Finding the optimal allocation to a health-care reimbursement account

A health-care reimbursement account offers tax savings to eligible employees who are willing to face a risk of losses due to unused account balances. Finding the optimal allocation to such an account requires decision making in the face of uncertain future medical expense. The allocation which minimizes expected total cost (sum of medical expense, excess taxes and forfeiture) is found for a known household expense distribution. This distribution would not be known in practice and must be estimated. Using the gamma density function as an approximation the variability in this distribution (specifically, the shape parameter α) is estimated using U.S. Consumer Expenditure Survey (CES) data. Separate estimates are provided for groupings by age and family size. It is assumed that the mean is unique to each household; thus each family in the CES represents a distinct population. A transformation to an F distribution is used to estimate the common value of α for each group. The optimal allocation is a multiple of the estimated mean: the multiplier depends on α, the probability that no expense will be incurred, the household marginal tax rate, and its risk preferences between the two losses. Values of the multiplier are provided for selected cases. It is shown that the results are relatively insensitive to moderate errors in the estimation process.