Stochastic Models for Continuing Care Retirement Communities

Abstract

Continuing care retirement communities (CCRCs) offer housing and a variety of services, including long-term care. Typically, the cost of this long-term care is wholly or partially covered by entry and/or periodic fees. Thus, CCRCs provide a long-term-care insurance benefit. For this and other reasons, actuarial involvement in the financial management of CCRCs is desirable. To carry out actuarial analyses of CCRCs, appropriate models are required to describe the status of individual residents and the CCRC population.

This paper presents models that assume that, at any time, a resident is in a given “state,” which is determined by the individual’s care requirements. The resident may make “transitions” between states at various times, and randomness is associated with both the transition times and the states entered. Actuarial calculations using such a model are discussed, and numerical illustrations are provided. A simple model is examined first; then generalizations are considered. The model for an individual resident can be embedded in a model for a CCRC population. This is explored with particular attention given to the “high-demand” situation in which potential residents are always waiting to enter the community. With this model, the goal is to analyze the future care requirements of the CCRC population.     

Methods for the Analysis of CCRC Data

Abstract

This paper presents an approach to analyzing continuing care retirement community (CCRC) data and demonstrates the methodology by using data from a CCRC. It is assumed that residents make "transitions" among a number of "states" that represent the levels of care required by residents. There is randomness associated with both the transition times and the states entered at these times. The model is conveniently characterized in terms of “transition intensity functions,” which represent the instantaneous rates of transition between pairs of states. Statistical methods for estimating these functions are discussed, and estimates are obtained from the data-set. A simulation approach for determining probabilities and other interesting quantities based on the estimated intensity functions is also described and illustrated.     

Balancing Retirement Security With the Needs of Frail Parents

Abstract

Caring for frail elderly parents can interfere with work responsibilities. People who provide care to their parents may need to take time off from work or retire altogether. However, reductions in labor supply at midlife can have serious implications for retirement wealth and, as a result, on economic well-being in later life. This paper examines how family support for the elderly can affect retirement savings by examining the relationship between labor supply, time help to parents, and financial assistance to parents. Using data from the Health and Retirement Study on a nationally representative sample of women ages 53–63, we found that women who helped their parents with personal care assistance worked significantly fewer hours than did those who did not help their parents, whereas those who provided financial assistance worked significantly more hours. Although few persons at midlife presently spend substantial amounts of time helping their elderly parents in any given year, for those who do, the costs can be high. Pressures on families are likely to mount in the near future as falling mortality and fertility rates continue to increase the proportion of the population that is very old and as women continue to play more important roles in the labor market.     

Genetics,Alzheimer’s Disease,and Long-Term Care Insurance

Abstract

This paper applies a model of Alzheimer’s disease (AD) developed by Macdonald and Pritchard (2000) to the question of the potential for adverse selection in long-term care (LTC) insurance introduced by the existence of DNA tests for variants of the ApoE gene, the ε4 allele of which is known to predispose one to earlier onset of AD. It computes the expected present values (EPVs) of model LTC benefits with respect to AD for each of five ApoE genotypes, weighted average EPVs with and without adverse selection, and sample underwriting ratings. The paper concludes that adverse selection could increase costs significantly in a small LTC insurance market only if current population genetic risk is not much smaller than that observed in case-based studies, and if carriers of the ε4 allele are very much more likely to buy LTC insurance. Finally, the paper considers the cost of a combined retirement package, providing both pension and LTC insurance, and shows that it can reduce adverse selection.     

Compound Poisson Model with Covariates

Abstract

Pet insurance in North America continues to be a growing industry. Unlike in Europe, where some countries have as much as 50% of the pet population insured, very few pets in North America are insured. Pricing practices in the past have relied on market share objectives more so than on actual experience. Pricing still continues to be performed on this basis with little consideration for actuarial principles and techniques. Developments of mortality and morbidity models to be used in the pricing model and new product development are essential for pet insurance. This paper examines insurance claims as experienced in the Canadian market. The time-to-event data are investigated using the Cox’s proportional hazards model. The claim number follows a nonhomogenous Poisson process with covariates. The claim size random variable is assumed to follow a lognormal distribution. These two models work well for aggregate claims with covariates. The first three central moments of the aggregate claims for one insured animal, as well as for a block of insured animals, are derived. We illustrate the models using data collected over an eight-year period.     

Markov Aging Process and Phase-Type Law of Mortality

Abstract

In this article, we propose a finite-state Markov process with one absorbing state to model human mortality. A health index called physiological age is introduced and modeled by the Markov process. Under this model the time of death follows a phase-type distribution. The model possesses many desirable analytical properties useful for mortality analysis. Closed-form expressions are available for many quantities of interest including the conditional survival probabilities of the time of death and the actuarial present values of the whole life insurance and annuity. The heterogeneity or frailty effect of a cohort can be expressed explicitly. The model is also able to explain some stylized facts of observed mortality data. We fit the model to some Swedish population cohort data and life tables compiled by the U.S. Social Security Administration. The fitting results are very satisfactory.     

