Finance

Formula Summary

1. Total Dividend

Total Dividend = Dividend Per Share × Number of Shares

Where:

• Dividend Per Share = dividend paid for one share
• Number of Shares = number of shares owned

2. Dividend Yield

Dividend Yield = D / pB

Where:

• D = Dividend Per Share
• pB = Beginning Price Per Share

3. Capital Gain or Loss

Capital Gain/Loss = (pE - pB) / pB

Where:

• pB = Beginning Price
• pE = Ending Price

If pE > pB → Capital Gain

If pE < pB → Capital Loss

4. Simple Return

R = ((pE - pB) + D) / pB

Where:

• R = Simple Return
• pB = Beginning Price
• pE = Ending Price
• D = Dividend Per Share

Another way to write it:

Simple Return = Capital Gain/Loss + Dividend Yield

5. Convert Simple Return to Continuously Compounded Return

r = ln(1 + R)

Where:

• r = Continuously Compounded Return
• R = Simple Return
• ln = Natural Logarithm

Easy memory:

Simple → ln → Continuous

6. Convert Continuously Compounded Return to Simple Return

R = e^r - 1

Where:

• R = Simple Return
• r = Continuously Compounded Return
• e ≈ 2.71828

Easy memory:

Continuous → e → Simple

7. Multiperiod Simple Return

For several periods:

R(T) = (1 + R1) × (1 + R2) × … × (1 + RT) - 1

Where:

• R(T) = Total Simple Return over T periods
• R1 = Return in Period 1
• R2 = Return in Period 2
• RT = Return in the final period
• T = Number of periods

Easy memory:

Simple Returns = MULTIPLY across periods

8. Multiperiod Continuously Compounded Return

r(T) = r1 + r2 + … + rT

Where:

• r(T) = Total Continuously Compounded Return over T periods
• r1, r2, … rT = Continuously compounded returns for each period

Easy memory:

Continuously Compounded Returns = ADD across periods

9. Convert Multiperiod Simple Return to Multiperiod Continuously Compounded Return

r(T) = ln(1 + R(T))

10. Convert Multiperiod Continuously Compounded Return to Multiperiod Simple Return

R(T) = e^r(T) - 1

11. Terminal Capital Using Individual Simple Returns

CT = C0 × (1 + R1) × (1 + R2) × … × (1 + RT)

Where:

• C0 = Initial Capital / Starting Money
• CT = Terminal Capital / Ending Money

12. Terminal Capital Using Total Simple Return

Because:

(1 + R1) × (1 + R2) × … × (1 + RT) = 1 + R(T)

we can shorten the formula to:

CT = C0 × (1 + R(T))

So these two formulas mean the same thing:

CT = C0 × (1 + R1) × (1 + R2) × … × (1 + RT)

and

CT = C0 × (1 + R(T))

13. Terminal Capital Using Continuously Compounded Returns

Using all individual continuously compounded returns:

CT = C0 × e^(r1 + r2 + … + rT)

Because:

r(T) = r1 + r2 + … + rT

we can shorten it to:

CT = C0 × e^r(T)

14. Capital When the Same Return Happens Every Period

If the same return R happens every period:

CT = C0 × (1 + R)^T

Where:

• C0 = Starting Capital
• CT = Ending Capital
• R = Return per period
• T = Number of periods

Example with a -9.7% return for 4 years:

CT = $100 × (1 - 0.097)^4

15. Ending Value Using One Continuously Compounded Return

Ending Value = Beginning Value × e^r

Or:

CT = C0 × e^r

If there are T periods and r(T) is the total continuously compounded return:

CT = C0 × e^r(T)

Arithmetic Mean Return

16. Arithmetic Mean Return

AM = (R1 + R2 + … + RT) / T

Where:

• AM = Arithmetic Mean Return
• R1, R2, … RT = Individual Period Returns
• T = Number of Returns

Easy meaning:

Arithmetic Mean = Add all returns ÷ Number of returns

Arithmetic mean answers:

“What was the average of the individual returns?”

Geometric Mean Return

17. Geometric Mean Return

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

Where:

• GM = Geometric Mean Return
• R1, R2, … RT = Individual Period Returns
• T = Number of Periods
• 1/T = Take the T-th root

Easy meaning:

Geometric Mean = the constant return per period that would produce the same actual ending capital

Geometric mean answers:

“What constant return every period would give me the same final amount?”

18. Geometric Mean Using Starting and Ending Capital

The same idea can also be written as:

GM = (CT / C0)^(1/T) - 1

Where:

• C0 = Starting Capital
• CT = Ending Capital
• T = Number of Periods

For example, if:

C0 = $100

CT = $66.50

T = 4

Then:

GM = ($66.50 / $100)^(1/4) - 1

GM = 0.665^(1/4) - 1

GM ≈ -0.097 = -9.7%

If Dividends Are Taken Out Instead of Reinvested

19. Total Ending Wealth

If dividends are withdrawn and kept separately:

Total Ending Wealth = Ending Value of Investment + Dividends Taken Out

20. Total Simple Return With Withdrawn Dividends

Total Simple Return = (Total Ending Wealth - Beginning Investment) / Beginning Investment

This allows you to count the dividends as part of your total wealth even though they were not reinvested.

Percentage and Decimal Conversion

21. Percentage to Decimal

Decimal = Percentage / 100

Examples:

29.5% = 0.295

20% = 0.20

5% = 0.05

-9.7% = -0.097

22. Decimal to Percentage

Percentage = Decimal × 100

Examples:

0.295 × 100 = 29.5%

0.20 × 100 = 20%

-0.097 × 100 = -9.7%

Master Notes

• R = Simple Return
• r = Continuously Compounded Return
• R(T) = Total Simple Return over T periods
• r(T) = Total Continuously Compounded Return over T periods
• AM = Arithmetic Mean Return
• GM = Geometric Mean Return
• C0 = Initial Capital / Starting Money
• CT = Terminal Capital / Ending Money
• pB = Beginning Price
• pE = Ending Price
• D = Dividend Per Share
• T = Number of Periods
• ln = Natural Logarithm
• e ≈ 2.71828

Simplest Memory Rules

Simple Return:

R = ((pE - pB) + D) / pB

Simple → Continuous:

r = ln(1 + R)

Continuous → Simple:

R = e^r - 1

Multiple Simple Returns:

MULTIPLY

Multiple Continuously Compounded Returns:

ADD

Arithmetic Mean:

ADD returns, then DIVIDE by number of returns

Geometric Mean:

MULTIPLY growth factors, take the T-th root, then subtract 1

Arithmetic Mean = Average Return

Geometric Mean = Constant Compounded Growth Rate

Terminal Capital with Simple Return:

CT = C0 × (1 + R(T))

Terminal Capital with Continuously Compounded Return:

CT = C0 × e^r(T)

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.

