General definition

A risk measure $\rho$ is usually defined as a mapping from a set of random variables (i.e. payoffs) $\mathcal{X}$ to the real numbers, that means

$$\rho: \mathcal{X} \longrightarrow \mathbb{R}$$ (7)

$$X \longmapsto \rho(X) .$$

The amount $\rho(X)$ is commonly interpreted as the minimum cash such that the ``risk'' of $X$ is ``acceptable'' to the holder of the payoff or portfolio whenever he/she has the additional amount $\rho(X)$ stored as risk capital (cf. Artzner et al., 1999).

Working with a portfolio base $B = (X_1,\ldots, X_n)$, a risk measure $\rho$ on the payoffs $\mathcal{X}$ implies a risk measure $\rho_B$ on the portfolios $u\in\mathbb{R}^n$ for which we have $X(u) \in \mathcal{X}$. In particular, if $X(\mathbb{R}^n) = \mathcal{X}$ we can define

$$\rho_B: \mathbb{R}^n \longrightarrow \mathbb{R}$$ (8)

$$u \longmapsto \rho(X(u)) .$$

We also write $\rho(u)$ for $\rho_B(u)$. Based on the context, no confusion can arise.

A performance measure can also be represented by functions as considered in (7) and (8). In contrast to risk measures, performance measures are ususally intended to describe ratios like the relation of the expected return to the risk capital or invested risk-adjusted capital. However, the concrete interpretation of such measures is postponed until we look at conrete examples, namely RORC and RORAC (cf. Section 2.3).