Asset Prices

Constant Proportion Portfolio Insurance (CPPI) is a dynamic portfolio man- agement strategy that is currently of popular interest in both industry and aca- demic research. The CPPI methodology is designed to guarantee, to the buyer, a minimum payoff at maturity using a portfolio comprised only of one risky asset and one riskless asset. The goal is to allow an amount of participation in capi- tal markets while removing downside risk. However, in the presence of realistic market assumptions risk to the seller exists in the form of gap risk. This gap risk accounts for the inability tomeet the guarantee atmaturity. There are many factors which contribute to the gap risk, including asset price behaviour and trading fre- quency. The effect of these factors are investigated in this paper within a discrete time framework. The results show that when considering realistic levels of volatility the CPPI does not perform well in comparison to a riskless investment and a gapless (buy- and-hold) p...

... Biographical notes: Leo MacDonald is an Assistant Professor in the Department of Economics, Finance and Quantitative Analysis in the Coles College of Business ... Prior to his appointment in 2006, he was a Faculty at the Ivey School of Business in London, Ontario, Canada. ...

This paper examines the difference in the net present values (NPV's) of North sea oil projects obtained using discounted cashflow methods based on the Weighted Average Cost of Capital (WACC) and a Modern Asset Pricing (MAP) method (Laughton and Jacoby, 1993; Jacoby and Laughton, 1992).

Empirically, co-skewness of asset returns seems to explain a substantial part of the cross-sectional variation of mean return not explained by beta. This finding is typically interpreted in terms of a risk averse representative investor with a cubic utility function. This paper questions this interpretation. We show that the empirical tests fail to impose risk aversion and the implied utility function takes an inverse S-shape. Unfortunately, the first-order conditions are not sufficient to guarantee that the market portfolio is the global maximum for this utility function, and our results suggest that the market portfolio is more likely to represent the global minimum. In addition, if we do impose risk aversion, then co-skewness has minimal explanatory power.

In spite of popularity and theoretical simplicity of the one-factor Capital Asset Pricing Model (CAPM) used in the valuation of financial assets, researchers are more concerned with the important extension proposed by , that is, the Three-Factor Pricing Model (TFPM). Alongside beta, average stock returns could be explained by some size and book-to-market supplementary effects. With these two complementary models, estimation of the cost of equity is carried out for the Tunisian banking sector. In order to account for inter-individual heterogeneity, estimation of parameters is conducted according to random coefficient specifications within the context of panel data analysis.

This paper studies the impact of ambiguity and ambiguity aversion on equilibrium asset prices and portfolio holdings in competitive financial markets. It argues that attitudes toward ambiguity are heterogeneous across the population, just as attitudes toward risk are heterogeneous across the population, but that heterogeneity of attitudes toward ambiguity has different implications than heterogeneity of attitudes toward risk. In particular, when some state probabilities are not known, agents who are sufficiently ambiguity averse find open sets of prices for which they refuse to hold an ambiguous portfolio. This suggests a different cross-section of portfolio choices, a wider range of state price/probability ratios and different rankings of state price/probability ratios than would be predicted if state probabilities were known. Experiments confirm all of these suggestions. Our findings contradict the claim that investors who have cognitive biases do not affect prices because they are infra-marginal: ambiguity averse investors have an indirect effect on prices because they change the per-capita amount of risk that is to be shared among the marginal investors. Our experimental data also suggest a positive correlation between risk aversion and ambiguity aversion that might explain the "value effect" in historical data.

A number of studies have separately identified mean reversion and momentum. This paper considers these effects jointly: potential for mean reversion and momentum is combined into one index, interpretable as an expected return. Combination momentum-contrarian strategies, used to select from among 18 developed equity markets at a monthly frequency, outperform both pure momentum and pure mean reversion strategies. A breakdown of the combination strategies indicates that both extreme high-expected-return and low-expected-return strategies rely heavily on country-indices with higher standard deviation of return, smaller firm size, and a lower number of firms. These observations suggest a mispricing rather than risk-compensation explanation of returns, with overreaction tied to information-based neglect of countries with fewer and smaller firms.

