Existence and Uniqueness of Equilibrium in a Reinsurance Syndicate

In this paper we consider a reinsurance syndicate, assuming that Pareto optimal allocations exist. Under a continuity assumption on preferences, we show that a competitive equilibrium exists and is unique. Our conditions allow for risks that are not bounded, and we show that the most standard models satisfy our set of sufficient conditions, which are thus not too restrictive. Our approach is to transform the analysis from an infinite dimensional to a finite dimensional setting.Existence of equilibrium; uniqueness of equilibrium; Pareto optimality; reinsurance model; syndicate theory; risk tolerance; exchange economy; probability distributions; Walras’ law

The investment horizon problem: A resolution

In the canonical model of investments, the optimal fractions in the risky assets do not depend on the time horizon. This is against empirical evidence, and against the typical recommendations of portfolio managers. We demonstrate that if the intertemporal coefficient of relative risk aversion is allowed to depend on time, or the age of the investor, the investment horizon problem can be resolved. Accordingly, the only standard assumption in applied economics/finance that we relax in order to obtain our conclusion, is the state and time separability of the intertemporal felicity index in the investor’s utility function. We include life and pension insurance, and we also demonstrate that preferences aggregate.The investment horizon problem; complete markets; life and pension insurance; dynamic programming; Kuhn-Tucker; directional derivatives; time consistency; aggregation

Long Dated Life Insurance and Pension Contracts

We discuss the "life cycle model" by first introducing a credit market with only biometric risk, and then market risk is introduced via risky securities. This framework enables us to find optimal pension plans and life insurance contracts where the benefits are state dependent. We compare these solutions both to the ones of standard actuarial theory, and to policies offered in practice. Two related portfolio choice puzzles are discussed in the light of recent research, one is the horizon problem, the other is related to the aggregate market data of the last century, where theory and practice diverge. Finally we present some comments on longevity risk and cohort risk.The life cycle model; pension insurance; optimal life insurance; longevity risk; the horizon problem; equity premium puzzle

The equity premium and the risk free rate in a production economy. A new perspective

We study a competitive equilibrium in a production economy, i.e., a system of prices at which firms’ profit maximizing production decisions and individuals’ preferred affordable consumption choices equate supply and demand in every market. We derive the equilibrium price of the firm and the equilibrium short term interest rate, the optimal consumption in society, as well as the risk premium on equity. Both a linear, and a nonlinear production technology are considered. For the linear one applied to the Standard and Poor’s composite stock price index for the last century, a risk premium of 0.062 corresponds to a relative risk aversion of 2.27. The model provides a riskfree interest rate for the period of 0.8%. The nonlinear model, however, highlights a hedging demand for the investors related to the real economy, which would, if taken into account, make the stock market of the last century less risky than it was perceived to be.Competitive equilibrium; production economy

The Nash Bargaining Solution vs. Equilibrium in a Reinsurance Syndicate

We compare the Nash bargaining solution in a reinsurance syndicate to the competitive equilibrium allocation, focusing on uncertainty and risk aversion. Restricting attention to proportional reinsurance treaties, we find that, although these solution concepts are very different, one may just appear as a first order Taylor series approximation of the other, in certain cases. This may be good news for the Nash solution, or for the equilibrium allocation, all depending upon one’s point of view. Our model also allows us to readily identify some properties of the equilibrium allocation not be shared by the bargaining solution, and vice versa, related to both risk aversions and correlations.Nash’s Bargaining Solution; Equilibrium; Pareto Optimal Risk Exchange; Reinsurance Treaties; Uncertainty; Risk Aversion; Correlations; Multinormal Universe

