Perhaps you are like me, and don’t. My excuse for not having one is that my medical expenses vary from year to year, and therefore I’m reluctant to set aside dollars into an FSA, only to lose those dollars if I don’t spend them.

Is that the situation you are in as well?

While conducting research on my part of the capstone project for completing the Executive MBA program at the Naval Postgraduate School, I had an epiphany that is changing my mind about the FSA. My new insight is that the FSA problem resembles in many respects the kind of resource management decision I encountered several times in the EMBA program. The decision of how much to invest in an FSA is related to a classical problem in resource management called the newsvendor problem.

The newsvendor problem is often described as the problem one faces when given a single buying opportunity where that choice maximizes profit in the face of uncertain future demand. This is often illustrated in the classroom with the situation faced by a paperboy, who buys a block of newspapers once per day (and only once per day) at a particular cost, and sells those papers for a profit. If he runs out, he is done for day and can’t buy any more papers until the next day, thus missing out on potential profit. Leftover papers are worthless, since they are yesterday’s news. How many papers should he buy at his one buying opportunity for the day if he wants to maximize his profit?

Some may say that he ought to buy the historical average. However, this assumption was shown in a different context in the late 1800’s not to provide the optimal solution. Instead, the best solution is found through a set of reasonable assumptions that the distribution of historical demand for papers (or whatever) follows the classical bell curve shape centered around an average value with a standard deviation that describes the width of the bell curve, and also by considering the relative cost of having too many versus the lost opportunity cost of not having enough.

The optimal solution for the newsvendor problem (which you can find on the internet, such as at Wikipedia) is found through a two-step process. First, we have to weigh the cost of overage (i.e., the cost of having too many) versus the cost of underage (i.e., the cost of having too little). The balancing probability P* that maximizes profit is found via the critical ratio:

P* = Cu / (Cu + Co)

where Cu = cost of underage and Co = cost of overage. You can see with a little algebra that the only case for when the probability is fifty percent is when the cost of overage is equal to the cost of underage. For most cases, the cost of overage is different from the cost of underage. Therefore, this tell us that to get the optimal order quantity, we need to adjust away from the average by taking into account how the two costs relate to each other. The optimal order quantity is found by the following relationship:

O* = average + z * standard deviation

where the average and standard deviation are determined from historical demand, and z is the number of standard deviations above the mean needed to meet all of the demand P* percent of the time, which we can find in a statistics table. Given that P* is a function of the cost of overage and underage, z represents the number of standard deviations we want to adjust from the average based on their relative weights.

Here is an example. Suppose that a paperboy buys a block of papers for 55 cents each, and sells them for $1.50 each. Suppose that history says that the average daily demand is 100 papers, with a standard deviation of 30 papers. To the paperboy, the cost of overage is 55 cents per paper – this represents the cost of not selling a paper, and thus what he has to cover from his profit for each paper he bought that was not sold. The cost of underage is 95 cents per paper – this represents the lost profit of not having a paper to sell to meet the demand for the paper he could have sold. Using the equation above,

P* = 0.95 / (0.95 + 0.55) = 0.633

Therefore, taking into consideration the cost of overage versus the cost of underage, the paperboy ought to plan on selling enough papers to meet all the demand 63.3 percent of the time. So, how many papers is that? For that, we need to look up the z-value for 0.633 in a statistics table, or use the Microsoft Excel NORMSINV function. Doing that yields z= 0.34. Therefore, the optimal number of papers he should order is:

O* = 100 + 0.34*30 = 110.2, rounded up to 111.

Hence, the paperboy ought to buy 111 papers to maximize his profit.

The situation of the paperboy is very analogous to what we face in deciding how much to set aside in an FSA. Similar to the paperboy, we are trying to maximize “profit” – in our case, it is tax savings. In contrast to the paperboy case, we aren’t concerned about having too many or few papers; instead, we are concerned about having too much in our FSA versus too little. How do we apply the newsvendor problem to this situation?

As an example, I pulled my financial records to see how much in medical and dental “eligible expenses” I spent each year for the last few years. (Eligible expenses are items such as doctor copays, prescription drugs, etc. Each FSA plan provides a detailed list of what are eligible expenses.) My historical average per year for the last 12 years is $678, with a standard deviation of $363. The highest I spent in any one year was a little over $1,300 and the lowest was a little less than $200. I point this out because that $1,100 spread was the basis of my concern as to why I chose not to invest in an FSA. My previous simple-minded thinking was telling me that if I saved more than $200 in my FSA, I risk losing the unspent funds. However, the newsvendor problem provides a different perspective, provided we can characterize the cost of overage versus the cost of underage. I’ll derive both for an FSA using marginal analysis.

