Sylvain Prado

Hedging residual value risk using derivatives

Keywords: residual value risk, credit risk, credit derivatives, factor modeling, copula. JEL Classi…cation: C10, G13

In the leasing industry the lessor faces a risk, at the end of the contract, in not recovering su¢ cient capital value from resale of the asset. We propose a model to hedge residual value risk using the Gaussian copula methodology. After discussing residual value risk and credit risk modelization, a new derivative product is introduced and analyzed; the Collateralized Residual Values (CRV). The model is applied to an European auto lease portfolio of operating lease contracts pertaining to a major company. Our results indicate that the …nancial product is easy to customize, and to implement through the contract characteristics and the level of correlation.

 [START_REF] Andersen | All your hedges in one basket[END_REF]Senior Asset Analyst at GE Capital Solutions Europe in London, UK. PhD Candidate at the University of Paris. Member of 'EconomiX'group in Paris, France. T +44 (0) 20 8754 2256 F +44 (0) 20 8754 2288 E sylvain.prado@ge.com sylvain.prado@gmail.com Meridian Trinity Square 23-59 Staines Road Hounslow, Middlesex TW3 3HF, UK GE Capital Solutions

Introduction

A lease is a contract in which one party transfers the use of an asset to another party for a speci…c period of time, at a predetermined rate. Leasing equipment is an important means of …nancing, and consequently represents a signi…cant part in many …nancial institutional portfolio. In 2006, leasing represented more than one-sixth of the world's annual equipment …nancing requirement [START_REF] Bluhm | An Introduction to Credit Risk Modeling[END_REF] . The value of the entire Global leasing market was estimated to be more than $633 billion [START_REF] Cherubini | Copula Methods in Finance[END_REF] . Academic results suggest that "leasing allows small …rms to …nance their growth, and/or survival while for large …rms, leasing appears to be a …nancial instrument used by sophisticated …nancial managers to minimize the after-tax cost of their capital" [START_REF] Copeland | A note on the evaluation of cancellable operating leases[END_REF] .

In the leasing industry, residual values are the forecasted prices of equipment in the second hand market.

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 money 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.

In the European auto lease market, most leases are closed-end leases: leasing companies assume the residual value risk. In 2001, car resale's price fell dramatically, as a result; US leasing companies su¤ered large losses, and some even dropped out of the business [START_REF] Cousseran | Le marché des CDO Modalités de fonctionnement et implications en termes de stabilité …nancière[END_REF] . Residual value risk is a key element in the leasing industry, however; there is minimal literature, and few developed models. The few studies were developed in three main areas; Operational Purpose [START_REF]DEMETRA 2.1 documentation and freeware available form circa[END_REF] ; The operational perspective aims to set the most accurate residual value; Basel 2 requirements [START_REF] Gibson | Understanding the risk of synthetic CDOs[END_REF] ; de…nes how to calculate reserves. Studies evaluate Basel 2 accuracy, and reserve calculation in relation to speci…c credit risk in the leasing industry, and Leasing Contract Valuation [START_REF] Gordon | Retreat is sounded as those leasing residual losses mount[END_REF] ; in the valuation analysis, the residual value risk is included through an American option. It allows a comparison of leasing (…nancial lease and operating lease) v.s. purchase decision. Unfortunately, it does not aim to hedge the speci…c Residual value risk, let-alone the correlation issue in a portfolio of equipment.

A lack of development on …nancial products hedging residual value risk, sent my research to credit risk. The recent important developments in …nance modelling and in new …nancial products were in credit derivatives. This implies a change in credit management involving banks and other …nancial institutions. A credit derivative is a contract between two parties that allows the use of a derivative instrument to transfer credit risk from one party to another. The risk seller, in form, has to pay a fee to the risk buyer who will take the risk. Over the last ten years, the credit derivative market has faced a substantial increase. A lot of credit risk models have been developed, therefore; increasing investor interest [START_REF] Grenadier | Valuing lease contracts A real-options approach[END_REF] .

In 1999, Li's Gaussian copula model 10 facilitated a dramatic success of this derivative sector. He proposed a fairly easy, and intuitive model depicting the payment default of a company like the survival probability of a human life [START_REF] Hull | Options, Futures, and Other Derivatives[END_REF] . It was also a new tool to evaluate the ongoing issue of credit risk; i.e. correlation. For instance, in a basket of loans there is an individual risk component. Each loan has a risk to default its payment. The systemic risk is the other component. An economic downturn could also impact the whole portfolio, and the systemic risk implies correlation.

Collateralized Debt Obligation (CDO) turns the correlation problem into a solution. It is a credit [START_REF] Gibson | Understanding the risk of synthetic CDOs[END_REF] Schmit produced several articles on Credit risk in leasing industry to analyse Basel 2 requirements accuracy. See Schmit (2003), Schmit (2004), Irotte, Schmit, and Vaessen (2004), Laurent and Schmit (2007). [START_REF] Gordon | Retreat is sounded as those leasing residual losses mount[END_REF] T. Copeland and J. Weston (1982) apply an American put with a decreasing exercise price and S. E Miller (1995) includes an American Call Option in a net present value formula to estimate the internal rate of return of the deal. S.R Grenadier (1995), focusing on the real estate arena, adds a residual value insurance that is equivalent to a put option on the underlying asset in the pricing of a varriety of leasing contracts. [START_REF] Grenadier | Valuing lease contracts A real-options approach[END_REF] "At the risky end of …nance" The economist (April 21st 2007) gives an up to date on the credit derivatives market: "According to the Bank for International Settlements, the nominal amount of credit-default swaps had reached $20 trillion by June last year. With volumes almost doubling every year since 2000, some reckon the CDS market will soon be worth more than $30 trillion".

1 0 See Li (2000). 1 1 In "Gaussian copula and credit derivatives" the Wall Street Journal (September 12, 2005) tells the story of David Li discovery and his impact on …nancial markets. derivative created from a portfolio of debt instruments [START_REF] Hull | Valuation of a CDO and an n-th to Default CDS Without Monte Carlo Simulation[END_REF] . The risk seller transfers the risk, therefore; the risk buyer takes the risk. Of course, the risk seller has to pay a fee to the risk buyer. The CDO became a successful product by allowing the credit risk division among di¤erent tranches. Synthetic CDOs [START_REF] Irotte | Credit risk mitigation evidence in Auto leases: LGD and residual value risk[END_REF] , in particular, were booming, improving liquidity, and allowing corporate bonds to be sliced and diced on the basis of risk. Investors were able to choose di¤erent levels of risk and returns [START_REF] Jost | Residual value analysis[END_REF] . The growth was so huge that it had a global macroeconomic impact, decreasing the risk of default impact, in ‡ating asset prices and narrowing credit spreads [START_REF] Laurent | An empirical approach to estimate residual value risk and its interconnection with credit risk the case of automotive leases[END_REF] . Prior to summer 2007, there was up to 30% of banking investment pro…t.

From that time, all products have slumped suddenly in value due to fraud, and low quality loan underwriting. The "credit crunch" allowed identi…cation of several weaknesses in the industry of loan securitization.

Severing the link between borrowers and risk takers, it promoted a lack of accountability. In addition, market protagonists contributed to a credit bubble [START_REF] Lasfert | The determinants of leasing decisions of small and large companies[END_REF] . Investor did not fully understand the products and had an over reliance on ratings provided by specialized agencies. Moreover, some securities were poorly structured.

Thereby, on a cleared market with more incentives, some experts are hoping for a recovery. Fortunately, securitization is not con…ned to consumer or corporate loans.

Residual value risk and credit risk have a clear analogy, constituting of units that are more or less risky.

A lease portfolio is similar to a loan portfolio, both could be divided into systematic and idiosyncratic risks. Losses occur when certain events happen, and again, the correlation risk has a huge impact. Hedging a portfolio of leasing equipment using derivative securities is attractive, and the idea to use some of the signi…cant developments in Credit risk modelling is attractive as well.

1 2 "Collateralized debt obligations divide the credit risk among di¤erent tranches: First senior tranches (rated AAA), second mezzanine tranches (AA to BB), and …nally equity tranches (unrated). Losses are applied in reverse order of seniority. Therefore junior tranches o¤er higher coupons to compensate for the added risk".

1 3 "Synthetic CDOs do not own cash assets like bonds or loans. Using credit default swaps (a derivatives instrument), synthetic CDOs gain credit exposure to a portfolio of …xed income assets without owning those assets". 1 4 See Hull (2005) 1 5 « La multiplication des émissions de CDO semble avoir contribué au resserrement prononcé de spreads intervenu au cours de ces eux dernières années sur l'ensemble des marché de crédit » . Cousseran and Rahmouni (2006). « This derivatives "money" is not being used to buy food, clothes or cars, which is why there has been no general pick-up in in ‡ation. But it has been used to in ‡ate asset prices, Mr Roche [Independent Strategy] argues » . "At the risky end of …nance" The economist (April 21st 2007).