Pareto Optimal Insurance Policies in the Presence of Administrative Costs

In his classical article in The American Economic Review, Arthur Raviv (1979) examines Pareto optimal insurance contracts when there are ex-post insurance costs c induced by the indemnity I for loss x. Raviv’s main result is that a necessary and sufficient condition for the Pareto optimal deductible to be equal to zero is c'(I) = 0 for all I >= 0. We claim that another type of cost function is called for in household insurance, caused by frequent but relatively small claims. If a fixed cost is incurred each time a claim is made, we obtain a non-trivial Pareto optimal deductible even if the cost function does not vary with the indemnity. This implies that when the claims are relatively small, it is not optimal for the insured to get a compensation since the costs outweighs the benefits, and a deductible will naturally occur. We also discuss policies with an upper limit, and show that the insurer prefers such contracts, but the insured does not. In Raviv’s paper it was also shown that policies with upper limits are dominated by policies with no upper limit, when there are ex-post costs to insurance. We show that the result is right, but the proof is wrong.Pareto optimal risk sharing; administrative costs in insurance; household insurance; XL-contracts

The long term equilibrium interest rate and risk premiums under uncertainty

Both the equilibrium interest rate and the equity premium, as well as risk premiums of risky investments are all important quantities in cost-benefit analyses. In the light of the current (2008 -) financial crisis, it is of interest to study models that connect the the financial sector with the real economy. The effects of climate change has, on the other hand, been the subject of extensive discussions, for example in connection with the Stern report. The paper addresses both these issues, first based on standard assumptions. In particular we investigate what is needed to have long-term interests lower than short term rates. Our model allows us to tell what happens to risk premiums in turbulent times, consistent with observations. Next we extend the pure exchange model to a production economy. As a result we obtain an equilibrium term structure of interest rates, as well as a model for the equity premium. We end by a discussion of risk adjustments of the discount factor. For projects aimed at insuring future consumption, the interest rate is smaller than the risk free rate. Mitigation can have the characteristics of such a project.Dynamic equilibrium; the Lucas model; term structure; CIR; pure exchange; production economy; equity premium puzzle; risk free rate puzzle; climate models; Stern Review

Empirical Tests of Models of Catastrophe Insurance Futures

The authors empirically investigate models of insurance futures derivatives contracts. In the fall of 1993 the Chicago Board of Trade (CBOT) started trading a contract designed to scrutinize catastrophic risk, which is currently done in the reinsurance markets. There are obvious advantages to trading on organized exchanges (standardization, liquidity, much reduced credit risk, etc.) as opposed to OTC markets. There has so far been little academic on these contracts. In this paper we look at the price history for the first two years within the context of a pricing model of Aase [1995]. This paper was presented at the Financial Institutions Center's May 1996 conference on "

Intuitive probability of non-intuitive events

Quantitative probability in the subjective theory is assumed to be finitely additive and defined on all the subsets of an underlying state space. Functions from this space into an Euclidian n-space create a new probability space for each such function. We point out that the associated probability measures, induced by the subjective probability, on these new spaces can not be finitely additive and defined on all the subsets of Euclidian n-space, for n ≥ 3. This is a consequence of the Banach-Tarski paradox. In the paper we show that subjective probability theory, including Savage’s theory of choice, can be reformulated to take this, and similar objections into account. We suggest such a reformulation which, among other things, amounts to adding an axiom to Savage’s seven postulates, and then use a version of Carathéodory’s extension theorem

Optimal spending of a wealth fund in the discrete time life cycle model

The paper analyses optimal spending of an endowment fund. We use the life cycle model for both expected utility and recursive utility in discrete time. First we find the optimal consumption and investment policies for both kinds of utility functions. This we apply to a sovereign wealth fund that invests broadly in the international financial markets. We demonstrate that the optimal spending rate, i.e., the consumption to wealth ratio, is significantly lower than the fund’s expected real rate of return. Using the expected return as the spending rate, implies that the fund’s value converges towards 0 with probability 1 and also in expectation, as time goes. For both kinds of long term convergence we find closed form threshold values. Spending below these values secures that the fund will last ”forever”. For reasonable values of the preference parameters, the optimal spending rate is demonstrated to satisfy these long term requirements