Author’s Reply: Statistical Independence and Fractional Age Assumptions - Discussion by Cecil Nesbitt; Elias S. W. Shiu; Serena Tiong,January 1997

Abstract

This paper addresses the problem of the sharing of longevity risk between an annuity provider and a group of annuitants. An appropriate longevity index is designed in order to adapt the amount of the periodic payments in life annuity contracts. This accounts for unexpected longevity improvements experienced by a given reference population. The approach described in the present paper is in contrast with group self-annuitization, where annuitants bear their own risk. Here the annuitants bear only the nondiversifiable risk that the future mortality trend departs from that of the reference forecast. In that respect, the life annuities discussed in this paper are substitutes for reinsurance and securitization of longevity risk.     

Zur Verteilung der Abschlusskosten in deutschen Lebensversicherungen

It will be shown that an amortization spread of the acquisition costs to the first N years of a classical german life insurance policy or lifelong annuity always leads to a non negative technical provision at the end of the first year if \( N\,{=}\,5 \) (and the technical provision starting with zero). For usual combinations of the age at entry and the policy duration even \( N\,{=}\,3 \) is sufficient for this purpose. The legal specification given by the new version of VVG in Germany concerning surrender value (i.e. positive value even in case of early lapse) is therefore consistent with the requirement of a amortization spread to 5 years. Additionally a recursion formula for the technical provision with N years spreaded acquisition costs is proven.     

Hedging Longevity Risk When Interest Rates are Uncertain

Abstract

This paper proposes an asset liability management strategy to hedge the aggregate risk of annuity providers under the assumption that both the interest rate and mortality rate are stochastic. We assume that annuity providers can invest in longevity bonds, long-term coupon bonds, and shortterm zero-coupon bonds to immunize themselves from the risks of the annuity for the equity holders subject to a required profit. We demonstrate that the optimal allocation strategy can lead to the lowest risk under different yield curves and mortality rate assumptions. The longevity bond can also be regarded as an effective hedging vehicle that significantly reduces the aggregate risk of the annuity providers.     

Structural properties of the price-to-earnings and price-to-book ratios

We examine the structural properties of a firm’s price-to-earnings (P/E) and price-to-book (P/B) ratios and the relation between these two ratios. A benchmark result is obtained under the hypothesis that firms use replacement cost accounting to value their operating assets, so that the P/B ratio coincides with Tobin’s q. The firm’s P/E ratio can then be expressed as a convex combination of the P/E ratios suggested respectively by the permanent earnings model and the Gordon growth model, with the relative weight to be placed on these two endpoints determined entirely by Tobin’s q. Under current financial reporting rules, the accounting for operating assets is likely to be more conservative than replacement cost accounting. Our findings characterize how the magnitude and behavior of the P/E and P/B ratios are jointly shaped by several key variables, including both past and anticipated future growth, economic profitability, and accounting conservatism     

Erklärung der privaten Krankenversicherungsnachfrage anhand ökonomischer und psychologischer Einflussfaktoren

To be able to plan measures insales planning successfully, informations are necessary about the expected trend of demand in the future. In this respecttheoretical funded andempirical relevant correlations between exogenous influencing factors and the process of demand must be systematically explored. Consequently for the private health insurance aglobal model was developed, which is the basis for presenting exemplarily an approach of explanation for the health cost insurance. Via this approach theshort-term fluctuations in the trend of demand, which were noticed in the past, can be traced satisfactorily. In this context besideseconomical alsopsychological influences like attitudes and expectations must be explicitly integrated in the approach of explanation. Naturally also the effects ofsociopolitical events, which are relevant to demand, were included in the analysis, especially as the process of demand for private health insurance is substantially marked by sociopolitical legislation.     

