On the economic risk capital of portfolio insurance

A formula for the conditional value-at-risk of classical portfolio insurance is derived and shown to be constant for sufficiently small loss probabilities. As illustrations, we discuss portfolio insurance for an equity market index using empirical data, and analyze the more general multivariate situation of a portfolio of risky assets. 1. Introduction. Portfolio insurance, introduced by Leland in the night of September 11, 1976, is a simple financial instrument used to protect capital against future adverse falls. A collection of seminal papers, which study some of its main properties, is Luskin [12]. Combined with classical actuarial contingencies like mortality risk, portfolio insurance leads to unit-linked insurance contracts, which under the economic risk capital viewpoint have been analyzed in Hürlimann [6]. In the present note, the focus on classical portfolio insurance reveals a new remarkable feature. It turns out that the economic risk capital as measured by value-at-risk or conditional value-at-risk (CVaR) remains constant, provided the loss probability is sufficiently small. In practice, confidence levels α around 70%–90% often suffice to guarantee this stability property. In Section 2, we derive a formula for the CVaR of portfolio insurance, and show that it is constant for small loss probabilities. Two specific examples illustrate our results. In Section 3, we discuss portfolio insurance for an equity market index on the basis of empirical data material. The more general multivariate situation of a portfolio of risky assets is exemplified in Section 4 in the bivariate case.