Statistical Measures of Asset Returns

Quantitative Methods. Worth 6 to 9 percent of the exam. One session: the lesson, the rules, the method, then the questions.

Quantitative MethodsStatistical Measures of Asset Returns
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The lesson

Runtime 18 minutes 50 seconds, measured from the published video.

The reading

Five to ten minutes on this one unit: what the exam wants, the idea in plain words, then straight into the trap and the practice.

The exam wants you to calculate and interpret the arithmetic, geometric, and harmonic means and know which one a given situation calls for, calculate sample and population variance and standard deviation, and calculate and interpret skewness, kurtosis, and the coefficient of variation. The question is rarely 'compute the mean'; it is 'which mean.'

A single word in the question stem decides which formula this module wants, more than any arithmetic does. The arithmetic mean, the sum of the observations divided by their count, answers one specific question: the best single estimate of next period's expected return, taken in isolation. It is the wrong tool for describing how an investment actually performed over several periods, because it ignores compounding. A return of plus 50 percent followed by minus 50 percent averages to zero arithmetically, but $100 that grows to $150 and then falls to $75 has actually lost a quarter of its value. The geometric mean, taking the product of (1 + each period's return) and finding its nth root, then subtracting 1, is the rate that actually compounds a starting value to the ending value observed, and it is always less than or equal to the arithmetic mean whenever returns vary at all. A third mean, the harmonic mean, answers a narrower question. It is n divided by the sum of the reciprocals of the observations. It gives the true average cost when an equal dollar amount, rather than an equal number of shares, is invested at each of several prices, the way regular monthly investing works. The three sit in a fixed order: harmonic at most geometric at most arithmetic, equal only when every value is identical.

A second decision, separate from which mean to use, is how to divide when computing variance. Population variance divides the sum of squared deviations by N, the full count, because the population mean is known exactly. Sample variance divides by n minus 1 instead. The sample mean used to measure those deviations was itself estimated from the same data. It fits the sample slightly too well and understates the true spread, unless that one degree of freedom is given back. The word 'sample' in a question stem is the trigger for n minus 1. 'Population' or 'all' is the trigger for N.

Shape matters as much as center and spread. Skewness describes an asymmetric distribution. Positive, or right, skew has a long tail of unusually large values. That tail pulls the mean above the median and the mode, so mean is greater than median is greater than mode. Negative, or left, skew reverses that order. Kurtosis measures how much probability sits in the tails relative to a normal distribution. Excess kurtosis is kurtosis minus the normal distribution's own value of 3. A positive excess kurtosis marks a leptokurtic distribution: more extreme events than a normal model would predict. That is the shape real asset returns tend to have. It is also the shape a risk model built on normality will get wrong.

Finally, the coefficient of variation, standard deviation divided by the mean, lets you compare risk per unit of return across investments whose return levels differ too much for their standard deviations alone to be compared fairly. A lower coefficient of variation means less risk taken on for each unit of return earned, whatever the absolute size of either number.

A return distribution with mean and one standard deviation marked mean return -1 SD +1 SD wider spread = higher variance
The return distribution's mean marks the center; its spread, measured by variance and standard deviation, is how wide the curve is drawn.

Worked in full

An investor earns annual returns of 15 percent, minus 8 percent, and 12 percent over three years. What is the geometric mean annual return, and how does it compare to the arithmetic mean? Geometric mean: multiply (1.15)(0.92)(1.12) = 1.1850, then take the cube root and subtract 1. 1.1850^(1/3) = 1.0582, so the geometric mean is 5.82 percent. The arithmetic mean is (15 - 8 + 12) / 3 = 6.33 percent. The geometric mean is lower, as it always is when returns vary: the negative year compounds against the base the following year builds on, a cost the arithmetic mean never sees.

The same problem, one step removed

Same three annual returns: 15 percent, minus 8 percent, 12 percent. Multiply (1 + each return) together to get the compound growth factor, then find its cube root and subtract 1 yourself to get the geometric mean.

The trap

Comparing two investments' risk by their standard deviation alone, and picking the one with the higher raw return without dividing by it, ignores that the higher-return choice may be carrying disproportionately more risk per unit of return; the coefficient of variation exists to catch exactly that comparison.