1 6 "Fear and loathing, and a hint of hope" The economist (February 14th 2008).

4

Therefore the aim of this research is to transfer a model from the Credit risk to the Residual risk. The one factor model is presented and modi…ed. This modi…cation allows the creation of a new product, the Collateralized Residual Value. Pykhtin and Dev (2003), …rst applied the one factor model to auto lease.

They calculated the economic loss associated to Residual risk, leading to an estimate on economic capital.

The model was constructed and modi…ed for …nancial lease with the option to buy out (the lessee has a purchase option at the end of the contract). Moreover, loss distribution was calculated for a …ne grained portfolio (speci…c to large portfolio without signi…cant individual exposure), as a result, the model was only driven by the systematic factor.

Our study di¤ers slightly, as we aim to hedge residual risk using a derivative …nancial product. This article is intended for people within the leasing industry interested by an innovative …nancial product, as well as people from the …nancial market concerned by leasing risk opportunities. More speci…cally, we aim to hedge risk for a classical European contract. The product should cover operating lease contracts on a de…ned number of units and de…ned characteristics equipment parameters. We complete this theoretical development by an empirical analysis in which we confront this new derivative with market reality. Research gathered is organized as follow: Sections 2 and 3 provide some backgrounds on residual value risk and CDO pricing; Section 4 describes the model and the …nancial product; Section 5 is an empirical analysis and Section 6 concludes.

Leasing

The initial idea of leasing is that it is the use of equipment in a business which produces bene…ts, not the ownership. One characteristic of ownership in leasing contract is residual value risk that generates competitiveness or losses.

Main characteristics

As previously mentioned, a lease 17 is a contract between two parties where a party (the lessor) provides equipment for usage on a speci…c period of time to another party (the lessee) for speci…ed payment. Three parties are involved in the process; equipment suppliers, lessors and lessees. The lessor is the party that grants the use of the asset to the lessee. The lessee is the party that obtains the use of the asset from the lessor. The lessor purchases the equipment to the supplier. All along the contract, the lessor has the legal ownership of the asset. To use the asset the lessee makes periodic payments to the lessor at an agreed rate of interest.

There are two families of lease contracts. An operating lease can be considered as a typical rental allowing the lessee to use an asset without owning it. A …nancial lease aims to transfer all risks and rewards of ownership to the lessee.

A lease is de…ned as a …nancial lease if it contains one of the following elements:

The ownership of the asset is transferred to the lessee by the end of the lease term.

The lessee has an end of contract option to buy the asset lower than the fair market value.

Whether the asset is transferred or not, the lease period is for a majority of the asset useful life.

Because of the specialized nature of the asset, the lessee only can use the equipment without major modi…cation.

Otherwise, it is an operating lease [START_REF] Lucko | Statistical Analysis and Model of Residual Value of di¤erent type of Heavy Construction Equipment[END_REF] .

A lease is a …nancial instrument for the procurement of equipment. Recovery rate on a lease is higher than on a standard loan. But why do enterprises lease ? Regarding large …rms, leasing minimizes the after tax cost of their capital. For small asset base companies, leasing increases access to equipment …nance. The inherent value of the purchased asset acts as collateral. The lessor is the owner of the equipment, and then is secured by the collateral. Another attractiveness is the leasing companies expertise. Leasing companies are not only intermediaries. Their expertise is a real value added in the leasing process. They have knowledge of the asset. They select the appropriate equipment based on the ability of the asset to contribute to cash ‡ow (through various parameters like equipment characteristics, economic life of the asset, taxes or residual value risk). Leasing companies have also skills in …nance, credit, equipment acquisition and dealing. All things considered, they facilitate the ‡ow between equipment suppliers and equipment users.

On lessor side, they are several key elements:

Asset leased: Used by the lessee for business purpose, it could be any kind of equipment (i.e. printers, trucks...)

Asset List price: The lessor is usually able to negotiate rebates and the lessee could be part of the acquisition process.

Lease period: It is a pre requisite agreement between the parties. According to the contract, it could be ‡exible .

End of term options: At the end of the contract, they are options allowed to the lessee; Lease period can be extended, lease can be renewed, equipment can be bought or returned.

Residual value: The lessor forecasts the market value of the asset at the end of the contract.

Depreciation: It might be seen as the variance between the List price and the Residual value all along the lease period.

Lease payment: As illustrated by Figure 1, several features are included in payments made by the lessee during the contract; depreciation of the asset (usually the larger component), interests on the lessor investment, servicing charges (including operation cost, insurances, counselling, repairs...).

Residual value risk versus competitiveness

The residual value risk is that the lessor faces the risk to not being able to recover su¢ cient capital value from the resale or disposal of the asset. As illustrated by Figure 2, the fair market value curve implies a gain on sale or a loss on sale depending on the level of residual value. Therefore the lessor has a dilemma: The higher the residual value, the lesser the risk of loss on sale.

But the higher the residual value, the lower the rental payment. At the same time, the lower the rental payment, the better the competitiveness. And conversely the less the residual value risk, the worse the competitiveness. Figure 3 displays the mechanism of competitiveness and sales results at the end of the contract.

Figure 3: Dynamical bene…ts

In others words, the lessor has to set a residual value to minimizing residual value risk and maximizing competitiveness. A solution would be the use of a …nancial product. Hedging residual value risk could be done through a security derivative. Security de…nition includes …nancial security (bond, stock) but also capital market securities (mortgage, long-term bonds). It is an investment instrument which o¤ers evidence of debt or equity. A security derivative is a …nancial security whose value is derived in part from the value and characteristics of another security, the underlying asset [START_REF] Lucko | Statistical considerations for predicting residual value of heavy equipment[END_REF] . It would allow the lessor to transfer the risk to a fourth party (i.e. insurance company, …nancial market...). 1 9 Securitization is the process of aggregating similar securities that can be transferred or delivered to another party.

Model pre requisites

Because it allows to create a link of two survival functions, Gaussian Copula is a key element in our analysis.

CDO pricing, default modeling, and the one factor model are also inherent to the …nancial product presented in Section 4.

CDO are a subclass of ABS

Asset Backed Securities are securities backed by a pool of assets. ABS include various subclasses ( Commercial Mortgage Backed Securities (CMBS) or credit card ABS...), depending on the underlying asset class.

Obligations are usually underlying Collateral Debt Obligation (CDO). The basic idea of CDO is to pool corporate bonds and selling o¤ pieces of the pool. A synthetic CDO replaces pool's bonds by speci…c credit derivatives, Credit Default Swaps (CDS).

All in all, CDS are triggered by a credit event. A credit event increases the likelihood that the rating of a bond decreases. Consequently, a credit event increases the risk that a bond issuer will default, by failing to repay principal and interest in a timely manner. The events triggering a credit derivative are de…ned in a bilateral swap con…rmation. It is a document that refers to an agreement between the two swap counterparts. There are several standard credit events that could be referred to in credit derivative transactions: Bankruptcy, Failure to Pay, Restructuring, Repudiation, Moratorium.

By selling a CDS, an investor can take exposure to an individual credit. He is receiving periodic payment from his client. At the same time, however, he has to pay contigent payment when default occurs. The client, conversely, can hedge individual credit by buying a CDS. He provides periodic payment to the client and receives contingent when default occurs.

3.2 Default, default, default....

Default modeling is about the expected default payment of an obligor in a bank credit portfolio. The obligor (or debtor) is an individual or company that owes debt to another individual or company (the creditor).

The obligor borrows or issues bonds.

The following framework de…nes our model. The model is underlaid by a probability space. This probability space is constituted of three parts. F, a Algebra, is the information available into a sample space called . Elements of F are the measurable events of the model. Events of default are measurable. For instance, the event that an obligor survives or defaults is a measurable event. The last element is Pr, a probability measure. Pr(default) is the probability of default. Finally, to summarize, the probability space ( ; F; Pr) is underlying our model.

In survival analysis, T is a random variable denoting the time of default and t are others di¤erent times.

If T > t, then the obligor defaults. The survival function, usually denoted S is de…ned as S(t) = Pr(T > t).

This function must be non increasing: S(t + 1) S(t).

We can now de…ne the complement of the survival function. Usually denoted F , it is a lifetime distribution function: F (t) = P r(T t) = 1 S(t). From this concept a default rate per unit time can be calculated, the event density. Usually denoted f , it is the derivative

f (t) = d dt F (t).
All of this allows the creation of an advanced function, the hazard function. The hazard function, usually denoted , is the event rate at time t conditional until time t or later. It is given by (t)dt = Pr(t

T < t + dt j T > t) = f (t)dt S(t) = S 0 (t)dt S(t) ( (t) 0 and R 1 0 (t)dt = 1 with no continuous or monotonic constraints). A cumulative hazard function is (t) = R t 0 (u)du.
Because (t) = S 0 (t) S(t) , then d dt (t) = S 0 (t) S(t) and (t) = log S(t).

Several distributions can be used in duration modeling (usually de…ne on R + ), the most common one being the exponential distribution (S(t) = e t ).

