Long-term care (LTC) costs and, in particular, those arising under an LTC insurance contract, are difficult to estimate. This is because of the complex effects of the processes of aging-disability and cognitive impairment. As disability is a gradual, as opposed to a discrete, process, and as the effects are sometimes reversible, a fairly complex model is necessary to capture its nature. This paper concentrates on modeling the disability process of aging only and, in particular, fully incorporates the recovery process as dictated by the data. With the recovery process modeled, the effect on the estimated model costs of disability of the common simplifying assumption that recoveries can be ignored is easily assessed.

This paper has twin objectives: (1) to present novel methodology, the penalized likelihood, for using interval-censored longitudinal data, such as the National Long-Term Care Study, to parameterize Markov models; and (2) to estimate the costs arising under an LTC insurance contract in respect of disability. The model is also used to show that ignoring recovery from disability can lead to significant overestimation of LTC insurance costs-suggesting that claims underwriting in LTC insurance may be an important factor in managing claims costs.

1.Modeling Long-Term Care Costs Modeling the processes that lead to claims under a lone-term care (LTC) insurance policy is complex. There are different underlying causes of claiming (e.g., physical versus mental deterioration), and the processes being modeled may involve progression through a number of states of health (e.g., progression through states of varying disability), rather than being binary as in life insurance (alive-dead), which, in turn, make it difficult to specify objective claims criteria. In addition, events leading to claiming are often reversible (e.g., people can recover from some types of disability)-a factor that can be difficult to allow for, unless a suitable modeling framework is used. Indeed, previous researchers have often assumed that recovery from disability is not possible, or their models incorporate it in a very approximate manner (Alegre et al. 2002; Dullaway and Elliot 1998; Haberman and Pitacco 1999; Nuttall et al. 1994; Rickayzen and Walsh, 2002). The continuous time model proposed in this paper has no such restriction, and recovery from disability is fully incorporated in the model. This allows the effect of ignoring recoveries to be quantified, as well as provides some insight into the importance of claims underwriting in LTG insurance.

In the United States, interval-censored longitudinal data on disability have been available since the mid-1980s, from the National LongTerm Care Study (NLTGS) in 1982, 1984, 1989, and 1994. Interval censored means individuals in the study are interviewed at fixed time points, resulting in their disability status being known at these points in time, with the number and timing of changes in some status of interest (e.g., disability status) unknown. These extensive studies, undertaken at great cost, include data on more than 35,000 lives-making them unique in scope. There were linked series of questionnaires establishing loss of activities of daily living (ADLs) and instrumental ADLs, and institutionalization. In a Markov framework, as used in this paper, interval-censored longitudinal data give rise to estimates of a transition probability matrix over an extended time period. There is then the problem of how to "convert" these probabilities, within the Markov framework, to realistic (positive and real) parameter estimates (transition intensities), which are more useful for actuarial applications. The method we propose is intuitive, produces reasonable estimates, and is flexible enough to work where other methods cannot be applied.

2.The goal of this paper is to describe and parameterize a continuous-time Markov model of the disability process and to look at the cost of disability under an LTG insurance contract. The data used to estimate the model parameters are from the 1982, 1984, 1989, and 1994 NLTCS in the United States (1997 NLTGS Public Use GD)-but this paper focuses on the work done with the 1982 and 1984 NTLGS in particular. Though not discussed in this paper, NLTCSs have also been undertaken in 1999 and 2004.

We start by introducing LTG insurance contracts in section 2. Then in section 3, after describing the data and some previous research that has used the data, the model and statistical framework are introduced. In section 4, after discussing the reasons standard maximum likelihood estimates cannot be calculated from the data, we look at how previous researchers have dealt with this problem. A novel method is then proposed and implemented to obtain estimates of the model's parameters. In section 5, confidence intervals for the parameters are estimated and then used in the graduation process. We use the parameterized model in section 6 to calculate the expected present value (EPV) of model LTG benefits in respect of disability. In particular, we look at single premiums for a range of sample policies and investigate the effect on model premiums of ignoring recovery from disability. Conclusions and discussion are provided in section 7.

The same measures of disability were used in all years and were defined by the inability to perform one or more of eight instrumental activities of daily living (IADLs, including light housework, laundry, meal preparation, grocery shopping, getting around outside, getting to places outside within walking distance, money management, using the telephone) or one or more of six ADLs (eating, getting in and out of bed, getting around inside, dressing, bathing, getting to the bathroom or using the toilet) without using personal assistance or special equipment.