5.2Lesson 5.2 · Module 5 — Risk and Return

Measuring Historical Return and Volatility

Arithmetic and geometric averages, volatility drag, sample variance and standard deviation, and annualization.

Arithmetic mean: average one-period return
Geometric mean: realized compound growth
Volatility drag: why gains and losses don't cancel
Standard deviation: typical size of fluctuations
Annualize volatility with √n, not ×n

Start

Learning objectives

By the end of this lesson, you should be able to:

1Compute the arithmetic mean return and interpret it as the average one-period outcome.
2Compute the geometric mean return and interpret it as the realized compound growth rate.
3Explain volatility drag and why gains and losses do not cancel.
4Compute sample variance and sample standard deviation of a return series.
5Interpret standard deviation as a typical fluctuation size in percentage points.
6Annualize volatility using the square-root-of-time rule, not simple multiplication.

5.2.1Section 1

Arithmetic average return

Suppose a stock returned +20%, −10%, and +15% over three years. The simplest summary is the arithmetic average — the plain mean of the yearly returns.

Arithmetic mean

\overset{ˉ}{R} = \frac{1}{T} t = 1 \sum T R_{t}

The arithmetic mean adds up each period's return and divides by the number of periods.

$R_{t}$: Return in period t.
$T$: Number of periods.
$\overset{ˉ}{R}$: Arithmetic mean return.

\overset{ˉ}{R} = \frac{0.20 + ( - 0.10 ) + 0.15}{3} = \frac{0.25}{3} \approx 0.0833

Arithmetic mean ≈ 8.33%

Interpret this as the average return in a randomly selected year — the typical one-period outcome you would have earned over this stretch.

5.2.2Section 2

Geometric average return

The arithmetic mean tells you the average year, but it does not tell you what actually happened to a dollar invested across all three years. For that we use the geometric mean — the constant growth rate that reproduces the actual wealth path.

Geometric mean

R_{G} = [t = 1 \prod T (1 + R_{t})]^{1/ T} - 1

Multiply one plus each return together (the total wealth ratio), take the T-th root, and subtract one. The result is the constant rate that reproduces the wealth path.

$(1 + R_{t})$: Growth factor in period t.
$T$: Number of periods.
$R_{G}$: Geometric mean return.

R_{G} = [(1.20) (0.90) (1.15)]^{1/3} - 1 = (1.242)^{1/3} - 1 \approx 0.0749

Geometric mean ≈ 7.49%

A constant 7.49% per year would have turned $100 into the same $124.20 that the actual sequence produced. Notice the geometric mean (7.49%) is below the arithmetic mean (8.33%) — this gap is volatility drag.

The wealth path makes this concrete. A $100 investment grows and shrinks each year:

Time	Start	Return	End
Year 1	$100.00	+20%	$120.00
Year 2	$120.00	−10%	$108.00
Year 3	$108.00	+15%	$124.20

5.2.3Section 3

Volatility drag

Here is the puzzle at the heart of volatility drag. A stock gains +20% one year, then loses −20% the next. The arithmetic mean is 0%. But did you break even?

Time	Start	Return	End
Year 1	$100	+20%	$120
Year 2	$120	−20%	$96

No. You end with $96. The geometric mean is approximately −2.02%. Why? Because the gain and the loss apply to different bases: +20% grows $100 to $120, but −20% shrinks $120 to $96. Returns compound multiplicatively, not additively — so equal-and-opposite percentages do not cancel.

The recovery math is stark. After a 20% loss you need more than a 20% gain to get back: you need +25%, because 1 / 0.80 − 1 = 0.25. The deeper the hole, the steeper the climb.

Try itCompounding worksheet

Trace the wealth path for +20% then −20%

Start with $100. Apply +20% then −20%. Compute the ending wealth, both averages, and the return needed to recover from the loss.

Work each value out by hand, then check. Hints appear when an answer is off.

Ending wealth from $100

Arithmetic mean of the two returns

Geometric mean of the two returns

Return needed to recover from a 20% loss

5.2.4Section 4

When to use each average

Definition · Arithmetic mean — use for

The average one-period outcome and as an estimator for the expected return in a typical future period. It answers: “what did an average year look like?”

Definition · Geometric mean — use for

The realized compound performance and long-term wealth growth. It answers: “what constant rate reproduces the actual wealth path?”

Warning: neither average is automatically the correct forecast of the future. The arithmetic mean is a reasonable estimate of next period's expected return; the geometric mean describes what compound growth you actually realized. Always be explicit about which one you are using and why.

5.2.5Section 5

Sample mean, variance, and standard deviation

To measure how spread out returns are, we use the sample variance and its square root, the standard deviation. Consider a stock with four annual returns: +10%, −5%, +20%, 0%. The mean is 6.25%.

t	Rₜ	Rₜ − R̄	(Rₜ − R̄)²
1	0.10	0.0375	0.001406
2	−0.05	−0.1125	0.012656
3	0.20	0.1375	0.018906
4	0.00	−0.0625	0.003906
	R̄ = 0.0625	sum →	0.036875

Sample variance

s^{2} = \frac{\sum _{t = 1}^{T} ( R _{t} - R ˉ ) ^{2}}{T - 1}

Square each deviation from the mean, sum them, and divide by T−1 (not T).

