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We solve the problem of an investor who maximizes utility but faces random preferences. We propose a problem formulation based on expected certainty equivalents. We tackle the time-consistency issues arising from that formulation by applying the equilibrium theory approach. To this end, we provide the proper definitions and prove a rigorous verification theorem. We complete the calculations for the cases of power and exponential utility. For power utility, we illustrate in a numerical example that the equilibrium stock proportion is independent of wealth, but decreasing in time, which we also supplement by a theoretical discussion. For exponential utility, the usual constant absolute risk aversion is replaced by its expectation.

We formalize a consumption–investment–insurance problem with the distinction of a state-dependent relative risk aversion. The state dependence refers to the state of the finite state Markov chain that also formalizes insurable risks such as health and lifetime uncertainty. We derive and analyze the implicit solution to the problem, compare it with special cases in the literature, and illustrate the range of results in a disability model where the relative risk aversion is preserved, decreases, or increases upon disability.

Optimal timing of retirement is an important part of retirement planning. We consider three types of individuals distinguished by the way they use information when deciding the retirement time. For each of these types, we analyze two elements influencing the decision, the market model and the mortality model, and we study the impact of working with one combination or another. Based on analytical solutions to almost all the combinations, we reach a conclusion, even relevant for practical advice: Young individuals must prioritize the market model over the mortality model while for older individuals, it is the other way around.

This chapter discusses both the overall design of pension plans and the design of specific pension products found in the Danish occupational pension system. From the perspective of designing a pension plan, or indeed a national pension system, the sponsors and the funding structure are the two most important top-level design choices. In Denmark, the occupational pension system was structured as a nonsponsored, defined contribution system centered on pension funds and life insurance companies. After introductory remarks in Section 4.1, Section 4.2 describes these design choices and how they affect benefit characteristics. Section 4.3 investigates optimal consumption and investment decisions over a lifetime in a theoretical framework. The analysis shows that a mandatory, defined contribution pension plan featuring low-cost annuities leads to significant welfare gains for procrastinators who are incapable of accumulating sufficient retirement savings on their own. Section 4.4 turns attention to a comparison of the theoretically optimal investment profiles with those found in marketed life-cycle products, while Section 4.5 discusses insurance of disability and death risk, which is often an integrated part of pension savings contracts.

In this paper we discuss the concept of the cost-of-capital (CoC) rate for an insurance company as an equilibrium in the economic triangle of policyholders, shareholders, and the regulator. This provides a possible rationalization and an economic foundation for a quantity that is widely used in practice but whose value is typically neither technically nor economically well justified. We show how it can be well founded in such a triangular equilibrium. Under a simple one-period model and a valuation procedure of a two-price economy for illiquid assets we provide a corresponding economic-theoretical quantification for the CoC rate. The resulting rates are illustrated by a number of concrete numerical examples., Journal of Risk and Insurance, 89 (4), ISSN:0022-4367, ISSN:1539-6975

The idea of forward rates stems from interest rate theory. It has natural connotations to transition rates in multi-state models. The generalization from the forward mortality rate in a survival model to multi-state models is non-trivial and several definitions have been proposed. We establish a theoretical framework for the discussion of forward rates. Furthermore, we provide a novel definition with its own logic and merits and compare it with the proposals in the literature. The definition turns the Kolmogorov forward equations inside out by interchanging the transition probabilities with the transition intensities as the object to be calculated., Revision of manuscript. The manuscript now contains a section on 'Forward-thinking and actuarial practice'. Furthermore, we have corrected typos and re-written certain sentences to improve readability and accuracy

On December 18, 2017, Ragnar Norberg died at the age of 72 in London after a long illness. Ragnar was born in Stockholm on March 15, 1945, and obtained his academic degrees from the Faculty of Math...

We consider portfolio optimization problems with expected loss constraints under the physical measure (Formula presented.) and the risk neutral measure (Formula presented.), respectively. Using Merton's portfolio as a benchmark portfolio, the optimal terminal wealth of the (Formula presented.) -risk constraint problem can be easily replicated with the standard delta hedging strategy. Motivated by this, we consider the (Formula presented.) -strategy fulfilling the (Formula presented.) -risk constraint and compare its solution with the true optimal solution of the (Formula presented.) -risk constraint problem. We show the existence and uniqueness of the optimal solution to the (Formula presented.) -strategy fulfilling the (Formula presented.) -risk constraint, and provide a tractable evaluation method. The (Formula presented.) -strategy fulfilling the (Formula presented.) -risk constraint is not only easier to implement with standard forwards and puts on a benchmark portfolio than the (Formula presented.) -risk constraint problem, but also easier to solve than either of the (Formula presented.) - or (Formula presented.) -risk constraint problem. The numerical test shows that the difference of the values of the two strategies (the (Formula presented.) -strategy fulfilling the (Formula presented.) -risk constraint and the optimal strategy solving the (Formula presented.) -risk constraint problem) is reasonably small.

