Variable annuities are life insurance products which are particularly attractive in a low interest rate environment. They allow the policyholder to participate in the growth of the economy through the performance of a reference fund, guaranteeing at the same time a minimum payment (GMAB) based on a fixed rate. A hybrid approach driven by time-inhomogeneous Lévy processes [1] is used to model the joint behavior of the underlying reference fund and the interest rate market. In addition to the choice at maturity between a guaranteed sum and the performance of a fund, the design of the contract incorporates a payment in case of death of the policyholder and surrender provisions. Following [2], the rate at which contracts are cancelled is modelled by a surrender intensity which depends on the performance of the reference fund, as well as the guaranteed rate, the interest rate prevailing at the time point of cancellation, and the surrender penalty function. Two versions of the surrender intensity are studied for valuation of the contract.

[1] Eberlein E. and Rudmann M. (2018), “Hybrid Lévy models: Design and computational aspects.” Applied Mathematical Finance, DOI: 10.1080/1350486X.2018.1536523.

[2] Escobar M., Krayzler M., Ramsauer F., Saunders D. and Zagst R. (2016), “Incorporation of stochastic policyholder behavior in analytical pricing of GMABs and GMDBs.” Risks, vol. 4(4), pp. 1-36.

]]>We discuss a portfolio management problem of the rational policyholder of a variable annuity (VA) with maturity guarantee who aims to maximize the utility of her terminal wealth. We consider a VA contract which allows the policyholder to modify her investment mix throughout the contract. This problem is formulated in terms of constrained optimal stochastic controland requires the maximization of a non-concave utility function. We solve the problem using a martingale approach and compare with existing results. In particular, we show that there exist different ways to set the guarantee fee, which impacts the policyholder's optimal investment strategy and the resulting cost to the insurer.

]]>The eastern states of Australia are supplied with electricity from the National Electricity Market, which is a grid that stores electricity generated by a variety of means. Electrical power blackouts occur when there is a disruption to supply to the grid from a power generator or when consumer demand exceeds supply plus any reserves. Employing data from the Australian Energy Market Operator (AEMO), a valuation method is proposed for power blackout insurance, which could be used by large-scale industrial users which seek compensation or by the AEMO when asking such users to power down during hours of peak demand.

]]>Annuities constitute a fundamental branch in the life insurance business. To obtain the cost of annuities, Financial and demographic factors are the key inputs for the related mathematical quantitative models. In this work [1], we adopt and compare different approaches to gain insights on the behavior of annuities, their structure and on the importance of their input factors. In particular, the sensitivity analysis is carried out at different location scales [2]. Moreover, we conduct the uncertainty quantification of the cost of annuities in case of dependence among inputs. Our results show that the contribution of financial risk to the cost of annuities plays a major role than the one of mortality risk, confirming with alternative methods the findings of [3].

[1] Borgonovo, E. and Rabitti, G. (2019), “Sensitivity Analysis of Annuity Models.” Working paper.

[2] Borgonovo, E. (2017), “Sensitivity Analysis. An Introduction for the Management Scientist.” Cham: Springer.

[3] Karabey, U., Kleinow, T., Cairns, A. J. G. (2014), “Factor risk quantification in annuity models.” Insurence: Mathematics and Economics, vol. 58, pp. 34-45.

]]>In this talk, I will discuss the economic approaches to evaluate the social cost of carbon, i.e., the present value of the flow of climate damages generated in the next few centuries by one more ton of CO2 emitted today. What discount rates should we use to perform this task? What is the risk profile of climate damages? I will combine standard asset pricing models and integrated assessment models to measure the impact of carbon emissions on intergenerational welfare, using the principles of utilitarian ethics.

]]>We investigate fair (market-consistent and actuarial) valuation of insurance liability cash-flow streams in continuous time. We first consider one-period hedge-based valuations, where in the first step, an optimal dynamic hedge for the liability is set up, based on the assets traded in the market and a quadratic hedging objective, while in the second step, the remaining part of the claim is valuated via an actuarial valuation. Then, we extend this approach to a multi-period setting by backward iterations for a given discrete-time step h, and consider the continuous-time limit for h à 0. We derive a partial differential equation for the valuation operator which satisfies the continuous-time limit of the multi-period, discrete-time iterations and prove that this valuation operator is actuarial and market consistent. We show that our continuous-time fair valuation operator has a natural decomposition into the best estimate of the liability and a risk margin. The dynamic hedging strategy associated with the continuous-time fair valuation operator is also established. Finally, the valuation operator and the hedging strategy allow us to study the dynamics of the net asset value of the insurer. Our results can be useful for understanding the valuation concepts in Solvency II and IFRS 17.