Basic elements on Copulas

Why do we use copula ? In a portfolio, credit risks are non independent. Copulas are a convenient approach to specify a joint distribution of survival times. Using a copula function, we are able to link the survival function of an obligor to the survival function of another obligor in a portfolio .

In our model, we use copula on a three dimensional perspective. For simpli…cation purpose, we will focus on the bivariate distribution function and the two dimensional copula. The following results, however, can be extended to the multivariate case (see Nelsen (2006) and Vershuere (2006)).

For a "rigorous" copula de…nition, we …rst have to de…ne the unit square and the concept of subcopula.

The unit square I2 is the product I I where

I = [0; 1].
A two dimensional subcopula is a function C 0 de…ned through the four following properties: 1_ DomC0 = S 1 S 2 (with S 1 and S 2 are subsets of I containing 0 and 1).

2_ C0 is grounded [START_REF] Meneguzzo | Copula sensitivity in collateralized debt obligations and basket default swaps[END_REF] .

3_ C0 is 2-increasing (for every x 1 x 2 and y 1 y 2 , H(x 1 ; y 1 ) H(x 2 ; y 2 )).

4_ For every u in S 1 and every v in S 2 , C0(u; 1) = u and C0(1; v) = v.

We are now able to de…ne a two dimensional copula: It is a two dimensional subcopula C whose domain is I 2 .

Figure 4 gives an intuitive notion of a two dimensional copula. The graph of a two dimensional copula is a continuous surface within the unit cube I 3 . In a few words, a copula function is a function that links univariate marginal to their full multivariate distribution: C(u; v) = Pr(u U; v V ). Therefore, using a copula function, we are able to link the survival function of a credit risk to the survival function of another credit risk in a portfolio.

Speci…c pre requisites, the Gaussian copula

In the model presented in this article, we use the Gaussian copula. Let be the bivariate normal distribution function with correlation coe¢ cient (0 1). The bivariate normal is a member of the family of elliptically contoured distributions. The density function of is (x; y) =

1 2 p 1 2 e ( 1 2(1 2 ) (x 2 +y 2 2 xy)) .
The densities for such distributions have contours that are concentric ellipses with constant eccentricity.

1 is the inverse of a normal distribution function.

Finally the Gaussian copula is

C(u; v) = 2 ( 1 (u); 1 (v); ).
Consequently variables are jointly elliptically distributed and we can set using a linear correlation as a measure of dependence: Let X and Y follow, respectively, the distribution F and G. They jointly follow the distribution function H. Then the linear correlation for X and Y is de…ned, using u = F (x) and v = G(y)

as = (X; Y ) = 1 p V ar(X) p V ar(Y ) R 1 0 R 1 0 [C(u; v) uv]dF 1 (u)dG 1 (v).
Another property of the bivariate normal distribution is radially symmetry. A bivariate normal distribution with parameters x , y , 2 x , 2 y and is radially symmetric about the point ( x , y ). It means that

H( x + x; y + y) = H( x x; y y).
Using copula, we are able to work on survival function. Indeed, for a pair of random variable with joint distribution function H(H(x; y) = P [X < x; Y < y]), the joint survival function copula is given by

H(x; y) = P [X > x; Y > y] = 1 H(x; y). The relationship is H(x; y) = 1 F (x) G(y) + H(x; y) = 1 + F (x) + G(y) + C(1 F (x); 1 G(y)).
In the next section, we assume that the correlation of default is driven by a common factor through a Gaussian copula.

The initial one factor model is used for CDO pricing

To resume the model in one sentence, a …rm defaults when its "asset value-like" stochastic process X, falls below a barrier. X is commonly identi…ed as the amount of asset and X the barrier as the amount of liabilities. The …rm defaults when the amount of asset is below the amount of liabilities. The idea was …rst introduced by Merton (1974). He transferred an option pricing model to the credit risk market. Then he applied the Black and Scholes model to credit risk. We present an alternative model using copula. Value added is in copula ‡exibility to dependent variables and copula ability to provide scale invariant measure of association between random variables. The intuitive aspect of this model contributed to the growth of credit risk market.

The model described below is the famous standard Gaussian copula developed by Li (2000) and exposed [START_REF] Merton | On the pricing of corporate debt: the risk structure of interest rates[END_REF] by Gibson(2004).

In a reference portfolio of i = 1; :::N credits, for each obligor, default payment occurs when x i (reference credit normalized asset value) falls below x i (the threshold).

x i = a i M + q (1 a 2 i )Z i (1)
x i has three main components: M , Z i , and a i .

M is the common factors a¤ecting all the credits, the systematic risk. Z i is the factor a¤ecting only credit i. a i is the correlation parameter (0 6 a i 6 1) and de…nes default dependency between companies in the economy. The correlation of asset value between credits i and j is equal to a i a j . The random variables are assumed to be independently distributed. Therefore unconditionally on the systematic risk, default payments are correlated but conditionally there are independent.

M , Z i , and x i are means-zero, unit variance random variables with distribution functions G(0; 1), H i (0; 1), 2 1 See also Meneguzzo and Vecchiato (2004) for an empirical study of credit derivatives within the copula framework. Cherubini, Luciano, and Vecchiato (2004) gives an overview of copula applications in Finance. and F i (0; 1). q i (t) is a risk neutral probability that credit i default before t. The default threshold x i is equal to F -1 i (q i (t)).

When does a default happen ?

A default happens when x i falls below x i .

But

x i falls below x i if F (x i ) < q i (t) , x i < F -1 i (q i (t)) , a i M + p 1 a 2 i Z i < F 1 (q i (t))
and …nally

Z i < F 1 (qi(t)) aiM p 1 a 2 i
Conditional on the value of the factor M , the probability of default is therefore

q i (t=M ) = H 1 ( F 1 (q i (t)) a i M p 1 a 2 i ) (2)
For any number of default in a portfolio of N obligors, we have to estimate the probability of default on time t and conditional on the common factor M .

Therefore, we set the number of default distribution using a binomial function [START_REF] Miller | Economics of automobile leasing: the call option value[END_REF] .

P N (l; t=M ) = l N q i (t=M ) (3)
Once we have the conditional default distribution, we estimate the distribution according the distribution of M .

The unconditional default distribution P N (l; t) can be calculated as

P N (l; t) = Z 1 1 P N (l; t=M )g(M )dM: (4)
In a CDO, the investor is responsible for the interval of loss [L; H]. The expected loss of a CDO is de…ned on [L; H]. We de…ne the loss for any default as A(1 R) with A the notional amount of credit and R the recovery rate.

The expected loss is

EL i = N X l=0 P N (l; T i ) max(min(lA (1 R); H) L; 0) (5)
with T i, i = 1; :::; n the periodic payment.

Now, how to price a CDO ?

A CDO contract speci…es two potential cash ‡ow streams: a Contingent leg and Fee leg.

On the contingent leg side, the protection seller makes one payment only if the references credit default.

The amount of a contigent payment is the notional amount multiplied by (1 R).

The contigent leg is

contigent = n X i=1 D i (EL i EL i 1 ) (6)
with D i the risk free discount factor for payment date i (e rt , with r the risk free rate). The risk free discount factor is usually derived from the risk free interest rate.

On the …xed leg side, the buyer of protection makes a series of …xed, periodic payments of CDO premium until the maturity, or until the reference credit defaults.

The expected present value of the Fee leg is

F ee = s n X i=1 D i i [(H L) EL i )] (7)
i is the accrual factor for payment date i and s is the spread per annum paid to the tranche investor

( i T i T i 1 ).
The value of the CDO contract to the tranche investor at any given point of time is the di¤erence between the present value of the contigent leg and the present value of the …xed leg. It is the di¤erence between the protection the buyer expects to pay, and the amount he expects to receive.

The Mark To Market value of the tranche, from the perspective of the tranche investor is

M T M = F ee Contigent (8)
At inception the mark to market is equal to 0, therefore the spread is 4.1 There is a similarity between credit risk and residual value risk. But there are also dissimilarities and speci…cally in Auto Lease.

s = Contigent n P i=1 D i i [(H L) EL i )] (9)
The main idea of the leasing model is that a portfolio of leased equipment is comparable to a portfolio of credit. A portfolio with losses on resales is equivalent to a portfolio of credit with companies defaulting.

As in a CDO, every unit into the lease portfolio, has an idiosyncratic and a systematic risks; asset speci…c characteristic impact is resale price (model type, obsolescence. . . ). At the same time, resale price of other assets has a signi…cant impact (bid and ask e¤ect, downturn on the resale price market, in ‡ation etc.. . . ).

There are also dissimilarities:

First of all, equipment units are resold only one time at the end of the contract, although for a CDO, there is a risk of default throughout the contract. Therefore, the model presented in next section is set for only one period.

Another dissimilarity (and not the least) is on correlation estimation. Di¤erence is not on calculation but on data source.

