Once the environment has been carefully and entirely characterized, the next step
in the model-building process is the identification and selection of a set of relevant
risk factors. Two distinct lines of inquiry can be followed.
The factors can be specified exogenously, as a result of experience, intuition, or
arbitrary decision. Such risk factors are typically fundamental or macro-economic
factors. For instance, when one attempts to build a model to explain stock returns,
factors such as the return on the stock market index, the market capitalization, the
dividend yield, the growth rate in industrial production, the return on default-free
securities, the oil price variations or the expected change in inflation should a priori
be considered good candidates.
The factors can also be determined endogenously by the use of statistical techniques
such as principal components and/or factor analysis. Typically, a succession
of models with an increasing number of factors is fitted to the data. Each factor is
not constructed according to any economic meaning, but statistically, according to
its incremental explanatory power with respect to the previous model. The goodness
of fit is monitored, with the idea that a strong increase in its value implies that the
last factor added is significant. The process stops when additional factors do not
measurably increase the goodness of fit of the model.
Each of the two methods has its own advantages and shortcomings. In the
exogenous specification, the resulting model is economically very tractable and is
expressed easily in a language with which most senior managers, brokers and
financial analysts are familiar. But omitting factors is frequent, as it is usually easier
to catalogue which risk factors are present rather than which are missing. With the
endogenous specification, the model includes the factors with the highest explanatory
power, but the next difficulty is to identify these statistical factors as relevant
economic variables. Of course, both techniques can be used simultaneously to
complement and validate their results.
As an example, on the equity side, the capital asset pricing model of Sharpe (1963)
has remained for a long time the unique model to explain expected returns. It is
nothing more than a single-factor model, with the market return taken as the only
explanatory variable. Although empirical evidence suggested the multidimensionality
of risk, it is only with the arbitrage pricing theory of Ross (1976) that multi-factor
models started to gain popularity. A significant enhancement was the three-factor
pricing model of Fama and French (1992), showing that adding size and book to
market value of equity as explanatory variables significantly improved the results of
the CAPM.
Another interesting example is the field of interest rate contingent claims. In this
case, the underlying variable is the entire interest rate term structure, that is, the
set of all yields for all maturities. How many factors should we consider to price
interest rate contingent claims, or equivalently, to explain yield curve variations?
The answers generally differ across models. Most empirical studies using principal
component analysis have decomposed the motion of the interest rate term structure
into three independent and non-correlated factors: the first is a shift of the term
structure, i.e. a parallel movement of all the rates. It usually accounts for up to
80–90% of the total variance (the exact number depends on the market and on the
period of observation); the second one is a twist, i.e. a situation in which long rates
and short-term rates move in opposite directions. It usually accounts for an additional
5–10% of the total variance. The third factor is called a butterfly (the intermediate
rate moves in the opposite direction of the short- and long-term rate). Its influence
is generally small (between 1% and 2% of the total variance).
Since the first statistical component has a high explanatory power, it may be
tempting to model the yield curve variations by a one-factor model. It must be
stressed at this point that this does not necessarily imply that the whole term
structure is forced to move in parallel, as this was the case in all duration-based
models commonly employed since the 1970s and still used by many asset and
liability managers. It simply implies that one single source of uncertainty is sufficient
to explain the movements of the term structure or the price of a particular interest
rate contingent claim. Most early interest rate models – such as Merton (1973),
Vasicek (1977), or Cox, Ingersoll and Ross (1985) – are single-factor models, and
they assume the instantaneous spot rate to be the single state variable.
One the one hand, single-factor models are generally adequate for securities which
are sensible to the level of the term structure of interest rates, such as short-term
bonds and their contingent claims. On the other, some securities such as swaptions
are sensible to the shape of the term structure (or to other aspects such as the
volatility term structure deformations) and not only to its level. They will require at
least a two-factor model. This explains why a second state variable, such as the longterm
interest rate, a spread or the rate of inflation was added in subsequent models,
even if it is cumbersome to handle.
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