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.