{"id":641,"date":"2024-12-06T22:03:30","date_gmt":"2024-12-06T22:03:30","guid":{"rendered":"https:\/\/wosscapital.com\/articles\/?p=641"},"modified":"2024-12-06T22:28:11","modified_gmt":"2024-12-06T22:28:11","slug":"a-case-for-expected-value-the-key-to-mathematical-edge","status":"publish","type":"post","link":"https:\/\/wosscapital.com\/articles\/a-case-for-expected-value-the-key-to-mathematical-edge\/","title":{"rendered":"A Case For Expected Value: The Key to Mathematical Edge"},"content":{"rendered":"\n
It took the better part of 18 months to write, edit and publish the book so, as one might imagine, my capacity for writing articles has been somewhat limited! With a refresh of the site and the hugely positive feedback received for “Escape The Wealth Illusion” it’s time to get back to some shorter form writing on a regular basis!<\/p>\n\n\n\n
I’ve plenty of subject matter on my to do list, but in a recent 1-2-1 where our objective was improving trading performance I was reminded that the importance of “expected value” alongside a strong risk plan, rules based and repeatable strategies and a consistency of application is often forgotten in trading circles. It’s a metric that I find super important in my own strategies and I often take for granted that the traders I help both understand and utilise it too.<\/p>\n\n\n\n
So, what is expected value and why do I consider it so important?<\/p>\n\n\n\n
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\n<\/figure>\n<\/div>\n\n\n\n
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“It’s just a big game of Maths!”<\/h3>\n\n\n\n
WoSS Capital Wierdo. Likes Maths. Thinks it matters (a lot)<\/p>\n<\/div>\n<\/div>\n\n\n\n
Well, it’s the mathematical calculation that when applied appropriately to probability defines the profit expectancy of any repeatable process over a long series of iterations. It’s the factual mathematics behind my often used quip above. I use that phrase in a humorous way more often than not, but the reality is I don’t use it hyperbolically. The mathematical identification of edge is one of the keys to profitability in my humble opinion and, without it, one has no idea if they should expect to profit from their trading strategies or not, no matter how well defined or complex said strategies are.<\/p>\n\n\n\n
Ever wondered just how the casinos consistently make money? This is your answer, every game is designed to carry a small positive expected value in their favour so they know, over the long series of iterations, they will make money. As long as they don’t change their processes, change the rules, change the terms of the game. If they’re consistent in applying the parameters of the mathematical edge they’ve built, they win. Simples. I’ll give an example of this later once we get into the meat of the content.<\/p>\n\n\n\n
Probability: A Working Understanding<\/h3>\n\n\n\n
In the trading world people talk about probability a lot. I often wonder though how many actually know what it means and how to apply it to our advantage.<\/p>\n\n\n\n
Simply put, probability measures the likelihood of any event<\/span><\/strong> occurring and thus can be used to helps us quantify “uncertainty”. From here, with the variables defined, we can predict outcomes. <\/p>\n\n\n\n
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To the right is the formula for calculating probability. It’s pretty simple and a fundamental requirement in going on to calculate expected value.<\/p>\n\n\n\n
Let’s look at a quick example to hammer this home, make sure you understand the point fully.<\/p>\n<\/div>\n\n\n\n
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P(A) = <\/p>\n<\/div>\n\n\n\n
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Number of favorable outcomes<\/p>\n\n\n\n
Total number of possible outcomes<\/p>\n<\/div>\n<\/div>\n<\/blockquote>\n<\/div>\n<\/div>\n\n\n\n
Example: Dice Rolls<\/h5>\n\n\n\n
Suppose you have a standard six numbered, six sided dice. You want to know the probability of rolling a 3 on that dice. To calculate the probability the formula is P(3) = 1\/6, or 16.67%. The same probability exists for any number on the dice, as there’s only 1 favorable outcome amongst the 6 possible outcomes. Further, we can calculate the probability of rolling two of the desired number in a row by chaining the results. So P(3) = 1\/6 x 1\/6 = 1\/12, or 8.33%.<\/p>\n\n\n\n
So many people in trading talk about probability in the abstract, inferring uncertainty in outcomes as a result of probability but without using it in its true mathematical context. I can be guilty of this too from time to time, where experience and having seen events play out before allow for a notional probability of outcomes to be applied. But in the online world of trading influencers this isn’t always the case, probability is just flat out used incorrectly.<\/p>\n\n\n\n