In credit risk, they are four main data sources available:

-Default events that are obviously concrete realization of credit risk. They are rare events and as a result there are usually few data available. Approximations and aggregation have to be made to constitute data bases.

-Companies credit ratings: They are provided by credit agencies and re ‡ect the credit risk of a company according to experts points of views. By and large, they are made through balance sheet and macroeconomic analysis.

-Credit spreads: They re ‡ect market perception of credit risk. A large amount of data is available. But spreads could be impacted by external elements like liquidity -Equity correlation: The factor model (cf Section 3), assumes a theoretical link between equity and credit risk. Correlations are then more easy to compute.

In residual risk, there is one main data source available: for a residual value calculation, inputs are observations from second hand markets. Correlation estimation of residual value is based on resale market statistics. Resale prices, asset characteristics and price index are used to set modelization variables. A large amount of data is available.

The last dissimilarity is on standard de…nitions. As multiple factors de…ne a resale, there are issues to de…ne resale asset classes or homogeneity prices.

Auto lease is an extreme illustration. The high price level in the automotive second hand market involves a high residual value level. Combined to a competitive leasing market, the level of price leads to high risks of loss on sale.

At the same time, automotive is a singular equipment. A car is not only a tool to go from a place to another. It is also a living place and a symbol. Automotive often re ‡ects driver's sociological characteristics.

The purchase of a vehicle is a sensitive act, even in business. Therefore, Auto lease is a wide area to analyze.

In

Homogeneous equipment type model

The initial idea is simple: we use the equipment resale value as the asset value-like x i and the probability of resale value below residual value (x i <x i ) as the probability of default.

In a reference portfolio of i = 1; :::N units (vehicles, equipment...), for each obligor, losses occur when x i (reference unit normalized asset value) falls below x i (reference unit normalized residual value).

x i = a i M + q (1 a 2 i )Z i (10)
-The correlation of resale's prices between unit i and j is equal to a i a j .

-M is the sectorial factor a¤ecting equipment units on resale's market and Z i is the risk of loss on resales on unit i -Xi, M , and Zi are means-zero, unit variance random variables with distribution functions F i (0; 1), G(0; 1), and H i (0; 1).The random variables are assumed to be independently distributed.

At that point, the construction is similar to the credit model, but we include residual risk. Resale's value can be lower than residual value. There is a risk of loss on sale.

Three new elements will have an impact on the leasing adjustment of the model.

V i is the residual value or in other words the expected fair market value. mF M V i is the historical average fair market value, eF M V i is the historical standard deviation.

mF M V i , eF M V i , and V i are set on a percentage of Lp, List price by unit. As an example, an asset bought e 10000 and leased for a Residual value of e 5000 has V i = 50%.

Then residual risk is added: Probability of loss at the end of the contract is q i (t). q i (t) is a variable with mean mF M V i , variance eF M V i , and distribution function

E i (mF M V i ; eF M V i )
The probability of loss depends on residual value:

q i = E i (V i ). So default threshold x i is equal to F -1 i (q i ).
Conditional on the value of the sectorial factor M , the probability of default is therefore

q i (M ) = H 1 ( F 1 (q i ) a i M p 1 a 2 i ) (11)
Again, conditional probability is P N (l; M ) = l N q i (M ) and the unconditional probability can be calcu-

lated as P N (l) = R 1 1 P N (l)g(M )dM .
As a result the recovery rate is equal to the probability of loss. By construction the recovery rate is R = qi. The loss on sale for any unit is (1 R)Lp. Finally, resale price becomes RLp.

Previous elements allow the creation of a …nancial product, inspired by Collateral debts obligations:

The Expected loss is

EL = N X l=0 P N (l) max(min(Lpl (1 R); H) L; 0) (12)
The contigent leg is

contigent = n X i=1 D i (EL) (13)
The premium leg is

F ee = s n X i=1 D i i [(H L) EL i )] (7)
The spread is

s = Contigent n P i=1 D i i [(H L) EL i )] (9)

Heterogeneous equipment type model: a portfolio of three di¤erent assets

The model is extended to a portfolio with non similar units. A company ‡eet is commonly constituted of various car model. In an European leasing contract for medium size European company, lessee usually request di¤erent categories of cars for an auto lease contract. The ‡eet is usually divided into three groups:

Executives'cars (usually high brand car), Employee cars (medium level cars) and Small cars.

Basically the construction is similar to the homogeneous equipment type model (cf Section 4.2).

Three representative's vehicles constitute the model: Ex, Em and Sm. Now there are di¤erent types of asset residual values, number of units, List price etc....

V 1 i , V 2 i ,V 3 i are residual values for group 1, 2, 3 mF M V 1 i ,mF M V 2 i , mF M V 3 i , are fair market values historical averages, eF M V 1 i ,eF M V 2 i ,eF M V 3 i are historical standard deviation historical averages. mF M V 1 i , eF M V 1 i
, and V 1 i are set on a percentage of List price. The recovery rate for group 1 is R1 = q1i, the loss on sale for any unit is R1(1 Lp1) with Lp1 unit list price. Indeed, resale price is R1Lp1.

The principle is the same for others groups.

For each vehicle, asset value is still

x i = a i M + p (1 a 2 i )Z i : For group 1, default threshold x i is equal to F -1 i (q i ) with q1 i = E i (V 1 i ) and E i (mF M V 1 i ; eF M V 1 i ):
The principle is the same for other groups.

The distribution of the number of default, conditional on the common factor M, is computed for each group as

P N1 (u; M ) = u N1 q1 i (M ), P N2 (v; M ) = v N2 q2 i (M ) , P N3 (w; M ) = w N3 q3 i (M ) with N 1 ; N 2 ; N 3
number of units, P N1 ; P N2 ; P N3 the conditional probabilities, and u; v; w number of defaults for group 1,

The probability of default is computed on the whole portfolio.

P N (u; v; w; M ) = P N1 (u; M )P N2 (v; M )P N3 (w; M )
The Expected loss is:

EL = N X l=0 P N (u; v; w; M ) max(min((Lp1u (1 R1))+(Lp2u (1 R2))+(Lp3u (1 R3)); H) L; 0) (14)
Premium leg, contigent leg and spread are calculated like in 4.2.

It is straightforward to generalize this approach to more than three vehicle type.

Collateralized Residual Values

We propose a …nancial product, the Collateralized Residual Values, that covers residual value risk. We display a sensitivity analysis of the CRV to the main characteristics of the leased asset.

A CRV is a new class of ABS

The Collateralized Residual Values (CRV) is a new class of Asset Backed Securities (ABS). The CRV is inspired by synthetic Collateralized Debt Obligations (CDO) structure. Like a CDO, CRV can be sliced and diced, and tranches can be sold. But CRV is not about credit risk. The purpose is to hedge residual value risk on a portfolio of leases. A credit derivatives, obviously, is more accurate to hedge credit risk in a portfolio of contract.

Sensitivity Analysis on a CRV

What is the sensitivity of a CRV to size, residual value, and fair market variance ?

In the following sensibility analysis, all underlying reference assets are cars. The portfolio is homogeneous.

List price (e 15000), Fair market value (e 4500) and Correlation ( p 0:3) by car are equal. Cars are leased on a three years contract.

We value four tranches of the CRV. The …rst tranche absorbs all losses until the …rst 25% of the portfolio, the second tranche until 50%, the third tranche until 75% and the fourth until 100%.

Impact of Fleet Size Table 1 shows that the buyer of protection on a ‡eet of 600 units should be willing to pay 125,21 basis points to hedge the …rst 50% losses using a CRV.

Table 1: Sensitivity to ‡eet size According to results, the spread is stable until 500 units. Then the increasing size reduces the cost of protection. An increase in size reduces idiosyncratic e¤ect. There is a diversi…cation of risk .

Table 2: Sensitivity to residual value

The higher the residual value, the higher the pricing of CRV (Table 2). As illustrated in Section 2.2, decreasing residual value reduces the risk of loss on sales.

Impact of Fair Market Value variance

Table 3: Sensitivity to fair market value variance Fair market value distribution tails depends of FMV variance (Table 3). For an higher variance, tail are larger. As a result, the spread is an increasing function of the variance.

Empirical Analysis

The model is applied to a six years historic resale's portfolio. The observations, between 2000 and 2008, are from a major European leasing company (General Electric Capital Solutions). We …rst estimate the correlation of assets to a common factor. Then fair market value parameters and residual value are estimated

Correlation to the one sector factor

According to section 4.2, we have to set the linear correlation (a i ) between the portfolio and a sectorial factor a¤ecting equipment units on resale's market (M ). The sectorial factor is assessed using Eurostat Harmonized Consumer Price index (HCPI [START_REF]An introduction to copulas, Second Edition[END_REF] ). The index Purchase of vehicles price allows comparison within European markets. Additionally, a portfolio index has to be created. The portfolio index provides a non-biased historical trend analysis and exposes portfolio sale price at di¤erent time.

