Sylvain Prado
email: sylvain.prado@ge.com

The European used-car market at a glance: Hedonic resale price valuation in automotive leasing industry

Keywords: Hedonic model, residual value, automotive market. JEL Classi…cation: C51, G12, G32

In the leasing industry, the risk of loss on sales at the end of the contract term, as well as pricing are critically impacted by the forecasted resale price of the asset (residual value). We apply the Hedonic methodology to European auto lease portfolios, in order to estimate the resale price distribution. The Hedonic approach estimates the price of a good through the valuation of its attributes. Following a discussion on Hedonic prices, we propose an operational model for the automobile resale market. The model is applied to four European countries (France, Germany, Spain and Great Britain), and distributions are calculated on two vehicle versions (Audi A4 & Ford Focus) allowing a comparison of market depreciation patterns and residual value risks.

Introduction

In the auto lease industry, a large part of the rent paid by the customer during a life contract is the di¤erence between the list price and the residual value. The leasing company makes or losses money depending on whether it accurately predicts the value of the asset at the end of the contract (fair market value). If residual values are forecasted to be higher than what the asset is actually worth at lease-end, then there will be a loss. At the opposite, if residual values are forecasted to be lower, then there will be a gain on resale. The estimated resale price of the car at the end of the contract term appears as a key component for the pricing, the risk of losses and the reserve calculation [START_REF] Ackerman | Used Cars as a Depreciating Asset[END_REF] .

Akerlof (1970) explained why used car valuation is so much lower than new car valuation. The automotive resale market is a¤ected by something called the 'lemon e¤ect'. A used car has the probability to be of a good quality or a bad one (i.e. lemon), and the uncertainty about quality implies a price adjustment [START_REF] Scott | The Valuation of Risks to Life: Evidence from the Market for Automobiles[END_REF] . The Akerlof theory helps to understand the large variance of prices between new and used markets, but it does not propose a methodology to calculate car depreciation.

Another way of looking at it is the Hedonic approach. The Hedonic theory provides solutions and estimates price-quality relation through a detailed calculation. A Hedonic model has been originally proposed by Waugh (1928) on vegetable products and by Court (1939) in the automobile industry. Hedonic models have been applied to a lot of commodities (mainly real estate and automobile but also fruits or vine [START_REF] Timothy | The Estimation of Demand Parameters in Hedonic Price Models[END_REF] ).

The automobile market itself has had di¤erent applications (quality corrected price index [START_REF] James | On the Estimation of Structural Hedonic Price Models[END_REF] , demand for fuel e¢ ciency, valuation of environmental and safety demand [START_REF] Court | Hedonic price indexes with automotive examples[END_REF] , test of the Akerlof e¤ect [START_REF] Couton | Hedonic Prices for Environmental and Safety Characteristics and the Akerlo¤ E¤ect in the French Car Market[END_REF] , behavior of the automobile market through price quality and competition [START_REF] Combris | Estimation of a Hedonic Price Equation for Bordeaux vine: Does Quality Matter ?[END_REF] ...). The main point underlying this paper is to apply the Hedonic methodology to estimate the resale price distribution of cars in a leasing perspective. This article is intended for people within the leasing industry interested by residual value risk, as well as academics concerned by a comparison of European markets.

We propose a methodology for operational applications to estimate the distribution of resale price. To this end, we apply a Hedonic model (a method of estimating value through constituent characteristics of the asset) on historical information from a major leasing company [START_REF] Keith | Price, Quality and Advertising Competition: An Econometric Investigation of the United Kingdom Car Market[END_REF] . Further to this, we estimate a value according to vehicle characteristics and country singularities. Resale price distributions of two vehicles (Ford focus C-max, Audi A4) are calculated in various European markets (France, Germany, Spain and Great Britain).

The paper is organized as follows. Section 2 discusses the Hedonic theory underlying our model. In section 3 some meaningful characteristics of the model are exposed. Section 4 presents our approach. Section 5 estimates the distributions and analyzes the results, …nally. Section 6 concludes.

The Hedonic theory underlies our model

The section discusses the Hedonic model approach, identi…cation issues and automotive assets speci…cities.

Goods attributes constitute the Hedonic theory.

To explain consumer behavior, Lancaster (1966) assumes that consumers get utility from goods attributes [START_REF] Keith | Hedonic Price Indexes for United Kingdom Cars[END_REF] .

Assuming that a car is the only good involved in the activity consumption of driving, it produces a …xed vector of attributes and the level of activity is a scalar associated with the vector (the relationship could be linear). The driver chooses a combination to maximize his utility function according to the characteristics of the goods under a constraint budget.

Inspired by Lancaster (1966), Pickering et al(1973) added an empirical perspective to the approach by conducting a survey in the UK. Following their results, they de…ned …ve groups of commodities (utilities, luxuries, leisure goods, central heating and automotive) and identi…ed eleven characteristics as signi…cant discriminators between groups. The principal attributes desired by car buyers were comfort, durability, economical, manoeuvrability, performance safety and style. They acknowledged that products and attributes may change groups through time because of product life cycle, di¤erent tastes between consumers, the growth of the market penetration (i.e. luxuries becoming utilities), complementarity or substitutability of goods [START_REF] John | Quality Change and Pricing Behaviour in the United Kingdom Car Industry 1956-1968[END_REF] .

They also …gured out that it could be relevant sometime to disaggregate a group (i.e. cars by makes).

The Hedonic model assumes that goods are valued for their utility-bearing attributes or characteristics.

In 1974, Rosen [START_REF] Dalen | Quality-corrected price indices: the case of the Dutch new passenger car market[END_REF] developed the framework of Hedonic models. The theory describes cars by n measurable characteristics (oil consumption, car size, power, technology...) and a vector Z(= z 1 ; z 2 ; :::; z n ) with z i measuring the amount of the i th characteristics. The existence of product di¤erentiation implies that a wide variety of alternative packages, completely described by numerical value of z, are available. Buyers and sellers locate in a spatial equilibrium. On one side, the consumption decision is made by a maximization of utility.

On the other side, the production decision is made by minimizing factor costs subject to a joint production function constraint relating to the number of units and factors of production. A price p(z) = p(z 1 ; z 2 ; :::; z n ) is de…ned at each point on the plane. Both consumers and producers are guided by prices through packages of characteristics bought and sold. Observations of p(z) represent a joint envelope of a family of value functions and another family of o¤er functions. At equilibrium, buyer and seller are perfectly matched when their demand and o¤er functions meet at eye level.

The approach consists in estimating the following model: P i(z) = F i(z i ; :::; z n ; y 1 ) (demand) P i(z) = Gi(z i ; :::; z n ; y 2 ) (supply) P i(z) is the implicit market price for attribute z i , y 1 and y 2 are vector of exogenous demand shift variables and a vector of exogenous supply shift variables, respectively. At equilibrium, market quantity demanded for products with characteristics z, (Q d (z i )) is equal to market quantity supplied with those attributes (Q s (z i )).

A P (z) function has to be found to make this equality possible. Unfortunately, di¤erential equations for setting (Q d (z i )) = (Q s (z i )) are not linear in most cases and closed solution are not always possible.

An identi…cation problem appears in Hedonic models.

There are identi…cation problems in the Rosen model : if p(z) is non linear, it may not be possible to …nd closed solutions. A lot of conditions must be imposed and partial di¤erential equations must be solved when there is more than one characteristic. Rosen believed that the form of the Hedonic function is an empirical matter and developed an empirical methodology to estimate demand and supply parameters (if no explicit solution for the Hedonic price function is available). Rosen solved the "garden variety identi…cation problem" by simultaneous identi…cation methods like 2SLS: p(z) is estimated by regressing all observed di¤erentiated product prices p on all their characteristics, z, using the best adaptable function. The estimated prices are then included in the complete formula as endogenous variables. Later, Brown and Rosen (1982) showed that this technique is not always possible (it still requires prior restriction on functional form) [START_REF] Ivar | Identi…cation and Estimation of Hedonic Models[END_REF] .

But Bartik (1987) argued that the econometric problem of estimating Hedonic demand parameters is not a standard identi…cation problem caused by demand-supply interactions: because a consumer decision cannot a¤ect an Hedonic function, it does not a¤ect the supplier. He pointed out another identi…cation problem: the Hedonic price function is not linear and the consumer can endogenously choose both quantities and marginal prices. Formulated through a characteristic bid equation [START_REF] William | Econometric Analysis[END_REF] it highlights the impact of consumer traits. Let p zj (Z i ) be the estimated Hedonic marginal price of characteristics z j = W ij , where Z i denotes a vector of observed characteristics of the product and W ij a consumer marginal bid for z j .

p zj (Z i ) = b 0 + b 1 Z i +b 2 X i +bD 0i +e ij .
X i is consumer expenditure on commodities others than Z. D 0i is a vector of observed demander traits a¤ecting the marginal bid. e ij is a disturbance term.

