Finance

Arithmetic Mean vs. Geometric Mean Return

This example shows why arithmetic mean return and geometric mean return can give very different answers even when they are calculated from the exact same annual returns.

The key idea is simple:

Arithmetic Mean Return = average of the yearly returns

Geometric Mean Return = the constant yearly return that would produce the same final investment value

These are two different questions, so they can give two different answers.

Brazil, 1995–1998

For learning purposes, suppose the Brazilian market had the following annual returns:

1995 = -27.1%

1996 = +152.9%

1997 = +112.1%

1998 = -83.0%

These figures are being used as an illustrative example to explain arithmetic and geometric mean returns. They should not be treated as actual historical Brazilian market returns.

Assume these returns include both capital gains or losses and dividends.

The returns were extremely different from year to year. Some years had very large gains, while other years had very large losses.

Now imagine that someone says:

“The mean annual return from 1995 to 1998 was 38.7%.”

That statement can actually be correct.

But another person could say:

“The mean annual return from 1995 to 1998 was -9.7%.”

That statement can also be correct.

How can one person say +38.7% while another says -9.7%, and both be correct?

The reason is that they are using two different types of mean return.

The 38.7% figure is the:

Arithmetic Mean Return

The -9.7% figure is the:

Geometric Mean Return

First: Arithmetic Mean Return

The arithmetic mean return, written as AM, is simply the normal average of the yearly returns.

The formula is:

AM = (R1 + R2 + … + RT) / T

For these four annual returns:

1995 = -27.1%

1996 = +152.9%

1997 = +112.1%

1998 = -83.0%

Add them together:

-27.1% + 152.9% + 112.1% - 83.0% = 154.9%

There are 4 annual returns.

So:

154.9% / 4 = 38.725%

Rounded:

Arithmetic Mean Return ≈ 38.7%

Therefore:

AM = 38.7%

This means that the ordinary average of the four annual returns was 38.7%.

That calculation is completely correct.

But there is a problem.

The arithmetic mean does not tell us the rate at which the actual money grew over those four years.

Why 38.7% Can Be Misleading

Suppose someone sees:

Arithmetic Mean Return = 38.7%

They might think:

“If I started with $100 and earned 38.7% every year for 4 years, how much would I have?”

If the return really were exactly 38.7% every year, the calculation would be:

Ending Capital = $100 × (1 + 0.387)^4

First:

1 + 0.387 = 1.387

So:

Ending Capital = $100 × 1.387^4

Approximately:

Ending Capital = $370.5

This would mean:

$100 → $370.50

But that is NOT what actually happened.

Why?

Because the investment did not earn 38.7% every year.

The actual annual returns in this example were:

-27.1%

+152.9%

+112.1%

-83.0%

Those returns must be applied one after another.

What Actually Happened to $100?

Suppose you invested:

$100

at the beginning of 1995.

We now follow the actual returns year by year.

1995: Lose 27.1%

Starting money:

$100

Return:

-27.1%

You keep:

100% - 27.1% = 72.9%

In decimal form:

72.9% = 0.729

So:

$100 × 0.729 = $72.90

After 1995:

$100 → $72.90

1996: Gain 152.9%

Now you start with:

$72.90

The return is:

+152.9%

A 152.9% gain means you keep your original 100% plus another 152.9%.

So the growth factor is:

1 + 1.529 = 2.529

Now calculate:

$72.90 × 2.529 ≈ $184.36

After 1996:

$72.90 → $184.36

1997: Gain 112.1%

Now you start with approximately:

$184.36

The return is:

+112.1%

Growth factor:

1 + 1.121 = 2.121

So:

$184.36 × 2.121 ≈ $391.02

After 1997:

$184.36 → approximately $391.02

1998: Lose 83.0%

Now you start with approximately:

$391.02

The return is:

-83.0%

If you lose 83%, you keep only:

100% - 83% = 17%

In decimal form:

17% = 0.17

So:

$391.02 × 0.17 ≈ $66.47

Rounded:

Ending Capital ≈ $66.50

Therefore:

$100 → approximately $66.50

So although the arithmetic mean return was:

+38.7%

the investor actually ended with less money than they started with.

