Austin Frakt at The Incidental Economist recently wrote an intriguing post using Charles B. Hatcher’s work to rationally set the size of an emergency fund.
The basic idea of the post is that the ratio of the annual opportunity cost of forgoing a higher investment return to the cost of borrowing should an emergency occur can be interpreted as a probability, and this probability can be used to estimate the needed size of the emergency fund.
Or, in plain English: It’s rational to have an emergency fund if the cost of the emergency fund should no emergency occur is less than or equal to the cost of borrowing should the emergency occur.
Explained Mathematically…
M = the size of the emergency fund, r2 = rate of return of investments, r1 = the rate of return of the emergency fund, rb = the borrowing rate, p = the probability of an emergency occurring in a given year.
Using the above variables,
- The cost of the emergency fund if no emergency occurs = (M)(r2 – r1)(1 – p)
- The cost of borrowing if the emergency occurs = (M)(rb)(p)
So it’s rational to have an emergency fund if:
- (M)(r2 – r1)(1 – p) <= (M)(rb)(p)
…which can be reworked in the following manner:
- (r2 – r1)- (p)(r2 – r1)<= (p)(rb)
- (r2 – r1) <= (rb)(p)+ (p)(r2 – r1)
- (r2 – r1) <= (p)(rb + (r2 – r1))
- (r2 – r1)/ (rb + (r2 – r1)) <= p
Since (r2 – r1)/ (rb + (r2 – r1)) <= (r2 – r1)/(rb) we can simplify the expression to
(r2 – r1)/(rb)<= p
Applied to Real Life
Following Mr. Frakt’s example and using the equation above, if the liquidity premium (i.e., the amount by which the rate of return on other investments exceeds the return on an emergency fund) is 2% and the interest rate for borrowing is 9%, then the probability of an emergency in a given year needs to be greater than 22% in order for it to be rational to have an emergency fund.
In other words, you’d have to believe you’re likely to have an emergency at least once in five years to justify the opportunity cost of the fund given today’s liquidity premium of 2% and borrowing rate of 9%.
Using the once-in-five-years probability, Mr. Frakt proposes that we can guess the necessary size of an emergency fund based on our experience. What kind of emergencies occur within a five-year period and how expensive are they? The catch is that if you already know from experience how much money is needed to cover emergencies occurring once every five years, I wouldn’t call these emergencies at all. They are periodic expected financial events that can be planned for via sinking funds.
The whole point of an emergency is that it’s an unexpected event, and it’s very difficult to assign probabilities to unexpected financial events. Sure, we know that over a very long time frame we’re bound to encounter an event we didn’t plan on, but we can’t really know the probability of such an event occurring this year, next year or in ten years.
It it, however, useful to consider the probability/frequency of emergencies in a general sense:
- The higher the liquidity premium relative to the cost of borrowing, the more frequently emergencies would have to occur in order for it to make sense to pay the premium to keep liquid funds available, and
- The lower the liquidity premium relative to the cost of borrowing, the less frequently emergencies would need to occur to justify the liquidity premium.
About the Author: Susan D. Tiner, financial organizer and consultant writes the blog Brain Dead Simple! Financial Organizing.
February 23, 2010 15 comments