What this unit turns on

Read these before the questions, not after them. Everything here traces to this module's own lesson and to the 2026 outline.

Multi-period returns are averaged geometrically; a single future period is estimated arithmetically

The geometric mean, [(1+r1)(1+r2)...(1+rn)]^(1/n) - 1, is the rate that actually compounds a starting value to the ending value observed over several periods, and it is always less than or equal to the arithmetic mean whenever returns vary. The arithmetic mean answers a different question: the best single estimate of next period's expected return, taken in isolation.

Equal-dollar averaging uses the harmonic mean, not the arithmetic mean

When the same dollar amount is invested at each of several prices, as in dollar-cost averaging or in averaging a P/E ratio across equal-dollar positions, the harmonic mean, n divided by the sum of the reciprocals, gives the true average cost, because more shares are bought at low prices and fewer at high ones. Harmonic mean is always less than or equal to geometric mean, which is always less than or equal to arithmetic mean, with equality only when every value is identical.

A skewed or outlier-heavy distribution is better described by its median than its mean

The median depends only on the middle-ranked observation and ignores how extreme the outliers are; the mean is pulled toward whichever tail is longer. When a distribution has a small number of extreme values, the median is the more representative measure of a typical outcome.

Population data divides by N; sample data divides by N minus 1

Sample variance divides the sum of squared deviations by n - 1, not n, because the sample mean used to compute those deviations is itself estimated from the same data, using up one degree of freedom; dividing by n would systematically understate the true population variance. Population variance, computed when every member of the population is observed, divides by N because the population mean is then known exactly, not estimated.

Standard deviation is never smaller than mean absolute deviation

Mean absolute deviation averages the unsigned distance of each observation from the mean; variance, and therefore standard deviation, squares those distances before averaging, which gives disproportionate weight to the largest deviations. For any dataset with more than one distinct value, standard deviation exceeds mean absolute deviation.

Coefficient of variation ranks risk per unit of return; a lower value is the more efficient investment

CV = standard deviation divided by mean expresses how much dispersion accompanies each unit of expected return, which makes assets with very different return levels comparable on a single scale. The Sharpe ratio applies the identical idea to excess return over the risk-free rate rather than to total return.

Excess kurtosis above zero means fatter tails, not a taller peak, is the risk that matters

Kurtosis measures how much probability sits in the tails relative to a normal distribution, whose own kurtosis is 3; excess kurtosis is kurtosis minus 3. A leptokurtic distribution, excess kurtosis above zero, has more probability in both the center and the extreme tails than the shoulders, meaning small moves are common but so are rare, severe ones, which is exactly the danger a risk manager cares about.

Skew direction follows the mean, not the bulk of the data

Positive, or right, skew pulls the mean above the median and mode, ordering them mean greater than median greater than mode, because a long right tail of large values drags the mean upward even though most observations sit below it. Negative, or left, skew reverses the order: mean less than median less than mode.

Correlation measures the strength and direction of a linear relationship, bounded between -1 and 1

A correlation near 1 or -1 indicates two variables move together closely in a straight-line relationship; a correlation near 0 indicates little or no linear relationship, though a strong nonlinear relationship can still exist and go undetected by correlation alone.

The trick

H is at most G is at most A

Harmonic mean, geometric mean, arithmetic mean: the three always sit in that order from smallest to largest, equal only when every observed value is identical. A question that gives all three and asks which is largest can be answered from this ordering alone, without recomputing anything.

The dollar-cost-averaging trigger phrase is equal dollar amount

Whenever a question says the same dollar amount was invested at different prices, or an equal dollar amount was placed in each stock, the harmonic mean is the answer, not the arithmetic mean of the prices or ratios.

Chebyshev's 75% and 88.9% versus the empirical rule's 68/95/99.7

Chebyshev's inequality, 1 minus 1 over k squared, gives 75% within 2 standard deviations and 88.9% within 3, for any distribution shape. The empirical rule's 68/95/99.7 figures apply only when the distribution is normal; a question that says unknown or non-normal distribution is signaling Chebyshev, not the empirical rule.