It becomes

p zj (Z i ) = b 0 + b 1 Z i +b 2 X i +bD 0i + D ui +r ij .
D ui is an unobserved taste component form. r ij is a random component and r ij + D ui = e ij . Therefore Z i and X i are correlated with unobserved tastes in the residual, leading to biased results (equivalent to di¤erent population of consumers).

In Bartik's article, the identi…cation problem is caused by the endogeneity of both prices and quantities when households face a nonlinear budget constraint (the distribution of income follows no simple law through its range making it di¢ cult to specify the problem entirely). An instrumental variable solution is suggested and applied (household example with addition of budget constraint). The implicit market price is estimated by regressing all observed di¤erentiated product prices p on all their characteristics by group of modality of D ui [START_REF] Raymond | Product Quality and Market E¢ ciency: The E¤ect of Product Recalls on Resale Prices and Firm Valuation[END_REF] . We follow on the Bartik critic in our analysis and propose a solution to manage the unobserved taste issue in Section 3.

Used cars are durable commodities.

Berndt (1983) provided general frameworks on Hedonic prices for durable goods. Assuming that the asset price of the n th capital good of vintage at time t is equal to the present value of its future services, we have:

q n;t; = s=Tn P s=0 1 1+r s V n;t+s; +s
where T n is the life time of the asset, r the interest rate, V n;t; the value of the asset at time of the ‡ow of services of the n th capital good of vintage . Berndt demonstrated that the Hedonic price equation can be expressed in terms of service prices in a single equation [START_REF] George | The E¢ ciency of Used Vehicle Markets: The Case of Technological Obsolescence and Di¤erential Repair Records[END_REF] .

This concept has been used originally in the second-hand automobile market analysis by Akerman (1973), who produced one of the …rst study on the rapid used car falling prices. The price of an automobile is evaluated as the discounted present value of its remaining services. The Akerman model included a Hedonic price, a repair cost, a service function and an expected gain on resale estimation. Akerman used a single equation and a regression to estimate the Hedonic price [START_REF] Thomas | Price-Quality Relations for Automobiles: A New Approach Applied Economics[END_REF] . Ho¤er and Pratt (1990), inspired by Akerman approach (the price of a resold vehicle as an implicit rental cost of holding a s year old automobile, including also automotive price less market price of interest) proposed a simpler model. A single equation, where depreciation declines with age at constant exponential rate, includes technological obsolescence, di¤erential repair record and fuel e¢ ciency as shift variables. The depreciated value (s; t) of an s-year old machine in

year t is ln( (s; t)) = A b 1 s + b 2 tech + b 3 maint + b 4 EP A: tech = 0 if the vehicle is not discontinued. maint = 1 if
the maintenance expenses are greater than average and EP A is a fuel e¢ ciency indicator.

All along the referenced studies, methodology moved from a remaining service approach to a Hedonic model including essentially the vehicle characteristics (physical or not). We acknowledge the "remaining service approach", however we believe that vehicle characteristics contain most of the information. As a consequence, we adjust the resale price from in ‡ation and we set a statistical model mainly through variables related to the vehicle characteristics.

3 Some characteristics of the model are discussed.

The model construction brings comments and discussions: The used market is (i)demand oriented and (ii)correlated with fuel price; (iii)Multicollinearity is a critical issue in Hedonic calculation; (iv)We have to choose a functional form, and (v)Heteroscedasticity impacts the model speci…cation.

Coe¢ cients interpretation depends on used market substitution to new market.

Berndt (1983) pointed out an argument against the Rosen identi…cation problem for the used market: under speci…c conditions, equation parameters can be directly interpreted as re ‡ecting demand (rather than cost or supply) and there is no identi…cation problem. If the supply curves of products are perfectly inelastic, then the market demand and supply curves would intersect at di¤erent levels of each combination of characteristics. The structure combination would be determined by the demand. The di¤erence of price level among products could be interpreted unambiguously as providing implicit measures of consumers'evaluation of the characteristic combinations. So coe¢ cients of the equation are well identi…ed, as well as estimates of the demand function parameters. Because the total quantities are …xed (assuming that there is a non signi…cant link with new market), the equation only re ‡ects demand in the used car market.

Hartman (1987) results validate that an application to the resale market avoids the identi…cation problem.

If the supply of the attributes embodied into used cars is almost perfectly inelastic, he states that simultaneity should not pose a problem in recovering Hedonic demand and supply parameters in new product market.

If the simultaneity is important, di¤erent assumptions about quantity of each make and model sold should generate di¤erent parameter estimates. Therefore the only question is : are the parameters statistically and economically signi…cant ? In his analysis, the resale value calculation was very robust to alternative sales assumptions. Hartman applied a single equation model [START_REF] Lancaster Kelvin | A New Approach to Consumer Theory[END_REF] to estimate the e¤ect of product recalls on resale prices and …rm valuation.

Two main conclusions can be stated: All referenced automotive studies use single equation techniques, and remarketing professionals usually believe in a substitution relation for young resale automotive market only which is a situation where demand and supply characteristics are quite similar. Therefore we apply a single equation and we exclude short term duration (less than 12 months age) vehicles.

Others products interact with price.

De…ning a framework on the demand analysis, Berndt (1983) discussed the input price-dependent quality adjustment case: the quality of a good (i.e. fuel e¢ ciency) is dependent on the quantity (or price) of another good (i.e fuel price). Berndt states that we could test the dependent [START_REF] Ohta | Automobile prices revisited: extensions of the Hedonic hypothesis[END_REF] (or independent) price hypothesis using classical testing procedures (i.e. economical and statistical signi…cance of fuel price on auto price).

Fuel price has a signi…cant part in the total cost of automobile usage, and then monthly fuel price constitutes our model.

Multicollinearity is a main issue in Hedonic models.

The econometrician walks between the two following risks while he selects the relevant variables for an Hedonic estimation: correlated explanatory variables and trickle down hypothesis.

In the automotive area, physical characteristics are often correlated (i.e. four wheels correlated to fuel capacities). According to the Gauss Markov theorem, OLS has the smallest variance. However, if explanatory variables are correlated, then small change in the data produces wrong sign, implausible magnitude and wide swings in parameter estimates [START_REF] Pickering | Are Goods Goods? Some Empirical Evidence[END_REF] . As a consequence, parameter correlations present a major issue for forecast applications. The simplest solution is to exclude variables at risk (i.e all variables related to the engine power, 1 8 See Appendix A4. 1 9 See Appendix A5. 2 0 But it also produces instability of coe¢ cient and higher standard errors, R 2 quite high, coe¢ cient with high standard error and low signi…cance levels (even if signi…cant). See Greene (2003) chapter 4 p57. number of cylinders, kilowatt, fuel consumption, fuel capacity...) in the case of non economical signi…cance [START_REF] Prado | Hedging residual value risk using derivatives[END_REF] .

Triplett (1969) highlights another problem. Because a small amount of variables is able to explain most of the variance (i.e. the weight of the vehicle correlated with engine power and price), there are some risk of biases in the Hedonic model and a substantial number of innovations are missed throughout this 'trickle down'hypothesis.

Therefore, the selected parameters of our model cover four axes of depreciation e¤ects: the level of usage, the original equipment cost, the market interactions and the pure physical characteristics of the vehicle.

Which functional form ?

Rosen (1974) states that the functional form is an empirical matter. In the same logic, Grilitch and Otha (1976) choose a semilogarithmic form for their regression because 'it provided a good …t of the data'. Most of the literature suggests the log form, others studies apply the log-log form [START_REF] Reis | Hedonic Price Indexes for New Passenger Cars in Portugal[END_REF] , and the Box Cox test [START_REF] Rosen Sherwin | Hedonic Prices and Implicit Markets: Product Di¤erentiation in Pure Competition[END_REF] has also been used to compare several functional forms. The Hedonic functional form problem constitutes a great discussion but it is not the main purpose of the article.

We followed the Grilitch and Otha position (1976) ('a good …t of the data') and empirical results lead us to the linear form of Cowling and Cubbin (1972). Their linear model includes multiple physical variables like horse power and length and to allow approximation to a non linear form, square transformation, cubic transformation and log transformation were applied to exogenous variables. Interactions terms were also included.

2 1 An advanced solution is the ridge regression estimator or principal component methodology. The problem is that we lost visibility on coe¢ cients meaning.

Unobserved tastes create heteroscedasticity.

As previously mentioned in Bartik's critics [START_REF] Jack | Automobiles and Hedonic Quality Measurement[END_REF] , because the choice for the studied commodities quality and other commodities is correlated with unobserved tastes in the residuals, then an heteroscedasticity issue appears. If residuals from the economic relation do not have constant variance, the model is not biased but the variance increases. Bartik states that any variable that exogenously shifts the budget constraint of the buyer will be an appropriate instrument: the budget constraint shift is correlated with the buyer choice of car attributes and the choice of other products yet uncorrelated with unobserved tastes.

