Expectancy – Millionaire Math That Converts Uncertainty into Profit
How To Take Control Of Your Financial Future And Engineer Your Success
Key Ideas
- Discover how mathematical expectancy converts the uncertainty of an unknowable future into a planned process that's scientific
- Surprise! How often your investments win is not the most important factor to your financial success (but this is…)
- 7 ways you can use expectancy to profit more consistently in your financial plan.
How can you reliably profit from investing when the future is unknowable and the markets appear to be random?
How can you consistently grow your career and improve your earning capacity when office politics and industry change get in the way?
How can you reliably grow your wealth so that your financial goals are simply a question of “when”, not “if”?
The answer to all of these questions is mathematical expectancy.
(Don’t worry if you’re math-phobic because the principles are what matter – not the math – and the principles are simple to understand.)
Mathematical expectancy gives you a set of proven rules that guide your plans and actions so that you can dramatically improve both the reliability and quality of your results.
It works in investing, career planning, health, finance, and much more. In fact, it’s such a fundamental truth that it’s hard to find any aspect of your life where expectancy analysis can’t improve your results.
Mathematical expectancy is how you convert an unknowable and uncertain future into statistical confidence. It’s how you convert doubt into a predictable outcome.
When you understand how mathematical expectancy works, it will change how you play the wealth building game forever.
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It’s All About Expectancy
Expectancy and the closely related principles of risk management and leverage are the three most important factors determining your future success.
In this article, I’ll explain expectancy using financial terms (since this site is “FinancialMentor”, after all), but please understand that what you’re learning is equally applicable to most other areas of your life. The financial world is just an obvious example because it can be easily quantified and proven.
That’s because all wealth is math, and there are two equations that govern how your wealth grows.
The mathematical expectancy equation determines your compound growth rate, and the future value equation determines what it will grow to, and by what date.
When you combine these two equations, you have a complete framework for understanding your wealth growth process. This applies to both your investing, and the comprehensive wealth plan that your investing fits into.
Unfortunately, most people have only a vague understanding of how expectancy works or what it means. Most people are turned off by math, so the topic is rarely discussed in the press or in bestselling business books (except my books here).
This is unfortunate. Readers are missing out, because mathematical expectancy proves that wealth planning is a rational, duplicable science that can be reduced to equations and principles that are safe and smart to use.
These equations define the scope and shape of the process by forming boundaries around the knowledge required, which then provides a clear direction for best practices.
More importantly, mathematical expectancy is particularly interesting because it converts the uncertainty of an unknowable future into a plannable process that is clear and scientific, and that has predictable outcomes.
How Expectancy and Probability Interact in the Real World
Expectancy goes by many names, including expectation, mathematical expectation, EV, average, mean value, mean, or first moment.
What it tells you is how much you can expect to make, on average, per dollar risked.
That definition clearly connects expectancy to your wealth growth, so let’s look at the formula:
Expected Value =
(Probability of Win * Average Win) – (Probability of Loss * Average Loss)
While that’s pretty straightforward, let’s make it even simpler and more intuitive by reducing it to just two variables: probability times payoff. It’s the probability of something occurring multiplied by the payoff when it occurs.
In other words, you already understand probability, which is the odds of something occurring. Everybody gets that. A fair coin has 50% odds of heads coming up on any flip.
Expectancy simply adds one more dimension by multiplying the probability of something occurring times the payoff you get when it occurs. That difference is critical.
For example, what happens if heads pays $5 and tails loses $2? And how does expectancy change when heads pays $7, but tails loses $8?
Those questions are answered by expectancy, not probability.
So what you should notice is how probability is the odds of something occurring, but expectancy tells you the financial impact those occurrences have, and that’s where leverage and risk management come into play.
The key thing to notice is how the payoff dimension completely changes the math. It converts the already intuitive odds of something occurring into something different – something that eludes most people, because we're not trained to think in terms of two dimensions with a payoff variable.
Expectancy is the result of how much you make when you’re right, minus how much you lose when you’re wrong, multiplied by how frequently you’re right or wrong.
That net number is the average amount you expect to make each time you put your capital at risk, which determines your return on investment in your future value equation.
How To Convert Uncertainty into Opportunity!
Now that you know how expectancy determines the growth of your wealth, let’s switch gears and connect all the logic blocks I’ve shared so far into a single picture that shows you how it all fits together and ties both leverage and risk management as essential disciplines in your wealth planning strategy.
(Note: The analysis below, while complete, is a high level overview (by necessity) so that it can fit in an already too-long article. For full development of each idea and the actionable steps to apply in your wealth plan, please see the full Expectancy Wealth Planning course here.)
