# Why stock returns are calculated in Log scale?

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Very interesting question Sam. Not many traders actually know this...

Consider this ... you invested Rs.100 in a stock in year 1.... it grows to Rs.200 in year 2 giving you a return of 100%. In year 3 the stock price comes back to Rs.100 from Rs.200 giving you a negative return of 50%. What do you think is the average return?

To calculate the average you are most likely going to do this... [+100 + (-50)] / 3 = 25%

So the arithmetic average is suggesting that you have earned an average of 25%. But in reality you have earned 0%.

Do you see the problem here? Arithmetic return has a positive bias. To eliminate this bias you take the log return. It removes this bias and puts all the numbers in a log plane, in the process you are also normalizing the returns to log normal.

For the same example see what the average is when you use log return... its 0% !

So next time when you want to calculate return on a time series data, especially while building trading strategies I'd suggest you use the log return, and not just the simple arithmetic return.

Good luck.

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but how to claculate log returns?

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amazing explanation…

In th above example, simple return can be calculated as (stock Price in Year 2/storck price in year 1)-1. To calculated the log return during the same period, you just have to take the natural log i.e. Ln (Year2/Year1).

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Kathik sir i have one doubt! If we only notice the one year return we got return of 69% in 2nd year by using log normal return. What sense does it make to have 69% return! isn’t it 100% return???

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Hi Karthik,

should the division not be by 2 to get 25% as answer.

The reason I came to reading this Q&A is because I have just started to read Varsity. In the chapter about "Volatility & Normal Distributions you explain
Converting Upper and Lower Range Log Returns to Simple Returns as " Spot * exponential( Average + SD)
and in the very next chapter “Volatility Applications” you explain the same conversion as "Spot * ( 1 + (Average + SD)) and Spot * ( 1 - (Average + SD)) for upper and lower range respectively.

Can you please explain why this difference. Thou I am late to start but I guess its never late to learn.

Thanks

same question in my mind , have u got answer, plz reply

you’re the kind of person who should write books. Thank you!