Detecting Premium Portfolios in Higher-order Moments CAPM

Tepmony Sim

E-mail: tepmonysim_rupp@yahoo.com

M.sc. “QEM ”, Ca ’ Thesis

Foscari University of Venice

June 2010,

Executive Summary

Since the first introduction by Sharpe (1964), the Capital Asset Pricing Model (CAPM) has then been one of the most popular models for asset pricing, and is a cornerstone of financial economics. However, the CAPM suffers from several restrictive hypotheses such as the normality of return distributions. It has been criticized by many empirical evidences and this has led for further considerations on the extensions within the model. An extension which we regard in this thesis is to insert higher-order moments other than variance into capital asset pricing relation. The resulting model is the Higher- order Moments CAPM. This approach is initially proposed by Rubinstein (1973) and sequentially developed by Kraus and Litzenberger (1976), Fang and Lai (1997), Athayde and Flôres (1997 and 2000), and Jurczenko and Maillet (2006b).

In this dissertation, the Higher-order Moment CAPM which takes into account up to the fourth moment is considered. In Four-moment CAPM, to arrive at the four- moment CAPM fundamental relation, two specific portfolios, besides the riskless asset and the market portfolio, are assumed to exist. Our main purpose is to confirm their existence and show that when they exist, they are not unique. From the validity of four- moment CAPM fundamental relation, the roles of the third moment (skewness) and the fourth moment (kurtosis) can be investigated. To detect the two specific portfolios above as well as to investigate the role of skewness and kurtosis in current financial data, the Seemingly Unrelated Regression (SUR) method by Zellner (1962) is carried out.  It is also our interest to confirm the validity of Four-moment CAPM in another way. Following Fang and Lai (1997), Hwang and Satchell (1999) and Galagedera et al (2004), we can use the Cubic Market Model as a proxy. Our purpose of doing so is to compare the performance of the two models with the in-hand data.

Before we can go through the empirical part, some theoretical foundations are provided. In this part, we basically follow the works by Jurczenko and Maillet (2006b). Several notations used to represent and to compute the higher-order moments are also given. Moreover, a generalization of the univariate higher-order C-(co)moments to multivariate higher-order C-(co)moments is introduced as well. The systematic risk, systematic skewness and systematic kurtosis can be calculated by several means. They can be obtained from the cubic market model, the four-moment CAPM fundamental relation, or by their own definitions. Besides the C-moments these risk factors are also calculated by L-moments. Finally, a comparison of these calculations can thus be made.  Detecting Premium Portfolios in Higher-order Moments CAPM*

Abstract

In Four-moment CAPM, the roles of skewness and kurtosis can be investigated under the validity of the so-called four-moment CAPM fundamental relation. This relation assumes that, besides the riskless asset and the market portfolio, another two specific portfolios exist. We are to show that these portfolios exist but they are not unique. We also confirm the validity of the four-moment CAPM in another direction. Following Fang and Lai (1997), Hwang and Satchell (1999) and Galagedera et al (2004), we use the Cubic Market Model as a proxy of Four-moment CAPM. In the theoretical framework, various notations to represent and to compute the higher-order C-moments of assets ’ returns are introduced. The generalization of the univariate C-(co)moments to the multivariate C-(co)moments is also provided. Besides C-moments, the systematic risk, the systematic skewness, and the systematic kurtosis are also calculated by using L-(co)moments.

Keywords: CAPM, Higher-order Moments, Kurtosis, L-moments, Premium Portfolios,  Skewness, SUR.

JEL Classification: C01, C10, G11, G12.

Introduction

Since it was first introduced by Sharpe (1964), Lintner (1965) and Mossin (1966), the Capital Asset Pricing Model (CAPM) has then been one of the most popular models for asset pricing, and is a keystone of financial economics. This particular theoretical framework relates the risk-return trade-off to a simple mean-variance relationship and/or to a quadratic utility function. However, the empirical evidence shows that the normality hypothesis, which it bases on, has to be rejected for many financial data. A quadratic utility function for an investor, furthermore, implies an increasing risk aversion. Instead, it is more reasonable to assume that risk aversion decreases with an increase in wealth. Due to several inadequacies revealed by empirical tests, CAPM has been considered for further extensions by taking into account more factors additional to mean and variance.  Amongst these extensions, multifactor CAPM is included. The most prominent one of multifactor framework is size effect of Banz (1981). He finds that the size of a firm and the return on its common stock are inversely related.

Later on, Fama and French (1992) suggest three-factor model, which includes the capital size and book-to-market value into classical CAPM. The findings of Fama-French in their three-factor model suggest that small cap and value portfolios have higher expected returns -- and arguably higher expected risk -- than those of large cap and growth portfolios. Carhart (1997), who aims to study the persistence of mutual fund returns, then proposes four-factor model, which is an extension of Fama-French three-factor model by adding a new factor, one-year momentum in stock returns. In short term at least, the results do not support the existence of skilled or informed mutual fund portfolio managers.  Another appropriate approach, which we regard as the center of our interests, is to insert higher-order moments than variance in a pricing relation. The main feature of these models is to obtain, for any risky asset, a linear equilibrium relation between the expected rate of return and higher-order moments systematic risk measures.