N minus 1 exists because the sample mean already used one degree of freedom

Bessel's correction, dividing by n - 1 instead of n, corrects for the fact that squared deviations measured from an estimated mean are systematically smaller than deviations from the true, unknown population mean.

The method

The order to work a question of this type in, every time, before you touch the numbers.

  1. Read the question for its trigger words first: sample versus population, multi-period versus single period, and equal-dollar amounts versus equal-share amounts.
  2. Choose the mean: geometric for compounded multi-period returns, harmonic for equal-dollar averaging, median for skewed or outlier-heavy data, arithmetic otherwise.
  3. Choose the variance divisor: N for a stated population, n - 1 for a stated sample.
  4. For a dispersion comparison across assets with different return levels, compute the coefficient of variation for each and select the lower one as more risk-efficient.
  5. For a shape question, locate the mean relative to the mode: mean above mode signals positive skew, mean below mode signals negative skew; excess kurtosis above zero signals fat tails (leptokurtic), below zero signals thin tails (platykurtic).
  6. [BA II Plus: enter each observation with the DATA worksheet (2ND DATA), then 2ND STAT to read x-bar, Sx (sample standard deviation) and sigma-x (population standard deviation) directly; confirm which one the question calls for before reading off a value]

Two worked examples, then you are on your own

The first is worked in full. The second gives you the setup and stops. After that the questions give you nothing, which is the point: the help fades on purpose, so the last thing you practise is the thing the exam actually asks of you.

Worked in full

An analyst observes annual returns of 10%, -5%, and 20% for an investment over three years. The geometric mean annual return is closest to:

Answer B. Geometric mean = [(1.10)(0.95)(1.20)]^(1/3) - 1 = [1.2540]^(0.3333) - 1 = 1.0784 - 1 = 7.84%. 33%, which overstates actual compound return.

Your turn, setup given

A sample of 10 monthly returns has a mean of 1.5%. The sum of squared deviations from the mean is 0.0045. The sample variance is closest to:

Read the question for its trigger words first: sample versus population, multi-period versus single period, and equal-dollar amounts versus equal-share amounts.

The practice run

Pick an answer, say how sure you are, then reveal. Being sure and wrong is the most useful thing that can happen in a session, so answer honestly: it sends the unit back to learning and puts it at the front of your revision queue.

Question 1Exam level

Portfolio A has an expected return of 12% and a standard deviation of 15%. Portfolio B has an expected return of 8% and a standard deviation of 9%. Which portfolio most likely has lower risk per unit of return?

How sure are you?

Correct: B. CV = standard deviation / mean. CV(A) = 15/12 = 1.25. CV(B) = 9/8 = 1.125. Lower CV means less risk per unit of return. Portfolio B has CV of 1.125 vs Portfolio A's 1.25. 889 which would be mean/stddev (incorrect formula direction).
A. Portfolio A has higher absolute return (12% vs 8%), so candidates assume it's the 'better' portfolio. Higher return does not mean lower risk per unit of return. CV measures how much risk you take for each percent of return earned.
C. Inverts the CV formula (mean/stddev instead of stddev/mean). CV = sigma/mu, not mu/sigma. Inverting gives the return per unit of risk (a different metric).

Unit: statistical-measures-of-asset-returns

Question 2Exam level

An analyst is calculating the average P/E ratio for a portfolio by investing an equal dollar amount in three stocks with P/E ratios of 10, 15, and 30. The most appropriate mean to use is:

How sure are you?

Correct: C. When averaging ratios under an equal-DOLLAR-amount investment (not an equal number of shares), the harmonic mean is the appropriate average. Harmonic mean = n / sum(1/xi) = 3 / (1/10 + 1/15 + 1/30) = 3 / (0.10 + 0.0667 + 0.0333) = 3 / 0.20 = 15.0.
A. Simple average of the three P/E ratios (10+15+30)/3 = 18.33. Arithmetic mean overweights the high P/E stock. Equal dollar investment means you buy more shares of low P/E stocks, so they should receive more weight.
B. Geometric mean is often used for averages, candidates may default to it. Geometric mean is appropriate for rates of return over time, not for averaging ratios with equal dollar investment amounts.