- Expectancy analysis is how you estimate outcomes that are uncertain. The fact that all your investments, business, and life plans are a bet on an unknowable future is, by definition, an uncertain outcome. That’s why expectancy analysis is required. It’s the scientific, reliable way to manage the risk of the unknown.
- Expectancy analysis is how you make smart financial decisions when all outcomes are uncertain. It gives you a scientific, rational way to reduce risk and maximize reward using leverage that converts unknowable outcomes into the closest thing to certainty you can get (without a crystal ball).
- The formula is really nothing more than probability times payoff. This stuff isn’t complicated, but it’s counterintuitive because we all think in terms of the odds of something occurring. Introducing payoff to the equation literally changes how you plan your investments, your life, and your financial future. Yes, it’s that important. It becomes a two-part, dynamic equation where unlikely events with very large payoffs, either negative or positive, have a disproportionately outsized influence on results.
- Disproportionate results have make-or-break impacts on the compound growth equation. The key principle of risk management is to control your plans so you can control outsized negative payoffs, commonly known as losses, from destroying your expectancy, and consequently, your wealth growth. Leverage is how you maximize the gains from your winning decisions.
- Designing your wealth plan to maximize gains through leverage while minimizing losses through risk management is how you tilt the payoff portion of the expectancy equation. If you can favorably tilt the payoff portion of the equation enough, then you can still profit even if you lose more often than you win. That’s how you create reliable profits out of unreliable, unpredictable future outcomes.
- Seek large, positive investment returns using leverage so you can win big when you succeed; but learn how to control risk for adverse losses during the inevitable failures by using risk management strategies. When you shoot for large positive outcomes when you’re right, while controlling risk to small negative outcomes when you’re wrong, you effectively tilt the expectancy equation to result in wealth (financial wealth, life “wealth,” time “wealth”). It’s literally as simple as that. Of course, the devil is in the details, which is what I explain in The Leverage Equation book. Your overriding goal for leverage is to tilt the payoff dimension of the expectancy equation, which is the dimension you have the most control over.
- Understanding all the implications of expectancy, and mastering the required skills of leverage and risk management as implied by expectancy analysis, are central to your success in all aspects of life, including money. It’s the single best way to take back control of your financial life from all the uncertainty inherent in putting capital at risk in an unknowable future.
I’m sure that’s a mouthful if you’re not familiar with these ideas. Again, the full course and the leverage book excerpted from that course develop these ideas fully.
However, I wanted to give you a step-by-step flow of how the logic connects – from uncertainty about an unknowable future to risk management and leverage strategies that control losses and maximize gains, thus tilting the payoff part of the equation to result in positive expectancy, or wealth growth.
Again, here’s the equation:
Expected Value =
(Probability of Win * Average Win) – (Probability of Loss * Average Loss)
Manage Your Payoff To Master Your Wealth Growth
The counterintuitive realization is that disproportionate payoffs can make you rich if you maximize gains through leverage when you’re right and manage the risk tightly when you’re wrong.
Even if you’re wrong 9 times out of 10, or even 99 times out of 100, you can still profit by tilting the payoff portion of the equation. This is how you can reliably achieve your financial goals when facing an uncertain future.
(In case you’re skimming, please stop and really think about the above paragraph before reading on. The implications are a complete game-changer to all of your life plans – financial and personal – when fully understood. The importance can’t be overstated.)
Equally as important, you’ll want to realize how a strategy that produces mostly winning outcomes can still be a loser with negative expectancy if the average loss is larger than the average win. In fact, many investing strategies are notorious for that problem.
The Trap of Needing to Win
But focusing on payoff is counterintuitive to most people because it’s not how we’re trained to think. I believe it’s a major reason that wealth eludes most people. We all have a natural bias toward winning with high reliability.
You want to be right. It feels good to win, and nobody likes to lose. We’re taught in school that high accuracy gets an A, and mediocre accuracy equals failure. Nothing below 70% correct is even acceptable, which is absurd. Even worse, many people mistakenly view failure as a measure of self-worth.
Everyone is looking for high reliability because we’re trained to think in terms of probability, but the percentage of winners versus losers is not the most important factor to your financial success, and it’s the thing you have the least control over.
The real key to expectancy is how you control losses and maximize gains – through risk management and leverage.
It’s irrational to focus on winning versus losing because, as I said earlier, the future is uncertain, so it’s not really within your control.
You should always try your best to win, but the reality is: if you play the game, losses are inevitable. It’s just a fact of life when the future is unknowable.
For example, I lose all the time. It’s a regular part of every week of my life. I never really get used to it because I’m human like everyone else, but I’ve trained myself to accept that putting capital at risk into an unknowable future means that losing is an inevitable part of the investment process, and I have to accept that.