In this dissertation, we consider some extensions of the traditional mean- variance framework that account for higher-order moment conditions and a more variegated structure of the risk premium concept. In particular, we examine the roles of skewness and kurtosis in pricing the recent financial data. Skewness characterizes the degree of asymmetry of a distribution around its mean. Negative (positive) skewness indicates a distribution with an asymmetric tail extending towards more negative (positive) values. Kurtosis characterizes the relative peakedness or flatness of a distribution compared with the normal distribution. In standard definition, kurtosis higher (lower) than three indicates a distribution which is more peaked (flatter) than a normal one. Similarly to the so-called systematic risk or beta, it is possible to test for a systematic skewness and systematic kurtosis. Systematic skewness and kurtosis are also known as co-skewness and co-kurtosis (Christie-David and Chaudry, 2001). Provided that the market has a positive skewness of returns, investors will prefer an asset with positive coskewness. Cokurtosis measures the likelihood that extreme returns jointly occur in a given asset and in the market; and thus investors prefer small co-kurtosis. The common characteristic of the models accounting for co-skewness and co-kurtosis is that they incorporate higher moments in the asset pricing framework. In the literature, two main approaches have been investigated: three- moment and four-moment CAPM. The theoretical higher-order moments CAPM is initially proposed by Rubinstein (1973), and, subsequently, developed by Ingersoll (1975), Kraus and Litzenberger (1976), Athayde and Flôres (1997 and 2000), and Jurcenzko and Maillet (2001 and 2006b). Other authors empirically study the validity of the higher-order moments CAPM such as, often, three- moment and four-moment CAPM. For three-moment CAPM, Barone-Adesi (1985) proposes a quadratic model to test the three-moment CAPM, while Harvey and Siddique (2000) find that the systematic skewness requires an average annual risk premium of 3.6% for US stocks. They also find that portfolios with high systematic skewness are composed of winner stocks (momentum effect).

Harvey (2000) shows that skewness, coskewness and kurtosis are priced in the individual emerging markets but not in developed markets. He observes that volatility and returns in emerging markets are significantly positively related. But the significance of the volatility coefficient disappears when co-skewness, skewness, and kurtosis are considered. Harvey ’s explanation for this phenomenon is the low degree of integration of the emerging markets. When accounting up to the fourth moment, Berényi (2002), Christie-David and Chaudry (2001), Chung, Johnson and Schill (2006), Fang and Lai (1997), Hwang and Satchell (1999), Galagedera, Henry and Silvapulle (2002) propose the use of the Cubic Market Model as a test for coskewness and cokurtosis. Berényi (2002) applies the four-moment CAPM to mutual fund and hedge fund data, and he then shows that volatility is an insufficient measure of risk for hedge funds and for medium risk averse agents.

Christie-David and Chaudry (2001) employ the four-moment CAPM on the future markets, where they find that systematic skewness and systematic kurtosis increase the explanatory power of the return generating process of future markets. Fang and Lai (1997), in purpose to corporate the effect of kurtosis, apply the four-moment CAPM on New York Stock Exchange (NYSE). They find that the expected rate of return is not only related to the systematic variance but also to the systematic skewness and systematic kurtosis. Hwang and Satchell (1999) investigate co-skewness and co-kurtosis in emerging markets. They show that systematic kurtosis is better than systematic skewness in explaining emerging market returns.

Following Jurzenko and Maillet (2006b), in Four-moment CAPM, we attempt to detect the two premium portfolios Z1m and Z2m introduced in the so- called Four-moment CAPM Fundamental Relation. These two portfolios are defined as such that: Z1m possesses zero-covariance and zero cokurtosis and has unitary coskewness with market portfolio, and Z2m  possesses zero-covariance and zero coskewness and has unitary cokurtosis with market portfolio. We are to show that these portfolios always exist, however, they are not unique. We propose some methods to elicit the appropriate ones. It is straightforward that when we can find these two premium portfolios, the effect of skewness and kurtosis can be examined. Besides, we also wish to test the validity of the four-moment CAPM in another way. We depart from testing the mean-variance CAPM, then the three-moment CAPM and finally the four-moment CAPM by  using, respectively, the linear market model, the quadratic market model and the cubic market model as the proxies.

In the theoretical framework, various notations for presenting and computing the portfolio returns are introduced. Moreover, a generalization of scalar C-moments of returns to multivariate case is also provided. In addition to conventional moments (C-moments) used in traditional way in higher-order moments CAPM, we also introduce robust moments -- called linear moments (L-moments). The main advantage of L-moments over C-moments is that L-moments, being linear functions of the data, suffer less from the effects of sampling variability: L-moments are more robust than C-moments to outliers in the data and enable more secure inferences to be made from small sample about an underlying probability distribution (see, for instance, Hosking, 1990; Hosking and Wallis, 1987; Ulrych et al, 2000). L-moments sometimes yield more efficient parameter estimates than the maximum likelihood estimates (see Hosking, 1990). Furthermore, L-moments exhibit some specifically fascinating features for financial applications. For example, they have abilities to reduce the so-called Hamburger moment problem (see Jondeau and Rockinger, 2003a; and Jurczenko and Maillet, 2006a); and they are also coherent shape measures of risk (see Artzner, Delbaen, Eber and Heath, 1999).

To obtain the estimations of all wanted parameters, we carry out throughout the thesis the so-called Seemingly Unrelated Regression (SUR) method by Zellner (1962). As pointed out by the author, it is only under special conditions that classical least-squares applied equation-by-equation yields efficient coefficient estimators. For conditions generally encountered, SURE are at least asymptotically more efficient than single-equation least-squares estimators.

The organization of the thesis is as follows. In Chapter 2, we discuss the literature and selected research papers of interest as well as the central theory behind CAPM. In Chapter 3, we present the theoretical framework used for the analysis. Chapter 4 presents the econometric methods used. We devote Chapter 5 for discussing the properties of the data material and for presenting the results obtaining by using econometric methods in previous part. The last part, Chapter 6, the conclusions are made; and the rests are for references and appendices.

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Market Frictions, Momentum and Asset Pricing

by

Lorenzo F. Naranjo

A dissertation submitted in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

Department of Finance

New York University

May 2009

Abstract

The first essay examines theoretically and empirically how borrowing and short-selling costs affect the pricing of derivatives contracts. In the presence of such costs, an agent trading a derivatives contract is unable to perfectly hedge the derivative if she is exposed to exogenous demand shocks. I derive an equilibrium model of the futures market where demand imbalances from traders affect the price of derivatives contracts. In the model, the price of a derivative depends on the risk- free rate and a latent demand factor. I estimate the model using S&P 500 index futures and the Kalman filter, and find that the latent demand factor is priced. I also find that the latent demand factor is closely related to proxies for demand pressure in the futures market, such as large speculator positions as well as investor sentiment. The demand factor is positive (negative) when buying (selling) pressure is high and is difficult to borrow (to short-sell the underlying). I also find that demand imbalances are correlated across different indexes, for both futures and options.

