Equity returns and volatility · RiskAtoms Academy

By the end you will be able to

1Calculate a share's total return, including dividends.
2Calculate log returns, convert them back to simple returns, and explain why log returns add up over time.
3Measure volatility as the standard deviation of returns, and scale it to a year.
4Explain why a loss needs a larger gain to recover.

Why it matters

A share falls 50% one year and rises 50% the next. The average return is zero. Yet you have lost a quarter of your money. Measuring returns properly is the first step in measuring risk.

Core explanation

Two ways to measure a return, and one way to measure risk

Total return. You buy a share for 50.00. A year later it is worth 54.00, and along the way it paid a dividend of 1.50. Your gain is the price rise plus the dividend: 5.50 on 50.00, a total return of 11%. It splits into a price return of 8% and a dividend yield of 3%. Quoting only the price return understates what shareholders earn.

Returns compound. Back to the hook. Invest 100. A −50% fall leaves 50. A 50% rise on 50 adds only 25, so you end at 75: a total of −25%. Simple returns over several periods must be multiplied, not added. And to recover from a 50% fall, you need a 100% gain.

Log returns. There is a measure that does add up: the log return, the natural logarithm of what you end with divided by what you started with. For the hook, the two log returns are −0.693 and 0.405. Add them and you get −0.288, which is exactly the log return over both years. Analysts and risk models use log returns because they add over time and because they treat a doubling and a halving symmetrically (+0.693 and −0.693). For daily moves of a percent or two, log and simple returns are almost identical.

Volatility. Two shares can both average 1% a month: one creeps up steadily, the other swings 10% either way. The second is riskier. Volatility captures the difference: it is the standard deviation of returns, which you met in L1.1.08. It is usually quoted for a year, even when measured from daily or monthly data.

Click each card to test your instincts.

The formal version

The simple total return, shown in full. With $P_0$ the starting price, $P_1$ the ending price and $D_1$ the dividend:

R = \frac{P_1 + D_1 - P_0}{P_0} = \underbrace{\frac{P_1 - P_0}{P_0}}_{\text{price return}} + \underbrace{\frac{D_1}{P_0}}_{\text{dividend yield}}

The log return uses the natural logarithm, which is new here, so let's build it step by step.

Why log returns add, on a few lines. Over two periods (ignoring dividends for clarity), the log of a product is the sum of the logs:

\begin{aligned} r_{0 \to 2} &= \ln\!\left(\frac{P_2}{P_0}\right) \\[2pt] &= \ln\!\left(\frac{P_1}{P_0} \times \frac{P_2}{P_1}\right) \\[2pt] &= \ln\!\left(\frac{P_1}{P_0}\right) + \ln\!\left(\frac{P_2}{P_1}\right) = r_1 + r_2 \end{aligned}

Volatility, on a few lines. From $n$ returns $r_t$ with average $\bar r$ , the sample standard deviation per period, then scaled to a year with $m$ periods a year (12 for months, about 252 for trading days):

\begin{aligned} \sigma_{\text{period}} &= \sqrt{\frac{1}{n-1}\sum_{t=1}^{n} \left(r_t - \bar r\right)^2} \\[2pt] \sigma_{\text{annual}} &= \sigma_{\text{period}} \times \sqrt{m} \end{aligned}

An assumption

The square-root rule assumes each period's return is independent of the last, with the same volatility. Real markets break this: calm and stormy periods cluster. It is still the standard convention for quoting volatility.

Try it

Start at −50% then +50%. Then try +100% then −50%: adding the simple returns says +50%, but you are back where you started. Watch how the log returns always add up to the right answer.

Try itSimple returns against log returns

Logarithm calculator →

Year 1 return−50%

Year 2 return+50%

Invest 100. After two years you have 75.00. Adding the simple returns says 0.0%; the truth is −25.0%. The log returns add up exactly, and e^−0.2877 − 1 = −25.0%.

	Simple	Log
Year 1	−50.0%	−0.6931
Year 2	+50.0%	0.4055
Just added	0.0%	−0.2877
Over both years	−25.0%	−0.2877

Simple returns do not add up; log returns do.

Log return against simple returnThe curve is ln(1 + R); the dashed line is “log = simple”. They agree near zero and part for big moves. The dots are your two years.

