Geometric average annual return - YouTube

Let's look at an example of a problem which asks to calculate the "geometric" average

annual return.

So, let's say we're given the following returns.

Let me just make something up.

Let's say, we have a 10% return, then we have a negative 15% return, and the third one is

20%.

Okay.

So, this is what we are given.

And now, how are we going to calculate their geometric average?

Let me pick the blue color...

So, here's...

So, there are two ways how you can do it.

One - you can do it by hand using the following formula: you open parentheses where you have

the product of three things which are linked to these three returns that are given for

each year.

So, 1 plus the first return.

Let's do it in decimals.

0.1.

Multiplied by 1 plus the second return.

However, because it's negative, we use the "minus" sign.

So, -0.15.

And then we multiply it by the third term, which is 1 plus the third return.

In decimals that's plus 0.2.

And then, the whole thing, the whole product of these three terms, is taken to the power

"one third", where 3 is always the number of the returns that are given to you.

And then you subtract 1.

So, if you do the math this way, you'll get 3.91%.

3.91%.

This is the answer.

But there's a second way of doing it using the financial calculator!

So, let's, let's do the following.

We have Year 0, Year 1, Year 2, and Year 3.

Right?

So, let's make something up for Year 0!

You just make it up.

Make up some dollar amount.

And the meaning of it will be - because we're talking about returns on stock investment,

for example, the meaning of Year 0 dollar amount would be how much you pay to buy, let's

say, one share of stock.

So, let's make it $100.

Right?

And I'm gonna, you know, add this note here: "make it up!"

Right?

So, you completely make it up.

It's completely up to you if it's $100 or $1 or $5 or anything else.

Okay.

So, if it's $100 that you're paying today to buy one share of stock, then after 1 year

what will be the price per share, if the return is 10%?

Well, it's gonna be 10% higher price.

So, we can do it in the financial calculator by using $100 from Year 0 as our Present Value,

and we would want to make it negative, then N=1 because we're doing it after just one

year, and I/Y, the interest rate, is what's given for the first year, 10%.

And we're computing the Future Value.

So, let me bring up the financial calculator.

Let's turn it on.

100...

Negative...

Is our Present Value...

1 is our N...

10 is our I/Y...

And we are computing...

Compute...

Future Value.

110!

Just like we expected.

Okay.

So, 110.

Then, the second year calculations of stock price in dollars are done the same way.

So, now the price is 110.

That's the PV, right?

And you make it negative again.

110.

Again, we want to see what the price will be 1 year later.

So, the N is still 1.

It's not 2, but it's still 1.

We're doing everything in one-year increments.

I/Y, the return on this stock investment, right? during the second year - that's a negative

15% return.

So, that's my I/Y.

And I'm computing the Future Value.

Okay.

Financial calculator again.

We can actually leave the answer on the display, change the sign of it to negative, and then

save it as PV.

So, now we're beginning the Year 2 calculations.

N=1.

So, I press "1", "N".

15, negative.

I'm doing "15", then "plus minus", then "I/Y".

And I am computing Future Value.

And, as expected, because of the negative return the price should be less than $110.

It's now $93.50.

Okay.

$93 and 50 cents is the new price.

And then we repeat the same thing one last time for Year 3!

So, PV is -93.5.

N is again 1, because we're looking at the change in stock price over 1 extra year.

I/Y is the third return that's given, which is 20%.

Okay.

So, I put 20.

And I'm again computing the Future Value.

Let's see what's that... what that's going to be.

Okay.

So this becomes my Present Value, and I need to make it negative first.

Negative ("+/-"), "PV"... "1", "N"... "20", "I/Y"...

Compute ("CPT"), Future Value ("FV")...

112.20.

Right?

$112 and 20 cents.

So, $112 and 20 cents.

And the last step is the following.

So, here is the last step: the last step is - we are taking this price from Year 0, our

starting point, and the price that we calculated for end of Year 3.

And now we are doing the following.

Present Value is $100 from Year 0, then Future Value is the last price that we found, 112.20,

they are 3 years apart, and we want to compute the implied interest rate, or growth, that

allows the $100 price per share to sort of gradually grow over the 3 years to $112.20.

Compute I/Y.

Okay.

Let's try that!

So, this is my Future Value.

I'm hitting "FV" again to save it.

Then, 100 - and I'm making this one negative - was my original Present Value.

"3", "N".

They are three years apart.

And I'm computing I/Y. 3.91, just like in the calculation I did first using this long

formula!

So, 3.91%.

3.91%.

Again, exactly the same answer.

So, it's up to you if you're comfortable with this complicated formula with double parentheses,

and the power which is a fraction, and so on.

If you know the order of operations and are sure you can do it right, then you can go

ahead and do it all in one step.

Or, you can do it and then just double-check your math using the financial calculator steps.

Now, let me give you one more explanation for the geometric average return.

So, we're basically saying that... so, this is "time", right?

So, we have "Year 0", 1, 2, and 3.

And so, we, we, let's say we're talking about buying shares of stock of some company.

So, instead of saying that... so, we start somewhere here...

I just randomly picked this point...

And then the price grows by 10%, right?

Over the first year.

So, this is +10%.

Then, what we are given is a 15% drop.

So, something like this, right?

It's -15%.

And then we have a 20% jump.

So, something like this, right?

So, +20%.

So, what we're doing now is - with the 3.91% geometric average we are kind of allowing

the price to slowly grow from the starting point, whatever that is, and we decided that

it's gonna be $100, for example, to something at the end of three years, which in our example

is $112.20.

Right?

So, we have +3.91% per year, and I'm gonna add something clarifying: "compounded annually".

Right?

And so the way we interpret the geometric average is - we would say something like this.

If you invest your money for 3 years (you're not saying 1 year, right?), for 3 years, the

entire investment period that we are given, then the money will be growing on average

by 3.91% per year, comma compounded annually.

With the arithmetic average you instead say that the return you earn on average every

year is so-and-so percent when the money is invested for just 1 year.

So, with the geometric average it's not a 1-year investment, it's a multi-year investment.

In our problem, it's a 3-year investment.