Coherent multiperiod risk measurement

Abstract. We explain why and how to deal with the definition, acceptability, computation and management of risk in a genuinely multitemporal way. Coherence axioms provide a representation of a risk-adjusted valuation. Some special cases of practical interest allowing for easy recursive computations are presented. The multiperiod extension of Tail VaR is discussed. 1. NEW QUESTIONS WITH MULTIPERIOD RISK RISK EVOLVING OVER SEVERAL PERIODS of uncertainty is different from one-period risk in many ways. An analysis of multiperiod risk requires consideration of new issues, since:-availability of information may require taking into account intermediate monitoring by supervisors or shareholders of a locked-in position,- the possibility of intermediate actions,availability of extraneous cash flows, of possible capital in- or outflows require handling sequences of unknown future “values”. A PORTFOLIO OR A STRATEGY built over several periods should be analyzed with respect to these issues. We attempt to:- distinguish models of future worth at the end of a holding period from models in which successive values or cash flows are examined, and are subject to some investment/financing strategy.- give some information about the necessity and/or availability of remedial funding at some intermediate date either in the case of sudden loss or in the case of insolvency of the firm, as urged for example in [Be] (notice that one-period models considered neither the source of (extra-) capital at the beginning of the holding period nor the actual consequences of a “bad event” at the end of the same period).- take into account the actual time evolution of risk and of available capital. Study whether a relevant risk-adjusted measurement should consider more than the distribution of final net worth of a strategy, to decide upon its acceptability at the initial date.- distinguish between the opinion of a risk manager on some strategy, and the attitude of a supervisor/regulator who, at any date, considers only the This research has benefited from support by PriceWaterhouseCoopers and by RiskLab, ET