relevant part of the joint distribution. Examining the corresponding conditional PP plots for each model in Fig. B.1 in the Appendix, it can be seen that the copula models that are chosen based on the least sum of squared values between the empirical and the stated copula seem to well capture the dependence structure for each company. 3.3. Comparing basis risk for the traditional and the EVT method To analyze the impact of the different methods on the assessment of basis risk (Step 4, see Table 2), we assume that each of

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R. Kellner, N. Gatzert / Journal of Banking & Finance 37 (2013) 4353–4367 Table A.1 Overview for the ARMAðp1 ; q1 Þ  GARCHðp2 ; q2 Þ models for each company and the S&P 500. Company

American Express Co. Aon Corp. Berkshire Hathaway Chevron Corp. Cisco Systems Coca Cola Co. ConocoPhillips Devon Energy Corp. eBay Inc. Exxon Mobile Corp. FedEx Corp. Hewlett–Packard Johnson & Johnson Kellog Co. March & McLennan Procter & Gamble SLM Corp. Southwest Airlines S&P 500 Wal-Mart Walt Disney Co. Yahoo Inc.

ARMAðp1 ; q1 Þ

GARCHðp2 ; q2 Þ

l

/1

/2

/3

h1

h2

a0

a1

a2

c1

0.05 0.01 0.11 0.00 0.04 0.00 0.00 0.02 0.05 0.01 0.00 0.03 0.01 0.01 – – – 0.00 0.14 0.00 0.00 0.09

0.42 0.50 1.31 0.06 0.57 0.60 0.69 – 0.10 0.10 0.02 0.05 0.01 0.11 – – – 0.08 0.09 – 0.06 0.38

0.60 0.91 0.78 0.07 0.17 – – – – 0.09 0.04 0.06 0.07 0.10 – – – – 0.09 – 0.07 –

 0.01

0.38 0.53 1.22 – 0.47 0.61 0.73 0.04 – – – – – – – – – – – 0.05 – 0.35

0.58 0.94 0.71 – 0.33 0.06 – 0.07 – – – – – – – – – – – 0.10 – 0.05

– 0.06 30.56 0.60 – 0.00 2.47 0.12 0.00 0.21 0.01 0.01 0.00 0.01 0.52 1.53 0.48 – 1.20 0.00 0.00 –

– 0.66 0.85 0.36 – 0.93 0.00 0.88 0.83 0.28 0.82 0.93 0.82 0.39 – 0.35 – – 0.94 0.96 0.92 –

– – – – – – – – – – – – – 0.52 – – – – – – – –

– 0.28 0.14 0.96 – 0.07 0.25 0.10 0.37 1.59 0.13 0.06 0.36 0.08 3.20 0.12 2.11 – 0.06 0.04 0.07 –

– – – – – – – – – – – – – – – – – – –

the companies purchases an index-linked instrument that is triggered if the loss of the S&P 500 (IL) exceeds the 92.5%-quantile SP L0:925 of the empirical data sample, while the companies’ critical loss level Lcritical ¼ Li;critical 0:975 ; i ¼ AE; . . . ; WD (‘‘AE’’ and ‘‘WD’’ denoting American Express and Walt Disney Co.) is given through the 97.5%quantile of the empirical data sample for the eleven companies’ daily losses.15 Table 4 compares basis risk when using the traditional and the EVT method (left and right column, respectively), whose procedure is identical except for the estimation of marginal probability distribution values. In addition, we compare the cases where the extreme value copula is available and where it is not available for estimating the dependence structure and only regular models are used that are estimated based on the whole marginal data (see Table 2). Due to the (almost) identical copula parameter values (if the same copula model is chosen for the traditional and the EVT method) and the same marginal distributions up to the thresholds, different values of basis risk in principle stem from differences in the tails of the marginal distributions. In general, the derivation of basis risk values can be explained in detail using Eq. (5), which shows that basis risk can be separated into three parts: (1) the probability that the negative daily change of the S&P 500 is below its trigger level F SPL ðSPL0:925 Þ, (2) minus the probability that the negative daily change of the S&P 500 is below its trigger level and that the company’s loss is below its critical loss level, CðF SPL ðSPL0:925 Þ; F L ðLi;critical 0:975 ÞÞ; i ¼ AE; . . . ; WD, and (3) divided by the probability the company’s loss exceeds its critical loss level, ð1  F L ðLi;critical 0:975 ÞÞ; i ¼ AE; . . . ; WD. Thus, the extent of basis risk depends on the relation between the three components as well as the relation between numerator and denominator in Eq. (5). Furthermore, basis risk increases if the marginal probability distribution values for F SPL ðSP L0:925 Þ (1) and F L ðLi;critical 0:975 Þ; i ¼ AE; . . . ; WD (3) increase. In the first case, the difference in the numerator is enlarged, and in the second case, the value in the denominator (3) of Eq. (5) is decreased. However, if both values increase, CðF SPL ðSP L0:925 Þ; F L ðLi;critical 0:975 ÞÞ (2) decreases, which in turn decreases the difference in the numerator and, thus, in tendency basis risk. Hence, basis risk values depend on the interaction of both effects, the increase in marginal probability distribution values and the decrease in the joint probability distribution value. In addition, the 15 The trigger and critical loss levels were subject to sensitivity analyses, which showed that the general results are robust.