Automotive Price Index

To set a common factor that would a¤ect the whole portfolio, several HCPIs are available ; HCPI all items, HCPI Energy, HCPI Petroleum products, HCPI Road transport equipment.

The Index "Purchase of vehicle" (Figure 6) appears to be the most relevant. It covers purchases of new vehicles and purchases of second-hand vehicles from other institutional sectors. It is available by country and on European level. This index is included in the modelization as the sectorial factor indicator (M ).

Vehicle customers have to choose between resale market and new market. As a consequence, resale market is strongly impacted by new vehicle market. Therefore, a positive or a negative correlation of the sectorial index with the portfolio can be expected. Calculation is a …ve-steps process:

1 Creation of buckets for Age and Mileage (details in appendix).

2.

Keys are created including the following components:

age=mileage=model=f iscalclass=f ueltype=country

3.

Keys with population of less than 100 units on the whole history are excluded e.g.:GBR=T oyota=previa=private=lightgoods=petrol=_6:33; 39month=_3:75000; 105000km is excluded.

During the last 8 years, less than 100 units in this bucket have been sold.

Representative and similar samples are created all along the history using the historical key frequency:

4.

A Random selection is processed by month through the following criteria: 1% of units by bucket are selected:

(e.g. for a bucket of 200 units, then 2 units are selected)

Restricted random sampling with replacement (SAS proc survey)

Priority levels: The sample is replicated on a monthly basis according to key frequency and by order of priority; selected month, semester of the selected month, whole history (e.g. If data are not available in the current month, then data are selected in the current quarter etc...).

5.

A monthly resale percentage is computed from the sample. The percentage of resale is resaleprice Listprice+OptionListprice .

It allows a comparison of resale performance between vehicles with di¤erent levels of price and option price.

The process is replicated several time to create several sample. 1000 random samples are created by month.

Portfolio Index Computation and results 565 representative buckets are selected from an initial pool of 38257 units. 97000 samples are calculated (97 period).

Among other perspective, it provides a graph distribution by month. Results are available by country and on European level. For instance, Figure 7

Fair Market Value and Residual Value setting

mF M V i , average fair market value, eF M V i , standard deviation and residual value (V i ) are parameters to include in the model.

Fair Market Value estimation is complex

We assess the Fair Market value at the end of the contract.

In others words, we estimate the depreciation of the asset for the next years. Since January 2004, the key average depreciation is 37.26% and the standard deviation is 46.76%.

35

Previous graphics illustrate the fact that FMV is not a constant value. There are trends and cycles that are not straightforward to identify. Depreciation in the value of a car occurs based on a range of factors.

The factors include cars condition, kilometers traveled and brand reputation. Moreover, brand reputation contains mechanics and popularity. Consequently, di¤erent methodologies are possible to forecast average fair market value.

Leasing industry usually works with internal modelization. Standard models have inside a model life cycle and a segment analysis. New legislations or macroeconomic impacts also are sometime included.

Additionally, External companies (Eurotax, X-ray, Cap) provides forecasted FMV. Forecasts are based on market data, modelization and expertise.

RV and FMV, a new perspective

In an operating lease contract, Residual value is de…ned as the forecasted fair market value. It is an input in rental calculation. It also drives the risk of loss on sales at the end of the contract. What is the impact of a CRV ?

Elements of the contract become di¤erent. Using securitization product, elements of the contract have to be rede…ned.

The fair market value still has to be forecasted. But residual value is now a threshold. As illustrated in Figure 12, the threshold is a level of risk chosen by the lessor and the lessee. In the model, mF M V is a forecasted average of fair market value at the end of the contract. And eF M V i is the estimated standard deviation of fair market value. So considered residual value is now an adjustment variable. Therefore the securitization product allows several choices within di¤erent levels of risk, di¤erent levels of rents, di¤erent market spreads, and di¤erent fair market value variance. Additionally, hedging can be made on speci…c tranches.

Six CRV Table 5: Pricing of six CRV

A CRV is built accordingly to leasing contract characteristic. As illustrated in Table 5 for six CRV, inputs in pricing are population size, list price, fair market value mean, fair market value variance and residual value. Through the selected residual value level and tranches limits, lessor and lessee can choose a level of risk. Moreover, in case of negative correlation parameter, CRV could go against a downturn in the sectorial market and create opportunities for risk diversi…cation. Like standards derivatives, CRV allows insurance or hedging for the lessor and, for the buyer taking opposite position in the …nancial market, speculation or arbitrary.

Conclusion

Li Gaussian copula model, initially used for credit risk, is transposed in residual value risk of the leasing industry. The Collateralized Residual Value (CRV), a new derivative product, is proposed. Pooling together a large portfolio of equipment that has been leased, the derivative converts end of contract risks into an instrument that may be sold in the capital market. As a standard derivative, it is a tool that transfers risk, and can be used for hedging or speculation. Moreover, it allows the lessor and lessee to select their degree of exposure to residual value risk and to improve competitiveness. As a result, the model is a contribution geared for people from the leasing industry interested by an innovative …nancial product, as well as people from the …nancial market concerned by leasing risk opportunities.

The present analysis could be extended in various ways. The accuracy of the correlation parameter can be improved by a complete macroeconomic analysis, and the fair market value parameter can also be improved.

Finally, other families of copula could be tested.

APPENDIX:

Figure 1 :

 1 Figure 1: Lease rental calculation

Figure 2 :

 2 Figure 2: Depreciation curve

Figure 4 :

 4 Figure 4: Two dimensional copula

4 A

 4 modi…ed model: The leasing model From equipment leasing speci…cities and the one factor model, we create a residual value risk model. A new product called Collateralized Residual Value (CRV) is adjusted through the leasing contract parameters.

  automotive market multiple factors in ‡uence resale price. A second hand vehicle price is impacted by age (time between registration date and resale date), mileage (number of kilometers at the end of the contract), damages (i.e. amount and type ), product life cycle (i.e. new model...), make (i.e. Toyota, Renault...), model (i.e. Yaris, Laguna...), version, body type (i.e. break, pick-up...), segment (i.e. small cars...) or external color. Figure5gives an overview. Choices have to be made to de…ne similar assets and prices (c.f Section 5).

Figure 5 :

 5 Figure 5: Multiple factors of automotive market

Figure 6 :

 6 Figure 6: European HCPI

  displays the simulation result for Portugal on January 2001. The percentage of resale distribution is on a range of [49%-72%].

Figure 7 :

 7 Figure 7: Portugal depreciation distribution on January 2001

Figure 8 :

 8 Figure 8: European HCPI and European portfolio YoY variance

  Resale percentage Mean and Variance of resale percentage are calculated from historical statistics. For simpli…cation purpose, resale price is computed through a percentage of List Price ( resaleprice Listprice+OptionListprice ).FMV subtlety To illustrate our presentation we focus on a speci…c Key: PEUGEOT 307 Tourisme Diesel _6.]33,39]month _4.]105000,145000]km _FRA.

Figure 9 :

 9 Figure 9: 36 months contracts / Peugeot 307 depreciation and historical average

Figure 10 :

 10 Figure 10: 24 months contracts and 36 months contracts depreciations

Figure 11 : 5

 115 Figure 11: Peugeot 306 and Peugeot 307

Figure 12 :

 12 Figure 12: Sale results through fair market value and residual value level

Figure 14 40Figure 15 :

 1415 Figure 14

Figure 16 :

 16 Figure 16: HCPI YoY variance

Percentage market penetrations are highly signi…catives in United states (27.7%), Germany (23.6%), and Spain (29.1%).

According to the White Clarke Global Leasing report (2008), "Globally, the industry contnued to growth robustly, with the top 50 countries increasing volume by 8.8%" between 2005-2006. The Percentage of the world market volume was respectively

41.1% and 38% for Europe and North America.[START_REF] Copeland | A note on the evaluation of cancellable operating leases[END_REF] See Lasfer and Levis (1998)[START_REF] Cousseran | Le marché des CDO Modalités de fonctionnement et implications en termes de stabilité …nancière[END_REF] See[START_REF] Gordon | Retreat is sounded as those leasing residual losses mount[END_REF] for a description of the 2001 Leasing industry crisis.[START_REF]DEMETRA 2.1 documentation and freeware available form circa[END_REF] Jost and Franke (2005) illustrate the use of a speci…c tool of statistical modelling to calculate residual value through a wide range of parameters. In[START_REF] Lucko | Statistical Analysis and Model of Residual Value of di¤erent type of Heavy Construction Equipment[END_REF] and Lucko, Anderson-cook, and Vorster (2006), residual values are set using regression methodology. Rode, Paul, and Dean (2002) outline a framework for analysing the uncertainty of residual value for assets, such as power generation facilities, for which few data points exists.

[START_REF] Gibson | Understanding the risk of synthetic CDOs[END_REF] Leasing de…nitions and legislations are quite di¤erent from a country to another. As we do not wish to focus on a speci…c legislation, de…nitions are made on an international common perspective.

