Optimal insurance under the insurer’s risk constraint

In this paper, we impose the insurer’s risk constraint on Arrow’s optimal insurance model. The insured aims to maximize his/her expected utility of terminal wealth, under the constraint that the insurer wishes to control the expected loss of his/her terminal wealth below some prespecified level. We solve the problem, and it is shown that when the insurer’s risk constraint is binding, the solution to the problem is not linear, but piecewise linear deductible. Moreover, it can be shown that the insured’s optimal expected utility will increase if the insurer increases his/her risk tolerance.     

Optimal insurance design with a bonus

This paper investigates an insurance design problem, in which a bonus will be given to the insured if no claim has been made during the whole lifetime of the contract, for an expected utility insured. In this problem, the insured has to consider the so-called optimal action rather than the contracted compensation (or indemnity) due to the existence of the bonus. For any pre-agreed bonus, the optimal insurance contract is given explicitly and shown to be either the full coverage contract when the insured pays high enough premium, or a deductible one otherwise. The optimal contract and bonus are also derived explicitly if the insured is allowed to choose both of them. The contract turns out to be of either zero reward or zero deductible. In all cases, the optimal contracts are universal, that is, they do not depend on the specific form of the utility of the insured. A numerical example is also provided to illustrate the main theoretical results of the paper.     

On expected utility for financial insurance portfolios with stochastic dependencies

The effect of background risks as human capital, market risks and catastrophic events has been considered in the literature in different contexts. In this note, we consider financial insurance portfolios with insurable risks and one background risk (uninsurable financial asset), such that the random losses and the background risk depend on environmental parameters. We study how dependencies between the risks influence the expected utility of the portfolio’s wealth distribution under risk aversion, when the environmental parameters are random. Stochastic bounds for the expected wealth are given from modeling the dependence between the parameters by different notions. Similar results are given for multivariate portfolios with n groups and multivariate risk aversion, besides an expected utility comparison result for the minimum and the total portfolio’s wealth.     

Portfolio selection and duality under mean variance preferences

This paper uses duality to analyze an investor’s behavior in a n-asset portfolio selection problem when the investor has mean variance preferences. The indirect utility and wealth requirement functions are used to derive Roy’s identity, Shephard’s lemma and the Slutsky equation. In our simple Slutsky equation the income effect is characterized by decreasing absolute risk aversion (DARA) and the substitution effect is always positive [negative] with respect to an asset’s holding if the asset’s mean return [risk] increases. Substitution effect and income effect work in the same direction presupposed mean variance preferences display DARA.     

Optimal insurance coverage in situations of pure and speculative risk and the risk-free asset

The insured's portfolio consists of an insurable (pure) risk, an uninsurable (speculative) risk, a (proportional) insurance policy and a risk-free asset. The optimal insurance policy (i.e., the proportion to be insured) is examined from the insured's point of view, using the reward to variability concept. The importance of the risk-free asset in reaching an exact and explicit solution is analyzed, while emphasizing the possibility of substitution of the risk-free investment and insurance mechanisms. The paper demonstrates possibilities of improving the insured's welfare by the use of the risk-free rate - which is sometimes less expensive than other risk reduction instruments. The analysis leads to a two-step solution, similar to the well- known Hirschleifer investment model and to the famous Capital Assets Pricing Model.     

Behavioral optimal insurance

The present work studies the optimal insurance policy offered by an insurer adopting a proportional premium principle to an insured whose decision-making behavior is modeled by Kahneman and Tversky’s Cumulative Prospect Theory with convex probability distortions. We show that, under a fixed premium rate, the optimal insurance policy is a generalized insurance layer (that is, either an insurance layer or a stop–loss insurance). This optimal insurance decision problem is resolved by first converting it into three different sub-problems similar to those in Jin and Zhou (2008); however, as we now demand a more regular optimal solution, a completely different approach has been developed to tackle them. When the premium is regarded as a decision variable and there is no risk loading, the optimal indemnity schedule in this form has no deductibles but a cap; further results also suggests that the deductible amount will be reduced if the risk loading is decreased. As a whole, our paper provides a theoretical explanation for the popularity of limited coverage insurance policies in the market as observed by many socio-economists, which serves as a mathematical bridge between behavioral finance and actuarial science.     

