Grading Investment Diversification Options in Presence of Non-Historical Financial Information

Clara Calvo,Vicente Liern,Carlos Ivorra,Blanca Pérez-Gladish

doi:10.3390/math9060692

Abstract

Modern portfolio theory deals with the problem of selecting a portfolio of financial assets such that the expected return is maximized for a given level of risk. The forecast of the expected individual assets’ returns and risk is usually based on their historical returns. In this work, we consider a situation in which the investor has non-historical additional information that is used for the forecast of the expected returns. This implies that there is no obvious statistical risk measure any more, and it poses the problem of selecting an adequate set of diversification constraints to mitigate the risk of the selected portfolio without losing the value of the non-statistical information owned by the investor. To address this problem, we introduce an indicator, the historical reduction index, measuring the expected reduction of the expected return due to a given set of diversification constraints. We show that it can be used to grade the impact of each possible set of diversification constraints. Hence, the investor can choose from this gradation, the set better fitting his subjective risk-aversion level.

Highlights

In modern portfolio theory (MPT), known as mean-variance theory, a portfolio of assets is selected such that the expected return is maximized for a given level of risk
In [22], we use the value of information to define an index measuring the financial impact of each possible set of diversification constraints when it is incorporated into a portfolio selection model, but, to this purpose, the above defined value of information has an obvious drawback, namely, that it takes as ideal reference an exact forecasting of the return of each asset, and the index defined in [22] does not take into account the possible effects that a good—but not exact—forecast would produce
In this paper we have considered the problem derived from the inclusion of additional non-historical information in the forecasting of the expected returns of financial assets, in portfolio selection problems

Summary

Value of Information

In this work emphasis is placed on the expected returns of the individual assets. Several authors as Chopra and Ziemba [16], Best and Grauer [17], DeMiguel and Nogales [18], Kan and Smith [19], or Siegal and Woodgate [20], among others, pointed out how errors in the variances and covariances tend to be smaller than the errors in expected returns. In [22], we use the value of information to define an index measuring the financial impact of each possible set of diversification constraints when it is incorporated into a portfolio selection model, but, to this purpose, the above defined value of information has an obvious drawback, namely, that it takes as ideal reference an exact forecasting of the return of each asset, and the index defined in [22] does not take into account the possible effects that a good—but not exact—forecast would produce This is not important in order to establish a relative ranking of several alternative sets of diversification constraints, as we did in [22], but in order to select the most adequate one in a particular context, it is desirable to somehow include into our analysis the reliability of the forecasted returns the investor is considering. We revisit the reduction index introduced in [22]

The Historical Reduction Index of a Set of Diversification Constraints

Empirical Analysis of the Behavior of the Index

Conclusions