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].     

Portfolio Optimization in Discontinuous Markets under Incomplete Information

We consider the problem of maximization of expected utility from terminal wealth for log and power utility functions in a market model that leads to purely discontinuous processes. We study this problem as a stochastic control problem both under complete as well as incomplete information. Our contribution consists in showing that the optimal strategy can be obtained by solving a system of equations that in some cases is linear and that a certainty equivalence property holds not only for log-utility but also for a power utility function. For the case of a power utility under incomplete information we also present an independent direct approach based on a Zakai-type equation.      

Optimal Portfolio Selection with Transaction Costs

Abstract

This paper examines the lifetime portfolio-selection problem in the presence of transaction costs. Using a discrete time approach, we develop analytical expressions for the investor's indirect utility function and also for the boundaries of the no-transactions region. The economy consists of a single risky asset and a riskless asset. Transactions in the risky asset incur proportional transaction costs. The investor has a power utility function and is assumed to maximize expected utility of end-of-period wealth. We illustrate the solution procedure in the case in which the returns on the risky asset follow a multiplicative binomial process. Our paper both complements and extends the recent work by Gennotte and Jung (1994), which used numerical approximations to tackle this problem.     

Optimal Investment Strategy to Minimize the Probability of Lifetime Ruin

Abstract

I study the problem of how individuals should invest their wealth in a risky financial market to minimize the probability that they outlive their wealth, also known as the probability of lifetime ruin. Specifically, I determine the optimal investment strategy of an individual who targets a given rate of consumption and seeks to minimize the probability of lifetime ruin. Two forms of the consumption function are considered: (1) The individual consumes at a constant (real) dollar rate, and (2) the individual consumes a constant proportion of his or her wealth. The first is arguably more realistic, but the second has a close connection with optimal consumption in Merton’s model of optimal consumption and investment under power utility.

For constant force of mortality, I determine (a) the probability that individuals outlive their wealth if they follow the optimal investment strategy; (b) the corresponding optimal investment rule that tells individuals how much money to invest in the risky asset for a given wealth level; (c) comparative statics for the functions in (a) and (b); (d) the distribution of the time of lifetime ruin, given that ruin occurs; and (e) the distribution of bequest, given that ruin does not occur. I also include numerical examples to illustrate how the formulas developed in this paper might be applied.     

A Note on Loadings and Deductibles: Can a Vicious Circle Arise?

In a recently reprinted paper Borch wonders whether an increase in insurance loadings, together with the consequent increase in customers' deductibles, may be the start of a vicious circle, in which higher deductibles produce higher loadings and vice versa, ad infinitum. This paper rules out the possibility of a vicious circle, in a model à la Borch. First of all, increases in costs of the type considered by Borch are not necessarily followed by increases in loadings. Second, increases in loadings are not necessarily followed by increases in deductibles, since in equilibrium insurance may be Giffen. Last but not least, loadings do not increase with deductibles, because the only viable equilibrium is a Stackelberg one.     

The econometrics of efficient portfolios

It is well known that the standard mean variance approach can be inappropriate when return distributions feature skewness, fat tails or multimodes. This is typically the situation for portfolios including derivatives. In this case, it can be necessary to come back to the basic expected utility approach. In this paper, an efficient portfolio maximizes the expected utility of future wealth. This paper presents an analysis of the efficiency frontier, formed by a set of efficient portfolios corresponding to a parameterized class of utility functions. First, we discuss the estimation of an efficient portfolio and introduce several tests of the efficiency hypothesis, depending on what is known about the utility function and the budget level. Next we analyse the shape of the frontier and develop a procedure for testing the separability of the efficiency frontier into K independent funds. The inference is semi-nonparametric because the return distribution is left unspecified. We illustrate our approach by an application to portfolios including derivatives.     

In which financial markets do mutual fund theorems hold true?

The mutual fund theorem (MFT) is considered in a general semimartingale financial market S with a finite time horizon T, where agents maximize expected utility of terminal wealth. The main results are:

Optimal asset allocation for participating contracts under the VaR and PI constraint

ABSTRACT

Participating contracts provide a maturity guarantee for the policyholder. However, the terminal payoff to the policyholder should be related to financial risks of participating insurance contracts. We investigate an optimal investment problem under a joint value-at-risk and portfolio insurance constraint faced by the insurer who offers participating contracts. The insurer aims to maximize the expected utility of the terminal payoff to the insurer. We adopt a concavification technique and a Lagrange dual method to solve the problem and derive the representations of the optimal wealth process and trading strategies. We also carry out some numerical analysis to show how the joint value-at-risk and the portfolio insurance constraint impacts the optimal terminal wealth.     

Explicit solutions to some single-period investment problems for risky log-stable stocks

Numerical approximations are presented for the expected utility of wealth over a single time period for a small investor who proportions her or his available capital between a risk-free asset and a risky stock. The stock price is assumed to be a log-stable random variable. The utility functional is logarithmic or isoeleastic ($\text{y}{}^{\mathrm{a}}\text{q}$, q < 0). Analytic results are presented for special choices of model parameters, and for large and small time periods.     

The relationship between expected utility and higher moments for distributions captured by the Gram–Charlier class

In this letter we derive the closed form solution for expected utility in terms of higher moments of the distribution of wealth when expected utility takes the CARA form and the distribution of wealth is captured by the Gram–Charlier class of distributions. We derive the condition under which positive skewness can be associated with a decrease in expected utility.     