Life Insurance Mathematics with Random Life Tables

Abstract

When the insurer sells life annuities, projected life tables incorporating a forecast of future longevity must be used for pricing and reserving. To fix the ideas, the framework of Lee and Carter is adopted in this paper. The Lee-Carter model for mortality forecasting assumes that the death rate at age x in calendar year t is of the form exp(α_x + (β_xK_t), where the time-varying parameter K_t reflects the general level of mortality and follows an ARIMA model. The future lifetimes are all influenced by the same time index K_t in this framework. Because the future path of this index is unknown and modeled as a stochastic process, the policyholders' lifetimes become dependent on each other. Consequently the risk does not disappear as the size of the portfolio increases: there always remains some systematic risk that cannot be diversified, whatever the number of policies. This paper aims to investigate some aspects of actuarial mathematics in the context of random life tables. First, the type of dependence existing between the insured life lengths is carefully examined. The way positive dependence influences the need for economic capital is assessed compared to mutual independence, as well as the effect of the timing of deaths through Bayesian credibility mechanisms. Then the distribution of the present value of payments under a closed group of life annuity policies is studied. Failing to account for the positive dependence between insured lifetimes is a dangerous strategy, even if the randomness in the future survival probabilities is incorporated in the actuarial computations. Numerical illustrations are performed on the basis of Belgian mortality statistics. The impact on the distribution of the present value of the additional variability that results from the Lee-Carter model is compared with the traditional method of mortality projection. Also, the impact of ignoring the dependence hat arises from the model is quantified.     

On the construction of tables for the calculation of certain survivorship benefits

Abstract

1. Benefits depending on two lives often involve the necessity for calculating tables with two arguments. There are, however, cases where this can be avoided, the chief example being the joint-life annuity when the mortality table is graduated by Makeham's formula. The question therefore arises whether there are other cases where a suitable graduation of the mortality table may render similar services.     

Auswirkungen von Enhanced Annuities auf den Bestand eines Versicherers

The value for money of a standard annuity is the higher, the longer the life expectancy of an insured, and therefore it is only acceptable for persons with an above average life expectancy. The discrepancy is intensified by tax regulations that favor lifelong annuity payments opposed to a lump sum. This discrimination of impaired insureds could be prevented if so-called enhanced annuities were offered, i.e. products where the annuity paid is the larger, the lower the person’s life expectancy. The article presents a quantitative comparison of the risk profile of insurance companies offering standard annuity contracts compared to enhanced annuities and an analysis of the impact of adverse selection on a standard insurer. By definition of individual mortality rates a heterogeneous insurance portfolio is specified. Besides we model the individual underwriting of enhanced annuities. A Monte Carlo Simulation provides results to compare the profit/loss situation of a portfolio of traditional annuity products and a portfolio of enhanced annuities with individual underwriting of different quality and to assess the impact of selection effects.     

On the properties of high-order non-monetary measures for risks

This paper investigates how welfare losses for facing high-order risk increases change when the risk environment of the decision maker is altered. To that aim, we define the nth-order utility premium as a measure of pain associated with facing the passage of one risk to a more severe one and we examine some of its properties. Changes in risk are expressed through the concept of stochastic dominance of order n. The paper investigates more particularly welfare changes of merging increases in risk, first ignoring background risks, then taking them into account. Merging increases in risk may be beneficial or not, depending on whether background risks are considered and how. The paper also provides conditions on individual preferences for superadditivity of the nth-order utility premium. The results confirm the importance and usefulness of two analytical concepts: mixed risk aversion and risk apportionment.     

Options pricing under the one-dimensional jump-diffusion model using the radial basis function interpolation scheme

This paper will demonstrate how European and American option prices can be computed under the jump-diffusion model using the radial basis function (RBF) interpolation scheme. The RBF interpolation scheme is demonstrated by solving an option pricing formula, a one-dimensional partial integro-differential equation (PIDE). We select the cubic spline radial basis function and adopt a simple numerical algorithm (Briani et al. in Calcolo 44:33–57, 2007) to establish a finite computational range for the improper integral of the PIDE. This algorithm reduces the truncation error of approximating the improper integral. As a result, we are able to achieve a higher approximation accuracy of the integral with the application of any quadrature. Moreover, we a numerical technique termed cubic spline factorisation (Bos and Salkauskas in J Approx Theory 51:81–88, 1987) to solve the inversion of an ill-conditioned RBF interpolant, which is a well-known research problem in the RBF field. Finally, our numerical experiments show that in the European case, our RBF-interpolation solution is second-order accurate for spatial variables, while in the American case, it is second-order accurate for spatial variables and first-order accurate for time variables.     