The Full Multiperiod Calculation

Instead of calculating every year separately, we can write the whole calculation in one line:

$100 × (1 - 0.271) × (1 + 1.529) × (1 + 1.121) × (1 - 0.830)

This becomes:

$100 × 0.729 × 2.529 × 2.121 × 0.170

Approximately:

$100 × 0.6648 = $66.48

Rounded:

Ending Capital ≈ $66.50

This is the actual result of applying all four yearly returns.

Why Didn’t the Arithmetic Mean Work?

Because simple returns compound.

Each year’s return applies to the amount of money remaining after the previous year.

The arithmetic mean simply does this:

Add the returns and divide by 4

It does not account for how your capital grows and shrinks from year to year.

But your actual money does.

For example, an 83% loss after large gains is devastating because that 83% loss is applied to a much larger amount of money.

That is why:

Arithmetic Mean Return = 38.7%

does not mean:

Your money grew at 38.7% every year

It only means:

The ordinary average of the four annual returns was 38.7%.

Now: The Geometric Mean Return

The geometric mean return, written as GM, answers a different question.

It asks:

“What single constant annual return would turn my starting money into the actual ending money over the same number of years?”

In this example:

Starting Capital = $100

Ending Capital ≈ $66.50

Number of Years = 4

So we want to find the single yearly return that would turn:

$100 → $66.50

over 4 years.

That annual return is approximately:

-9.7%

This is the geometric mean return.

What Does -9.7% Mean?

It does NOT mean the market actually returned exactly -9.7% in each of the four years.

It did not.

The annual returns in this example were:

1995 = -27.1%

1996 = +152.9%

1997 = +112.1%

1998 = -83.0%

Instead, -9.7% means:

If the investment had earned the same return every year for 4 years, a return of about -9.7% per year would have produced the same final value of approximately $66.50.

That is the key meaning of the geometric mean.

Geometric Mean Formula

The geometric mean return is:

GM = [(1 + R1) × (1 + R2) × … × (1 + RT)]^(1/T) - 1

Do not let the formula look scary.

It simply does three things:

1. Multiply all the growth factors together.

2. Find the T-th root.

3. Subtract 1.

For four years:

T = 4

So we use the 4th root.

Geometric Mean Calculation

The four growth factors are:

1995:

1 - 0.271 = 0.729

1996:

1 + 1.529 = 2.529

1997:

1 + 1.121 = 2.121

1998:

1 - 0.830 = 0.170

Multiply them:

0.729 × 2.529 × 2.121 × 0.170 ≈ 0.6648

This means that after all four years, the investment is worth about:

66.48% of its original value

Now find the 4th root of 0.6648:

0.6648^(1/4) ≈ 0.903

Now subtract 1:

0.903 - 1 = -0.097

Convert to percentage:

-0.097 × 100 = -9.7%

Therefore:

Geometric Mean Return ≈ -9.7%

Why Do We Use the 4th Root?

Because there are 4 years.

We want to find one constant annual growth factor that, when multiplied by itself 4 times, gives the same total result.

We know the total growth factor is approximately:

0.6648

So we are asking:

“What number multiplied by itself 4 times equals approximately 0.6648?”

That number is approximately:

0.903

So:

0.903 × 0.903 × 0.903 × 0.903 ≈ 0.6648

And:

0.903 = 1 - 0.097

Therefore:

Geometric Mean Return ≈ -9.7%

Why $100 × (1 - 0.097)^4 = $66.5?

Now the formula should make much more sense.

The geometric mean return is:

-9.7% per year

Convert it into decimal form:

-9.7% = -0.097

The annual growth factor is:

1 - 0.097 = 0.903

There are 4 years.

