On the Validity of Risk Measures over Time: Value-at-Risk, Conditional Tail Expectations and the Bodie-Merton-Perold Put
Over the past decade, risk measurement has received a much needed amount of attention from the .nancial community. Risk measures based on .xed quantiles un- der the actual probability distribution, especially Value-at-Risk and its re.nement the Conditional Tail Expectation, were instrumental in capturing the attention of .nancial decision-makers. However, these were developed in a way that is inconsistent with eco- nomic theory. Consequently, these instruments possess characteristics that make them invalid risk measures for the purposes they intend to serve, be it informing life-cycle investors or guaranteeing the .rm.s capital adequacy through regulation. In particular, in addition to failing to guarantee the intregity of .nancial .rms when used for capital adequacy, these measures can eventually decrease with the investment horizon. Risk-neutral .xed-quantile measures are valid for framing life-cycle decisions because of their economic content. When endowed with a dynamic replication technology, Q- measure .xed-quantile risk measures become least-cost insurance contracts that may be used for capital adequacy considerations. However, no single quantile of the risk-neutral distribution can be used for the procurement of risk capital at all horizons. A risk-neutral varying-quantile instrument is needed. This unique instrument is a put option proposed by Merton-Perold (1993) and Bodie (1995). The Bodie-Merton-Perold Put is universally valid for both risk disclosure to investors and for the regulatory provision of risk capital at all horizons. It is a natural candidate for an industry standard in risk measurement.
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