A Note on a "Square-Root Rule" for Reinsurance

In previous work, the current authors derived a mathematical expression for the optimal (or "saturation") number of reinsurers for a given number of primary insurers (see Powers and Shubik, 2001). In the current paper, we show analytically that, for large numbers of primary insurers, this mathematical expression provides a "square-root rule"; i.e., the optimal number of reinsurers in a market is given asymptotically by the square root of the total number of primary insurers. We note further that an analogous “fourth-root rule” applies to markets for retrocession (the reinsurance of reinsurance).Primary insurance, Reinsurance, Market size, Square-root rule

A Sequence of Nested Exponential Random Variables with Connections to Two Constants of Euler

We investigate a recursively generated sequence of random variables that begins with an Exponential random variable with parameter (i.e., inverse-mean) 1, and continues with additional Exponentials, each of whose random parameter possesses the distribution of the prior term in the sequence. Although such sequences enjoy some (limited) applicability as models of parameter uncertainty, our present interest is primarily theoretical. Specifically, we observe that the implied sequence of distribution functions manifests a surprising connection to two well-known mathematical quantities first studied by Leonhard Euler: the Euler-Gompertz and Euler-Mascheroni constants. Through a close analysis of one member of this distribution-function sequence, we are able to shed new light on these constants

Market Bubbles and Wasteful Avoidance: Tax and Regulatory Constraints on Short Sales

Although short sales make an important contribution to financial markets, this transaction faces legal constraints that do not govern long positions. In evaluating these constraints, other commentators, who are virtually all economists, have not focused rigorously enough on the precise contours of current law. Some short sale constraints are mischaracterized, while others are omitted entirely. Likewise, the existing literature neglects many strategies in which well advised investors circumvent these constraints; this avoidance may reduce the impact of short sale constraints on market prices, but may contribute to social waste in other ways. To fill these gaps in the literature, this paper offers a careful look at current law and draws three conclusions. First, short sales play a valuable role in the financial markets; while there may be plausible reasons to regulate short sales-- most notably, concerns about market manipulation and panics -- current law is very poorly tailored to these goals. Second, investor self-help can ease some of the harm from this poor tailoring, but at a cost. Third, relatively straightforward reforms can eliminate the need for self-help while accommodating legitimate regulatory goals. In making these points, we focus primarily on a burden that other commentators have neglected: profits from short sales generally are ineligible for the reduced tax rate on long-term capital gains, even if the short sale is in place for more than one year.Short sales, Momentum traders, Value investors

Toward a Theory of Reinsurance and Retrocession

There is a natural tradeoff between the benefits of increasing the number of competitors in an insurance market and the drawback to the weakening of the law of large numbers due to the diminishing of average reserves. In this investigation we consider the possibility for optimal layers of reinsurance and retrocession in the design of the insurance industry. A general question which may be asked of all financial institutions is what factors limit the number of layers of paper which can be constructed

Cooperative and Noncooperative Solutions, and the \u27Game within a Game\u27

In a previous essay, we developed a simple (in)efficiency measure for matrix games. We now address the difficulties encountered in assessing the usefulness and accuracy of such a measure

A Note on a ‘Square-Root Rule’ for Reinsurance

In previous work, the current authors derived a mathematical expression for the optimal (or “saturation”) number of reinsurers for a given number of primary insurers (see Powers and Shubik, 2001). In the current paper, we show analytically that, for large numbers of primary insurers, this mathematical expression provides a “square-root rule”; i.e., the optimal number of reinsurers in a market is given asymptotically by the square root of the total number of primary insurers. We note further that an analogous “fourth-root rule” applies to markets for retrocession (the reinsurance of reinsurance)

The Best and Worst of All Possible Worlds: Some Crude Evaluations

The 2 x 2 matrix game plays a central role in the teaching and exposition of game theory. It is also the source of much experimentation and research in political science, social psychology, biology and other disciplines. This brief paper is addressed to answering one intuitively simple question without going into the many subtle qualifications that are there. How efficient is the non-cooperative equilibrium? This is part of a series of several papers that address many of the qualifications concerning the uses of the 2 x 2 matrix games

Insurance Market Games: Scale Effects and Public Policy

We propose a game-theoretic model to study various effects of scale in an insurance market. After reviewing a simple static model, we present a one-period game in which both the buyers and sellers of insurance make strategic bids, and show that, under reasonably broad conditions, market equilibrium exists. For a special case, we then consider how both the price and quantity of insurance, as well as other quantities of interest to public policy decision makers, are affected by the number of insurance firms, the number of customers, and the total amount of capital provided by investors.Insurance market games, scale effects, public policy