The second essay studies the stochastic behavior of implied interest rates from derivatives contracts. I start from the observation that there are many proxies for the short-term interest rate that are used in asset pricing. Yet, they behave differently, especially in periods of economic stress. Derivatives markets offer a unique laboratory to extract a short-term borrowing and lending rate available to all investors that is relatively free from liquidity and credit effects. Interestingly, implied interest rates do not resemble benchmark interest rates such as the three- month T-bill rate or LIBOR, but instead are much more volatile. I argue that the volatility in the implied short-term rate in futures and option markets is due to frictions arising from borrowing and short-selling costs. By using the techniques developed in the first essay of this dissertation, I propose a methodology to filter the “true” risk free rate from noisy implied interest rates. The risk-free rate that results from this estimation has time-series properties similar to Treasury and LIBOR rates, and anticipates interest rate changes. The spread between the risk-free and the T-bill rate correlates with liquidity proxies of the Treasury market, while the spread between LIBOR and the risk-free rate is related to economic distress.

The third essay examines the effect of improvements in relative rankings among stocks. The fact that the momentum effect survived the widely cited paper by Jegadeesh and Titman (1993) is itself evidence in favor of a behavioral explanation of this hypothesis. The experimental design of Jegadeesh and Titman (2001) excludes the possibility that the results are some artifact of thin trading or small stocks. However, to validate the behavioral hypothesis we must consider other possible implications. If the momentum effect is indeed under-reaction and delayed overreaction to news events as suggested in Jegadeesh and Titman (2001), then we should anticipate an even greater effect when there are improvements in relative rankings. We refer to this as the acceleration hypothesis and find that it is a significant and distinct factor that has interesting implications for the cross section of security returns.

1.1 Introduction

In this paper I study theoretically and empirically the effect of borrowing and short-selling costs in the pricing of derivatives contracts. In the presence of such costs, an arbitrageur that provides liquidity in a particular derivatives market will be unable to perfectly hedge the derivative if she is exposed to exogenous demand shocks. Thus, demand imbalances for the derivative affect prices, in a similar manner as in the model originally studied by Garleanu, Pedersen, and Poteshman (2007) for options, and Vayanos and Vila (2007) for fixed-income markets.

In perfect markets, it is well-known that the cost of carrying forward a hedged derivatives position should be equal to the risk-free rate minus the dividend yield of the underlying asset. Thus, in frictionless markets it is possible to use derivatives contracts to borrow or lend funds at the risk-free rate. Borrowing funds is accomplished by selling the derivative and buying the risky asset, whereas lending funds is accomplished by buying the derivative and short-selling the risky asset.

In general, though, there are costs faced by agents when they want to borrow funds or short-sell the risky asset. As a result, if a group of agents borrow funds using the derivatives market, the equilibrium derivative’s price should reflect the costs associated with borrowing to preclude arbitrage opportunities. Agents borrowing through the derivatives market generate demand pressure for the derivative, affecting its price because of the borrowing costs. A similar effect follows if a group of agents generate demand pressure in the derivatives market to short-sell the risky asset. Thus, in the presence of borrowing and short-selling costs the equilibrium price of a derivatives contract depends on its demand imbalance.

Even though the intuition can be applied to price any derivative contract, in this paper I specialize the analysis to the pricing of futures contracts. There are several reasons for doing this. First, the pricing formulas can be derived in closed-form, which is convenient for empirical tests of the model. Second, futures contracts are very simple instruments with linear payoffs, which makes their valuation more robust to model assumptions. Third, futures contracts have been trading for a long time, which provides with a long time series of observations to use in empirical tests. Fourth, futures contracts are very liquid instruments, which makes their prices less susceptible to deviate from fair value for liquidity reasons.

The model setup is described in Section 1.2. There are two types of agents: arbitrageurs and traders. Arbitrageurs are rational agents that maximize their expected utility and operate in a competitive market in which there are no-arbitrage opportunities. Traders can be speculators or hedgers, although I do not distinguish between them and assume that all traders have an exogenous demand for the derivative. Arbitrageurs take the opposite position in the derivatives market and hedge their position using the risky asset and risk-free bonds.

In the model arbitrageurs pay borrowing and short-selling costs. One way to motivate this is to assume that arbitrageurs trade with a bank that provides brokering services. In order to derive a closed-form solution for the model, I make the simplifying assumption that borrowing costs increase linearly with demand. The assumption is consistent with the fact that borrowing rates may differ from lending rates, and that it is more expensive to borrow larger amounts of the riskless or risky asset. If traders take a long position in futures contracts, arbitrageurs will take a short position in the futures market. In order to hedge the additional risk carried by the futures, arbitrageurs will simultaneously take a long position in the risky asset. If this position is sufficiently large, they will have to borrow and pay borrowing costs. Equivalently, if traders short the futures, arbitrageurs will have to take the opposite position in the futures market and hedge it by selling the risky asset. If the shorting demand by traders is sufficiently large, arbitrageurs will have to sell short the risky asset, incurring in short-selling costs.

In Section 1.3 I derive an equilibrium model of the futures market where demand imbalances from traders affect the price of futures contracts. The mechanism is as follows. If traders’ demand is positive, then arbitrageurs short the derivative, buy the risky asset and borrow. In equilibrium, arbitrageurs borrow when traders’ demand is positive, which is precisely when borrowing costs are high. In a competitive market, arbitrageurs set the price of the derivative such that they are indifferent between taking the opposite side of the trade or doing nothing. This equilibrium price will be higher than the price obtained in an otherwise equivalent frictionless economy. A similar logic applies if traders want to short the derivative. In that case, arbitrageurs buy the derivative, short the underlying asset and pay short-selling costs. In equilibrium, the price of the derivative should be set lower than in an otherwise equivalent frictionless economy in order to motivate arbitrageurs to take the long position.

In this economy the futures price in the presence of borrowing and short-selling costs depends on the risk-free rate and a latent demand factor. Essentially, the model predicts that the difference between the usual cost-of-carry formula and observed prices is given by the latent demand factor. Thus, derivatives “mispricings” could just be proxying for demand pressure and borrowing and short-selling costs. Moreover, the futures price is equal to the expected spot price under a “local” equivalent risk-neutral measure, implying that arbitrageurs receive a time-varying risk premium for providing liquidity to traders.

