Asset Pricing Zheng Zhenlong Chapter 9: Factor pricing models

Asset Pricing Zheng Zhenlong Contents Introduction CAPM ICAPM Comments on the CAPM and ICAPM APT APT vs. ICAPM

Asset Pricing Zheng Zhenlong Brief introduction

Asset Pricing Zheng Zhenlong Brief introduction More directly, the essence of asset pricing is that there are special states of the world in which investors are especially concerned that their portfolios not do badly. The factors are variables that indicate that these “bad states” have occurred. Any variable that forecasts asset returns (“changes in the investment opportunity set”) or macroeconomic variables is a candidate factor. Such as :term premium, dividend/price ratio, stock returns

Asset Pricing Zheng Zhenlong Should factors be unpredictable over time? Factors that proxy for marginal utility growth, though they don’t have to be totally unpredictable, should not be highly predictable. If one chooses highly predictable factors, the model will counterfactually predict large interest rate variation. In practice, this consideration means that one should choose the right units: Use GNP growth rather than level, portfolio returns rather than prices or price/dividend ratios, etc.

Asset Pricing Zheng Zhenlong The derivations of factor pricing model Determine one particular list of factors that can proxy for marginal utility growth Prove that the relation should be linear. Remark: all factor models are derived as specializations of the consumption-based model.

Asset Pricing Zheng Zhenlong guard against fishing One should call for better theories or derivations, more carefully aimed at limiting the list of potential factors and describing the fundamental macroeconomic sources of risk, and thus providing more discipline for empirical work.

Asset Pricing Zheng Zhenlong Capital Asset Pricing Model (CAPM) wealth portfolio return. In expected return / beta language, CAPM can be derived from consumption-based model by different assumption.

Asset Pricing Zheng Zhenlong Different assumption 1) two-period quadratic utility 2) exponential utility and normal returns, 3) Infinite horizon, quadratic utility and i.i.d. returns 4) Log utility. Same assumption: no labor income

Asset Pricing Zheng Zhenlong Two-period quadratic utility ， no labor income Investors have quadratic preferences and only live two periods, marginal rate of substitution is thus

Asset Pricing Zheng Zhenlong the budget constraint is

Asset Pricing Zheng Zhenlong Just as

Asset Pricing Zheng Zhenlong Exponential utility, normal distributions, no labor income If consumption only in the last period and is normally distributed, we have a is the coefficient of absolute risk aversion.

Asset Pricing Zheng Zhenlong the budget constraint is

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Asset Pricing Zheng Zhenlong Quadratic value function, dynamic programming first order condition So,

Asset Pricing Zheng Zhenlong suppose the value function were quadratic, Then, Some addition assumptions: –The value function only depends on wealth. –The value function is quadratic. It needs the following assumptions: the interest rate is constant, returns are iid, no labor income.

Asset Pricing Zheng Zhenlong the existence of value function (Proof ) Suppose investors last forever, and have the standard sort of utility function Define the value function as the maximized value of the utility function in this environment.

Asset Pricing Zheng Zhenlong Value functions allow you to express an infinite period problem as a two period problem

Asset Pricing Zheng Zhenlong Why is the value function quadratic? Remark: quadratic utility function leads to a quadratic value function in this environment Specify: Guess: Thus,

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Asset Pricing Zheng Zhenlong Log utility, no labor income

Asset Pricing Zheng Zhenlong Log utility has a special property that “income effects offset substitution effects,” or in an asset pricing context that “discount rate effects offset cash flow effects.”

Asset Pricing Zheng Zhenlong How to linearize the model? The twin goals of a linear factor model derivation are to derive what variables derive the discount factor, and to derive a linear relation between the discount factor and these variables. This section covers three tricks that are used to obtain a linear functional form. Taylor approximation the continuous time limit normal distribution

Asset Pricing Zheng Zhenlong Taylor approximation The most obvious way to linearize the model is by a Taylor approximation

Asset Pricing Zheng Zhenlong Continuous time limit If the discrete time is short enough, we can apply the continuous time result as an approximation For a short discrete time interval,

Asset Pricing Zheng Zhenlong Normal distribution in discrete time Stein’s lemma : If f and R are bivariate normal, g(f) is differentiable and,then

Asset Pricing Zheng Zhenlong Remark: If m=g(f), if f and a set of the payoffs priced by m are normally distributed returns, and if, then there is a linear model m=a+bf that prices the normally distributed returns.

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Asset Pricing Zheng Zhenlong Similar,it allows us to derive an expected return-beta model using the factors

Asset Pricing Zheng Zhenlong Two period CAPM Stein’s lemma allows us to substitute a normal distribution assumption for the quadratic assumption in the two period CAPM. Assuming R W and R i are normally distributed, we have:

Asset Pricing Zheng Zhenlong Log utility CAPM Stein’s lemma cannot be applied to the log utility CAPM because the market return cannot be normally distributed. For log utility CAPM, g(f)=1/R W, so If R W is normally distributed, E(1/R W2 ) does not exist. The Stein’s lemma condition is violated.