Therefore:

$100 × 0.903^4

which is the same as:

$100 × (1 - 0.097)^4

Approximately:

$100 × 0.665 = $66.50

So:

$100 → $66.50

This matches the actual result produced by the four annual returns.

That is why the geometric mean properly describes the constant annual rate at which the investment would have grown or declined over the whole period.

Arithmetic Mean vs. Geometric Mean

The arithmetic mean asks:

“What is the average of the individual yearly returns?”

Answer:

38.7%

The geometric mean asks:

“What constant annual return would have produced the same actual final investment value?”

Answer:

-9.7%

Both calculations are correct.

They simply answer different questions.

Why Can One Be Positive and the Other Negative?

This is probably the strangest part at first.

How can:

Arithmetic Mean = +38.7%

while:

Geometric Mean = -9.7%?

Because the yearly returns were extremely volatile.

There were huge gains:

+152.9%

+112.1%

But there was also a devastating loss:

-83.0%

An 83% loss destroys most of the capital remaining at that point.

The arithmetic mean treats all four percentages like ordinary numbers and averages them.

But the geometric mean considers how those returns actually compound on the investment.

That is why geometric mean is much more useful when asking:

“How did my money actually grow over several years?”

A Very Simple Example of the Same Idea

Suppose you invest:

$100

Year 1:

+50%

Year 2:

-50%

Arithmetic Mean:

(50% - 50%) / 2 = 0%

But actual money:

Year 1:

$100 × 1.50 = $150

Year 2:

$150 × 0.50 = $75

So:

$100 → $75

You actually lost:

25%

Therefore, an arithmetic mean of 0% does NOT mean your money stayed unchanged.

The geometric mean would show the constant annual rate that takes $100 to $75 over two years.

That is why the geometric mean is better for describing actual compounded growth over time.

The Simplest Way to Remember the Difference

Think:

Arithmetic Mean = Average Return

Geometric Mean = Growth Rate

Or even simpler:

Arithmetic Mean asks: “What was the average percentage?”

Geometric Mean asks: “What constant yearly rate matches what actually happened to my money over time?”

Notes

• AM = Arithmetic Mean Return
• GM = Geometric Mean Return
• Arithmetic mean is the ordinary average.
• Arithmetic Mean formula:

AM = (R1 + R2 + … + RT) / T

• In the illustrative example, the four annual returns are:

-27.1%, +152.9%, +112.1%, -83.0%

• Arithmetic Mean:

(-27.1% + 152.9% + 112.1% - 83.0%) / 4

= 38.7% approximately

• Arithmetic Mean Return = +38.7%
• This does NOT mean the investment actually grew at 38.7% every year.
• Actual investment growth must use the yearly returns one after another.
• Starting with $100:

$100 × 0.729 × 2.529 × 2.121 × 0.170 ≈ $66.50

• Therefore:

$100 → approximately $66.50

• The investor actually lost money over the full period.
• Geometric Mean Return = constant annual return that would produce the same ending capital.
• Geometric Mean formula:

GM = [(1 + R1) × (1 + R2) × … × (1 + RT)]^(1/T) - 1

• In this example:

GM ≈ -9.7%

• This means a constant return of approximately -9.7% per year for 4 years would turn $100 into approximately $66.50.
• Calculation:

$100 × (1 - 0.097)^4 ≈ $66.50

• ^4 means the annual growth factor is applied for 4 periods.
• Arithmetic mean answers:

“What was the average of the yearly returns?”

• Geometric mean answers:

“What constant annual return matches the actual compounded growth of my money?”

• Arithmetic mean can be positive even when the investment actually loses money over the whole period.
• Large gains and large losses can create this situation because returns compound.
• Geometric mean takes compounding into account.
• The Brazil example above is illustrative only and is not being presented as actual historical Brazilian market data.
• Easy memory:

Arithmetic Mean = Average

Geometric Mean = Actual Growth Rate

• Most important idea:

+38.7% arithmetic mean and -9.7% geometric mean can both be correct because they measure different things.