The theory has several testable implications. First, the model predicts the existence of a risk premium for the demand factor. Intuitively, if borrowing and short-selling costs are high when demand is high, and vice-versa, then the arbitrageur is exposed to an additional source of risk when hedging her portfolio and commands a premium. Second, the latent demand factor should be related to proxies for demand pressure in the futures market. In other words, proxies for demand pressure in the futures market should be priced factors in the term-structure of futures prices. Third, the demand factor in equilibrium should be related to the borrowing and short-selling costs of the marginal trader. Thus, the model predicts that the latent demand factor is positive when buying pressure is high and it is difficult to borrow irrespective of short-selling costs. Similarly, the latent demand factor is negative when selling pressure is high and it is difficult to short-sell the underlying asset regardless of borrowing costs. Fourth, if demand pressure for derivatives is driven by a common factor, deviations from fair-value should be correlated across markets and derivative instruments.

I estimate the model parameters and state variables using S&P 500 index futures and the Kalman filter in Section 1.4. This is possible because the model delivers a closed-form solution for the valuation of futures contracts and the state- variables follow a system of Gaussian processes. I estimate sixteen parameters by maximizing the likelihood function of price innovations.

The estimation reveals that the demand factor that enters the pricing formula is significantly priced. Thus, engaging in index futures arbitrage is risky and should be compensated. This provides a rationale for the existence of a large industry dedicated to arbitrage deviations from fair-value in derivatives markets. Any agent who has a competitive advantage in borrowing cheaper than others enjoys an economic surplus.

I also verify that the latent demand factor is related to proxies for demand pressure in the futures market, such as large speculators positions in S&P 500 futures and market sentiment for large institutional investors (Han, 2008). If on average the pressure is generated by large speculators, it follows that hedgers are actually paid a premium for the services they provide.

Moreover, I find that the latent demand factor is positive when buying pressure is high and it is difficult to borrow independently of short-selling costs. The opposite effect is also true. The latent demand factor is negative when selling pressure is high and it is difficult to short-sell the risky asset, independently of borrowing costs. This implies that borrowing costs affect the pricing of derivatives only when there is demand pressure to buy the derivative but not to sell it. Similarly, short-selling costs are relevant only when when there is demand pressure to short the derivative but not to buy it.

Finally, I look at whether latent demand imbalances co-move across instruments and markets. Since the demand factor is essentially capturing deviations from fair value in a market without frictions, it is possible to obtain estimates of this factor for other indexes and also put-call parity relations on stock index options. I find that mispricings in futures and put-call parity relations in three indexes (S&P 500, DJIA and Nasdaq 100) are significantly correlated with each other and also with the sentiment proxy, suggesting co-movement in demand imbalances across instruments and markets.

This paper is closely related to two papers that show how demand imbalances for a derivative affect its price. Garleanu et al. (2007) show how demand for options affect options prices, while Vayanos and Vila (2007) analyze a similar effect for fixed-income markets. The main difference with these two papers is that I analyze both theoretically and empirically the effect of borrowing and short-selling costs in the pricing of derivatives.

Also, this study contributes to the literature that studies index futures arbitrage. The main focus of this literature has been to understand absolute mispricings, whereas this paper contributes understanding the sign and magnitude of mispricings.

Finally, this paper fits into a broader literature that looks at how market frictions prevent arbitrageurs in some circumstances from profiting of arbitrage opportunities. Effectively, I show that borrowing costs open a channel through which demand imbalances in the derivatives market can affect no-arbitrage prices, even when the payoff is linear in the underlying asset.

Chapter 2

Implied Interest Rates in a Market with Frictions

2.1 Introduction

Measuring and understanding the risk-free rate is a fundamental question in financial economics. The risk-free rate is a required input in many theories that are extensively used by academics and practitioners in finance, like the CAPM, the APT and the no-arbitrage valuation of derivatives products. The valuation of real and financial assets depends crucially on using the correct risk-free rate. In complete markets, the risk-free rate is uniquely determined as the conditional expectation of the inverse of the stochastic discount factor. In incomplete markets, however, the risk-free rate might not trade at all. If that is the case, there might be several ways of defining what we understand for the risk-free rate (see e.g. Cochrane, 2005, Section 6.5). In this paper, I define the risk-free rate for a particular maturity to be the yield of a zero-coupon bond that is free of liquidity and credit risk effects.

In Section 2.2 I start analyzing the two main proxies for the short-term interest rate that are commonly used in asset pricing. On the one hand, it is common for empirical researchers in finance to use the yield on three-month T-bills as a proxy for the risk-free rate. The intuition for this practice is simple: T-bills are backed by the full faith and credit of the U.S. government. As such, they are the safest investment available for an investor whose consumption is denominated in U.S. dollars. One problem with using T-bill yields as a proxy of the risk-free rate is that only the U.S. government can borrow at this rate. Also, Treasury rates can exhibit periods of flight-to-liquidity during which investors are willing to pay more for the benefit of holding a liquid security (Longstaff, 2004). On the other hand, it is common for practitioners to use LIBOR rates as a proxy for the risk-free rate when valuing derivatives contracts. The intuition for this practice is also simple: practitioners regard LIBOR as their opportunity cost of capital. However, LIBOR represents the interest rate charged on an uncollateralized loan between banks and hence is subject to credit risk.

As a result, these interest rates behave differently, especially in periods of economic stress. As an example, Figure 2.1 shows how Treasury and LIBOR rates have being drifting apart during the 2007-08 credit and liquidity crisis. The so- called TED spread, defined as the difference between LIBOR and Treasury rates, was at an all time high (461 bp) during October 2008.