critical amount by which CðF SPL ðSP L0:925 Þ; F L ðLi;0:975 ÞÞ; i ¼ AE; . . . ; WD decreases is strongly influenced by the type of dependence structure captured through the copula C. To gain further insight into the drivers of basis risk, Fig. 1 displays the QQ plots in the upper tail for the distributions of the S&P 500 and American Express Co, using the fitted logistic and t distribution, respectively, as displayed in Table 4, and the GPDs (see Table A.2, Appendix). The dashed lines represent the empirical trigger (S&P 500) and critical loss (American Express Co.) level. As can be seen, in both cases the quantile values according to the traditional approach (using the logistic and t distribution) are overestimated, which leads to higher estimates b F SPL ðSP L0:925 Þ ¼ 0:9343 and b F AE ðLAE;critical Þ ¼ 0:9771 than the under 0:975 the EVT approach, yielding to b ðSP L0:925 Þ ¼ 0:9259 and F GPD SP L AE;critical b F GPD ðL Þ ¼ 0:9762. Accordingly, the joint probability distriAE 0:975 bution value through the traditional approach, which is equal b ðb F SPL ðSP L0:925 Þ; b F AE ðLAE;critical to C ÞÞ ¼ 0:9228, is higher than under 0:975 GPD b b the EVT approach, C ð F L ðSP L Þ; b F GPD ðLAE;critical ÞÞ ¼ 0:9147. With SP

0:925

SP

0:975

respect to the assessment of basis risk for the S&P 500 and American Express Co., the effect of higher marginal probability distribution values outweighs the decrease in the enumerator by virtue of the higher joint probability distribution value, so that basis risk is higher under the traditional approach than under the EVT approach (0.50 > 0.47).16 The results in Table 4 illustrate that this observation is true for all cases with identical copula models and, thus, the traditional method seems to overestimate basis risk. We next focus on the impact of the dependence structure and compare the basis risk assessment for the two methods with and without allowing for using the extreme value copulas as an alternative to the regular copula models only. The results in Table 4 (see rows ‘‘(regular copula only)’’) show that in those cases

16 While basis risk values for the other companies under consideration can be explained similarly, further interaction effects occur in the case of companies for which the traditional method leads to lower estimates for the marginal probability distribution value for the critical loss level. Here, the denominator in Equation (5) is higher than under the EVT approach, while the higher value for the estimate of F SPL ðSP L0:925 Þ under the traditional approach increases the numerator. Furthermore, if one marginal probability value is higher, while the other is lower, it depends on the respective amount, which determines whether the joint probability distribution is higher or lower under the traditional approach,

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Table A.2 Overview for estimated parameters of the marginal distributions and GPDs for each company and the S&P 500 (standard errors in parantheses). Company

Decision

Marginal Distributions logistic

l American Express Co.

t

Aon Corporation

t

Berkshire Hathaway

t

Chevron Corp.

t

Cisco Systems

t

Coca Cola Co.

t

ConocoPhillips

t

Devon Energy Corp.

Logistic

eBay Inc.

t

Exxon Mobile Corp.

t

FedEx Corp.

t

Hewl-Packard

t

Johnson & Johnson

t

Kellog Co.

