Utility Functions

Abstract

This article is a self-contained survey of utility functions and some of their applications. Throughout the paper the theory is illustrated by three examples: exponential utility functions, power utility functions of the first kind (such as quadratic utility functions), and power utility functions of the second kind (such as the logarithmic utility function). The postulate of equivalent expected utility can be used to replace a random gain by a fixed amount and to determine a fair premium for claims to be insured, even if the insurer’s wealth without the new contract is a random variable itself. Then n companies (or economic agents) with random wealth are considered. They are interested in exchanging wealth to improve their expected utility. The family of Pareto optimal risk exchanges is characterized by the theorem of Borch. Two specific solutions are proposed. The first, believed to be new, is based on the synergy potential; this is the largest amount that can be withdrawn from the system without hurting any company in terms of expected utility. The second is the economic equilibrium originally proposed by Borch. As by-products, the option-pricing formula of Black-Scholes can be derived and the Esscher method of option pricing can be explained.     

Risk-adjusted credibility premiums using distorted probabilities

Abstract

Denneberg (1990) and Wang (1996a) propose that one calculate risk-adjusted insurance premiums as the expectation with respect to a distorted probability measure, a non-additive set function. This premium principle is supported by the theories of decision making of Yaari (1987) and of Schmeidler (1989). Denneberg (1994a) presents three conditioning rules for updating non-additive set functions in light of available information. In this work, we show how to apply these three update rules to calculate a risk-adjusted credibility premium and, thereby, combine credibility theory with this relatively new premium principle. Our main result is that, for some pairs of distortion function and update rule, one gets the same risk-adjusted credibility premium by distorting the predictive probability distribution, as required by the theory of Yaari, or by updating the distorted probability, as required by the theory of Schmeidler.     

Equity and Credibility

We build on previous work concerned with measuring equity and consider the problem of using observed claim data or other information to calculate premiums which maximize equity. When these optimal premiums are used, we show that gathering more information or refining the risk classification always increases equity. We study the case for which the premium is constrained to be an affine function of the claim data and obtain results analogous to classical credibility theory, including the inhomogeneous and homogeneous cases of the Bu¨hlmann-Straub model. We derive formulas for the credibility weights in certain cases.     

Utilitarianism and risk

Abstract

In day-to-day life, we are continuously exposed to different kinds of risk. Unfortunately, avoiding risk can often come at societal or individual costs. Hence, an important task within risk management is deciding how much it can be justified to expose members of society to risk x in order to avoid societal and individual costs y – and vice versa. We can refer to this as the task of setting an acceptable risk threshold. Judging whether a risk threshold is justified requires normative reasoning about what levels of risk exposure that are permissible. One such prominent normative theory is utilitarianism. According to utilitarians, the preferred risk threshold is the one that yields more utility for the most people compared to alternative risk thresholds. In this paper, I investigate whether and the extent to which utilitarian theory can be used to normatively ground a particular risk threshold in this way. In particular, I argue that there are (at least) seven different utilitarian approaches to setting an acceptable risk threshold. I discuss each of these approaches in turn and argue that neither can satisfactorily ground an acceptable risk threshold.     

Characterizations of optimal reinsurance treaties: a cost-benefit approach

This article investigates optimal reinsurance treaties minimizing an insurer’s risk-adjusted liability, which encompasses a risk margin quantified by distortion risk measures. Via the introduction of a transparent cost-benefit argument, we extend the results in Cui et al. [Cui, W., Yang, J. & Wu, L. (2013). Optimal reinsurance minimizing the distortion risk measure under general reinsurance premium principles. Insurance: Mathematics and Economics 53, 74–85] and provide full characterizations on the set of optimal reinsurance treaties within the class of non-decreasing, 1-Lipschitz functions. Unlike conventional studies, our results address the issue of (non-)uniqueness of optimal solutions and indicate that ceded loss functions beyond the traditional insurance layers can be optimal in some cases. The usefulness of our novel cost-benefit approach is further demonstrated by readily solving the dual problem of minimizing the reinsurance premium while maintaining the risk-adjusted liability below a fixed tolerance level.     

Knowledge Elicitation of Gompertz' Law of Mortality

A new formulation of Gompertz' law of mortality is proposed. Explicit formulas for moment generating function and moments of this formulation are derived. The new formulation is applied to the theory of heterogeneous populations and formulas for stop-loss transformations are derived.     

Reinsurance premium principles based on weighted loss functions

ABSTRACT

In this paper, we propose new reinsurance premium principles that minimize the expected weighted loss functions and balance the trade-off between the reinsurer's shortfall risk and the insurer's risk exposure in a reinsurance contract. Random weighting factors are introduced in the weighted loss functions so that weighting factors are based on the underlying insurance risks. The resulting reinsurance premiums depend on both the loss covered by the reinsurer and the loss retained by the insurer. The proposed premiums provide new ways for pricing reinsurance contracts and controlling the risks of both the reinsurer and the insurer. As applications of the proposed principles, the modified expectile reinsurance principle and the modified quantile reinsurance principle are introduced and discussed in details. The properties of the new reinsurance premium principles are investigated. Finally, the comparisons between the new reinsurance premium principles and the classical expectile principle, the classical quantile principle, and the risk-adjusted principle are provided.     