Unit: statistical-measures-of-asset-returns

Question 3Exam level

A return distribution has a mean of 8%, median of 9%, and mode of 11%. This distribution is most likely described as:

How sure are you?

Correct: B. Negative skew: mean < median < mode. Here: 8% < 9% < 11%. The distribution has a long left tail (large negative outliers pulling the mean down). This is typical of strategies that have frequent small gains but occasional large losses (e.g., writing options).
A. You might confuse the direction. 'Positive skew' sounds like positive returns, and the distribution here has positive values. Positive skew (right skew) has mean > median > mode. Here mean (8%) is LESS than mode (11%), indicating leftward skew.
C. All three values are positive, candidates may assume this means a normal/symmetric distribution. Symmetry requires mean = median = mode. The 3-percentage-point gap between mean (8%) and mode (11%) clearly indicates asymmetry.

Unit: statistical-measures-of-asset-returns

Question 4Exam level

A fund with excess kurtosis of +2.5 is most likely described as:

How sure are you?

Correct: C. Excess kurtosis (kurtosis - 3) > 0 means leptokurtic: fat tails, peaked center, higher probability of extreme outcomes than the normal distribution. Excess kurtosis = +2.5 means kurtosis = 5.5, well above the normal distribution's kurtosis of 3. This is characteristic of hedge fund returns and financial asset returns generally.
A. Platykurtic starts with 'plat' (flat), and excess kurtosis sounds like 'extra flat' to some candidates. Platykurtic means NEGATIVE excess kurtosis (thinner tails, flatter peak). Positive excess kurtosis = fat tails = leptokurtic.
B. You might know mesokurtic = normal distribution, and may think +2.5 is 'close to normal'. Mesokurtic has excess kurtosis = 0 (kurtosis = 3). Any deviation from zero is either lepto (+) or platy (-).

Unit: statistical-measures-of-asset-returns

Question 5Exam level

The Sharpe ratio of a portfolio is 0.65. The portfolio return is 12%, the risk-free rate is 4%, and the portfolio standard deviation is 12.3%. Which of the following best describes the Sharpe ratio?

How sure are you?

Correct: B. Sharpe ratio = (Rp - Rf) / σp = (12% - 4%) / 12.3% = 8% / 12.3% = 0.65. The numerator is EXCESS return (return above risk-free rate), not total return. The denominator is total portfolio standard deviation. So the ratio measures excess return per unit of total risk.
A. Ignores that the numerator uses excess return (Rp - Rf), not total return. Using total return (12%) would give 12%/12.3% = 0.98, not 0.65. The Sharpe ratio uses risk-free adjusted return.
C. Confuses Sharpe ratio with information ratio or alpha. The Sharpe ratio is a ratio (excess return per unit of risk), not an absolute excess return. 0.65 does not represent a percentage point return above anything.

Unit: statistical-measures-of-asset-returns

Question 6Exam level

For a dataset with unknown distribution, Chebyshev's inequality states that at least what percentage of observations lie within 2 standard deviations of the mean? The value is closest to:

How sure are you?

Correct: B. Chebyshev's inequality: for any distribution, at least (1 - 1/k²) of observations lie within k standard deviations. For k=2: 1 - 1/4 = 75%. Key: this is a minimum guarantee for ANY distribution. The 68% and 95% figures are from the normal distribution specifically (empirical rule: 68-95-99.7).
A. 68% is the well-known 1-standard-deviation figure from the normal distribution. 68% applies to the normal distribution's 1-sigma rule. Chebyshev for k=1 gives only 0% minimum (useless for k=1). For k=2, Chebyshev gives 75%.
C. 95% is the normal distribution's 2-sigma rule (empirical rule). You might confuse this with Chebyshev's 2-sigma result. 95% applies to normal distributions only. Chebyshev's 75% is the minimum for ANY distribution. Including highly skewed ones. Chebyshev is more conservative and universally applicable.

Unit: statistical-measures-of-asset-returns

Question 7Exam level

An analyst calculates the mean absolute deviation (MAD) of a return series as 3.2%. The standard deviation of the same series is most likely:

How sure are you?

