Lecture notes, Index Models and the Arbitrage Pricing Theory PDF

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Investment Management Chapter VIII – Index Models and the Arbitrage Pricing Theory 

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Arbitrage pricing theory (APT) – an asset pricing theory that is derived from a factor model, using diversification and arbitrage arguments. The theory describes the relationship between expected returns on securities, given that there are no opportunities to create wealth through risk free arbitrage opportunities. The APT predicts a security market line linking expected returns to risk, through a different path. The theory relies on three main propositions: o Security returns can be described by a factor model; o There are sufficient securities to diversify away idiosyncratic risk; o Well-functioning security markets do not allow for the persistence of arbitrage opportunities. Law of one price – the rule stipulating that securities or portfolios with equal returns under all circumstances must sell at equal prices to preclude arbitrage opportunities. Revenues and expenses are listed and their difference is calculated as net income. The critical property of a risk-free arbitrage portfolio is that any investor, regardless of risk aversion or wealth, will want to take an infinite position in it. There is an important difference between arbitrage and risk-return dominance arguments in support of equilibrium: o A dominance argument holds that when an equilibrium price relationship is violated, many investors will make limited portfolio changes, depending on their degree of risk aversion. Aggregation of these limited portfolio changes is required to create a large volume of buying and selling, which in turn restores equilibrium prices. o By contrast, when arbitrage opportunities exist each investor wants to take as large a position as possible; hence it will not take many investors to bring about the price pressures necessary to restore equilibrium. Therefor, implications for prices derived from no-arbitrage arguments are stronger than implications derived from a risk-return dominance argument. Risk arbitrage – speculation on perceived mispriced securities, usually in connection with merger and acquisitions targets. If beta isn’t the only source of risk that is priced, then CAPM-alphas should persists. A stock might be “cheap” for a good reason. When you compare two assets, even if they aren’t identical, their prices should still be highly correlated; i.e. a stock that is highly correlated to small stocks should generate positive alpha. In other words, any stock that shares the same risks as small-caps should have similar pricing. The set of portfolios for which the nonsystematic variance approaches zero as n gets large consists of more portfolios that just the equally weighted portfolio. Well-diversified portfolio – a portfolio spread out over may securities in such a way that the weight in any one security is close to zero. Imagine a single factor market where the well-diversified portfolio, M, represents the market factor, F, the excess return on any security is given by: Ri=αi +β i R M +e i

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Investment Management Chapter VIII – Index Models and the Arbitrage Pricing Theory and that of a well-diversified (therefor zero residual) portfolio, P, is: E ( R P )=α P + β P E ( R M )  Although the APT is built on the foundation of well-diversified portfolios, we’ve seen that even large portfolios may have non-negligible residual risk.  Since arbitrage activity will quickly pin the risk premium of any zero-beta well-diversified portfolio to zero, then that for any well-diversified P, E ( R P )=β P E ( R M )

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In other words, the risk premium (expected excess return) on portfolio P is the product of its beta and the market-index risk premium. The previous equation this establish that SML of the CAPM applies to well-diversified portfolios simply by virtue of the “no-arbitrage” requirement of the APT. If residual risk is sufficiently high and the obstacles to complete diversification are considerably difficult, we cannot have full confidence in the APT and the arbitrage activities that rely on it. The previous equation might also be used to predict the risk premiums of portfolios with residual risk; the higher the residual risk is the less accurate is the approximation. The APT is more efficient than the CAPM in terms that it does not require that all investors must be mean-variance optimisers. It is sufficient that a small number of sophisticated arbitrageurs scour the market for arbitrage opportunities. This alone produces a mean return-beta (previous equation) that is a good and unbiased approximation for all assets but those with significant residual risk. The reason APT is not fully superior to the CAPM is that at the level of individual assets and high residual risk, pure arbitrage may be insufficient to enforce the expected risk premium equation. Therefore, we need to run to the CAPM as the theoretical construct behind equilibrium risk premiums. Comparing the APT arbitrage strategy to maximisation of the Sharpe ration in the context of an index model may well be the more useful framework for analysis. The APT is couched in a single-factor market, and applies with perfect accuracy to well-diversified portfolios. It shows arbitrageurs how to generate infinite profits if the risk premium of a well-diversified portfolio deviates from the risk-premium equation. The trades executed by these arbitrageurs are the enforcers of the accuracy of this equation. The APT ignores the fact that an increase in a position of a not fully risk-free (well-diversified) will increase the risk of the “arbitrage” position, potentially without bound. But if you limit the scale of the risky arbitrage, the composition of your overall risky portfolio then becomes relevant. There are multiple factors that may influence the expected returns on stock, such as the interest rate fluctuation, inflation rates and so on. Exposure to any of these factors will affect a stock’s risk and hence its expected return. We can derive a multifactor version of the APT to accommodate these multiple sources of risk, a two factor model would be: Ri=E ( Ri ) + β1 i F 1 + β 2i F 2+ei Each factor has zero expected value because each measures the surprise in the systematic variable rather than the level of the variable. Similarly, the 2