Potential measures and expected present value of operating costs until ruin in renewal risk models with general interclaim times

In this paper, a Sparre Andersen risk process with arbitrary interclaim time distribution is considered. We analyze various ruin-related quantities in relation to the expected present value of total operating costs until ruin, which was first proposed by Cai et al. [(2009a). On the expectation of total discounted operating costs up to default and its applications. Advances in Applied Probability 41(2), 495–522] in the piecewise-deterministic compound Poisson risk model. The analysis in this paper is applicable to a wide range of quantities including (i) the insurer's expected total discounted utility until ruin; and (ii) the expected discounted aggregate claim amounts until ruin. On one hand, when claims belong to the class of combinations of exponentials, explicit results are obtained using the ruin theoretic approach of conditioning on the first drop via discounted densities (e.g. Willmot [(2007). On the discounted penalty function in the renewal risk model with general interclaim times. Insurance: Mathematics and Economics 41(1), 17–31]). On the other hand, without any distributional assumption on the claims, we also show that the expected present value of total operating costs until ruin can be expressed in terms of some potential measures, which are common tools in the literature of Lévy processes (e.g. Kyprianou [(2014). Fluctuations of L'evy processes with applications: introductory lectures, 2nd ed. Berlin Heidelberg: Springer-Verlag]). These potential measures are identified in terms of the discounted distributions of ascending and descending ladder heights. We shall demonstrate how the formulas resulting from the two seemingly different methods can be reconciled. The cases of (i) stationary renewal risk model and (ii) surplus-dependent premium are briefly discussed as well. Some interesting invariance properties in the former model are shown to hold true, extending a well-known ruin probability result in the literature. Numerical illustrations concerning the expected total discounted utility until ruin are also provided.     

Optimal investment and risk control for an insurer with partial information in an anticipating environment

Abstract

This paper considers an optimal investment and risk control problem under the criterion of logarithm utility maximization. The risky asset process and the insurance risk process are described by stochastic differential equations with jumps and anticipating coefficients. The insurer invests in the financial assets and controls the number of policies based on some partial information about the financial market and the insurance claims. The forward integral and Malliavin calculus for Lévy processes are used to obtain a characterization of the optimal strategy. Some special cases are discussed and the closed-form expressions for the optimal strategies are derived.     

Pricing Dynamic Insurance Risks Using the Principle of Equivalent Utility

We introduce an expected utility approach to price insurance risks in a dynamic financial market setting. The valuation method is based on comparing the maximal expected utility functions with and without incorporating the insurance product, as in the classical principle of equivalent utility. The pricing mechanism relies heavily on risk preferences and yields two reservation prices - one each for the underwriter and buyer of the contract. The framework is rather general and applies to a number of applications that we extensively analyze.     

Catastrophe Risk Bonds

Abstract

This article examines the pricing of catastrophe risk bonds. Catastrophe risk cannot be hedged by traditional securities. Therefore, the pricing of catastrophe risk bonds requires an incomplete markets setting, and this creates special difficulties in the pricing methodology. The authors briefly discuss the theory of equilibrium pricing and its relationship to the standard arbitrage-free valuation framework. Equilibrium pricing theory is used to develop a pricing method based on a model of the term structure of interest rates and a probability structure for the catastrophe risk. This pricing methodology can be used to assess the default spread on catastrophe risk bonds relative to traditional defaultable securities.     

Risk-adjusted value allocation for (non-traded) assets with performance ratios

We propose a new valuation principle for possibly non-traded assets based on an implicit definition of a benchmark. The valuation principle allows taking (default and shortfall) risk constraints explicitly into account. The resulting risk-adjusted value functional is monotonic, positively homogeneous, partially concave and allows for an additive allocation of risk-adjusted values of non-traded assets in a portfolio. The valuation principle is applied to the problem of hedging and pricing in incomplete markets. Furthermore, accounting for non-traded assets is considered and we derive a risk-adjusted balance sheet for non-deterministic cash streams.     

Asymptotic results for the risk process based on marked point processes

Abstract

In this paper asymptotic properties for the risk process will be studied when the number of risk units tends to infinity. The paper extends asymptotic properties for the classical risk process to more general processes. In the classical risk process the claim amounts are assumed independent and identically distributed, and the claim number process is a homogeneous Poisson process.

The key tool is point process theory with associated martingale theory. The results are illustrated by examples.     

