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Using Probability to Set the Size of An Emergency Fund

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:

  1. (r2 – r1)- (p)(r2 – r1)<= (p)(rb)
  2. (r2 – r1) <= (rb)(p)+ (p)(r2 – r1)
  3. (r2 – r1) <= (p)(rb + (r2 – r1))
  4. (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.

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  1. Interesting idea, but I see three significant assumptions here if one is to borrow as their emergency fund:
    1) Someone will be willing to lend you money in your time of need
    2) You’ll be able to get the cash in a timely manner
    3) Given the “emergency” circumstances you’ll get the expected interest rate.

    What’s more, depending on the emergency, do you really want to be trying to get a loan during a likely stressful time. I think for most people, they would be best served with a reasonable sized pile of cash.

  2. This post was a surprise!

    I am intrigued by the resistance in general to viewing EFs rationally (not necessarily in expressed in this post, depending on interpretation of the penultimate paragraph or two) . Note that doing so does not presuppose making a decision based on other criteria. It is a bit like some folks’ feeling that the more you know about electrodynamics the less beautiful a sunset. Hogwash! Science and math need not detract from other ways of viewing things. They only add.

    The point of my posts on the matter is that one can use a bit of math and some reasonable assumptions to get a handle on the rough size of the issue. I don’t see how doing that can be any worse than the guesstimates one hears about elsewhere (6 months, 1 year of salary, etc.). Upon what are those based anyway? Time until one might find a new job after layoff? That’s not the only emergency one should consider.

    All my posts on this acknowledge that ultimately I don’t make my decision on the math. There is tremendous value to sleeping at night, even if it means being irrationally conservative. One should not be ashamed to admit that! And I am not, despite the math.

  3. Kevin raises some good points. Also missing from the analysis is the size of the emergency. If you have an emergency once every five years but only need a portion of your emergency fund each time, the math changes. The nature of the emergency may also vary the time it takes to either rebuild the fund or pay off the debt following the emergency.

    Emergency funds are not supposed to be a break even endeavor. It’s a form of insurance and has a cost, a small one.

    As for size, if you aim for six month’s worth of necessary expenses I like to apply the Pareto Principal. That is, 80% of your emergencies will likely only require 20% of you fund. So, even if your just getting started building an emergency fund, one month’s expenses will still cover most situations.

  4. Thanks for stopping by to reply. 🙂

    Interesting, by the way, that you said this: “Upon what are those based anyway? Time until one might find a new job after layoff?”

    That’s very close to what my own is based upon. It’s set to 6 months of our family expenses, with the reasoning that, should my business collapse and my wife get laid off (at the same time), 6 months should be enough time to find a full-time job doing some sort of accounting work.

    …and yes, that may well be “irrationally conservative.” (At least, I hope it is…)

  5. What happened to the ability to edit posts and remove typos?

  6. It’s the Pareto Principle not Pareto Principal 🙂

  7. Turned it off last week when I was updating WordPress and getting some errors. Then I forgot to turn it back on. It’s back on now and appears to be working for me.

    Thanks for the reminder.

  8. I too use a “months of salary” approach. But honestly I have little idea how long it would take to find a job. And that does vary with the economy considerably (!!!).

    My meta-point here is that there is a lot of huffing and puffing over this issue but very little actual quantitative analysis or sensible, rigorous theory. I don’t see the point in all the strong opinions if they can’t be backed up by something other than “sleeping at night.” That is how I decide this, and I freely admit it is a weak basis. It isn’t how I decide many other things for which more empirical evidence exists. (I don’t choose index funds because they help me sleep at night. It is the evidence that they work well that is what helps me sleep.)

    I would like to see more studies of emergencies and emergency funds. What size(s) are really necessary and for whom and why? Hatcher’s work is the first I’m aware of to attempt a rational theory. Good for him. We need more.

  9. @Austin Frakt, I certainly wasn’t suggesting that taking a rational approach to estimating the EF size is a bad idea, just that the probability approach is best used in a general sense, to estimate “rough size” and cost, and that the more one knows about the frequency and size of specific kinds of emergencies, the easier it is to plan for these for via sinking funds.

    I think @Dylan’s point comparing the EF to insurance is a good one, because it’s a similar transfer-of-risk argument, i.e., there’s always a cost involved in transferring risk, and the cost goes up/down depending on the probability of the event insured.