Pricing Annuity Guarantees Under a Regime-Switching Model

Abstract

We consider the pricing problem of equity-linked annuities and variable annuities under a regimeswitching model when the dynamic of the market value of a reference asset is driven by a generalized geometric Brownian motion model with regime switching. In particular, we assume that regime switching over time according to a continuous-time Markov chain with a finite number state space representing economy states. We use the Esscher transform to determine an equivalent martingale measure for fair valuation in the incomplete market setting. The paper is complemented with some numerical examples to highlight the implications of our model on pricing these guarantees.     

A Statistical Method for Monitoring a Change in the Rate of Nonacceptable Inpatient Claims

Abstract

This paper is an extension of earlier work (Rosenberg 1998; Rosenberg, Andrews, and Lenk 1999; Rosenberg and Griffith 2000) that introduced a statistical control model to supplement current efforts inexpensively to help reduce unnecessary expenditures. The application of the study was to predict the rate of nonacceptable inpatient claims (NACs). In that work, a statistical model was proposed to link information obtained through an expensive audit with inexpensive information that is readily available to estimate the probability that a claim is a NAC. The premise was that a statistical system can be developed to supplement the expensive audit for additional control between audits.

Estimates of the NAC rate obtained from the statistical model are used as input in a statistical monitor to assess whether the NAC rate had changed over time. The statistical monitor is the subject of this paper. The idea is that subgroups of claims can be analyzed inexpensively with the statistical monitor to determine whether any current intervention is required prior to the time of the next scheduled audit, or whether adjustments are needed in the determination of claims to be sampled for the audit. In this study, the estimate for the NAC rate at t ₀ is compared against the estimate of the NAC rate at some later time t ₁. A decision rule is proposed to assess whether a change in the NAC rate has occurred for that subgroup. The methodology is also applicable to other health care measurements.     

Hedging Equity-Linked Life Insurance Contracts

Abstract

This paper examines a portfolio of equity-linked life insurance contracts and determines risk-minimizing hedging strategies within a discrete-time setup. As a principal example, I consider the Cox-Ross-Rubinstein model and an equity-linked pure endowment contract under which the policyholder receives max(S_T , K) at time T if he or she is then alive, where S_T is the value of a stock index at the term T of the contract and K is a guarantee stipulated by the contract. In contrast to most of the existing literature, I view the contracts as contingent claims in an incomplete model and discuss the problem of choosing an optimality criterion for hedging strategies. The subsequent analysis leads to a comparison of the risk (measured by the variance of the insurer’s loss) inherent in equity-linked contracts in the two situations where the insurer applies the risk-minimizing strategy and the insurer does not hedge. The paper includes numerical results that can be used to quantify the effect of hedging and describe how this effect varies with the size of the insurance portfolio and assumptions concerning the mortality.     

Intraday Empirical Analysis and Modeling of Diversified World Stock Indices

This paper proposes an approach to the intraday analysis of diversified world stock accumulation indices. The growth optimal portfolio (GOP) is used as reference unit or benchmark in a continuous financial market model. Diversified portfolios, covering the world stock market, are constructed and shown to approximate the GOP, providing the basis for a range of financial applications. The normalized GOP is modeled as a time transformed square root process of dimension four. Its dynamics are empirically verified for several world stock indices. Furthermore, the evolution of the transformed time is modeled as the integral over a rapidly evolving mean-reverting market activity process with deterministic volatility. The empirical findings suggest a rather simple and robust model for a world stock index that reflects the historical evolution, by using only a few readily observable parameters. Mathematics Subject Classification: (1991) primary 90A12, secondary 60G30,62P20 JEL Classification: G10, G13     

Timing and diversification: A state-dependent asset allocation approach

Abstract

The influence of changing economic environment leads the distribution of stock market returns to be time-varying. A conditionally optimal investment hence requires a dynamic adjustment of asset allocation. In this context, this paper examines the improvement in portfolio performance by simulating portfolio strategies that are conditioned on the Markov regime switching behaviour of stock market returns. Including a memory effect eliminates the empirical shortcoming of discrete state models, namely that they produce a standard and an extreme state in stock returns. So far, this has prevented the regimes from being used as a valuable conditioning variable. Based on a discrete state indicator variable, is presented evidence of considerable performance improvement relative to the static model due to optimal shifting between aggressive and well diversified portfolio structures.     

Deriving the arbitrage pricing theory when the number of factors is unknown

This paper examines the use of proxies (or reference variables) for the true factors in the arbitrage pricing theory (APT). It generalizes other authors' existing work and shows that, when there are more reference variables than the true factors, the APT still holds. The possibility of fewer reference variables than the true factors is also considered, but the APT is not shown to hold, in the same sense, for this case. This work builds on an earlier paper by Ingersoll (Ingersoll J 1984 J. Finance 39 1021-39), and our propositions can be thought of as specializations of his theorems. Similar to Nawalkha (Nawalkha S 1997 J. Financial Economics 46 357-81), our work does not use the mathematics of Hilbert and Banach spaces and, thus, is open to a much wider audience. The practical implication of our results is that model builders should be generous with the number of factors they use, as excessively parsimonious models suffer from inaccuracy.     