Logistic

March & McLennan

T

Procter & Gamble

t

SLM Corp.

t

Southwest Airli

t

S&P 500

Logistic

Wal-Mart

t

Walt Disney Co.

logistic

Yahoo Inc.

t

t

;l

0.039 (0.016) 0.020 (0.015) 0.015 (0.019) 0.040 (0.015) 0.049 (0.010) 0.009 (0.020) 0.052 (0.014) 0.032 (0.017) 0.029 (0.015) 0.055 (0.015) 0.042 (0.018) 0.048 (0.018) 0.017 (0.016) 0.001 (0.021) 0.015 (0.011) 0.011 (0.010) 0.008 (0.015) 0.005 (0.007) 0.038 (0.021) 0.021 (0.021) 0.001 (0.020) 0.137 (0.016)

GPDs

1

logistic

t

;1

0.576 (0.018) 0.567 (0.017) 0.722 (0.020) 0.601 (0.015) 0.342 (0.011) 0.794 (0.021) 0.543 (0.014) 0.453 (0.008) 0.565 (0.015) 0.568 (0.016) 0.717 (0.018) 0.704 (0.019) 0.599 (0.016) 0.537 (0.010) 0.394 (0.013) 0.370 (0.011) 0.541 (0.017) 0.258 (0.007) 0.554 (0.010) 0.852 (0.022) 0.530 (0.010) 0.518 (0.018)

t

n

b

2.079 (0.123) 2.880 (0.211) 4.152 (0.406) 5.369 (0.570) 1.420 (0.066) 5.652 (0.670) 6.039 (0.723) –

0.516 (0.104) 0.413 (0.098) 0.144 (0.077) 0.284 (0.074) 0.305 (0.090) 0.227 (0.081) 0.240 (0.065) 0.232 (0.063) 0.366 (0.0820) 0.261 (0.074) 0.299 (0.074) 0.314 (0.084) 0.266 (0.071) 0.059 (0.067) 0.456 (0.108) 0.436 (0.089) 0.379 (0.095) 0.260 (0.070) 0.088 (0.067) 0.034 (0.067) 0.182 (0.073) 0.6945 (0.132)

0.677 (0.081) 0.4580 (0.053) 0.539 (0.056) 0.390(0.039)

3.704 (0.309) 3.626 (0.310) 5.688 (0.655) 4.231 (0.397) 4.178 (0.392)

1.859 (0.104) 2.649 (0.186) 2.587 (0.182) 3.319 (0.270) – 6.902 (0.983) – 1.247 (0.053)

1.002 (0.111) 0.467(0.050) 0.364(0.034) 0.387 (0.036) 0.340 (0.035) 0.531 (0.053) 0.412 (0.041) 0.406 (0.043) 0.439 (0.043) 0.538 (0.052) 0.569 (0.071) 0.354 (0.039) 0.489 (0.056) 0.221 (0.021) 0.499 (0.048) 0.564 (0.055) 0.470 (0.047) 1.720 (0.238)

Note: llogistic ; lt and 1logistic ; 1t denote the location and scale parameters for the logistic and non-central t distribution, while t represents the degrees of freedom in case of the non-central t distribution.

Table A.3 Independence test, Kendall’s tau and test for asymptotic dependence for each company and the S&P 500. Company

Kendall’s tau

Independence

Asymptotic dependence (traditional)

Asymptotic dependence (EVT)

American Express Co. Aon Corp. Berkshire Hathaway Chevron Corp. Cisco Systems Coca Cola Co. ConocoPhillips Devon Energy Corp. eBay Inc. Exxon Mobile Corp. FedEx Corp. Hewlett–Packard Johnson & Johnson Kellog Co. March & McLennan Procter & Gamble Corporation Southwest Airlines Wal-Mart Walt Disney Co. Yahoo Inc.

