(Mathematician needed) Continuous vs. discrete returns?

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(Mathematician needed) Continuous vs. discrete returns?

Hi!

I am entering the returns of my trades into an excel sheet, to calculate statistical data. I calculate the returns in percentages (closing price/ opening price)-1 and came across a problem: Do i need to calculate the returns as continuous or discrete returns in order to get the whole sum of all returns? In some books they prefer calculating the continuous returns (natural logarithm(closing price/ opening price), but I donīt understand the whole concept about it..

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I do not think that there is something like a continuous return. Real returns are always discrete.

From real returns you can calculate a rate of returns (similar to an interest rate). And that rate of returns could be discrete of continuous. For fancy option formulas like the Black-Scholes formula continuous interest rates are used. However, I do not think that it is necessary to calculate a continuous rate of return for a number of trades.

For trade evalaution it is sufficient to calculate simple returns or logarithmic returns. The main difference is the way you compound returns. For simple returns compounding is achieved by multiplying returns, logarithmic returns can be compounded by adding them up.

If you feel uneasy with logarithmic returns, just use simple returns.

The following 2 users say Thank You to Fat Tails for this post:

I don't understand a lot of what you are saying but it reminded me of a presentation i saw, given by a relatively young kid who won the world cup of trading.

I didn't totally understand what he was saying (over my head), but from what i gathered his strategy is successful almost entirely because of how he compounded returns.

Did i misunderstand, or can compounding returns (and adjusting position sizes accordingly (for instance)) really make that much of difference?

I apologize in advance if im not making sense...most of this is over my head but i just had to ask.

Mathematically, all you need is a strategy that has a positive expectancy.

Once you have identified a small edge, compounding can turn that edge into giant profits.

For a beginning trader it is important

(1) to understand the markets and identify an approach to trading with a PROVEN edge
(2) to develop the skills - mainly discipline and mindset - to trade that system

The advanced trader should focus on position sizing and compounding. The most important task becomes

(3) to keep the risk of ruin low and at the same time compound profits

Yes, I totally agree that compounding of returns makes a difference.

But the edge comes first, discipline and mindset come second, and then position sizing and compounding make the difference between just an above average and a top tier trader.

The following 2 users say Thank You to Fat Tails for this post:

Thanks for that extremely clear and
understandable (not over my head) explanation.

"Mathematically, all you need is a strategy that has a positive expectancy."
I have been exploring ways to get similar results to the ones VanTharp said he got in one of his books...he said he and a buddy worked out a system that was profitable using random entries and exits. I completely believe he did although he didn't go into details...a buddy of mine thinks he exaggerating or lying. What are your thoughts on that?

"Once you have identified a small edge, compounding can turn that edge into giant profits."
When you say compounding are you saying in the 'financial formula' sort of way or do you mean compounding through other means?

"...(3) to keep the risk of ruin low and at the same time compound profits"
I recently went over the 'turtles' position sizing formula...very interesting and useful i think...the author said he had moved on to much more sophisticated formulas ... as of now my position sizing is strictly within the realms of a series of trades, not in the bigger picture of a basket of strategies trading a basket of assets in a basket of markets, etc. if possible, would you be able to give me an idea of how i can start learning to use the benefits of more sophisticated position sizing?

With random entries and exits you will not be able to generate profits. The expectancy for each trade is zero before commissions. If you deduct commissions you will have a negative expectancy. Appropriate position sizing can only improve returns of a strategy with a positive expectancy.

Compounding is achieved by adjusting the trade size as a function of your capital. Basically you reduce your position after a loss and you increase it after a win. There are several ways of doing this. One is known as fixed-fractional betting. There are a few threads in this forum which elaborate on this subject.

The turtles were trend followers. They used volatility adjusted stops. Their position sizing was limiting the loss to 1% of account equity. This means that if a trader had a trading capital of $ 100,000, he was allowed to risk $ 1,000 per trade. Whenever the capital increases due to winning trades, the risk is increased, as it always stays at 1% of the trading capital. After a losing trade the capital is decreased to maintain the risk of ruin at a constant level.

The following 3 users say Thank You to Fat Tails for this post:

i understand ... i guess i assumed (wrongly im sure) that a truly random entry and exit algorithm (using a 1:1 risk reward) would have a 50% win loss ratio over many many trades...

recently, after fumbling around with contract size betting 'algorithm' spreadsheets for hours upon hours i happened to stumble upon the usefulness of that phenomena "reduce your position after a loss and you increase it after a win" ... i will definitely look into fixed-fractional betting .. thank you

are there any other simple techniques (like fixed-fractional betting) that improve returns or reduce risk on a trader's 'edge'?