We study a classical continuous-time consumption-investment problem of a power utility investor with deterministic labor income with the important feature that the consumption-investment process is...

We consider orderings between one-dimensional r.v.s X, Y (risks). An obvious example is a.s. ordering, X ≤a.s.Y . We shall, however, mainly be concerned with orderings which only involve the distributions, i.e., which are law invariant.

We consider an insurance policy issued at time 0 and terminating at a fixed finite time n (not necessarily an integer!). There is a finite set of states of the policy, \( \mathcal {J}= \{ 0,\ldots ,J \} \), and Z(t) denotes the state of the policy at time t ∈ [0, n].

In an abundance of settings, one will as a first modeling attempt assume independence of the r.v.s involved. This is mathematically very convenient but extremely dangerous: in many situations where a disastrous event occurred despite its risk being calculated to be so low that it could be neglected for any practical purpose, a back mirror analysis has revealed that the reason was precisely the simultaneous occurrence of events assumed to be independent when doing the risk calculation. The study of dependence and how it may be modeled is therefore crucial, and this is the topic of this chapter.

In most of the insurance models considered so far, strategies have been static: premiums, arrangements for premiums, reinsurance, dividends or investment, etc. have been fixed once and for all at t = 0.

Predicting the typical order of minima or maxima is important in a number of applications. In agriculture, we may want to say something about the maximal content of a pesticide in a series of samples to check if the series passes the standards.

In this section, we study two extensions to the general Markov framework studied in Chap. V. The two extensions draw to some extent on the same mathematical generalization, but arise from two practically very different sources. One extension is an extension to duration-dependent intensities.

The last decades have seen the areas of insurance mathematics and mathematical finance coming closer together. One reason is the growing linking of pay-outs of life insurances and pension plans to the current value of financial products, another that certain financial products have been designed especially to be of interest for the insurance industry (see below). Nevertheless, some fundamental differences remain, and the present section aims at explaining some of these, with particular emphasis on the principles for pricing insurance products, resp. financial products.

A multi-state life insurance model is described naturally in terms of the intensity matrix of an underlying (time-inhomogeneous) Markov process which specifies the dynamics for the states of an insured person. Between and at transitions, benefits and premiums are paid, defining a payment process, and the technical reserve is defined as the present value of all future payments of the contract. Classical methods for finding the reserve and higher order moments involve the solution of certain differential equations (Thiele and Hattendorff, respectively). In this paper we present an alternative matrix-oriented approach based on general reward considerations for Markov jump processes. The matrix approach provides a general framework for effortlessly setting up general and even complex multi-state models, where moments of all orders are then expressed explicitly in terms of so-called product integrals of certain matrices. Thiele and Hattendorff type of theorems may be retrieved immediately from the matrix formulae. As a main application, methods for obtaining distributions and related properties of interest (e.g. quantiles or survival functions) of the future payments are presented from both a theoretical and practical point of view, employing Laplace transforms and methods involving orthogonal polynomials.

Let X = (X1, …, Xn) be a vector of r.v.s describing the outcome of a statistical experiment. For example, in the insurance context, n can be the number of persons insured for losses due to accidents in the previous year, and Xi the payment made to the ith.

In this chapter, we study the often encountered problem of assessing the order of magnitude of \(\Pi _n(x)={\mathbb P}(S_n>x)\) for some sequence X1, X2, … of r.v.s and Sn = X1 + ⋯ + Xn. In many applications, n is an r.v. N rather than deterministic, and the object of interest is then \(\Pi _N(x)={\mathbb P}(S_N>x)\).

In this chapter, we study products offered by insurance companies that, unlike the pure insurance policies in Chap. V, gives investors both insurance and investment under a single integrated plan.

Stochastic control first appeared in life insurance mathematics as an application of the so-called linear regulator. The idea of linear regulation was part of the origin of stochastic control theory developed in the 1950s and 1960s and has frequently been applied in engineering since then.

In most of the book, it has been assumed that a claim is paid out immediately to the insured. In practice, delays in reporting a claim and/or covering often arise, and the total claim amount may not be known at the time the claim occurs. One typical example is accident insurance, where the degree of disability is not a priori known and treatment takes place over a period.

In the classical portfolio optimization problem considered by Merton, the resulting constant proportion investment plan requires a diffusive trading strategy. This means that, within any arbitrarily small time interval, the investor must impractically both buy and sell stocks. We study the problems of a mean-square and a power utility investor for whom the trading strategy is constrained to be smooth, i.e. nondiffusive. This means that over sufficiently small time intervals, the investor is either a seller or a buyer of stocks. The mathematical framework is built around quadratic objectives such that trading activity is punished quadratically. Mean-square utility is quadratic, and power utility is covered by quadratic punishment of distance to Merton’s power utility portfolio. We present semi-explicit solutions and, in a series of numerical illustrations, show the impact of trading constraints on the portfolio decision over the investment horizon.