0.47 0.01 0.15 0.31 0.00 0.30 0.00 0.22 0.01 0.36 0.81 0.40 0.27 0.07 0.03 0.01 0.01 0.01 0.39 0.41 0.02

No Yes No No Yes No Yes No Yes No No No No No No Yes Yes Yes No No Yes

No – No Yes – Yes – No – Yes Yes No Yes – – – – – Yes Yes –

No – Yes No – Yes – No – No Yes No No – – – – – Yes Yes –

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0.8 0.6 0.4 0.0

0.2

Empirical cumulative probability

0.8 0.6 0.4 0.2 0.0

Empirical cumulative probability

1.0

S&P 500, American Express Co. - upper tail(EVT)

1.0

S&P 500, American Express Co. - upper tail

0.0

0.2

0.4

0.6

0.8

1.0

0.0

Theoretical cumulative probability

0.6

0.8

1.0

1.0 0.8 0.6 0.4 0.0

0.2

Empirical cumulative probability

0.8 0.6 0.4 0.2 0.0

Empirical cumulative probability

0.4

S&P 500, Berkshire Hathaway - upper tail(EVT)

1.0

S&P 500, Berkshire Hathaway - upper tail

0.0

0.2

0.4

0.6

0.8

1.0

0.0

Theoretical cumulative probability

0.2

0.4

0.6

0.8

1.0

Theoretical cumulative probability S&P 500, Chevron Corp. - upper tail

0.8 0.6 0.4 0.2 0.0

0.0

0.2

0.4

0.6

0.8

Empirical cumulative probability

1.0

1.0

S&P 500, Chevron Corp. - upper tail(EVT)

Empirical cumulative probability

0.2

Theoretical cumulative probability

0.0

0.2

0.4

0.6

0.8

1.0

Theoretical cumulative probability

0.0

0.2

0.4

0.6

0.8

1.0

Theoretical cumulative probability

Fig. B.1. Conditional PP plots for the copula models according to Table 4 (left: traditional method; right: EVT method).

where an extreme value copula fits the data best, basis risk seems to be higher than when using a regular copula model. Even though these results deviate from the observations made with respect to the marginal distribution functions, this behavior can be explained through upper tail dependence. In particular, in all cases, the coefficient of upper tail dependence is higher in

case of the EVT copula models as compared to the regular copula models. This leads to higher values for the joint probabilities of the trigger and the critical loss levels under the regular copula models, which in turn results in a stronger reduction of the numerator and thus to lower basis risk values as compared to the EVT copula models.

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S&P 500, Coca Cola Co. - upper tail(EVT) 1.0 0.8 0.6 0.4 0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.4

0.6

0.8

1.0

S&P 500, Devon Energy Corp. - upper tail

S&P 500, Devon Energy Corp. - upper tail(EVT)

0.8 0.6 0.4 0.2 0.0

0.2

0.4

0.6

0.8

Empirical cumulative probability

1.0

Theoretical cumulative probability

0.0 0.0

0.2

0.4

0.6

0.8

1.0

0.0

Theoretical cumulative probability

0.2

0.4

0.6

0.8

1.0

Theoretical cumulative probability S&P 500, Exxon Mobile Corp. - upper tail

0.8 0.6 0.4 0.2 0.0

0.0

0.2

0.4

0.6

0.8

Empirical cumulative probability

1.0

1.0

S&P 500, Exxon Mobile Corp. - upper tail(EVT)

Empirical cumulative probability

0.2

Theoretical cumulative probability

1.0

0.0

Empirical cumulative probability

0.2

Empirical cumulative probability

0.8 0.6 0.4 0.2 0.0

Empirical cumulative probability

1.0

S&P 500, Coca Cola Co. - upper tail

0.0

0.2

0.4

0.6

0.8

1.0

0.0

Theoretical cumulative probability

0.2

0.4

0.6

0.8

1.0

Theoretical cumulative probability Fig. B.1 (continued)

Hence, even though basis risk can be higher or lower depending on the concrete company and the interaction effects explained above, the results indicate that measuring basis risk through the EVT method is more accurate than using the traditional approach. The main reasons for this conclusion are that in

the present setting higher p-values for all marginal distributions are achieved under the EVT approach (see Table 3), i.e. if the tail above the threshold is estimated through a GPD. Furthermore for more than half of the considered examples exhibiting asymptotic dependence, the inclusion of extreme value copulas lead to an

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S&P 500, FedEx Corp. - upper tail

0.6

0.8

1.0 0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

Theoretical cumulative probability

S&P 500, Hewlett-Packard - upper tail

S&P 500, Hewlett-Packard - upper tail(EVT)