Finance

Converting Simple Return to Continuously Compounded Return

The easiest way to understand the relationship between a simple return and a continuously compounded return is to remember that they describe the same investment performance, but they express it in different mathematical ways.

A simple return, written as R, tells you how much your investment gained or lost compared with the amount you started with.

A continuously compounded return, written as r, expresses that same gain or loss using a different mathematical method.

The conversion formula is:

r = ln(1 + R)

Where:

r = Continuously Compounded Return

R = Simple Return

ln = Natural Logarithm

The most important thing is that R must be entered as a decimal, not as a percentage.

For example:

20% = 0.20

10% = 0.10

5% = 0.05

29.5% = 0.295

Example 1: Converting a 20% Simple Return

Suppose:

Simple Return = 20%

First convert 20% into decimal form:

20% = 0.20

Use the formula:

r = ln(1 + R)

Insert R = 0.20:

r = ln(1 + 0.20)

r = ln(1.20)

Using the ln button on a calculator:

ln(1.20) ≈ 0.1823

Convert the result back into a percentage:

0.1823 × 100 = 18.23%

Therefore:

Simple Return = 20%

Continuously Compounded Return = 18.23%

They are different percentages, but they represent the same investment performance.

Example 2: Including a Dividend

Suppose:

Beginning Price = $50 per share

Ending Price = $60 per share

Dividend = $2 per share

First calculate the simple return.

The simple return formula is:

R = ((Ending Price - Beginning Price) + Dividend) / Beginning Price

Put in the numbers:

R = (($60 - $50) + $2) / $50

First calculate the price gain:

$60 - $50 = $10

Then add the dividend:

$10 + $2 = $12

Now divide by the beginning price:

$12 / $50 = 0.24 = 24%

Therefore:

Simple Return = 24%

Now convert the 24% simple return into a continuously compounded return.

First convert 24% into decimal form:

24% = 0.24

Use:

r = ln(1 + R)

So:

r = ln(1 + 0.24)

r = ln(1.24)

Using a calculator:

ln(1.24) ≈ 0.2151

Convert to percentage:

0.2151 × 100 = 21.51%

Therefore:

Simple Return = 24%

Continuously Compounded Return ≈ 21.51%

An important point is that the dividend was already included when calculating the 24% simple return.

Therefore, you do not add the dividend again when calculating the continuously compounded return.

If you added the dividend again, you would count the same dividend twice.

The correct process is:

Price Change + Dividend → Simple Return → Continuously Compounded Return

Example 3: Coca-Cola

Suppose:

Beginning Price = $45.27

Ending Price = $57.00

Dividend Per Share = $1.64

First calculate the simple return.

The formula is:

R = ((pE - pB) + D) / pB

Put in the numbers:

R = (($57.00 - $45.27) + $1.64) / $45.27

First calculate the price increase:

$57.00 - $45.27 = $11.73

Then add the dividend:

$11.73 + $1.64 = $13.37

Now divide by the beginning price:

$13.37 / $45.27 ≈ 0.295

Convert into a percentage:

0.295 × 100 = 29.5%

Therefore:

Simple Return ≈ 29.5%

Now convert that simple return into a continuously compounded return.

Convert 29.5% into decimal form:

29.5% = 0.295

Use:

r = ln(1 + R)

So:

r = ln(1 + 0.295)

r = ln(1.295)

Using a calculator:

ln(1.295) ≈ 0.259

Convert to percentage:

0.259 × 100 = 25.9%

Therefore:

Continuously Compounded Return ≈ 25.9%

So:

29.5% Simple Return = 25.9% Continuously Compounded Return

These percentages are different, but they represent the same investment performance.

How to Convert Back

If you already know the continuously compounded return and want to find the simple return, use:

R = e^r - 1

Where:

R = Simple Return

r = Continuously Compounded Return

e ≈ 2.71828

For example:

Continuously Compounded Return = 25.9%

First convert 25.9% into decimal form:

25.9% = 0.259

Then use:

R = e^0.259 - 1

Using a calculator:

e^0.259 ≈ 1.295

Subtract 1:

1.295 - 1 = 0.295

Convert into a percentage:

0.295 × 100 = 29.5%

Therefore:

Continuously Compounded Return = 25.9%

Simple Return = 29.5%

So you can move in both directions.

Simple Return → Continuously Compounded Return

Use:

r = ln(1 + R)

Continuously Compounded Return → Simple Return

Use:

R = e^r - 1

Notes — Simple Return

• Symbol = R
• Simple return tells you the gain or loss compared with the amount you started with.
• Simple return is usually easier to understand because it directly compares beginning money with ending money.
• For a stock, simple return can include both a capital gain or loss and a dividend yield.
• Simple Return formula:

R = ((pE - pB) + D) / pB

• pB = Beginning Price
• pE = Ending Price
• D = Dividend Per Share
• Simple return can also be understood as:

Simple Return = Capital Gain/Loss + Dividend Yield

• Capital Gain/Loss formula:

(pE - pB) / pB

• Dividend Yield formula:

D / pB

• Example:

Beginning Price = $50

Ending Price = $60

Dividend = $2

Price increase:

$60 - $50 = $10

Add dividend:

$10 + $2 = $12

Simple Return:

$12 / $50 = 0.24 = 24%

• Therefore:

Simple Return = 24%

• For multiple periods, simple returns are multiplicative.
• This means you normally multiply the growth factors, not add the simple returns.
• Multiperiod simple return formula:

