Sunday, June 26, 2022

Geometric Mean Vs. Arithmetic Mean

ladies and gentlemen welcome back i already introduced you to the concept of arithmetic mean this one has some shortcomings when you are dealing with percentile changes not percentages but percentile changes the word changes is very very important here here i provide you with a case to.

Show you why it has this shortcoming and why it's not suitable as a central tendency measure or analytic when you're dealing with percentage changes here suppose that you have a hundred euros which you invest in the stock market now uh you do that at the beginning of.

2021 and then in 2021 the stock market does great you get a hundred percent more so your money basically doubles your 100 euro becomes 200 euro now you have to 200 euros at the end of 2021 and then you keep the mark you keep the money in the stock market but in 2022 the stock market does even better and.

Now you have 200 return just in 2022 and in 2023 whatever money you had um you get 50 more of data so you get a 50 return in the stock market and then in 2024 whatever money you had uh you make a 90 loss there now if we calculate just a normal arithmetic mean then we would.

Take the hundred percent increase pla and then add the 200 increase the 50 increase and then the 90 decrease so because it's a decrease it's minus 90 so it's plus minus 90 you could also eliminate this plus sign and just write minus 90 but just for the sake of clarity i put it there and then you.

Divided by the number of observations which in this case is four which gives you as an average 65 so if we want to know the central tendency with the arithmetic mean then it tells us hey on average you made 65 return per year which is really really good those of you who invest in the.

Stock market would say wow that's a great number 65 but i want you to realize what happened here so i made an extra table how much will you have if you started with 100 euros at the beginning of 2021. well then at the end of 2021 because you had a hundred percent increase 100 return on your money you started with.

100 euros and now you have 100 percent more so you doubled your money now you have 200 euros at the end of 2022 because you made 200 uh profit so basically you not only double but triple your money now you have 600 euros.

At the end of 2023 because you made 50 percent extra return now you're you're you're 600 you you made 50 extra on that now it becomes 900. but then in 2024 when you lose 90 of your money your 900 becomes now 90 euros okay so let's look at what happened here at the beginning of 2021 i had 100 euros and.

After all these years i'm left with 90 euros but the arithmetic mean tells me that hey you did pretty good you had 65 percent return on average well this is not really a good picture of reality because i have now 10 euros less after all these years than what i started out with so.

It's not a very accurate representation of what the average performance was and the reason for that is because the arithmetic mean is just not suitable for percentile changes what is suitable for central changes is the geometric mean now this is the formula for the geometric mean don't be intimidated by it it's actually quite.

Simple to understand what the geometric mean formula this this very complex looking formula basically says is hey take the percentile changes multiply them all together so the one from 2021 multiplied with the one from 2022.

Multiply it with the one from 2023 and then multiply that with the one from 2024 multiply them all together and then raise whatever you have to the power of one divided by n and it's just the number of observations here we have four so it would be raising to the power of one divided by four.

But but but but we cannot just take these percentile changes that you have here and plug them in the formula you cannot do that um and the reason for that is because we sometimes have these negative percentile changes and you just cannot plug them in this formula it's not the way to do it if you want to.

Know why then there's a text attached to this video which you can download and read about it but for most of you just know that you cannot put negative numbers there so we have to find a way around it what is the way around it we have to go through some steps basic steps and that is that first.

We divide each percentile change by 100 so if we have a 50 percent change that becomes 50 divided divided by 100 becomes 0.5 if you have a 5 percent will change a positive 5 percent change that becomes 0.05 if we have a 50 decrease then that becomes if we divide it by 100.

Minus 0.5 etc etc the second step is to add that number to one before you enter it in the formula and then you multiply the numbers in step two and then you raise that to the power of one divided by n and again is the number of observations which in this case was four now let's have a look.

Step one so we divide each percent percentile change of the table that we just saw by 100 so plus 100 percent would become one plus 200 percent becomes two plus fifty percent becomes zero point five and minus ninety percent becomes minus zero point nine then step two is to add.

The number one to your answer from step one so one plus one becomes two two plus one becomes three um 0.5 plus 1 becomes 1.5 and minus 0.9 plus 1 becomes 0.1 now step 3 is to multiply the numbers that you found in step 2 which is what we do here 2 times 3 times 1.5 times 0.1 gives.

Us 0.9 now we have to raise that to the power of 1 divided by n n is 4 which is 1 divided by 4 which is also 0.25 and that gives us as an answer 0.97 don't worry i'm going to show you exactly in a second how to.

Do this raise the power to 0.25 don't worry about that but whatever the answer is the answer is 0.97 now how do we interpret this 0.97 so what does that mean on average we lost three percent per year on average.

We lost three percent per year now let's go back to this situation don't you agree that the minus three percent so the three percent loss per year is a better representation of the central tendency of what happened in these years than what the arithmetic mean.

Says because their arithmetic means says hey you did pretty good 65 increase per year good job but you know the geometric means as poor you you lost some money on average three percent per year so now you know when to use the geometric mean and when to use the.

Arithmetic mean use the geometric mean when you're dealing with percentile changes not percentages but percentile changes now the last thing that i have to show you is how to raise something to the power uh off for instance one divided by n that's very simple now let's see how to.

Uh raise something to the power offerings is one divided by n in excel so you start with typing the equal sign in any random cell doesn't matter i just choose this one now let's say i want to raise 100 to the power of 0.25 so i use this little hat here this little hat you find usually on top of the number six on most laptops it could also be in.

Another position and then just type in 0.25 and then click on enter so there you have your answer if i want to have for instance 50 and i want to raise that to the power of i don't know 0.9 just do it like this that's your answer.

Okay you can also do it on a calculator but excel works fine that's it for this session hope it was valuable for you thank you

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