Fortunately, there are other markets that investors can use to lend or borrow from which it is possible to infer short-term interest rates. For example, an investor can borrow by entering into a long position in a forward contract and selling the underlying asset. Similarly, an investor can lend by shorting a forward and buying the underlying asset. If this transaction is performed through an organized exchange, standard features such as margin requirements and the existence of a clearing corporation significantly reduce the credit risk of the transaction. Also, since derivatives contracts are in zero net-supply, flight-to-liquidity problems are mitigated. Thus, derivatives markets offer a unique laboratory to extract a short- term borrowing and lending rate available to all investors that is free from liquidity and credit risk effects.

In perfect markets, it is well-known that the cost of carrying a forward position should be equal to the risk-free rate minus the underlying asset’s dividend yield. Thus, in frictionless markets2 it is possible to use forward and futures contracts as substitutes for risk-free bonds to derive risk-free rate estimates.

In Section 2.4 I study the implied risk-free rate obtained from futures contracts and put-call parity relations written on major indexes: S&P 500, Nasdaq 100 and Dow Jones Industrial Average (DJIA). The data used in this section is described  in Section 2.3. I find that on average implied interest rates from both futures and options lie between Treasury and LIBOR rates. From January 1998 to December 2007, three-month rates implied from futures prices are on average 48 bp above Treasury and 5 bp below LIBOR, whereas implied interest rates from options are on average 50 bp above Treasury and 3 bp below LIBOR. Thus, implied interest rates are very similar regardless of whether they are extracted from futures or options, and are on average much closer to borrowing (LIBOR) rather than lending (Treasury) rates. This result is consistent with the common industry practice of using LIBOR rates as a proxy for the risk-free rate when valuing derivatives contracts, and also with previous findings in the literature (Brenner and Galai, 1986).

Interestingly, the time-series of implied interest rates do not resemble that of benchmark interest rates such as the three-month T-bill rate or LIBOR, but instead is much more volatile. As discussed in Chapter 1, the phenomenon is expected if we account for market frictions. In the presence of borrowing and short-selling costs, an arbitrageur that provides liquidity in a particular derivatives market will  be unable to hedge the derivative perfectly if she is exposed to exogenous demand shocks. In this case demand imbalances for the derivative will affect prices, making implied interest rates to be correlated with demand, increasing their volatility and affecting their level.

However, it is possible to use the methodology outlined in Chapter 1 to estimate the risk-free rate implied in derivatives prices. Using the Kalman filter, it is possible to infer the time-series of the risk-free rate from the price of a derivative contract. The main identifying assumption used in the estimation is that the risk-free rate is much more persistent than the demand factor driving the “mispricing” of the derivative.

I restrict the estimation of the risk-free rate to S&P 500 futures contracts because they provide with the longest time-series of observations, and they are one of the most liquid derivatives contracts available. In Section 2.5 I show that the risk-free rate that results from this estimation has similar time-series properties as Treasury and LIBOR rates.

The estimated spot rate is on average 15 bp above the fed funds rate, and short-term rates are on average lower than implied rates computed in Section 2.4. Most interestingly, the spot rate seems to be forecasting the Federal funds rate, suggesting that futures markets anticipate changes in short-term interest rates. Also, the spread between LIBOR and the risk-free rate is high in periods of market stress, as proxied by the VIX and the credit spread between AAA and BAA bonds. The spread between the risk-free rate and three-month T-bill rates is high in periods of market illiquidity. These results suggest that the estimated risk-free rate is less affected by liquidity and credit risk than Treasury rates and LIBOR, respectively.

While there is a huge literature on dynamic term structure modeling3 using Treasury bonds and LIBOR rates, the use of derivatives for extracting information about the risk-free rate has been mostly ignored by financial economists. A notable exception is Brenner and Galai (1986), who are probably the first to estimate implied risk-free rates from put-call parity relations on stock option prices. In related work, Brenner et al. (1990) also look at implied risk-free rates using Nikkei index futures data. Liu, Longstaff, and Mandell (2006) obtain implied risk-free rates from plain-vanilla swap contracts, but they use the three-month General Collateral (GC) repo rate as a proxy for the three-month risk-free rate. I make no initial assumptions about what the risk-free rate should be. Feldhütter and Lando (2008) also use swap data to estimate the risk-free rate. However, I use a much larger time-series and I cross-validate my implied rate with different assets. I also account for the fact that demand pressure can affect derivatives prices, distorting implied risk-free rate estimates in significant ways.

Chapter 3

Momentum and the Acceleration Hypothesis (joint with Stephen Brown )

Introduction

A seminal paper by Jegadeesh and Titman (1993) showing that there is strong evidence of serial dependence in the return rankings of stocks has been highly influential in subsequent research for both by academics and practitioners. To date there have been 305 citations of this research in refereed journal publications, and altogether 1188 citations including citations in unpublished working papers that have been electronically circulated1. The results of this research are widely accepted among practitioners and have been used to develop trading strategies both in the United States and abroad. The analysis has been extended in a number of ways to examine whether it applies in other countries (for example Rouwenhorst, 1998) and to other financial statistics (for example, earnings reports). Despite the influence of this research and the importance of its application, the momentum phenomenon remains mysterious. What is the source of this momentum and is it likely to persist in the context of numerous trading strategies designed to exploit it?

One curious finding difficult to reconcile with rational market behavior is the result that the strength of the momentum effect is if anything greater after the effect was described and published in 1993 (Jegadeesh and Titman, 2001). While the results of the 1993 paper appear to diminish and even disappear when the 1993 study is extended to 2004 and to NASDAQ listed stocks (Table 3.1), Jegadeesh and Titman (2001) exclude microcap stocks from the analysis. The results are if anything much stronger once we exclude from the analysis stocks trading under $5.00 (Table 3.2). This result is important, not only as a post sample test of the momentum hypothesis, but also because it excludes the possibility that the momentum effect is an artifact of thin trading and/or small stocks in the sample.

These results leave standing the behavioral hypothesis of Jegadeesh and Titman (2001). These results alone however cannot exclude other possible explanations for the observed momentum effect. One approach is to examine other possible implications of the behavioral hypothesis. The purpose of this paper is to examine one such implication. If the momentum effect arises from delayed overreactions that are eventually reversed, then we should anticipate that improvements in the relative ranking of stocks have an even more extreme effect. We identify this additional implication as the acceleration hypothesis and find that it is a significant and distinct factor that has implications that differ for small and large traded equities. This would appear to be confirmation of the behavioral hypothesis.