We follow Bartik's approach including the index of industrial production [START_REF] Waugh | Quality Factors in ‡uencing vegetables prices[END_REF] as a proxy of the economic situation of the buyer (we propose a temporal budget constraint shifter). Because most of buyers are professionals impacted by a market seasonality, we include a seasonality variable on a quarterly basis. Finally, in order to manage unobserved characteristics (i.e. brand name perception and reputation...), we also insert a manufacturer e¤ect [START_REF] David | Used Car Markets: Reliability Does Matter, but Do Consumer Reports?[END_REF] . 4 We use the Hedonic model to estimate the distribution of resale price.

We apply the straightforward regression approach of Otha and Grilitch (1976). Removing the impact of uncertain variables and using the classical OLS properties, resale price distributions are calculated.

2 4 See Section 2. 2 5 Excluding energy and construction. 2 6 We do not work with a model car type level, because our goal is to apply a methodology ‡exible enough to include new cars and non exhaustive data. Moreover, our model does not include the life cycle of vehicles ('honey moon'e¤ect for new models...) because of the di¢ culty to collect and to standardize the information. In the list of unobserved characteristics, there is also the remarketing performance. The value could be impacted by the remarketing team in charge of the resale process. Finally, we do not include macroeconomic impacts (which need a proper analysis). Therefore unobserved e¤ects mentioned above constitute the random variable of the statistical model.

Ohta and Griliches have an empirical approach.

Regarding theoretical issues (including the one discussed in Sections 2 and 3), Ohta and Griliches state that Hedonic model usage 'has an air of "measurement without theory", but one should remember the limited aspirations of the Hedonic approach and not confuse it with attempts to provide a complete structural explanation of the events in a particular market' 27 . They exposed a strong empirical criterion for hypothesis testing 28 . They included a make e¤ect as a proxy of unmeasured characteristics. A real e¤ect linked to unmeasured physical characteristics and a putative one (linked to prestige, service availability...) constitute the make e¤ect.

In their model, the price of model k of make i and age s at time t is P kits = f ct(M i ; P t ; D s ; e

P aij x kivj )
with M i the e¤ect of the i th make, P t the pure Hedonic price index at time t, D s the e¤ect of age s (depreciation). a ij are parameters re ‡ecting the imputed price of physical characteristic j at time t. x kivj is the level of the physical characteristic j embodied in model k of make i and vintage v (v = t s).

They applied their models on new and used cars and tested di¤erent hypotheses 29 (i.e geometric depreciation held separately from makers). Otha and Grilitch approach is now a standard. As a consequence, Yerger (1995) used this method to discuss an article written by Ho¤er and Pratt (1990) which was inspired by Akerman approach 30 . Following these authors, our approach is mainly empirical. We select the model structure that best …ts to reality and choose exogenous variables with a statistical and economic signi…cance.

2 7 Ohta and Griliches (1976) p326. 2 8 "The rejection or acceptance of an hypothesis should depend on the researcher's interests and his loss function"(p 339). Grilitch and Otha put in perspective the statistical and economic signi…cance. Instead of following a formal Fisher test, they use the di¤erence in the standard errors of the unconstrained and constrained regressions as a relevant measure of the priceexplanatory power of a particular model. They do not reject null hypothesis if di¤erences between the standard errors of the unconstrained and constrained regressions are less than or equal to 0:01.

2 9 Their results on the US market are worth to mention: no gains to move to performance variables (so we can only use the vehicle characteristics); geometric depreciation is an adequate approximation but it is not constant accross time and manufacturers; New and used car market can be analysed jointly. Unfortunately, because of the rise of fuel cost (1973) they aknowledged that their analysis was already obsolete.

3 0 See Appendix A6.

Statistical models are slightly di¤erent by country.

Our analysis includes four countries (France, Germany, Spain, Great Britain) and we de…ne a model for each of them 31 . The real resale price is explained by a …rst group of variables indicating the level of usage: age and mileage are in logarithm due to the well known non linearity property of car depreciation. An indicator of usage intensity, the mileage per month, is also included and signi…cant. The second group of variables is related to the list price. A cubicle variable of list price is added (high initial price increase devaluation). The make e¤ect is introduced through a dummy variable of manufacturer multiplied by the list price. Variables bringing market information contitute the third group: the diesel pump price, the industrial production index and the quarter sale date. The last group includes pure physical characteristics that are slightly di¤erent from a country to another (average fuel consumption, body type, number of seats, engine power, number of cylinders, automatic transmission, number of doors).

We estimate the distribution of resale price.

We wish to calculate the distribution of y 0 (the resale price) for a regressor vector x 0 (group of variables explaining the resale price). The usual regression formula is y 0 = a 0 b 0 x 0 . X and y 0 denote the full data matrices. b 0 is the coe¢ cient vector. We assume 32 that y 0 follows a normal distribution 33 equal to

N (x T 0 b 0 + e; s 2 [1 + x T 0 (X T X) 1 x 0 ]).
The con…dence interval is calculated with

x T 0 b 0 t =2(n p 1) p M SE[1 + x T 0 (X T X) 1 x 0 ] 1=2 : 3 1 See models details in Appendix B. 3 2
Data are composed of subgroups by models, age, mileage and physical vehicle characteristics. Normality hypothesis test are possible on subgroups with a signi…cant amount of data. For models with same age and mileage, H0 is not rejected. The test of normality on the two analyzed vehicles (Ford focus and the Audi A4) is not rejected.

3 3 See Appendix A7.

An adjustment removes uncertain variables e¤ects.

All the exogenous variable values are known with certainty 34 , except the fuel pump price and the production distribution index. We aim to remove the product interaction e¤ect (Dp) and the temporal budget constraint (Ip) in order to focus on the vehicle valuation. Assuming that the diesel price and the production price follow a normal distribution, we calculate the mean and the variance from 2004 to 2008 and we estimate a risk neutral distribution of the resale price.

The unconditional resale price distribution of x T 0 b e0 can be solved as

N ( 1 R 1 1 R 1 m(x 0 =Dp; Ip) g(Dp) j(Ip) dDp dIp; 1 R 1 1 R 1 n(x 0 =Dp; Ip) g(Dp) j(Ip) dDp dIp)
Where m() = x T 0 b 0 + e and n()

= s 2 [1 + x T 0 (X T X) 1 x 0 ]
. g() and j() are the probability density of the fuel pump price and the production distribution index. The integrals are calculated with numerical integration.

5 We apply the methodology to four European countries.

Models by country, regression results and graphical illustrations, through two vehicle versions, provide an insight of European markets.

5.1 Models are created according to the information usually available in the leasing industry.

In order to quantify the Hedonic price, we apply the model to four European markets (France, Germany, Spain, and Great Britain 35 ) using internal sales data from January 1st 2004 to December 31st 2008 of a major leasing company. Statistics are based on random samples of cars sold in various channels (auction, dealers, private sellers, etc). Vehicle age samples range from 1 to 10 years, and have mileage ranging from 1,000 to 400,000 km. As expected for leasing companies resale statistics, a concentration of vehicles with high mileage and short age spans (concentration on 24, 36 and 48 months of age with a mileage between 80000 km and 120000 km) constitutes a large part of our sample. All monetary values (sales prices, diesel prices) are adjusted according to the in ‡ation. We aim to create a tool allowing a leasing company to catch all the available Hedonic information of the car activity from it's historical sales. According to the company position in markets, the amount of data is signi…cantly di¤erent by country but su¢ cient for calculation (Fr: 112,875 units, Ger: 7,398 units, Sp: 14,674 units; Gb: 33,506 units). Contrary to some referenced studies applying the Hedonic model to the car market, we do not limit our analysis to a segment or version of cars. As a consequence, the explained variance (R 2 ) is slightly lower (and even more to applied studies on the much more stable new car market). To estimate the manufacturer e¤ect, statistics include several manufacturer names (Fr: 9, Ger: 4, Sp: 6, Gb: 8) by country.

5.2

The regression provides a Hedonic price assessment of the European markets.

All variables have a signi…cant economic value 36 . The explained variances of the OLS regressions are between 0.75 and 0.8. Characteristics adding quality to the car (engine power, number of seats, etc) as well as the industrial production index (as a proxy of budget variation) have a positive sign. According to the Hedonic theory, the price of fuel is an additional cost of the driving activity and has therefore a negative e¤ect. The variables of age, mileage and usage intensity (mileage per month) reduce the resale price, there are parameters correlated to obsolescence and wear. A slight seasonal e¤ect exists in all markets. The well known and better valuations of German manufactured cars (positive make e¤ect) are veri…ed in all countries.

