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.