Moment Problem and Its Application to Tail Risk Assessment

The paper discusses how to assess risk by computing the best upper and lower bounds on the expected value E[φ(X)], subject to the constraints E[Xⁱ] = µ_i for i = 0, 1, 2, . . . , n. φ(x) can take the form of the indicator function φ(x) = 𝕀_(−∞,K](x) in which the bounds on Pr(X ≤ K) are calculated and the form φ(x) = (ϕ(x)−K)₊ in which the bounds on financial payments are founds. We solve the moment bounds on E[𝕀_(−∞,K](X)] through three methods; namely, the semidefinite programming method, the moment-matching method, and the linear approximation method. We show that for practical purposes, these methods provide numerically equivalent results. We explore the accuracy of bounds in terms of the number of moments considered. We investigate the usefulness of the moment method by comparing the moment bounds with the “point” estimate provided by the Johnson System of distributions. In addition, we propose a simpler formulation for the unimodal bounds on E[𝕀_(−∞,K](X)] compared to the existing formulations in the literature. Furthermore, for those problems that could be solved both analytically and numerically given the first few moments, our comparisons between the numerical and analytical results call attention to the potential differences between these two methodologies. Our analysis indicates the numerical bounds could deviate from their corresponding analytical counterparts. The accuracy of numerical bounds is sensitive to the volatility of X. The more volatile the random variable X is, the looser the numerical bounds are compared to their closed-form solutions