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On the Interaction Between Firm Level Variables, the CAPM Beta, and Stock Returns

Thesis by

Laura Panattoni

In Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

California Institute of Technology

Pasadena, California

2009

Abstract

In Chapter 1, I conduct a theoretical study of how horizontal industry concentration affects a firm’s market capitalization and systematic risk. I first develop a method for incorporating an equilibrium theory of the firm, drawn from industrial organization, into a single period version of the Capital Asset Pricing Model (CAPM). This extension establishes the microeconomic determinants of systematic risk by relating firm specific variables to Beta.

Unlike the previous literature, I add local product market shocks to a general, deterministic profit function and use an orthogonal decomposition of the market return to endogenize the Cov[Ri,RM]. I also use this method with standard Hotelling and Cournot models of firm behavior and with different sources of uncertainty to provide examples of how increasing concentration can increase, decrease, and be independent of Beta. In Chapter 2, I exploit a natural experiment afforded by the announcement of ‘Paragraph IV’ patent infringement decisions. These judgments have two unique features. They create an exogenous change in industry concentration, since they determine whether the corporate owner of a brand name prescription drug will maintain or lose monopoly marketing rights. They also satisfy the methodological requirements to use a short window event study. Against a backdrop of contradictory empirical evidence, this experiment provides a clean test to empirically determine the sign of how a change in horizontal industry concentration affects stock returns. For a sample of 38 District Court decisions between 1992 and 2006, I find that the announcement return is between [1.24%, 2.83%] if the brand firm ‘wins’ the case and between [-5.24%, -5.82%] if the brand ‘loses’. Finally, I use these returns to construct the first market valuation of the monopoly rents for brand name pharmaceutical firms. I find that the value to a brand firm of maintaining marketing exclusivity for 1 ‘average’ drug for 92 months is between [6.48%, 8.65%]. In Chapter 3, I explore the cross-sectional determinants of Beta.

The two main goals of this exercise is to understand the explanatory power of popular asset pricing variables and firm level variables, such as the coefficient of variation of profit. The estimation relies on a minimum distance approach that reduces to the familiar least squares estimators. This approach permits the estimation of a dataset where the number of cross sectional observations is larger than the number of time period and accounts for the measurement error in Beta. I use two different sets of variables where one is weighted by assets, referred to as ‘Book’ variables and the other is weighted by market capitalization, referred to as ‘Market’ variables. I include two robust checks, one of which includes adding industry fixed effects. I find some striking results with respect to both the two asset pricing variables and the coefficient of variation of profit proxy. Since my statistics are pooled over different time periods, I cite the statistics from the 2001 sub-period because it has three times as many observations as the rest of the periods combined. Turnover has the largest magnitude and t-statistics in both sets of regressions. In 2001, the means of Beta A and Beta were .94 and 1.2 respectively. I found that a one standard deviation change in turnover increased the magnitude of Beta A by .22 and Beta by .25. The bid ask spread percentage had a larger magnitude coefficient in the ‘Market Regressions’, which indicated that a one standard deviation change in this variable increased Beta by .08. On the other hand, I found that ln(assets), ln(size), and book-to-market had the smallest magnitudes and t-statistics. Finally, both regressions indicate that as the proxy for the coefficient of variation of profit variable increases (decreases) for firms with a positive (negative) expected profit, Beta increases. For the 2001 sub-period in the ‘Market’ regressions, a one standard deviation change in the absolute value of this proxy, increases Beta by a magnitude of and .15 for firms with positive and negative ‘earnings’. Finally, these results are robust to industry fixed effects.

Introduction

Asset pricing models have been developed, somewhat myopically, with little reference to the product market and therefore to industrial organization. The best known asset pricing models base their predictions on variations in investor preferences, behavioral biases in financial markets, or each asset’s covariance with a market portfolio. Remarkably, these models disregard how economic fundamentals in the product market, such as firm specific or industry wide characteristics, may affect financial equilibria. Fama [17] acknowledged this omission when he advocated that researchers should either relate the behavior of expected returns to “the real economy in a rather detailed way” [p1610] or establish that no such relationship exists.

The Capital Asset Pricing Model (CAPM) provides an excellent example of an equilibrium financial model in which asset prices are derived independent of the ‘real economy’. The CAPM’s main insight is that an asset’s expected excess return is determined by its covariance with a market or aggregate portfolio. The covariance is included in an assets’s Beta term, which within the capital budgeting framework, may be interpreted as a risk adjusted discount rate. While the CAPM relates a firm’s Beta term1 to its expected return, the model provides no relationship between a firm’s profits and its Beta term. Therefore, the model cannot predict which type of firm strategies, types of competition, or industry characteristics create more systematic risk. In terms of capital budgeting, there is no way to update the risk adjusted discount rate in response to product market changes.

In this chapter, I conduct a theoretical investigation of how industry concentration2 affects a firm’s market capitalization and systematic risk within the CAPM framework. I first incorporate an equilibrium theory of firm behavior into the CAPM. This extended version of the CAPM can combine models of both perfect and imperfect industry competition with a competitive model of security pricing. Therefore, I contribute to the emerging theoretical literature establishing a micro-foundation for asset pricing models. I then use this extension of the CAPM with two standard models of firm behavior, the Hotelling model and the Cournot model, to test the effects of concentration.4 These examples illustrate how product market factors, such as different types of firm competition or different sources of uncertainty, influence the way in which concentration affects financial outcomes.

Although firm level variables, such as risky cash flows and the capital structure, have been related to the CAPM since the early 1970’s, (Rubinstein), the literature incorporating an equilibrium theory firm behavior into the CAPM has been sparse. Subrahmanyam and Thomadakis used a quantity choosing model of firm behavior and the Lerner Index to coincidentally study the effect of industry concentration. They found that for a given capital labor ratio, decreasing concentration increased systematic risk. Bhattacharyya and Leach applied the CAPM valuation formula to firm profit and found conditions under which Beta is independent of the quantity chosen. Kazumori used the consumption CAPM and the idea of consumption risk, the cost of switching products if the product fails, to find that increasing market share increased systematic risk. As an increasingly asymmetric distribution of market shares also represents increased concentration, Kazumori’s results contradicts the sign of the results found by Subrahmanyam and Thomadakis.

