Ludovic Moreau
email: lumoreau@u-paris10.fr

M Aglietta

V Bignon

M Boutillier

M Flandreau

R Sylla

L J White

A Century of Bond Ratings as a Business 1

Keywords: industry study, bond ratings, financial regulation JEL codes: L84, G18, G24

Historical accounting datasets about a leader of the bond rating industry have been gathered in order to provide an unprecedented long term view on this business. To better judge of the dynamics at play, similar data for representatives of older and broader business fields is also introduced. Overall, this empirical discussion plays down the importance of regulatory « licenses » given to bond rating firms and puts forward the coming of a « modern » business model where issuers pay for ratings.

Résumé

En utilisant l'historique comptable d'une grande agence de notation, on fournit une vision sur le long-terme du dynamisme de cette activité. Par souci de comparaison, on utilise des données semblables pour des activités connexes à la fois plus anciennes et plus courantes. Cela conduit à relativiser l'importance de l'utilisation réglementaire des notes et à se concentrer sur la mise en place d'un business model « moderne » où les émetteurs paient pour l'activité de notation des titres de dettes.

Mots clés: étude industrielle, notations d'obligations, réglementation financière.

JEL codes: L84, G18, G24

In the early decades of the past century, a couple of American firms started to provide bond ratings and managed to run a lasting business out of it. In the 1930's, American Financial authorities introduced rules relying on these privately issued opinions. This particular way of dealing with regulation was then left unchallenged and even extended, particularly over the last quarter of the twentieth century. Once authorities had started using bond ratings straightforwardly, Partnoy (1999, pp. 681-703) argued that private third party certifiers relevant to the marketplace were bound to end up mostly providing regulatory benefits or "licenses".

To provide a ground for such a "regulatory license" theory, one may want to draw a parallel between the profits of bond raters and the extent of statutory rules relying on bond ratings. This is required to provide more than yet another statement on how the rating business might have been regulatory inflated. Yet relevant data is scarce… This paucity of hard information has not fostered the coherence of the views on the bond rating business. For example, Partnoy (2006) also argued that what made the rating firms attractive to investors was their involvement in the structured finance issuance boom and the fact that they had repeatedly escaped liability in courts.

What did really matter? More broadly, does the rating business truly appear so special that one needs to turn to such "exogenous" factors? In an attempt to deal with these questions, the present paper introduces a number of genuine datasets. Moody's is the rating firm that has managed to stay independent for the longest time and to go public on the stock market on its own. Historical accounting datasets about this corporation then provide an unprecedented long term view on the dynamics of the rating business. In particular, since Moody's changed its business model in the late 1960's, this allows an historical discussion on the role of the fee structure chosen by rating firms.

Part 1 of this paper gives background information about the development of the rating business. Part 2 introduces an empirical discussion using accounting data unavailable to date. The two main results are the followings. First, the pattern of earnings of the early Moody's corporation does not fit the regulatory license theory well: it becomes less sustainable on the wake of the first financial rulings using ratings as straightforward inputs. Not surprisingly, equity investors also do not appear to value that much the coming of a regulatory rent. Secondly, when rating firms exhibited impressive profitability measures over the early 2000's, these remain in line with a trend that can be traced back to the mid 1970's and to the transition to an "issuer pay" business rating model. Part 3 provides a discussion of these results by looking at the structure of the American bond market since 1945. After 1970, "modern" (publicly available and issuer-paid) ratings were of particular interest for foreign corporate bond issuers increasingly entering the American bond market and for the new major buyers of non-Treasury bonds (households and foreign holders). Concluding remarks comes back on the issue of dealing with two rating business models in regulatory discussions on the wake of the structured finance crisis.

-Some Remarks on the History of the Bond Rating Business

Credit reporting or mercantile credit agencies have sometimes been introduced as the main precursors to bond rating firms (see Cantor &Packer (1994, p.1 col b, §1) and Partnoy (1999, §2 p.636- §1 p.637)). Sylla (2002, pp. 19-25) introduced a broader perspective by raising a simple question: why did rating firms develop that late in bond market history? Indeed, investors had been buying bonds since the sixteenth century centuries and a meaningful experience of rating securities came as late as the early twentieth century 1 . The main reason is that bond markets remained mostly sovereign debt markets: businesses in Europe met most of their capital needs thanks to bank loans and stock issues. Dating back to the 1850's and focusing on railroads in its early decades, the corporate bond market can be considered as an American financial innovation that later spread to the rest of the world. The bond rating business was born out the needs coming from this innovation. Before the creation of any rating firm, the American corporate bond market experienced half a century of spectacular growth. This growth went along with 3 historical developments that provide a picture of the boundaries of the rating business:

(i) the credit-reporting or mercantile credit agencies: The critical role played by this intermediary in the development of credit lines among American business networks is studied in full length by [START_REF] Olegario | A culture of credit: embedding trust and transparency in American business[END_REF]. Precedents to these institutions developed well before the rise of the American corporate bond market, the business however especially took hold over the second half of the nineteenth century. Founded in 1841 by L. Tappan, the Mercantile Agency gathered information on the business standing and creditworthiness of businesses all over the United States (ii) the specialized financial press: Over the second half of the nineteenth century, journalists created the business of supplying comparative information on the assets and earning power of the companies. As the editor of The American Railroad Journal, H. V. Poor gathered and published systematic information on the property of railroads, their assets, liabilities and earnings. In 1868, he started a firm to publish yearly his Manual of the Railroads of the United States, which reported financial and operating statistics for most of the major American railroads and was widely recognized as the authoritative source of such information for several decades (see Chandler (1956, chapters 9 and 11)). In 1906, J. Moody entered this business and underwent solvency on the wake of the 1907 panic. Reincorporating his company in 1908, he innovated with the publication of a rating manual in 1909 (see Harold (1938, pp. 9-12)). In 1916, the Poor Company entered this bond rating business as a natural outgrowth. In 1922, the second new entrant, Standard Statistics, was also a financial information company. In 1924, the third and last firm to enter the early rating business was the Fitch Publishing Company, a security quotation publisher since 1913. In 1941, two of these rating firms merged to form Standard &Poor's (S&P). In 1966, S&P was taken over by the publishing giant McGraw &Hill (MG&H).

(iii) the investment bankers [START_REF] Flandreau | Bonds and Brands: Intermediaries and Reputation in Sovereign Debt Markets 1820-1830[END_REF] studies sovereign bond markets over the 1820's and concludes that the hierarchy of underwriters was a proxy for the one of issuers. As financial intermediaries for the railroad securities, American great investment bankers had a strong reputation incentive to monitor every deal. Their access to the suppliers of capital through a vast network, often international, was at stake. They did act as investors' insiders, insisting that issuers provide all relevant information related to company operations on an ongoing basis and sometimes requiring seats on the board of directors of corporations. At the turn of the twentieth century, the size of the investing class started to grow and this pivotal role of investment bankers started to be questioned. The broad public's request for more publicly available information on the quality of investments first reflected in State "Blue Sky" laws and would later bring both mandatory disclosure laws for issuers of securities and the Securities and Exchange Commission (SEC). Embedding trust in a number of key individuals was certainly better suited to closed networks of wealthy individuals (Sylla (2002, 1 p. 34)). Especially with the 1912-1913 Money Trust investigation in mind, the rise of the rating business looks quite contemporaneous to the weakening of great investment bankers over the 1920's. Harold (1938, 2-3 p.16) states that "in no circles has the attitude toward bond ratings been more hostile than among investment bankers" and mentions a number of attempts to influence ratings by some investment banking houses.

Reviewing the three historical precedents, the rating business can be considered at the border of two different activities. Historical ties with (i) and (ii) portrays this business as a mere development of financial journalism, the purpose of which is to provide relevant information to investors. With (iii), however, private certification is stepping in. Great American investment bankers did not give reviews on the deals they were selling; they did act as quality certifiers who ended up placing people on the board of companies... Sylla (2002, pp. 25;33) mentions a transfer of reputational capital from these quality certifiers to the rating firms and focuses on the historical forces that may have led to it. But the extent of this transfer remains unclear… Since the development of rating firms came to a standstill with the 1940's, Partnoy (1999, p. 646) argued that any reputation that might have been gained over the 1920's had quickly gone away on the wake of the 1929 crisis. Remember that the market structure remained quite concentrated (4 established firms after 1924, back to 3 following the 1941 merger that created S&P). Furthermore, weakened market leaders of the rating industry became external growth opportunities for related business over the 1960's. This may be interpreted as a poor ability to deal with the post World War II (WWII) American capital markets. In any case, at the turn of the 1970's, rating firms changed their business model: when they had formerly been selling their ratings to investors in every manner manageable, they started charging issuers 2 . The reasons for this change still remain to be established (see White (2009, pp. 6-7)). Existing comments can be sorted in two broad views of rating industry development:

 Financial market development: With the post-WWII decades, growth brought more stable and higher corporate earnings and this allowed the use of internal funds for investment purposes. Market finance was also replaced by other institutional means: over 1948-1965, about half of debt securities were privately placed; furthermore, commercial banks introduced term loans as an alternative to bond financing (see Atkinson (1967, Chapter II); Sylla (2002, 5 p. 30) and (Kemmerer (1952, pp. 459-481)). The change in the fee structure can then be presented as a natural outcome of both the poor profitability of a business model dating back to the 1910's and a kind of risk free environment. Yet, explaining the pressure on a business model is hardly explaining the birth of a new one3 . With the "functional" view introduced above in mind, rating firms may be reviving their links with private certification at the onset of a new era of market finance. The issuer-pay ratings were first introduced in 1968 by S&P for sub-sovereign issues.

The increased role of ratings in this field would soon bring public attention (see NYT (1972) andTwentieth Century Fund (1974)). Furthermore, following the Penn central default in 1970, commercial paper issuers began to solicit ratings to lower their capital costs and revitalize confidence. This succeeded in changing market perception, a result that certainly expanded the market niche for charging issuers (Cantor &Packer (1994, p.4)). Last but not least, comparing nineteenth, early twentieth century and modern (1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006)(2007) sovereign bond underwriting practices, Flandreau et al. (2009b) provides results tending to show that modern investment bankers outsourced certification tasks to rating firms.