Anyway, hopefully this example conveys to you the knowledge on what probability actually is in a mathematical context as it’s one of the key variables we need when considering expected value.<\/p>\n\n\n\n
Edge<\/h3>\n\n\n\n
Before we move on to EV, I want to frame its use in creating edge over markets, the aim of any profitable trader. Edge is also an often confused concept, but it’s simply a defined method of positive statistical weighting over a long series of consistently repeatable events (steps)<\/span>.<\/strong> That’s to say that it’s the process we identify and repeat consistently that means the probability of profit is always greater than the probability of loss. A system where expected value is ALWAYS<\/span><\/strong> positive.<\/p>\n\n\n\n
This is one of the reasons I bang on about consistency of risk management, position sizing and rule application. About being robotic and not letting emotion interfere with your process once you’ve identified and confirmed your edge. So many traders I speak to have chopped and changed strategy during periods of drawdown, tricked themselves into “refining” their strategy because it had a series of losers in a row and in the process, mostly unbeknown to them, changed the EV from positive to negative without even knowing it existed.<\/p>\n\n\n\n
Using our casino example again, ask yourself how many times in your lifetime any casino game has changed its rules. Can you think of one? Me either, but casinos go through periods of drawdown too? yeah, they do, but they know the maths and they are playing the long game. They know that over the long series of events repeated the same way robotically, they win. This is also, incidentally, why they get so upset when people find ways around their edge (like counting cards, for example). The act of counting cards at the Blackjack table turns their positive EV negative, in favour of the gambler. So their annoyance, in a mathematical context at least, is understandable!<\/p>\n\n\n\n
Expected Value<\/h3>\n\n\n\n
Ok, here we are, expected value. Let’s jump straight into the formula and start from there.<\/p>\n\n\n\n
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The formula, to the right, is again pretty straightforward and hopefully you can see from it why I spent the opening part of this article defining the mathematical calculation for probability: we need it here to calculate expected value.<\/p>\n\n\n\n
Again, let’s look towards an example to make this understandable.<\/p>\n<\/div>\n\n\n\n
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EV = <\/p>\n\n\n\n
EV =<\/p>\n<\/div>\n\n\n\n
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(Profit Scenario) + (Loss Scenario)<\/p>\n\n\n\n
(Profit x Profit Probability) + (Loss x Loss Probability)<\/p>\n<\/div>\n<\/div>\n<\/blockquote>\n<\/div>\n<\/div>\n\n\n\n
Example: Flipping a Coin<\/h5>\n\n\n\n
I use this example intentionally as it’ll both allow me to show the formula at work, but also make another point in a trading context about some influencers out there.<\/p>\n\n\n\n
In the coin toss scenario, the probability of flipping heads (or tails) from our earlier formula is P(Heads) = 1\/2, or 50%. Now, let’s imagine that we strike a deal where every head that’s flipped, you get $1 and every tails I get $1, or a 1:1 risk to reward ratio. <\/p>\n\n\n\n
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So, in this scenario our expected value (or Profit Expectancy) is zero. If we repeat the same steps over and over again there’s absolutely no edge against the game. It is, pun intended, a coin flip over the long series of iterations. Sure, you may win or lose a few in a row in a short series, but statistically speaking the more iterations the closer this number trends to zero.<\/p>\n<\/div>\n\n\n\n
What if we changed the variables though? Let’s say we get $1 for a heads but only $0.50 for tails (a 1:2 risk to reward ratio). Well now, we have a positive profit expectancy from this new game. Over the long term this would be expected to return a profit. No longer a coin flip, even though it’s a literal coin flip!<\/p>\n<\/div>\n\n\n\n
I hope you can see where I’m going with this point, but when you see gurus promoting a 1:1 risk reward ratio on strategies that produce a 50\/50 (or worse) win rate in testing, you’re literally deploying a strategy that will lose. Over the long term, over the series of trades, you will not be profitable. The maths says so. This is why EV calculation is always<\/span><\/strong> the last step in my process of deploying a strategy. Once I have the backtest results, I know the target R:R based on them and I’ve confirmed the steps are repeatable on a long enough timescale I calculate EV. If it’s not positive, then no matter how much time was spent on that strategy, it’s a losing horse. It goes no further.<\/p>\n\n\n\n