Explicit solutions of optimal consumption,investment and insurance problems with regime switching

We consider an investor who wants to select his optimal consumption, investment and insurance policies. Motivated by new insurance products, we allow not only the financial market but also the insurable loss to depend on the regime of the economy. The objective of the investor is to maximize his expected total discounted utility of consumption over an infinite time horizon. For the case of hyperbolic absolute risk aversion (HARA) utility functions, we obtain the first explicit solutions for simultaneous optimal consumption, investment, and insurance problems when there is regime switching. We determine that the optimal insurance contract is either no-insurance or deductible insurance, and calculate when it is optimal to buy insurance. The optimal policy depends strongly on the regime of the economy. Through an economic analysis, we calculate the advantage of buying insurance.     

Optimal risk retention under partial insurance

This paper examines the situation where a risk-averse insured determines the optimal amount of deductible (or stop-loss) insurance. The insurer uses two different premium principles, the expected value principle and the exponential principle. The insured has an exponential utility function. Specific numerical results are obtained for the optimal stop-loss limit in the case of a group life insurance plan. The exact results are contrasted with those obtained by using the normal approximation instead of the exact distribution of aggregate claims.     

A simple insurance model: optimal coverage and deductible

An insurance model, with realistic assumptions about coverage, deductible and premium, is studied. Insurance is shown to decrease the variance of the cost to the insured, but increase the expected cost, a tradeoff that places our model in the Markowitz mean-variance model.     

Production under uncertainty with insurance or hedging

This paper examines the output decision of a risk-averse producer facing profit risk in the presence of insurance or hedging. Conditions under which the producer’s output increases upon the introduction of generic insurance are derived, giving rise to conditions for deductible insurance (commodity call options), coinsurance-type insurance (commodity futures), and restricted deductible insurance, respectively. This paper improves upon the literature by considering general profit risk, possibly revenue risk or cost risk, that may not be multiplicative. Moreover, unlike Machnes and Wong’s [Geneva Pap. Risk Insurance Theory 28 (2003) 73–80] condition on the loading factor that may not lead to an explicit and unique value, the condition derived in this paper gives rise to a unique upper bound for the loading factor. Finally, their assumptions on the utility function, such as quadratic utility and constant absolute risk aversion for the case of restrictive deductible insurance and zero-loading are made substantial less restrictive.     

Optimal multi-period insurance contracts

In multi-period insurance contracts (such as automobile insurance contracts), unlike single-period ones, the premiums that the insured must pay increase whenever he files a claim. Hence, the buyer faces a problem that is absent in one-period models, namely: he must determine for which damages he should file a claim and for which he should not.The optimal claims policy of the buyer is presented for a large class of insurance contracts. It is shown that the buyer will file a claim only if it is larger than some critical value. Based on this it is shown that the buyer prefers a contract that provides full coverage above a deductible for damages that exceed his critical value. In this case the optimal contract is not unique since the buyer is indifferent to the form of the contract for damages below his critical value. It is shown, however, that as in one-period models (Arrow (1963, 1974)) there exists an optimal contract that provides full coverage above a deductible. In multi-period setting, however, the buyer will file a claim only if the damage is sufficiently higher than the deductible.It is also shown that the buyer prefers a strictly positive deductible. Unlike the one-period case (Mossin (1968)), this result holds true even if the premium rates equal the expected payments.     

Financial Giffen goods: Examples and counterexamples

In the basic Markowitz and Merton models, a stock’s weight in efficient portfolios goes up if its expected rate of return goes up. Put differently, there are no financial Giffen goods. By an example from mortgage choice we illustrate that for more complicated portfolio problems Giffen effects do occur.     

Behavioral optimal insurance

The present work studies the optimal insurance policy offered by an insurer adopting a proportional premium principle to an insured whose decision-making behavior is modeled by Kahneman and Tversky’s Cumulative Prospect Theory with convex probability distortions. We show that, under a fixed premium rate, the optimal insurance policy is a generalized insurance layer (that is, either an insurance layer or a stop–loss insurance). This optimal insurance decision problem is resolved by first converting it into three different sub-problems similar to those in Jin and Zhou (2008); however, as we now demand a more regular optimal solution, a completely different approach has been developed to tackle them. When the premium is regarded as a decision variable and there is no risk loading, the optimal indemnity schedule in this form has no deductibles but a cap; further results also suggests that the deductible amount will be reduced if the risk loading is decreased. As a whole, our paper provides a theoretical explanation for the popularity of limited coverage insurance policies in the market as observed by many socio-economists, which serves as a mathematical bridge between behavioral finance and actuarial science.     