 Regulation: in the 1930's, American authorities began to use bond ratings for the purpose of regulating bank and insurance investments (see, infra, box 1 p. 12). On the wake of the financial difficulties of the early 1970's, the practice of incorporating ratings in regulatory procedures was revived by the Securities Exchange Commission (SEC) in its regulation of broker dealer in 1975. This rule referred to "Nationally Recognized Statistical Ratings Organizations" (NRSRO), a new category that paved the way for the use of ratings by other official bodies (Partnoy (1999, pp. 693-698;and 2001, pp. 16-17) provide descriptive statistics on how regulatory uses mushroomed over the last quarter of the twentieth century in America; JFRAC (2009, pp. 87-118) gives a thorough global picture of financial regulations using ratings). As regulatory references to ratings multiply, ratings starts deriving value from ensuring that a security complies with these regulations. Shifting away from the sale of information to investors, bond rating firms may end up merely providing "regulatory licenses" (see (Partnoy (1999, pp. 681-703)). In any case, regulation first increased demand with regulatory procedures starting in the 1930's and then contracted supply with the NRSRO category as a barrier to entry from 1975 (White ( 2002)). The 2006 Credit Rating Agency Reform Act targeted the supply side: entry was made easier by working on the NRSRO designation process. A focus on regulatory interferences tends to downplay the coming of the modern business model. If anything, this evolution may be interpreted as a move to better extract a regulatory rent.

These two broad views of rating development remain open to debate and pervade most discussions on the rating business. As can be seen in appendix A, a recognised source of investor research considers the modern Moody's as a part of the publishing industry (broken down as follows: magazines (Martha Stewart Living), financial information publishers (McGraw-Hill, FactSet, IHS Inc., Moody's, Thomson) and internet directories (R.H.-Donnelley, Monster Worldwide)). On the contrary, Partnoy (2006, p. 67) argued that Moody's had financial ratios in no comparison to major financial publishers and a stock price evolution in no comparison to major broker dealers. Partnoy (2006) further put forward three factors to explain the impressive business record of the modern Moody's: the critical role that rating firms played in structured products issuance; regulatory benefits or "licenses" and an immunity from civil and criminal liability for malfeasance. Which role do these factors really play? for example, Hill (2004, pp. 67-68) noted evidence of a "sticky institutional norm" favouring two ratings from S&P and Moody's that could not clearly be related to existing regulations. The following empirical analysis aims at contributing to this debate.

-Empirical Analysis

Since most rating firms are subsidiaries, access to relevant data is a major issue. This issue is obvious in existing attempts to run "traditional" industry analyses following the "Structure Behavior Performance" breakdown (see White (2002, pp. 44-51) and Smith &Walter (2002, pp. 293-305)). Similarly, Flandreau et al. (2009a, pp. 11-12 and table 1) provided by an interesting discussion of the interwar rating business that ended up focusing on output and signals (number, frequency and price of publications, branching). Overall, this difficulty has brought a pervading use of proxies in analyses of the rating business. Musing on the impact of regulatory licenses, Partnoy (1999, pp. 649-650) looks at the workforce and the number of outstanding ratings as proxies for business activity. Musing on the impact of competition, Becker &Milbourn (2008) looks at outstanding American corporate ratings to judge on how market shares evolved over 1998-2006. This use of proxy has also been endorsed by regulators. In its first report on NRSRO as required by the 2006 Credit Rating Agency Reform Act, the Security and Exchange Commission (SEC) produced a view on competition by looking at outstanding ratings (see SEC (2008, pp. 34-36)).

This paper aims at adding to these analyses by keeping the discussion on financials.

Consider that Moody's is the only rating firm that went public as an independent corporation and that it managed this for quite limited periods of time (1929-1962; 2000-today). For the first time, the resulting public accounting information has been compiled into single datasets. Note that Moody's has always been one of the very few leaders of the rating industry. Furthermore, the "old" Moody's follows an investor-paid business model while the "modern" one follows an issuer-paid business model. This makes this historical discussion valuable since modern analyses of the two business models are bound to compare small competitors to market leaders. To better judge of the dynamics at play and to deal with periods when Moody's is not public, similar datasets have been gathered for Dun &Bradstreet (D&B) and McGraw &Hill (MG&H) (that is, respectively, the credit reporting firm that controlled Moody's from 1962 to 2000 and the publishing giant that counts S&P as one of its business segment since 1966). These may not be the best business peers but they make sense considering the discussion on the history of the rating business introduced in section 1 4 .

The bottom line: power in earnings

For any firm, looking at net income brought the greatest historical range. To get a middle term view, graph 1 does not display annual figures but rolling 3 year Compound Average Growth Rates (CAGR). These rates may be compared to the ones of the business peers evolve and to how the profits of all American firms and of all American Finance &Real Estate corporations evolve.

Graphically, bear in mind that any positive figure means a positive growth: whatever the volatility, a lasting stage of positive figures means good times. 4 Take the example of the interwar period. Flandreau et al. (2009, 4 p. 10) mentioned that business peers could be found in other financial information businesses developments "tying advice to data" (business forecasts, for instance (see [START_REF] Favero | Weather forecast or rain-dance? On inter-war business barometers[END_REF]). Yet finding public data over decades from a typical business entity of this kind is quite a challenge (Flandreau et al. (2009, 1 p. 11)). Here, one can still regret that McGraw &Hill (MG&H) is not that associated with financial press at this time. Still, this company is dealing with specialized press publishing and will soon step in the financial press. Needless to say, a true empirical follow-up to [START_REF] Sylla | An Historical Primer on the Business of Credit Rating[END_REF] would have required continuous accounting data over the 1920's from a credit reporting firm, from a financial press publisher and from an investment banking house.

From 1925 to 1930, the early Moody's corporation has an impressive pattern of earnings as compared to the average American firm and even to the average American financial firm. After the Great Crash, the investor paid rating firm recovers with the real economy (dotted line). This means earlier than the average financial company and than MG&H. This recovery does not last long though. Bad times start at the end of the 1930's first in the real economy and then in finance.

As compared to MG&H, Moody's first resists better but then it is unable to profit from growth in real activity over the early 1940's. Positive signs appear at the end of the 1940's but business growth is brought to a halt at the turn of the 1950's. This slowdown is weathered out by MG&H. Both Moody's and D&B suffer the blow, but the rating company shows difficulties earlier and to a more preoccupying extent than the mercantile agency. Over the early 1950's, Moody's recovers yet MG&H and D&B seem well more attractive. This said, for the first time since 1929, a truly sustainable pattern of earnings is managed by Moody's over 1951Moody's over -1960. . This allows the rating company to face the slowdowns in real activity and finance occurring at the end of the 1950's.

When D&B settled to buy Moody's in the early 1960's, the investor paid rating company was indeed an attractive business opportunity. This business record hardly fit the extent of the regulatory rent coming from the use of ratings in financial regulations [START_REF] Partnoy | The Siskel and Ebert of Financial Markets?: Two Thumbs Down for the Credit Rating Agencies[END_REF]). As can be seen in box 1, the first rules came at the beginning of the 1930's and the regulatory reliance on ratings reached its full extent at the end of the decade. Yet graph 1 shows that Moody's does not find a sustainable pattern of earnings before the early 1950's and, further, that recovery occurred earlier in related activities. These findings are quite unsurprising considering that the "regulatory licenses" came as American regulators enacted the Securities and Exchange Act (1934). This Act codified mandatory financial information by corporate issuers: a quite valuable service of early investor-paid rating firms would now be obsolete... From 1961 to 1998, graph 1 displays data only for the business peers. MG&H suffered at the turn of the 1970's and of the 1990's but each time quite quickly recovered. In contrast, D&B did not meet troubles before the early 1990's but then encountered more structural difficulties.

From 1999, the modern Moody's power in earnings has proved impressive with most of the 3 year CAGR well above 20%. This is in sharp contrast with D&B's experience, but MG&H's earnings have also shown a nice pattern since the second half of the 1990's. To some extent, the expansion of the modern Moody's is coherent with the pattern of financial profits. This said, the rating company has the up side, especially in weathering out a downward trend starting in 2003.

However, the last observation shows that D&B is the only one to face the bust of the structured finance episode. 12 * : this box introduces only the first regulations in History, see supra, section 1 p. 7-8 for how the regulatory rent evolved through the second half of the twentieth century

Box 1 -The Coming of a Regulatory Rent* Insurance:

In an answer to the 1931 crisis, the New York State Insurance Department ruled in 1932 that bonds rated in the first five rating grades by one of the rating agencies would be considered eligible for amortization on a cost basis. The decision was criticized over the 1930's: extensive use of amortization led to dubious valuations in front of quite low market prices. In 1940, the National Association of Insurance Commissioners (NAIC), the follower of the NCIC, however stood by this practice and stated that amortization would be given to bond rated: i) in the first four grades by two rating agencies, ii) in the first five grades by three agencies or iii) in the first five grades by two agencies plus a pricing requirement (priced at 55 or better in September, October, and November). In 1953, the NAIC reformulated the eligibility criteria in two tests. "Test 1" was a rating from the first four rating grades of one of the accredited agencies or a number of balance sheet requirements. "Test 2" mainly dealt with earnings requirements (see [START_REF] Atkinson | Credit Ratings and Complementary Sources of Credit Quality Information[END_REF])

Banking

By 1930, the Federal Reserve had begun using bond ratings in their examination of member bank portfolios. This use could be considered "informal". In 1931, the Comptroller of the Currency officially adopted ratings to measure national banks' bond quality: bonds rated Baa/BBB or above would be carried at cost; bonds with lower ratings would be marked to market thanks to fractional write-offs. In tune with previous insurance practices, this ruling was well received at the time (see WSJ (1931a &b); Harold (1938, p.27)). During the following years, many State banking superintendents adopted the Comptroller's plan (see Harold (1938, pp. 27-28)).