Application of Relational Models in Mortality Immunization

The prediction of future mortality rates by any existing mortality models is hardly exact, which causes an exposure to mortality (longevity) risk for life insurers (annuity providers). Since a change in mortality rates has opposite impacts on the surpluses of life insurance and annuity, hedging strategies of mortality and longevity risks can be implemented by creating an insurance portfolio of both life insurance and annuity products. In this article, we apply relational models to capture the mortality movements by assuming that the realized mortality sequence is a proportional change and/or a constant shift of the expected one, and the size of the changes varies in the length of the sequences. Then we create a variety of non-size-free matching strategies to determine the weights of life insurance and annuity products in an insurance portfolio for mortality immunization, where the weights depend on the sizes of the proportional and/or constant changes. Comparing the hedging performances of four non-size-free matching strategies with corresponding size-free ones proposed by Lin and Tsai, we demonstrate with simulation illustrations that the non-size-free matching strategies can hedge against mortality and longevity risks more effectively than the size-free ones.     

Redistribution and Insurance: Mandatory Annuitization With Mortality Heterogeneity

This article examines the distributional implications of mandatory longevity insurance when mortality heterogeneity exists in the population. Previous research has demonstrated the significant financial redistribution that occurs under alternative annuity programs in the presence of differential mortality across groups. This article embeds that analysis into a life‐cycle framework that allows for an examination of distributional effects on a utility‐adjusted basis. It finds that the degree of redistribution that occurs from the introduction of a mandatory annuity program is substantially lower on a utility‐adjusted basis than when evaluated on a purely financial basis. In a simple life‐cycle model with no bequests, complete annuitization is welfare enhancing even for those with higher‐than‐average expected mortality rates, so long as administrative costs are sufficiently low. These findings have implications for policy toward annuitization, particularly as part of a reformed Social Security system.     

Mobile banking innovators and early adopters: How they differ from other online users?

The rapid pace of technological development has created opportunities for financial service providers to offer their services via multiple electronic channels. The mobile phone is one of the most promising but so far marginally adopted channel for using financial services by consumers. Earlier literature on innovation adoption argued that those among the first to adopt new innovations possess unique characteristics compared to the majority of consumers. This paper aims to study how mobile banking innovators and early adopters differ from other users of online banking services. An internet survey was conducted among customers of a large Scandinavian bank in Finland yielding 2,675 responses. Logistic regression was used to identify variables differentiating between users of mobile banking and other online banking services. Somewhat contradictory to earlier findings the results indicate that only age (p<0.0005) and gender (p=0.010) differentiate these two groups of customers, while education (p=0.957), income (p=0.624), occupation (p=0.596) and size of the household (p=0.151) were found to be insignificant in differentiating the groups. The results offer service providers better knowledge of the typical mobile banking user thus adding value to their marketing actions in the field of electronic banking.     

The impact of SEC investigations and accounting and auditing enforcement releases on firms’ cost of equity capital

I study the impact of an SEC investigation (as captured by accounting and auditing enforcement releases) on a firm’s cost of equity capital. AAERs are often used in accounting literature as a proxy for fraudulent financial reporting. Fraudulent financial reporting should lead to an increase in cost of equity capital as a firm’s future cash flows become less certain. Overall, this study provides evidence of changes in cost of equity capital for firms targeted by an SEC AAER on the date the investigation is first made public.     

Decision usefulness of whole-asset operating lease capitalizations

Monson (2001) and Hepp and Scoles (2012) argue that some leased assets should be capitalized at the assets' purchase price (whole-asset value) rather than at the present value of future minimum lease payments (right-of-use asset value). The argument is based in part on the notion that the assets under lease generate future income not the obligation related to future lease payments. To test the notion we compare associations between capitalizations representing whole asset values and current and future return on assets (ROA) and return on equity (ROE) with capitalizations representing right-of-use asset values and current and future ROA and ROE. Our results indicate that the whole-asset annuity values are incrementally associated with future ROA and ROE over right-of-use asset values. We interpret our results to suggest that the current practice of capitalizing future lease payments does not fully reflect the income generation provided by leased assets.     