Optimal insurance design in the presence of exclusion clauses

The present work studies the design of an optimal insurance policy from the perspective of an insured, where the insurable loss is mutually exclusive from another loss that is denied in the insurance coverage. To reduce ex post moral hazard, we assume that both the insured and the insurer would pay more for a larger realization of the insurable loss. When the insurance premium principle preserves the convex order, we show that any admissible insurance contract is suboptimal to a two-layer insurance policy under the criterion of minimizing the insured’s total risk exposure quantified by value at risk, tail value at risk or an expectile. The form of optimal insurance can be further simplified to be one-layer by imposing an additional weak condition on the premium principle. Finally, we use Wang’s premium principle and the expected value premium principle to illustrate the applicability of our results, and find that optimal insurance solutions are affected not only by the size of the excluded loss but also by the risk measure chosen to quantify the insured’s risk exposure.     

An optimal insurance strategy for an individual under an intertemporal equilibrium

In this paper, we discuss how a risk-averse individual under an intertemporal equilibrium chooses his/her optimal insurance strategy to maximize his/her expected utility of terminal wealth. It is shown that the individual’s optimal insurance strategy actually is equivalent to buying a put option, which is written on his/her holding asset with a proper strike price. Since the cost of avoiding risk can be seen as a risk measure, the put option premium can be considered as a reasonable risk measure. Jarrow [Jarrow, R., 2002. Put option premiums and coherent risk measures. Math. Finance 12, 135-142] drew this conclusion with an axiomatic approach, and we verify it by solving the individual’s optimal insurance problem.     

Effects of insurance premium and deductible on production

We consider a risk-averse firm bearing the revenue risk and fuzzy production cost. Using the quadratic utility function the sufficient conditions for a deductible insurance to increase the output are derived and found to be the functions of insurance premium and deductible. We also show that the optimal production for a firm in the fuzzy environment is less than that in the crisp environment.     

Background risk,bivariate risk attitudes,and optimal prevention

This paper extends Eeckhoudt et al.’s (2012) results for precautionary effort to bivariate utility function framework. We establish an equivalence between the agent’s precautionary effort motive and the signs of successive cross-derivatives of the bivariate utility function. We show that the introduction (or deterioration) of an independent background risk induces more prevention to protect against wealth loss provided the individual exhibits correlation aversion of some given order. The conditions on the individual’s risk preferences are given to generate some specific prevention behaviors in the univariate framework with multiplicative risks. Our conclusion also indicates that an increase in the correlation between wealth risk and background risk leads to a reduction in optimal prevention.     

Optimal proportional reinsurance and investment with transaction costs, I: Maximizing the terminal wealth

We consider a problem of optimal reinsurance and investment with multiple risky assets for an insurance company whose surplus is governed by a linear diffusion. The insurance company’s risk can be reduced through reinsurance, while in addition the company invests its surplus in a financial market with one risk-free asset and n risky assets. In this paper, we consider the transaction costs when investing in the risky assets. Also, we use Conditional Value-at-Risk (CVaR) to control the whole risk. We consider the optimization problem of maximizing the expected exponential utility of terminal wealth and solve it by using the corresponding Hamilton-Jacobi-Bellman (HJB) equation. Explicit expression for the optimal value function and the corresponding optimal strategies are obtained.     

Optimal investment of DC pension plan under short-selling constraints and portfolio insurance

In this paper we investigate an optimal investment problem under short-selling and portfolio insurance constraints faced by a defined contribution pension fund manager who is loss averse. The financial market consists of a cash bond, an indexed bond and a stock. The manager aims to maximize the expected S-shaped utility of the terminal wealth exceeding a minimum guarantee. We apply the dual control method to solve the problem and derive the representations of the optimal wealth process and trading strategies in terms of the dual controlled process and the dual value function. We also perform some numerical tests and show how the S-shaped utility, the short-selling constraints and the portfolio insurance impact the optimal terminal wealth.     

Optimal Cross Hedging of Insurance Derivatives

Abstract

We consider insurance derivatives depending on an external physical risk process, for example, a temperature in a low dimensional climate model. We assume that this process is correlated with a tradable financial asset. We derive optimal strategies for exponential utility from terminal wealth, determine the indifference prices of the derivatives, and interpret them in terms of diversification pressure. Moreover, we check the optimal investment strategies for standard admissibility criteria. Finally, we compare the static risk connected with an insurance derivative to the reduced risk due to a dynamic investment into the correlated asset. We show that dynamic hedging reduces the risk aversion in terms of entropic risk measures by a factor related to the correlation.