Academic journal article Journal of Risk and Insurance

Bounds for Right Tails of Deterministic and Stochastic Sums of Random Variables

Academic journal article Journal of Risk and Insurance

Bounds for Right Tails of Deterministic and Stochastic Sums of Random Variables

Article excerpt


We investigate lower and upper bounds for right tails (stop-loss premiums) of deterministic and stochastic sums of nonindependent random variables. The bounds are derived using the concepts of comonotonicity, convex order, and conditioning. The performance of the presented approximations is investigated numerically for individual life annuity contracts as well as for life annuity portfolios, where mortality is modeled by Makeham's law, whereas investment returns are modeled by a Brownian motion process.


An insurance risk is described by a random variable (rv) X that represents the (discounted) value of future claims of an individual insurance contract or the aggregate claims of an insurance portfolio over a given reference period. One of the main tasks of the actuary is to assess the "dangerousness" of insurance risks, either by determining their distribution functions (df) or by summarizing their characteristics quantitatively by means of one or more risk measures.

An important class of risk measures related to an insurance risk X consists of the expectations E[[(X - d).sub.+]] = E[max (0, X - d)] for different values of d. In a reinsurance context, [(X - d).sub.+] can be interpreted as the liability of the reinsurer in a stop-loss reinsurance contract and E[[(X - d).sub.+]] is the associated net reinsurance premium, called the stop-loss premium at retention d. More generally, the quantity E[[(X - d).sub.+]] can be interpreted as a measure for the right tail of the df of X, beyond outcome d. Intuitively, a rv with larger right tails has more probability mass concentrated in the right tail of the df and hence, is "more dangerous." In the sequel, we will often call the stop-loss premium E[[(X - d).sub.+]] the right tail at level d of (the df of) X.

As applications we consider the problem of measuring the right tail of a single life annuity (cash flows with a stochastic time horizon) and of a diversified portfolio of life annuities (cash flows with a deterministic time horizon). Using our results for compound sums we obtain very precise bounds. We provide a number of numerical illustrations which reveal a significant improvement compared with the bounds obtained by traditional comonotonic approximations.

An rv Y that has uniformly larger right tails than X is said to be "larger in the increasing convex order sense," notation X [less than or equal to] [sub.icx] Y. In an actuarial context, the increasing convex order is also called stop-loss order (see, e.g., Denuit et al., 2005, and the references therein). In terms of utility theory, X [less than or equal to] [sub.icx] Y means that any risk-averse decision maker prefers risk X over risk Y. Hence, the calculation of right tails of insurance risks makes sense in order to evaluate the "dangerousness" of the risks at hand. In case X [less than or equal to] [sub.icx] Y and in addition E[X] = E[Y], one says that X is smaller than Y in the convex order sense, notation X [less than or equal to] [] Y.

In practice, it is not always straightforward to compute right tails or stop-loss premiums. In the actuarial literature a lot of attention has been devoted to determine bounds for stop-loss premiums in case only partial information about the distribution is available (see, e.g., De Vylder and Goovaerts, 1982; Jansen, Haezendonck, and Goovaerts, 1986; Hurlimann, 1996, 1998, 2002).

A particular type of problems arises when determining right tails of a sum S = [X.sub.1] + ... + [X.sub.n] when full information about the distributions of the [X.sub.i] is available but the dependence structure between these [X.sub.i] is not known or too cumbersome to work with. In Dhaene et al. (2002a, b) it is shown that in this context the least upper bound of S in the increasing convex order sense is obtained by replacing the unknown copula of the random vector ([X.sub.1], [X.sub.2], ..., [X.sub.n]) by the comonotonic copula. …

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