0.4

0.6

0.8

1.0

1.0

0.0

0.2

Empirical cumulative probability

0.8 0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

Theoretical cumulative probability

S&P 500, Johnson & Johnson - upper tail(EVT)

S&P 500, Johnson & Johnson - upper tail

1.0

0.0

0.8 0.6 0.4 0.0

0.2

0.4

0.6

0.8

Empirical cumulative probability

1.0

1.0

Theoretical cumulative probability

0.2

Empirical cumulative probability

0.4 0.0

0.2

Theoretical cumulative probability

1.0

0.0

Empirical cumulative probability

0.2

Empirical cumulative probability

0.8 0.6 0.4 0.2 0.0

Empirical cumulative probability

1.0

S&P 500, FedEx Corp. - upper tail(EVT)

0.0

0.2

0.4

0.6

0.8

1.0

0.0

Theoretical cumulative probability

0.2

0.4

0.6

0.8

1.0

Theoretical cumulative probability Fig. B.1 (continued)

improvement in the empirical estimation of the dependence structure.17 17

The improvement of the marginal distributions’ fit through the usage of the GPD in the tail is in line with previous analyses using backtesting methods (see, e.g. Ghorbel and Trabelsi, 2009). However, p-values are chosen for comparing the distribution’s fit since the scaling between 0 and 1 provides more intuitive and comparable interpretations than absolute values of, e.g. value at risk violations.

4. Summary and implications This paper deals with the empirical estimation of basis risk associated with index-linked hedging strategies. In our analysis, we compared two approaches for empirically estimating basis risk in the case of U.S. stock prices and used the concept of copulas for modeling the dependence structure in two different ways. In addi-

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S&P 500, Walt Disney Co. - upper tail(EVT)

0.8 0.6 0.4 0.0

0.2

Empirical cumulative probability

0.8 0.6 0.4 0.2 0.0

Empirical cumulative probability

1.0

1.0

S&P 500, Walt Disney Co. - upper tail

0.0

0.2

0.4

0.6

0.8

1.0

0.0

Theoretical cumulative probability

0.4

0.6

0.8

1.0

S&P 500, Wal-Mart - upper tail(EVT)

0.8 0.6 0.4 0.2 0.0

0.0

0.2

0.4

0.6

0.8

Empirical cumulative probability

1.0

1.0

S&P 500, Wal-Mart - upper tail

Empirical cumulative probability

0.2

Theoretical cumulative probability

0.0

0.2

0.4

0.6

0.8

1.0

0.0

Theoretical cumulative probability

0.2

0.4

0.6

0.8

1.0

Theoretical cumulative probability Fig. B.1 (continued)

tion to a traditional approach, we specifically focused on the application of extreme value theory (EVT), whose estimation procedure is dedicated to the tail of a distribution. In this way, EVT aims to overcome the potential drawback of the traditional method, where the whole probability distribution is estimated, which can generally lead to a misestimation of tail probability values (even though the stated distribution is not rejected). The empirical analysis was based on stock price data from the S&P 500 (used as the underlying index) and eleven selected companies that are listed in the S&P 500 (assumed as hedgers) and exhibit a sufficiently high dependence with the index. First, in case of the traditional approach, the whole marginal probability distributions for stock price losses were estimated using the t and logistic distributions, and the dependence structure among the two risk processes was modeled using Archimedean and elliptical copulas. Second, an EVT method was examined that substituted the probability values in the upper tail of the marginal distributions by estimates using a generalized Pareto distribution (GPD). This way, the impact of a misestimation in the tails of the marginal probability distributions on the quantification of basis risk could be analyzed in an isolated way, as the dependence structure between the two risk processes remained almost unchanged. In this setting, the Kolmogorov–Smirnov test showed that the overall fit for the marginal probability distributions could be increased if a GPD is used to estimate probability values in the upper tail. In addition, we integrated extreme value copulas as an alternative to the regular copula mod-