Asset Pricing Zheng Zhenlong Intertemporal Capital Asset Pricing Model (ICAPM) The ICAPM generates linear discount factor models in which the factors are “state variables” for the investor’s consumption-portfolio decision.

Asset Pricing Zheng Zhenlong the value function depends on the state variables so we can write

Asset Pricing Zheng Zhenlong Start from We have

Asset Pricing Zheng Zhenlong Define the coefficient of relative risk aversion, Then we obtain the ICAPM,

Asset Pricing Zheng Zhenlong Thus, in discrete time

Asset Pricing Zheng Zhenlong Is the CAPM conditional or unconditional?

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Asset Pricing Zheng Zhenlong The log utility CAPM expressed with the inverse market return is a beautiful model, since it holds both conditionally and unconditionally. There are no free parameters that can change with conditioning information. Finally it requires no specification of the investment opportunity set, or no specification of technology. However, the expectations in the linearized log utility CAPM are conditional.

Asset Pricing Zheng Zhenlong Should the CAPM price options? the quadratic utility CAPM and the nonlinear log utility CAPM should apply to all payoffs: stocks, bonds, options, contingent claims, etc. However, if we assume normal return distributions to obtain a linear CAPM, we can no longer hope to price options, since option returns are non-normally distributed

Asset Pricing Zheng Zhenlong Why bother linearizing a model?

Asset Pricing Zheng Zhenlong What about the wealth portfolio? To own a (share of) the consumption stream, you have to own not only all stocks,but all bonds, real estate, privately held capital, publicly held capital (roads, parks, etc.), and human capital. Clearly, the CAPM is a poor defense of common proxies such as the value-weighted NYSE portfolio.

Asset Pricing Zheng Zhenlong Implicit consumption-based models

Asset Pricing Zheng Zhenlong Ex-post returns The log utility model also allows us for the first time to look at what moves returns ex-post as well as ex-ante. Aggregate consumption and asset returns are likely to be de-linked at high frequencies, but how high (quarterly?) and by what mechanism are important questions to be answered.

Asset Pricing Zheng Zhenlong Identity of state variables

Asset Pricing Zheng Zhenlong Arbitrage Pricing Theory (APT) The intuition behind the APT is that the completely idiosyncratic movements in asset returns should not carry any risk prices, since investors can diversify them away by holding portfolios. Therefore, risk prices or expected returns on a security should be related to the security’s covariance with the common components or “factors” only.

Asset Pricing Zheng Zhenlong The APT models the tendency of asset payoffs (returns) to move together via a statistical factor decomposition Define So,

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Asset Pricing Zheng Zhenlong Thus, with N= number of securities, the N(N-1)/2 elements of a variance-covariance matrix are described by N betas, and N+1 variances. With multiple (orthogonalized) factors, we obtain

Asset Pricing Zheng Zhenlong If we know the factors we want to use ahead of time, we can estimate a factor structure by running regressions. If we don’t, we use factor analysis to estimate the factor model.

Asset Pricing Zheng Zhenlong Exact factor pricing

Asset Pricing Zheng Zhenlong Approximate APT using the law of one price There is some idiosyncratic or residual risk; we cannot exactly replicate the return of a given stock with a portfolio of a few large factor portfolios. However, the idiosyncratic risks are often small. There is reason to hope that the APT holds approximately, especially for reasonably large portfolios.

Asset Pricing Zheng Zhenlong Suppose Again take prices of both sides,

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Asset Pricing Zheng Zhenlong Limiting arguments

Asset Pricing Zheng Zhenlong These two theorems can be interpreted to say that the APT holds approximately (in the usual limiting sense) for either portfolios that naturally have high R 2, or well-diversified portfolios in large enough markets.

Asset Pricing Zheng Zhenlong Law of one price arguments fail

Asset Pricing Zheng Zhenlong Remark: the effort to extend prices from an original set of securities (f in this case) to new payoffs that are not exactly spanned by the original set of securities, using only the law of one price, is fundamentally doomed. To extend a pricing function, you need to add some restrictions beyond the law of one price.

Asset Pricing Zheng Zhenlong the law of one price: arbitrage and Sharpe ratios The approximate APT based on the law of one price fell apart because we could always choose a discount factor sufficiently “far out” to generate an arbitrarily large price for an arbitrarily small residual. But those discount factors are surely “unreasonable.” Surely, we can rule them out.

Asset Pricing Zheng Zhenlong

Asset Pricing Zheng Zhenlong Theorem

Asset Pricing Zheng Zhenlong APT vs. ICAPM Factor structure can imply factor pricing (APT), but factor pricing does not require a factor structure. High R 2 in time-series regressions of the returns on the factors may imply factor pricing (APT), but again are not necessary (ICAPM).

Asset Pricing Zheng Zhenlong The biggest difference between APT and ICAPM for empirical work is in the inspiration for factors. The APT suggests that one start with a statistical analysis of the covariance matrix of returns and find portfolios that characterize common movement. The ICAPM suggests that one start by thinking about state variables that describe the conditional distribution of future asset returns and non-asset income.

Asset Pricing Zheng Zhenlong

Asset Pricing Zheng Zhenlong