This article studies the consumption-investment-insurance problem of a family. The wage earner faces the risk of a health shock. The family can buy long-term life insurance that can only be revised at significant costs. A revision is only possible as long as the insured person is healthy. The combination of unspanned labor income and the stickiness of insurance decisions reduces the long-term insurance demand significantly. Since such a reduction is costly and families anticipate these potential costs, they buy less protection at all ages. In particular, young families stay away from long-term life insurance markets altogether.

We view the retail non-life insurance decision from the perspective of the insured. We formalize different consumption–insurance problems depending on the flexibility of the insurance contract. For exponential utility and power utility we find the optimal flexible insurance decision or insurance contract. For exponential utility we also find the optimal position in standard contracts that are less flexible and therefore, for certain nonlinear pricing rules, lead to a welfare loss for the individual insuree compared to the optimal flexible insurance decision. For the exponential loss distribution, we quantify a significant welfare loss. This calls for product development in the retail insurance business.

We investigate the conditions under which life-cycle investment strategies based on age may be ‘near enough’ to optimal, focusing on the treatment of the pension account balance and assumptions about risk aversion. We show that dynamically adjusting the strategy in response to fluctuations in balance as well as age can lead to moderate improvements over product designs currently seen in the market; although most of the potential gains might be captured by specifying the glide path with reference to measures reflecting the projected balance over time. The risk aversion assumption emerges as a far more important consideration, with much greater reductions in expected utility arising from mismatches between the risk aversion of the investor and that underpinning the glide path design. Our analysis suggests possibilities for improving life-cycle or target date funds, and highlights the benefit of offering a suite of such funds that cater for members with differing risk aversion.

We study the modelling and valuation of surrender and other behavioural options in life insurance and pension. We place ourselves in between the two extremes of optimal and arbitrary interventions by the policyholders. We present a model where one single parameter reflects the extent of rationality among policyholders. This presentation includes conditions which ensure that when the parameter goes to infinity contract values converge to the values corresponding to policyholders exhibiting optimal behaviour. When expenses are taken into account we lose the duality between the policyholder’s valuation of the contract and what we speak of as the market reserve. We include this in our model, and we give an upper bound for the difference between the value when the policyholder behaves optimally from her own point of view and the worst case market reserve from the pension fund point of view. In a series of numerical examples we illustrate the impact of the rationality parameter on the contract values.

In a classical Black–Scholes market, we establish a connection between two seemingly different approaches to continuous-time utility optimization. We study the optimal consumption, investment, and life insurance decision of an investor with power utility and an uncertain lifetime. To separate risk aversion from elasticity of inter-temporal substitution, we introduce certainty equivalents. We propose a time-inconsistent global optimization problem, and we present a verification theorem for an equilibrium control. In the special case without mortality risk, we discover that our optimization approach is equivalent to recursive utility optimization with Epstein–Zin preferences in the sense that the two approaches lead to the same result. We find this interesting since our optimization problem has an intuitive interpretation as a global maximization of certainty equivalents and since recursive utility, in contrast to our approach, gives rise to severe differentiability problems. Also, our optimization approach can there be seen as a generalization of recursive utility optimization with Epstein–Zin preferences to include mortality risk and life insurance.

In this paper, we consider the optimal dividend payment strategy for an insurance company that has two collaborating business lines. The surpluses of the business lines are modeled by diffusion processes. The collaboration between the two business lines permits that money can be transferred from one line to another with or without proportional transaction costs, while money must be transferred from one line to another to help both business lines keep running before simultaneous ruin of the two lines eventually occurs.

We derive worst-case scenarios in the case where the interest rate and the various transition intensities in a life insurance model are mutually dependent. Examples of this dependence are that surrender intensities and interest rates are high at the same time, that mortality intensities of a policyholder as active and disabled, respectively, are low at the same time, and that mortality intensities of the policyholders in a portfolio are low at the same time. The set from which the worst-case scenario is taken reflects the dependence structure and allows us to relate the worst-case scenario-based reserve, qualitatively, to a Value-at-Risk-based calculation of solvency capital requirements. This brings out perspectives for our results in relation to qualifying the standard formula of Solvency II or using a scenario-based approach in internal models. The fact that our worst-case scenario is deterministic and not adapted to the development of the portfolio makes the results particularly powerful for certain applications but also draws on methodological advancements. The formalistic results are exemplified in a series of numerical studies.