The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 437 Wednesday, February 4, 2004 1:08 PM

Arbitrage Pricing: Finite-State Models 437

f = ( f 1 , ,...fk) , K << N

fi = α

si

∑ sris

s = 1

where ris = zis– aisand αs are the weights of those portfolios that iden-

tify factors.

APT requires that the constants a, when the model is formulated in

excess returns, are zero. To test APT the model parameters have first to

be estimated. Suppose that returns are normal IID variables and that the

multifactor model is unconstrained. Model estimation can be done by

Maximum Likelihood methods which are, in this case, identical to Ordi-

nary Least Square (OLS) estimates. The model parameters are then

obtained as the empirical moments, as follows:

aˆ = μˆ– Bˆμˆ

K

T

∑(zt–^ μˆ^ )(zKt –^ μˆK^ )

Bˆ = -------------------------------------------------------------t =^1 -

T

∑(zKt – μˆK^ )(zKt–^ μˆ^ K)

t = 1

T

∑

ˆ^1

μ = ---- zt

Tt = 1

T

∑

ˆ^1

μ (^) K = ---- zKt
Tt = 1
Now suppose that there is a risk-free asset and that the model is
constrained by the APT constraints. In this case, we can still use MLE
estimation which yields a zero intercept and the following sensitivities: