# How much should I put in my FSA?

## Optimizing the average case with a probabilistic approach; a.k.a. The real nerdwallet

Flexible spending accounts (FSAs) are an instrument for saving taxes on medical expenses in the USA. At the beginning of the year (or when you start a job) you can specify how much money you will contribute to the FSA.

The money assigned to the FSA has 3 properties:

1. It can only be spent on “qualified medical expenses”
2. No tax is deducted on money allocated to the FSA.
3. You will not receive any money back from the FSA even if your medical expenses are smaller than the amount you allocated to the FSA. I.e. if you put 100 dollars in the FSA but you only have medical expenses of 60 dollars then you will lose the remaining 40 dollars as well.
A common question that people ask is, “How much money should I put in my FSA account?” In this article, I propose to decide the FSA allocation by minimizing the expected total expense by treating the annual medical expense as a random variable. Keep reading for details.

Let x be FSA allocation. let y be the annual medical expense. Let f(y) be the probability distribution of y. Assume a taxation rate of r, e.g. if the tax rate is 25% then r=0.25. A reasonable way for choosing the FSA is to minimize the total expense that you have to incur. We can write our total expense s as a function of x and y as follows:

𝑠(x,y) =𝑥−𝑟𝑥 If 𝑥>𝑦 else 𝑦+(𝑦−𝑥)𝑟−𝑟𝑥

Let me first unpack this function in words. If our FSA allocation, x, is greater than the medical expenses, y, then we will mandatorily lose our FSA deposit x but we will save taxes at the rate of rx. So our total expense will be x - rx. On the other hand if our medical expense is larger than FSA then we will have to pay our medical expense, y, and tax on the difference between the two, i.e. r(y-x), but, we will still save tax on the FSA allocation so the total spending s(x,y) is y(1+r)-2rx.

Now we can chose the optimal FSA amount x by minimizing the expected value of 𝔼[s(x,y)] where the expectation is over the random medical expense.

x* = argminₓ 𝔼[s(x,y)].

Let us consider a simple numerical example. Also say that you are considering getting a new eye lenses this year but you are not sure whether you will go ahead or not. Without this procedure your medical expense will be \$100 and with this expense it will be \$200, and you have a 50/50 chance of doing the procedure, then 𝔼[s(x,y)] = (s(x,100) + s(x,200))/2. Then 𝔼[s(x,y)] is a piecewise linear function of x with kinks at 100 and 200.

If your tax rate is 33% then the middle case simplifies to a constant value of \$133.33 and the expense is higher than \$133.33 if x<100 or x>200. So in this example, the optimal amount of money to put in the FSA is anywhere between \$100 to \$200. But if your tax rate is 25% then you should put only 100 dollars in the FSA so that your expected expense is \$137.5. If you put \$200 in the FSA then your expected expense will be \$150. But if you put 0 dollars in FSA then with a tax rate of 0.25 you would have spent \$187.5 so the FSA can still help save a lot of money when used judiciously. In general it seems that people with lower taxes are better off putting lesser amount in their FSA account but not 0. This flies in the face of advice given at nerdwallet.

Note the paradoxical phenomenon that a person in a lower tax bracket (25% compared to 33%) appears to spend more in expectation than a person in higher tax bracket because of the FSA! This paradox occurs because we are measuring the effect of the tax savings due to FSA in alleviating an absolute medical cost. Obviously people with higher taxes get more benefits from tax savings.

In conclusion, I proposed a reasonable method to optimally allocate money to your FSA account. For better results a more careful model of the probability distribution of the medical expense f(y) should be made by smoothing the distribution. E.g. with a uniform distribution between two extreme values. Simple numerical integration methods can be used to plot 𝔼[s(x,y)] as a function of x.