]]>Insurance premium principle includes the expected loss plus a risk loading factor to cover the loss from adverse claim experience and generate a profit. This paper considers a stochastic differential game with multiple insurers who are competing to sell insurance contracts by controlling their insurance premium. The existing works on this competitive premium problem mainly follow the development of retail pricing models, but fail to consider the randomness of payoffs by selling insurance contracts. The present paper aims to fill in this gap with the large body of literature on insurance surplus process modeling. Specifically, we model the surplus per policy unit by the diffusion approximation to the classical Cramér-Lundberg model. The risk exposure of an insurer (i.e., the number of policies) is assumed to be impacted (linearly) by all insurers in the market. Closed-form Nash equilibrium premium strategies are solved for insurers who are aiming to maximize their expected terminal exponential utilities. To investigate the robustness of equilibrium premium strategies, we further allow insurers to perceive different levels of ambiguity towards the underlying aggregate claim amount process. Closed-form expression for such robust premium strategies are obtained as well, and comparative statics analysis for the model parameters is implemented.

]]>Risk measurement models for financial institutions typically focus on the net portfolio position and thus ignore distinctions between 1) assets and liabilities and 2) uncollateralized and collateralized liabilities. However, these distinctions are economically important. Liability risks affect the total amount of claims on the institution, while asset risks affect the amount available for claimants. Collateralization also affects the amounts recovered by different classes of claimants. We analyze a model of a financial institution with risky assets and liabilities, with potentially varying levels of collateralization across liabilities, showing that correct economic risk capital allocation requires complete segregation of asset, uncollateralized liability, and collateralized liability risks, with different risk measures for each. Our numerical analyses suggest that the conventional approach frequently yields over-investment in risky assets.

]]>This paper studies an equilibrium model between an insurance buyer and an insurance seller, where both parties' risk preferences are given by convex risk measures. The interaction is modeled through a Stackelberg type game, where the insurance seller plays rst by oering prices, in the form of safety loadings. Then the insurance buyer chooses his optimal proportional insurance share and his optimal prevention eort in order to minimize his risk measure. The loss distribution is given by a family of stochastically ordered probability measures, indexed by the prevention eort. We give special attention

to the problems of self-insurance and self-protection. We prove that the formulated game admits a unique equilibrium, that we can explicitly solve by further specifying the agents criteria and the loss distribution. In self-insurance, we consider also an adverse selection setting, where the type of the insurance buyers is given by his loss probability, and study the screening and shutdown contracts. Finally, we provide case studies in which we explicitly apply our theoretical results.

Prima facie, valuing a longevity-contingent claim that provides guaranteed income-for-life should be a relatively straight forward operation. One selects a mortality basis with a proper discount curve and the remainder is left to expectations. And yet, history is strewn with examples of mispriced annuities. From early attempts by the English King Henry VIII to securitize cash flows from the dissolution of the monasteries in the 16th century, to European tontines in the 17th and 18th centuries and of course North American variable annuity companies who teetered on the precipice of bankruptcy in the early 21st century, it seems annuities can be a very tricky business. Motivated by these examples and recent controversies, in this talk I will review the pricing of plain, compound and decorative annuities, discuss the economic rationale for their existence and conclude with some observations on why Mother Nature would prefer that retirees pool longevity risk.

]]>In our recently submitted paper we employ a lifecycle model that uses utility of consumption and bequest to determine an optimal Deferred Income Annuity (DIA) purchase policy. We lay out a mathematical framework to formalize the optimization process. The method and implementation of the optimization is explained, and the results are then analyzed. We extend our model to control for asset allocation and show how the purchase policy changes when one is allowed to vary asset allocation. Our results indicate that (i.) refundable DIAs are less appealing than non-refundable DIAs because of the loss of mortality credits; (ii.) the DIA allocation region is larger under the fixed asset allocation strategy due to it becoming a proxy for fixed-income allocation; and (iii.) when the investor is allowed to change asset-allocation, DIA allocation becomes less appealing. However, a case for higher DIA allocation can be made for those individuals who perceive their longevity to be higher than the population.

[1] Habib, F., Huang, H., Mauskopf, A., Nikolic, B. and Salisbury, T.S. (2018), “Optimal Allocation to Deferred Income Annuities” Insurance Mathematics and Economics, submitted

]]>In this paper, we investigate a class of robust non-zero-sum reinsurance-investment stochastic differential games between two competing insurers under the time-consistent mean-variance criterion. We allow each insurer to purchase a proportional reinsurance treaty and invest his surplus into a financial market consisting of one risk-free asset and one risky asset to manage his insurance risk. The surplus processes of both insurers are governed by the classical Cramér-Lundberg model and each insurer is an ambiguity-averse insurer (AAI) who concerns about model uncertainty. The objective of each insurer is to maximize the expected terminal surplus relative to that of his competitor and minimize the variance of this relative terminal surplus under the worst-case scenario of alternative measures. Applying stochastic control theory, we obtain the extended Hamilton-Jacobi-Bellman (HJB) equations for both insurers. We establish the robust equilibrium reinsurance-investment strategies and the corresponding equilibrium value functions of both insurers by solving the extended HJB equations under both the compound Poisson risk model and its diffusion-approximated model. Finally, we conduct some numerical examples to illustrate the effects of several model parameters on the Nash equilibrium strategies.