R(T) = (1 + R1) × (1 + R2) × … × (1 + RT) - 1

• Easy memory rule:

Simple Returns → MULTIPLY across periods

Notes — Continuously Compounded Return

• Symbol = r
• A continuously compounded return is another way of expressing the same investment gain or loss.
• It uses a mathematical function called a natural logarithm.
• It can also be called:
• Log Return
• Logarithmic Return
• These names mean the same thing:

Continuously Compounded Return = Log Return = Logarithmic Return

• Formula for converting from a simple return:

r = ln(1 + R)

• ln = Natural Logarithm
• Use the ln button on a scientific calculator.
• Always convert the simple return percentage into a decimal first.
• Example:

Simple Return = 20%

20% = 0.20

r = ln(1.20)

r ≈ 0.1823

r ≈ 18.23%

• Therefore:

20% Simple Return = 18.23% Continuously Compounded Return

• To convert back into a simple return:

R = e^r - 1

• e ≈ 2.71828
• Use the e^x or exp function on a calculator.
• For multiple periods, continuously compounded returns are additive.
• This means you can simply add the individual continuously compounded returns.
• Multiperiod formula:

r(T) = r1 + r2 + … + rT

• Easy memory rule:

Continuously Compounded Returns → ADD across periods

Notes — Difference Between Simple Return and Continuously Compounded Return

• Simple Return symbol = R
• Continuously Compounded Return symbol = r
• Simple return expresses the investment gain or loss in the normal percentage form.
• Continuously compounded return expresses the same investment gain or loss using logarithms.
• Simple return is generally easier for beginners and investors to understand.
• Continuously compounded return is commonly used in financial analysis, statistics, and financial models.
• Simple return for a stock may directly include:
• Capital gain or loss
• Dividend yield
• Continuously compounded return is normally calculated after the simple return has already been calculated.
• Simple Return formula:

R = ((pE - pB) + D) / pB

• Continuously Compounded Return formula:

r = ln(1 + R)

• Example:

Simple Return = 29.5%

Continuously Compounded Return = 25.9%

• These percentages are different, but they represent the same investment performance.
• A dividend should be included when calculating the simple return if the dividend was received during the holding period.
• Once the dividend has already been included in the simple return, do not add it again when converting to the continuously compounded return.
• For multiple periods:
• Simple Returns = Multiply
• Continuously Compounded Returns = Add
• Simple returns are multiplicative across periods.
• Continuously compounded returns are additive across periods.
• Simple multiperiod formula:

R(T) = (1 + R1) × (1 + R2) × … × (1 + RT) - 1

• Continuously compounded multiperiod formula:

r(T) = r1 + r2 + … + rT

• To convert from simple return to continuously compounded return:

r = ln(1 + R)

• To convert from continuously compounded return back to simple return:

R = e^r - 1

• Easy memory:

Simple → ln → Continuous

Continuous → e → Simple

• Easy multiperiod memory:

Simple = MULTIPLY

Continuous = ADD

• Most important idea:

Simple return and continuously compounded return are not two different profits. They are two different ways of expressing the same investment performance.

Finance

Multiperiod Returns

A multiperiod return is simply the return you earn over more than one period.

A period could be one day, one month, or one year. So if you hold an investment for 5 years, you have a 5-year multiperiod return.

The most important rule is:

Simple returns are multiplicative.

Continuously compounded returns are additive.

Those two sentences sound difficult, but the idea is actually very simple.

First: What Does “Multiplicative” Mean?

Multiplicative simply means we multiply the returns from each period together.

Suppose you invest:

$100

Your investment earns:

Year 1 = +10%

Year 2 = +20%

You might think:

10% + 20% = 30%

But that is not the correct total simple return.

Why?

Because after Year 1, you no longer have $100.

You have:

$100 × 1.10 = $110

Then the 20% return in Year 2 is earned on $110, not the original $100.

Year 2:

$110 × 1.20 = $132

So you started with:

$100

and ended with:

$132

Your total gain is:

$132 - $100 = $32

Therefore:

2-Year Simple Return = $32 / $100 = 32%

So:

Year 1 = 10%

Year 2 = 20%

but:

Total Simple Return = 32%, not 30%

This happens because simple returns compound.

The Simple Return Multiperiod Formula

For several periods:

R(T) = (1 + R1) × (1 + R2) × … × (1 + RT) - 1

Where:

R(T) = total simple return over all periods

R1 = return in Period 1

R2 = return in Period 2

RT = return in the final period

T = total number of periods

The important part is:

Multiply, then subtract 1.

Easy 2-Year Example

Suppose:

Year 1 Simple Return = 10%

Year 2 Simple Return = 20%

First convert percentages into decimals:

10% = 0.10

20% = 0.20

Then:

R(2) = (1 + 0.10) × (1 + 0.20) - 1

R(2) = 1.10 × 1.20 - 1

R(2) = 1.32 - 1

R(2) = 0.32

R(2) = 32%

Therefore:

2-Year Simple Return = 32%

Why Do We Add 1?

This is very important.

Suppose your return is 10%.

You do not multiply your money by 0.10 because that would only calculate the profit.

Instead:

1 + 0.10 = 1.10

The 1 represents your original 100% of money.

The 0.10 represents your extra 10% return.

So:

1.10 = original money + 10% gain

Similarly:

20% return → 1.20

5% return → 1.05

30% return → 1.30

What If You Lose Money?

The same rule works with negative returns.

Suppose:

Year 1 Return = -20%

Year 2 Return = +10%

Convert them to decimals:

-20% = -0.20

10% = 0.10

Then:

R(2) = (1 - 0.20) × (1 + 0.10) - 1

R(2) = 0.80 × 1.10 - 1

R(2) = 0.88 - 1

R(2) = -0.12

Therefore:

2-Year Simple Return = -12%

Let’s see this using money.

Start with:

$100

After losing 20%:

$100 × 0.80 = $80

Then gain 10%:

$80 × 1.10 = $88

You finish with:

$88

You lost:

$100 - $88 = $12

So:

Total Return = -12%

Notice:

-20% + 10% = -10%

But your actual total return is:

-12%

This is another reason why you should not simply add simple returns across periods.