Credit market imperfections have been blamed for the depth and persistence of the Great Depression in the USA. Could similar mechanisms have played a role in ending the East Asian miracle? After a brief account of the nature of the recent crises, we use a model of highly levered creditconstrained firms due to Kiyotaki and Moore (1997) to explore this question. As applied to land-holding property companies, it predicts greatly amplified responses to financial shocks-like the ending of the land price bubble or the fall of the exchange rate. The initial fall in asset values is followed by the 'knock-on' effects of the scramble for liquidity as companies sell land to satisfy their collateral requirements-causing land prices to fall further. This could lead to financial collapse where-like falling dominoes-prudent firms are brought down by imprudent firms. Key to avoiding collapse is the nature of financial stabilisation policy; in a crisis, temporary financing can prevent illiquidity becoming insolvency and launching 'lifeboats' can do the same. But the vulnerability of financial systems like those in East Asia to short-term foreign currency exposure suggests that preventive measures are also required.

In this note, we prove that in asset price models with lognormal stochastic volatility, when the correlation coecien t between the Brownian motion driving the volatility and the one driving the actualized asset price is positive, this price is not a martingale.

Using a large international equity market database that has not been previously used for such a purpose, this paper documents that value (i.e., high book-to-market ) stocks outperform growth (i.e., low book-to-market ) stocks, on average, in most countries during the January 1975 -December 1995 period, both absolutely and after adjusting for risk. The international evidence confirms the findings of previous work reported for the U.S.. For 1975-1995, the annual difference between the average returns on portfolios of high and low book-to-market stocks is 12.94% in North America, 10.42% in Europe, 17.26% in Pacific-Rim per year, and value stocks outperform growth stocks in 17 out of 18 national capital markets. Our analysis also shows that a three-factor model explains most of the cross-sectional variation in average returns on industry portfolios across countries and that the superior performance of the value investing strategy, documented in this study, is a manifestation of size and book-to-market effects. These results are consistent with those reported by French (1994, 1996) that show that the value-growth pattern in stock returns is largely explained by a three-factor asset pricing model. Our results suggest that the three-factor asset pricing model is not limited to the U.S. stock market.

We introduce banks, modeled as in Diamond and Rajan (JoF 2000 or JPE 2001), into a standard DSGE model and use this framework to study the role of banks in the transmission of shocks, the effects of monetary policy when banks are exposed to runs, and the interplay between monetary policy and Basel-like capital ratios. In equilibrium, bank leverage depends

This Article, a sequel to earlier articles by the author on financial regulation and global finance, retrieves and updates J. M. Keynes’s original International Clearing Union plan for what ultimately became the International Monetary Fund (“IMF,” “Fund”). Its motives are not antiquarian. The Article argues that our current domestic and global financial troubles alike are best viewed as products of our not having gone something more like that original Keynesian route with the Fund. Part I tells the tale of Keynes’s original Clearing Union plan for what was to become the IMF. It emphasizes the plan’s basic structure and motivations, as decisively rooted in Keynes’s financially oriented re-conception of what subsequently came to be known as “macroeconomic” theory. It highlights in particular the Clearing Union’s projected role as an institutional lever through which trading nations could jointly prevent financial-stability-imperiling global imbalances in the form of persistent trade s...