Worked example

A year of Fjord Shipping

Fjord Shipping is a made-up Norwegian company. You hold it for a year, from a price of NOK 200.00 to NOK 230.00. Returns are monthly, from month-end prices in Norwegian kroner; a dividend of NOK 8 is paid in June.

Total return, log returns and volatility

Step 1 of 5

1

Monthly returns

Month	Start	End	Dividend	Simple	Log
Jan	200	206		+3.00%	0.0296
Feb	206	198		−3.88%	−0.0396
Mar	198	210		+6.06%	0.0588
Apr	210	215		+2.38%	0.0235
May	215	203		−5.58%	−0.0574
Jun	203	196	8	+0.49%	0.0049
Jul	196	205		+4.59%	0.0449
Aug	205	212		+3.41%	0.0336
Sep	212	220		+3.77%	0.0370
Oct	220	214		−2.73%	−0.0277
Nov	214	225		+5.14%	0.0501
Dec	225	230		+2.22%	0.0220

Regulatory lens · looking ahead

No rules apply yet. Log returns and volatility are the raw material of market risk measurement. Value at Risk and the Basel market risk framework (FRTB) are built on them (L4.3), and so is the equity shock in Solvency II for insurers (L4.6). Volatility also drives option prices (L1.7.09).

Sparks

SparkThe worst day

On 19 October 1987, Black Monday, the Dow Jones Industrial Average fell 22.6% in one day, its largest one-day percentage fall. As a log return that is −0.256. If daily returns were normally distributed with typical volatility, a day like that should essentially never happen. Fat tails, the extra risk of extreme days, return in later levels.

SparkA fear gauge

The VIX index, published by Cboe, measures the volatility the options market expects for the S&P 500 over the next 30 days. It usually sits somewhere between 10 and 30. It closed above 80 in both November 2008 and March 2020.

Key takeaways

Total return = (price change + dividends) ÷ starting price. It is price return plus dividend yield.
Simple returns compound: multiply (1 + R) across periods. Adding them overstates the result.
The log return is r = ln(1 + R). Log returns add up over time, and R = eʳ − 1 converts back.
Volatility is the standard deviation of returns. To annualise, multiply by the square root of the periods in a year.
Losses need bigger gains to recover: after a fall of L, you need 1 ÷ (1 − L) − 1.

Reaction

Check what stuck

Question 1 · objective 1

You buy a share at 80.00. A year later it is 84.00, and it paid a dividend of 2.00 during the year. What is your total return, in per cent?

%

Question 2 · objective 2

A share returns 10% (simple) over a year. What is its log return, to four decimal places?

Question 3 · objective 4

A share rises 20% in year 1 and falls 20% in year 2. What is the total return over both years, in per cent?

%

Question 4 · objective 3

A share's monthly returns have a standard deviation of 5%. Assuming months are independent, what is its annualised volatility, in per cent, to one decimal place?

%

Question 5 · objective 4

A portfolio falls 20%. What gain does it need to get back to where it started?

A20%B25%C40%DIt depends on the volatility

Glossary

Total return: The gain from holding a share over a period, counting both the price change and any dividends, as a fraction of the starting price.
Price return: The change in the share price alone, as a fraction of the starting price.
Dividend yield: The dividend received over the period divided by the starting price.
Log return: The natural logarithm of (ending value ÷ starting value). Also called the continuously compounded return.
Natural logarithm: The natural logarithm: the power you must raise e (about 2.718) to, to get a number. ln(e) = 1, ln(1) = 0.
Standard deviation: A measure of spread: the typical distance of the values from their average.
Volatility: The standard deviation of a share's returns: a measure of how widely they swing around their average.
Annualise: Convert a figure for a shorter period (a day, a month) into its equivalent for a year.

Deep Dives & Origins

Deep Dives

J. C. Hull, Options, Futures, and Other Derivatives, Pearson, the section on estimating volatility from historical data. Intermediate.
J. Y. Campbell, A. W. Lo and A. C. MacKinlay, The Econometrics of Financial Markets, Princeton University Press, 1997, chapter 1. Advanced.

Origins

J. C. Hull, Options, Futures, and Other Derivatives, Pearson.
Bodie, Kane and Marcus, Investments, McGraw Hill, the chapter on risk, return and the historical record.
Cboe Global Markets, VIX index methodology and historical data.

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