Continuously Compounded Returns Are Different

A continuously compounded return, written as r, works differently over multiple periods.

Instead of multiplying the yearly continuously compounded returns, we simply add them.

This is why continuously compounded returns are called additive.

The formula is:

r(T) = r1 + r2 + … + rT

Where:

r(T) = total continuously compounded return

r1 = continuously compounded return in Period 1

r2 = continuously compounded return in Period 2

rT = continuously compounded return in the final period

So the rule is extremely simple:

Simple Returns → Multiply

Continuously Compounded Returns → Add

Easy Continuously Compounded Example

Suppose:

Year 1 continuously compounded return = 8%

Year 2 continuously compounded return = 12%

The total continuously compounded return is simply:

8% + 12% = 20%

Therefore:

2-Year Continuously Compounded Return = 20%

There is no need to multiply the individual log returns.

You simply add them.

Why Is This Useful?

Imagine you have returns for 10 years.

With simple returns, you have to multiply all 10 growth factors:

(1 + R1) × (1 + R2) × … × (1 + R10)

With continuously compounded returns, you can simply do:

r1 + r2 + … + r10

That is one reason log returns are useful in finance.

Coca-Cola Multiperiod Example

Suppose someone bought Coca-Cola shares at the end of 2000 and held them until the end of 2009.

That is a nine-year investment period covering the annual returns from 2001 through 2009.

Assume that all dividends received were reinvested.

Reinvested dividends means that instead of taking the dividend cash and spending it, the investor puts that money back into the investment.

This allows the dividend money to also participate in future investment growth.

Coca-Cola Using Simple Returns

To calculate the total nine-year simple return, all the yearly simple-return growth factors are multiplied together.

The calculation begins like this:

R(9) = (1 - 0.214) × (1 - 0.053) × … × (1 + 0.295) - 1

The … simply means there are other yearly returns between those shown.

After multiplying all nine annual return factors:

R(9) = 0.154

Convert it into a percentage:

0.154 × 100 = 15.4%

Therefore:

9-Year Simple Return = 15.4%

This means that over the entire nine-year period, the investment grew by 15.4% overall.

It does not mean 15.4% every year.

It means 15.4% for the whole nine-year period combined.

Coca-Cola Using Continuously Compounded Returns

Now we can describe the exact same nine-year investment using continuously compounded returns.

Instead of multiplying the yearly returns, we add them.

The calculation is:

r(9) = -0.241 - 0.055 + … + 0.259

After adding all nine continuously compounded annual returns:

r(9) = 0.143

Convert it into a percentage:

0.143 × 100 = 14.3%

Therefore:

9-Year Continuously Compounded Return = 14.3%

So the same nine-year investment can be described as:

9-Year Simple Return = 15.4%

or

9-Year Continuously Compounded Return = 14.3%

These percentages are different, but they describe the same investment performance.

You Can Convert Between Them

Just like with a one-period return, you can convert a multiperiod simple return into a multiperiod continuously compounded return.

The formula is:

r(T) = ln(1 + R(T))

For Coca-Cola:

Simple Return = 15.4%

Convert to decimal:

15.4% = 0.154

Then:

r(9) = ln(1 + 0.154)

r(9) = ln(1.154)

r(9) ≈ 0.143

Convert to percentage:

0.143 × 100 = 14.3%

So:

15.4% Simple Return = 14.3% Continuously Compounded Return

They describe the same total growth.

Converting Back to Simple Return

You can also go backwards.

The formula is:

R(T) = e^r(T) - 1

We know:

r(9) = 14.3%

Convert it into decimal form:

14.3% = 0.143

Then:

R(9) = e^0.143 - 1

R(9) ≈ 1.154 - 1

R(9) ≈ 0.154

Convert to percentage:

0.154 × 100 = 15.4%

So again:

14.3% Continuously Compounded Return = 15.4% Simple Return

Same investment performance.

Different way of expressing it.

Now Let’s Use Actual Money

Suppose you invested:

$100

in Coca-Cola at the end of 2000 and held the investment through the end of 2009, while reinvesting all dividends.

The total nine-year simple return was:

15.4%

So how much would your $100 become?

Beginning Capital = $100

Simple Return = 15.4%

Ending Capital:

$100 × (1 + 0.154)

= $100 × 1.154

= $115.40

Therefore:

$100 became $115.40

Your total profit was:

$115.40 - $100 = $15.40

The Same Result Using Continuously Compounded Return

The continuously compounded return was:

14.3%

or:

r(9) = 0.143

To calculate the ending amount using a continuously compounded return:

Ending Capital = Beginning Capital × e^r(T)

So:

Ending Capital = $100 × e^0.143

e^0.143 ≈ 1.154

Therefore:

$100 × 1.154 = $115.40

Again:

Ending Capital = $115.40

So both methods produce exactly the same ending amount.

See the Important Point

Using simple returns:

$100 × (1 + 15.4%) = $115.40

Using continuously compounded returns:

$100 × e^0.143 = $115.40

Both give:

$115.40

Therefore:

15.4% Simple Return

and

14.3% Continuously Compounded Return

describe the same growth from $100 to $115.40.

What Is Capital?

The term capital simply means the amount of money you have invested.

C0 means your initial capital, or the amount you start with.

CT means your terminal capital, or the amount you have at the end.

So:

C0 = Starting Money

CT = Ending Money

For example:

C0 = $100

CT = $115.40

Terminal Capital Using Simple Returns

If you start with capital C0 and invest it for several periods, the ending capital can be calculated using simple returns as:

CT = C0 × (1 + R1) × (1 + R2) × … × (1 + RT)

Because all those yearly returns together give the total multiperiod simple return, we can also write:

CT = C0 × (1 + R(T))

For the Coca-Cola example:

C0 = $100

R(9) = 15.4%

So:

CT = $100 × (1 + 0.154)

CT = $100 × 1.154

CT = $115.40

Terminal Capital Using Continuously Compounded Returns

We can also calculate ending capital using continuously compounded returns.

First, add all the individual continuously compounded returns:

r(T) = r1 + r2 + … + rT

Then:

CT = C0 × e^r(T)

For Coca-Cola:

C0 = $100

r(9) = 0.143

Therefore:

CT = $100 × e^0.143

CT = $100 × 1.154

CT = $115.40

Again, both methods give exactly the same result.

