This paper investigates an operation mechanism for mutual aid platforms to develop more sustainably and profitably. A mutual aid platform is an online risk-sharing platform for risk-heterogeneous participants, and the platform extracts revenues by charging participants commission and subscription fees. A modeling framework is proposed to identify the optimal commissions and subscriptions for mutual aid platforms. Participants are divided into different types based on their loss probabilities and values derived from the platform. We present how these commissions and subscriptions should be set in a mutual aid plan to maximize the platform’s revenues. Our analysis emphasized the importance of accounting for risk heterogeneity in mutual aid platforms. Specifically, different types of participants should be charged different commissions/subscriptions depending on their loss probabilities and values on the platform. Participants’ shared costs should be determined based on their loss probabilities. Adverse selection occurs on the platform if participants with different risks pay the same shared costs. Our results also show that the platform’s maximum revenue will be lower if the platform charges the same fee to all participants. The numerical results of a practical example illustrate that the optimal commission/subscription scheme and risk-sharing rule result in considerable improvements in platform revenue over the current scheme implemented by the platform.

Modeling taxation of Variable Annuities has been frequently neglected, but accounting for it can significantly improve the explanation of the withdrawal dynamics and lead to a better modeling of the financial cost of these insurance products. The importance of including a model for taxation has first been observed by Moenig and Bauer (2016) while considering a Guaranteed Minimum Withdrawal Benefit (GMWB) Variable Annuity. In particular, they consider the simple Black–Scholes dynamics to describe the underlying security. Nevertheless, GMWB are long-term products, and thus accounting for stochastic interest rate has relevant effects on both the financial evaluation and the policyholder behavior, as observed by Goudenège et al. (2018). In this paper, we investigate the outcomes of these two elements together on GMWB evaluation. To this aim, we develop a numerical framework which allows one to efficiently compute the fair value of a policy. Numerical results show that accounting for both taxation and stochastic interest rate has a determinant impact on the withdrawal strategy and on the cost of GMWB contracts. In addition, it can explain why these products are so popular with people looking for a protected form of investment for retirement.

This paper addresses the Bayesian estimation of the shape parameter of Pareto distributions, and its application to premium calculation of large claims excess of loss (XL) reinsurance contracts. It studies the use of the generalized inverse Gaussian (GIG) as a Pareto prior conjugate, a family that contains as a particular case the gamma distribution. An exact credibility formula is deduced allowing the calculation of individual reinsurance premiums. These are premiums suited to the excesses history of a sole portfolio. A family of predictive distributions for the excesses is derived. We apply our exact credibility model to a sample of excesses arisen in ten Spanish portfolios of liability motor insurance from year 1992 to year 2001.

A great attention has been devoted, in the actuarial literature, to premium calculation principles and it has been often emphasized that these principles should not only be defined in strictly actuarial terms, but should also take into account the market conditions (Bühlmann (1980), de Jong (1981)).

In this paper we propose a decision model to define the pricing policy of an insurance company that operates in a market which is stratified in k risk classes .

It is assumed that any class constitutes a homogeneous collective containing independent risks Sj(t) of compound Poisson type, with the same intensity λj. The number nj of risks of that are held in the insurance portfolio depends on the premium charged to the class by means of a demand function which captures the concept of risk aversion and represents the fraction of individuals of , that insure themselves at the annual premium xj.

With these assumptions, the return Y on the portfolio is a function of the vector x = (x1, x2, …, xk) of the prices charged to the single classes (and of the time) and x is therefore the decision policy instrument adopted by the company for the selection of the portfolio, whose optimal composition is evaluated according to a risk-return type performance criterion.

As a measure of risk we adopt the ultimate ruin probability q(w) that, in the assumptions of our model, can be related to a safety index τ, by means of Lundberg-de Finetti inequality. Even though it has been widely debated in the actuarial field, the use of q(w) offers undeniable operational advantages. In our case the safety index τ can be expressed as a function of x and therefore, in the phase of selecting an efficient portfolio, it becomes the function to be maximized, for a given level M of the expected return.

