Factor Models Flashcards
Priced Factors
Non-zero expected excess return–lambda_a,k not beta_a,k
Arbitrage portfolio (2 properties)
A set of weights such that
- price weakly less than zero, probability of non-zero payoff is positive.
- Price is strictly less than zero and payoff is non-negative (but can be zero) with probability one.
Arbitrage portfolio in factor model must have
beta_pk=0 for all k and zero initial cost.
APT portfolio weights, what is alpha
w_p,N = \frac{\alpha_N}{||\alpha_N||\sqrt N}
alpha is residual of projection of excess returns into the betas.
What is lambda_k,N in APT?
It is the set of factor prices given loadings, recovered from projecting excess returns into betas.
To avoid arbitrage, equals the expected return of the tradable factor. If non-zero then it is “priced”. Note that beta_a,k can be non-zero while the factor has 0 expected return!
What does PCA recover?
alpha_i, \beta_a,k, \epsilon_a,t–lambda_a,t comes from an excess return regression, using cross-sectional stock data.
What does APT imply about CAPM?
1) full frontier constructed from K factor portfolios;
2) Some portfolio is always ex post mean-variance efficient, can always get 1 factor model to fit the data.
Write down conditional CAPM (unconditional ave)
RP = E[beta_amt][R_m,t+1-R_f,t]+Cov(\beta_amt,E_t[R_m,t+1-R_f,t])
What does conditional CAPM tell us?
An asset can have a higher unconditional average return if its beta moves with the market risk premium.
Factors can mean one of two things:
1) Common factors in ex-post returns across assets
2) Factors for which exposure determines expected returns.
APT links two what ideas:
1) If a factor is priced but idiosyncratic, you can get high SR by diversifying (Gramm Schmidt procedure);
2) Not every common factor in returns is priced (if lambda_pk=0)