]]>We focus on the initiation option featured in many Guaranteed Lifelong Withdrawal Benefit variable annuity contracts, granting their owner the right to decide the age at which lifetime withdrawals should begin. Such contracts have been successfully analysed using a PDE approach, see Huang et al. (IME, 56(2014), 102-111). While the latter method is elegant, it becomes less viable when the valuation model is more involved and other guarantees are considered. We exploit the Least Square Monte Carlo method and explore the interaction of the initiation option with lapses and other riders, and the effect of stochastic volatility, interest rates and mortality.

]]>In a previous article published in Insurance: Mathematics and Economics [2], we explored a multiperiod cost-of-capital valuation of a liability cashflow subject to repeated capital requirements. The resulting liability value is given by a backward recursion, mathematically similar to that of dynamic risk measures described in for instance [1]. In a Markovian setting, the value at each time is given by some function of the state variables. Evaluating this function is a problem that suffers from the curse of dimensionality: If the state space is too high-dimensional, direct calculation becomes unfeasible, even when allowing for discretization of continuous state variables. To tackle this problem, we seek to approximate the value as a function of state variables via a linear regression on basis functions in each step of the backward recursion. This approach is somewhat famously applied in [3] for the purpose of valuing complex American options, where this method is referred to as least squares Monte Carlo. In this talk, based on ongoing research, we explore the use of least squares Monte Carlo approach to calculate the cost-of-capital value described in [2], where we will provide some convergence results as well as numerical results.

[1] Artzner, P., Delbaen, F., Eber, J.-M., Heath, D. and Ku, H. (2007), “Coherent multiperiod risk adjusted values and Bellman's principle.” Annals of Operations Research, 152, 5-22.

[2] Engsner, H., Lindholm, M. and Lindskog, F. (2017), “Insurance valuation: A computable multiperiod cost-of-capital approach.” Insurance Mathematics and Economics, 72, 250-264.

[3] Longsta_, F.A. and Schwartz, E.S. (2001), “Valuing American Options by Simulation: A Simple Least-Squares Approach.” The Review of Financial Studies, 14 (1),113-147.

]]>Combining the best of drawdown and annuity, the investment returns and the longevity credits, tontines offer a great alternative to current pension products. Tontines are well-understood under the assumption of constant market returns and a perfect pool. However the literature lacks in understanding of the impact of fluctuating market returns and realized mortality rates. We present a numerical study that analyzes how many members a tontine need in order to deliver the outcome that idealized results suggest. We assume a heterogeneous member profile, an explicit tontine scheme, and a Black-Scholes market. We give a practical answer that shall be used in the future for pension products in the UK market. In the end, we compare our result with the 1000-member rule by Qiao and Sherris for Group-Self-Annuitization-Schemes from 2013.

[1] Qiao, C. and Sherris, M. (2013), “Managing Systematic Mortality Risk with Group Self-Pooling and Annuitization Schemes.” Journal of Risk and Insurance, vol. 80(4), pp. 949_974.

]]>This paper studies the portfolio management problem for an individual with a non-exponential discount function and habit formation in finite time. The individual receives a deterministic income, invests in risky assets, and consumes and buys insurance continuously. The utility of the individual comes from excessive consumption, legacy and terminal wealth. The non-exponential discounting makes the optimal strategy taken by a naive person time inconsistent which means traditional optimal control strategy will change from time to time. Nash subgame perfect theory is used to derive the equilibrium strategy for a sophisticated person, therein being time consistent. We derive both a time-inconsistent strategy for the naive person and a time-consistent strategy for a sophisticated person by the method introduced in [5]. We calculate the analytical solution under the naive strategy and the equilibrium strategy in the CRRA case, and compare the results of the two strategies. By numerical simulation, we find that the sophisticated individual will spend less for consumption and insurance and save more compared with the naive person. The difference in strategies of the naive and sophisticated person decreases over time. In addition, if the individual of either type is more patient in the future or has a higher tendency to form a habit, he will consume less and buy less insurance, and the difference between two strategies will also be increased. The sophisticated person's consumption and habit level are initially lower than that of a naive person but are higher at later times.