In practical statistical work one frequently meets certain problems. For instance, we may have the following data about loss ratios in certain insurance companies and corresponding, numbers of insurance in force:

I assume further that we have no reason to believe that the companies, their loss ratios and their structure of insurances in force differ in any other way than by the size of companies. The problem is how to get quick estimates of mean losses and their variances in different companies?

A straitforward way to estimate the mean loss ratio would be to compute the usual mean of numbers pi, (Σpi/6) = 14; its standard deviation is 6,5. As this procedure of the “first statistician” seems to be too simple and naive, a “more cautious” statistician would compute the weighted mean loss ratio

The “more cautious” statistician would arque that his result is much better than the other result 14. But what would be the variance of the estimate 7,1, and what is the variance in the different companies?

The transition from defined benefit to defined contribution (DC) pension schemes has increased the interest in target annuitization funds that aim to fund a minimum level of retirement income. Prior literature has studied the optimal investment strategies for DC funds that provide minimum guarantees, but far less attention has been given to portfolio insurance strategies for DC pension funds focusing on retirement income targets. We evaluate the performance of option-based and constant proportion portfolio insurance strategies for a DC fund that targets a minimum level of inflation-protected annuity income at retirement. We show how the portfolio allocation to an equity fund varies depending on the member’s age upon joining the fund, displaying a downward trend through time for members joining the fund before ages in the mid-30s. We demonstrate how both portfolio insurance strategies provide strong protection against downside equity risk in financing a minimum level of retirement income. The option-based strategy generally leads to higher accumulated savings at retirement, whereas the constant proportion strategy provides better downside risk protection robust to equity market jumps/volatilities.

The estimation of outstanding claims is one of the important aspects in the management of the insurance business. Various methods have been widely dealt with in the actuarial literature. Exploration of the inaccuracies involved is traditionally based on a post-facto comparison of the estimates against the actual outcomes of the settled claims. However, until recent years it has not been usual to consider the inaccuracies inherent in claims reserving in the context of more comprehensive (risk theoretical) models, the purpose of which is to analyse the insurer as a whole. Important parts of the technique which will be outlined in this paper can be incorporated into over-all risk theory models to introduce the uncertainty involved with technical reserves as one of the components in solvency and other analyses (Pentikäinen et al. (1989)).

The idea in this paper is to describe a procedure by which one can explore how various reserving methods react to fictitious variations, fluctuations, trends, etc. which might influence the claims process, and, what is most important, how they reflect on the variables indicating the financial position of the insurer. For this purpose, a claims process is first postulated and claims are simulated and ordered to correspond to an actual handling of the observed claims of a fictitious insurer. Next, the simulation program will ‘mime’ an actuary who is calculating the claims reserve on the basis of these ‘observed’ claims data. Finally, the simulation is further continued thus generating the settlement of the reserved claims. The difference between reserved amounts and settled amounts gives the reserving (run-off) error in this particular simulated case. By repeating the simulation numerous times (Monte Carlo method) the distribution of the error can be estimated as well as its effect on the total outcome of the insurer.

Merton and Perold (1993) offered a framework for determining risk capital in a financial firm based on the cost of the implicit guarantee the firm provides to its subsidiaries to make up any operating shortfall. Merton and Perold assume the price of such guarantees is observable from the market at large. For an insurer, this may not be a realistic assumption. This paper proposes an insurance-specific framework for determining the cost of those parental guarantees, and utilizing that cost in pricing and portfolio mix evaluation. An insurer’s capital is treated as a shared asset, with the insurance contracts in the portfolio having simultaneous rights to access potentially all that shared capital. By granting underwriting capacity, an insurer’s management team is implicitly issuing a set of options to draw upon the common capital pool — similar in structure to letters of credit (LOC), except they are not loans but grants. The paper will (i) discuss the valuation of parental guarantees, beginning with Merton and Perold; (ii) treat insurer capital as a shared asset and explore the dual nature of insurer capital usage; (iii) offer a method for determining insurer capital usage cost; and (iv) demonstrate how this method could be used for product pricing and portfolio mix evaluation using economic value added concepts.