In 1935, Amendments to the Federal Banking Act specified that all national banks were subject to the orders of the Comptroller's Office as for the securities they might purchase for their own accounts. On February, 15th 1936, the Comptroller issued a new ruling stating that "the purchase of investment securities in which the investment characteristics are distinctly and predominantly speculative, or investment in securities of a lower designated standard than those which are distinctly and predominantly speculative, is prohibited". A footnote added that "the terms applied herein may be found in recognized rating manuals" (see Harold, (1938 p. 30)). This more radical decision spurred unprecedented hostility about the use of bond ratings as tools to influence the structure of commercial banks portfolios (see WSJ (1936a &b)). It also created confusion about what the footnote exactly meant because it was relying on an unsettled market convention: Moody's kept interpreting the ruling as pointing to Baa/BBB as a cutoff but the American Banker considered A/A (see MIS (2004, pp. 1-2)). The Comptroller refused to make this point clear and then stated that ratings were not "the sole criterion, or even a necessary criterion, for judging whether or not a particular bond was eligible for purchase by a national bank" (see WSJ (1936c &d)). Nonetheless, controversies did not quiet down (see WSJ (1936e)). On June 27 th , 1938, all the federal banking authorities published a joint statement more in tune with the 1931 ruling: bonds would be divided into groups and that bond issues bearing any of the first four rating grades were to be given a privileged status (by being valued at their purchase price or at par and by being therefore insulated from day-to-day price fluctuations).

The use of ratings by all banking authorities was now clearly set. For individual banks, this meant that informational requirements and uncertainties would be minimized for investments in the top four rating categories while lower rated or even unrated bonds would require an added burden of justification. In 1949, the Executive Committee of the National Association of Supervisors of State Banks joined the Federal authorities in reaffirming the process outlined by the 1938 statement (see Federal Reserve (1949)). 13

Financial ratios

Using accounting data, a number of financial ratios can be drawn over the past 80 years.

A natural starting point is to look at basic profitability measures. operating margins completely change the picture of graphs 2 and 3 to the advantage of MG&H. This is quite welcome since this new picture is more in tune with the net earnings dynamics displayed in graph 1. Turning to the right-hand side of the graph, the modern Moody's corporation again appears to be something else by extracting between 30 to 60 cents in operating income out of every dollar of sales. The modern Moody's started with an operating margin well above the ones D&B's and MG&H's. Furthermore, this margin grew at a fast pace when the two others stayed between 15% and 30%.

Looking at raw profits, segment reporting from D&B and MG&H can be used in an attempt to overcome Moody's accounting data limitations. Graph 4b then shows operating margins for the modern Moody's next to the business lines incorporating rating firms. By starting from the right, note how the profitability of MG&H's financial services division shows a trend keen to the one of the modern Moody's. Furthermore, the financial service division of MG&H displays an impressive trend line going back to 1975. Such a trend could hardly be achieved without a steady and long term profitability of the modern rating business. This interpretation is further confirmed by the operating margins of the D&B business line incorporating Moody's from 1969 to 1988.

The opinion of the stock market

When each of the ratios used above has some advantages, it is generally held that stock markets generate better estimates than any model using financials as inputs. For this reason, graph 5 provides historical Price to Earnings (P/E) ratios. This measure has some disadvantages. First, low or negative earnings per share bring hardly informative extreme values. Secondly, this measure may be manipulated by managers. Looking at horizontal patterns is a way of overcoming these issues.

The old Moody's corporation went public in October 1928. Over 1929, Moody's stock was on average bought at more than 8 times its earnings, as compared to more than 10 for MG&H's stock and for the SP500 average. Moody's stock endured the market crash yet equity investors' confidence never came back to pre-crisis levels: the P/E moved around 5 up to 1939. As for the comparison to the business peers, extreme values aside, the reference point for MG&H remained somewhere around 10 until 1938. Note that confidence fades from 1937 to 1941 and that this holds for Moody's, MG&H and the SP500. It seems that equity investors do not see a particular windfall in the final rulings standing by the use of ratings as inputs and establishing the regulatory license for an undetermined period of time (see, supra, box 1: 1938 for banking and 1940 for insurance). From 1941 on, the SP500 bounces back and MG&H quickly recovers to reach the 10 P/E reference point in 1945. Moody's P/E also recovers but stagnates; thankfully it does not suffer the market slowdown occurring over the second half of the 1940's. Looking at 1949 and[START_REF] Guthmann | The Movement of Debt to Institutions and Its Implications for the Interest Rate[END_REF] is still an open shot between Moody's, MG&H and D&B. An upward trend starts in 1950, and Moody's appears standing aside from it by remaining around a 5 reference point. By contrast, D&B early takes the up side to the SP500 while MG&H exhibit volatility before consistently beating the market at the end of the decade.

The SP500 P/E stagnates over the 1960's as compared to both MG&H and D&B. Things do not change for the best with the 1970's. A couple of years aside, MGH's P/E ratio remained aligned to the SP500 average over the last quarter of the century. Considering the last decade, the reference point for this stock would be somewhere between 20 and 25. D&B managed the slowdown of the 1970's better than MG&H. These good times ended in the late 1980's: D&B de-correlated from the SP500 average and exhibited volatility. Over the last decade, a reference point for this stock would be around 20. In September 2000, the modern Moody's corporation concluded its spin-off from D&B.

Confidence in the future of this independent rating entity was high among investors: over 2000, they proved on average ready to pay more than 50 times Moody's earnings to become shareholders. The P/E ratio then evolved but the reference point for this company could still be thought around 50 until 2004. Confidence has then strikingly faded… Moody's line joined the one of MG&H in 2006 and the three lines became one over 2007-2008. Bear in mind that equity investor trust in a dedicated rating entity like Moody's eroded well before any public concern over structured finance securities.

-Result Discussion: the Structure of the American Bond Market since 1945

Is there something so special about the rating business that one needs to turn to exogenous factors? To deal with this question, a narrow business analysis is hardly enough. Musing on the historical development of rating firms Sylla (2002, 3 p. 33) pointed to macroeconomic factors and mentioned "historical rhymes" between the 1920's and the last quarter of the twentieth century.

In an attempt to further link the history of rating firms to the one of the American corporate bond market, this section provides descriptive statistics allowing a discussion of the microeconomic figures presented above. These statistics rely on a normalized body of datasets going to date (the Federal Reserve flow of funds). This is not the case for information on earlier periods. For the sake of comparison, some data from the most common sources on 1900-1945 are given in appendix C but the discussion here will focus on the post-World War II decades.

The first matter of interest is whether the coming of financial globalization and financial deregulation brought unprecedented levels of bonded debt after 1970. On graph 6, a striking feature is the decrease of Federal bonded debt which has steadily freed capital over the post world-war II era.

Another important point is that other areas of the American bond market have not that benefited of this structural change. American sub-sovereign government and corporate bond market issues have remained around a 10% level achieved by the late 1950's. Municipals then settled to a 8% level after 1995. The American corporate bond market share proves a little more volatile: there is a 12% peak over 1969-1971, a 8 % low over the mid-1980's and a 12% peak over the early 2000's. The rise of American corporate bond levels over the second half of the 1990's seems to benefit from the decreasing share of Treasuries over 1995-2001. Overall, the main long term trend of interest is the rising share of foreign corporation bonds since the 1990's: when this share had been below 2% since 1945, a steady growth brought it around 4% since 2003. This means good business for modern issuer paid rating firms: is there a better way than buying the services of this intermediary to manage a bond issue on the American field? By providing a perspective on the relative importance of three fields at the core of rating activities, graph 6 mostly evidences a straightforward consequence of financial globalization as a structural factor behind "modern" rating success.

When graph 6 provided a look on shares in the supply of credit market debt, Graphs 7a &b turn to the demand side: who are the key holders of American &Foreign corporate bond issues and of American sub-sovereign issues? These are the ones truly benefiting of the existence of a pervading bond rating system. Graph 7a gives the holdings of sub-sovereign American credit market debt. At the beginning of the 1970's starts a dramatic decrease of commercial bank holdings. The share of insurance companies then also starts decreasing. This structural change reaffirms households and non-profit organizations as major holders of municipal debt. Furthermore, over the 1980's, the share of mutual funds rises from negligible to more than 20%. Overall, these changes portray a shift from traditional institutional investors to the investing public. While ratings may have some coordinating values for institutional investors, households and individuals may be more eager to check these specialized third party opinions.

Concluding Remarks: Two Businesses and One Issue for Regulatory Policy

Ratings were for long an American bond market feature. Thanks to the financial globalization starting in the 1970's, they have now become a key institution of the global bond market. Over recent years, the role of major rating firms in the structured finance episode has been discussed in news editorials and political circles. On the wake of heavy market disruptions, this heated debate reached an unprecedented scale when basic knowledge on the rating business is still missing. This is a concern when regulators are bound to act against such a background. There was little detailed discussion on competition in the rating business when the 2006 Credit Rating Agency Reform Act was passed as a "duopoly relief" regulation. Opening public registration to new entrants was certainly a good thing; yet it was bound to do little about market shares since market leaders enjoy strong reputational advantages (see Hill (2004, 4-6 p. 85)). The policy move did little more than bringing major rating firms under the Securities and Exchange Commission (SEC) authority and requiring a number of key disclosures (information on interest conflicts management, performance statistics, methodologies, etc.).

In December 2008, the SEC adopted new rules of this kind as a policy response to the role of ratings in structured finance. The agency had proposed to force full disclosure of rating histories to allow better performance assessments. This was hardly manageable for investor-paid rating firms and the final rule settled for a limited and random disclosure (see SEC (2009) on rule 17g-2). This episode clearly shows that dealing with two businesses in one remains an issue for current regulatory initiatives. Furthermore, sketching a competition policy remains a challenge for regulators. A former insider of the rating industry links troubles with structured finance ratings to competitive pressures allowing arrangers to take advantage of "rating shopping" (Fons (2008, p. 7)). After a year-long inquiry and campaign by the New York State Attorney, the "Cuomo agreement" precisely targeted related bad practices (see [START_REF] Cuomo | Attorney General Cuomo Announces Landmark Reform Agreements with the Nation's Three Principal Credit Rating Agencies[END_REF]). Moody's (2009) publicly acknowledged that the credit rating industry needed third-party help in order to properly deal with competition and information disclosure. In December 2008, an ambitious move was set aside by the SEC. Re-proposed in February 2009, this proposal would force information sharing between an issuer-paid rating firm and its NRSRO competitors (both issuer-paid and investor-paid, see SIFMA ( 2009) commenting rule 17g-5). This would certainly fundamentally change the shape of competition in the rating business. This initiative aside, an open question is whether competition policy should discriminate between the two existing rating business models. Drawing on Moody's history, this paper provided an unprecedented discussion on this issue. While this company has always been one of the few leaders of the rating industry, it was paid by investors until the early 1970's and has then been paid by issuers.