A decision theoretic formulation of insurance risk theory

Abstract

In a number of papers Borch has shown how certain insurance problems can be formulated using the concept of utility. (See Borch [3], [4], [5], [6], [7] and [8].) Borch's work is used as a building block in Part I of this report, which presents a Bayesian decision theoretic formulation of some of the main aspects of insurance risk theory. Part I makes use of the concepts of utility and subjective probability. It is admitted that these concepts are more commonly associated with individuals rather than groups of individuals such as insurance companies. However, in this report, we will refer to an insurance company as an individual (albeit a neuter one) and assume that it can quantify its preferences for consequences and its opinions about the occurrence of events. Further, we assume that a company “behaves” according to certain rules of consistent behavior which imply that when presented with several risky courses of action, the company will take the action which has the greatest expected utility. Formal treatments of assumptions that lead to this mode of behavior can be found in Savage [17] and Pratt, Raiffa, and Schlaifer [15].     

Multiperiod Optimal Investment-Consumption Strategies with Mortality Risk and Environment Uncertainty

Abstract

In this article we investigate three related investment-consumption problems for a risk-averse investor: (1) an investment-only problem that involves utility from only terminal wealth, (2) an investment-consumption problem that involves utility from only consumption, and (3) an extended investment-consumption problem that involves utility from both consumption and terminal wealth. Although these problems have been studied quite extensively in continuous-time frameworks, we focus on discrete time. Our contributions are (1) to model these investmentconsumption problems using a discrete model that incorporates the environment risk and mortality risk, in addition to the market risk that is typically considered, and (2) to derive explicit expressions of the optimal investment-consumption strategies to these modeled problems. Furthermore, economic implications of our results are presented. It is reassuring that many of our findings are consistent with the well-known results from the continuous-time models, even though our models have the additional features of modeling the environment uncertainty and the uncertain exit time.     

Trend-following hedge funds and multi-period asset allocation

Abstract

Selected hedge funds employ trend-following strategies in an attempt to achieve superior risk-adjusted returns. We employ a lookback straddle approach for evaluating the return characteristics of a trend-following strategy. The strategies can improve investor performance in the context of a multi-period dynamic portfolio model. The gains are achieved by taking advantage of the funds' high level of volatility. A set of empirical results confirms the advantages of the lookback straddle for investors at the top end of the multi-period efficient frontier.     

Minimizing CVaR in global dynamic hedging with transaction costs

This study develops a global derivatives hedging methodology which takes into account the presence of transaction costs. It extends the Hodges and Neuberger [Rev. Futures Markets, 1989, 8, 222–239] framework in two ways. First, to reduce the occurrence of extreme losses, the expected utility is replaced by the conditional Value-at-Risk (CVaR) coherent risk measure as the objective function. Second, the normality assumption for the underlying asset returns is relaxed: general distributions are considered to improve the realism of the model and to be consistent with fat tails observed empirically. Dynamic programming is used to solve the hedging problem. The CVaR minimization objective is shown to be part of a time-consistent framework. Simulations with parameters estimated from the S&P 500 financial time series show the superiority of the proposed hedging method over multiple benchmarks from the literature in terms of tail risk reduction.     

Asymptotic Theory for a Risk Process with a High Dividend Barrier

The only way to avoid ruin in the classical model of the collective risk theory is that the surplus increases to infinity. We consider a modified model with a dividend barrier that prevents this behavior. It is shown that there is a simple approximation formula for the time of ruin when the level of the dividend barrier is high and the Cramér-Lundberg condition is satisfied. A numerical example is presented in the case when the claims are exponentially distributed. The relation to queuing theory is used to derive the proportion of time the surplus is below some given level.     

Combining the Effects of OLS and Spread on Futures Hedging: Evidence from the Taiwan Stock Index

Abstract:

This study proposes a dynamic hedge ratio, the combined ordinary least squares spread (COLSS), which combines the hedge ratio of ordinary least squares and the value of spread. Using this dynamic ratio for hedging with futures contracts, one can replace spot risk with spread risk. The COLSS captures not only the long-run equilibrium between spot and futures returns, but also the short-run deviation from equilibrium. The spread is forecast by one-period lagged stock market factors and high-order moments that are estimated by an options model. In the in-sample and out-of-sample tests, the COLSS strategy achieves significant risk reduction and outperforms the alternative models by a large utility improvement.     

Who gains and who loses on stock markets? Risk preferences and timing matter

This paper uses an agent-based multi-asset model to examine the effect of risk preferences and optimal rebalancing frequency on performance measures while tracking profit and risk-adjusted return. We focus on the evolution of portfolios managed by heterogeneous mean-variance optimizers with a quadratic utility function under different market conditions. We show that patient and risk-averse agents are able to outperform aggressive risk-takers in the long-run. Our findings also suggest that the trading frequency determined by the optimal tolerance for the deviation from portfolio targets should be derived from a tradeoff between rebalancing benefits and rebalancing costs. In a relatively calm market, the absolute range of 6% to 8% and the complete-way back rebalancing technique outperforms others. During particular turbulent periods, however, none of the existing rebalancing techniques improves tax-adjusted profits and risk-adjusted returns simultaneously.