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.