5.3 The analysis on Ford focus and Audi A4 give additional informations.

France, Germany and Spain share the same currency (Euro) and results estimate the resale price distribution of a vehicle, according to the amount of information available from historical sales. The samples of the four countries have two manufacturers in common: Audi and Ford. We choose the characteristics of the Ford Focus (C-max 1800 TDCI 115 Ghia 5P) and Audi A4 (1.9 Tdi 130 Pack 4P) as a basis to compare the four markets. The information provided by the model could be summarized by two elements. On one hand, a higher valuation of car at the end of the contract reveals better opportunities for leasing business. On the other hand, a higher volatility implies uncertainty on the resale price, and therefore a higher risk of loss on sale.

Bucket results: A …rst analysis approach 37 on the bucket of a 36 month age group, and 90000 kilometers emphasizes three points. First, the Audi A4 has a better valuation than the Ford Focus in every country. As mentioned previously, German cars bene…t from a 'positive make e¤ ect'; they are objects of prestige and share a reputation of good quality cars. Secondly, the high level of standard deviation in all markets reveals a huge volatility. Acknowledging that the second hand market is not as liquid as a …nancial market, it illustrates that a car, as an asset in a balance sheet of a company, constitutes a signi…cant risk.

Thirdly, in Germany, cars get a better valuation. A high resale price constitutes a good element for a leasing business; however the German market also has a higher standard deviation, and therefore a higher risk of loss on sales.

Conclusion and extensions

The Hedonic theory has been widely used for the automotive market analysis. We discuss and propose an application to second-hand vehicles in the leasing industry, where the residual value is a critical parameter (residual value risk). The model is based on attributes in order to estimate the resale price distribution. A product interaction e¤ect (fuel price), and a temporal budget shifter (industrial production index) are also included. The methodology applied to four European countries provides a perspective of the automotive resale markets. Focusing on the pattern of depreciation of two vehicles (Ford Focus and Audi A4), the approach illustrates the di¤erent levels of probability of losses according to the resale information available by a leasing company. The approach also allows the comparison of market opportunities, through pricing analysis and risk. Our study can be extended in several ways. The leasing industry includes all types of equipment and the application of the Hedonic valuation would be ‡exible enough to be extended to assets other than automotive. Moreover, our analysis could also be extended to contracts with a purchase option or a rewrite option on age and mileage (i.e. customers can choose to extend or interrupt their contract(s)).

Two other elements in the area of residual value risk should be included to complete the analysis: the vehicle life cycle impact and the macroeconomic impact (the general market depreciation). The macroeconomic element would need a more thorough study. A 5: Berndt de…nes a general framework on commodities.

X(= x 1 ; x 2 ; :::x n ) a vector of commodity, B(= b 1 ; b 2 ; :::; b n )a vector of qualities for each commodity, Z(= z 1 ; z 2 ; :::; z i ; z n ) a vector of physical characteristics for each commodity and P (= p 1 ; p 2 ; :::; p n ) a vector of price for each commodity. Moreover we have an utility function u = F (x; b) and B n = H n (Z).

x n = f (u; x 1 ; x 2 ; :::; x n 1 ; b n )

For a new quality level from b n0 to b n1 under the assumption of a log-log form;

In the case called the "simple packaging hypothesis" (or input price-independant quality adjustment), b n is only dependant of z i , we have a quality function b i = h i (z i );

x n must be equalized at the margin: pn0 pn0 = pn1 pn1 = p n where p n is a base price constant re ‡ecting the price of the standardized unit.

2 2

 2 See Hogarty (1975). 2 3 See Atkinson and Halvorsen (1990), Van Dalen and Bode (2004).

A 4 :

 4 Hedonic price, repair cost, service function and expected gain on resale estimation are components of the model. Akerman use a single equation and a regression to estimate the Hedonic price:logA(v; m) = C + E(v; m) + C(v; M ) + W (v; M ) + L(v; M ) + H(v; M ).with A(v; m)=new car list price including federal tax and handling and transportation charges. E(v; m)=1 if car has height cylinders. C(v; M )=1 if compact. W (v; M ), L(v; M ) and H(v; M ) are weight, length and horse power. v is model year and m is the model. Hartman Equation inspired by Grilitch and Otha equation:LogP kit = Bo + B 1 M K i + B 2 M D k + B 3 AGE s + B 4 jA kij + B 5j R jkP kit is the resale price in period t for a car of make i and model k.M Ki and M D k are dummy variables indication Make and model. Age is the age of the car in t. A kij is the level of attribute j embodied in model k and make i. R jk summarize the car recalls history indicating cumulative recalls of type j for model k.

Through a log transformation of 1 )Ford

 1 , then lnp n1 = lnp n0 + lnh n1 (z n1 ) and an assumption of log-log form of the quality conversion function lnh n1 (z n ) = k P k=1 b nk ln z n1;k :

  Graphical results: The graphics of distribution through age and mileage give an additional perspective of the depreciation 38 . The variance is not economically di¤erent when we modify age and mileage parameters (whatever the currency, the age and the mileage, the standard deviation does not exceed two Euros). Age and Mileage do not increase the volatility. Regarding average depreciation, German vehicles are highly correlated with mileage, but Spanish cars are not. Surprisingly, the graphical analysis of age impact on vehicles, reveals that British cars are heavily impacted by the level of usage (kilometer per month variable coe¢ cient) and as a consequence, 12 month age vehicles have a resale price equivalent to 24 age month vehicles. The last two points indicate that Hedonic valuations are signi…cantly di¤erent by country. European markets are not homogeneous, and residual value distributions are always singular. On a business perspective, leasing contracts would be impacted by country speci…cities.

This article is part of a general study on resale market hedging[START_REF] Prado | Hedging residual value risk using derivatives[END_REF]. We aim to estimate the distribution of the resale price in order to include the depreciation behavior in a derivative product.

In the resale market, there is an asymetry of information; the car owner has a better knowlegde of the probability of bad lemons. If Second hand vehicles were valued like as new vehicles, then it would attract lemons (lemons'sellers would have the opportunity to sale their vehicles and buy a new one on the new vehicle market) and it would create an arbitraging opportunity. Akerlof used the automotive market as a best illustration and extended his idea to other markets (the cost of deshonesty...).

[START_REF] Combris | Estimation of a Hedonic Price Equation for Bordeaux vine: Does Quality Matter ?[END_REF].

[START_REF] Keith | Hedonic Price Indexes for United Kingdom Cars[END_REF] and[START_REF] Dalen | Quality-corrected price indices: the case of the Dutch new passenger car market[END_REF].

[START_REF] Scott | The Valuation of Risks to Life: Evidence from the Market for Automobiles[END_REF].

Couton, Gardes, and Thepaut (1995).

[START_REF] Keith | Price, Quality and Advertising Competition: An Econometric Investigation of the United Kingdom Car Market[END_REF] and[START_REF] John | Quality Change and Pricing Behaviour in the United Kingdom Car Industry 1956-1968[END_REF].

In Europe, statistics on resale prices are not as abundant as in …nancial markets and leasing companies often have to use internal data to forecast the market value. External information is usually not available on line, costly and time consuming to collect. Morevover, there is a non homogeneous information and format by country. Therefore we use the internal resale information (GE resale data warehouse).

0 Similar attributes or characteristics could be shared by di¤erent goods. Usually goods have several characteristics, and a combination of goods may have attributes di¤erent to goods used separately.1 1 We could also add the technology obsolescence to the list. 1

See Appendix A1.

[START_REF] Scott | The Valuation of Risks to Life: Evidence from the Market for Automobiles[END_REF] Ekeland, Heckman and Nesheim (2004) reconsidered the identi…cation and estimation of the hedonic model. They show that most of empirical studies are based on arbitrary linearisation. Two new estimations procedures are proposed: a non parametric transformation method and instrumental variables in a general nonlinear setting.1 4 See Appendix A1.1 5 Bartick made an adjustment by group of cities.

6 See Appendix A2.1 7 See Appendix A3.

[START_REF] Timothy | The Estimation of Demand Parameters in Hedonic Price Models[END_REF] We limit our analysis to …xed contract with no purchase option and no rewrite, therefore age, mileage and sale dates are known with certainty.3 5 Great Britain has a sterling pound currency and very limited cross bordering sales with others european countries because of the right hand side weel of the car. Therefore, GB statistics add an original perspective of european markets analysis.

[START_REF] Court | Hedonic price indexes with automotive examples[END_REF] See models results in Appendix C. An indicator of automatic transmission was tested and statistically signi…cant for France. Because the coe¢ cient sign was negative, we removed it.

[START_REF] Couton | Hedonic Prices for Environmental and Safety Characteristics and the Akerlo¤ E¤ect in the French Car Market[END_REF] See Appendix D.3 8 See Appendix E.