Looking at a number of financial ratios, the modern issuer paid rating firm indeed appears to be something else. A number of arguments have been put forward to explain this profitability:

(i) the pervasive use of ratings in financial regulations, (ii) the little success of legal cases brought against rating firms (iii) the involvement in the structured finance issuance boom.

This paper provided a new perspective on these arguments. First, the business record of the early Moody's corporation shows no particular windfall in the coming of the first American regulations using ratings as a straightforward input (i). Note, by the way, that Moody's has for long publicly supported the withdrawal of ratings from financial regulations (see, for example, MIS (1995)). Secondly, Moody's recent profitability is in line with profitability trends that started in the 1970's and this suggests paying more attention to the coming of a modern business model (iii).

Whence comes "modern" ratings success? Musing on the factors listed above provides little insight if the state of fixed income markets is missing. Ratings are nothing but an institution of capital markets and the discussion must somehow turn to macroeconomics (Sylla (2002, 3 p. 33)). Reviewing the structure of the American credit market since 1945, two points may help to understand the success of modern ratings. On the one hand, American issuer-paid ratings were particularly interesting for a rising star of credit market borrowing over the last quarter of the twentieth century: foreign corporate bond issuers. On the other hand, the holdings of non-Treasury bonds show a decrease in the share of major institutional investors compensated by an increase in the one of households and foreign holdings. These holders are great beneficiaries of the public good feature of issuer-paid ratings (commented by Schwarcz (2002, 3 p.8)). These two points are first steps in building a better picture of the interplay between credit market and rating business development.

  through a network of agents and sold reports to subscribers. It became R.G. Dun & Co in 1859. The company's subscribers grew from 7,000 in the 1870's to 40,000 in the 1880's, and by 1900 its reports covered more than a million businesses. In 1849, J. Bradstreet founded a similar firm, which by 1857 was publishing what apparently was the world's first commercial credit rating book. In 1933, the Dun and the Bradstreet companies merged to form Dun & Bradstreet (D&B). In 1962, Dun & Bradstreet (D&B) acquired Moody's Investors Service, the bond rating firm that J. Moody had created in 1909.

First

  figures around 10-15%. Plotted against MG&H, D&B has the downside from the early 1940's to the early 1960's. The situation is then strikingly reversed and this occurred at the time when Moody's was incorporated as a business line... The modern issuer paid Moody's displays very high and volatile ROA. A sharp upward trend over 1996-1999 is brought to a halt in 2000. The recoveries are always ended by further declines but the ROA remains very impressive (>25%):this pattern may be more linked to asset management than to external shocks.

Graph 6 -

 6 Debt Securities as a Share of Total Credit Market Borrowing Graph 7b gives the holdings of corporate and foreign bonds. A major feature is the long term decrease in the share of life insurance companies. This decrease is first compensated by a rising share of pension and government retirement funds. But this share stagnates from the mid-1960's to the early 1980's. The structural decrease in life insurance holdings is then eventually compensated by two structural First, there is a recovery in household holdings starting in the mid 1960's. Brought to a halt in the late 1970's, it bounces back over the 1980's as pension and government retirement fund holdings start dropping. Secondly, there is an impressive growth in foreign holdings. The growth starts Graph 7a -Share of Total Municipal Securities &Loans Holdings Graph 7b -Share of Total Corporate &Foreign Bonds Holdings in the late 1960's yet the share of foreign holdings remains under 8% until 1981. This share then increased at a fast pace and dominates since 2006. Again, these remarks point to a shift from traditional institutional investors to holders that greatly benefit from publicly available third-party rating on bond quality.

This does not mean that there were no previous experiences with rating securities; but their scope remained limited. According to the founder of the first rating firm: "I cannot claim much credit for creating the idea, and certainly I think the general use of commercial and credit ratings had something to do with bringing the idea of possible bond ratings to my mind. While no one in this country had attempted such a thing as investment ratings by means of symbols, yet even in those days bonds were classified into groups according to quality and salability, especially by large investment institutions, such as insurance companies. Moreover, there had existed for a considerable time, I think, a bond rating system in Vienna and also I believe in Berlin. These foreign systems had been developed by symbols, and the Austrian Manual of Statistics, which carried these symbols, was quite well known in Europe, although not at all in this country." (seeHarold (1938, p. 11)).

S&P began charging municipal bond issuers in 1968 and most other issuers in 1971, Fitch and Moody's began charging corporate issuers in 1970 (seeCantor &Packer (1994, p. 4)).

There have been organizations following the "old" business model since the coming of the modern rating business (seeBIS (2000, p.25)). Overall, the global rating industry is however heavily weighted toward the modern rating business model with an oligopoly allowing the 3 "modern" rating firms(Moody's, S&P and Fitch) to own more than 80% of the global market.

Such an interpretation tends to downplay numerous statements made on the wake of the structured finance crisis. Most commentators had ended up associating the impressive profitability measures from rating firms to their involvement in structured finance securities issuance during the speculative episode of the early 2000's. Going into more details, the SEC showed that staffing among the three leaders of the rating business followed the rise in activity and in revenues in the RMBS field but not in the CDO field (see Appendix B). This certainly raises issues about business management. This said, overall, the recent profitability of the leaders of the rating industry remains in line with a trend that can be traced back to the mid 1970's. This trend occurred on the wake of the transition to the modern "issuer pay" business rating model in the early 1970's.

Turning to the demand side, the most impressive figures are the total share of financial intermediaries, which, at the turn of the 1940's, rises above 60% on both graphs. This intermediation of credit market borrowing is a key feature that will last for decades after World War II (see, supra, section 1, p. 7).

Linking the fate of the early rating business to this structural change is an avenue for future work.

For corporate and foreign holdings, the share of all financial intermediaries rises steadily. As for the 1920's, the main channel is the traditional banking system something that can more easily seen in the figures reported by [START_REF] Guthmann | The Movement of Debt to Institutions and Its Implications for the Interest Rate[END_REF]. On the wake of the 1929 market crash, the decrease in these holdings leaves the field open for a complete predominance of life insurers' holdings. For subsovereign holdings, savings institutions were the major intermediary at the beginning of the century.

Their share has then decreased dramatically. Note also the increasing share of commercial banks starting at the end of the 1920's, which correspond to the decrease in their corporate and foreign holdings documented corporate and foreign holdings.


Sabrina Khanniche
email: skhanniche@groupama-am.fr

Groupama Asset Management

Economix

Evaluation of Hedge Fund Returns Value at Risk Using GARCH Models

Keywords: JEL Classification: G11, G12, G23 Keywords: Hedge Fund, Value at Risk, GARCH models

The aim of this research paper is to evaluate hedge fund returns Value-at-Risk by using GARCH models. To perform the empirical analysis, one uses the HFRX daily performance hedge fund strategy subindexes and spans the period March 2003 -March 2008. I found that skewness and kurtosis are substantial in the hedge fund returns distribution and the clustering phenomenon is pointed out. These features suggest the use of GARCH models to model the volatility of hedge fund return indexes. Hedge fund return conditional variances are estimated by using linear models (GARCH) and non-linear asymmetric models (EGARCH and TGARCH). Performance of several Value at Risk models is compared; the Gaussian VaR, the student VaR, the cornish fisher VaR, the normal GARCH-type VaR, the student GARCHtype VaR and the cornish fisher GARCH-type VaR. Our results demonstrate that the normal VaR underestimates accurate hedge fund risks while the student and the cornish fisher GARCH-type VaR are more reliable to estimate the potential maximum loss of hedge funds.

Since the 1990's, a substantial growth of hedge funds has been observed. According to HedgeFund Intelligence, assets under management accounted for $2, 65 trillion in 2007.

Since 2000, investors have been looking for new investments. Fuelled by the prospect of double and triple-digit returns and the losses caused by the 2000 "dot-com bubble", a wide range of them have sought exposure to hedge funds. They have been getting more and more importance within the financial community since the beginning of this decade. It is thus important to keep an eye on hedge funds particularly because they have attracted a growing share of institutional investors capital. This share has been constantly progressing as a result of the growth of incoming liquidity under their management, then delegate to hedge funds.

The institutionalization of hedge funds raises the issue of a good risk management because public savings are concerned.

In the same way, as hedge funds seek positive absolute returns they therefore engage in aggressive strategies leading to extreme losses. Their potential impact on systemic risk is real. This is particularly true in regard of the leverage hedge funds undertake. It may raise financial stability issues. Its use has been drawn to the attention since the beginning of the nineties. In 1993/94, several highly-leveraged funds were said to have amplified volatility in the US bond market because of the forced liquidation of their positions. When the Federal Reserve unexpectedly raised interest rates, funds were forced to deleverage causing bond prices to fall.

Additionally, to meet margin calls, these funds had to sell off their long positions in European securities markets, transmitting the disturbance to European markets. In the same way, in 1997/98, hedge funds had been able to take substantial short positions in Asian markets as a result of the leverage provided by their counterparties. Those episodes fuelled complaints that hedge funds using leverage destabilize the markets. Concerns about hedge fund leverage reach its highest point with the collapse of Long-Term Capital Management in 1998. In order to deliver attractive returns, the fund had highly leveraged positions as price discrepancies in US securities market were small. In September 1998, LTCM's leverage was very high. It had 600 million in capital to offset positions worth $1 billion. With a high probability of failure, fears arose that the distressed sales of its positions in US securities markets might destabilize financial markets. These fears led to the rescue of LTCM by the FED. In 2007, hedge funds own most of the risky CDO tranches (around 46%, OECD). As they are very big players, the counterparty credit exposures are very large. Problems arose when hedge funds started failing.

Hedge funds have been experiencing extreme funding liquidity risk since banks were short on capital, especially because they relied on leverage. They then faced higher margins. The interaction between the liquidity risk and leverage lead to extreme losses.