The literature in this field can be compared according to two salient features of the models. The first feature is the structure of uncertainty. In their one period model, Subrahmanyam and Thomadakis use an additive shock to the demand function and to the labor supply. Bhattacharyya and Leach use state probability pricing. Finally, Kazumori uses a continuous time stochastic calculus framework with shocks to consumption at each period. The structure of uncertainty in these papers makes it difficult to study the effect of a different parameter on systematic risk, add uncertainty to a different parameter, or change the character of firm competition within the framework of these models.

On the other hand, I add local product market shocks to a general profit function which provides a more general framework. Product market shocks are simply a shock added to any primitive of a profit function, such as costs or consumer preferences. This method characterizes the effect of a small amount of uncertainty by linearly approximating a random profit function. Many deterministic models from industrial organization can easily be used within this framework. Therefore, unlike the previous literature, my method is not dependent on the exact model of firm behavior or which parameters are shocked.

I chose to study industry concentration because aside from the above two conflicting theoretical results, the previous work on industry concentration has been contradictory and mostly empirical. The empirical work supports the three contradictory conclusions that in- creasing industry concentration decreases, increases, and does not affect a stock’s expected returns. In support of the first conclusion, Hou and Robinson [31] estimate that firms in the quintile of the most competitive industries have returns nearly four percent greater than firms in the most concentrated quintile. These authors argue that competitive industries are riskier because they are more likely to face change from innovation, Schumpeter’s creative destruction, and that they are more sensitive to demand shocks due to lower barriers to entry. However, they do not provide a formal model that links these arguments to their econometric work, and they do not look at share prices.

In contrast, Lustgarten and Thomadakis and Melicher, Rush, and Winn support the other two conclusions. Lustgarten and Thomadakis found that announced changes in a firm’s accounting earnings led to a greater change in the market capitalization in more concentrated industries. If future prices are treated as exogenous, this weakly implies that the stock returns increased with concentration. Finally, Melicher, Rush, and Winn look at 495 manufacturing firms and find that industry concentration has no effect on a stock return.

Using the Hotelling and the Cournot model, this chapter will provide theoretical examples of how the standard deviation of profit, the market capitalization, and the expected return approximately change with N when there are different sources of uncertainty. First, the standard deviation of profit decreases in N except in the case of a Cournot firm is facing a shock to costs. The market capitalization always decreases in N for sufficiently small shocks. Finally, the expected return either increases in or is independent of N, except in the case of a Hotelling firm with a relatively large market share facing a shock to the intensity of product differentiation. The expected return increases in N when firms face shocks to costs, when a Hotelling firm with a relatively small market share is facing a shock to the intensity of product differentiation, and when a Cournot firm is facing a shock to the slope of the demand function. The expected return is independent of N when the shock simply rescales the profit function, when there is a monopoly, and when there are perfectly competitive firms.

This chapter will be divided into three main sections. The first section will incorporate an equilibrium theory of the firm from industrial organization into CAPM. The second section will use the Hotelling model of firm behavior to study how industry concentration affects a firm’s market capitalization and systematic risk. Finally, the third section uses a Cournot and competitive model to study how industry concentration affects these financial outcomes.

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EMPIRICAL TESTS OF ASSET PRICING MODELS IN FINNISH STOCK MARKET

Mauri Paavola

LAPPEENRANTA UNIVERSITY OF TECHNOLOGY

School of Business Finance

Helsinki, October 30th, 2007

Porvoonkatu 1 D 133

00510 Helsinki

+358 50 588 0826

ABSTRACT

This study investigates the relationship between different sorts of risk and return on six Finnish value-weighted portfolios from the year 1987 to 2004. Furthermore, we investigate if there is a large equity premium in Finnish markets. Our models are the CAPM, APT and CCAPM. For the CCAPM we concentrate on the parameters of the coefficient of the relative risk-aversion and the marginal rate of intertemporal substitution of consumption, whereas for the CAPM we estimate the market beta and for the APT we will select some macroeconomic factors a priori.

The main contribution of this study is the use of General Method of Moments (GMM). We implement it to all of our models. We conclude that the CAPM is still a robust model, but we find also support for the APT. In contradiction to majority of studies, we are able to get theoretically sound values for the CCAPM’s parameters. The risk-aversion parameters stay below two and the marginal rate of intertemporal substitution of consumption is close to one. The market beta is still the most dominant risk factor, but the CAPM and APT are as good in terms of explanatory power.

Key words:

Stochastic Discount Factor (SDF), Capital Asset Pricing Model (CAPM), Arbitrage Pricing Theory (APT), Consumption-based Capital Asset Pricing Model (CCAPM), Equity premium puzzle, General Method of Moments (GMM)

INTRODUCTION

The economic theory of capital asset pricing relies heavily on the principles of present value calculations and the hypothesis of efficient capital markets. The former tells us that the price of an asset is a function of the expected future yields discounted to the current date. This should apply to all assets, such as stocks, land, houses and durables, since they are alternative investment objects. (Takala & Pere, 1991) In particular, modern financial theory is founded on three central assumptions: markets are highly efficient, investors exploit arbitrage opportunities and investors are rational. (Dimson & Mussavian, 1999).

An important body of research in financial economics has been the behavior of asset returns and especially the forces that determine the prices of risky assets. There are also a number of competing theories of asset pricing. These include the original capital asset pricing models (hereafter CAPM) of Sharpe (1964), Lintner (1965) and Black (1972), the intertemporal models of Merton (1973), Long (1974), Rubinstein (1976) and Cox et al. (1985), the consumption-based asset pricing theory (hereafter CCAPM) of Breeden (1979); Lucas (1978) and the arbitrage pricing theory (hereafter APT) of Ross (1976).