Casino Games Example: Roulette<\/h3>\n\n\n\n
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\n<\/figure>\n<\/div>\n\n\n\n
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I’ve chosen roulette as the example as I had an image from a previous article, a handy coincidence!<\/p>\n\n\n\n
In this example we’ll be concentrated on the most common way gamblers play roulette, thinking it’s their best chance of winning as it’s “50\/50”, the gamblers fable of probability.<\/p>\n\n\n\n
On a standard roulette wheel there are 18 red numbers, 18 black numbers and, importantly, zero (usually green). Ever wondered why that zero exists? Well, you’re gonna learn today! We’ll use $100 as our bet and we’re gonna go for black. <\/p>\n\n\n\n
So, who has the edge here? I bet you can guess (and I did rather give it away earlier in the article, but let’s do the maths anyway). I’m sure some famous singer the kids like said maths was fun and it’s probably the only thing I’d ever agree with them about.<\/p>\n<\/div>\n<\/div>\n\n\n\n
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To the right is the profit expectancy (or expected value) calculation on betting red or black on the roulette wheel. As you might expect, it’s negative in favour of the player, meaning positive in favour of the house. This is why zero exists and this, dear friends, is why “the house always wins” over a long enough time period and why you never see a poor casino owner!<\/p>\n<\/div>\n\n\n\n
So, what’s the basic casino business model then and how the hell does this apply to the trading conversation we’re having here? Well, in a nutshell:<\/p>\n\n\n\n
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Have strict rules, follow them explicitly every time and NEVER<\/span><\/strong> change them, not even in periods of drawdown. If they lose 20 hands in a row they don’t rush in to change the rules, they know the long term series WILL<\/span><\/strong> trend positive because they’ve done the maths (unless a lot of people are cheating!)<\/li>\n\n\n\n
Participate in as many possible iterations that follow the rules as possible.<\/li>\n\n\n\n
Accept that you can never know the outcome of a single game, so the only thing you care about in any single game is were the rules followed. The outcome is irrelevant.<\/li>\n\n\n\n
Only deploy games that carry a positive EV and accept that over the long term these will return a profit if all other conditions are met.<\/li>\n<\/ul>\n\n\n\n
Applications in Trading<\/h3>\n\n\n\n
If you’ve not figure it out already – this applies directly to trading and to the pursuit of and edge that brings profitability, certainly in my opinion:<\/p>\n\n\n\n
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Have rules based, repeatable trading strategies that are followed without discretion over the long term and never let emotion or drawdowns convince you to change the rules (or “optimise”).<\/li>\n\n\n\n
Accept the only thing that matters in any individual execution of this strategy is following all the rules (including position sizing and risk management, which make up part of the rules based strategy).<\/li>\n\n\n\n
Test the strategy with historic data and calculate the win rate and target expectation in multiples of R. Use this to calculate EV and if it’s not positive, discard the strategy.<\/li>\n\n\n\n
If it is positive, then repeat it, religiously, over the long term. Mathematically your strategy has an edge, you just need to give it enough time to play out.<\/li>\n<\/ul>\n\n\n\n
So, don’t forget about expected value. Don’t go blindly into trading strategies that don’t carry a positive EV and don’t think of probability outside of the mathematical context that will help you in your pursuit to become profitable.<\/p>\n\n\n\n
As I said above, it’s just a big game of maths. For those of you who have heard me say this repeatedly over the years, this gives you a little insight into the inner workings of my brain and hopefully helps you realise that it’s not just a humorous quip.<\/p>\n\n\n\n
As always, thanks for reading and I’m looking forward to getting more of these out in the coming months.<\/p>\n\n\n\n
It took the better part of 18 months to write, edit and publish the book so, as one might imagine, my capacity for writing articles has been somewhat limited! With a refresh of the site and the hugely positive feedback received for “Escape The Wealth Illusion” it’s time to get back to some shorter form […]<\/p>\n","protected":false},"author":1,"featured_media":650,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"nf_dc_page":"","om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[13,10,12],"tags":[],"class_list":["post-641","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-educational","category-market","category-trading"],"yoast_head":"\n
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