Risk and return: measuring volatility with real market history
Risk is uncertainty about what an investment will return, and history lets us measure it. This lesson explains total return, the difference between the simple average and the compounded return, and standard deviation as a measure of volatility. A calculator then compares the S&P 500 with the 10-year US Treasury bond across periods from 2000 to 2024, showing that the same asset can look excellent in one decade and poor in the next.
- Level
- Beginner
- Reading time
- About 24 minutes
- Know this first
- Stocks, bonds and funds
By the end you will be able to:
- calculate a total return and explain why the compounded return is below the simple average
- compute a standard deviation and say what it measures
- compare the return and risk of stocks and bonds over different periods
- explain why a short period can contradict the risk-return trade-off
- name the main types of investment risk
Return, and the uncertainty around it
Every investment is a bet on a future nobody can see. The SEC's investor site defines risk in finance as the degree of uncertainty and potential financial loss inherent in an investment decision, and notes that in general, as investment risks rise, investors seek higher returns to compensate themselves for taking them. This lesson makes both halves of that sentence measurable: the return an investment earns, and the volatility with which that return swings from year to year.
The previous lesson showed that stocks, bonds and funds are different claims. Here the question is what history says about how those claims behave. The data are real: annual returns of US large-company stocks and of 10-year US Treasury bonds from 2000 to 2024, published by Professor Aswath Damodaran of NYU Stern. The first lesson in this module introduced the instruments, and the lesson on the cost of equity shows how the required return is estimated. Here the focus is measurement.
Measuring a return
A total return counts everything an investment pays: the change in price plus any dividends or interest. If a share rises from $100 to $108 and pays $2 in dividends, the total return is (108 − 100 + 2) ÷ 100 = 10%. The stock series used below includes dividends, as its source states.
Over several years there are two ways to summarise a series of annual returns, and they can give different answers:
The simple average ignores that gains and losses build on each other. Suppose an investment gains 50% in one year and loses 50% the next. The average is 0%. But $10,000 becomes $15,000 and then $7,500. The compounded return is √(1.5 × 0.5) − 1 = −13.40% a year. Volatility drags down the compounded return, which is why the compounded figure is the honest measure of growth. The lesson on compound interest covers the growth mechanics.
Measuring volatility
The most common measure of volatility is the standard deviation of returns: roughly, how far a typical year's return falls from the average. It is computed by taking each year's distance from the average, squaring it, averaging the squares (dividing by the number of years minus one for a sample) and taking the square root.
Take four years of invented returns: +10%, −5%, +20% and +7%. The average is 8%. The distances are +2, −13, +12 and −1, which square to 4, 169, 144 and 1, a total of 318. Dividing by 3 gives 106, and the square root of 106 is 10.30%. A higher figure means returns scatter more widely, so the future is harder to predict. The worst year is a simpler and more intuitive risk measure, since it shows what the investor would have had to sit through.
Stocks against Treasury bonds
Set the last year of the period and how many years to look back. The left page shows the S&P 500 including dividends and the right page the return on a 10-year US Treasury bond, both from the same dataset. At the starting settings the period is the ten years from 2015 to 2024. Each page shows the simple average, the compounded return, the standard deviation, the worst year, and what $10,000 would have become.
From 2015 to 2024, stocks returned 12.98% a year compounded and bonds 0.26%. Stocks' yearly returns had a standard deviation of 16.22% against 8.05% for bonds. The worst year was −18.04% for stocks and −17.83% for bonds. $10,000 became $33,874 in stocks and $10,267 in bonds. Stocks returned 12.72 percentage points a year more, and their yearly swings were larger.
Four experiments. First, at the starting settings stocks compounded at 12.98% a year against 0.26% for bonds, and their standard deviation was about twice as large. That is the risk-return trade-off in one picture. Second, set the last year to 2009 and the history to 10 years, the period 2000 to 2009. Now stocks compounded at −0.95% and bonds at 6.26%, and $10,000 in stocks ended at $9,086, below the amount invested. Higher risk does not guarantee higher return in any one period. Third, compare the simple average and the compounded figure for stocks in that period: the average is positive, 1.16%, but the compounded return is negative, which is volatility drag. Fourth, set the last year to 2022 and the history to 5 years. The worst year is almost the same for stocks and bonds, because 2022 was a bad year for both.
The ten years behind the starting settings:
| Year | US stocks | 10-year Treasury bond |
|---|---|---|
| 2015 | 1.38% | 1.28% |
| 2016 | 11.77% | 0.69% |
| 2017 | 21.61% | 2.80% |
| 2018 | −4.23% | −0.02% |
| 2019 | 31.21% | 9.64% |
| 2020 | 18.02% | 11.33% |
| 2021 | 28.47% | −4.42% |
| 2022 | −18.04% | −17.83% |
| 2023 | 26.06% | 3.88% |
| 2024 | 24.88% | −1.64% |
| Compounded a year | 12.98% | 0.26% |
What the history says, and what it does not
Across the 25 years from 2000 to 2024 in this dataset, US stocks lost money in 6 years and 10-year Treasury bonds in 6 years. The SEC's investor site says large company stocks as a group have lost money about one year in three on average over the long run, so the share in this window is smaller than that long-run rule of thumb. The window is short, and one run of years can look very different from another. In 2000 to 2009 stocks went nowhere, and in 2015 to 2024 they compounded at 12.98%.