[1] Detemple, J.B., Zapatero, F. (1992) “Optimal consumption-portfolio policies with habit formation.” Mathematical Finance, 2(4), 251-274

[2] Ekeland I., Mbodji O., Pirvu T.A. “Time-Consistent Portfolio Management.” SIAM Journal of Financial Math, 2012, 3, 1-32

[3] Björk T., Murgci A., Zhou, X.Y. (2014) “Mean-variance portfolio optimization with state-dependent risk aversion.” Mathematical Finance, 24(1), 1-24

[4] Wei J.Q., Li D.P., Zeng Y., Yang H.L.(2018) “Robust optimal consumption-investment strategy with non-exponential discounting.” Journal of Industrial And Management Optimization, In press

[5] Björk T., Khapko M., Murgoci A. “On time-inconsistent stochastic control in continuous time (2017)” Finance Stoch, 21, 331-360

]]>This paper sets out a market-consistent valuation methodology for insurance liabilities with nonreplicable cash flows. An explicit allowance for risks associated with such cash flows is common in modern accounting standards, as well as statutory solvency regimes. The paper also seeks to justify such an allowance from an agency perspective, in the interest of consumer protection. We aim to show that the agency perspective can provide clearer guidance as to the purpose of an explicit buffer for non-replicable cash flows, as well as how it may be valued. We define the market-consistent value of a liability cash flow in discrete time subject to repeated capital requirements in accordance with the principles of the Solvency II regulation by considering a hypothetical transfer of the liability to an external insurer, similarly to what was considered in [4], [1] and [2]. We focus both on theoretical aspects of the valuation concerning market-consistency and the effects of filtrations, quantifying the information flows, and practical aspects such as computability for an insurer who is only able to simulate liability cash flows without being able to express the cash flow model analytically.

[1] Engsner, H., Lindholm, M. and Lindskog, F. (2017), “Insurance valuation: a computable multiperiod cost-of-capital approach.” Insurance: Mathematics and Economics, vol. 72, pp. 250-264.

[2] Engsner, H., Lindensjö, K. and Lindskog, F. (2018), “The value of a liability cash flow in discrete time subject to capital requirements.” ar Xiv:1808.03328v1.

[3] Engsner, H., Lindskog, F. and Skrutkowski, M. (2019), “Valuation of insurance liability cash flows from an agency perspective.” Current work.

[4] Möhr, C. (2011) “Market-consistent valuation of insurance liabilities by cost of capital.” ASTIN Bulletin, vol. 41, pp. 315-341.

]]>We show that a law-invariant pricing functional defined on a general Orlicz space is typically incompatible with frictionless risky assets in the sense that one and only one of the following two alternatives hold: either every risky payoff has a strictly-positive bid-ask spread, or the pricing functional is given by an expectation and hence every payoff has zero bid-ask spread. In doing so we extend and unify a variety of “collapse to the mean” results from the literature such as those obtained in [1] and [3], and we highlight the key role played by law invariance in determining such a collapse. As a byproduct, we derive several applications to law-invariant acceptance sets, risk measures and Schur-convex functionals. Finally, we discuss the relevance of these results in the framework of market-consistent pricing of insurances, as outlined in [2].

[1] Castagnoli, E., Maccheroni, F. and Marinacci, M. (2004), “Choquet insurance pricing: a caveat.” Mathematical Finance, vol. 14 (3), pp. 481-485.

[2] Dhaene, J., Stassen, B., Barigou, K., Linders, D. and Chen, Z. (2017), “Fair valuation of insurance liabilities: Merging actuarial judgement and market-consistency” Insurance: Mathematics and Economics, vol. 78, pp. 14-27.

[3] Frittelli, M. and Rosazza Gianin, E. (2005), “Law invariant convex risk measures.” Advances in Mathematical Economics, vol. 7, pp. 33-46.

]]>Phase-type (PH) distributions are defined as distributions of lifetimes of finite continuous-time Markov processes. Their traditional applications are in queueing, insurance risk, and reliability, but more recently, also in finance and, though to a lesser extent, to life and health insurance. The advantage is that PH distributions form a dense class and that problems having explicit solutions for exponential distributions typically become computationally tractable under PH assumptions. The fitting of PH distributions to human lifetimes is considered, and some new software is developed. The pricing of life insurance products such as guaranteed minimum death benefit and high-water benefit is treated for the case where the lifetime distribution is approximated by a PH distribution and the underlying asset price process is described by a jump diffusion with PH jumps. The expressions are typically explicit in terms of matrix-exponentials involving two matrices closely related to the Wiener-Hopf factorization, for which recently, a Lévy process version has been developed for a PH horizon. The computational power of the method of the approach is illustrated via a number of numerical examples.

[1] Asmussen, S., Laub, P.J. and Yang, H. (2019), “Phase-Type Models in Life Insurance: Fitting and Valuation of Equity-Linked Benefits.” Risks, vol. 7(1), 17, pp. 1-22.

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