How much money to put into a flexible spending account revisited

This past week I got my first notice from the Northwestern Benefits office that Open Enrollment will be coming up soon. That got me thinking about the question of how much money to put into my flexible spending account (FSA).

For those unfamiliar with FSAs, they are quirk of the US tax code. They allow people to have money taken out of their paycheck on pre-tax basis and then use that cash for qualifying medical expenses such as health insurance co-pays, uninsured dental work or new glasses. Users typically have to specify how much to put into their FSA in the fall. They then get to use the cash over the coming calendar year. The kicker is that FSAs have a use-it-or-lose-it property: If you don’t spend your entire FSA contribution, the leftover cash is gone into the ether.

I wrote about putting money into FSAs back in December — mostly to take exception to a Forbes assertion that people do not put enough money into FSAs. The gist of that post was that one could think of funding an FSA as a newsvendor problem. The newsvendor problem is the most basic inventory problems with uncertain demand. It determines the stocking level when the seller has only one chance to order before learning demand. In the FSA context, it says that if you know the distribution of your annual spending on medical items and services, you should fund your FSA so that you cover all of your spending with a probability equal to your marginal tax rate. Thus if your marginal tax rate is, say, 30%, then 30% of the time you should finish the year with extra cash in your account and 70% you should spend it all without going out and buying some more contact lenses. (If you’re not tracking all of that, go back and look at the old post which goes through it all in more detail.)

So what else is there to say? Two words: Risk Aversion.

The analysis I posted in December assumed that the decision maker was only concerned with her average return and didn’t care about any variance in her spending. It turns out that if the decision maker does not like uncertainty, she has an incentive to increase her FSA contribution. For a little intuition, consider covering 25% of your spending vs covering 99% of your spending. In the former case, 75% of the time you don’t know what your total spending will be. In the latter, 99% of the time all of your spending is covered.

An example might help. Suppose that our decision maker’s qualified spending averages $1,000 per year but is either $1,000-Δ with probability 1/2 or $1,000+Δ with probability 1/2 for some Δ between zero and 1,000. This is a simplistic set up but it gives an idea of how variability matters. The bigger Δ is, the more risk the decision maker faces.

Suppose that the decision maker puts Q≥ $1,000-Δ pre-tax dollars in her FSA and that her marginal tax rate is τ. If we look at her average spending at the end of the year (in post-tax dollars), we get:

RN(Q)=-(1/2)(1-τ)Q-(1/2)[(1-τ)Q+((1,000+Δ)-Q)],

where RN stands for “risk neutral.” The first term corresponds to a low spending year. Q pre-tax dollars translates to (1-τ)Q post-tax dollars and that covers all the spending. If spending is high, our decision makers needs to come up with an extra (1,000+Δ)-Q post-tax dollars. A bit of algebra lets us rewrite RN(Q) as

(τ-1/2)Q-(1/2)(1,000+Δ).

So in the risk neutral case, utility is increasing in Q if the tax rate is above 50% and she should set her FSA contribution to $1,000+Δ. If τ is less than 50%, she should it to 1,000-Δ. Let me assert that this is what I stated above simplified to this simple, two-outcome distribution.

But how does this change if our decision maker is risk averse? Specifically, let’s endow her with a CARA utility function, U(x)=1-Exp(-βx). This is a common assumption and allows a clear measure of risk aversion: a higher β corresponds to a higher level of risk aversion.

I’ll spare you the math, but the optimal FSA contribution now has a surprisingly simple form:

Q*=min{$1,000+Δ-Log[(1/τ)-1]/β,$1,000+Δ}

There are a few things to note here. First, Log[(1/τ)-1] is negative if the tax rate is greater than one half. Thus if the risk neutral decision maker were to cover all her spending with her FSA, our risk averse decision maker will as well. Otherwise, the optimal stocking level is a discount off the maximum spending of $1,000+Δ. How big the discount is depends on the tax rate and our measure of risk aversion β. the more risk averse the decision maker is, the smaller is the discount. Similarly, the higher the tax rate, the smaller the discount. As long as the tax rate is not zero and β is not too small (specifically, greater than Log[(1/τ)-1]/(2Δ)), the risk averse decision maker puts more money in her FSA than the risk neutral decision maker.

So do Americans put too little money in their FSAs? If you believe that Americans are very risk averse in spending in this regard, maybe not. Given a sufficient level of risk aversion, the chance of having excess cash in one’s FSA could be well above the marginal tax rate.