APPENDIX

APPENDIX A: Methodological aspects.

A 1: In Rosen Hedonic framework, we can de…ne the marginal price on a characteristic level. For any vector of observed characteristics Z (of the car), the Hedonic marginal price for a characteristic z (i.e. fuel consumption) is an estimate of both the marginal bid for z of the household purchasing Z and the marginal o¤er for z of the …rm producing z. Linear version of these marginal bid and marginal o¤er function are de…ned through two equations: Estimated Hedonic marginal price of characteristics p zj (Z i ) = A 3: Akerman model estimates used car value.The price of an Automobile of a given age, K, can be expressed as the discounted present value of its remaining services: Using this framework we are now able to calculate a price according to the physical characteristics of a commodity.

In the case called the "variable packaging hypothesis" (or input price-dependant quality adjustment), for

Using this formula, in an empirical analysis, we could test the simple versus the variable repackaging hypothesis using classical hypothesis testing procedures. (i.e. fuel price on auto price).

A 6: Yerger(1995) applied Grilitch and Otha method to discuss an article written by Ho¤er and Pratt which was inspired by Akerman approach.

For a model i and at trend variable, time t ((t = 1; :::; 12), the price P it is LogP it = 0 + 1 t + By his model, Yerger tested and approved the market e¢ ciency in Automotive market.

A 7: Ordinary Least Square allows prediction interval calculation 39 .

The Ordinary Least Square Method estimate a and b by minimizing the sum of squared error

An unbiased estimate s 2 of e 2 is given by mean squared error: Age is number of month between the registration and the sale date.

Dis is the distance travelled, including any distance done on an odometer that has been changed.

Kpm is the distance travelled per month.

Lp 2 is the cubic of the real least price (including option price).

APPENDIX C: Regression results:

France Results

Germany Results

APPENDIX E: Graphical Analysis:

Ford Focus age impact: France Germany Audi A4 age impact:

France Germany


Rihab Bedoui
email: rihab.bedoui@u-paris10.fr.

† Makram

Ben Dbabis

Copulas and bivariate risk measures : an application to hedge funds

Keywords: JEL classification : C13, C14, C15, G23 Hedge fund strategies, share index, dependence, copula, tail dependence, bivariate Value at Risk

With hedge funds, managers develop risk management models that mainly aim to play on the effect of decorrelation. In order to achieve this goal , companies use the correlation coefficient as an indicator for measuring dependencies existing between (i) the various hedge funds strategies and share index returns and (ii) hedge funds strategies against each other. Otherwise, copulas are a statistic tool to model the dependence in a realistic and less restrictive way, taking better account of the stylized facts in finance. This paper is a practical implementation of the copulas theory to model dependence between different hedge fund strategies and share index returns and between these strategies in relation to each other on a "normal" period and a period during which the market trend is downward. Our approach based on copulas allows us to determine the bivariate VaR level curves and to study extremal dependence between hedge funds strategies and share index returns through the use of some tail dependence measures which can be made into useful portfolio management tools.

Introduction

Academic research began to focus on sources of returns and risk of hedge funds in 1997 with the pioneering work of Fung and Hsieh (1997). Brown, Goetzmann and Ibbotson (1999) studied a sample of offshore hedge funds between 1989 and 1995 and found a positive risk adjusted. Their results did not highlight neither the effect of managers talent, nor the persistence of the performance of some others. Other studies have focused on the performance of hedge funds without taking into account the different factors, styles and characteristics associated with abnormal returns. [START_REF] Capocci | Introduction aux hedge funds[END_REF] argued that many hedge fund strategies are designed to be weakly correlated with share index returns and bond index returns and he also studied the correlation between various hedge fund strategies. He showed that the correlation between hedge fund strategies and share and bond indexes varies greatly from one strategy to another and between the various strategies. The high yield index is more strongly correlated with most indexes. To investigate the role of the period under study, [START_REF] Capocci | Introduction aux hedge funds[END_REF] studied the correlations on the sub-period January 2000-December 2002 (bear market) and he noticed that correlations generally tend to decrease comparing to the "normal" period even if this is not true for all indexes ( for instance short selling or emerging market, see Capocci and Hübner (2003) and Capocci and Mahieu (2003)). [START_REF] Capocci | Introduction aux hedge funds[END_REF] found that the short selling index is negatively correlated with all other strategies when it was not with the share index. Moreover, the correlations between individual strategies remain relatively the same for the two periods. All strategies (without considering the short selling strategy) are positively correlated with the share index but the mortgage backed indexes, equity and multi-market strategy are only very weakly correlated as well as these strategies have little or no correlation with the bond index and the raw material index. However, this dependence coefficient which is often used by practitioners has several disadvantages. Indeed, the correlation coefficient is not defined if the second order moments of random variables are not finite. Besides, a correlation coefficient of zero does not necessarily imply independence between the variables studied. In addition, we can measure only linear correlation using this coefficient. Furthermore, the linear correlation coefficient is not invariant by a continuous and increasing function ( such as the logarithm function, Embrechts et al (1999)) and it does not take into account the dependence of extremal values.

The study of comovements between hedge fund strategies on the one hand and comovements between them and the share index on the other hand, has been made using the correlation coefficient. This paper proposes to use a more appropriate methodology based on copulas theory. Our aim is firstly to model the structure of dependence between the returns of different hedge fund strategies and secondly between returns of each strategy and the market on two periods: a "normal" period and a period representing the occurrence of a rare and extreme event (when the market trend is downward) as well as modelling tail dependence between these variables through the use of extremal dependence coefficients. Moreover, we are interested in determining the level curves of the bivariate Value at Risk (VaR) between (i) different hedge funds strategies and the marginal rate of substitution (MRS) between the VaR of two hedge funds strategies and (ii) the VaR of a particular hedge funds strategy and the share index for a given risk level by using copulas theory. Hence, our methodology enables us specifically to take into account the extreme values and to study the impact of dependence on various measures of risk without making restrictive assumptions about the linearity and monotonocity of these series.

Copulas functions are a statistical tool which has many advantages. First, copulas make it possible to determine the nature of dependence of the series, be it linear or not, monotone or not. In addition to the fact that they offer a great flexibility in the implementation of the multivariate analysis, copulas authorize a wider selection of the marginal distributions of the financial series. Second, they allow a less banal representation of the statistical dependence in finance based on the traditional correlation measure ( see Embrechts et al. (1999)). Third, they authorize less restrictive univariate probability distributions which make it possible to better accounting for the stylized facts in finance (leptokurticity, asymmetry, tail dependence). Fourth, they consider very general multivariate distributions, independently of the laws of the marginal ones which can have different laws and be unspecified. Furthermore, the copulas approach enables us to ease the implementation of multivariate models. Indeed, this approach allows the decomposition of the multidimensional law into its univariate marginal functions and a dependence function that would make possible extensions of some results obtained in the univariate case to the multivariate case. Hence, copula is an exhaustive statistic of the dependence.

The structure of the paper is as follows. Section 2 describes the concept of copulas and their basic properties. Section 3 presents the empirical aspect of our work. We present the hedge funds data and highlight a certain number of dependencies between the share index and the various hedge funds strategies, as well as dependences existing between the various strategies. We identify the copulas that model these dependences over two different periods: one "normal" period and a period during which the market trend is downward. Moreover, this modeling of the dependences will enable us to determine the level curves of the bivariate VaR between the hedge funds strategies. Lastly, we evaluate tail dependences between the market index, the hedge funds index and the strategy dedicated shorts through the use of various extremal dependence coefficients. Section 4 is devoted to the conclusion.

Dependent Models : a copula approach

A copula is a multidimensional uniform distribution. It is a relatively old statistical tool introduced by [START_REF] Sklar | Fonctions de répartitionà n dimensions et leurs marges[END_REF], brought up to date by Genest et Mackay (1986). As defined by Nelsen (1998), "Copulas are functions that join or couple multivariate functions to their one dimensional margins. Copulas are distribution functions whose one dimensional margins are uniform". In what follows and with reference to the work of Nelsen (1998), Genest & MacKay (1986), Denuit & Charpentier (2004),we attempt to give a more precise definition to copulas and to present some main properties.