  10. If one doesn’t have a rough sense of the size of the emergency one is planning for one can’t justify any amount, whether months of salary or otherwise. We are all making assumptions based on experience. I am merely suggesting we do so after being informed with the best possible tools, techniques, and evidence (which I admit is not all that good, yet).

    Taking the insurance point of view, well, sometimes it is worth buying extra insurance and sometimes it is not. How does one decide in the EF case? Should you “buy” an extra month’s worth or not? How did you decide how to answer that? What are your assumptions? Why is that the right answer?

    That’s really what I’m getting at.

  11. With insurance, I don’t believe there is a “right” answer to the appropriate amount. Consider LTC, if you get hit by a bus and never use the policy, does that mean the “right” amount would have been $0 coverage. I don’t believe so.

    I’d argue an EF is similar. There is no “right” amount. The best amount for any one person/family can only be known in retrospect as is the case with every type of insurance I can think of.

    With EF and insurance questions, comfort level of the person in question is hugely significant. Discounting the “sleep at night” argument here is dangerous because all the analysis in the world can’t produce the “right” answer for each individual–maybe what would be “right” for each person on average, but not for each individual.

    All that said, I applaud efforts to narrow the range of what would make sense for most people as an EF. At the very least, it’s a place to start.

  12. Great post, I’ve seen this discussed previously but it’s not mentioned a lot.

    One things to consider the purpose of emergency savings fund is not have to cash out of other investments at an inopportune time. Once you get past a certain level of savings, at least from my experience, you no longer really need specific emergency savings allocation. You can take it from your fixed income portfolio and liquidate your other assets when you need to.

  13. 1.
    You write, “I would like to see more studies of emergencies and emergency funds.” No doubt these studies would show that, in the past, some have put too much into their EFs, and some, too little. But this assumes past performance is a guarantee of the future. It also assumes every individual’s circumstances will mimic the average. I think instead we should counsel people more towards not fussing if that mother of an emergency does not occur and instead celebrate the /peace of mind/ they get over the years by having a good emergency fund. Compute the value of that peace of mind please. It is no small number. Like one of the other posters says, this is about insurance.

    Those who worked on risk models for the mortgage securities etc. markets came up with seemingly pretty mathematical algorithms, too. But the assumptions were so large that the models in application were extraordinarily flawed. It seems to me that this is the case here, too. Add in: What kind of errors should one attribute to one’s estimate of the probability of an emergency? If one computes 1/5 using the approach above, to me 1/2 is not that far away, considering the errors in estimation. The three rates of return named by themselves may vary a lot from one year to the next. In other words, a factor of safety (well known in engineering circles, for one) should be used. This is maybe more famously known as a huge “fudge factor,” because uncertainties are so large and compound when combined in an algorithm.

    I think people should continue to plan their EF around their sense of the cost of a worst case scenario. Using a certain number of months of current paycheck makes more sense to me than arriving at a hard number as the alleged algorithm above seems to push. To me, the algorithm is a flimsy disguise for garbage-in, garbage out computation. I profane passing this off as good applied mathematics or even good finance guidance. Instead, there should be a follow-up to this article noting how so many got suckered into such thinking with the recent bubble, and look where it got us.

  14. There is probably no end to this debate. But I think those on the “other” side misunderstand my perspective. I like mathematical analysis because it helps me understand the problem more fully and does give some broad guidance as to what the solution should be. It is not the final word, nor should it be.

    One can charge “garbage in, garbage out” but one should apply that equally to all methods. I find the “Oh, I dunno, a year seems good enough” type approach no less garbage-y. It is just far less explicit about assumptions. But those assumptions are still there. They must be.

    Let me put it this way. I’m not denying the value of a good night’s sleep. As I’ve said (on my blog, if not here), that’s ultimately how I do it too. But I also think there is value in mathematical analysis. It is just another perspective. I can keep several in mind at once. Maybe that’s unusual.

  15. Austin Frakt, I agree with your statement, “I like mathematical analysis because it helps me understand the problem more fully and does give some broad guidance as to what the solution should be.” My view is that such disclaimers need to be much more prominent. I mean, it is okay to write, “This is really (mathematical) art and just for art’s sake.” 🙂

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