must hold, which yields a significant improvement in the constrained bound (5), which was previously based solely on knowledge of the second moment of the portfolio, without accounting for marginal information (see Table 5 for details).
In particular, [mathematical expression not reproducible] In what follows we assume that S belongs to the moment space [U.sub.s]([a, b]; [[micro].sub.1], [d.sub.2],..., [d.sub.s]-1) in which the constraints [d.sub.k] (k = 1,2,..., s - 1) are consistent with the marginal information (18).
in which [mathematical expression not reproducible] In fact, the bounds A and B can be interpreted as the mass points of the sum [mathematical expression not reproducible]having components that are consistent with the marginal information ( 18) and that are given as
At this point it is then no longer clear whether this bound (which is sharp in a moment space) remains sharp when the marginal information is added; that is, it is uncertain whether we can construct random variables [X.sub.i] (i = 1,2,..., n) that are consistent with the marginal information (18) such that their sum takes the values [mathematical expression not reproducible] and [[micro].1] + with probabilities p and 1--p, respectively.
Assumption 1: Knowledge of marginal distributions: When only marginal information is used, we can apply the RA of Embrechts, Puccetti, and Ruschendorf (2013) to estimate bounds on Cap[S].