PORTFOLIO MATH

The Average Return Can Be Misleading.

Two portfolios can have the exact same average annual return and produce completely different outcomes for the investor.

START WITH THE NUMBER EVERYONE KNOWS

“The stock market returns an average of 7% a year.”

That line gets thrown around a lot, but the number may tell you surprisingly little about how much money you actually made.

Consider two hypothetical investors. Each starts with $100,000 and invests for 12 years.

What if both portfolios have the exact same 7% average annual return?
PORTFOLIO A

7% every single year.

AVERAGE ANNUAL RETURN 7%

Portfolio A earns 7% every single year—an unlikely outcome for an equity portfolio, but useful for illustration.

YEAR 1+7%
YEAR 2+7%
YEAR 3+7%
YEAR 4+7%
YEAR 5+7%
YEAR 6+7%
YEAR 7+7%
YEAR 8+7%
YEAR 9+7%
YEAR 10+7%
YEAR 11+7%
YEAR 12+7%

The arithmetic average annual return is 7%.

STARTING VALUE $100,000
VALUE AFTER 12 YEARS $225,219
PORTFOLIO B

Now change the sequence of returns.

AVERAGE ANNUAL RETURN 7%

Portfolio B experiences a very different series of gains and losses:

YEAR 1+5%
YEAR 2−2%
YEAR 3+4%
YEAR 4+49%
YEAR 5+45%
YEAR 6+40%
YEAR 7−50%
YEAR 8+35%
YEAR 9−47%
YEAR 10+32%
YEAR 11−50%
YEAR 12+23%

Add those 12 annual returns together and divide by 12:

ARITHMETIC AVERAGE 7%

Exactly the same average annual return as Portfolio A.

Yet the investor in Portfolio B finishes with only approximately $94,007.

STARTING VALUE $100,000
VALUE AFTER 12 YEARS $94,007
YEAR-BY-YEAR RETURNS

A 7% average can hide a lot of volatility.

Portfolio B's arithmetic average is 7%, but the experience behind that average includes several large gains and several very large losses.

+50% 0% −50%
+5% YR 1
−2% YR 2
+4% YR 3
+49% YR 4
+45% YR 5
+40% YR 6
−50% YR 7
+35% YR 8
−47% YR 9
+32% YR 10
−50% YR 11
+23% YR 12
Portfolio B annual returns. Gains extend above the 0% line and losses extend below it. The arithmetic average of all 12 annual returns is 7%.
THE RESULT

Same starting balance. Same 7% average return. Two completely different outcomes.

STARTING BALANCE $100,000 Both portfolios
ARITHMETIC AVERAGE 7% Both portfolios
PORTFOLIO A $225,219 Ending value
PORTFOLIO B $94,007 Ending value
Why?
ARITHMETIC VS. GEOMETRIC RETURN

Your wealth does not compound at the arithmetic average.

It compounds according to the actual gains and losses experienced by the portfolio over time.

The arithmetic average simply adds the annual returns together and divides by the number of observations.

ARITHMETIC AVERAGE Sum of Annual Returns ÷ Number of Years

But your money does not reset to $100,000 at the beginning of every year.

Each new percentage return is applied to the portfolio value that remains after all of the gains and losses that came before it.

Your wealth therefore compounds according to the geometric return—the actual compounded experience of gains and losses over time.

The magnitude of losses matters because future returns are earned on whatever capital remains after the loss.
THE MATH OF LOSSES

A loss requires a larger gain just to get back to even.

Percentage gains and losses are not symmetrical.

PORTFOLIO LOSS −10%
GAIN REQUIRED TO RECOVER +11.1%
PORTFOLIO LOSS −20%
GAIN REQUIRED TO RECOVER +25%
PORTFOLIO LOSS −30%
GAIN REQUIRED TO RECOVER +42.9%
PORTFOLIO LOSS −50%
GAIN REQUIRED TO RECOVER +100%

A portfolio that loses 50% does not need a 50% gain to recover.

It needs a 100% gain.

That is why large losses after strong gains can erase years of progress and significantly hinder long-term wealth building.

PORTFOLIO VALUE OVER TIME

Follow the money—not the average.

Portfolio B initially grows dramatically faster than Portfolio A. By the end of Year 6, it has grown to more than $323,000. But repeated large losses eventually reduce the account below its original starting value.

Portfolio A versus Portfolio B over 12 years Both portfolios start at $100,000. Portfolio A grows steadily to approximately $225,219. Portfolio B grows to more than $323,000 before repeated large losses reduce it to approximately $94,007. $0 $100k $200k $300k Portfolio B peak: ≈ $323,691 Portfolio A: $225,219 Portfolio B: $94,007 START 1 2 3 4 5 6 7 8 9 10 11 12 YEAR
Portfolio A
Portfolio B
Hypothetical illustration assuming the stated annual returns and no contributions, withdrawals, investment advisory fees, transaction costs, or taxes.
WHAT HAPPENED TO PORTFOLIO B?

The investor was winning—until the losses changed the math.

START $100,000
AFTER YEAR 4 $159,454
AFTER YEAR 5 $231,208
AFTER YEAR 6 $323,691
AFTER −50% $161,846
AFTER −47% $115,801
AFTER SECOND −50% $76,428
FINAL VALUE $94,007

At the end of Year 6, Portfolio B is worth approximately $323,691—far more than Portfolio A at the same point.

Then comes a 50% loss.

The portfolio immediately falls to approximately $161,846.

Later, a 47% loss and another 50% loss further reduce the amount of capital available to participate in subsequent gains.

THE BIGGER LESSON

Average return does not tell you the whole story.

Portfolio A and Portfolio B both report an arithmetic average annual return of 7%.

But Portfolio A compounds steadily while Portfolio B experiences repeated large drawdowns.

PORTFOLIO A $225,219 Ending value
PORTFOLIO B $94,007 Ending value

Because your wealth does not compound at the arithmetic average of your annual returns. It compounds according to the actual compounded experience of gains and losses over time.

And when large losses follow strong gains, they can erase years of progress and significantly hinder long-term wealth building.

PORTFOLIO RISK

How much downside risk is actually hiding inside your portfolio?

With markets near all-time highs, this may be an especially important time to understand how much downside risk is actually inside the investments you own.

Strong markets can make risk easy to overlook. But as the example above shows, avoiding large losses can matter just as much as capturing gains.

Analyze Your Portfolio
WRITTEN BY
AJ Blackstone
President, PortfolioLab

This material is provided for educational and informational purposes only and should not be construed as individualized investment, tax, or legal advice or as a recommendation to purchase or sell any security.

Portfolio A and Portfolio B are hypothetical illustrations and do not represent the performance of any actual investment, account, strategy, or client. The examples assume the stated annual returns and do not include investment advisory fees, transaction costs, taxes, contributions, withdrawals, or other expenses that would affect actual results.

Past performance does not guarantee future results. Investing involves risk, including possible loss of principal. Arithmetic average return, compounded return, volatility, and drawdown measure different aspects of investment performance and should not be considered in isolation.