Investment Management Chapter VIII – Index Models and the Arbitrage Pricing Theory firm-specific component of unexpected return, e i , also has zero expected value.  Factor portfolio – a well-diversified portfolio constructed to have a beta of 1.0 on one factor and a beta of zero on any other factor.  The returns on the factor portfolio track the evolution of particular sources of macroeconomic risk, but are uncorrelated with other sources of risk. Factor portfolios will serve as the benchmark portfolios for a multifactor security market line. β P 1 and  The factor exposures of any portfolio, P, are given by its betas β P 2 . A competing portfolio, Q, can be formed by investing in factor portfolios with the following weights: β P 1 in the first factor portfolio, β P 2 in the second factor portfolio, and 1− β P 1 − β P 2 in T-bills. By construction, portfolio Q will have betas equal to those of portfolio P and expected return of: E ( r Q ) =r f +β P 1 [ E ( r 1) −r f ] +β P 2 [ E ( r 2) −r f ] 

When choosing factors we need to follow these two guidelines: o First, we want to limit ourselves to systematic factors with considerable ability to explain security returns. o Second, we wish to choose factors that seem likely to be important risk factors, that is, factors that concern investors sufficiently that they will demand meaningful risk premiums to bear exposure to those sources of risk.  One example of the multifactor approach is the work of Chen, Roll, and Ross, who chose the following set of factors on the basis of the ability of these factors to paint a broad picture of the macro economy: IP=% change ∈industrial production EI =% change∈expected inflation UI=% change ∈unanticipated inflation bonds −term government bonds CG=Excess return of long term corporate long bonds GB=Excess return of long term government Tbills This list gives rise to the following five-factor model of security returns during holding period t as a function of the change in the set of macroeconomic indicators: r it =α i + β iIP IP t + β iUI UI t + β iCG CGt + βiGB GBt +e it

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To estimate the betas of a given stock we would use a multiple regression of the returns of the stock in each period on the five macroeconomic factors. The residual variance of the regression estimates the firm-specific risk.  An alternative approach to specifying macroeconomic factors as candidates for relevant sources of systematic risk uses firm characteristics that seem on empirical ground to proxy for exposure to systematic risk. In other words, the factors are chosen as variables that on past evidence seem to predict high average returns and therefore may be capturing risk premiums. One example of this approach is the so-called Fama and French (FF) three-factor model, r it =α i + β ℑ R Mt + βiSMB SMB t + β iHML HML t + eit 3

Investment Management Chapter VIII – Index Models and the Arbitrage Pricing Theory Where SMB=Small minus big , that is , the return of a portfolio of small stocks ,∈excess of the return on a portfolio o HML=High minus low , that is , the returnof a portfolio of stocks with a highbook ¿−market ratio∈excess of   

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In this model, the market index does play a role and is expected to capture systematic risk originating from macroeconomic factors. These two firm-characteristic variables are chosen because of long-standing observations that corporate capitalisation (firm size) and book-to-market ratio seem to be predictive of average stock returns. It is important to distinguish the multifactor APT from the multi-index CAPM. In the latter, the factors are derived from a multiperiod consideration of stream of consumption as well as randomly evolving investment opportunities pertaining the distributions of rates of return. Hence, the hedge index portfolios must be derived from considerations of the utility of consumption, non-traded assets and changes in investment opportunities. A mult-index CAPM will therefore inherit risk factors from sources that a broad group of investors deem important enough to hedge. In contrast, the APT is largely silent on where to look for priced sources of risk. This lack of guidance is problematic, buy by the same token, it accommodates a less structured search of relevant risk factors. APT just prices assets relative to other assets. APT can be useful in better spotting a misprice and/or identify the fundamental risk factors. If assets tend to be well priced with respect to the APT-factors, we can sumply focus on understanding where the RP of the factors comes from. If part of a factor’s RP is unexplained by any model, we can still use it as a proxy for the RP to the mysterious source of risk.

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