Thus, the use of leverage by hedge funds to a considerable extent raises the debate about the implications of hedge fund operations for systemic stability and increases their exposure to market risk. The objective of absolute performance conducts them to such a fragile situation.

And the fact that they implicitly manage public savings justifies this need of a good risk management that has never been more pressing.

All these worries highlight the importance of a reliable way to evaluate hedge fund risks.

However, the traditional measures of risk are not suitable for hedge funds as they not deal with some of hedge funds specificities. Indeed, under the mean-variance approach, hedge funds are very attractive as they generate high returns with low risk. But, the expansive literature about hedge fund returns analysis concludes that the mean-variance approach is not adequate to investigate hedge funds risk. [START_REF] Fung | Is mean-variance analysis applicable to hedge funds?[END_REF], Brook and [START_REF] Brooks | The statistical properties of hedge fund index returns and their implications for investors[END_REF] or Amenc, Curtis and Martellini [2003] demonstrated that the monthly returns distribution of hedge fund strategies present negative skewness and excess kurtosis. Thus, according to [START_REF] Kat | The dangers of mechanical investment decision-making: The case of hedge funds[END_REF], volatility underestimates hedge fund risks since volatility does not give any information about asymmetric distribution of returns and extreme losses. But, as noticed by [START_REF] Scott | On the direction of reference for moments of higher order than the variance[END_REF], investors are interested in higher moments of the distribution function. These drawbacks underline the importance of an adequate measure of hedge funds risk that accounts for extreme losses in order to capture as much as possible their characteristics. This paper is focused on this concern. One aims to go further the traditional measures of risk and introduce a more appropriate method to hedge funds.

While we point out that positions undertake by directional hedge funds strategies 3 increase the market risk and extreme losses have to be considered, we decided to focus on Value at Risk (VaR). VaR is widely used in the practice of risk management within the financial industry.

Its popularity is mainly due to the expression of the market risk in only one figure. For a methodology to be considered sound, the probability of loss should be reflected accurately.

Different types of VaR have been introduced. Riskmetrics assumes that the continuously compounded daily return of a portfolio follows a conditional normal distribution. But in the case of skewed and fat-tails returns, estimates result in an underestimation of the true risk. [START_REF] Favre | Mean-modified Value-at-Risk optimization with hedge funds[END_REF] introduced the modified VaR that is based on the cornish fisher expansion quantile in order to take into account the asymmetry and the kurtosis of the distribution. The historical estimation methodology in turn provides a non-parametric estimate of VaR. It does not make any assumption about the distribution of the portfolio 3 Fund positions are based on the evolution of the market as a whole.

-4 -return. It assumes that the distribution of returns will remain the same in the past and in the future. While these methods have a real contribution, they are not forward-locking and not adapted to capture hedge fund specificities. And as we seek to predict VaR, GARCH-type VaR appears to be more appropriate. In the same way, hedge fund returns exhibit nonlinear dependencies [START_REF] Fung | Is mean-variance analysis applicable to hedge funds?[END_REF]). These nonlinearities are caused by the use of derivatives, leverage and illiquid assets. It is thus important to capture time-variation in hedge fund returns. GARCH-type VaR allows the inclusion of time-varying conditional volatility in the VaR and deals with asymmetric and leptokurtosism phenomenon. Nonlinearities also often arise from the presence of dynamic strategies as hedge fund managers are able to shift their exposures rapidly. To account for the nonlinearity aspect and to analyze the dynamics of hedge fund risk, one also considers asymmetric GARCH models to trace hedge fund returns volatility process more effectively.

On the other hand, because of their private nature, hedge funds do not have to disclose their results. Most of researches on hedge funds are based on monthly data. The "democratisation" of hedge funds has encouraged them to provide data with higher frequency. Another contribution of this paper is the use of daily data to measure all hedge fund strategies risks. This article is organised as follows: Firstly, I present hedge fund strategies according to Hedge Fund Research data provider. Secondly, one introduces the Value at Risk concept and the conventional variance models considered. While the normal distribution is widespread, it however cannot describe fat-tails returns. Hence, the student-t distribution of returns is also applied dealing with leptokurtosis. GARCH, EGARCH and TGARCH process take into account the observed volatility clustering of returns. Thirdly, the GARCH models are applied to forecast a 1 day-ahead Value at Risk for various thresholds (5%, 2.5% and 1%). The forecasts are compared with the VaR based on the standard deviation. Finally, one backtests the different VaR approaches to test the relevance of the VaR models considered.

Hedge fund Strategies Description

Hedge fund strategies may be grouped in three different areas; directional, arbitrage and specific situations. Directional strategies entail a bet on the direction of the overall market.

They involve taking positions on forward and option markets, and in global markets.

Arbitrage strategy managers seek to exploit price discrepancies. Finally specific situations strategy managers tend to benefit from events affecting companies such a merger arbitrage and a restructuration.

Directional strategies

Arbitrage strategies Specific situation strategies

Equity hedge

Macro

Convertible arbitrage

Equity market neutral

Relative value arbitrage

Distressed securities Event driven

Merger arbitrage

To perform the empirical analysis, one uses the HFRX hedge fund indexes. The description of hedge fund strategies is directly taken from HFR documentation.

Global Index

The HFRX Global Hedge Fund Index is designed to be representative of the overall composition of the hedge fund universe. It is comprised of eight strategies; convertible arbitrage, distressed securities, equity hedge, equity market neutral, event driven, macro, merger arbitrage, and relative value arbitrage. The strategies are asset weighted based on the distribution of assets in the hedge fund industry.

Convertible arbitrage

Convertible Arbitrage involves taking long positions in convertible securities and hedging those positions by selling short the underlying common stock. A manager will, in an effort to capitalize on relative pricing inefficiencies, purchase long positions in convertible securities, generally convertible bonds, convertible preferred stock or warrants, and hedge a portion of the equity risk by selling short the underlying common stock.

Distressed Securities

Distressed Securities managers invest in, and may sell short, the securities of companies where the security's price has been, or is expected to be, affected by a distressed situation.

Managers will seek profit opportunities arising from inefficiencies in the market for such securities and other obligations.

Equity Hedge

Equity Hedge, also known as long/short equity, combines core long holdings of equities with short sales of stock or stock index options. Equity hedge portfolios may be anywhere from net long to net short depending on market conditions. Equity hedge managers generally increase net long exposure in bull markets and decrease net long exposure or even are net short in a -6 -bear market

Equity Market Neutral

Equity Market Neutral strategies strive to generate consistent returns in both up and down markets by selecting positions with a total net exposure of zero. Trading managers will hold a large number of long equity positions and an equal, or close to equal, dollar amount of offsetting short positions for a total net exposure close to zero. By taking long and short positions in equal amounts, the equity market neutral manager seeks to neutralize the effect that a systematic change will have on values of the stock market as a whole.

Event driven

Event Driven investment strategies or "corporate life cycle investing" involves investments in opportunities created by significant transactional events, such as spin-offs, mergers and acquisitions, industry consolidations, liquidations, reorganizations, bankruptcies, recapitalizations and share buybacks and other extraordinary corporate transactions. Event driven trading involves attempting to predict the outcome of a particular transaction as well as the optimal time at which to commit capital to it.

Macro

Macro strategies attempt to identify extreme price valuations in stock markets, interest rates, foreign exchange rates and physical commodities, and make leveraged bets on the anticipated price movements in these markets. Profits are made by correctly anticipating price movements in global markets and having the flexibility to use any suitable investment approach to take advantage of extreme price valuations

Merger Arbitrage

Merger Arbitrage, also known as risk arbitrage, involves investing in securities of companies that are the subject of some form of extraordinary corporate transaction, including acquisition or merger proposals, exchange offers, cash tender offers and leveraged buy-outs. Typically, a manager purchases the stock of a company being acquired or merging with another company, and sells short the stock of the acquiring company. A manager engaged in merger arbitrage transactions will derive profit (or loss) by realizing the price differential between the price of the securities purchased and the value ultimately realized when the deal is consummated.

Relative Value Arbitrage

Relative Value Arbitrage is a multiple investment strategy approach. The overall emphasis is on making "spread trades" which derive returns from the relationship between two related securities rather than from the direction of the market. Generally, trading managers will take offsetting long and short positions in similar or related securities when their values, which are mathematically or historically interrelated, are temporarily distorted. Profits are derived when the skewed relationship between the securities returns to normal.

Methodology

VaR is defined by the following relationship:

  level confidence t time at asset on return r VaR r P t t t t r         1 ) 1 ( ) ( 1 
It describes the estimated maximum potential loss of an asset not exceeded with a given probability defined as the confidence level, over a given period of time.

It is computed as follows:

  ) ) ( ) ( ) ( ) 2 ( ) ( ) ( ) ( ) ( 1 volatility of estimate type GARCH of case in h volatility r h confidence of level the on depending quantile z return average r E horizon time T r h T z r E T VaR t t t t t               
The VaR is the result of a product of the scalar denoted ) ( z representing a quantile depending on the level of confidence with the volatility in 1  t at horizon . t According to [START_REF] Wilmott | Derivatives[END_REF], the assumption of zero mean is valid over short-term horizons.

This assumption is based on the conjecture that the magnitude of mean is substantially smaller than the magnitude of the standard deviation and therefore can be ignored.

-8 -Therefore, the VaR formula considered is:

) 3 ( ) ( ) ( ) ( 1 r h T z VaR t t t       1.
2.

To evaluate hedge fund risks, one considers different types of VaR. The difference is made upon the quantile and the volatility model considered.

1. Types of quantile.

 Normal quantile  Student quantile. The Student's t distribution deals with the phenomenon of excess kurtosis by modelling tail thickness by a parameter called "degree of freedom".

 Cornish fisher quantile. [START_REF] Favre | Mean-modified Value-at-Risk optimization with hedge funds[END_REF] introduced the modified VaR that is based on the cornish fisher expansion quantile. It is an expansion around the normal distribution in order to take into account the asymmetry and the fat tails of the distribution.

t coefficien kurtosis K t coefficien skewness S on distributi normal the of quantile z S z z K z z S z z z normal normal normal normal normal normal normal CF            ) ( )) ( 5 ) ( 2 ( 36 1 ) 3 ))( ( 3 ) ( ( 24
1 ) 1 ) ( ( 6 1 ) ( ) ( 2 3 3 2    
time series of returns R t , three distinct specifications have to be provided; one for the conditional mean equation, one for the conditional variance, and one for the conditional error distribution, conditional on 1   t , the information set available at time t-1.

GARCH models assume that the conditional mean equation is modelled as follows:

) 1 , 0 ( ) 4 ( N iid c R t t t t t t        
However, when returns exhibit serial dependence structure, the mean equation is modelled as an AR, MA or ARMA process.

-9 -We first focus on GARCH model introduced by [START_REF] Bollerslev | Generalized autoregressive conditional heteroscedasticity[END_REF]:

p j and q i j i p t p t q t q t t         1 0 1 0 ) 5 ( ² ² 1 1 ² ² 1 1 0 ²                           
We also considered [START_REF] Nelson | Conditional heteroskesdasticity in asset returns: a new approach[END_REF] exponential GARCH model (EGARCH) in which the logarithm of conditional variance is specified as:

) 6 ( log log ² 1 1 1 0 ² j t p j j k t k t r k k q i i t i t i t i t i E                                      
Finally we deal with Zakoian [1990] threshold GARCH model (TGARCH) in which the conditional variance is specified as follows:

                    0 0 0 1 ) 7 ( 1 1 1 1 2 1 ² 1 1 ² 0 ² t t t p j j t j t t i q i i t i t if if d d          



An assumption about the conditional distribution of the error term

  t is required.
While the normal (Gaussian) distribution is widespread; it however cannot describe fat-tails returns. Hence the student's t distribution is also applied, modelling tail thickness by a parameter called "degree of freedom".

To sum up, 21 VaRs are computed; those based on standard deviation and computed with normal, student and cornish fisher quantile and those based on conditional volatility models (GARCH, TGARCH and EGARCH under the assumption of both normal and student assumption) computed with the same quantiles.

Empirical Analysis of Hedge Fund Return Indexes Volatility Data

Hedge fund providers licence their indexes to partners who can then create investable products. These products track the index by investing in a weighted portfolio of its constituents. To this end, only a limited numbers of liquid hedge funds are selected which leads to a sub-representativity bias.

Investable indexes providers have very strict hedge fund selection criterion in order to end up with a product easy to manage. However, very few hedge funds fulfil them. Consequently, the whole universe of hedge funds is not represented. Investable indexes are less representative than non investable indexes.

To perform the empirical analysis, one uses the HFRX (investable Hedge Fund Research

Summary Statistics

Globally it can be noticed that hedge fund index returns 4 are quite favourably compared to stocks and bonds (appendix A). HFRX global index has an average mean return of 5.73 %.

Amongst investment styles, event driven exhibits the highest average mean return with 7.78 % followed by macro (7.58 %). Convertible arbitrage and equity market neutral strategies exhibit the lowest returns.

Appendix A shows that standard deviations of stock market indexes and JP Morgan EMU Bond Index are much higher than those of hedge funds. Russell 2000 has an average standard deviation of 18.50%, followed by NASDAQ (16.59%) and Dow Jones EURO STOXX 50 (16.21%), while HFRX global index exhibits a weak average standard deviation of 3.74%. 4 Returns of hedge fund indexes at time t are computed as follows: None of the hedge fund returns distributions seem to be symmetric. As a matter of fact, their skewness coefficients are different from zero and negative. The same conclusion is reached when we look at the stock and bond market indexes.

1 ) 8 ( ln 1 1               t
Globally, hedge funds, stocks and bonds show evidence of fat tails, since the kurtosis exceeds 3, which is the normal value. It means that extreme returns (either losses or gains) are more likely than they are with the normal distribution. The returns distribution is leptokurtic. The combination of negative skewness and excess kurtosis denote a high probability of negative returns. Equity market neutral strategy exhibits the highest kurtosis coefficient with 38.03 followed by relative value arbitrage with 14.62 and merger arbitrage with 13.96.

According to those findings, hedge fund returns distribution seems to be far from normally distributed. According to the Jarcque-Bera test of normality, the null hypothesis of normally distributed returns is not accepted for none of the indexes. Indeed, the results show that the statistic is higher than theoretical value read in Chi-Square table with two degrees of freedom at the significant value of 5% (5.99) for all hedge fund return indexes and stock and bond market indexes. These results demonstrate that measures based on mean-variance approach are not reliable. On the other hand, table 1 below shows the existence of ARCH effects for most hedge fund strategies. This might be linked to the non-normality of the returns but also to the conditional distribution. That's why, one decided to go further the normal distribution and consider the Skewness one and the Cornish-Fisher expansion.

) ( 2 prob R T  ) ( ) 5 ( prob Q ) ( ) 10 ( prob Q -12 -
Volatility clustering is exhibited in appendix B. We can see that periods of high and low volatility are grouped together. These stylized facts are typical features of financial time series and demonstrate that the volatility of hedge funds varies through time. This suggests the use of non-linear time series structures to model the volatility of hedge fund returns.

Volatility Forecasting Using GARCH Models

As hedge fund strategies exhibit serial correlation induced by discontinuous trading (table 1), we represent the return generating process in the mean equation of the GARCH-type model with an ARMA-like component5 in order to desmooth hedge fund returns. A lag structure of p=1 and q=1 seems to be the most suitable (for equity market neutral, merger arbitrage and relative value arbitrage, the appropriate lag structure is p=2 and q=2). The autocorrelation and partial autocorrelation functions are used to identify the form of the mean equation. The choice of the adequate mean equation is based on selection criterion6 .

Then, one models the hedge fund strategies conditional variance as a GARCH, TGARCH and EGARCH process as they deal with nonlinearities and asymmetries. The conditional variances were estimated under the assumption that residuals  t follow the normal and the student law. A lag structure of p=1 and q=1 is also considered. -13 -According to these findings, all hedge fund strategies present asymmetric effects. This feature is in line with the negative skewness, nonlinearities and dynamic trading strategies that characterise hedge fund returns. Convertible Arbitrage -0,529% -0,656% -0,678% -0,703% Distressed Securities -0,346% -0,350% -0,351% -0,366% Event Driven -0,647% -0,646% -0,619% -0,676% Equity Hedge -0,842% -0,949% -1,008% -0,910% Equity Market Neutral -0,604% -0,739% -0,796% -0,738% Macro -1,094% -1,574% -1,897% -1,856% Merger Arbitrage -0,521% -0,334% -0,343% -0,349% Relative Value Arbitrage -0,512% -0,816% -0,876% -0,816% Global Index -0,548% -0,653% -0,637% -0,599%

Student quantiles

1% Standard Deviation GARCH TGARCH EGARCH VaR VaR VaR VaR
Convertible Arbitrage -0,562% -0,702% -0,723% -0,753% Distressed Securities -0,394% -0,429% -0,424% -0,430% Event Driven -0,705% -0,704% -0,688% -0,733% Equity Hedge -0,886% -1,014% -1,080% -0,978% Equity Market Neutral -0,646% -0,808% -0,854% -0,802% Macro -1,207% -1,743% -2,131% -2,098% Merger Arbitrage -0,583% -0,378% -0,397% -0,400% Relative Value Arbitrage -0,573% -0,898% -0,971% -0,919% Global Index -0,591% -0,701% -0,718% -0,639% -14 - Convertible Arbitrage -0,690% -0,857% -0,885% -0,918% Distressed Securities -0,404% -0,408% -0,409% -0,427% Event Driven -0,990% -0,988% -0,947% -1,034% Equity Hedge

-1,125% -1,268% -1,346% -1,216% Equity Market Neutral -2,774% -3,395% -3,660% -3,392% Macro -2,207% -3,173% -3,825% -3,743% Merger Arbitrage -1,171% -0,750% -0,771% -0,786% Relative Value Arbitrage -1,136% -1,811% -1,945% -1,812% Global Index -0,862% -1,027% -1,001% -0,942%
Table 2 reports results of 1 day-ahead VaR forecasts computed for 1 % threshold using normal, student and cornish fisher quantiles.

Among investment styles, equity hedge, merger arbitrage and relative value exhibit the highest VaR for almost all methods. These strategies seek to benefit from a spread which leads to a significant use of leverage increasing the level of risk. This is particularly true for relative value arbitrage (cf LTCM). Managers experience small price discrepancies and use a high level of leverage in order to generate returns. In the same way, merger arbitrage managers are exposed to market risk. Indeed when the market is down, merger arbitrage activity suffers.

Results show that VaR based on the normal quantile underestimate market risk. Those based on student and cornish fisher quantiles seem to be more relevant methods as their values are higher.

Among VaR methods, whatever the quantile may be, TGARCH and EGARCH-type VaR exhibit the highest values.

If one has a glance to the appendixes G, H and I, we can see that the GARCH/TGARCH and EGARCH-type VaR do react immediately to small or large price changes.

Backtesting VaR Models

To support the conclusions found above, it is necessary to test these VaR models. The quality of the VaR forecasts depends on the quality of the VaR method. One has to judge whether our VaR forecasts are consistent with subsequently realized returns given the confidence level.

When the number of realized observations falling outside VaR predictions is in line with the confidence level, the VaR model is adequate.

The tests start from a hit sequence function. It describes whether or not a loss in excess of the reported VaR has been realized. The function is defined as follows:

forecats Risk at Value ante ex VaR returns post ex daily R VaR R if VaR R if I t t t t t t t t t                ) ( ) 19 ( ) ( 0 ) ( 1 ) ( 1 1 1      
A VaR model will be accurate if and only if the hit sequence function satisfies both the unconditional coverage property and the independence property8 [START_REF] Christoffersen | Evaluating interval forecasts[END_REF]).

