If you and I sat down to flip a coin, and bet $5 on each flip, would you expect to make or lose money after 1 flip? After 100? After 1,000,000?

For most of you the answer is probably yes to a profit or loss on one flip. but breakeven or close to it for 100 and 1,000,000. 

But the way to prove that mathematically is using Expected Value.

Expected Value (EV) is the expected average return on any probability based event. 

To know the expected return on an event, you need to know three things.

1. The probability of winning

2. The return on a win

3. The cost of a loss

The coin flip gives us a great example to aid in understanding how EV works.

In the coin flip scenario, the probability of winning is 50% or 0.5 when expressed as a decimal.

There are two outcomes, heads or tails, and both have an equal chance of winning.

The return on a win is $5.

The cost of a loss is $5

We can use these to calculate our expected value on the coin toss, using the expected value formula

Expected Value = Probability of a win x Return on a win - Probability of a loss x Cost of a loss

Probability of a loss is just 100% - Probability of a win, so 100% - 50% = 50% or 0.5

So the formula becomes:

EV = 0.5 x $5 - 0.5 x $5

 EV = $2.50 - $2.50

EV = 0

Which tells us that in the long run, you and I are wasting each others time with these bets. Over a large enough series of bets, we'd expect to break even.

I like this example, because it shows that EV isn't necessarily going to tell you the return you'll get on any one bet.

In our coin flip scenario, it's literally impossible to have a $0 profit or loss after a single coin flip. Someone will make $5, the other person will lose it. 

But the EV comes out at zero, because EV looks at the expected return across a large number of events.

The more events, the more accurate EV becomes. 

You might win two in a row, or even ten in a row. But as the number of flips approaches infinity, the odds become exponentially higher that we will return to a zero dollar P&L.

So, what if we change the game?

I'm a degenerate gambler, and you're not that interested in gambling.

So let's say I offer you $10 if you win, but you only have to pay me $5 if you do, as a way to incentivise you to gamble.

You'd be mad (although possibly morally right if I'm a problem gambler) not to take me up, and EV is how we prove that.

In this case, the probability of the win and loss are still exactly the same, at 50%.

But now, for you, the return on a win becomes $10.

So your EV formula becomes:

Expected Value = Probability of a win x Return on a win - Probability of a loss x Cost of a loss

EV = 0.5 x $10 - 0.5 x $5

 EV = $5 - $2.50

EV =+ $2.50

What this means is that every time we play this game, you could expect to make $2.50.

Again, obviously no single game will net you $2.50, you either lose $5 or win $10. But if did 100 flips, you'd expect your return to be somewhere reasonably close to $250, making this a game you'd do quite well out of playing.

For me on the other hand, my return on a win remains at $5, but the cost of a loss is now $10.

So my EV formula is:

Expected Value = Probability of a win x Return on a win - Probability of a loss x Cost of a loss

EV = 0.5 x $5 - 0.5 x $10

 EV = $2.50 - $5

EV = - $2.50

Meaning in the long run, I'd expect to lose money to you.

As a general rule, Positive EV (Such as you got in the weighted coin flip) is what you're aiming for when gambling, or when trading. 

Negative EV is to be avoided at all costs, as it's going to lose you money in the long run.

How does that translate to trading?

Trading is also a series of probabilistic events with a known return, and with enough data, a decent estimate of the probability of a win.

Many traders have seen tables such as this one, which help you recognise the win rate needed to breakeven for a given risk to reward, or vice versa.

But this only tells you if a strategy is profitable or not by having one of the two be higher than the necessary value for breakeven, it tells you nothing of how profitable your strategy or how much you can expect to make over a series of trades.

That's where Expected Value comes in.

EV is a single number you can look at that instantly tells you whether a strategy is profitable or not, and by how much. It tells you the average return on your risk you can expect for every trade you place, once averaged out over a large enough sample size.

When trading, we can convert the EV formula to more familiar terms.

Probability of a win = Win Rate

Return on a win = Average R:R achieved on winning trades

Probability of a loss = 100% - Win Rate

Cost of a loss = Average R:R achieved on losing trades

so then 

Expected Value = Probability of a win x Return on a win - Probability of a loss x Cost of a loss

becomes

Expected Value = Win Rate x Average R:R achieved on winning trades - [(100% - Win Rate) x Average R:R achieved on losing trades]

At VEMA, these are all figures we track for you, so we can easily calculate the EV for any analysis card you might want to investigate.

 

Are Fees included in the calculation?

Yes. At VEMA, we try and focus on the numbers that really matter. 

As such, we've included order fees in the EV calculations, to make sure it represents as close to a true representation of your expected outcomes as we can deliver.

 

So I've got my Expected Value, now what is that actually telling me?

The first thing to see is whether the value is positive or negative. 

Like in gambling, positive EV means a trading strategy is profitable in the long run based on the data collected.

Negative EV means it's losing on average.

You can then use this information to decide whether a given strategy is worth pursuing, as a general rule, if a strategy is positive EV, you want to do more of it, and if negative EV, cut it out because it's costing you money.

That said, it's always important to remember EV in trading is a probability calculation, meaning it needs a significant number of data points before it can be considered to have a degree of accuracy. In "Trading in the Zone" Mark Douglas recommends trying a strategy for at least 20 trades before making tweaks, and I think that's a decent rule of thumb here too.

The next thing to understand is what the number is telling you.

You can consider the number a multiplier similar to R:R.

Every trade you take using that strategy, you'd expect to make whatever you set your account risk percentage at multiplied by your EV calculation.

In practice, what that means is if I now take a trade with my 4h bias being bearish, I can expect to make 0.54 x my account risk percent per trade.

So if I risk 1%, over 100 trades I'd expect to make 54% on my account.

If I risked 2% of my account over the same number of trades, I'd expect to make around 108% on my account. 

This figure isn't compounded, so the real results would actually likely be much better than that as my account grows and the P&L's become larger.

But we can see here how EV gives you some measure of predicting a strategies performance. 

If you know a strategy has an EV of +1, and you have a frequency of 10 trades a month, you can expect to make 10% a month on your account using that strategy.

Of course, just like in the gambling example before, my EV of 0.54 doesn't mean I actually expect to make 0.54 x my account risk percentage on any given trade. But it means on average, over a sufficient number of trades, I should expect to see roughly that value achieved on a per trade basis.

 

 

How can I Improve my Expected Value on Trading Strategies?

 

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Conclusion:

Expected Value is a one glance number for knowing if a given trading strategy is profitable or not, and by how much over a large number of trades.

At VEMA, we're all about focusing on what really matters, and that's why we've introduced Expected Value to our analyse cards.