Factor pricing models.pptVIP

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Define So, We can construct the factor decomposition as a regression equation. Define the as regression coefficients, and then the are uncorrelated with the factors The factor structure is thus a restriction on the covariance matrix of payoffs. For example,if there is only one factor, then 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 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. Exact factor pricing Suppose that there are no idiosyncratic terms. This is called an exact factor model. payoff xi can be synthesized as a portfolio of the factors and a constant (risk-free payoff). Thus, the price of xi can only depend on the prices of the factors f specialized to returns xi = Ri and of course p(Ri) = 1. Use p(1) = 1/Rf and we can solve for expected return as 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. Suppose Again take prices of both sides, Limiting arguments Theorem: Fix a discount factor m that prices the factors. Then, as The factor model is defined as a regression, so The theorem says that as , the price of the residual goes to zero. Theorem: As the number of primitive assets increases, the R2 of well-diversified portfolios increases to 1. These two theorems can be interpreted to say th