Portfolio optimisation under non-linear drawdown constraints in a semimartingale financial model

A drawdown constraint forces the current wealth to remain above a given function of its maximum to date. We consider the portfolio optimisation problem of maximising the long-term growth rate of the expected utility of wealth subject to a drawdown constraint, as in the original setup of Grossman and Zhou (Math. Finance 3:241–276, 1993). We work in an abstract semimartingale financial market model with a general class of utility functions and drawdown constraints. We solve the problem by showing that it is in fact equivalent to an unconstrained problem with a suitably modified utility function. Both the value function and the optimal investment policy for the drawdown problem are given explicitly in terms of their counterparts in the unconstrained problem.     

Risk-Adjusted Economic Value Analysis

Abstract

The ability of commonly used profitability measures to reflect risk exposure appropriately is evaluated and found lacking. As an alternative, a modern portfolio theory approach, based on utility theory, is recommended. Generalized formulas for calculating risk-adjusted economic values by deriving risk adjustments from certainty equivalents are developed by using the Markowitz expected utility maxim. Practical applications are described. Where appropriate, simplifying assumptions are shown to result in closed-form solutions, thereby reducing the need for extensive, stochastic cashflow simulations. The resulting formulas can be used to measure financial performance on a risk-adjusted basis consistently across different lines of business or to evaluate risk exposures in strategic alternatives.     

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.     

Time to wealth goals in capital accumulation

This paper considers the problem of investment of capital in risky assets in a dynamic capital market in continuous time. The model controls risk, and in particular the risk associated with errors in the estimation of asset returns. The framework for investment risk is a geometric Brownian motion model for asset prices, with random rates of return. The information filtration process and the capital allocation decisions are considered separately. The filtration is based on a Bayesian model for asset prices, and an (empirical) Bayes estimator for current price dynamics is developed from the price history. Given the conditional price dynamics, investors allocate wealth to achieve their financial goals efficiently over time. The price updating and wealth reallocations occur when control limits on the wealth process are attained. A Bayesian fractional Kelly strategy is optimal at each rebalancing, assuming that the risky assets are jointly lognormal distributed. The strategy minimizes the expected time to the upper wealth limit while maintaining a high probability of reaching that goal before falling to a lower wealth limit. The fractional Kelly strategy is a blend of the log-optimal portfolio and cash and is equivalently represented by a negative power utility function, under the multivariate lognormal distribution assumption. By rebalancing when control limits are reached, the wealth goals approach provides greater control over downside risk and upside growth. The wealth goals approach with random rebalancing times is compared to the expected utility approach with fixed rebalancing times in an asset allocation problem involving stocks, bonds, and cash.     

Stability of utility maximization in nonequivalent markets

Stability of the utility maximization problem with random endowment and indifference prices is studied for a sequence of financial markets in an incomplete Brownian setting. Our novelty lies in the nonequivalence of markets, in which the volatility of asset prices (as well as the drift) varies. Degeneracies arise from the presence of nonequivalence. In the positive real line utility framework, a counterexample is presented showing that the expected utility maximization problem can be unstable. A positive stability result is proved for utility functions on the entire real line.     

Fallacy of the log-normal approximation to optimal portfolio decision-making over many periods

The fallacy that a many-period expected-utility maximizer should maximize (a) the expected logarithm of portfolio outcomes or (b) the expected average compound return of his portfolio is now understood to rest upon a fallacious use of the Law of Large Numbers. This paper exposes a more subtle fallacy based upon a fallacious use of the Central-Limit Theorem. While the properly normalized product of independent random variables does asymptotically approach a log-normal distribution under proper assumptions, it involves a fallacious manipulation of double limits to infer from this that a maximizer of expected utility after many periods will get a useful approximation to his optimal policy by calculating an efficiency frontier based upon (a) the expected log of wealth outcomes and its variance or (b) the expected average compound return and its variance. Expected utilities calculated from the surrogate log-normal function differ systematically from the correct expected utilities calculated from the true probability distribution. A new concept of ‘initial wealth equivalent’ provides a transitive ordering of portfolios that illuminates commonly held confusions. A non-fallacious application of the log-normal limit and its associated mean-variance efficiency frontier is established for a limit where any fixed horizon period is subdivided into ever more independent sub-intervals. Strong mutual-fund Separation Theorems are then shown to be asymptotically valid.     

Job search,unemployment and savings

This paper examines optimal search policies on the basis of two alternative assumptions, first that individuals consume income as it is received, and second, that individuals can save and subsequently draw from their savings. In this model workers search for heterogeneousjobs, which are characterized by fixed wages and random duration of employment spells. Workers choose jobs which will maximize the total expected lifetime utility of consumption. The optimal steady-state job acceptance policy in both cases takes the form of a fixed partition of the set of all job offers into acceptable and unacceptable ones. In the absence of a capital market, employment duration appears to be irrelevant for the marginal job offer and all jobs offering wages which exceed the marginal one are also acceptable independent of the distribution functions of employment duration. Nevertheless, the dispersion of employment duration at inframarginal acceptable jobs affects job choice. It is shown that when individuals can save, and if current utility and probability density functions are exponential, then steady-state rates of saving during periods of employment, and of dissaving during periods of unemployment, are independent of wealth and constant. The model developed is then used to examine the determinants of unemployment and the properties of optimal savings rates, and to obtain a number of testable hypotheses about savings behavior and job security.     

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.