Attributes and their Characteristic Portfolios

Abstract

The financial instruments, or in general the assets, of a portfolio have many characteristics or attributes such as expected return, market capitalization, beta with respect to an index, membership in a certain economic sector, etc. If we denote a certain attribute by a _i for asset i (having value V _i ) then the exposure of a portfolio V (with weights w _k for k = 1,..., M) to this particular attribute is defined as

$$\matrix{ {{a_V} \equiv \sum\limits_{k\, = \,1}^M {{w_k}{a_k}} \, = \,{{\bf{w}}^T}{\bf{a}}} \cr}$$

where we have used the obvious notation a^T =(a₁,…,a _M )- If, for instance the characteristics a _i are measures of how strongly the assets i = 1,…, M belong to the automotive industry then a _V is the exposure of the portfolio to the automotive industry. Another example: If the characteristics a _i are the asset returns R _i then the exposure a _V is simply the portfolio return R _V . From these examples one can already guess that characteristics (or attributes) are a very general and rather abstract concept with broad applications.¹