ROBUST PARAMETER ESTIMATION
The most commonly used approach for estimating security expected returns, covariances, and other parameters that are inputs to portfolio optimization models is to calculate the sample analogues from historical data. These are sample estimates for the parameters we need. It is important to remember that when we rely on historical data for estimation purposes, we in fact assume that the past provides a good representation of the future.
It is well-known, however, that expected returns exhibit significant time variation (referred to as nonstationarity). They are impacted by changes in markets and economic conditions, such as interest rates the political environment, consumer confidence, and the business cycles of different industry sectors and geographical regions. Consequently, extrapolated historical returns are often poor forecasts of future returns.
Similarly, the covariance matrix is unstable over time. Moreover, sample estimates of covariances for portfolios with thousands stocks are notoriously unreliable, because we need large data sets to estimate them, and such large data sets of relevant data are difficult to procure. Estimates of the covariance matrix based on factor models are often used to reduce the number of statistical estimates needed from a limited set of data.
In practice, portfolio managers often alter historical estimates of different parameters subjectively or objectively, based on their expectations and forecasting models for future trends. They also use statistical methods for finding estimators that are less sensitive to outliers and other sampling errors, such as Bayesian and shrinkage estimators. A complete review of advanced statistical estimation topics is beyond the scope of this chapter. We provide a brief overview of the most widely used concepts.34
Shrinkage is a form of averaging different estimators. The shrinkage estimator typically consists of three components: (1) an estimator with little or no structure (like the sample mean); (2) an estimator with a lot of structure (the shrinkage target); and (3) a coefficient that reflects the shrinkage intensity. Probably the most well-known estimator for expected returns in the financial literature was proposed by Jorion.35 The shrinkage target in Jorion's model is a vector array with the return on the minimum variance portfolio, and the shrinkage intensity is determined from a specific formula.36 Shrinkage estimators are used for estimates of the covariance matrix of returns as well37), although equally weighted portfolios of covariance matrix estimators have been shown to be equally effective as shrinkage estimators as well.38
Bayesian estimation approaches, named after the English mathematician Thomas Bayes, are based on subjective interpretations of the probability that a particular event will occur. A probability distribution, called the prior distribution, is used to represent the investor's knowledge about the probability before any data are observed. After more information is gathered (e.g., data are observed), a formula (known as Bayes' rule) is used to compute the new probability distribution, called the posterior distribution.
In the portfolio parameter estimation context, a posterior distribution of expected returns is derived by combining the forecast from the empirical data with a prior distribution. One of the most well-known examples of the application of the Bayesian framework in this context is the Black-Litterman model,39 which produces an estimate of future expected returns by combining the market equilibrium returns (i.e., returns that are derived from pricing models and observable data) with the investor's subjective views. The investor's views are expressed as absolute or relative deviations from the equilibrium together with confidence levels of the views (as measured by the standard deviation of the views).
The ability to incorporate exogenous insight, such as a portfolio manager's opinion, into quantitative forecasting models is important; this insight may be the most valuable input to the model. The Bayesian framework provides a mechanism for forecasting systems to use both important traditional information sources such as proprietary market data, and subjective external information sources such as analyst's forecasts.
It is important to realize that regardless of how sophisticated the estimation and forecasting methods are, they are always subject to estimation error. What makes matters worse, however, is that different estimation errors can accumulate over the different activities of the portfolio management process, resulting in large aggregate errors at the final stage. It is therefore critical that the inputs evaluated at each stage are reliable and robust, so that the aggregate impact of estimation errors is minimized.