Final_1.pdf

(3.19)

The formula to evaluate the variance may be easily adapted to evaluate the
covariance between two portfolios. We will leave it to the reader to reason
out that the covariance between two portfolios YandZis given by

(3.20)

In any case, the formula for variance in Equation 3.20 can be used to calcu-
late the risk in a portfolio.
Assuming normality, a two-standard-deviation movement on either side
of the daily mark to market should be able to catch the price movement for
the next day, 95 percent of the time. Yet in practice that may not turn out
to be the case. Let us therefore examine the points of failure of the model.
We start by listing the inputs to the risk calculation and then examine the
different scenarios. The key inputs are the factor exposures and the covari-
ance matrix.
When a big news event occurs with respect to a particular stock, the fac-
tor exposures that we use in the model for that stock are no longer valid. The
market now trades on the expectation that the factor exposures after the news
event are likely to be a lot different. The covariance structure between the fac-
tors is still intact. In any case, since the factor exposure input to the risk model
is no longer valid, the calculation breaks down for that particular stock.
The second scenario is the occurrence of a scenario-altering, huge macro-
economic event, for example, relating to interest rates. The big event typi-
cally manifests itself in the form of a liquidity crisis. In these situations, the
covariance structure breaks down, leading to the breakdown of the model.
Another explanation for observing price moves of more than two stan-
dard deviations of that expected by the Gaussian assumption is attributed to
the nonnormal fat-tailed distribution of asset returns observed in practice.
This can be somewhat addressed by calibrating the number of standard de-
viations to use in our assessment of the range of price movement. It is there-
fore important that users of the multifactor technology also be aware of the
potential points of failure in the model.

Application: Calculation of Portfolio Beta

The topic of this section is more of a misnomer, and it likely to strike the
reader as an anomaly. We had earlier stated that the APT is a more

cov()r rYZ, =+∆h XVX hY T ZT h hYZT

σσσportfolio^222 =+specific

=+

cf

hXVX hTT h h∆ T

Factor Models 47