Backtesting Results

The number of observations for each hedge fund index is 1241. One uses the last 250 observations for out-of-sample forecasting. For each model, the first 991 daily returns are used to form a VaR forecast for day 992. Then, data from day 2 to day 992 are used to form a VaR forecast for day 993 and so on. 250 out-of-sample forecasts are generated recursively by moving the estimation-window forward through time. Convertible Arbitrage -0,53% -0,66% -0,68% -0,70% Distressed Securities -0,35% -0,35% -0,35% -0,37% Event Driven -0,65% -0,65% -0,62% -0,68% Equity Hedge -0,84% -0,95% -1,01% -0,91% Equity Market Neutral -0,60% -0,74% -0,80% -0,74% Macro -1,09% -1,57% -1,90% -1,86% Merger Arbitrage -0,52% -0,33% -0,34% -0,35% Relative Value Arbitrage -0,51% -0,82% -0,88% -0,82% Global Index -0,55% -0,65% -0,64% -0,60% Convertible Arbitrage -0,56% -0,70% -0,72% -0,75% Distressed Securities -0,39% -0,43% -0,42% -0,43% Event Driven -0,71% -0,70% -0,69% -0,73% Equity Hedge -0,89% -1,01% -1,08% -0,98% Equity Market Neutral -0,65% -0,81% -0,85% -0,80% Macro -1,21% -1,74% -2,13% -2,10% Merger Arbitrage -0,58% -0,38% -0,40% -0,40% Relative Value Arbitrage -0,57% -0,90% -0,97% -0,92% Global Index -0,59% -0,70% -0,72% -0,64%

Cornish fisher quantiles

1% Standard Deviation GARCH TGARCH EGARCH VaR VaR VaR VaR
Convertible Arbitrage -0,69% -0,86% -0,89% -0,92% Distressed Securities -0,40% -0,41% -0,41% -0,43% Event Driven -0,99% -0,99% -0,95% -1,03% Equity Hedge

-1,13% -1,27% -1,35% -1,22% Equity Market Neutral -2,77% -3,40% -3,66% -3,39% Macro -2,21% -3,17% -3,83% -3,74% Merger Arbitrage -1,17% -0,75% -0,77% -0,79% Relative Value Arbitrage -1,14% -1,81% -1,95% -1,81% Global Index -0,86% -1,03% -1,00% -0,94%
A glance to table 3 shows that the ex-post violations rate is larger than the initial coverage rate. For most of the models, losses in excess of the reported VaR occurred more frequently, no matter the threshold. This suggests that VaR understates the actual level of risk.

Under the cornish fisher GARCH-type (2.5% and 1%), losses in excess occurred less frequently for several hedge fund strategies. This means that VaR is too conservative. 9 No reported conditional variance for several GARCH and TGARCH estimations -17 - VaRs models pass all the tests except the GARCH and TGARCH type VaR (it fails the unconditional coverage for equity market neutral and merger arbitrage. The EGARCH-type VaR does not pass the conditional coverage for merger arbitrage).

When VaRs are computed using the cornish fisher quantile, the worst performer model is once more the one based on the standard deviation. The GARCH, TGARCH and EGARCHtype VaR pass all tests.

Conclusion

This research paper aimed to investigate hedge funds market risk. One demonstrates that daily hedge fund return distributions are asymmetric and leptokurtic. Furthermore, volatility clustering phenomenon and the existence of ARCH effects demonstrate that hedge funds volatility varies through time. These features suggest the modelisation of their volatility using symmetric (GARCH) and asymmetric models (EGARCH and TGARCH).

The conditional variances were estimated under the assumption that residuals  t follow the normal and the student law. The conditional variance of hedge fund strategies exhibits asymmetric effects and mean reversion among investment styles.

-19 -

The knowledge of the conditional variance was used to forecast 1-day-ahead ahead VaR. The estimations are compared with the Gaussian, the student and the modified VaR. The results demonstrate that VaR models based on normal quantile underestimate risk while those based on student and cornish fisher quantiles seem to be more relevant measurements. GARCH-type

VaR are very sensitive to changes in the return process.

Backtesting results show that the choice of the model used to forecast volatility is important.

Indeed, the VaR based on standard deviation is not relevant to measure hedge funds risks as it fails the appropriate tests. On the opposite side, GARCH, TGARCH and EGARCH-type VaR are accurate as they pass successfully the backtesting tests. The quantile used has also an impact on the relevance of the VaR models considered. GARCH-type VaR computed with the student and especially cornish fisher quantiles lead to better results.

                             ) ( ˆ) ( ) ( ) 21 ( 1 1 log 2 1 ) ( ) (     ˆ) 22 ( ) 1 ( ) 1 ( log 2 1 0 1 0 n n n sample the in ones of number n n T n p p LR n n n n uc                     The likelihood ratio uc LR has an asymptotic ) 1 ( 2  distribution.

Test of Independence

VaR violations at various periods must be independent over time. [START_REF] Christoffersen | Evaluating interval forecasts[END_REF] introduced the likelihood ratio of independence which examines the serial independence of the hit sequence function given the confidence level. It is expressed as follows:

                                 
The LR test of independence is asymptotically distributed as a )

.

Test of Conditional Coverage

Christoffersen [1998]

  indices) daily performance subindexes split by an investment style and an aggregate index which encompasses all hedge fund strategies and spans the period March 31 st 2003 -March 3 rd 2008 (1241 observations). The HFRX indexes are based on the Hedge Fund Research (HFR) database. The indexes measure the net of fee returns denominated in US dollar. Funds included must be currently open to new transparent investment, have at least $50 Million under management and meet 24 month track record. The HFRX indexes consist of eight single strategies presented above; convertible arbitrage, distressed securities, equity hedge, equity market neutral, event driven, macro, merger arbitrage and relative value arbitrage. The HFRX global hedge fund index encompasses over 55 funds.

  Appendixes D and E present the hedge fund strategies parameters estimates. Whatever the distribution followed by residuals t  , the conditional variance parameters are significant in all cases. The Ljung-Box and the ARCH-LM tests confirm that models considered take the heteroscedasticity into account for almost all hedge fund strategies. macro strategy), which indicates that bad news have larger impact on the volatility of the returns.We observe that EGARCH(1;1) process display a value of 0

Table 1 :

 1 ARCH Test of Hedge Fund Return Indexes

	Convertible Arbitrage	28.25 (0.00)	19.28 (0.00)	33.76 (0.00)
	Distressed Securities	5.36 (0.25)	84.75 (0.00)	115.70 (0.00)
	Event Driven	122.75 (0.00)	30.00 (0.00)	38.65 (0.00)
	Equity Hedge	202.79 (0.00)	47.18 (0.00)	58.46 (0.00)
	Equity Market Neutral	429.76 (0.00)	43.86 (0.00)	49.68 (0.00)
	Macro	191.88 (0.00)	42.57 (0.00)	51.25 (0.00)
	Merger Arbitrage	269.26 (0.00)	14.42 (0.01)	28.85 (0.00)
	Relative Value Arbitrage	282.26 (0.00)	11.49 (0.04)	24.21 (0.01)
	Global	240.90 (0.00)	79.74 (0.00)	94.30 (0.00)

Table 2 :

 2 Value at Risk Forecasts Results

	Normal quantiles
	Standard	GARCH	TGARCH	EGARCH
	Deviation
	VaR	VaR	VaR	VaR

Table 3 :

 3 Number and Proportion of Hedge Funds VaR Failures

	Normal quantiles
		1%
	Standard Deviation GARCH TGARCH EGARCH
	VaR	VaR	VaR	VaR

Table 4 :

 4 Unconditional Coverage, Independence and Conditional Coverage TestThe first table exhibits results when VaRs are computed using the normal quantile. The worst VaR model is performed by the one based on the standard deviation. It fails the unconditional and conditional coverage tests for all the hedge fund strategies (except distressed securities).On the opposite side, all GARCH-type VaR models pass successfully all the tests for most of the hedge fund strategies. GARCH and TGARCH-type VaR do not pass the unconditional and conditional coverage for equity market neutral and merger arbitrage. EGARCH-type VaR fails these tests for equity hedge, equity market neutral, merger arbitrage and global index.

	Normal quantiles
					1%
		Standard Deviation	GARCH	TGARCH	EGARCH
		Nb of F	POF	Nb of F	POF	Nb of F	POF	Nb of F	POF
	Convertible Arbitrage	11	4.40%	6	2.40%	6	2.40%	7	2.80%
	Distressed Securities	4	1.60%	3	1.20%	3	1.20%	3	1.20%
	Event Driven	17	6.80%	7	2.80%	8	3.20%	7	2.80%
	Equity Hedge	23	9.20%	5	2.00%	7	2.80%	11	4.40%
	Equity Market Neutral	15	6.00%	11	4.40%	10	4.00%	10	4.00%
	Macro	16	6.40%	5	2.00%	4	1.60%	4	1.60%
	Merger Arbitrage	16	6.40%	12	4.80%	12	4.80%	12	4.80%
	Relative Value Arbitrage	22	8.80%	8	3.20%	9	3.60%	9	3.60%
	Global Index	24	9.60%	8	3.20%	11	4.40%	10	4.00%
	Student quantiles 10
					1%
		Standard Deviation	GARCH	TGARCH	EGARCH
		Nb of F	POF	Nb of F	POF	Nb of F	POF	Nb of F	POF
	Convertible Arbitrage	9	3,60%	5	2,00%	6	2,40%		2,00%
	Distressed Securities	3	1,20%	-	-	-	-		0,80%
	Event Driven	11	4,40%	6	2,40%	4	1,60%		1,60%
	Equity Hedge	21	8,40%	4	1,60%	6	2,40%		2,40%
	Equity Market Neutral	13	5,20%	10	4,00%	9	3,60%	10	4,00%
	Macro	15	6,00%	4	1,60%	3	1,20%		1,60%
	Merger Arbitrage	15	6,00%	-	-	-	-	11	4,40%
	Relative Value Arbitrage	19	7,60%	-	-	4	1,60%		2,40%
	Global Index	21	8,40%	2	0,80%	4	1,60%		2,00%
	Cornish fisher quantiles 11
					1%
		Standard Deviation	GARCH	TGARCH	EGARCH
		Nb of F	POF	Nb of F	POF	Nb of F	POF	Nb of F	POF
	Convertible Arbitrage	6	2,40%	3	1,20%	2	0,80%		0,80%
	Distressed Securities	2	0,80%	3	1,20%	2	0,80%		1,20%
	Event Driven	4	1,60%	2	0,80%	1	0,40%		0,80%
	Equity Hedge	8	3,20%	1	0,40%	1	0,40%		0,00%
	Equity Market Neutral	1	0,40%	0	0,00%	0	0,00%		0,00%
	Macro	4	1,60%	0	0,00%	0	0,00%		0,00%
	Merger Arbitrage	4	1,60%	0	0	0	0		0,40%

10

No reported conditional variances for several GARCH and TGARCH estimations 11 GARCH, TGARCH and EGARCH parameters are estimated under the assumption that residuals follow the normal law -18 -When VaRs are computed using the student quantile, the best performers models are still GARCH-type VaR. Again the VaR model based on standard deviation fails almost all tests (except for distressed securities and independence test for several hedge fund strategies).