Definition of a bivariate copula

A bivariate copula C is 2increasing function on the unit 2-cube I 2 = [0, 1] × [0, 1]. Formally, it is a function C : I 2 → I with the following properties: :

• For all u, v ∈ I, C (u, 0) = 0 = C (0, v) and C (u, 1) = u, C (1, v) = v • For all u 1 , u 2 , v 1 and v 2 ∈ I such as u 1 ≤ u 2 and v 1 ≤ v 2 : C (u 2 , v 2 ) -C (u 1 , v 2 ) -C (u 2 , v 1 ) + C (u 1 , v 1 ) ≥ 0 Thus: ∆ u 2 u 1 ∆ v 2 v 1 C x, y ≥ 0

Basic properties of Copula function 2.2.1 Theorem of Sklar

Let H be a bivariate cumulative density function with univariate marginal cumulative density functions F and G. There exists a copula C such that for all x, y ∈ R :

H x, y = C F (x) , G y
If F are G continuous, then C is unique. This theorem shows that any bivariate cumulative density function H can be written in the form of a copula function. Therefore, it is possible to construct a wide range of multivariate distributions by choosing the marginal distributions and an apropriate copula.

Property of Invariance

Let X 1 and X 2 be two continuous random variables with margins F 1 et F 2 , linked by a copula C. Let h 1 and h 2 two strictly increasing functions,then :

C (F 1 (h 1 (x 1 )) , F 2 (h 2 (x 2 ))) = C (F 1 (x 1 ) , F 2 (x 2 ))

Theorem 1 (Fréchet bounds)

Let C be a bivariate copula, then for all (u, v) ∈ Dom C :

max (u + v -1, 0) ≤ C (u, v) ≤ min (u, v)
Fréchet (1951) shows that there exists upper and lower bounds for a copula function. In two dimensions, both of the Fréchet bounds are copulas themselves, but as soon as the dimension increases, the Fréchet lower bound is no longer n-increasing function.

Theorem 2

Let C be a copula, for all u 1 , u 2 , v 1 , v 2 ∈ Dom C :

| C (u 2 , v 2 ) -C (u 1 , v 1 ) |≤| u 1 -v 2 | + | u 1 -v 1 |
for all a ∈ I : ♣ t → C (t, a) is non -decreasing: Horizontal section of C in a ♣ t → C (a, t) is non-decreasing : Vertical section of C in a

hedge funds and study of dependences

This section is devoted to the empirical part of our work. After presenting the data, we seek to highlight a number of dependencies between the share index and the hedge fund index on the one hand, and the different hedge funds strategies on the other hand. We select suitable copulas representing these dependencies over two different periods: a "normal" period and a period during which the market trend is downward. Besides, the copulas approach enables us not only to determine the level curves of the bivariate VaR between the hedge funds strategies but also to evaluate tail dependences between the market index, the hedge funds index and the strategy dedicated short via extremal dependence coeficients.

Data

The period under study extends from January 1994 to December 2006. Regarding the data of alternative assets, we use the historical returns of the 13 monthly indexes divided into 2 categories: an overall index of hedge funds for all strategies, and a set of indexes representing the 12 strategies of hedge funds that make up the database CSFB / TREMONT.

According to Fung and Hsieh (2000), these indexes are less affected by the survivor bias than the individual funds data. Unlike other indexes, CSFB / Tremont indexes take into account the return of the committee weighted by the size of funds in the basket of funds. To characterize the share market, we use the international index share MSCI World whose returns are taken from the TASS database.

Modeling dependence over the entire period

We carry out the study of the dependence structure between the hedge fund indexes and share index. We consider two periods, a "normal" period and a period during which the market trend is downward, in order to analyse the deformation of the structure of dependence. A goodness of fit test allows us to validate the choice of copula selected. The search of dependencies focuses on both hedge funds strategies and share index as well as on hedge fund strategies in relation to each other. Through the application of various tests of adjustment, it appears that the normal distribution provides the best adjustment for the considered variables. Figure 1 shows the adjustment of the distribution of share and hedge funds indexes to a Gaussian distribution using two graphic approaches : respectively hitograms'goodness of fit and Quantile-Quantile Plot (QQPlot). We Figure 1: Adjustement of the share index distribution and the hedge funds distribution to a Gaussian distribution remark more "picked" central values and heavier left tails of share and hedge funds indexes distributions than the Gaussian density function. In fact, the empirical distribution presents rare observations with a decrease slower than the exponential decrease of the normal distribution. However the use of the Jarque-Bera test of the null hypothesis that the sample in the empirical vector comes from a normal distribution with unknown mean and variance, against the alternative that it does not come from a normal distribution, leads us to assume a normal distribution for the share and hedge funds indexes.

Graphical analysis

The choice of the suitable copula is the first difficulty in the implementation of modeling dependence.

Figures 2, 3 and 4 make it possible to apprehend the form of the dependences which exist on the one hand between the share index, the hedge funds index and the hedge funds strategies and the structure of dependence between the different strategies hedge funds on the other hand.

In addition to the sign and the intensity of the dependences, these graphs provide us a first indication on the tail dependence which we will treat thereafter. The set of points is very close to the second bisector and the upper tail of the distribution is highly concentrated. The largest values in the upper tail for one variable coincide with large values of the same sign for the other variable. We note that the dependence between the share index and the dedicated short index is negative. This is not surprising since this strategy aims to maintain a short position on assets. Figure 4 shows that there exists positive dependences between share index and the Equity Market strategy, the Multi-strategy and the Emerging Market strategy as well as a negative dependence between the Dedicated Short Bias strategy and the Equity Market Strategy.

Relying on this graphical approach (Figures 2, 3 and 4) as a first step, the normal copula seems to be the most suitable copula that models the form of dependence between the share index and hedge fund index and between hedge fund strategies in relation to each other. We will use different goodness-of-fit tests to choose the adequate copula to our variables.

Estimation of copulas parameters

The Normal copula is defined as follows. Let Φ -1 be the inverse fonction of a standard normal distribution and H α the distribution function of the bivariate normal distribution with correlation coefficient α.

The normal copula C N α is : C N α (u, v) = H α Φ -1 (u) , Φ -1 (v) f or u, v ∈ [0, 1] , α ∈ [-1, 1]
. There is no explicit form for the distribution function of the bivariate normal law.

To estimate the parameters of copulas which model the dependencies between these variables, we use the parametric method Inference Functions for Margins (IFM). This method was introduced by Shih & Louis (1995). It can reduce the estimation problem in two steps : -The estimation of the parameters θ 1 , . . . , θ n of the margins.

-The estimation of the parameter θ C of the copula.

Let us denote :

θ = (θ 1 , . . . , θ n , θ C )
We begin by determining the maximum likelihood estimators of margins:

θ i = argmax θ i n k=1 f i x k i , θ i
We then introduce these estimators in the copula part of the log-likelihood function, which leads to:

θ C = argmax θ C n k=1 ln c F 1 x k 1 , θ 1 , . . . , F 1 x k n , θ n , θ C
Other procedures can be applied, such as non-parametric estimation of margins followed by maximum likelihood estimation for the parameter of the copula. We refer the readers to Genest et al. (1995) and Shih and Louis (1995) procedures. There are other non-parametric estimation methods such as empirical copula (see Deheuvels (1979)) and parametric estimation methods like the moments method, the maximum likelihood method and the CML Canonical Maximum Likelihood (see Bouye and al. (2000)).

The IFM method we use has the advantage of being based on calculations less cumbersome than maximum likelihood procedure. Nevertheless, the determination of information matrix of Godambe can be very complicated because it creates multiple derived calculations. For the same reasons as for the maximum likelihood method, a possible error in margins estimation in this method may lead to erroneous estimator of the copula parameter.

Tables 1 to 3 include the values of the estimated parameters of copulas. We note a positive dependence between the share index and the hedge funds index on the one hand and the share index with the equity market strategy and multi-strategy on the other hand. However, the strength of this dependence differs from one strategy to another. There is a negative dependence between the share index and dedicated short strategy. The upper tail of the distribution is very concentrated. Our results also show a negative dependence between dedicated short strategy and the other hedge funds strategies and a positive dependence between the equity market strategy and multi-strategy. This means that returns of share index coincide with the returns of the same sign for the hedge funds index, the equity market strategy and multi-strategy. Therefore losses and gains in these variables occur simultaneously. By contrast, returns of the dedicated short strategy coincide with the returns of the opposite sign for the share index, hedge funds index, the equity market strategy and multi-strategy. Thus, losses of the dedicated short strategy occur simultaneously with gains of the share index, hedge funds index, the equity market strategy and multi-strategy. These observations offer useful portfolio management tools.

Goodness of fit tests

Choosing the suitable copulas that model the dependencies between our random variables is of paramount importance. The question is the following : what is the best structure of dependence that can be adapted to the phenomenon studied? Goodness of fit tests for copulas are relatively recent. There are few papers on this issue, but the field is in constant development.