This article attempts to assess to what extent the central bank or the government should respond to developments that can cause financial instability, such as housing or asset bubbles, overextended budgetary policies or excessive public and household debt. To analyse this question, we set up a simple reduced-form model in which monetary and fiscal policy interact, and imbalances (bubbles) can occur in the medium-run. Considering several scenarios with both benevolent and idiosyncratic policy-makers, the analysis shows that the answer depends on a number of characteristics of the economy, as well as on the monetary and fiscal policy preferences with respect to inflation and output stabilisation. We show that socially optimal financial instability prevention should be carried out by: (i) both monetary and fiscal policy (sharing region) under some circumstances; and (ii) fiscal policy only (specialisation region) under others. There is, however, a moral hazard problem: both policy-makers have an incentive to be insufficiently pro-active in safeguarding financial stability and shift the responsibility to the other policy. Specifically, under a range of circumstances, we obtain a situation in which neither policy mitigates instability threats (indifference region). These results can be related to the build-up of the current global financial crisis, and have strong implications for the optimal design of the delegation process.

The article analyses the Swedish banking crisis in the early 1990s. Newly deregulated credit markets after 1985 stimulated a competitive process between financial institutions where expansion was given priority. Combined with an expansive macro policy, this contributed to an asset price boom. The subsequent crisis resulted from a highly leveraged private sector being simultaneously hit by three major exogenous events: a shift in monetary policy with an increase in pre-tax interest rates, a tax reform that increased after tax interest rates, and the ERM crisis. Combined with some overinvestment in commercial property, high real interest rates contributed to breaking the boom in real estate prices and triggering a downward price spiral resulting in bankruptcies and massive credit losses. The government rescued the banking system by issuing a general guarantee of bank obligations. The total direct cost to the taxpayer of the salvage has been estimated at around 2 per cent of GDP.

, and seminar participants at the University of Toronto and the Wharton Brown Bag seminar for helpful comments. We also thank Huarong Tang and Sehyun Yoo for excellent research assistance. Xia gratefully acknowledges the NASDAQ fellowship from the Rodney L. White Center at the Wharton School.

This paper studies an application of a Darwinian theory of portfolio selection to stocks listed in the Dow Jones Industrial Average (DJIA). We analyze numerically the long-run outcome of the competition of fix-mix portfolio rules in a stock market with actual DJIA dividends. In the model seemingly rational strategies can do very poorly against seemingly irrational strategies. Moreover, the interaction of strategies can lead to stochastic time series of asset prices that do not converge. The simulations also show that the evolutionary portfolio rule discovered in Hens and Schenk-Hopp´e (2004) will eventually hold total market wealth in competition with fix-mix portfolio rules derived from mean-variance optimization, maximum growth theory and behavioral finance. According to this evolutionary rule, portfolio weights should be proportional to the expected relative dividends of the assets. As an implication asset prices converge to expected relative dividends.

We turn our attention to the role of money for determining nominal magnitudes. Using US data, we find that the aggregate “nominal output plus and stock market capitalisation” is closely related to the money stock, lending support to one of Milton Friedman’s key monetarist propositions. This finding should be particularly important for ECB monetary policy: an inflation-free euro plays a

... This is the message of a paper prepared for the Jackson Hole conference (August 1999) by ... for the relationship between (broad) money and in£ation over short-and medium-term horizons. ... It becomes clear that, when predicting future in£ation, current in£ation plays a crucial role ...

Financing Constraints and Corporate Investment EMPIRICAL models of business investment rely generally on the assumption of a "representative firm" that responds to prices set in centralized securities markets. Indeed, if all firms have equal access to capital markets, firms' responses to changes in the cost of capital or tax-based investment incentives differ only because of differences in investment demand. A firm's financial structure is irrelevant to investment because external funds provide a perfect substitute for internal capital. In general, with perfect capital markets, a firm's investment decisions are independent of its financial condition. An alternative research agenda, however, has been based on the view that internal and external capital are not perfect substitutes. According to this view, investment may depend on financial factors, such as the availability of internal finance, access to new debt or equity finance, or the functioning of particular credit markets. For example, a firm's internal cash flow may affect investment spending because of a "financ-We are grateful to members of the Brookings Panel for helpful comments and suggestions and to Charles Himmelberg and Jaewoon Koo for excellent research assistance. Financial support from the Federal Reserve Bank of Chicago is acknowledged. Hubbard acknowledges support from a John M. Olin Fellowship at the National Bureau of Economic Research. The views expressed here are not necessarily those of the Federal Reserve Bank of Chicago.