1 day-ahead Equity Hedge VaR

								1 day-ahead Convertible Arbitrage VaR
								1.00%
								0.50%
								0.00%
								1	51	101	151	201
								-0.50%
			combined the unconditional coverage test and the independence test to -1.00%
	form a test of conditional coverage. It is written as follows: -1.50%
	Returns			Normal VaR		Returns	Normal VaR
	Normal GARCH type VaR	Normal TGARCH type VaR	Normal GARCH type VaR	Normal TGARCH type VaR
	Normal EGARCH type VaR					Normal EGARCH type VaR
			LR		LR		LR	(	) 24
				cc		uc		ind
	The distribution of the conditional coverage test is asymptotically 2  with two degrees of 1.50%
	1.00%
	freedom. 0.50%
	0.00%
	1	51	101		151		201
	-0.50%
	-1.00%
	-1.50%
	-2.00%
	Returns			Normal VaR
	Normal GARCH type VaR		Normal TGARCH type VaR
	Normal EGARCH type VaR

1 day-ahead Equity Market Neutral VaR

	1 day-ahead Event Driven VaR Appendix H:			1 day-ahead Macro VaR
	1.50%								2.50%
									2.00%
	1.00%								1.50%
	0.50%						0.50% 1 day-ahead Global Index VaR 1.00%
									0.00%
	0.00%				1.00%				-0.50%	1	51	101	151	201
	-0.50%	1	51	101	0.50%	151	201		-1.50% -1.00%
	-1.00%				0.00%	1	51	101	151 -2.50% -2.00%	201
	-1.50%				-0.50%				-3.50% -3.00%
					-1.00%				-4.00%
	-2.00%								-4.50%
			Returns		-1.50%	Normal VaR			Returns	99% VaR
			-2.00% Normal GARCH type VaR	Normal TGARCH type VaR		Normal GARCH type VaR	Normal TGARCH type VaR
			Normal EGARCH type VaR				Normal EGARCH type VaR
						Returns		Normal VaR
						Normal GARCH type VaR	Normal TGARCH type VaR
						Normal EGARCH type VaR
			1 day-ahead Merger Arbitrage VaR			1 day-ahead Relative Value Arbitrage VaR
	1.50%								2.00%
	1.00%								1.50%
	0.50%								1.00%
									0.50%
	0.00%
		1	51	101		151	201		0.00%
	-0.50%								1	51	101	151	201
									-0.50%
	-1.00%
									-1.00%
	-1.50%								-1.50%
	-2.00%								-2.00%
	-2.50%								-2.50%
			Returns			Normal VaR			Returns	Normal VaR
			Normal GARCH type VaR	Normal TGARCH type VaR		Normal GARCH type VaR	Normal TGARCH type VaR
			Normal EGARCH type VaR				Normal EGARCH type VaR
									3.50%
									3.00%
									2.50%
									2.00%
									1.50%
									1.00%
									0.50%
									0.00%
									-0.50%	1	51	101	151	201
									-1.00%
									-1.50%
									-2.00%
									-2.50%
									-3.00%
									-3.50%
									-4.00%
									-4.50%
									Returns	Normal VaR
									Normal GARCH type VaR	Normal TGARCH type VaR
									Normal EGARCH type VaR
									-34 --35 -

Hedge Fund Historical Returns and VaR Forecasts (student quantile, 1% confidence level) 1 day-ahead Global Index VaR

				1,00%
				0,50%
				0,00%
					1	51	101	151	201
				-0,50%
				-1,00%
				-1,50%
				-2,00%
						Returns	Student VaR
						Student GARCH type VaR	Student TGARCH type VaR
						Student EGARCH type VaR
			1 day-ahead Convertible Arbitrage VaR	1 day-ahead Distressed Securities VaR
	1,30%						0,50%
	0,80%
	0,30%						0,00%
							1	51	101	151	201
	-0,20%	1	51	101	151	201
	-0,70%						-0,50%
	-1,20%
			Returns Student GARCH type VaR	Student VaR Student TGARCH type VaR	-1,00%
			Student EGARCH type VaR			Returns	Student VaR	Student EGARCH type VaR

1 day-ahead Equity Hedge VaR

	1 day-ahead Event Driven VaR Appendix I:				1 day-ahead Macro VaR
	1,50%						2,50%
							2,00%
	1,00%						1,50%
	0,50%						0,50% 1,00%
	0,00%						0,00%
		1	51	101	151	201	-0,50%	1	51	101	151	201
	-0,50%						-1,00%
							-1,50%
	-1,00%						-2,00%
	-1,50%						-3,00% -2,50%
	-2,00%						-3,50%
							-4,00%
	-2,50%						-4,50%
			Returns		Student VaR				Returns		Student VaR
			Student GARCH type VaR	Student TGARCH type VaR			Student GARCH type VaR	Student TGARCH type VaR
			Student EGARCH type VaR					Student EGARCH type VaR
			1 day-ahead Merger Arbitrage VaR				1 day-ahead Relative Value Arbitrage VaR
	1,50%						2,00%
	1,00%						1,50%
							1,00%
	0,50%						0,50%
	0,00%						0,00%
		1	51	101	151	201	-0,50%	1	51	101	151	201
	-0,50%
							-1,00%
	-1,00%						-1,50%
	-1,50%						-2,00%
							-2,50%
	-2,00%
									Returns		Student VaR
			Returns	Student VaR	Student EGARCH type VaR			Student TGARCH type VaR	Student EGARCH type VaR
	1,50%						3,00% 3,50%
	1,00%						2,00% 2,50%
	0,50%						1,00% 1,50%
	-0,50% 0,00%	1	51	101	151	201	-1,00% 0,50% 0,00% -0,50%	1	51	101	151	201
	-1,00%						-2,00% -1,50%
	-1,50%						-3,00% -2,50%
	-2,00%						-3,50%
							-4,00%
	-2,50%						-4,50%
			Returns		Student VaR				Returns		Student VaR
			Student GARCH type VaR	Student TGARCH type VaR			Student GARCH type VaR	Student TGARCH type VaR
			Student EGARCH type VaR					Student EGARCH type VaR
												-36 --37 -
									1 day-ahead Equity Market Neutral VaR

Hedge Fund Historical Returns and VaR Forecasts (cornish fisher quantile, 1% confidence level) 1 day-aheadGlobal IndexVaR

					1.50%
					1.00%
					0.50%
					0.00%
					-0.50%	1	51	101	151	201
					-1.00%
					-1.50%
					-2.00%
					-2.50%
					-3.00%
							Returns			CF VaR
							CF GARCH type VaR		CF TGARCH type VaR
							CF EGARCH type VaR
			1 day-ahead Convertible Arbitrage VaR			1 day-ahead Distressed Securities VaR
	1,00%								0.40%
	0,50%
	0,00%
	1		51	101	151	201		-0.10%	1	51	101	151	201
	-0,50%
	-1,00%
	-1,50%								-0.60%
			Returns		CF VaR				Returns	CF VaR
			CF GARCH type VaR	CF TGARCH type VaR			CF GARCH type VaR	CF TGARCH type VaR
			CF EGARCH type VaR						CF EGARCH type VaR
			1 day-ahead Equity Hedge VaR				1 day-
	2,10%
	1,60%
	1,10%
	0,60%
	0,10%
	-0,40%	1	51	101	151	201
	-0,90%
	-1,40%
	-1,90%
	-2,40%
	-2,90%
			Returns			CF VaR
			CF GARCH type VaR		CF TGARCH type VaR
			CF EGARCH type VaR

ahead Equity Market Neutral VaR

			1 day-ahead Event Driven VaR				1 day-ahead Macro VaR
	1.50%						4.00%
	0.50% 1.00%						2.00%
	0.00%						0.00%
	-0.50%	1	51	101	151	201	-2.00%	1	51	101	151	201
	-1.00%
	-1.50%						-4.00%
	-2.00%						-6.00%
	-3.00% -2.50%						-8.00%
	-3.50%						-10.00%
			Returns		CF VaR				Returns		CF VaR
			CF GARCH type VaR	CF TGARCH type VaR			CF GARCH type VaR	CF TGARCH type VaR
			CF EGARCH type VaR					CF EGARCH type VaR
			1 day-ahead Merger Arbitrage VaR
	2.00%
	1.00%
	0.00%
	-1.00%	1	51	101	151	201
	-2.00%
	-3.00%
	-4.00%
	-5.00%
	-6.00%
			Returns		CF VaR
			CF GARCH type VaR	CF TGARCH type VaR
			CF EGARCH type VaR
							0,00%
								1	51	101	151	201
							-5,00%
							-10,00%
							-15,00%
							-20,00%
									Returns		CF VaR
									CF GARCH type VaR	CF TGARCH type VaR
									CF EGARCH type VaR
												-38 -

1 day-ahead Relative Value Arbitrage VaR

	3.00%
	2.00%
	1.00%
	0.00%
	-1.00%	1	51	101	151	201
	-2.00%
	-3.00%
	-4.00%
	-5.00%
			Returns		CF VaR
			CF GARCH type VaR	CF TGARCH type VaR
			CF EGARCH type VaR

EconomiX, Université Paris X Nanterre, 200 avenue de la République, 92000 Nanterre, E-mail: sabrina_khanniche@yahoo.fr

Either an AR, a MA or an ARMA.

R², AIC, SIC and Log Likelihood.

GARCH, TGARCH and EGARCH parameters are estimated under the assumption that residuals follow the normal law

These tests are presented in appendix G.

** no significant at 5% confidence level

** no significant at 5% confidence level

** no significant at 5% confidence level

** no significant at 5% confidence level

** no significant at 5% confidence level