Why Is One Multiplicative and the Other Additive?

This is the main idea of the entire topic.

Simple returns are multiplicative because your money changes after every period.

If you gain 10%, your money becomes:

Original Money × 1.10

If you then gain another 20%, you multiply the new amount by:

1.20

Therefore:

1.10 × 1.20

That is why simple returns are multiplicative.

Continuously compounded returns are additive because log returns can simply be added across time.

If:

Year 1 log return = r1

Year 2 log return = r2

then:

Total Log Return = r1 + r2

That is why continuously compounded returns are additive.

The Simplest Possible Example

You invest:

$100

Year 1 return = +10%

Year 2 return = +20%

Using Simple Returns

$100 × 1.10 = $110

$110 × 1.20 = $132

So:

Ending Money = $132

Total Simple Return:

($132 - $100) / $100 = 32%

Therefore:

Total Simple Return = 32%

Using Continuously Compounded Returns

Convert Year 1:

ln(1.10) ≈ 0.0953 = 9.53%

Convert Year 2:

ln(1.20) ≈ 0.1823 = 18.23%

Now simply add them:

9.53% + 18.23% = 27.76%

So:

Total Continuously Compounded Return = 27.76%

Convert it back:

e^0.2776 - 1 ≈ 0.32 = 32% Simple Return

Both describe the same movement:

$100 → $132

The Most Important Rule

If you remember only one thing from this topic, remember:

Simple Returns = MULTIPLY

Continuously Compounded Returns = ADD

For example:

Simple returns:

(1 + R1) × (1 + R2) × (1 + R3) - 1

Continuously compounded returns:

r1 + r2 + r3

Notes

• Multiperiod Return = return over more than one period.
• A period can be a day, month, year, or another chosen length of time.
• T = total number of periods.
• R(T) = total simple return over T periods.
• r(T) = total continuously compounded return over T periods.
• Simple returns are multiplicative.
• Continuously compounded returns are additive.
• Do not normally add simple returns from different periods.
• To combine simple returns: multiply the growth factors.
• Simple multiperiod formula: R(T) = (1 + R1) × (1 + R2) × … × (1 + RT) - 1
• To combine continuously compounded returns: add them together.
• Continuously compounded multiperiod formula: r(T) = r1 + r2 + … + rT
• To convert total simple return to total continuously compounded return: r(T) = ln(1 + R(T))
• To convert total continuously compounded return to total simple return: R(T) = e^r(T) - 1
• C0 = initial capital, or starting money.
• CT = terminal capital, or ending money.
• Using simple returns: CT = C0 × (1 + R(T))
• Using continuously compounded returns: CT = C0 × e^r(T)
• Both methods give the same terminal capital.
• Coca-Cola nine-year simple return = 15.4%.
• Coca-Cola nine-year continuously compounded return = 14.3%.
• $100 invested over that period became approximately $115.40.
• Reinvesting dividends means putting dividends back into the investment instead of taking the cash out.
• Reinvested dividends can then participate in future investment growth.
• A total nine-year return of 15.4% means 15.4% over the whole nine years, not 15.4% every year.
• Easy memory rule: Simple = Multiply. Continuous = Add.
• Simplest idea: different calculations, same investment, same final money.

Finance

Continuously Compounded Returns

A continuously compounded return, written as r, is simply another way of expressing the gain or loss from an investment. It describes the same investment result as a simple return, but it expresses that result in a different way.

Think about measuring distance. The distance between Miami and Chicago could be described as approximately 1,200 miles or approximately 1,900 kilometers. The actual distance has not changed. We are simply expressing the same distance using two different units.

Returns work in a similar way. An investment’s gain or loss can be expressed using a simple return or a continuously compounded return. The numbers will usually be different, but they describe the same investment performance over the same period.

For example, Coca-Cola’s return during 2009 can be expressed as a 29.5% simple return. The same investment performance can also be expressed as a 25.9% continuously compounded return.

So:

Coca-Cola’s 2009 performance:

Simple Return = 29.5%

Continuously Compounded Return = 25.9%

These numbers are different, but they describe the same gain during the same year.

Simple Return vs. Continuously Compounded Return

We already know that a simple return, written as R, tells us how much our investment gained or lost compared with the amount we originally invested.

For example, if we invest $100 and finish with $120, our gain is $20.

Simple Return = Gain / Beginning Investment

Simple Return = $20 / $100

Simple Return = 0.20 = 20%

So the simple return is 20%.

A continuously compounded return, written as r, takes that same simple return and expresses it using a natural logarithm.

The formula is:

r = ln(1 + R)

Where:

r = Continuously Compounded Return

R = Simple Return

ln = Natural Logarithm

Do not let the term natural logarithm make this look difficult. For what we are doing here, ln is simply a button on a scientific calculator. You do not need to understand all the mathematics behind logarithms yet to calculate the return.

The basic process is:

Step 1: Take the simple return.

Step 2: Convert the percentage into a decimal.

Step 3: Add 1.

Step 4: Press ln on the calculator.

That gives you the continuously compounded return.

Coca-Cola Example

Coca-Cola had a simple return of 29.5% during 2009.

We want to convert this simple return into a continuously compounded return.

The formula is:

r = ln(1 + R)

First, convert 29.5% into decimal form:

29.5% = 0.295

Therefore:

R = 0.295

Now put it into the formula:

r = ln(1 + 0.295)

Add 1:

r = ln(1.295)

Using the ln button on a calculator:

ln(1.295) ≈ 0.259

Convert the decimal back into a percentage:

0.259 × 100 = 25.9%

Therefore:

Continuously Compounded Return = 25.9%

So Coca-Cola’s 2009 return can be expressed in two ways:

Simple Return = 29.5%

Continuously Compounded Return = 25.9%

Again, this does not mean Coca-Cola produced two different investment gains. It is the same investment result expressed in two different ways.

Think again about distance:

1,200 miles ≈ 1,900 kilometers

They look like different numbers, but they describe the same distance.

Similarly:

29.5% Simple Return = 25.9% Continuously Compounded Return

They look like different percentages, but they describe the same investment performance.