But payoffs are different. I actively manage my payoffs because that’s the part of the equation that’s controllable; and fortunately, the math is clear: if you do a good job of controlling losses, you can get rich relatively easily. It’s just a question of sample size.
The bottom line is: successful wealth builders are fine with losing more often than they’d like, but they’re very attached to the relative size of those wins and losses because that’s what’s most important to your financial outcome in life.
Think of risk management as the defensive half of your wealth plan to tilt payoff in the expectancy equation, and think of leverage as the offensive half of your wealth plan to tilt payoff. They each tilt payoff favorably, but in opposite directions.
When you put both leverage and risk management together in your wealth plan, the net effect is to radically tilt your payoff to such an extreme degree that your success becomes a matter of sample size. It’s not a question of if; it’s a question of when. All you have to do is implement both disciplines with persistence.
In Summary
Mathematical Expectancy can be somewhat counterintuitive because most people are conditioned to think in terms of probability, not expectancy.
Expectancy is probability times payoff, and adding that payoff component to the equation changes everything.
Your wealth compounds according to expectancy, not probability. Introducing the payoff component to the equation emphasizes the essential role that risk management and leverage both play in your wealth growth.
Risk management minimizes losses, and leverage maximizes gains. Together, they can create positive expectancy and wealth growth even if you lose far more often than you win (low probability of success).
Payoff is particularly important because the future is unknowable, so controlling probability is difficult. You can guesstimate probability, but it’s ultimately unknowable. However, you can control payoff.
Smart wealth builders focus on those things they can control so they can produce a predictably profitable outcome regardless of circumstances. Mathematical expectancy gives you the framework to achieve that objective, and leverage is the tool that you use to create large wins, thus tilting the payoff equation to create positive expectancy.
If you found this analysis interesting, then the low-risk next step is to purchase a copy of The Leverage Equation book so you can learn how to apply expectancy analysis to produce large wins. The price of the book is small, but the value of the information makes it a no-brainer risk-versus-reward investment.
Additionally, if you can see the importance of this strategy to all your future plans, then consider leveling up to the full Expectancy Wealth Planning course that The Leverage Equation (and this article) were excerpted from.
It’s fully guaranteed because it’s knowledge that pays.
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Appreciable, commendable and helpful towards economic growth. Though I have not gone deeply. At present I am in tremendous financial crisis and hope I will be benefitted from the article and also sought further help to improve. Thanks a lot.
There are a lot of free resources and education on this web site to help you turn things around. I hope it helps.
This sounds like a feedback loop to me. I can smell systems theory here!
I am a statistician who runs training workshops about statistics.
First, this is a great example of applying expected value. This is exactly why the only time it’s rational to play the lottery is when the pot gets enormous. The probability of winning is so tiny that on average everyone loses.
I would like to point out a small wording choice, though, that is not quite accurate. I’m sure it will go over the heads of just about everyone else who reads this and it doesn’t affect your point. It’s a common misuse of odds in everyday English that makes my job harder because I have to unteach this to my students. And since you are using the technical definitions, I suspect you want to be accurate and will take this as a friendly suggestion and not a criticism.
You are using probability and odds interchangeably but they are different scales for measuring likelihood. “A fair coin has 50% odds of heads coming up on any flip” is not true. That’s the probability. The odds are 1:1. Odds are the ratio of heads to tails. Probability is the ratio of heads to all possible outcomes.
Change all your “odds” to either “likelihood” or “probability” and you will be good to go.
Upvote!
Great article. How would you apply this concept as an investor? Something like this?
Investment A with estimated investment returns of 25%, 10%, 0%, -5% with corresponding probabilities of 10%, 50%, 30%, 10%
1000 x .25 x .1 = 25
1000 x .1 x .5 = 50
1000 x 0 x .3 = 0
1000 x -.05 x .1 = -5
Expectancy = 70 = (25+50+0-5)
Vs
Investment B with estimated investment returns of 50%, 25%, -25%, -50% with corresponding probabilities of 10%, 50%, 30%, 10%
1000 x .5 x .1 = 50
1000 x .25 x .5 = 125
1000 x -.25 x .3 = -75
1000 x -.5 x .1 = -50
Expectancy = 50 = (50+125-75-50)
An investor who based their investments upon expectancy would go with investment A.
Thanks
Ben, I have never once worked through the calculation you provided because it’s not practical from a real world perspective. The assumption behind the calculation is that such probabilities and outcomes could actually be estimated accurately, which they can’t. There’s a different way to apply expectancy analysis as taught in my books and courses that is practical and works. The key is put it to work for you in a way that produces real results. Unfortunately, the real world doesn’t provide us with scientifically accurate probabilities and payoffs so you have to approach the process differently. Hope that helps.