The present study adds to the sparse published Australian literature on the size effect, the book to market (BM) effect and the ability of the Fama French three factor model to account for these effects and to improve on the asset pricing ability of the Capital Asset Pricing Model (CAPM). The present study extends the 1981–1991 period examined by Halliwell, Heaney and Sawicki (1999) a further 10 years to 2000 and addresses several limitations and findings of that research. In contrast to Halliwell, Heaney and Sawicki the current study finds the three factor model provides significantly improved explanatory power over the CAPM, and evidence that the BM factor plays a role in asset pricing.

COMMON PERSISTENCE IN CONDITIONAL VARIANCES' BY TIM BOLLERSLEV AND ROBERT F. ENGLE Since the introduction of the autoregressive conditional heteroskedastic (ARCH) model in Engle (1982), numerous applications of this modeling strategy have already appeared. A common finding in many of these studies with high frequency financial or monetary data concerns the presence of an approximate unit root in the autoregressive polynomial in the univariate time series representation for the conditional second order moments of the process, as in the so-called integrated generalized ARCH (IGARCH) class of models proposed in Engle and Bollerslev (1986). In the IGARCH models shocks to the conditional variance are persistent, in the sense that they remain important for forecasts of all horizons. This idea is readily extended to a multivariate framework. Even though many time series may exhibit persistence in variance, it is likely that several different variables share the same common long-run component. In that situation, the variables are naturally defined to be co-persistent in variance, and the co-persistent linear combination is interpretable as a long-run relationship. Conditions for co-persistence to occur in the multivariate linear GARCH model are presented. These conditions parallel the conditions for linear co-integration in the mean, as developed by Engle and Granger (1987). The presence of co-persistence has important implications for asset pricing relationships and in optimal portfolio allocation decisions. An empirical example relating to the time series properties of nominal U.S. dollar exchange rates for the deutschemark and the British pound provides a simple illustration of the ideas.

This paper assesses how market fundamentals affect asset return volatility by drawing on evidence from the U.S. natural gas futures market. One of the novel features of this paper is the use of the deviation of temperatures from normal (weather surprise) as a proxy for demand shocks and a determinant of the conditional volatility of natural gas futures returns. I estimate a GARCH model using daily natural gas futures data from January 1997 to December 2000. The empirical result shows that the weather surprise variable has a significant effect on the conditional volatility of natural gas prices and the inclusion of the weather surprise variable in the conditional variance equation reduces volatility persistence. Combined with the evidence that volatility is considerably higher on Monday and the day when natural gas storage report is released, these results show that information about market fundamentals are important determinants of natural gas price volatility. Aside from these findings, I also document that returns of the first month futures are more volatile than those of the second month futures, which is consistent with Samuelson's (1965) hypothesis that commodity futures price volatility declines with contract horizon. Acknowledgement This paper is based on a chapter of my dissertation. I am grateful to Timothy Dunne, Aaron Smallwood and Daniel Sutter for their guidance. Also I thank Dennis O'Brien and the Institute for Energy Economics and Policy for financial support to acquire natural gas futures trading data, Peter Lamb, Mark Richmond and Reed Timmer for help on weather data. All errors remain mine.

The paper provides a survey of the work that has been done in financial econometrics in the past decade. It proceeds by first establishing a set of stylized facts that are characteristics of financial series and then by detailing the range of techniques that have been developed to model series which possess these characteristics. Both univariate and multivariate models are considered.