What Does “ln” Mean?

The symbol ln means natural logarithm.

For a beginner, the easiest way to think about it is:

ln is a mathematical function used to convert a simple return into a continuously compounded return.

You normally do not calculate ln manually. You use the ln button on a scientific calculator.

For example:

ln(1.295) ≈ 0.259

That is all you need to do for this calculation.

Remember:

ln does NOT mean divide.

ln does NOT mean multiply.

It is its own mathematical function.

Another Simple Example

Suppose an investment has a 10% simple return.

We want to find its continuously compounded return.

First convert 10% into decimal form:

10% = 0.10

Use the formula:

r = ln(1 + R)

Put in R = 0.10:

r = ln(1 + 0.10)

r = ln(1.10)

Using a calculator:

ln(1.10) ≈ 0.0953

Convert this into a percentage:

0.0953 × 100 ≈ 9.53%

Therefore:

Simple Return = 10%

Continuously Compounded Return ≈ 9.53%

Both numbers represent the same investment performance.

Example Using Money

Suppose you invest $1,000 and your investment grows to $1,100.

Your gain is:

$1,100 - $1,000 = $100

Your simple return is:

$100 / $1,000 = 0.10 = 10%

So:

R = 10%

Now calculate the continuously compounded return:

r = ln(1 + 0.10)

r = ln(1.10)

r ≈ 0.0953

r ≈ 9.53%

Therefore, the same investment can be described as:

Simple Return = 10%

or

Continuously Compounded Return = 9.53%

The amount of money you actually have at the end is still $1,100. We have simply changed the way we express the return.

Converting Back to a Simple Return

We can also go in the opposite direction.

If we already know the continuously compounded return, we can convert it back into a simple return.

The formula is:

R = e^r - 1

Where:

R = Simple Return

r = Continuously Compounded Return

e ≈ 2.71828

The number e is a special mathematical number, just like π is a special mathematical number.

You do not normally need to calculate powers of e manually. A scientific calculator usually has an e^x button or an exp function.

Coca-Cola Example: Converting Back

We already know that Coca-Cola’s continuously compounded return was approximately:

r = 25.9%

First convert 25.9% into decimal form:

25.9% = 0.259

Use the formula:

R = e^r - 1

Put in r = 0.259:

R = e^0.259 - 1

Using a calculator:

e^0.259 ≈ 1.295

Then subtract 1:

1.295 - 1 = 0.295

Convert it into a percentage:

0.295 × 100 = 29.5%

Therefore:

Simple Return = 29.5%

So we went from:

29.5% Simple Return → 25.9% Continuously Compounded Return

and then back from:

25.9% Continuously Compounded Return → 29.5% Simple Return

They are simply two different ways of expressing the same return.

The Two Conversion Formulas

To convert a simple return into a continuously compounded return:

r = ln(1 + R)

To convert a continuously compounded return into a simple return:

R = e^r - 1

A very easy way to remember the direction is:

Simple → use ln → Continuous

Continuous → use e → Simple

Why Are the Two Returns Different?

You might wonder why a simple return of 29.5% becomes a continuously compounded return of only 25.9%.

The reason is that the two methods measure the same gain using different mathematical systems.

You do not need to worry too much about the deeper mathematics yet. The important point is:

Simple returns and continuously compounded returns are two different ways of expressing the same investment performance.

The continuously compounded return will not normally have exactly the same percentage as the simple return.

For Coca-Cola:

Simple Return = 29.5%

Continuously Compounded Return = 25.9%

Do not accidentally treat these as the same number.

What Happens When Returns Are Small?

There is an important relationship between simple returns and continuously compounded returns.

When the simple return is small, the difference between the simple return and the continuously compounded return is also small.

Mathematically, when R is small:

r ≈ R

The symbol ≈ means approximately equal to.

In simple words:

When the return is small, simple return and continuously compounded return are very close to each other.

For example, suppose the simple return is only 2%.

Convert 2% into decimal form:

2% = 0.02

Now calculate the continuously compounded return:

r = ln(1 + 0.02)

r = ln(1.02)

r ≈ 0.0198

Convert it to a percentage:

0.0198 × 100 ≈ 1.98%

So:

Simple Return = 2.00%

Continuously Compounded Return ≈ 1.98%

The difference is tiny:

2.00% - 1.98% = 0.02 percentage points

Therefore, when returns are small, it usually makes very little difference whether we look at the simple return or continuously compounded return.

What Happens When Returns Are Larger?

When the return becomes larger, the difference between the two measures can become more noticeable.

For example, suppose the simple return is 50%.

Convert it into decimal form:

50% = 0.50

Calculate the continuously compounded return:

r = ln(1 + 0.50)

r = ln(1.50)

r ≈ 0.4055

Convert it into a percentage:

0.4055 × 100 ≈ 40.55%

Therefore:

Simple Return = 50%

Continuously Compounded Return ≈ 40.55%

Now there is a much bigger difference between the two percentages.

This means that in some periods the simple return and continuously compounded return may be very close, while in other periods the difference may be substantial.

What About a Loss?

Continuously compounded returns can also describe a loss.

Suppose an investment has a simple return of -20%.

The minus sign tells us that the investment lost value.

Convert -20% into decimal form:

-20% = -0.20

Use the formula:

r = ln(1 + R)

r = ln(1 - 0.20)

r = ln(0.80)

Using a calculator:

ln(0.80) ≈ -0.2231

Convert it into a percentage:

-0.2231 × 100 ≈ -22.31%

Therefore:

Simple Return = -20%

Continuously Compounded Return ≈ -22.31%

Again, these are not two separate losses. They are two different ways of describing the same loss.

Why Do Investors Often Prefer Simple Returns?

For many ordinary investors, simple returns are easier to understand.

Suppose you start with:

$1,000

and finish with:

$1,200

You gained:

$1,200 - $1,000 = $200

Your simple return is:

$200 / $1,000 = 20%

That is very straightforward.

The simple return directly answers:

“How much money did I gain or lose compared with the amount I started with?”

This is why simple returns are extremely useful for investors.

Why Learn Continuously Compounded Returns Then?

