Returns and risk: a first look · RiskAtoms Academy

By the end you will be able to

1Calculate the simple return on an investment, and describe its volatility in words.
2Split a return into its price part and its income part.
3Explain why returns over several periods compound rather than add.
4Explain why two investments with the same average return can carry very different risk.

Why it matters

“This fund returned 8% a year.” It sounds like all you need to know. But 8% from a fund that rose gently every month and 8% from one that halved and then doubled are not the same experience, and not the same risk.

Core explanation

How much you made, and how bumpy the ride was

Return. If you put in 100 and end up with 110, you made 10. But 10 on 100 is very different from 10 on 10,000. So we state the gain as a percentage of what you put in. That percentage is the return, always measured over some holding period: a day, a month, a year.

A return has two sources:

The price return: the price went up or down while you held it.
The income return: cash paid to you along the way, such as a dividend on a share or a coupon on a bond.

Add the two and you have the total return. Ignoring the income is a common mistake: over long periods, dividends have made up a large part of what shareholders earned.

Returns compound. A gain of 10% followed by a loss of 10% does not bring you back to the start. 100 becomes 110, and 10% off 110 is 11, so you end at 99: a return of −1.0%. Each return acts on what you have at that moment, not on what you started with.

Quick intuition checks

Risk: how uncertain is the outcome?

In everyday speech, risk means the chance of something bad. In finance, risk is broader: the chance that the outcome differs from what you expected, in either direction. The simplest measure of it is volatility: how much returns typically swing around their average.

The explorer below shows two illustrative funds. Both averaged 1.00% a month over the year. The calm fund's months ranged from 0.6% to 1.4%. The bumpy fund's ranged from −7.0% to 9.0%, with 5 losing months. Same average, very different ride.

ExploreSame average, different ride

Two illustrative funds, twelve monthly returns each.

Calm fund

Average month: +1.00%
Best month: +1.4%
Worst month: +0.6%
Months with a loss: 0 of 12
Volatility (std. dev.): 0.25%
100 grows to: 112.68

Bumpy fund

Average month: +1.00%
Best month: +9.0%
Worst month: −7.0%
Months with a loss: 5 of 12
Volatility (std. dev.): 5.31%
100 grows to: 110.98

Monthly returns

100 invested at the start

Volatility is usually measured by the standard deviation of returns. You do not need the formula yet; think of it as the size of a typical swing. The calm fund's is about 0.25% a month and the bumpy fund's about 5.31%. L1.5.06 shows how to calculate it.

Notice one more thing. Despite the same average, the bumpy fund ends the year lower: 100 grew to 110.98 against 112.68. Big swings cost something even when they average out, for the same reason that +10% and −10% leave you at 99.

Describing volatility in words

Low volatility: returns stay close to their average, and the outcome over a period is fairly predictable. High volatility: returns swing widely, losing periods are common even when the average is positive, and the outcome is much less certain.

The formal version

Write $P_0$ for the price at the start, $P_1$ for the price at the end, and $D$ for any income received in between. This is new notation, so let's build the formula one piece at a time.

Over several periods, returns compound. With returns $R_1, R_2, \dots$ written as decimals, start from the plain statement and expand it:

\begin{aligned} \text{value at the end} &= \text{start} \times (1 + R_1) \times (1 + R_2) \\ \text{return over both} &= (1 + R_1)(1 + R_2) - 1 \end{aligned}

Conventions

All returns here are simple returns over the stated period, not annualised and not continuously compounded. Income is counted when it is received and is not reinvested. Taxes and fees are ignored. The percentage maths is in the Math Atom on percentages.

Try it

The tool starts with the worked example below. Drag the end price below the start price and watch the dividend soften the loss. Then set the dividend to zero and see the price return on its own.

Try itReturn calculator

Open the full ROI calculator →

Price you paid (P₀)A$40.00

Price at the end (P₁)A$43.00

Income received per share (D)A$1.20

Total return

+10.50%

Price return+7.50%

Income return+3.00%

Gain per shareA$4.20

Where the return came from price income

−60%0+60%

Worked example

A year in an Australian bank share

You buy 100 shares listed on the Australian Securities Exchange at A$40.00 each. During the year the company pays dividends of A$1.20 a share. A year later the price is A$43.00. Amounts in Australian dollars; taxes and fees ignored. Try each step before revealing it.

Working out the return

Step 1 of 5

1
Price return
$(43.00 - 40.00) \div 40.00 = 0.075$ , so 7.5%.

For a batch of returns, the descriptive statistics calculator gives the average and standard deviation in one go.

Regulatory lens · looking ahead

Volatility is the raw material of market risk measurement. Banks turn it into Value at Risk and Expected Shortfall under the Basel market risk rules, and insurers into the market risk part of their Solvency II capital. Those measures return in later levels; L1.5.06 first puts numbers on volatility.

Sparks

SparkDown 89%, and a long way back

From its peak in September 1929 to its low in July 1932, the Dow Jones Industrial Average lost about89% of its value. A fall that large needs a rise of more than 800% to recover, and the index did not regain its 1929 peak until 1954.

SparkRisk cuts both ways

To a risk manager, an unexpected gain is as much a sign of risk as an unexpected loss. A trading desk that makes far more than its models predicted is taking more risk than anyone thought, and is often questioned as closely as one that loses.

Key takeaways

Simple return = (end price − start price + income) ÷ start price, usually stated as a percentage.
Total return = price return + income return. Leaving out income understates what investors earn.
Returns compound: +10% then −10% leaves you at 99, and a 50% fall needs a 100% rise to recover.
Risk is the uncertainty of outcomes. Volatility, how much returns swing around their average, is its simplest measure.
Two investments with the same average return can have very different volatility, and the bumpier one tends to end lower.

Reaction

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Glossary

Return: The gain or loss on an investment over a period, as a percentage of what was put in.
Holding period: The length of time over which a return is measured, such as a day, a month or a year.
Price return: The part of the return that comes from the change in price.
Income return: The part of the return that comes from cash paid out while you hold the asset, such as dividends or coupons.
Dividend: Cash a company pays to its shareholders out of its profits.
Risk: In finance, the chance that outcomes differ from what you expect, for better or worse.
Volatility: How much an asset's returns typically swing around their average. Measured by the standard deviation.
Standard deviation: A measure of how spread out numbers are around their average. Roughly, the size of a typical swing.

Deep Dives & Origins

Deep Dives

Bodie, Kane and Marcus, Investments, McGraw-Hill, the chapter on risk, return and the historical record. Intermediate.
Dimson, Marsh and Staunton, Triumph of the Optimists: 101 Years of Global Investment Returns, Princeton University Press, 2002. Beginner, history.

Origins

Bodie, Z., Kane, A. and Marcus, A. J., Investments, McGraw-Hill.
Dimson, E., Marsh, P. and Staunton, M., Triumph of the Optimists, Princeton University Press, 2002.
S&P Dow Jones Indices, Dow Jones Industrial Average historical closing levels.

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