We propose a framework for studying optimal market making policies in a limit order book (LOB). The bid-ask spread of the LOB is modelled by a Markov chain with finite values, multiple of the tick size, and subordinated by the Poisson process of the tick-time clock. We consider a small agent who continuously submits limit buy/sell orders at best bid/ask quotes, and may also set limit orders at best bid (resp. ask) plus (resp. minus) a tick for getting the execution order priority, which is a crucial issue in high frequency trading. By trading with limit orders, the agent faces an execution risk since her orders are executed only when they meet counterpart market orders, which are modelled by Cox processes with intensities depending on the spread and on her limit prices. By holding non-zero positions on the risky asset, the agent is also subject to the inventory risk related to price volatility. Then the agent can also choose to trade with market orders, and therefore get immediate execution, but at a least favorable price because she has to cross the bid-ask spread.

This paper will discuss the current research in building models of conditional variances using the Autoregressive Conditional Heteroskedastic (ARCH) and Generalized ARCH (GARCH) formulations. The discussion will be motivated by a simple asset pricing theory which is ...

The beta value, an indicator of systematic risk, is used to estimate the costs of equity and the evaluation of a stock's reasonable price. It is useful to airlines because their capital assets and operations are relatively sensitive to systematic risks. To obtain better estimates, it is useful to identify the causes for fluctuations in beta values. This examines the volatility of airline betas in the capital asset pricing and three-factor pricing models, as well as exploring the potential factors affecting their values. The findings suggest that airline betas are volatile over time and that crashes and stock market trends may also impact them. The business cycle, operating and financial leverage, and capital structure all positively influence the sample airlines' betas as well. The effects of return on equity on betas are negative, and liquidity might also affect betas. r

This paper deals with the foundations of the stochastic mathematical finance, and it has three main purposes: 1. We present a self-contained construction of the vector stochastic integral HX with respect to a d-dimensional semimartingale X = (X1 t ,...,Xd t ). This notion is more general than the componentwise stochastic integral Pd i=1 H iXi. 2. We show that

We propose a general class of analytically tractable models for the dynamics of an asset price leading to smiles or skews in the implied volatility structure. The considered asset can be an exchange rate, a stock index, or even a forward Libor rate. The class is based on an explicit SDE under a given forward measure. The models we propose feature i) explicit asset-price dynamics, ii) virtually unlimited number of parameters, iii) analytical formulas for European options.

This paper provides a review of the main features of asset pricing models. The review includes single-factor and multifactor models, extended forms of the Capital Asset Pricing Model (CAPM) with higher order co-moments, and asset pricing models conditional on time-varying volatility.

A new method for pricing lookback options (a.k.a. hindsight options) is presented, which simplifies the derivation of analytical formulas for this class of exotics in the Black-Scholes framework. Underlying the method is the observation that a lookback option can be considered as an integrated form of a related barrier option. The integrations with respect to the barrier price are evaluated at the expiry date to derive the payoff of an equivalent portfolio of European-type binary options. The arbitrage-free price of the lookback option can then be evaluated by static replication as the present value of this portfolio. We illustrate the method by deriving expressions for generic, standard floating-, fixed-, and reverse-strike lookbacks, and then show how the method can be used to price the more complex partial-price and partial-time lookback options. The method is in principle applicable to frameworks with alternative asset-price dynamics to the Black-Scholes world.

We explore whether a Financial Transactions Tax (FTT) is likely to correct the market failures that have contributed to the financial crisis, to what extent FTT succeeds in raising revenues, and how the FTT compares to alternative taxes in terms of efficiency. We find little evidence that the FTT will be effective in correcting market failures. Taxing of transactions is not well targeted at behaviour that leads to excessive risk and systemic risk creation. The empirical evidence does not suggest that the introduction of an FTT reduces volatility or asset price bubbles. An FTT will likely raise significant revenues and we estimate those revenues for the Netherlands. In the short term, the incidence of the tax will be chiefly on the current holders of securities. Ultimately, the tax will be borne in part by end users, and we estimate the likely effects on economic growth. When compared to alternative forms of taxation of the financial sector, the FTT is likely less efficient given the amount of revenues. In particular, taxes that more directly address existing distortions, such as the current VAT exemption for banks, and the bias towards debt financing, provide more efficient alternatives.

