Conventional wisdom is that we all should have an emergency fund (EF), a chunk of cash (or equivalent) set aside for use when financial trouble strikes. When employment prospects are good, I’ve seen recommendations for EFs as low as three months of salary. In times of high unemployment, some recommend EFs as large as one year’s worth of salary.

These are seat-of-the-pants estimates. If anything they’re based on how long it might take you to regain employment after loss of a job. But income replacement isn’t the only purpose of an EF. One might tap it in the event of any financial emergency (e.g. unexpectedly high health care costs, urgent home repair, etc.). My interest in this post is to explore a rational means by which to set one’s EF size. It is based on the work by Charles Hatcher, summarized in a prior post. (See also the Bogleheads Forum discussion of Hatcher’s methods.)

In Should Households Establish Emergency Funds? Hatcher compares the opportunity cost per year of having an emergency fund (the difference in rates of return between a more aggressive investment and that of the EF cash equivalent holding, denoted by the liquidity premium) with its per-year benefits if an emergency occurred (the avoided borrowing costs). The result is a simple expression for the minimum probability, *p*, of an emergency in a given year such that holding an EF is rational: *p=(r _{2}-r_{1})/r_{b}*, where

*r*is the rate of return on investments,

_{2}*r*is the rate of return of one’s EF, and

_{1}*r*is the borrowing rate (Hatcher suggests a APR/2 is a good estimate of the borrowing rate).

_{b}Hatcher assumes that the EF is equal in value to the cash needed in an emergency so the expression of the probability of rationally holding an EF given above is independent of EF size. That provides an opportunity to use it to determine a rational EF size as follows. If we interpret *p* as the lower bound on the annual rate of emergency such that an EF is rational we can ask: what is a typical size for an emergency that occurs with at least that frequency? The answer to that question provides some guidance to the rational EF size.

Let’s use a concrete example. Suppose borrowing costs are 18% APR, which is typical of some credit cards but much higher than a home equity loan. Then *r _{b}* is about 18%/2 = 9%. Suppose also that the liquidity premium is

*r*= 2% then the probability of an emergency must exceed about 22% to justify an EF. That’s about one emergency every five years. Based on one’s own experience one might have a general sense of the size of an emergency that’s likely to occur at that rate. I certainly do. For me this would suggest that an EF no larger than two months of salary ought to be sufficient. Of course, my borrowing costs may be even lower and liquidity premium may be higher. So maybe I could get buy with an even smaller EF.

_{2}-r_{1 }Instead of relying on my own intuition of typical emergency size over some time period, I’d love to see a study of such a thing. For example, an analysis of variation in monthly household outflow relative to income as a function of important variables like household size, age of head of household, education, and other relevant economic and demographic variables would be useful. From such a study one could estimate various measures of the likely largest emergency that would occur over an interval of time.

I think the foregoing approach is one reasonable place to start in determining a rational EF size. But I wouldn’t end here. I’m not convinced one can know one’s borrowing costs and liquidity premium at the time of an emergency. The former could be higher and the latter lower, suggesting an even higher EF is rational. That’s one reason to increase one’s EF beyond the size suggested above. Another reason is peace of mind. There is no law against being irrationally conservative. Sleeping well at night is worth something.