[START_REF] Gordon | Retreat is sounded as those leasing residual losses mount[END_REF] In next sections, we propose a model to hedge residual value risk on an operating lease that is the most common contract in Europe for Auto Lease. The model can be extended and modi…ed for a …nancial lease (see[START_REF] Pykhtin | Residual risk in auto leases[END_REF]).

0 A function H from S 1 S 2 is grounded if H(x; a 2 ) = 0 = H(a 1 ; y) for all (x; y) in S 1 S 2 with a 1 and a 2 last elements of S 1 and S 2 .

[START_REF] Bluhm | An Introduction to Credit Risk Modeling[END_REF] The number of default distribution is usually computed through a recursion method[START_REF] Andersen | All your hedges in one basket[END_REF] or[START_REF] Hull | Valuation of a CDO and an n-th to Default CDS Without Monte Carlo Simulation[END_REF]). In the case of homogeneous credits and for simpli…cation purpose, a binomial function is simpler and lead to similar results.

[START_REF] Cherubini | Copula Methods in Finance[END_REF] Graphics of HCPI series are reported in Appendix.

[START_REF] Copeland | A note on the evaluation of cancellable operating leases[END_REF] TRAMO-SEATS: They consist of new versions of programs TRAMO, "Time series Regression with ARIMA noise, Missing values and Outliers", and SEATS, "Signal Extraction in ARIMA Time Series", created by Gómez and Maravall in 1996, of program TERROR, "TRAMO for Errors", and program TSW, a Windows version of TRAMO-SEATS with some modi…cations and additions, developed by G. Caporello and A. Maravall at the Banco de España.


Taoufik Bouraoui
email: t.bouraoui@u-paris10.fr

The impact of stock spams on volatility

Keywords: stock spam, event studies, volatility, penny stock JEL classification: G14, D84, C12

This paper is dedicated to study the impact of stock spams through the analysis of the variations of volatility. We use the methodology of event studies on a sample of hundred ten firms.

The results show positive and significant changes in volatility during 12 days of the event window; a widening of the variation [lowest price -highest price] was noticed following the consignment of messages by the spammers. The sending of stock spams affected the behaviour of investors, indicating thus that the spamming activity is a lucrative business.

I. Introduction

Spam is not only a means used to send massively unsolicited advertising messages.

Hackers, now, use this practice to influence stock prices and push up the values of certain securities. The spammer launches a campaign to promote the stocks of a society by sending emails massively: he purchases stocks of a society for which the price is low, sends spams to artificially increase the stock value and then resells them with profit. The stock spam targets securities whose share price is relatively low; the targeted society generally is not conscious of the abusive use of its mark or its social denomination in the spams for speculative purpose.

Stock spams are increasing on Internet. So, it's interesting to wonder whether this phenomenon affect really the volatility of prices. To do this, we are going to use the event studies methodology. It is a method which allows analysing the reactions of market to a given event. Since Fama, Fisher, Jensen and Roll (1969), event studies have become a reference method in finance. Today, this methodology is fluently applied to test the informational impact of different events, notably the announcements of alliances or mergers and acquisitions [Hubler and Meschi (2000), Guards (2003), Woolridge and Snow (1990)],

announcements of annual earnings [Mignon, Lardic (2003)], stock repurchase [Mai, Tchemeni (2000)], etc… In this work, the event is stock spams. To our knowledge, only three studies are available on this topic: Bohme and Holz (2006), Frieder and Zitterain (2007) and Hanke and Hauser (2008). Bohme and Holz (2006) and Frieder and Zitterain (2007) were interested in studying the impact on volumes and returns. Whereas, in the study of Hanke and Hauser (2008), the authors highlight the impact on volatility, but by using a panel regression.

Our main contribution in this paper is threefold. Firstly, we study the impact on volatility while using the event studies methodology. Secondly, we take into account the assumptions associated with the implementation of the method of event studies which are not always verified empirically, such as normality, independence and homogeneity of variances between securities, and stability of variance over time. Thirdly, we employ an updated database which contains firms recently targeted by the campaigns of spam in order to know if spammers always succeed in affecting the behaviour of investors.

The impact of the occurrence of new information on the price of financial assets has already been the subject of considerable attention for more than forty years. However, the effect of financial informations on the second moment of the conditional distribution of returns (volatility) is very little approached.

The purpose of this paper is to provide an answer to the following question: does stock spam have a significant effect on volatility? If so, does it increase the volatility or decreases it? To this end, this article unfolds as follows. Section II examines the origin as well as the working of stock spams. In section III, we present the methodology of event studies. In the fourth section, we set our data. Empirical results are reported and discussed in section V.

Finally, section VI concludes.

II. Stock Spams

Since the existence of Word Wide Web, resources are democratized and the flow of information circulating on the networks has been increasing. However, the content of information didn't always evolve in the right sense and various people understood quickly how to use these resources abusively.

The spams, called as spamming or mail-rubbish, are used to designate the nonsolicited electronic mails having an advertising character. This expression comes from a Monty Python's sketch (name of troop of English comedies) in which the word spam (contraction of "spice ham", English brand of sausage) is repeated constantly in order to incite the listeners to become consumers. The first goal of spam is to make advertisement at low cost by massive dispatches of electronic messages. Frieder and Zittrain (2007) note that this curse represents more than 65% of e-mail traffic.

The National Commission of Computer science and Freedom of France performed a study in which it tried to classify spams according to two different classifications: the first one according to the target: it found that 85 % of spams aim at individuals, while 15 % only are intended for firms. The second classification is according to the language in which spams are written: 84.8% of spams are written in English against only 8% in Asian and 7% in French origin. The proportion of spams in other languages is negligible. Later, spams written in English were classified according to several themes; and it proved that messages advertising stock exchange and financial products occupy the second place with a percentage of 40% behind messages with pornographic character or proposing formulas of meeting that reach 42%. In the same context, Sophos, a specialist in protection of corporations against spams, established a classification of twelve main issuing countries of spams between July 2006 and October 2006. The table 1 reveals the important place occupied by USA with a percentage of 21,6%, practically twice of China which follows with only 13,4%. The complete list of the twelve countries is the following: The spread of wrong information about stock exchange is an efficient means to act on the value of securities for dishonest aims of speculation, and with the development of Internet, it becomes simple and easy to reach a big number of investors.

Stock spams are based on a simple principle; the spammer starts by buying gradually a big number of stocks. Then, he sends false information about the share prices by mail in order to encourage potential investors on a bad way. Unfortunately, investors believe in such information and buy securities with significant amounts. As a result, brutal increases in share prices take place. Finally, the dishonest speculator, the originator of all these activities, sells stocks at higher prices. The following figure illustrates an example of a stock spam encouraging investors to buy securities of Diamond Film, a company specializing in environmental protection in Canada. exchange security mentioned in a spam increased 215.2% on average. This number falls to 154.1% when the message is transmitted before the opening hours of the market. The impact on returns was also studied; by implementing the methodology of event studies, they note that prices climb of +1,7% the first day of the campaign. Frieder and Zittrain (2007) led the same type of survey for the period of January 2004 until July 2005; they showed that a spammer makes in two days a medium benefit of 4,9 % of the share value, while the investor sees his investment, in two days, falling of 8 %. They also noted an increase of volumes and positive returns of the stocks touched by spams. Similarly, Hanke and Hauser (2008) were also interested in studying the effect of stock spams on return, volatility and turnover. They constituted a sample of 235 firms that were the subject of spam during 2005. Besides the presence of significant and positive impact on all the three variables during the first day of the event, the authors emphasize two results. Firstly, they show that lack of liquidity has a strong link with the presence of impact; more the stock is illiquid, more the impact observed is important. Secondly, they find that repeated spamming on successive days generates an additional demand on behalf of investors for targeted securities.

This leads us to conclude that spams can affect and mark the presence of an abnormal activity on market. In order to study the impact on volatility, we implement the event studies methodology.

III. Methodology of event studies

Event studies enable to measure the informative relevance of an event, notably the analysis of the behaviour of share prices at the arrival of information. They are based on the idea according to which financial markets react immediately to new information susceptible to affect the future profitability of the society. [Hubler and Meschi (2000)]. Empirically, an event study consists in determining an abnormal volatility at the date of announcement of the event. This abnormal volatility is interpreted as the measurement of the impact of the event on share prices.

Mackinlay (1997) identifies seven stages for the implementation of this methodology.

III.1 Stage 1: Definition of the event

The first stage of an event study consists in defining the event and identifying the period during which this event will be studied, called « event window » or « period of test ».

In this paper, as mentioned in the introduction, the event is stock spams. Regarding the event window, and unlike others papers which take a period of test centred around the date of event [Hubler and Meschi (2000)], we choose a period of test of length 15 days which starts at the date of sending spam and spreads until the fourteenth day. Indeed, the stock spam is an advertising message which brings a private and little known information. So, we cannot fear flight of information of the type of those that can precede the official announcement of merger and acquisitions or earnings. Bohme and Holz (2006) led the same type of reasoning on an event study by returns; they chose an event window which begins at the date of announcement and extends until the fourth day.

