On market value margins and cost of capital

The valuation of insurance liabilities plays a central role in the design of any solvency framework. We investigate the notion of “fair value of liabilities” at a conceptional level and compare several implementations which are currently discussed in the Solvency II project. Our focus is on the cost of capital approach based on market information. In particular, we discuss the applicability of arguments borrowed from financial mathematics.     

Biometric worst-case scenarios for multi-state life insurance policies

It is common actuarial practice to calculate premiums and reserves under a set of biometric assumptions that represent a worst-case scenario for the insurer. The new solvency regime of the European Union (Solvency II) also uses worst-case scenarios for the calculation of solvency capital requirements for life insurance business. Surprisingly, the actuarial literature so far offers no exact method for the construction of biometric scenarios that let premiums and reserves be always on the safe side with respect to a given confidence band for the biometric second-order basis. The present paper partly fills this gap by introducing a general method that allows one to construct such scenarios for homogenous portfolios of life insurance policies. The results are especially informative for life insurance policies with mixed character (e.g. survival and occurrence character). Two examples are given that illustrate the new method, demonstrate its usefulness for the calculation of premiums and reserves, and show how the new approach could improve the calculation of biometric solvency reserves for Solvency II.     

Hannover Morbiditäts- und Mortalitäts-Pflegestudie (HMMPS): Langzeitverläufe, Pflegestufenübergänge und Reaktivierungen in der gesetzlichen Pflegeversicherung

After determining the initial care level corresponding to the regulations of the German Institutional Long-Term Care Insurance, the care level may change over the course of time. Care assistance may even be cancelled owing to a reduction in necessity for care or due to the death of the recipient. From a cohort of 88,575 long term care patients of the statutory health insurance ‘‘Deutsche BKK’’ this report describes the probabilities for a change or loss of care level and death with regard to age, gender and the length of care in each level. This study thus provides a basis for actuarial calculations of long-term care insurance based on empirical data.     

Assessing the cost of capital for longevity risk

The cost of capital is a key element of the embedded value methodology for the valuation of a life business. Further, under some solvency approaches (in particular, the Swiss Solvency Test and the developing Solvency 2 project) assessing the cost of capital constitutes a step in determining the required capital allocation.Whilst the cost of capital is usually meant as a reward for the risks encumbering a given life portfolio, in actuarial practice the relevant parameter has been traditionally chosen, at least to some extent, inconsistently with such risks. The adoption of market-consistent valuations has then been advocated to reach a common standard.A market-consistent value usually acknowledges a reward to shareholders’ capital as long as the market does, namely if the risk is systematic or undiversifiable. When dealing with a life annuity portfolio (or a pension plan), an important example of systematic risk is provided by the longevity risk, i.e. the risk of systematic deviations from the forecasted mortality trend. Hence, a market-consistent approach should provide appropriate valuation tools.In this paper we refer to a portfolio of immediate life annuities and we focus on longevity risk. Our purpose is to design a framework for a valuation of the portfolio which is market-consistent, and therefore based on a risk-neutral argument, while involving some of the basic items of a traditional valuation, viz best estimate future flows and allocated capital. This way, we try to reconcile the traditional with a market-consistent (or risk-neutral) approach. This allows us, in particular, to translate the results obtained under the risk-neutral approach in terms of a properly redefined embedded value.     

On the lifetime and one-year views of reserve risk,with application to IFRS 17 and Solvency II risk margins

This paper brings together analytic and simulation-based approaches to reserve risk in general (P&C) insurance, applied to the traditional actuarial view of risk over the lifetime of the liabilities and to the one-year view of Solvency II. It also connects the lifetime and one-year views of risk. The framework of the model in Mack (1993) is used throughout, although the results have wider applicability.The advantages of a simulation-based approach are highlighted, giving a full predictive distribution, which is used to estimate risk margins under Solvency II and risk adjustments under IFRS 17. We discuss methods for obtaining capital requirements in a cost-of-capital risk margin, and methods for estimating risk adjustments using risk measures applied to a simulated distribution of the outstanding liabilities over their lifetime.     

Characterization of Two-Scale Gradient Young Measures and Application to Homogenization

This work is devoted to the study of two-scale gradient Young measures naturally arising in nonlinear elasticity homogenization problems. Precisely, a characterization of this class of measures is derived and an integral representation formula for homogenized energies, whose integrands satisfy very weak regularity assumptions, is obtained in terms of two-scale gradient Young measures.     