Breeden (1979) and Lucas (1978) took a different approach in defining equilibrium in capital markets. They are able to show, under certain assumptions, that return on assets should be linearly related to the growth rate in aggregate consumption if the parameters of the linear relationship can be assumed to be constant over time (Elton et al., 2007). Breeden (1979) and Lucas (1978) models are so called “representative” agent models of asset returns in which per capita consumption is perfectly correlated with the consumption stream of a typical investor. In this type of models, a security’s risk can be measured using the covariance of its return with per capita consumption (Kocherlakota, 1996).

The CAPM is by far the most famous asset pricing model. It is widely used and examined both in literature and in practice. However, the CAPM is only a description of the reality. By this we mean that the CAPM does not help us understand what the ground factors are and how they affect the risky returns. If we want to go deeper and try to understand what the affecting forces are, how the investors define the returns for the risky assets, we have to start from the basic utility theory and try to find the solution from there.2 The intuitive model to examine is the CCAPM, where the return is given with the covariance of investor’s marginal utility. Moreover, in contrast to the CAPM, intertemporal general-equilibrium models identify clearly the economic forces that influence the risk-free real interest rate and the compensation that investors earn by accepting risk.3 (Carmichael, 1998).

Objectives and methodology

The purpose of this thesis is to find out what the affecting forces behind the stock returns are, and which of these risk factors are significant. We will test the traditional market beta of the CAPM and some other macro-economic risk factors employed in the APT. For the CAPM and APT our focus is on the risk factors (betas), whereas for the CCAPM we will focus on the risk-aversion and discount factor parameters, γ and β. Our purpose is to compare all of these models. We will also try to find reasonable values for the CCAPM’s parameters that have failed numerous times in empirical studies. Our focus will be on the CCAPM, because it is the least known from these asset pricing models. The CAPM and APT serve more as a benchmark models, although we are going to present them in detail. The results of testing the different models are quite inconclusive. CAPM is widely used and its functions are well documented. On the other hand, there is a big group of researchers5 that say that the CCAPM and its consumption beta should be preferable on theoretical grounds, although its empirical testing has failed numerous times.

We will employ Lucas (1978) study for testing the CCAPM with the standard Constant Relative Risk-aversion (hereafter CRRA) power utility function. We will examine a developed stock market of Finland and try to explain the differences in the returns of value-weighted portfolios. The other purpose of this thesis is to examine the equity-premium puzzle emerged from the fact that the consumption tends to be too smooth. Mehra & Prescott (1985) presented the equity-premium puzzle and this rock solid evidence against the CCAPM is still unsolved.

These differences in our value-weighted portfolio returns should be explained only by the different risk factors and the sensitivities of returns to the risk factors according to the theory. In the CAPM the expected equity premium (excess return) is proportional to market beta. The APT relates the expected rate of return on a sequence of primitive securities to their factor sensitivities, suggesting that factor risk is of critical importance in asset pricing. In comparison, the standard CCAPM measures the risk of a security by the covariance of its return with per capita consumption. (Elton et al., 2007)

Most of the empirical tests of these models have been conducted for developed markets, e.g., U.S. and Germany. This study will examine the stock market of Finland. To the best of our knowledge, the Finnish stock market has not been examined this way. There are tests of the CCAPM and of course of the CAPM and APT, but no comparisons of these models in the same data set. We will compare the realized asset returns within these models to see which model provides the best results of explaining the time-series variation of value-weighted stock portfolios.

One of the main contributions of this thesis is the use of the Generalized Method of Moments (hereafter GMM) method. Again, to the best of our knowledge, this method has not been used in this way for the Finnish data, i.e., comparing these asset pricing models in the same data set. We will employ the GMM to all of our empirical tests and make comprehensive conclusions of the asset pricing models’ ability to explain the portfolio returns. The GMM is a general statistical method for obtaining estimates of parameters of statistical models and it is widely used in the finance literature. All the empirical tests are done with Matlab.

Limitations and motivations

This study is performed from the European investor’s point of view so that the currency used in this study is euro and also risk-free rate is quarterly Euribor. In selecting the factors for our different models we will choose them a priori, as in Chen et al. (1986). The data used in this study is gathered from ETLA, Research Institute of the Finnish Economy, and Data- stream. The research period will be from the beginning of the year 1987 to the end of year 2004.

We will test the asset pricing models in their purest form. This means that, e.g., the CCAPM is tested as it was presented in Lucas (1978). Thus, we will not use any other implications of the CCAPM that are, e.g., the habit formation of Constantinides (1990), the “non-expected utility” preferences of Epstein and Zin (1989) or the investment-based asset pricing model of Cochrane (1996), to name a few. However, these studies are important because they had had some success in solving the problems of the CCAPM and the key results are presented in this thesis. We are well aware that doing this thesis in this way, without any further assumptions or modifications of the CCAPM, may lead to a rejection of the CCAPM. This is probably the case, because we know that Finnish stock market has a relatively high equity premium, especially in our research period.9 How- ever, we also know that other models that we presented above have not had success in solving the equity premium puzzle, at least not in different markets, data sets, etc. Thus, right now we do not have an explicit solution to the equity premium puzzle, which we will show in further chapters.

Structure

The thesis is structured as follows. The next chapter introduces the basics of the utility theory and the concept of Stochastic Discount Factors (SDF). The third and fourth chapter presents the different asset pricing models in great detail and also one of the biggest debate issues in earlier studies, the determination of relevant factors, is represented. We will separate the discussion between the CAPM/APT and the CCAPM, because our main focus in on the less known CCAPM. Furthermore, the CAPM and APT are quite alike models, but the CCAPM comes from totally different grounds. In the fifth chapter we will go through some of the basic problems associated with the asset pricing models and go through an extensive amount of previous empirical studies. The sixth chapter describes the data for our purposes. We will especially concentrate on the research methodology, because we have found out that there are a lot of different methods to choose from. Furthermore, the GMM is explained in a difficult way and also quite irrationally in many parts of the literature, although it is a quite simple and effective method. The seventh chapter reports the empirical results and findings. The final chapter is for conclusions and suggestions for further research.

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