III.2 Stage 2: Selection criterion

Once the event is defined, it is necessary to determine a selection criterion, i.e. a criterion on which the event study will be based. The majority of works on this topic have used either the volumes or the returns. In this paper, we chose volatility as criterion.

The volatility of a stock exchange security indicates in which amplitude the price of this security can vary, to the rise as to the fall, relative to its average price, over a period of time. The volatility of assets is all the stronger as the market prices are unstable. This is in particular observed following an event concerning the security in question. However, assuming that volatility is constant over time amounts to suppose that the event specific to the firm does not affect the risk of its security.

Volatility must be estimated because it is not directly observable. For that, several methods can be used whose principal ones are:

• Squared return (Harris (1987), Dravid (1987)) :

) ( ) ( ) ( it it it it it B Ln H Ln B H Ln VT - = =
Where it H and it B are respectively the highest and the lowest prices of security i on date t.

The first two measurements are rather adapted for high frequency data (intra-day data), which is not the case for our study 3 . In addition, Parkinson (1980) and Alizadeh, Brandt and Diebold (2002) show that the use of the highest and the lowest prices of the same day, in comparison with the first two measurements, gives a better estimation of the true volatility.

For these reasons, we adopt the third method to measure volatility.

III.3 Stage 3: Normal volatility -Abnormal volatility

To assess the impact of an event, it is necessary to calculate an abnormal volatility or an excess of volatility due to the event. The abnormal volatility is the difference between the observed volatility and the normal or theoretical volatility. The last one is the volatility that we would normally have observed in the absence of event; it must be modelled over a period preceding the period of test called "the estimation window".

III.4 Stage 4: Estimation window

The estimation window precedes the event window. It is much longer than the period of test; generally it has a length equal at least three times the length of event window in order to have enough number of observations for estimation. In our survey, we choose 146 10 observations 4 preceding the date of event. We must in particular ensure that the two windows do not overlap to prevent that the impact of the event is not found in the estimator and to avoid, thus, that the study is skewed.

The estimation window (L 1 ) and the event window (L 2 ) can be schematized as follows:

Figure 2: Estimation window and event window

(L 1 ) (L 2 ) t = -146 t = 0 t = 14

III.5 Stage 5: Test of hypothesis

After having identified the estimation window, the abnormal volatility can be calculated. At this stage, we set up a test of hypothesis, i.e. a null hypothesis H 0 against an alternate hypothesis H 1 , in order to see if stock spams have an effect or not on volatility.

III.6 Stage 6: Empirical results

It is the stage of analysis of abnormal volatilities by implementing the appropriate statistical tests.

III.7 Stage 7: Interpretations and conclusions

At this stage, we conclude if stock spams affected or not the volatility of targeted securities.

If the event studies methodology has the advantage of being validated and tested on various works, it supposes, however, some statistical properties which, unfortunately, are not always checked empirically. Theoretically, the method supposes that:

• The data are distributed according to a normal law.

• The volatilities of securities are independent and identically distributed (iid).

• The variance is constant over time.

In this paper, we consider each of these three hypotheses in applying the methodology on our data.

IV. Data

The data used to lead our empirical study are extracted from the website <http://www.spamnation.info/stocks/>. This website lists all firms targeted by stock spams since 1999. But, to have the history of daily volumes for each stock, we used the Datastream database. In the beginning, we constituted a sample of 180 firms. However, the unavailability of historical prices for some companies, considering the majority of them have just been created, led us to remove them from the sample. Moreover, other securities had missing quotations on several days. These securities were also excluded from the sample. Finally, we kept only 110 firms. These firms fulfill the following criteria:

• They were targeted by spams after January 2006 in order to obtain the largest possible number of data for the estimation window.

• The availability of at least 100 historical prices starting from the date of sending the first spam.

• The number of missing quotations should not exceed 10.

The sample thus formed contains firms which were targeted by stock spams during the period from February 2006 to October 2008. For each firm, we have 161 daily volatility measurements (event window (15) + estimation window (146)). These firms belong to varied sectors of activity; so we find companies specialized in multimedia, energy, biology, international distribution, telecommunications… Also, they are not all American; they come from different countries (Canada, China…). Nevertheless, the common point between these companies is that they are known under the name of penny stocks companies.

The penny stock term designates the stocks whose share price is extremely low.

Generally, the share price is below 5 dollars, and firms which are targeted are very small and not commonly known. Another common point between these firms is that their securities are traded in OTC markets, notably the Over-The-Counter Bulletin Board (OTCBB) and the Pink Sheets, which are less controlled than the main stock exchanges. These markets do not have a physical place as the NYSE or the AMEX; they are only represented by a computer network that displays in real time the share prices. Firms quoted on these markets are speculative and highly illiquid; it is the reason for which they are targeted by advertising campaigns.

The abnormal volatility of stock i on day t is given by:

t i t i t i KT VT AVT , , , - =
Where, AVT it: the abnormal volatility of stock i on day t.

VT it: the real or observed volatility of stock i on day t.

KT it: the theoretical volatility of stock i on day t.

To estimate the theoretical volatility, we use the stock's average volatility over the estimation window (

∑ = = 146 1 , , 146 1 t t i t i VT KT
). The choice of this method is justified by its simplicity of implementation. Moreover, May and Tchemeni (1996), in a simulation study, underline that the use of the historical average of a variable gives better specified and more appropriate results than the use of the market model or the standardized model.

With this method, the calculation of abnormal volatility obeys the following expression:

∑ = - = 146 1 , ,
θ t = ) (MAVT MAVT t t σ ~ T N-1 (1)

V. Results

V.1 Cross-sectional Student test

In order to justify the use of this test, we have tested the heteroscedasticity of the series of volatilities. Our results 5 show that 60 securities among 110 are heteroscedastic, i.e. their variances vary over time. To have unbiased results, it is necessary to take into account this fact. So, we implement the cross-sectional Student test which enables to calculate a variance for each date of the event window. In order to apprehend better the impact of spams, we represent graphically the evolution of the mean abnormal volatilities during the period of test.

According to table 2, we note that the sending of stock exchange spams generated an increase in the volatility of securities on the entire event window. This increase is significant over the first three days of the event and from t = 5 to t = 13. Furthermore, we record the biggest abnormal variation of volatility (+8.79 %) in t = 0. The evolution of the mean abnormal volatilities (fig. 3) shows clearly this impact on the first day, then its progressive reduction until the fifth day where we note the weakest rise of volatility (+0.8%).

Nevertheless, this last increase is not significant, and given that the mean abnormal volatility of the previous day (4th day) is also not significant, it lets us think that the impact has lasted only during the first three days. However, we realize from the 6th day the appearance again of a significant impact which continued until the date t = 13. But, this impact on the second interval of the event window [t=5; t=13] is less important than the one observed on the interval [t=0; t=2]; the increase in volatility over the second period varies between +3.77% (14th day) and +4.67% (6th day).

It should be noted that this increase in volatility is associated with a rise of volumes on the one hand, and a rise of returns followed by a fall, on the other hand 6 . Indeed, volumes and volatilities have evolved in the same sense. Increased movements of transaction (purchases and/or sales) on securities targeted by stock spams have led to a widening of the range [lowest price -highest price]. These results corroborate the works of Crouch (1970), Harris (1987) and Jain and Joh (1988) who showed a positive relationship between volume and volatility.

On the other hand, the increase in volatility is put in parallel with as well an increase as a reduction of returns. This can be interpreted as follows:

• If the increase of volatility is accompanied by an increase in returns: the answer of investors to the messages of spammers by purchasing massively securities raises the prices. Consequently, the difference between the highest and the lowest price of the day emphasizes an important variation. In this case, the widening of the range is rather from the side of the highest price. This seems to corroborate the work of Gallant, Rossi and Tauchen (1992) and Hanke and Hauser (2008) who show a positive relationship between return and volatility.

• If the increase of volatility is accompanied by a decrease in returns: investors, having a very modest budget, cannot invest in shares quoted on known stock exchange as the NYSE or the NASDAQ. When they receive the message from the spammer, they believe in the information contained there in the hope of becoming rich and making fortunes. Hence, they respond positively to the request of the spammer by buying securities with large quantities. However, when they realize the next days that prices did not climb as that was promised in messages, they try to get rid of securities by selling them at low prices. The movement of sale with significant quantities leads to an increase in the fluctuation in prices. The widening of the difference between the highest and the lowest price, in this case, is generated by a reduction in the lowest price.

These results are consistent with those of Pindyck (1984) who attributes the decline in return of the NYSE market index during the period 1965-1981 to an increase in volatility. Similarly, French, Schwert and Stambaugh (1987) find that the volatility of S&P is negatively related to return.