We consider a simple pure exchange economy with two assets, one riskless, yielding a constant return on investment, and one risky, paying a stochastic dividend. Trading takes place in discrete time and in each trading period the price of the risky asset is fixed by imposing market clearing condition on the sum of traders individual demand functions. We assume that agents' strategies are consistent with maximizing a CRRA utility, so that the individual demand for the risky asset is expressed as a fraction of the agent's wealth and the evolution of price and wealth distribution is described by means of a dynamical system. We consider agents whose individual demand function is based on future prices forecasts obtained on the basis of past market history and we analyze two cases.

A considerable part of the vast development in Mathematical Finance over the last two decades was determined by the application of stochastic methods. These were therefore chosen as the focus of the 2003 School on "Stochastic Methods in Finance". The growing interest of the mathematical community in this field was also reflected by the extraordinarily high number of applications for the CIME-EMS School. It was attended by 115 scientists and researchers, selected from among over 200 applicants. The attendees came from all continents: 85 were Europeans, among them 35 Italians.

The views expressed in this Working Paper are those of the author(s) and do not necessarily represent those of the IMF or IMF policy. Working Papers describe research in progress by the author(s) and are published to elicit comments and to further debate.

Asset pricing models predict relations between assets' expected rates of return and their risks or behavioral attributes. However, almost all tests of these models use realized return as a proxy for expected return. This paper extracts two sets of estimates of expected returns from Wall Street analysts (for more than 7,000 stocks over a five-year period) and Value Line analysts (for approximately 2,900 stocks for 15 years), and tests the relation between these ex-ante expected return estimates and factors proposed in the literature to explain them. Consistent with the predictions of the CAPM, we find that market beta is positively related to expected returns. As with tests that employ ex-post realized returns, we find that expected return is negatively related to the firm's market value of equity. Contrary to the implication of Fama and French (1993) and others, we find that using the expectations of Wall Street analysts, book-to-market ratio is priced, but its sign is negative, indicating that low book-to-market firms are expected to earn higher returns than high book-to-market firms. Value Line analysts have expectations of returns that are not affected by book-to-market ratio. Finally, prior one-year return is negatively related to expected returns. J a n -8 7 J u l -8 7 J a n -8 8 J u l -8 8 J a n -8 9 J u l -8 9 J a n -9 0 J u l -9 0 J a n -9 1 J u l -9 1 J a n -9 2 J u l -9 2 J a n -9 3 J u l -9 3 J a n -9 4 J u l -9 4 J a n -9 5 J u l -9 5 J a n -9 6 J u l -9 6 J a n -9 7 J u l -9 7 J a n -9 8 J u l -9 8 J a n -9 9 J u l -9 9 J a n -0 0 J u l -0 0 J a n -0 1 J u l -0 1

A model for a financial asset is constructed with two types of agents, who differ in terms of their beliefs. The proportion of the two types changes over time according to stochastic processes which model the interaction between the agents. Agents do not persist in holding "wrong" beliefs and bubble-like phenomena in the asset price occur. We consider tests for detecting bubbles in the conditional mean and multiple changes in the conditional variance of the process. A wavelet analysis of the series generated by our models shows that the strong persistence in the volatility is likely to be the outcome of a mix of changes in regimes and a moderate level of long-range dependence. These results are consistent with what has been observed by and .