In a recent paper Seal (1980) calculated numerically survival probabilities based on Pareto claim distributions.

The Pareto density may be written as

Generalizing, the Pareto distribution may be regarded as a special case of the so-called beta-prime distribution (Keeping, 1962, p. 83) with density function

where B(p, q) = is the beta function.

In his paper Seal (1980, Appendix 1) arrived at a contradiction concerning this beta-prime distribution. He found on one side that all derivatives of the characteristic function exist at the origin and on the other side that only the moments of order n < q exist. In this note we will show that this contradiction is due to the use of an incorrect expression for the characteristic function of the beta-prime distribution, which was taken over from Johnson and Kotz (1970, Ch. 26) and Oberhettinger (1973, Table A).

Motivated by insurance applications, we propose a new approach for the validation of real-world economic scenarios. This approach is based on the statistical test developed by Chevyrev and Oberhauser ((2022) Journal of Machine Learning Research, 23(176), 1–42.) and relies on the notions of signature and maximum mean distance. This test allows to check whether two samples of stochastic processes paths come from the same distribution. Our contribution is to apply this test to a variety of stochastic processes exhibiting different pathwise properties (Hölder regularity, autocorrelation, and regime switches) and which are relevant for the modelling of stock prices and stock volatility as well as of inflation in view of actuarial applications.

The prominence of the Euler allocation rule (EAR) is rooted in the fact that it is the only return on risk-adjusted capital (RORAC) compatible capital allocation rule. When the total regulatory capital is set using the value-at-risk (VaR), the EAR becomes – using a statistical term – the quantile-regression (QR) function. Although the cumulative QR function (i.e., an integral of the QR function) has received considerable attention in the literature, a fully developed statistical inference theory for the QR function itself has been elusive. In the present paper, we develop such a theory based on an empirical QR estimator, for which we establish consistency, asymptotic normality, and standard error estimation. This makes the herein developed results readily applicable in practice, thus facilitating decision making within the RORAC paradigm, conditional mean risk sharing, and current regulatory frameworks.

We consider exponential smoothing Yn = αXn + (1 − α)Yn−1, 0 < α < 1, in experience rating. Here the premium Yn is determined by the policy's own claims history (Xn). In order to uniformize the fluctuation of premiums, it is appropriate to use a bigger α for the big policies than for the small ones. When the size of the policy changes with time, a need arises to change α correspondingly. It has recently been shown that changing based on the size of the premiums Yn may lead to too low a tariff level. This result is presented here and illustrated by means of simulation. Further, some general results are given how the changing can be made without a decline in the tariff level. The results are applied to a tariff system in which the linking of the smoothing parameter to the size of the policy is particularly motivated.

Target date funds (TDFs) are becoming increasingly popular investment choices among investors with long-term prospects. Examples include members of superannuation funds seeking to save for retirement at a given age. TDFs provide efficient risk exposures to a diversified range of asset classes that dynamically match the risk profile of the investment payoff as the investors age. This is often achieved by making increasingly conservative asset allocations over time as the retirement date approaches. Such dynamically evolving allocation strategies for TDFs are often referred to as glide paths. We propose a systematic approach to the design of optimal TDF glide paths implied by retirement dates and risk preferences and construct the corresponding dynamic asset allocation strategy that delivers the optimal payoffs at minimal costs. The TDF strategies we propose are dynamic portfolios consisting of units of the growth-optimal portfolio (GP) and the risk-free asset. Here, the GP is often approximated by a well-diversified index of multiple risky assets. We backtest the TDF strategies with the historical returns of the S&P500 total return index serving as the GP approximation.