The concept behind an FSA is to set aside pre-tax dollars to spend on eligible medical and dental expenses. In my case, I’m in the 25 percent marginal tax bracket. Let O* represent the ideal number of dollars I should set aside in an FSA to maximize my outcome. If I let D represent the total amount of eligible medical and dental expenses in a given year, I note the following:

If O* is greater than D, I didn’t spend all the FSA dollars I had set aside. This is the cost of overage case (i.e., I had too many dollars in my FSA account). What is the overage? If I would have reduced O* by one dollar, that would have been a pre-tax dollar I could have kept, which converts to 75 cents after taxes because I’m in the 25 percent tax bracket. Therefore, the cost of overage is 75 cents – I lose 75 cents for each dollar I contribute but don’t spend in my FSA. (Said another way, you don’t lose a whole dollar in your unspent FSA since they are pre-tax dollars; instead, you lose 1 minus your marginal tax bracket for each unspent dollar.)

If O* is less than D, I didn’t have enough in my FSA account and thus am paying for expenses with after tax dollars over and beyond what was covered by my FSA account. This is the cost of underage case. What is the underage? If I would have increased O* by one dollar, I would have lost 75 cents in after tax dollars because it is now in my FSA account (again because I’m in the 25 percent tax bracket), but save the one dollar of pre-tax dollars I would have had to spend otherwise. Therefore, the cost of underage is 25 cents – I lose 25 cents for each dollar I don’t contribute to the FSA that I had to cover with after tax dollars, because I’m paying for that expense with after-tax dollars instead of pre-tax dollars from my FSA.

As you can surmise from the above, if your marginal tax bracket is different than mine, your cost of overage is found by taking 1 minus your tax bracket, and your cost of underage is your tax bracket.

In my case, the probability that balances the cost of overage and cost of underage in is found by plugging each into the P* equation from earlier. In my case, it results in the following:

P* = Cu / (Cu + Co) = 0.25 / (0.25 + 0.75) = 0.25

This means that I will have leftover dollars in my FSA account 25 percent of the time, and that I will burn through my entire FSA allotment 75 percent of the time. Algebra reveals that no matter what your tax bracket is, Cu + Co always = 1. Therefore, the probability that balances the cost of overage and cost of underage is always equal to your marginal tax bracket.

Looking up the z-value for 0.25 yields -0.674. Yes, it is a negative number. That means that I want less in my FSA account than the historical average of my eligible expenses. The amount I want is therefore

O* = $678 + (-0.674)*$363 = $433

Again, this means my chances are 25 percent that I’ll not spend the entire $433 in my FSA account in a given year and will have leftover dollars I lose. However, 75 percent of the time, I will burn through the $433 in my FSA account. Setting aside $433 in pre-tax dollars results in a tax savings of $108 in the 25 percent marginal tax bracket.

What about that other 25 percent of the time when I don’t use all my FSA dollars? To look at this situation, I’ll do a breakeven analysis. In my case, I’ll break even between what I gain in tax savings versus what I lose in unused FSA contributions if I have at least $325 in eligible expenses ($433 – $108 = $325). In other words, this is the point at which the amount I would lose in unspent FSA dollars equals the tax savings of $108. I want to find the z value that corresponds with $325, so that I can find the probability that I’ll spend less than $325 dollars. Rearranging the O* equation, dropping the * designation and solving for z yields

z = (O – average) / standard deviation

With my average and standard deviation,

z = (325 – 678) / 363 = -0.97

Looking up -0.97 in a statistic table for the corresponding probability, or using the Microsoft Excel NORMSDIST function, yields 0.165. This means 16.5 percent of the time, I will spend less than $325. A 16.5 percent chance is equivalent to a one-in-six chance of occurring. (This is corroborated from my previous expenses – I spent less than $325 per year twice in the twelve years comprising my data set). I’m willing to tolerate that risk.

I’ve provided a spreadsheet that will do the calculations for you.

Therefore, given these odds and what the newsvendor problem is telling me, the tax saving is worth the trade to me. I’ll be signing up next year.

## Leave a Reply

You must be logged in to post a comment.