The simple formula to improve your decision making

I just watched The Big Short last night and became inspired to write this post. Having read the book years ago, I didn’t know what to expect, but I’m happy to say they’ve done an exceptional job. The movie offers the most accurate depiction of the years leading up to the 2008 financial crisis that I’ve seen. It doesn’t mean it didn’t take some artistic liberties, but far less so than many so-called “reputable” news outlets. I watched it with my wife and was impressed how it managed to keep her attention despite stories about my day job generally being her favorite natural sleeping aid. I highly recommend the movie to everyone, but I digress.

The Formula

I think the movie illustrates that we as people are very bad at estimating and properly weighting risk, both in our day-to-day lives, and in our investing. Therefore I want to introduce a simple and powerful way of thinking about risk through the following formula:



Beyond finance

I recently had a conversation with a friend about whether cycling in downtown Toronto is more dangerous than driving in downtown Toronto. His argument rested on the fact that there are fewer bike accidents than car accidents per year, making biking safer. He focused in on “# of times bad things happen” part of the formula.

Why is this incorrect? There are even fewer “accidents” resulting from getting trashed and trying to frogger yourself across the QEW.  It doesn’t mean it’s a safer thing to do.

In order to truly assess risk you need to know more than the number of accidents cyclists got into (# of times bad things happened). You also need to know how many total bike trips have been taken as well as how badly the cyclists were hurt when they got into accidents (impact of outcome).

Let’s say that cyclists get into accidents every 100 trips they take, but cars get into accidents every 50 trips taken. Which is safer?

Probability of bike accident = 1/100 =1%
Probability of car accident = 1/50 = 2%

Cars are riskier right? Actually the above tells us nothing about the relative riskiness of the two methods of transportation. We need to consider the impact of having an accident. Let’s say that cyclist’s get three times as badly hurt (on average) as car drivers when they do get into an accident, what is less risky in that case?

EVbike = 1/100 * 3 = 3%
EVcar = 1/50 * 1 = 2%

When taking into account and quantifying the impact, it is now cycling that appears to be the more dangerous of the two modes of transportation.

Before I get a bicycle helmet thrown at my head I just want to point out these are not real stats. I actually have no clue which mode of transportation is safer. In fact I doubt anyone really does because it’s hard to get a reliable estimate on the number of cycling trips taken. I’m just debunking the idea that comparing the number of accidents gives us any indication of risk.


While sometimes difficult to use in everyday life, this formula is a great starting point of any investment analysis. To illustrate let me go back to a scene in the “Big Short” movie when the Cornwall Capital guys decide to short (that means profit from the demise of) AA mortgage bonds (relatively stable) versus BBB mortgage bonds (far worse quality). They make this bet despite the fact that the latter are far more likely to fail. Why did they make a bet on the far less likely outcome, rather than take what seemed to be the sure thing? The movie does not address this well, but it comes down to the EV formula above.

The guys would get back 5 times their money if BBB bonds failed, but they would get 20 times the money if AA bonds failed. If the probability of BBB bonds failing was assessed at 90% and probability of the AA bonds failing at 40%, which bet would you take?

EVaa = 40% * 20 = 8
EVbbb = 90% * 5 = 4.5

According to EV logic you should actually take the AA bet, despite the fact that you are less likely to be right, because it has almost double the expected value of the alternative.

Investing with non-binary outcomes

The above analysis works because there are only two possible outcomes and one of them involves losing everything. If the guys are wrong about the mortgage market, and none of the bonds fail, they lose their entire investment regardless of the bet taken. This is what’s meant by a binary outcome, you either win or lose everything, but nothing in between. In reality most investments offer a fluid set of possibilities. However, in many cases an initial analysis can still be done using a multi-part EV.

EV of investment = EV of good outcome – EV of bad outcome

Let’s say you believe all the news stories about the housing market in Toronto flattening out. This means you would expect prices to start stagnating or rising very slowly for a number of years. In this scenario, after taking into account mortgage and other ownership costs, you should expect real returns on house investments to be somewhere between 0 and 1% a year. You assign this scenario a 95% probability.

EVgood = 95% * 1% = 0.95%

Let’s say that you also believe there is a very slight possibility that there is a bubble and it’s going to burst. Given that rates cannot go below 0 (at least not much), the Bank of Canada would be powerless to stop the decline of prices by lowering rates, likely leading to at least a 30% decline. You assign this scenario a 5% probability.

EVbad = 5% * -30% = -1.5%

You can then combine the two events to determine whether you should invest:

EV = 0.95% + (-1.5%) = -0.55%

Since the overall EV is less than 0 you should not make this investment even if you think there is no bubble and prices will not fall. The near certainty of modest returns in the future is more than offset by the small probability of a severely bad outcome.

While I used only two scenarios, you can use this same process to come up with as many as you like, and add them together to come up with an EV value for the investment you plan to make. You can also calculate EV values for a number of competing investment options to help you decide which offers the best risk/reward balance.


I know it’s hard or near impossible to accurately quantify the impact and/or probability of an outcome in many cases. This is why assessing risk is something that requires years of practice and a good intuition. However, making decisions without taking into account all the factors represented in the formula is extremely dangerous. The expected value formula is simple starting point to anchor my thinking. It is the minimum that I consider when making important decisions under uncertainty.

I hope the above is relatively clear and you’ll find it helpful in your decision making. I know it has helped me make the right decision many times when the choice that seemed superficially obvious was the wrong one. Not to mention, thinking this way also has the added benefit of completely frustrating my wife and friends. Can’t wait until my boys are old enough! They’re going to love this!

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2 thoughts on “The simple formula to improve your decision making

  1. Really

    Dear Sir,

    Some people understand quickly. I am not one of them.
    It take me a great deal of effort and I mean an extraordinary amount of time to understand anything.
    This article is so beautifully and simply written – I am extremely grateful for your explanation.
    The same goes for the bitcoin article. There is remarkable clarity of thought compared to the blogs I read these days. It is could very well be part of a book and I believe you should author books. Thank you !

  2. Elicia Smith

    The tradtional path to success is typically described as working hard in an area of expertise in order to rise to the next job, position or career acomplishment.

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