Coherent Multiperiod Risk Measurement

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 inor 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, ETH Zürich. 1 current portfolio, refusing to take into account future possible changes in the composition of the portfolio (see the example in Section 8). Remark. With one period of uncertainty, capital appeared both as a buffer at the initial date and as wealth at the final date. Intermediate dates raise the question of the nature of capital (valued in a market or accounting way) at such dates. 2. REVIEW OF ONE PERIOD COHERENT ACCEPTABILITY COHERENT ONE PERIOD RISK ADJUSTED VALUES’ theory is best approached (see [ADEH1], p. 69, [ADEH2], Section 2.2, as well as [He]) by taking the primitive object to be an “acceptance set”, that is a set of acceptable future net worths, also called simply “values”. This set is supposed to satisfy some “coherence” requirements. If we assume here (as well as in following sections) a zero interest rate for simplicity, the representation result states the following: for any acceptance set, there exists a set P of probability distributions (called generalised scenarios or test probabilities) on the space Ω of states of nature, such that a given position, with future (random) value denoted by X, is acceptable if and only if: For each test probability P ∈ P, the expected value of the future net worth under P, i.e. EP[X], is non-negative. The risk-adjusted value π(X) of a future net worth X is defined as follows: compute, under each test probability P ∈ P , the average of the future net worth X of the position, in formula EP [X], take the minimum of all numbers found above, which corresponds to the formula π(X) = infP∈P EP [X]. The axioms of coherent risk measures, well known by now (see[ADEH1]), translate for coherent risk-adjusted values into: monotonicity: if X ≥ Y then π(X) ≥ π(Y ), translation invariance: if a is a constant then π(a · 1 + X) = a + π(X), positive homogeneity: if λ ≥ 0 then π(λ ·X) = λ · π(X), superadditivity: π(X + Y ) ≥ π(X) + π(Y ). Remark. The risk measure ρ(X) for X studied in [ADEH1] and [ADEH2], is simply the negative of the risk adjusted value π(X) for X. The change of sign will simplify the treatment of measures of successive risks. 3. COHERENT MULTIPERIOD RISK-ADJUSTED VALUE THE CASE OF T PERIODS OF UNCERTAINTY will be described here in the language of trees. As noted already by one of the authors, they allow for some things “more easily done than said”, and we first need to define a few terms. We represent the availability of information over time by the set Ω of “states of nature” at date T and, for each date t = 0, ... , T , the partition Nt of Ω consisting of the set of smallest events which by date t are declared to obtain or not. These events are “tagged” by the date t and are called the nodes of the tree at date t. We use for such a node n the notation (n, t(n)) or n× {t(n)}. The partition Nt+1 is a refinement of the partition Nt and this provides the ancestorship relation of (m, t) to (n, t + 1) by means of the inclusion n ⊂ m. 2 For example, the “three period (four date) binomial tree” can be described in two ways (see Figure 1) by Ω = N3 = {[uuu], [uud], [udu], [udd], [duu], [dud], [ddu], [ddd]}, N2 = {[uu], [ud], [du], [dd]},N1 = {[u], [d]}, N0 = {[]}. The ancestorship relation amounts to suppress the right hand letter in each word based on u and d and the tagging amounts to count the number of letters within the brackets. From now on we shall most of the time neglect to write the brackets [ and ]. {ω1, ... , ω8} × {0} {ω1, ω2, ω3, ω4} × {1} t t t t t t t t t t t t {ω1, ω2} × {2} j j j j j j {ω1} × {3} d d d