In the recent subprime mortgage crisis, which has caused banks and insurance companies to go bankrupt or into acquisition, the lender and insurer have exhibited not only correlated defaults when exposed to common risk factors but also counterparty default risk, which is triggered by mortgage defaults. Given the correlated defaults and the counterparty default risk, we use the reduced-form approach to derive the closed-form formulas of mortgage insurance contracts with premium refunds, annual premiums and upfront premiums. Regardless of the nature of the premium structures, the numerical analysis with parameter calibration demonstrates that both the correlated defaults and the counterparty default risk significantly impact mortgage insurance premiums, particularly in long-term mortgage loans.

My assistants at ETH have a wall calendar—not with the usual pictures of Swiss mountains, hills and lakes, but with “quotations for intelligent people”. Recently, the quotation for the week read as follows: “Even the future is no longer what it used to be in the past”.

Observe that also in this supposedly intelligent approach it seems impossible to speak about the future without referring to the past. I shall not deviate from this rule. Of course, my task is greatly simplified by the fact that Paul Johansen has just entertained you in a charming way about the past 25 years of ASTIN and the earlier endeavors leading to the foundation of ASTIN.

In the year 1693, Edmond Halley constructed the first mortality table based on mortality data from Breslau which he had obtained through the intervention of Leibniz. This can be regarded as the starting point of actuarial science. In my opinion it can however not be considered as the starting point of the actuarial profession. Why? Yes, Halley's table was used for some eighty years because subsequent information coincided with his estimate of mortality. Yes, De Moivre, in his classic textbook of 1725, performed ingenious calculations of annuities, based on the same table. Yes, Süssmilch published the first basic and substantial work of demography in 1741 but—here comes the big but—no government (and nobody else sold annuity insurance at that time) made use of the available scientific method to calculate annuities. Perhaps the first statistical results to be taken seriously were the Northampton tables of 1780, devised by Richard Price. Incidentally, this date coincides reasonably well with the first valuation by William Morgan in 1786. Hence, I think that either of these dates may be taken, at the earliest, as the start of the actuarial profession, a profession being by definition a dedicated group of people accepted by society for the performance of a particular skill. Let me make my point explicit: We have historical evidence of the existence of actuarial science about 90 years prior to the emergence of the actuarial profession.

We extend the Annually Recalculated Virtual Annuity (ARVA) spending rule for retirement savings decumulation (Waring and Siegel (2015) Financial Analysts Journal, 71(1), 91–107) to include a cap and a floor on withdrawals. With a minimum withdrawal constraint, the ARVA strategy runs the risk of depleting the investment portfolio. We determine the dynamic asset allocation strategy which maximizes a weighted combination of expected total withdrawals (EW) and expected shortfall (ES), defined as the average of the worst 5% of the outcomes of real terminal wealth. We compare the performance of our dynamic strategy to simpler alternatives which maintain constant asset allocation weights over time accompanied by either our same modified ARVA spending rule or withdrawals that are constant over time in real terms. Tests are carried out using both a parametric model of historical asset returns as well as bootstrap resampling of historical data. Consistent with previous literature that has used different measures of reward and risk than EW and ES, we find that allowing some variability in withdrawals leads to large improvements in efficiency. However, unlike the prior literature, we also demonstrate that further significant enhancements are possible through incorporating a dynamic asset allocation strategy rather than simply keeping asset allocation weights constant throughout retirement.

We construct a double common factor model for projecting the mortality of a population using as a reference the minimum death rate at each age among a large number of countries. In particular, the female and male minimum death rates, described as best-performance or best-practice rates, are first modelled by a common factor model structure with both common and sex-specific parameters. The differences between the death rates of the population under study and the best-performance rates are then modelled by another common factor model structure. An important result of using our proposed model is that the projected death rates of the population being considered are coherent with the projected best-performance rates in the long term, the latter of which serves as a very useful reference for the projection based on the collective experience of multiple countries. Our out-of-sample analysis shows that the new model has potential to outperform some conventional approaches in mortality projection.

This study examines the practical application of a system for the derivation of member utility functions for the purpose of recommending investment-channel choice to members of a defined-contribution retirement fund. The utility functions of post-retirement benefits from members of a defined-contribution fund are elicited. The risk aversion of each member is measured and the results are compared with a standard risk-tolerance assessment method.