els that are estimated based on the tail data only, whereas regular copula models make use of the whole distribution. This procedure allowed an examination of the impact of misestimating the dependence structure in the tail on the assessment of basis risk in case of index-linked hedging instruments. Our analysis showed that with respect to the assessment of the marginal distributions, the traditional method seemed to overestimate distribution values, which lead to higher basis risk values. Further interaction effects occurred when comparing regular and EVT copula models, which emphasized that the degree of asymptotic dependence is a crucial driver for basis risk. In particular, basis risk values were higher when allowing for extreme value copulas as compared to regular copula models. Thus, an adequate assessment of the marginal distributions as well as the dependence in the tails is very important when estimating basis risk and should be conducted for each firm individually. Our results indicated that an estimation procedure that is focused on the part of the (joint) distribution relevant for basis risk may allow a more adequate assessment and can help avoid potential misestimation problems. These occur if the applied method provides estimates for the whole distribution, even though only a specific part of the distribution is of interest for the analysis. Considering the improvement in the mixed marginal distributions’ fit, the inclusion of the GPDs to fit the upper tail appeared to be very beneficial for the accurateness of basis risk assessment, even though standard errors of the GPD estimates tend to be higher by virtue of reduced available data in

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the tails. Moreover, with respect to capturing the dependence structure among risk processes, using extreme value copulas as an alternative to regular copulas and to approximate the dependence structure in the tails implied a better fit of the dependence structure in several cases and should thus be taken into account as a viable alternative to classical dependence models. These considerations are especially relevant in the context of basis risk in order to gain an adequate picture of the risk of non-payment associated with an index-linked hedging instrument. The problem of estimating the tail of a distribution is thereby not only of relevance for measuring basis risk in the context of index-linked hedging strategies, but also when quantifying other downside risk measures such as the value at risk or quantiles in general. Thus, companies and regulators should be careful when estimating basis risk. Our results suggest that fitting exclusively the part of the probability distribution, which is of interest for the respective risk measure in principle leads to more accurate estimates of basis risk. Appendix A. Appendix Tables A.1–A.3 and Fig. B.1. References Balkema, A., de Haan, L., 1974. Residual life time at great age. Annals of Probability 2 (5), 792–804. Brodin, E., Rootzén, H., 2009. Univariate and bivariate GPD methods for predicting extreme wind storm losses. Insurance: Mathematics and Economics 44 (3), 345–356. Castelino, M.G., 1992. Hedge effectiveness: basis risk and minimum-variance hedging. Journal of Futures Markets 12 (2), 187–201. Cébrian, A.C., Denuit, M., Lambert, P., 2003. Analysis of bivariate tail dependence using extreme value copulas: an application to the SOA medical large claims database. Belgian Actuarial Bulletin 3, 33–41. Chollete, L., de la Pena, V., Lu, C.C., 2012. International diversification: an extreme value approach. Journal of Banking and Finance 36 (3), 871–885. Coles, S., Heffernan, J., Tawn, J., 2000. Dependence measures for extreme value analyses. Extremes 2 (4), 339–365. Cotter, J., 2001. Margin exceedances for european stock index futures using extreme value theory. Journal of Banking and Finance 25 (8), 1475–1502. Cummins, J.D., Lalonde, D., Phillips, R.D., 2004. The basis risk of catastrophic-loss index securities. Journal of Financial Economics 71 (1), 77–111. Dupuis, D.J., Jones, B.L., 2006. Multivariate extreme value theory and its usefulness in understanding risk. North American Actuarial Journal 10 (4), 1–27. Embrechts, P., Lindskog, F., McNeil, A., 2003. Modeling dependence with copulas and applications to risk management. In: Rachev, S.D. (Ed.), Handbook of Heavy Tailed Distributions in Finance. Elsevier/North-Holland, Amsterdam. Figlewski, S., 1984. Hedging performance and basis risk in stock index futures. Journal of Finance 39 (3), 657–669. Gatzert, N., Kellner, R., 2011. The influence of non-linear dependencies on the basis risk of industry loss warranties. Insurance: Mathematics and Economics 49 (1), 132–144. Genest, C., Rémillard, B., 2004. Tests of independence and randomness based on the empirical copula process. Test 13 (2), 335–369. Genest, C., Segers, J., 2009. Rank-based inference for bivariate extreme value copulas. Annals of Statistics 37, 2990–3022. Genest, C., Rémillard, B., Beaudoin, D., 2009. Goodness-of-fit tests for copulas: a review and power study. Insurance: Mathematics and Economics 44 (2), 199– 213. Ghorbel, A., Trabelsi, A., 2009. Measure of financial risk using conditional extreme value copulas with EVT margins. Journal of Risk 11 (4), 51–85.

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