Even though simple returns are easier to understand, continuously compounded returns are also important and widely used in finance.

They are useful in many financial calculations, especially when working with returns over time, statistics, financial models, asset prices, and other more advanced topics.

At this stage, you mainly need to understand that continuously compounded returns are another way of expressing periodic gain or loss.

You do not need to think of them as replacing simple returns.

Both are useful.

Other Names for Continuously Compounded Returns

A continuously compounded return can also be called a:

Logarithmic return

or

Log return

These three terms mean the same thing:

Continuously Compounded Return = Logarithmic Return = Log Return

So if you later see the term log return, do not think it is a completely new type of return. It is simply another name for a continuously compounded return.

What Does “Return” Mean When the Type Is Not Specified?

There is one final rule that is important to remember.

If the word “return” is used without telling you whether it means a simple return or a continuously compounded return, it means:

Simple Return

So:

“Return” by itself → Simple Return

If continuously compounded return, logarithmic return, or log return is meant, it will normally be specified.

One Full Example From Start to Finish

Suppose you invest $2,000 in a stock.

At the end of the period, your investment is worth $2,400.

Your gain is:

$2,400 - $2,000 = $400

Now calculate the simple return:

Simple Return = $400 / $2,000

Simple Return = 0.20 = 20%

Therefore:

R = 20%

Now convert that simple return into a continuously compounded return.

Convert 20% into decimal form:

20% = 0.20

Use:

r = ln(1 + R)

r = ln(1 + 0.20)

r = ln(1.20)

Using a calculator:

r ≈ 0.1823

Convert it into a percentage:

0.1823 × 100 ≈ 18.23%

Therefore, the same investment performance can be expressed as:

Simple Return = 20%

or

Continuously Compounded Return ≈ 18.23%

Now suppose we only know the continuously compounded return of 18.23% and want to get back to the simple return.

Convert it into decimal form:

18.23% = 0.1823

Use:

R = e^r - 1

R = e^0.1823 - 1

R ≈ 1.20 - 1

R ≈ 0.20

Convert it into a percentage:

0.20 × 100 = 20%

So we are back to:

Simple Return = 20%

The Simplest Way to Think About It

Imagine two people describing exactly the same distance.

One says:

1,200 miles

The other says:

1,900 kilometers

Different numbers. Same distance.

Now imagine two finance students describing exactly the same investment gain.

One says:

29.5% simple return

The other says:

25.9% continuously compounded return

Different numbers. Same investment performance.

That is the main idea.

Notes

• Simple Return = R
• Continuously Compounded Return = r
• A continuously compounded return is another way of expressing an investment’s periodic gain or loss.
• Simple returns and continuously compounded returns describe the same investment performance, but use different calculations.
• Think of it like miles and kilometers: different numbers, same distance.
• Simple → Continuously Compounded formula: r = ln(1 + R)
• Continuously Compounded → Simple formula: R = e^r - 1
• ln = natural logarithm.
• e ≈ 2.71828
• Use the ln button on a calculator to calculate a natural logarithm.
• Use the e^x or exp function on a calculator when converting back to a simple return.
• Always convert a percentage into a decimal before putting it into the formulas.
• Example: 29.5% = 0.295
• Example: 25.9% = 0.259
• Coca-Cola 2009 Simple Return = 29.5%.
• Coca-Cola 2009 Continuously Compounded Return = 25.9%.
• The two percentages are different but represent the same investment performance.
• Continuously Compounded Return = Logarithmic Return = Log Return.
• These three names refer to the same concept.
• When simple returns are small, simple returns and continuously compounded returns are very close.
• When R is small: r ≈ R
• The symbol ≈ means “approximately equal to.”
• Example: 2% simple return ≈ 1.98% continuously compounded return.
• Larger returns can create a larger difference between the two measures.
• Continuously compounded returns can also be negative when an investment loses money.
• Simple returns are often easier for investors because they directly compare beginning money with ending money.
• Continuously compounded returns are still important and widely used in finance.
• If only the word “return” is used without specifying the type, it means simple return.
• Easy memory rule: Simple → ln → Continuous.
• Easy memory rule: Continuous → e → Simple.
• Simplest idea to remember: simple return and continuously compounded return are two different ways of expressing the same gain or loss.

Finance

Simple Returns

A simple return, written as R, tells us how much an investment gained or lost during a certain period compared with the amount invested at the beginning. In very simple terms, it answers the question: “How much did I make or lose compared with what I originally paid?” A simple return can come from two places. The first is the change in the price of the asset, called a capital gain or loss. The second is any cash received while owning the asset, such as a dividend from a stock or an interest payment from a bond.

For a stock, the simple return formula is:

R = ((pE - pB) + D) / pB

In this formula, R means simple return, pB means the price per share at the beginning of the period, pE means the price per share at the end of the period, and D means the dividend per share received during the period.

The easiest way to understand the formula is to read it as: take the change in price, add the dividend received, and divide everything by the price you originally paid.

A share is one unit of ownership in a company. The price per share tells us how much one share costs. For example, if a stock has a price per share of $40, buying one share costs $40. Buying 5 shares costs $200, while buying 20 shares costs $800. The number of shares changes the total amount invested, but the price per share is still $40.

A dividend is cash paid by a company to its shareholders. Dividends are usually stated as a dividend per share. For example, if a company pays a dividend of $1.50 per share, a person who owns 1 share receives $1.50. Someone who owns 10 shares receives $15, and someone who owns 100 shares receives $150.

The total dividend can therefore be calculated as:

Total Dividend = Dividend Per Share × Number of Shares

For example, suppose a company pays a dividend of $1.50 per share and you own 30 shares. Your total dividend is:

$1.50 × 30 = $45

Therefore, you receive $45 in total dividends.

The dividend yield is different from the dividend itself. A dividend is an amount of money, while the dividend yield is a percentage. Dividend yield tells us how large the dividend is compared with the beginning price of the share.

For the holding period used in the simple-return calculation:

Dividend Yield = D / pB

Suppose a share costs $40 at the beginning of the period and pays a dividend of $2 per share. The dividend yield is:

$2 / $40 = 0.05 = 5%

Therefore, the dividend yield is 5%. This means that the dividend received is equal to 5% of the amount originally invested.