Let consider a sample of random vectors iid X i ∈ R d as X i = X 1,i , . . . , X d,i ∼ X iid. Let H be the distribution function of X and C the copula of X, thus :

H (x 1 , . . . , x d ) = C (F 1 (x 1 ) , . . . , F d (x d ))
Generally, a statistic test distinguishes between two assumptions :

Null hypothesis H = H 0 or C = C 0 Alternative hypothesis H H 0 or C C 0 A first solution is to compare the empirical copula defined as :

C k 1 n , . . . , k p n = 1 n card i | R 1,i ≤ k 1 , . . . , R p,i k p
to the estimated parametric copula C, where R j,i , j = 1, . . . , p & i = 1, . . . , n denotes the rank of X j,i among the observations X j,1 , . . . , X j,n . Dependency accepted is one that ensures that C is as close as possible to C.

Let { x k , y k } n
k=1 be a sample of size n of pairs of random variables. The empirical copula the function C n defined as :1

C n i n , j n = Number of pairs x, y in the sample as x ≤ x i , y ≤ y j n
The empirical density function of the copula C is given by:

c n i n , j n =          1 n if x i
, y j is an element of the sample.

otherwise

The link between C n and c n is defined as follows :2 :

C n i n , j n = i p=1 j q=1 c n p n , q n c n i n , j n = C n i n , j n -C n i -1 n , j n -C n i n , j -1 n + C n i -1 n , j -1 n
The empirical copulas are useful to provide non-parametric estimators of measures of dependence such as the ρ of Spearman, and the τ of Kendall. Indeed, these two measures are determined empirically as follows:

ρ = 12 n 2 -1 n i=1 n j=1 C n i n , j n - i n - j n τ = 2n n -1 i-1 p=1 j-1 q=1 c n i n , j n c n p n , q n -c n i n , q n c n p n , j n
Figure 5 presents the empirical copula that models the dependence between the hedge fund index (IHF). It also reports the parametric copula allowing the adjustment of the empirical copula to the normal copula of parameter alpha = 0.5516 estimated by the IFM method. We note a good fit between the empirical copula and the normal one. This validates the choice of the normal copula to model dependence between share and hedge fund indexes.

Figure 5: Empirical copula versus parametric copula for the hedge funds index (IHF) and the share index Figure 6 shows the density of the copula that models dependence between the hedge funds index and the share index. There is a strong concentration in the upper tail of the distribution, and to a lesser extent in the lower tail distribution which indicates that positive largest values from the share index and hedge funds index occur together . By contrast, the probability of the simultanous occurence of negative largest values is smaller than the probability of the positive ones.

In addition to these graphical methods, we use the goodness of fit test of Genest, Quessy and Remillard (2006) to validate the choice of copula selected. Genest, Quessy & Rémillard (2006) have expanded the work of [START_REF] Wang | Model selection and semiparametric inference for bivariate failuretime data[END_REF] providing alternative statistics given by :

S n = 1 0 | K n (t) | 2 k (θ n , t) dt where k (θ n , t) is thedensity f unction associated to K (θ, t) and T n = sup 0≤t≤1 | K n (t) |
S n is based on the distance of Cramèr Von Mises, while the statistic T n is based on that of Kolmogorov Smirnov. It is important to note that these statistics have several advantages:

-Some simple formulas are available for S n and T n in terms of the rank of obsevations, which -The selection procedures are not influenced by an external constant whose selection and influence on the limit distribution of the test statistic were not considered by [START_REF] Wang | Model selection and semiparametric inference for bivariate failuretime data[END_REF].

-S n and T n distributions can be determined not only for archimedian bivariate copulas but also for copulas with dimension greater than 2 and for copulas satisfying the general condition of regularity.

-The parametric bootstrap method is valid and can be used to approximate the P-values associated with functional K n and in particular with S n and T n .

Performing a parametric bootstrap for statistics S n and T n , it is possible to obtain approximate thresholds associated with these statistical assumptions of type C ∈ (C θ ), ie : the copula C associated to F belongs to the family of copulas (C θ ). In the paper of [START_REF] Genest | Goodness-Of-Fit Procedures For Copula Models Based On The Probability Integral Transformation[END_REF], one confirms the validity of this bootstrap approach for both types of the most common goodness of fit tests : (i) tests where we compare the distance between a multivariate empirical distribution and parametric estimation under the null hypothesis or (ii) those where we compare the distance between the empirical estimators and parametric univariate pseudoobservations such as W i , obtained through the integral transformation of probability. [START_REF] Genest | Goodness-Of-Fit Procedures For Copula Models Based On The Probability Integral Transformation[END_REF] showed that :

S n = n 3 + n n-1 i=1 K 2 n j n K θ n , j + 1 n -K θ n , j n -n n-1 j=1 K n j n K 2 θ n , j + 1 n -K 2 θ n , j n
and

T n = √ n max i=0,1 & 0≤ j≤n-1 | K n j n -K θ n , j + 1 n |
The test concludes to the rejection of the null hypothesis H 0 : C = C 0 when the observed values S n and T n are above the quantile of order 1 -α of their distributions under the assumption H 0 .

After applying this test, the normal copula is the suitable copula modeling the dependence between the share index and the different hedge funds strategies and between these strategies in relation to each other.

Note that there exists other goodness of fit tests : the parametric bootstrap method in the approach of [START_REF] Wang | Model selection and semiparametric inference for bivariate failuretime data[END_REF], the parametric bootstrap method based on C θ n ( see [START_REF] Genest | Parametric bootstrap and pseudoobservations[END_REF]), the goodness of fit test of copulas based on the transformation of Rosemblatt (old fashion) (see [START_REF] Breymann | Dependence structures for multivariate highfrequency data in finance[END_REF]) and the goodness of fit test of copulas based on the transformation of Rosemblatt (new fashion) (see [START_REF] Klugman | Fitting bivariate loss distributions with copulas[END_REF], [START_REF] Ghoudi | Empirical processes based on pseudo-observations, II. The multivariate case[END_REF], [START_REF] Genest | Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models[END_REF].

Modeling dependence on the subperiod January 2000-December 2002

In order to try to answer the question of the dependence structure variation when the market trend is downward, a first possibility is to estimate the parameters of bivariate copulas existing between the share index, hedge funds index and hedge funds strategies via the Inference Functions for Margins method.

As previously, we find that the normal copula is the suitable copula that models the dependence structure between the share index, hedge funds strategies and hedge fund index. Moreover, this modeling of dependence provides extra information on the evolution of the degree of dependence in relation with the market trends. Tables 4, 5 and These results show that the dependence parameter has increased for the normal copulas that model (i) dependence between the share index and the strategies dedicated short , multi strategy and emerging market and (ii) the dependencies between the hedge funds strategies. In other words, more the copula parameter is high, the greater the dependency is. This means that the share index, the dedicated short strategy and emerging market strategy have become more dependent when the market trend is downward.

However, during the "normal" period, the dependence between the share index and the equity market strategy was positive but the sign of dependence has becoming negative, when an extreme event had occurred ( September 11, 2001). It is the same for the sign of dependence between the equity market strategy and the multi-strategy, which indicates that losses and, respectively, gains of the share index have a higher probability to coincide with gains and, respectively, losses of the dedicated short strategy and emeregent market strategy when the market trend is downward than in the "normal" period. While in the "normal" period, the returns of the equity market strategy coincide with returns of the same sign for the share index and the multi-strategy, when the market trend is downward the returns of the equity market strategy occur simultanously with returns of the opposite sign for the share index and the multi-strategy.

Copulas and bivariate VaR

Value at Risk (VaR) has become the standard measure that financial analysts use to asses market risk. The VaR is defined as the maximum potential loss due to adverse market movements for a given probability.

The use of copulas allows us to determine the level curves of the bivariate VaR and examine for a given threshold level, the marginal rate of substitution between the VaR of two univariate risks .

Indeed, since we have marginal distributions of returns of different hedge fund strategies, and the share index, it is possible to trace the level curves corresponding to the minimum copula (antimonotonicity), maximum copula (comonotonicity) and the independence copula. Let r A and r B be the returns of the series A and B. Let F A and F B be the univariate distribution functions of returns of respectively r A and r B . We have, for any threshold α ∈ [0, 1] :

(r A , r B ) ; max (F A (r A ) + F B (r B ) -1, 0) = α} , anti -monotonicity; {(r A , r B ) ; F A (r A ) .F B (r B ) = α} , independence; {(r A , r B ) ; min (F A (r A ) , F B (r B )) = α} , comonotonicity.
The level curves from the empirical copula are given by :

{(r A , r B ) ; C (F A (r A ) , F B (r B )) = α}
The level curves are used to determine the marginal rate of substitution between the two univariate VaR. The more the empirical curve is high approaching the case anti-monotonicity, the more is the dependence between the returns A et B and the more is the compensation effect. However, the more the curves are close to their lower limit, corresponding to the case of comonotonicity or positive dependence, the more the returns tend to move in the same direction, the dependence between losses (correlation) is therefore very high. Regarding the curves of multiplication, they correspond to diversification (see Cherubini and Luciano (2001)).