Inside the Solvency 2 Black Box: Net Asset Values and Solvency Capital Requirements with a least-squares Monte-Carlo approach

The calculation of Net Asset Values and Solvency Capital Requirements in a Solvency 2 context–and the derivation of sensitivity analyses with respect to the main financial and actuarial risk drivers–is a complex procedure at the level of a real company, where it is illusory to be able to rely on closed-form formulas. The most general approach to performing these computations is that of nested simulations. However, this method is also hardly realistic because of its huge computation resources demand. The least-squares Monte Carlo method has recently been suggested as a way to overcome these difficulties. The present paper confirms that using this method is indeed relevant for Solvency 2 computations at the level of a company.     

Longevity risk,cost of capital and hedging for life insurers under Solvency II

The cost of capital is an important factor determining the premiums charged by life insurers issuing life annuities. This capital cost can be reduced by hedging longevity risk with longevity swaps, a form of reinsurance. We assess the costs of longevity risk management using indemnity based longevity swaps compared to costs of holding capital under Solvency II. We show that, using a reasonable market price of longevity risk, the market cost of hedging longevity risk for earlier ages is lower than the cost of capital required under Solvency II. Longevity swaps covering higher ages, around 90 and above, have higher market hedging costs than the saving in the cost of regulatory capital. The Solvency II capital regulations for longevity risk generates an incentive for life insurers to hold longevity tail risk on their own balance sheets, rather than transferring this to the reinsurance or the capital markets. This aspect of the Solvency II capital requirements is not well understood and raises important policy issues for the management of longevity risk.     

The Advanced Australian Shuffle

The “classical” Australian under-down shuffle starts with a deck of n cards. Then one proceeds as follows: one card under the deck, one on the table, one under the deck, one on the table, etc. One continues until only one card remains. There is an explicit formula to calculate the number of the card in the original deck that survives, and this is the basis of several mathematical magic tricks.     

Theoretical solution versus industry standard: Optimal leverage function for CPDOs

In this paper, we derive an optimal leverage function for Constant Proportion Debt Obligations (CPDOs) by using stochastic control techniques. The investor’s goal is to maximise redemption of capital at maturity. The control variable of the problem is the leverage process, i.e. the time dependent notional exposure to the underlying risky index/portfolio. The control problem is solved explicitly with the help of the Legendre transform applied to the HJB equation of stochastic control. A closed form solution is given for the optimal leverage. Contrary to the industry practise, the optimal leverage derived in this paper is a non-linear, bell-shaped function of the CPDO assets value.     

Two-level value function approach to non-smooth optimistic and pessimistic bilevel programs

ABSTRACT

The authors' paper in Dempe et al. [Necessary optimality conditions in pessimistic bilevel programming. Optimization. 2014;63:505–533], was the first one to provide detailed optimality conditions for pessimistic bilevel optimization. The results there were based on the concept of the two-level optimal value function introduced and analysed in Dempe et al. [Sensitivity analysis for two-level value functions with applications to bilevel programming. SIAM J. Optim. 22 (2012), 1309–1343], for the case of optimistic bilevel programs. One of the basic assumptions in both of these papers is that the functions involved in the problems are at least continuously differentiable. Motivated by the fact that many real-world applications of optimization involve functions that are non-differentiable at some points of their domain, the main goal of the current paper is to extend the two-level value function approach by deriving new necessary optimality conditions for both optimistic and pessimistic versions in bilevel programming with non-smooth data.     

The Solvency II square-root formula for systematic biometric risk

In this paper, we develop a model supporting the so-called square-root formula used in Solvency II to aggregate the modular life SCR. Describing the insurance policy by a Markov jump process, we can obtain expressions similar to the square-root formula in Solvency II by means of limited expansions around the best estimate. Numerical illustrations are given, based on German population data. Even if the square-root formula can be supported by theoretical considerations, it is shown that the QIS correlation matrix is highly questionable.     

A goal programming approach for developing pre-harvest forecasts of crop yield

In this paper, an attempt has been made to develop a pre-harvest forecast of sugarcane yield. The forecast is based on plant biometrical characteristics such as plant height, girth of cane, number of canes per plot and width of third leaf from the top. Some of these characteristics are correlated and hence the conventional practice of fitting a regression model using least square technique for estimating the parameters does not lead to satisfactory pre-harvest forecasts. Keeping in view the difficulties relating to the violation of the assumptions of normality, independence and homoscedasticity of classical multivariate regression analysis, the present study proposes an alternative approach, which is free from assumptions. It employs a goal programming formulation to estimate the pre-harvest yield of sugarcane on the basis of measurements on biometrical characters of the plant. In order to assess the quality of forecasts, variance of residuals obtained from the proposed method has been compared with that obtained from the conventional regression analysis. The study reveals that there is no significant difference (P-value=0.43461) in the variances of the two residual series. Thus, without compromising the quality of forecast, the proposed alternative methodology can be adopted to estimate the sugarcane yield 3 months before harvest in situations, where the assumptions of conventional regression analysis are violated.     

