RORC and RORAC

We define a performance measure

$$\varphi(X) := \frac{\mathbf{E}[X]}{\rho(X)} .$$ (11)

$\varphi$ is called the RORC, i.e. the Return On Risk Capital. In contrast to a risk measure, this performance measure does not care about the absolute value of the risk capital, but of its proportion to the mean return which is gained on it. For $\rho =$   ES$_{\alpha}$, i.e.

$$\varphi_{\alpha}(X) := \frac{\mathbf{E}[X]}{\text{ES}_{\alpha}(X)} ,$$ (12)

we talk of the ES-RORC. Some authors (cf. Tasche, 2000) call (11) the RORAC. We think that the ``Return On Risk-Adjusted Capital'' should be defined as in (14).

Working on RORC optimization, one might face the problem that the optimal portfolio (although the portfolio value is constant) implies a huge amount of risk capital together with a huge expected return. However, practical reasons might imply an upper bound for the risk capital. So we need a constraint $\rho(u) \leq \rho_{\text{max}}$, i.e. in case of the ES-RORC

$$_{\alpha}(u) \leq _{\text{max}} ,$$ (13)

might be imposed.

The performance measure

$$\psi(X) := \frac{\mathbf{E}[X]}{V+\rho(X)}$$ (14)

is called the RORAC, i.e. the Return On Risk-Adjusted Capital. Indeed, $\psi$ measures the mean (or expected) return per unit engaged capital, since $V+\rho$ is the value of the invested capital plus the costs of risk (cf. (5)). Hence, in contrast to RORC (11), RORAC considers not only the risk capital but the risk-adjusted investment capital and therefore seems to be a more sophisticated performance measure. For $\rho =$   ES$_{\alpha}$, i.e.

$$\psi_{\alpha}(X) := \frac{\mathbf{E}[X]}{V+\text{ES}_{\alpha}(X)} ,$$ (15)

we talk of the ES-RORAC.

As in the case of ES-RORC, the additional constraint (13) might be imposed in the case of ES-RORAC optimization.