Using Hong Kong equity data, this study examines the pricing effects of beta, firm size, and book-to-market equity, conditional on market situations, that is, whether the market is up or down. Evidence supports the hypothesis that, if the risk variable is priced by the market, then there exists a systematic but conditional relationship between the risk variable and average return, and this relationship takes on opposite directions during up and down markets. However, the significance of the relations is often affected by the changing values of the risk variables that result from changes in market conditions. Specifically, it is found that all three risk variables, beta, size and book-to-market equity, exhibit conditional pricing effects. This is the first comprehensive study of its kind on the Hong Kong market, which provides out-of-sample evidence relative to earlier tests on US data. The findings give important insights into capital market behaviour and of value in investment management and corporate financial decisions. JEL classification: G1; G12; G15

This paper derives an after tax vc'rslon of the Capital Asset Pricing Model. The model accounts for a progressive tax scheme and for wealth and mcome related constramts on borrowmg. The equdibrmm relatIonshIp inchcates that before-tax expected rates of return are linearly related to systematic risk and to dlvldend yield The sample estimates of the variances of observed betas are used to arrive at maximum likelihood estimators of the coefficients. The results Indicate that, unlike prior studies, there is a strong positive relatIonship between dividend yield and expected returu for NYSE stocks. Evidence IS also presented for a chentele effect *We thank Roger Clarke, Tom Foregger, Bdl Schwert, Wdliam Sharpe, and the referee, Mchael Brennan, for helpful comments, and Jim Starr for computatIona assistance. Any remammg errors are the authors' responsibility. R.H. Litzenberger and K. Ramaswamy, Taxes, dividends and capital asset prices J F.E D

This paper expresses the price of a spread option as the sum of the prices of two compound options. One compound option is to exchange vanilla call options on the two underlying assets and the other is to exchange the corresponding put options. This way we derive a new analytic approximation for the price of a European spread option, and a corresponding approximation for each of its price, volatilty and correlation hedge ratios. Our approach has many advantages over existing analytic approximations, which have limited validity and an indeterminacy that renders them of little practical use. The compound exchange option approximation for European spread options is then extended to American spread options on assets that pay dividends or incur carry costs. Simulations quantify the accuracy of our approach; we also present an empirical application, to the American crack spread options that are traded on NYMEX. For illustration, we compare our results with those obtained using the approximation attributed to , which is commonly used by traders.

We review the theories on how liquidity affects the required returns of capital assets and the empirical studies that test these theories. The theory predicts that both the level of liquidity and liquidity risk are priced, and empirical studies find the effects of liquidity on asset prices to be statistically significant and economically important, controlling for traditional risk measures and asset characteristics. Liquidity-based asset pricing empirically helps explain (1) the cross-section of stock returns, (2) how a reduction in stock liquidity result in a reduction in stock prices and an increase in expected stock returns, (3) the yield differential between on-and off-the-run Treasuries, (4) the yield spreads on corporate bonds, (5) the returns on hedge funds, (6) the valuation of closed-end funds, and (7) the low price of certain hard-to-trade securities relative to more liquid counterparts with identical cash flows, such as restricted stocks or illiquid derivatives. Liquidity can thus play a role in resolving a number of asset pricing puzzles such as the small-firm effect, the equity premium puzzle, and the risk-free rate puzzle. * The authors thank Joel Hasbrouck for helpful comments.

Although stock prices fluctuate, the variations are relatively small and are frequently assumed to be normal distributed on a large time scale. But sometimes these fluctuations can become determinant, especially when unforeseen large drops in asset prices are observed that could result in huge losses or even in market crashes. The evidence shows that these events happen far more often than would be expected under the generalized assumption of normal distributed financial returns. Thus it is crucial to properly model the distribution tails so as to be able to predict the frequency and magnitude of extreme stock price returns. In this paper we follow the approach suggested by and combine the GARCH-type models with the Extreme Value Theory (EVT) to estimate the tails of three financial index returns S&P 500, FTSE 100 and NIKKEI 225 representing three important financial areas in the world. Our results indicate that EVT-based conditional quantile estimates are more accurate than those from conventional GARCH models assuming normal or Student's t distribution innovations when doing not only in-sample but also out-of-sample estimation. Moreover, these results are robust to alternative GARCH model specifications. The findings of this paper should be useful to investors in general, since their goal is to be able to forecast unforeseen price movements and take advantage of them by positioning themselves in the market according to these predictions. JEL classification: C52; C53; D46 ; G15