Dynamic risk measures under model uncertainty

In this paper we consider the notion of dynamic risk measures, which we will motivate as a reasonable tool in risk management. It is possible to reformulate an example of such a risk measure in terms of the value functions of a Markov decision model (MDM). Based on this observation the model is generalized to a setting with incomplete information about the risk distribution which can be seen as model uncertainty. This issue can be incorporated in the dynamic risk measure by extending the MDM to a Bayesian decision model. Moreover, it is possible to discuss the effect of model uncertainty on the risk measure in binomial models. All investigations are illustrated by a simple but useful coin tossing game proposed by Artzner and by the classic Cox–Ross–Rubinstein model.     

On finite completely simple semigroups having no finite basis of identities

The purpose of this note is to provide a correct proof of the fact that there exist finite completely simple semigroups having no finite basis of identities.     

Corporate valuation,capital structure and risk management: A stochastic DCF approach

In this article we discuss a general stochastic framework for designing corporate investment, financing and risk management strategies for financially constrained firms. The strategy entailing the highest benefits for shareholders is considered to be the optimal strategy. This paper focuses on a simulation of present value distributions of the capital positions of a company, explicitly taking into account the risk of fluctuations in future cash flow as well as the risk of insolvency. The present value distribution of equity is used as a central instrument for evaluation of shareholder benefits. Expected present values are also computed. The investment and financing policy of the company pursued at the time of the valuation is reflected in certain global model parameters, which themselves influence the future profit distribution policy of the company. The main parameters are the extent of debt, the annual debt funding requirements, the average earnings power of the company – expressed as an expected annual return on total capital – and the risk of annual earnings – expressed as the standard deviation of the annual return on total capital. An explicit illustration of the volatility risk and default risk seems not only to be a suitable way of illustrating the impact of capital structure on corporate value. Such an depiction may also provide answers to the question of the link between hedging and enterprise value. This paper highlights the fact that investment, finance and hedging strategies should go hand in hand.     

Kronecker’s density theorem and irrational numbers in constructive reverse mathematics

To prove Kronecker’s density theorem in Bishop-style constructive analysis one needs to define an irrational number as a real number that is bounded away from each rational number. In fact, once one understands “irrational” merely as “not rational”, then the theorem becomes equivalent to Markov’s principle. To see this we undertake a systematic classification, in the vein of constructive reverse mathematics, of logical combinations of “rational” and “irrational” as predicates of real numbers.     

Axioms for a theory of semantic equivalence

Gödel-type semantic completeness theorems are established for a formal theory of semantic equivalence based on L.A. Zadeh's concept of a linguistic variable. The linguistics that is employed allows for the expression of propositions such as “it is not the case that ‘young’ is semantically equivalent with ‘not old’”, or, in symbols (young(x) ≅ ∼old(x)).The result is a two-leveled semantics which, at the lower level, is a multivalent interpretation of fuzzy assertions (e.g., ∼old(x)) in terms of fuzzy subsets of a given universe and, at the upper level, is a two-valued (classical) interpretation based on the fact that two closed fuzzy assertions either do or do not have the same truth value. The main proof is of the Henkin variety, employing adaptations of the algebraic methods found in Rasiowa [9] and Rasiowa and Sikorski [10].     

Eta products of weight one,and decomposition of primes in non-abelian extensions

This article is a slightly extended and revised version of a conference talk at “Arithmetik an der A7” in Würzburg, June 23rd, 2017. We present a conjecture on the coincidence of Hecke theta series of weight 1 on three distinct quadratic fields. Then we discuss a special instance of the Deligne–Serre Theorem, implying that the decomposition of prime numbers in a certain extension of the rationals is governed by the coefficients of the eta product \(\eta^{2}(z)\).     

Parity in partition identities

This paper considers a variety of parity questions connected with classical partition identities of Euler, Rogers, Ramanujan and Gordon. We begin by restricting the partitions in the Rogers-Ramanujan-Gordon identities to those wherein even parts appear an even number of times. We then take up questions involving sequences of alternating parity in the parts of partitions. This latter study leads to: (1) a bi-basic q-binomial theorem and q-binomial series, (2) a new interpretation of the Rogers-Ramanujan identities, and (3) a new natural interpretation of the fifth-order mock theta functions f ₀(q) along with a new proof of the Hecke-type series representation.