We determine the 95% level curves of the bivariate VaR between the share index and the hedge funds index ( Figure 7) and the level curves of bivariate VaR of different hedge funds strategies ( Figures 8 and9).

We note from the Figure 7 that the 95% level curve of the empirical copula is closer to Figure 7: Bivariate VaR of share index and hedge funds index that corresponding to the case co-monotony (the lower limit) or positive dependence. It follows that the returns of the share index and the hedge funds index operate in the same direction,"the correlation"between the losses is therefore high. As a consequence, it is preferably not to put these two elements into a single portfolio.

We remark according to the Figure 8 that the 95% level curve of the empirical copula is closer to the level curve of multiplication or independence which is the case of diversification: the losses of the Dedicated short strategy and the Equity Market Neutral strategy are not correlated. In order to guarantee diversification, it is best to put these two strategies in the same portfolio.

Finally, we note from Figure 9 that the 95% level curve of the empirical copula is high and closer to that of the anti-monotonicity case. Consequently, the dependence between the returns of the Dedicated Short strategy and the Long Short Equity strategy is negative and the effect of adjustment and compensation will take place. To ensure a better allocation in its portfolio, it is advisable to combine these two strategies in the same portfolio of funds.

We note that the maximum potential loss, calculated through the risk measure VaR is higher in the dependence case compared to the case of independence. This means that assuming the non-correlation between hedge fund strategies and share index underestimate the portfolio risk measured by VaR.

Tail dependence : extremal coefficients χ and χ

The study of the tail dependence allows us to describe the dependence in the tails of distribution and to examine the simultaneous occurrence of extreme values. We use two coefficients of tail dependence enabling us to measure the asymptotic dependence between hedge funds strategies and the share index.

The dependence measures χ and χ were introduced by Coles et al. (1999). After transformation of (X,Y)which are the two series to study into (U, V) having uniform marginal distributions, we get :

Pr(V > u U > u) = 2 - 1 -Pr (U < u, V < u) 1 -Pr (U < u) ≈ 2 - logC (u, u) log (u) f or 0 ≤ u ≤ 1.
The dependence measure χ (u) is defined as

χ (u) = 2 - logC (u, u) log (u) f or 0 ≤ u ≤ 1.
The function χ (u) is thus a quantile-dependent measure of dependence. The sign of χ (u) determines whether the variables are positively or negatively associated to the quantile level u.

χ (u) is bounded as follows :

2 - log (2u -1) log (u) ≤ χ (u) ≤ 1
The lower bound is interpreted as -∞ for u ≤ 1/2, and 0 for u = 1.

A single parameter measure of extremal dependence is given by

χ = lim u→1 χ (u)
Loosely stated, χ is the probability of one variable being extreme given that the other is extreme.

-In the case χ = 0, the variables are asymptotically independent. Thus, we require a complementary dependence measure to assess extremal dependence within the class of asymptotically independent variables. By analogy with the definition of χ (u), comparaison of joint and marginal survivor functions of (U, V) leads to :

χ (u) = 2logPr (U > u) logPr (U > u, V > u) -1 = 2log (1 -u) logC (u, u) -1 f or 0 ≤ u ≤ 1,
where -1 < χ (u) < 1 for all 0 ≤ u ≤ 1.

To focus on extremal characteristics, we also define :

χ = lim u→1 χ (u)
The measures χ and χ are related to the Ledford & Tawn (1996Tawn ( , 1998) characterisation of the joint tail behavior, through η, the tail dependence coefficient and L (t), the relative strength of limiting dependence : χ = 2η -1 χ = c i f χ=1 and L(t)→c>0 as t→∞, 0 i f χ=1 and L(t)→0 as t→∞, and i f χ<1.

We have the following classification : -χ ∈ [0, 1]: the set (0, 1] corresponds to asymptotic dependence; -χ ∈ [-1, 1] ; the set [-1; 1) corresponds to asymptotic independence. Thus the complete pair (χ, χ) is required as a summary of extremal dependence: -(χ > 0; χ = 1) implies asymptotic dependence, in which case the values of χ and χ determine the strength of dependence within the class;

-(χ = 0; χ < 1) implies asymptotic independence.

In practice, we first assess χ : -χ < 1 ⇒ asymptotic independence ; -χ = 1 ⇒ asymptotic dependence, here we must also estimate χ.

We calculate the extremal dependence coefficient χ in order to examine the simultaneous occurrence of extreme values and to measure the asymptotic dependence of the minimum and the maximum between the share index and the hedge funds index as well as the asymptotic dependence between the share index and the strategy Dedicated Short. The results which are grouped in Table 7, lead us to conclude that the value of χ is lower than 1 for the share index and the hedge funds index and for the share index and strategy Dedicated Short. Therefore, there is an asymptotic independence between the share index and the hedge funds index on the one hand and between the share index and strategy Dedicated Short on the other hand. Loosely speaking, knowing extreme loss for the share index, there is zero probability that loss with a comparable intensity to take place concurrently for the hedge fund index and the Dedicated Short strategy.

Conclusion

There is a regular expansion of the hedge funds debate towards the financial institutions. Faced with the ever-changing environment, managers of hedge funds are expected to adapt by modeling the structure of dependence between different hedge fund strategies in relation to each other and in relation to the stock market.

This paper provides elements and techniques on this debate by relying on the theory of copulas. We have proposed a study of dependence and tail dependence of the different hedge funds strategies and the share index on one hand and of the different hedge funds strategies in relation to each other on the other hand. We also assess the risk of different hedge funds strategies in various cases of dependence through the use of bivariate Value at Risk.

Contrary to what the funds managers preconize, our results highlighted a certain number of dependences between the share index and some hedge funds strategies and between these various strategies themselves. The sign and the intensity of dependence increase for most hedge funds strategies in times of crisis . The share index, the dedicated short strategy and the emerging market strategy are more dependent when the market is bear. For some other strategies, the sign of dependence changes from a positive into a negative value in a crisis period. This is the case for the equity market strategy and the share index on the one hand, and the equity market strategy and multi-strategy on the other. A note worthy finding is that taking into account the dependencies between the different hedge funds strategies via the copula theory has a significant impact on the risk measures such as VaR. Furthermore, the determination of the level curves of the bivariate VaR and study of the simultaneous occurrence of extreme values using the coefficients of tail dependence could provide to alternative fund managers a more precise estimation of risk and a better allocation of their portfolio in both the absence and presence of extreme movements.

Figure 2 :

 2 Figure 2: Scatter plot of the hedge fund index returns and the share index returns

Figure 3 :

 3 Figure 3: Scatter plot of the Dedicated Short strategy and the share index

Figure 4 :

 4 Figure 4: Scatter plot of the share index, the Dedicated Short strategy and other hedge funds strategies

Figure 6 :

 6 Figure 6: Density function of the copula hedge funds index (IHF)-share index

  [START_REF] Costinot | Revisiting the dependence between financial markets with copulas, Groupe de Recherche Opérationnelle[END_REF] show the different values of the estimated copulas parameters in a crisis period (January 2000-December 2002).

Figure 8 :

 8 Figure 8: Bivariate VaR of Dedicated short strategy and Equity Market Neutral strategy

Figure 9 :

 9 Figure 9: Bivariate VaR of Dedicated Short strategy and the Long Short Equity strategy

Table 1 : Estimated copulas parameters of the share index and hedge funds strategies

 1

		The share index
	hedge funds index	0.5516
	Dedicated short strategy	-0.7421
	Equity market strategy	0.3161
	Muti Strategy	0.2560
	emerging markets strategy 0.5781

Table 2 :Estimated copulas parameters of the dedicated short strategy and the other hedge funds strategies

 2

		Dedicated short strategy
	Equity market strategy	-0.2976
	Muti Strategy	-0.111
	emerging markets strategy -0.5522

Table 3 : Estimated copulas parameters of hedge funds strategies

 3

		Muti Strategy
	Dedicated short strategy -0.5522
	Equity market strategy	0.3170

Table 4 : Estimated copulas parameters of the share index and hedge funds strategies in a crisis period

 4

		Share index
	Hedge fund index	0.4010
	Dedicated short strategy	-0.8819
	Equity market strategy	-0.1517
	Muti Strategy	0.5166
	emerging markets strategy 0.7949

Table 5 : Estimated copulas parameter of the dedicated short strategy and the emerging market strategy in a crisis period

 5

	Dedicated short strategy
	emerging markets strategy -0.80

Table 6 : Estimated copulas parameter of the equity market strategy and the multi strategy in a crisis period

 6

	Equity market strategy
	Multi Strategy -0.3571

Table 7 :

 7 Measure χ

		Share index
	hedge funds index	-0.015
	dedicated short Strategy -0.0568

The empirical copulas were originally introduced by (Deheuvels (1979)).

To demonstrate this proposition, see Nelsen (1998).