Optimized Flexible Spending

How to Maximize a Flexible Spending Account – Modeling After the Classic News-Vendor Problem.

Flexible Spending Accounts (FSAs) are becoming more popular among organizations and individuals as the stress on health-care has been discussed by the media. The simple advantage of an FSA is that the way that funds are deducted from your paycheck allow you to spend pre-tax income on health-care related expenses. This means you’ll save an amount equal to the taxes you would have paid on the money you set aside. You can use funds in your FSA to pay for certain medical and dental expenses, including co-payments and deductibles. Sometimes even travel and parking related expenses can be included in an FSA. Yet, as expected, the adoption of such account requires the account holder to reasonably predict how much they will spend on health-care during a 12 month period. Depending on your employer, a 2.5 month grace period, or a $500 carryover allowance may be given. Note that these possible options are mutually exclusive and not always included. At the end of a year (or period that includes the grace period) the funds in your FSA are removed. Some who are familiar with game theory may have already realized the connection to widely studied News-Vendor Problem. To formulate this problem as a News-Vendor problem, it is realized that the FSA customer has the option to purchase a limited number of “doctor visits” at the beginning of the season. As these funds are added to the account in a lump sum, it parallels a news vendor purchasing papers each day. To complete the analogy, we must consider the entire FSA period as a single day, as compared to the News-Vendor Problem. Indeed, it is also realized that in this case we are not maximizing profit, but total potential tax savings.

Knowing that the expected profit change with the purchase of an additional paper in the traditional News-Vendor model is:

Where y is the number of copies sold, p is the selling price of one copy, h is the cost to the publisher per copy, and F ̅ is the Complementary Cumulative Distribution Function (CCDF) of selling y copies. We can transfer this intuition to the FSA model by representing the expected tax savings seen with each additional doctor visit as:

Where (1/(1-t))φ is the additional tax savings with each additional visit, and φ is the cost of each visit. P(X≥k) is the probability of the individual/family visiting the doctor more than k times. It represents the CCDF of the family going to the doctor's at least k times. Note that the optimal number of visits would be k=y/φ, when y is the lump sum amount invested in the FSA. Now, a little algebra gives:

In order for the additional office visit cost to not outweigh the tax savings it must be that:

Combining the previous two equations we get:

Considering the base case of never visiting the doctor (k = 0), we would never claim any tax deductions. Thus, it is seen that we want to maximize the number of visits without violating the above criteria. Seeing this, the optimal amount we should invest into the fund (purchasing k visits) is:

Note that this makes sense as P(X≥k) is a decreasing function (see example graph below). To extend the model even further, consider using the following Poisson probability mass function:

Let λ=12 , φ=25, and t=0.22. These values represent the Poisson parameter Lambda, the cost per visit, and the effective tax bracket percentage that the account holder falls into, respectively. In this case, k is the number of doctor visits purchased. We know that y*, the optimal amount to deposit in the FSA, is described as:

We also know that the CCDF for our model is:

Plugging in our values we get:

Using Excel’s cumulative probability capabilities, the value of k that makes the above inequality false is 10. Thus, taking k = 9 gives max{k:P(X≥k)≥1-t} to be correct. Multiplying the number of visits, k, by the cost per visit, φ, we get a total amount of $225 that should be invested in the FSA.

Of course, this amount to deposit only represents the optimal funds to carry due to doctor visits. To consider deductibles, medication, travel, and parking costs, similar models with different, logical probability mass functions can be combined with the given example to give an even more robust justification of FSA deposit amounts.