Two lessons follow. First, risk is not only the chance of a loss in a typical year, it is also the chance that a whole period turns out badly. Second, bonds are not safe from loss. The SEC lists several types of risk, including inflation risk, which reduces the purchasing power of a fixed payment, and interest rate risk, where rising interest rates make newly issued bonds more appealing. Treasury bond prices can fall when rates rise, which fits the table's figures for 2022. Stocks and bonds fell together that year, a fact the next lesson takes up when it looks at how holdings combine.
What this lesson cannot tell you
- History is not a forecast. The two decades above differ sharply, and neither predicts the next.
- One country, two assets. The data cover US large-company stocks and 10-year US Treasury bonds. Other markets and other assets behave differently.
- Standard deviation treats gains and losses alike. A year of unusually large gains raises it as much as a year of large losses, though investors care far more about the losses.
- Annual data hide the path. Within a year a holding can fall far and recover.
- Returns here are before costs and taxes. A real investor pays fees and may pay tax, which a later lesson covers.
- The calculator uses ten years at most. Short windows swing widely, so conclusions from five or ten years are weak.
Common mistakes
Using the simple average to describe growth
The simple average overstates how fast money grew when returns are volatile. Use the compounded return.
Reading volatility as the only risk
It measures how much returns scatter, not the chance of losing all the money, being unable to sell, or failing to keep up with inflation.
Assuming the past decade will repeat
The same asset can look excellent in one period and poor in the next. A short window proves little.
Calling bonds risk free
Bond prices can fall when interest rates rise, and fixed payments lose buying power in inflation.
Ignoring the worst year
An average hides what an investor had to live through. Ask whether you could hold through the worst year without selling.
Try it yourself
1. An investment returns +50% in one year and −50% in the next. What is the simple average, and what does $10,000 become?
Show the working
The average is (50 − 50) ÷ 2 = 0%. But 10,000 × 1.5 = 15,000, and 15,000 × 0.5 = $7,500. The compounded return is √(1.5 × 0.5) − 1 = −13.40% a year.
2. Four years of returns are +10%, −5%, +20% and +7%. What is the sample standard deviation?
Show the working
Average = 8%. Squared distances: 4, 169, 144, 1, total 318. Divide by (4 − 1) = 3 to get 106, and the square root is 10.30%.
3. US stocks returned −18.04% in 2022. What does $10,000 become after that year?
Show the working
10,000 × (1 − 0.1804) = $8,196. To get back to $10,000 it would then need a gain of 10,000 ÷ 8,196 − 1 = 22.01%, which is more than the loss.
4. In the ten-year table for 2015 to 2024, in how many years did stocks lose money, and in how many did Treasury bonds?
Show the working
Stocks lost in 2018 and 2022, so 2 years. Bonds lost in 2018 (−0.02%), 2021, 2022 and 2024, so 4 years. Bonds had more losing years but much smaller losses in most of them.
5. Stocks and Treasury bonds both lost about 18% in 2022. What does that suggest about holding both as protection against each other?
Show the working
It shows that bonds do not always rise when stocks fall. In some years the two move together. That is why a mix of assets reduces risk by an uncertain amount, which the next lessons examine.
Where this is used
- An investor compares the compounded return and the worst year of different assets before choosing how to split savings.
- A fund fact sheet reports annualised return and standard deviation for the fund and its benchmark.
- A pension fund tests whether it could survive a bad stretch like 2000 to 2009 and still pay members.
- An analyst uses historical volatility as one input when estimating the risk of a stock.
Key terms
- Total return
- The gain from price changes plus income such as dividends or interest, as a share of the amount invested.
- Simple average return
- The sum of yearly returns divided by the number of years.
- Compounded return
- The constant yearly rate that would produce the same final value, also called the annualised or geometric return.
- Volatility
- How widely an investment's returns vary from period to period.
- Standard deviation
- A measure of how far returns typically fall from their average.
- Volatility drag
- The amount by which swings reduce the compounded return below the simple average.
- Interest rate risk
- The risk that rising interest rates reduce the price of existing bonds.
- Inflation risk
- The risk that rising prices reduce the purchasing power of a fixed payment.
Questions people ask
Why does a higher return usually come with more risk?
Investors will not hold a risky asset unless they expect to be paid for it. If an asset were both riskier and expected to pay less, nobody would buy it. The payment is only an expectation and not a promise.
Is a standard deviation of 16% large?
It means that in a typical year the return fell about 16 percentage points above or below the average in that period. Compared with the 8% for the bonds above, it is large.
Why not just look at the worst year?
It is easy to understand but depends on one observation. Standard deviation uses every year, so the two together give a fuller picture.
Does a long time reduce the risk?
It reduces the chance of a bad average, but the 2000 to 2009 window shows that a decade can still end in a loss for stocks.
Sources and review
- US Securities and Exchange Commission, Investor.gov, What is risk?, for the definition of risk, the link between risk and the return investors seek, and the types of risk including inflation, interest rate and liquidity risk.
- Aswath Damodaran, NYU Stern, Historical returns on stocks, bonds and bills: 1928-2024 (last updated 5 January 2026), for the annual returns of the S&P 500 including dividends and of the 10-year US Treasury bond for 2000 to 2024. The page names its data source as multiple data services.
- Aswath Damodaran, NYU Stern, Annual returns on stock, T.bonds and T.bills: 1928 - current (last updated 5 January 2019), an older edition of the same dataset, used to check the 2000 to 2018 values.
How the numbers were checked: the annual returns were typed from the dataset and the 2000 to 2018 values were compared with an older edition of the same dataset. The averages, compounded returns, standard deviations and growth of $10,000 were recomputed in plain code, and the calculator, which runs separately, was compared with that code for every period it allows. The four-year example is invented for teaching.