Risk Aversion,Downside Risk Aversion and Paying for Stochastic Improvements

This paper considers the relationship between risk preferences and the willingness to pay for stochastic improvements. We show that if the stochastic improvement satisfies a double-crossing condition, then a decision maker with utility v is willing to pay more than a decision maker with utility u, if v is both more risk averse and less downside risk averse than u. As the condition always holds in the case of self-protection, the result implies novel characterizations of individuals’ willingness to pay to reduce the probability of loss. By establishing a general result on the correspondence between an individual's willingness to pay, and his optimal purchase of stochastic improvement when there is a given relationship between stochastic improvements and the amount paid for them, we further show that all results on the willingness to pay can be applied directly to characterize the conditions under which a more risk averse individual will optimally choose to buy more stochastic improvement. Generalizations of existing results on optimal choice of self-protection can be obtained as corollaries.     

Immediate Annuity Pricing in the Presence of Unobserved Heterogeneity

Abstract

One of the acknowledged difficulties with pricing immediate annuities is that underwriting the annuitantis life is the exception rather than the rule. In the absence of underwriting, the price paid for a life-contingent annuity is the same for all sales at a given age. This exposes the market (insurance company and potential policyholder alike) to antiselection. The insurance company worries that only the healthiest people choose a life-contingent annuity and therefore adjust mortality accordingly. The potential policyholders worry that they are not being compensated for their relatively poor health and choose not to purchase what would otherwise be a very beneficial product.

This paper develops a model of underlying, unobserved health. Health is a state variable that follows a first-order Markov process. An individual reaches the state “death” either by accident from any health state or by progressively declining health state. Health state is one-dimensional, in the sense that health can either “improve” or “deteriorate” by moving farther from or close to the “death” state, respectively. The probability of death in a given year is a function of health state, not of age. Therefore, in this model a person is exactly as old as he or she feels.

I first demonstrate that a multistate, ageless Markov model can match the mortality patterns in the common annuity mortality tables. The model is extended to consider several types of mortality improvements: permanent through decreasing probability of deteriorating health, temporary through improved distribution of initial health state, and plateau through the effects of past health improvements.

I then construct an economic model of optimal policyholder behavior, assuming that the policyholder either knows his or her health state or has some limited information. the value of mortality risk transfer through purchasing a life-contingent annuity is estimated for each health state under various risk-aversion parameters. Given the economic model for optimal purchasing of annuities, the value of underwriting (limited information about policyholder health state) is demonstrated.     

The Hierarchical Repeat Sales Model for Granular Commercial Real Estate and Residential Price Indices

This paper concerns the estimation of granular property price indices in commercial real estate and residential markets. We specify and apply a repeat sales model with multiple stochastic log price trends having a hierarchical additive structure: One common log price trend and cluster specific log price trends in deviation from the common trend. Moreover, we assume that the error terms potentially have a heavy tailed (t) distribution to effectively deal with outliers. We apply the hierarchical repeat sales model on commercial properties in the Philadelphia/Baltimore region and on residential properties in a small part of Amsterdam. The results show that the hierarchical repeat sales model provides reliable indices on a very detailed level based on a small number of observations. The estimated degrees of freedom for the t-distribution is small, largely rejecting the commonly made assumption of normality of the error term.     

Small-time ruin for a financial process modulated by a Harris recurrent Markov chain

We consider a nonstandard ruin problem where: (i) the increments of the process are heavy-tailed and Markov-dependent, modulated by a general Harris recurrent Markov chain; (ii) ruin occurs when a positive boundary is attained within a sufficiently small time. Our main result provides sharp asymptotics for the small-time probability of ruin, viz., P(sup? _{n≤δ u} S _n ≥u), where {S _n } denotes the discrete partial sums of the process and δ∈(0,1/μ), where μ is the mean drift. We apply our results to obtain risk estimates which quantify, e.g., repetitive operational risk losses or the extremal behavior for a GARCH(1,1) process.     

Prognose des Abwicklungsergebnisses mittels Chain-Ladder-Verfahren für abhängige Run-Off-Portfolios

In the present paper we analyse how the estimators from Merz u. Wüthrich (2007) could be generalised to the case of N correlated run-off triangles. The simultaneous view on N correlated subportfolios is motivated by the fact, that in practice a run-off portfolio often has to be divided in subportfolios, so that the homogeneity assumption of the claims reserving method on each subportfolio is satisfied. We derive an explicit formula for the process-variance, the estimation-error and the prediction error made by the forecast for the claims development result with the Chain-Ladder method. We illustrate the results by an example.