18-MultiFactorModels Page 539 Wednesday, February 4, 2004 1:10 PM
Multifactor Models and Common Trends for Common Stocks 539exponentials. If returns are jointly normally distributed, then log prices
are jointly normally distributed while the real prices are lognormally
distributed. Suppose that a variable X is normally distributed with
expected value and variance μ, σ^2. The variable ex is lognornomally dis-
tributed with the following expected value and variance:μ +^1 ---σ^2
2
e , σ
2
(e
σ^2- 1 )
If returns are Independent and Identical Normal (IIN) variables, lin-
ear factor models imply that prices with different factor sensitivities will
have different average returns, an expression of the risk-return trade-off
required by investors. In addition, the variance of log prices will grow
linearly with time at different rates for each process. The means that dif-
ferent price processes will therefore evolve exponentially at different
rates and diverge exponentially. Under the assumption that factors
behave as a stationary Vector Auto Regressive (VAR) Model as in the
state space-models, the dependence is more complex but there is still an
exponential divergence of prices.
An exponential divergence of prices is not sustainable in the long
run. Clearly corrective phenomena are at work in financial markets,
though exactly how corrections are made is the subject of different
hypotheses. It has been hypothesized that stock price processes are sub-
ject to discrete regime-changes; this assumption, widely studied in the
literature, leads to nonlinear models. It has also been hypothesized that
disruptive phenomena are at work, so that the price of a firm’s stock
might grow rapidly but then the firm is subject to phenomena such as
bankruptcy, merger, acquisition or corporate restructuring; this links
financial theory to macroeconomics and is beyond the scope of this
book. A third hypothesis is that correction phenomena—and perhaps
discrete changes—are always at work in markets; these phenomena can
be modeled within the domain of linear models with the techniques of
cointegration. The fact that portfolio separation in a fixed and closed
economy implies collinearity lends additional theoretical support to the
cointegration of asset prices. Bossaert showed how cointegration natu-
rally arises if one slightly relaxes the assumption of separation.^9
Cointegration (see Chapter 12 can be modeled in two different but
equivalent ways, using either state-space models or Error Correction
Models (ECMs). ECMs are VAR models with restrictions. Consider that
it is always possible to write a VAR model in ECM form:(^9) Peter Bossaerts, “Common Nonstationary Components of Asset Prices,” Journal of
Economic Dynamics and Control 12 (1988), pp. 347–364.