The optimal commodity tax structure in a four-good model

This paper examines the structure of optimal commodity tax rates in a four-good model with nonvanishing cross-elasticities between commodities. The following optimal tax rules are obtained: (i) at the optimum, the tax rate on the commodity that is the least substitutable for leisure is higher than that on the commodity that is the most substitutable for leisure; (ii) the greater the cross-substitutability between two commodities, the closer to uniformity are the optimal commodity tax rates on those two commodities. In addition, the condition under which all commodity tax rates are ranked is identified. JEL Classification H21     

Some comments on the APT

Abstract

The paper reviews some aspects of arbitrage pricing theory (APT). It derives an improved version of the model and examines it in view of the APT debate, adding some new observations in favour of the model. The topics examined include: (a) model testability; (b) implications of approximate APT pricing to Fama-Macbeth testing methodology; and (c) a comparison between APT pricing and approximate exact pricing.     

Equity Risk Premium

Abstract

The equity risk premium (ERP) is an essential building block of the market value of risk. In theory, the collective action of all investors results in an equilibrium expectation for the return on the market portfolio excess of the risk-free return, the ERP. The ability of the valuation actuary to choose a sensible value for the ERP, whether as a required input to capital asset pricing model valuation, or any of its descendants, is as important as choosing risk-free rates and risk relatives (betas) to the ERP for the asset at hand.

The historical realized ERP for the stock market appears to be at odds with pricing theory parameters for risk aversion. Since 1985, there has been a constant stream of research, each of which reviews theories of estimating market returns, examines historical data periods, or both. Those ERP value estimates vary widely from about ?1% to about 9%, based on a geometric or arithmetic averaging, short or long horizons, short- or long-run expectations, unconditional or conditional distributions, domestic or international data, data periods, and real or nominal returns.

This paper examines the principal strains of the recent research on the ERP and catalogues the empirical values of the ERP implied by that research. In addition, the paper supplies several time series analyses of the standard Ibbotson Associates 1926–2002 ERP data using short Treasuries for the risk-free rate. Recommendations for ERP values to use in common actuarial valuation problems also are offered.     

Estimating population mean,variance, and percentage points from truncated data

Abstract

Consider a set of observations which are obtained by truncating a sample of known size. The truncation procedure consists of deleting a known number of the largest sample values and a known number of the smallest sample values. One problem considered is the use of this data to estimate certain of the population percentage points for which the corresponding sample data was deleted. Another problem is to estimate the population mean and standard deviation. This paper presents solutions to these problems which are valid for a rather general class of continuous statistical populations. The results obtained should be applicable to most practical cases of a continuous type. A sample analog of the percentage point estimation procedure has interesting uses for life testing situations. Namely, the first time at which a specified number of additional items of a sample will have failed can be predicted from the values of the items which have already failed.     

Skewness in individual stocks at different investment horizons

Abstract

This paper examines the (a)symmetry of several individual stock returns at different investment horizons: daily, weekly and monthly. While some asymmetries are observed in daily returns, they disappear almost completely in weekly and monthly returns. The explanation for this fact lies in the convergence to normality that takes place when the investment horizon increases. These features allow one to question several financial models; in particular, they question the preference for positive skewness as a factor for investments in stock markets.     

Random matrix models for datasets with fixed time horizons

This paper examines the use of random matrix theory as it has been applied to model large financial datasets, especially for the purpose of estimating the bias inherent in Mean-Variance portfolio allocation when a sample covariance matrix is substituted for the true underlying covariance. Such problems were observed and modeled in the seminal work of Laloux et al. [Noise dressing of financial correlation matrices. Phys. Rev. Lett., 1999, 83, 1467] and rigorously proved by Bai et al. [Enhancement of the applicability of Markowitz's portfolio optimization by utilizing random matrix theory. Math. Finance, 2009, 19, 639–667] under minimal assumptions. If the returns on assets to be held in the portfolio are assumed independent and stationary, then these results are universal in that they do not depend on the precise distribution of returns. This universality has been somewhat misrepresented in the literature, however, as asymptotic results require that an arbitrarily long time horizon be available before such predictions necessarily become accurate. In order to reconcile these models with the highly non-Gaussian returns observed in real financial data, a new ensemble of random rectangular matrices is introduced, modeled on the observations of independent Lévy processes over a fixed time horizon.     

Collaborative Commissioning of Secondary Care Services by Primary Care Trusts

This article examines collaborations between primary care trusts in the commissioning of secondary care services in England. It applies principal-agent theory qualitatively to two case studies. The theory suggests that collaboration should take place if organizations share relevant information and agree joint objectives. The study findings show that sharing information is not a major problem for these case studies, but that agreeing joint objectives is.