The consignment of stock spams has generated positive and significant mean abnormal volatilities over 12 days. So, we reject the null hypothesis H 0 . The appearance of new information, which is in our case the messages of spam, increased uncertainty about the penny stock securities. This uncertainty resulted in a rise of the volatility of share prices following the increase in the movement of transaction. The first two limits of the event studies methodology are rather associated with the implementation of the Student test described according to equation [START_REF] Alizadeh | Range-Based Estimation of Stochastic Volatility Models[END_REF]. This parametric test assumes that the data are distributed according to a normal law, on the one hand, and they are independent and identically distributed (iid), on the other hand. However, these two assumptions are not checked on our data 7 , which is a general characteristic of financial series.

So, the use of this first test can not reflect the real effect of stock spams on volatility. In order to improve and to give more robustness to our results, we apply now a second statistical test which enables to cure these limits. However, the test of Cowan (1992), to our knowledge, has never been applied.

V.II Cowan rank test

The statistic of the test is given by the following formula:

( )

∑ = - - = L t t D Cowan K E K L K E K L Z 1 2 2 ) ( 1
) ( ~ N (0,1) (2)
Where, L 2 : length of the event window. Under the null hypothesis of no abnormal volatilities, the test of Cowan allows to compare the average ranks of each date of the event window with the expected average rank calculated over the complete period of study. To implement it, we have to firstly transform, for each firm, the series of abnormal volatility into their respective ranks. These ranks are defined in ascending order: rank 1 and 161 correspond respectively to the lowest and the highest abnormal volatility in the series.

The results of this test are reported in table 3.

We realize that results are not sensitive to the used statistical test. As we have found previously in the cross-sectional Student test, volatility, here, is also positively and significantly affected during 12 days of the period test (from t = 0 to t = 2 and from t = 5 to t = 13). The most important variation is observed during the first day of the event (100.1) where all securities are assigned by high ranks above the average rank. This reaction consists in a positive response from investors who believed in the information contained in spams. The next two days (t =1 and t =2) show a fall in volatility expressed by a reduction in the value of the average rank. This demonstrates that the effect starts to disappear gradually, especially when this degradation finishes by non significant average ranks; such is the case of days 4 and 5. However, dice the sixth day (t =5), we observe that the increase in volatility comes back to become significant; the average rank of volatilities during this day (6 th day) exceeds the expected average rank and amounts to 87,7. This significant impact was continued until the 13 th day.

VI. Conclusion

This paper has focused on the impact of stock spams on the volatility of penny stock's securities. For this purpose, we constituted a sample of 110 companies which were targeted by spams between February 2006 and October 2008. After we calculated the mean abnormal volatilities over the event window of 15 days by using the event studies methodology, we set up two statistical tests: a parametric test (cross-sectional Student test) and a nonparametric test (Cowan rank test).

The results of both tests show that stock spams affect positively and significantly the volatility of prices: a widening of the variation [lowest price -highest price] was noticed following the consignment of messages by the spammers. This seems to corroborate the works of Koski (1998) and Hanke and Hauser (2008) who also found an increase in volatility following respectively the announcement of stock splits and stock spams. We can conclude that the spamming activity is a very lucrative business which continues to affect the behaviour of investors who still believe in wrong information in the hope to accomplish profits.

However, if significant increases in volatility are observable, the effect cannot be generalized to all securities in the sample. So, it would be interesting to detail the results by studying the impact on each security. Moreover, the number of spams received per day during the duration of the advertising campaign varies from one security to another. Thus, we record for some stocks 3 or 4 messages received throughout the period of the campaign, whereas we note for other stocks hundreds of messages received during only one day. In this context, it would be also very convenient to study the extent of the impact according to the number of messages received by security.

Figure 1 :

 1 Figure 1: Example of stock spam

•

  Absolute value of return (Crouch (1970), Teiletche and Lespagnol (2005)) : it it R VT = • The difference between the highest price and the lowest price (Parkinson (1980), Alizadeh, Brandt and Diebold (2002)) :

DK

  : average rank of all stocks on date D ; ∈ D [0, 14] E(K) : expected average rank : length of the period of analysis (= estimation window (L1) + event window (L2)). t K : average rank of all stocks on date t ; ∈ t [-146, 14]

Table 1 :

 1 The main issuing countries of spams

	Rank	Country	Percentage
	1	USA	21.6 %
	2	China (& Hong Kong)	13.4 %
	3	France	6.3 %
	3	South Korea	6.3 %
	5	Spain	5.8 %
	6	Poland	4.8 %
	7	Brazil	4.7 %
	8	Italy	4.3 %
	9	Germany	3.0 %
	10	Taiwan	2.0 %
	11	Israël	1.8 %
	12	Japan	1.7 %
		Other countries	24.3 %
			Source : Sophos 1

  In order to appraise the informative content of the stock spam in terms of volatility, we test the null hypothesis H 0 against the alternative hypothesis H 1 at the 5% level:

						AVT	i	,	t	VT	i	t	146 1	t	VT	i	t	; = t	0	,....,	14
						H O : Absence of abnormal volatilities
						H 1 : Presence of abnormal volatilities
	We consider the following variables:
	-MAVT t : The mean abnormal volatility of all stocks for every date of the event
	window:
													MAVT	t	=	110 1	∑ = 110 1 i	AVT	i	,	t
	-	σ	t	(MAVT	)	: standard deviation of mean abnormal volatilities. It is calculated for
	every date of the event window:
						σ	t	(	MAVT	)	=	110 1	-	1	110 1 ∑ = i	(	110 . t i AVT	-	t MAVT	)	2

-θ t : cross-sectional Student test; it is given by :

Table 2 :

 2 Mean abnormal volatilities (%) and statistics of Student

	Date	MAVT t (%)	θ t
	0	+8.79	2.980**
	1	+8.31	3.764***
	2	+5.55	3.012**
	3	+1.84	1.249
	4	+0.8	0.603
	5	+4.67	2.213*
	6	+5.28	2.454*
	7	+3.92	1.994*
	8	+5.73	3.124**
	9	+4.01	2.555*
	10	+5.79	2.857**
	11	+3.94	2.612**
	12	+3.43	2.226*
	13	+3.77	2.240*
	14	+3.01	1.693
	* significant at 5%
	** significant at 1%
	*** significant at 0,1%
	Fig. 3: Evolution of mean abnormal volatilities during the event window

  This second test is used in order to lift completely the hypothesis not checked by the

cross-sectional Student test. It is a nonparametric test for which it is not necessary to specify the conditions that the sample has to fill. Nonparametric tests such as the sign test, the generalized sign test or the Wilcoxon signed rank test were already largely used

(Berry, Gallinger and Henderson (1990)

,

Giaccotto and Sfiridis (1996)

,

Campart and Pfister (2008)

).

Table 3 :

 3 Cowan rank test

	Date	K	D	Z Cowan
	0	100.1	5.587***
	1	99.8	5.518***
	2	95.4	4.215***
	3	83.7	0.813
	4	85.5	1.321
	5	87.7	1.962*
	6	92.9	3.481***
	7	89.7	2.550*
	8	90.9	2.906**
	9	90.8	2.880**
	10	91.4	3.047**
	11	92.1	3.249**
	12	90.4	2.755**
	13	90.5	2.802**
	14	87.3	1.858
	* significant at 5%
	** significant at 1%
	*** significant at 0,1%

http://www.sophos.fr/pressoffice/news/articles/2006/11/dirtydozq306.html

http://www.infos-du-net.com/image/Spam-image,0101-5735-712----jpg-.html#

We have daily data. These will be presented in the section IV.

It is the maximum number of observations that we retained following the unavailability of historical stock quotes for some securities.

Results are not reported in this paper. Nevertheless, they are available from the author.

In previous papers respectively relating to the impact of stock spams on volumes and on returns, we have obtained positive and significant variations in volume over the entire period of test. However, returns were affected positively the first day of the event and negatively the following days.

Results are not reported in this paper. Nevertheless, they are available from the author.

Buyers and sellers of penny stock's securities, by their movements of transaction, contribute to increase volatility. However, the rise of volatility during the second period from t = 5 to t = 13 is less pronounced than the effect observed during the first three days. This can be explained by the fact that change in volatility during the first days is generated by a widening of the gap [lowest price -highest price] from only one side (the side of the highest price) insofar as the investors respond to spams by massive purchases of stocks, which increases the share prices. While in the second period (from t =5 to t =13), the widening of the range is rather done on both sides because the investors who were purchasers during the first period are very quickly transformed into sellers when they realize that the information to which they have responded is a swindle.

Finally, the use of the cross-sectional Student test as well as the Cowan rank test gives us the same result: stock spams have a positive and significant impact on the volatility of penny stock's securities. This finding leads us to record that the business of spamming is flourishing and continues to make money for spammers. Indeed, with the use of data more recent compared to those of the works of Bohme and Holz (2006), Frieder and Zittrain (2007) and Hanke and Hauser (2008), we expected that investors have realized that these campaigns of stock spams are scams, and therefore, no impact will be observed on volatilities. However, we find that investors still continue to believe in such information in the hope to become rich.