$R_{t} - \overset{ˉ}{R}$: Deviation of period t from the mean.
$T - 1$: Degrees of freedom (sample correction).

s^{2} = \frac{0.036875}{3} \approx 0.012292

s² ≈ 0.012292

Standard deviation

s = s^{2}

The square root of the variance returns the spread to the same units as the returns themselves.

s = 0.012292 \approx 0.1109

s ≈ 11.09%

Interpreting the result: the stock's average annual return was 6.25%, while its standard deviation was 11.09%. This means an individual year's return typically moved by roughly 11 percentage points away from the average. A typical ±1σ range is 6.25% ± 11.09%, i.e. about −4.84% to +17.34%.

Be careful with units. The 11.09 is percentage points of return — not 11.09% of the mean, and not an expected loss. It is a measure of how widely individual yearly returns scattered around the average.

Try itVolatility worksheet

Compute mean, variance, and standard deviation

Returns are +10%, −5%, +20%, 0%. Enter each as a percentage (e.g. 10), except variance which is a decimal.

Mean return R̄

Sample variance s² (enter as a decimal, e.g. 0.0123)

Sample standard deviation s

5.2.6Section 6

Why square the deviations?

Why not just average the raw deviations? Because positive and negative deviations would cancel — their sum is always zero by construction of the mean. Squaring every deviation solves two problems at once:

It makes everything positive. A squared number is never negative, so gains and losses both add to the spread.
It weights large movements more. A small deviation like 3.75 contributes 3.75² = 14.06, but a large one like 13.75 contributes 13.75² = 189.06 — over thirteen times more. Big swings dominate the measure.
The square root returns to original units. Variance is in “percent-squared”; taking the root brings it back to plain percentage points.

Note: standard deviation is not the average distance from the mean (that quantity is the mean absolute deviation, MAD). Standard deviation is the root-mean-square deviation — it punishes large swings more heavily.

5.2.7Section 7

Sample variance versus population variance

Population variance

σ^{2} = \frac{\sum _{t = 1}^{T} ( R _{t} - R ˉ ) ^{2}}{T}

Divide by T when you have the entire population of returns.

Sample variance

s^{2} = \frac{\sum _{t = 1}^{T} ( R _{t} - R ˉ ) ^{2}}{T - 1}

Divide by T−1 (Bessel's correction) when your data is a sample of a larger process.

Historical investment data is almost always a sample — a finite window drawn from an ongoing, unknown return-generating process. Dividing by T−1 corrects the downward bias that comes from using the sample mean (which sits closer to the data than the true mean). For the rest of this module, always use the sample formula.

5.2.8Section 8

Annualizing returns and volatility

Returns and volatility scale differently when you move between periods. A monthly average return of 1% annualizes arithmetically to 12%, but compounds to (1.01)¹² − 1 ≈ 12.68%. Volatility, however, scales with the square root of time.

Annualizing volatility

σ_{annual} = σ_{period} \times n

To annualize volatility, multiply the periodic standard deviation by the square root of the number of periods per year. Do NOT multiply by n.

$σ_{period}$: Standard deviation over one period (e.g. one month).
$n$: Number of periods per year (12 for monthly, 252 for daily).

σ_{annual} = 0.04 \times 12 \approx 0.04 \times 3.464 \approx 0.1386

4% monthly → 13.86% annual

The square-root rule comes from the fact that independent variances add. With 12 independent months, total variance is 12× a single month, so the standard deviation grows by √12 ≈ 3.464 — not by 12.

Two quick examples at monthly frequency:

Monthly σ	× √12	Annual σ
4%	0.04 × 3.464	13.86%
5%	0.05 × 3.464	17.32%

The common mistake is to multiply volatility by 12 (giving 48% or 60%). That overstates risk by treating consecutive months as perfectly correlated. The square-root rule reflects how independent fluctuations accumulate.

Try itAnnualization challenge

Annualize a monthly volatility

A stock has a monthly standard deviation of 5%. What is its annualized volatility? Enter the answer as a percent (e.g. 17.32).

Annualized volatility σ_annual

Attempt 1 of 3

5.2.9Section 9

Estimation uncertainty

A historical average of 12% does not establish that the true future expected return is exactly 12%. Historical estimates are just estimates, and they are plagued by several problems:

Limited sample. A few years of data produce a noisy estimate of the true mean.
Unusual periods. A single boom or crash can dominate a short window.
Extreme observations. Outliers inflate or deflate both mean and volatility.
Changing models. The process generating returns can shift over time.
Survivorship bias. We only see the companies and funds that survived.
Regime changes. Inflation, policy, and structure evolve, so the past may not resemble the future.

Treat historical means and volatilities as noisy estimates with a range of plausible true values — not as exact future forecasts.

Try itMastery check

Pass with 4 of 6 correct

Answer all questions, then check your work. You can retry any time — mastery is based on correctness, not speed.

01
Returns: 20%, −10%, 15%. What is the arithmetic mean?
02
Same returns. What is the geometric mean (approximately)?
03
+20% then −20%. What is the ending wealth from a $100 start?
04
Returns: 10%, −5%, 20%, 0%. What is the sample standard deviation (approximately)?
05
Monthly volatility is 5%. What is the annualized volatility (approximately)?
06
Which statement about standard deviation is correct?

Lesson summary

1Arithmetic mean answers 'what was the average annual return?'
2Geometric mean answers 'what constant rate reproduces the wealth path?'
3Volatility drag: gains and losses don't cancel because they apply to different bases.
4Sample variance divides by T−1; standard deviation is its square root.
5Standard deviation measures typical fluctuation size — in percentage points, not as a fraction of the mean.
6Annualize volatility by multiplying by √n, not n.
7Historical averages and volatilities are estimates, not exact future forecasts.

Replay this lesson Continue to Covariance, Correlation, and Diversification