Buying more shares changes the total dollar amount of the dividend, but it does not normally change the dividend yield percentage if every share was bought at the same price and receives the same dividend per share. For example, if one share costs $40 and pays a $2 dividend, the dividend yield is 5%. If you buy 10 shares, you invest $400 and receive $20 in total dividends:

$20 / $400 = 0.05 = 5%

If you buy 100 shares, you invest $4,000 and receive $200 in total dividends:

$200 / $4,000 = 0.05 = 5%

The dollar amounts become larger because you own more shares, but the dividend yield remains 5%.

A capital gain happens when the price of an asset increases. A capital loss happens when its price decreases. The capital gain or loss, expressed as a percentage of the beginning price, is:

Capital Gain/Loss = (pE - pB) / pB

For example, suppose you buy a share for $40 and later sell it for $50. The price increased by:

$50 - $40 = $10

The capital gain is:

$10 / $40 = 0.25 = 25%

Therefore, you earned a 25% capital gain from the increase in the share price.

If instead you bought the share for $40 and later sold it for $35, the price decreased by $5. That would be a capital loss rather than a capital gain.

A simple return combines the capital gain or loss with the dividend yield. Therefore:

Simple Return = Capital Gain or Loss + Dividend Yield

Example: One Share

Suppose you buy one share for a beginning price of:

pB = $50

One year later, the share price has increased to:

pE = $60

During the year, the company also pays:

D = $2

in dividends per share.

The simple return is:

R = (($60 - $50) + $2) / $50

First, calculate the price change:

$60 - $50 = $10

Then add the dividend:

$10 + $2 = $12

You made a total gain of $12 from an original investment of $50:

R = $12 / $50

R = 0.24 = 24%

Therefore:

R = 24%

The 24% simple return has two components. The capital gain is:

($60 - $50) / $50 = $10 / $50 = 0.20 = 20%

The dividend yield is:

$2 / $50 = 0.04 = 4%

Adding them together gives:

20% + 4% = 24%

Therefore:

Simple Return = 24%

Example: Different Number of Shares

Now suppose you buy 25 shares instead of one share. The beginning price is still $50 per share, so the total amount invested is:

$50 × 25 = $1,250

At the end of the period, each share is worth $60. Therefore, the total value of the 25 shares is:

$60 × 25 = $1,500

The increase in the value of the shares is:

$1,500 - $1,250 = $250

The dividend is $2 per share, and you own 25 shares, so the total dividend received is:

$2 × 25 = $50

Your total gain is therefore:

$250 + $50 = $300

Compared with your original investment of $1,250:

R = $300 / $1,250

R = 0.24 = 24%

The number of shares increased your total dollar profit from $12 to $300, but the percentage return stayed at 24%.

This shows an important difference between amounts and percentages. The price per share tells us the price of one unit. The dividend per share tells us how much dividend is paid for one unit. The number of shares tells us how many units we own. The total dividend tells us how much dividend money we receive from all our shares together. The dividend yield expresses the dividend as a percentage of the beginning price. The simple return expresses the total gain or loss as a percentage of the beginning investment.

The Coca-Cola example in the original notes follows exactly the same idea. A share was bought at the end of 2008 for $45.27 and sold at the end of 2009 for $57.00. During 2009, a dividend of $1.64 per share was received. The simple return was therefore:

R = (($57.00 - $45.27) + $1.64) / $45.27

First, calculate the increase in the share price:

$57.00 - $45.27 = $11.73

Then add the dividend:

$11.73 + $1.64 = $13.37

Now compare the total gain of $13.37 with the original price of $45.27:

R = $13.37 / $45.27

R = 0.2953 ≈ 29.5%

Therefore:

Simple Return ≈ 29.5%

This return can again be separated into two components. The increase in Coca-Cola’s share price produced a capital gain of approximately 25.9%.

Capital Gain:

($57.00 - $45.27) / $45.27

$11.73 / $45.27 = 0.259 ≈ 25.9%

The dividend produced a dividend yield of approximately 3.6%.

Dividend Yield:

$1.64 / $45.27 = 0.0362 ≈ 3.6%

Adding the two together gives:

25.9% + 3.6% = 29.5%

Therefore, the total simple return was approximately 29.5%.

The idea of a simple return is not limited to stocks. It can be used for other assets as well. For example, if you buy a bond, you may receive an interest payment instead of a dividend. You can still calculate the simple return by taking the change in the bond’s price, adding the interest payment received, and comparing the total with the bond’s beginning price.

In general, the simple return of an asset can be understood as:

Simple Return = (Capital Gain or Loss + Cash Flow Received) / Beginning Price

For a stock, the cash flow may be a dividend. For a bond, the cash flow may be an interest payment. The basic idea stays the same: compare everything you gained or lost during the period with the amount you had invested at the beginning.

Finally, simple returns may also be called arithmetic returns or holding-period returns. In this context, these terms refer to the same basic concept.

Notes

• Share = one unit of ownership in a company.
• Price per share = the price of one share.
• Number of shares = how many shares you own.
• Dividend = cash paid by a company to shareholders.
• Dividend per share (D) = dividend paid for each individual share.
• Total Dividend = Dividend Per Share × Number of Shares.
• Dividend Yield = dividend received compared with the beginning price.
• Dividend Yield formula = D / pB.
• Capital Gain = increase in an asset’s price.
• Capital Loss = decrease in an asset’s price.
• Capital Gain/Loss formula = (pE - pB) / pB.
• pB = price at the beginning of the period.
• pE = price at the end of the period.
• D = dividend per share received during the period.
• R = simple return.
• Simple Return formula = ((pE - pB) + D) / pB.
• Simple Return = Capital Gain/Loss + Dividend Yield.
• More shares = larger total dollar investment.
• More shares = larger total dollar dividend.
• More shares do not automatically mean a higher percentage return.
• For stocks, cash flow is commonly a dividend.
• For bonds, cash flow is commonly an interest payment.
• Simple return, arithmetic return, and holding-period return are names for the same concept in this discussion.
• The easiest way to remember simple return is: price change + cash received, divided by beginning price.