Risk (pain) vs Reward (gain)

[nonlinear5]

Suppose you have the risk-reward curve as shown below. Intuitively, I feel that the optimal point on that curve is somewhere at around risk=0.2, as labeled. My question is, what the mathematical way to determine this point? My guess is that it would involve calculating some kind of tangent line to the curve. The simple ratio R=reward/risk does not help, because it peaks infinitely close to risk=0. Thanks.

 

[K-Pia]

Great.
Reward is dependant on Risk.
You've got an asymmetry between risk and reward.
However your ratio is concave ... Can't you make it convex ?

I don't get your optimal point ... Optimal according to what ?
According to the fact that the derivative is decreasing ?
Well ... The optimal point could be the max Ratio.

But an optimization is done with regard to some criteria.
What are these creterias ? Max Ratio ? Min Ruin ? ...
Looks like you'd like to refer to the derivative.
But how slow its increase should be for you,
To consider the increase in risk useless ?

The simple ratio reward/risk does not help, because it peaks infinitely close to risk=0

Looks like reward tends to 3.75 towards infinity.
Whereas your risk isn't bounded.
So R:R tends towards 0. Yes

What I work on is to make:
- Risk bounded and concave.
- Reward infinite and convexe.

Then as Kelly would tell you,
Bet if Ratio > P(Risk)/P(Reward)

However it's a lot of known Unknowns.

 

[nonlinear5]

I don't get your optimal point ... Optimal according to what ?

I should have posted the entire curve, see below. The reward drops sharply and irreversibly to the right of the red point. Thus, a small error around the red point would cause the reward/risk to collapse. So, we want to be somewhere to the left of the red point, but how far to the left? The green point is where I intuitively feel we are "safe enough", or "optimal enough". What I want to know is how to sensibly quantify this intuition.

 

[eganon69]

I don't think it is as simple as what you are drawing. You have to take into account your systems win probability and max drawdown you are willing to take before you can achieve an optimal risk. This same curve may produce a different result for different traders all trading the same system. For example, if your system has 80% wins and 20% losses then IN GENERAL over 10 trades you will have 8 rewards won and 2 risks lost so you could trade that system with more risk because your win rate far exceeds your loss rate and the reward could be maximized more. However, you also have to take into account MAX Drawdown you can tolerate. In this 20% losses system you could have as many as say 10 losses in a row. If you are risking 3% per trade that's 30% drawdown. Can you tolerate that?? If not then let's say you think 15% is acceptable you may want to max out risk at 1.5%. But now what is you had 20% winners and 80% losses on the other extreme. This system would have to have a very high reward ratio to make all the losses minimal and yet allow you to continue trading. This system would achieve bankruptcy with 1.5% risk per trade. So you need to be able on the lower end of the curve with higher reward and lower risk. You could have as many as 35 losses in a row or more in this system over several hundred trades. So risk of something like 0.5% might make more sense.

You need to take this curve in context with your win rate and max drawdown you are willing to tolerate to answer this for yourself.

 

[K-Pia]

I should have posted the entire curve, see below. The reward drops sharply and irreversibly to the right of the red point. Thus, a small error around the red point would cause the reward/risk to collapse. So, we want to be somewhere to the left of the red point, but how far to the left? The green point is where I intuitively feel we are "safe enough", or "optimal enough". What I want to know is how to sensibly quantify this intuition.

I have been a turkey to extrapolate the way I did !
Well, it depends the kind of error you get.
What's the kind of deviations you get ?

Then according to it's PDF, You estimate...
The max deviation that you should ever experience ?
You can add redundancy, enlarge it for safety against outliers.
Then you find the optimal point. But without the error analysis ...
We can't really help to establish the point that won't lead to the breakdown.

But Yes this kind of risk is additive.
So it will also depends on the Accuracy.
You can derive analytically Max Drawdown from it.

 

[Xela]

What I want to know is how to sensibly quantify this intuition.

It's subjective: it depends on both your trading-frequency and your degree of risk-aversion.

My own perspective is that trading is about risk management far more significantly than it's about profit maximization. Many people disagree with this perspective. Whether they're actually making a living I have no idea.

 

[nonlinear5]

After looking at my chart and thinking about it, I realized that what I want to do is to identify the segment of the curve which corresponds to the most extreme North-West. So, my thoughts are now back to the tangent line of some sorts.

 

[Xela]

what I want to do is to identify the segment of the curve which corresponds to the most extreme North-West.

You could do an opinion poll in Alaska, perhaps?

 

[nonlinear5]

I don't think it is as simple as what you are drawing. You have to take into account your systems win probability and max drawdown you are willing to take before you can achieve an optimal risk.

The chart reflects the performance of the system as a function of risk taken. This already accounts for the win/loss probabilities and drawdowns.

 

[nonlinear5]

Okay, I have an idea, illustrated below.

The steps to find the optimal point are:

1. Make an identity line (thick red line) from [0,0] to the point of max gain.
2. From the risk/reward (blue) curve, drop the perpendiculars down to the identity line.
3. Find the longest perpendicular (thin red line). This will identify the optimal point on the curve (the green point).

Now that the geometry is in place, I'll try to formalize it as an equation.