Correct: C. Standard deviation is always greater than or equal to MAD for any dataset with more than one observation. This is because standard deviation squares the deviations before averaging, which amplifies large deviations. MAD = mean of |xi - x̄|. Variance = mean of (xi - x̄)². The squaring in variance (and thus standard deviation) gives disproportionate weight to outliers, making standard deviation ≥ MAD always.
A. You might may think MAD is more 'severe' because it captures all deviations. MAD uses absolute values (no squaring). Standard deviation uses squared deviations which magnify the impact of outliers, always resulting in a larger value.
B. You might may think both measures of dispersion should yield the same result. Equal only in the impossible case of zero variance. In practice, squaring always produces a larger average than taking absolute values.

Unit: statistical-measures-of-asset-returns

Question 8Exam level

An analyst wants to measure the dispersion of returns for a portfolio that has known population data (all 240 monthly returns for the past 20 years). The appropriate formula for variance divides the sum of squared deviations by, most likely:

How sure are you?

Correct: B. When the FULL POPULATION is known, divide by N (240). Population variance σ² = Σ(xi - μ)² / N. Sample variance s² = Σ(xi - x̄)² / (n-1). The N-1 correction (Bessel's correction) is needed only when the mean is estimated from the sample. Because the sample mean itself uses one degree of freedom. When the population is completely known, μ is exact, and N is the correct denominator.
A. Students memorize 'divide by N-1' as the rule without understanding when it applies. N-1 applies only for samples. Using N-1 on population data would underestimate the true population variance.
C. For n=240, the difference between 240 and 239 is small (0.4%), so candidates think it's negligible. The CFA exam tests the conceptual distinction, not the numerical difference. The correct formula depends on whether you have a population or a sample, not on sample size.

Unit: statistical-measures-of-asset-returns

Question 9Exam level

Which measure of central tendency is most appropriate for describing the 'typical' return when a return distribution is highly positively skewed with a few extreme positive outliers?

How sure are you?

Correct: B. The median is resistant to outliers because it depends only on the middle value of the sorted dataset, not the magnitude of extreme values. In a positively skewed distribution, the arithmetic mean is pulled upward by the extreme positive values, making it an unrepresentative measure of the 'typical' or 'central' return. Median is the preferred measure when distributions are skewed or contain outliers.
A. Arithmetic mean is the most commonly used average in everyday finance. In skewed distributions, the mean is distorted by outliers. A few extremely high returns pull the mean above what most observations experience.
C. Mode represents the most frequent return, which sounds like 'typical'. Mode may not be unique (bimodal distributions) and may not be near the center. It ignores most of the data. Mode is rarely the best measure of central tendency for return data.

Unit: statistical-measures-of-asset-returns

Question 10Above the exam

A portfolio has an annual return distribution with a mean of 9% and a standard deviation of 14%. An analyst also finds the distribution has positive excess kurtosis. Combining the normal-distribution 1-standard-deviation rule with the kurtosis finding, the analyst should most likely conclude that the TRUE probability of a return falling within one standard deviation of the mean (roughly -5% to 23%) is:

How sure are you?

Correct: B. The 'about 68% within one standard deviation' rule is a property of the NORMAL distribution specifically. Positive excess kurtosis means the actual distribution has fatter tails (more extreme outcomes) AND typically a higher peak around the mean than a true normal distribution, so the actual share of observations within one standard deviation will differ from the textbook 68% figure. The direction and size of the difference depends on the specific distribution, but it should not be assumed to be exactly 68% once kurtosis has been shown to deviate from the normal benchmark.
A. The 68% figure is specifically a normal-distribution property; it does not hold exactly for every distribution shape. A distribution with confirmed positive excess kurtosis has already been shown to deviate from normality, so applying the normal rule without adjustment ignores the very finding the question describes.
C. Kurtosis directly describes the shape of a distribution's tails and peak relative to normal, which is exactly what determines how the probability mass is distributed around the mean; it is not unrelated to this question, it is the central fact the question is testing.

Unit: statistical-measures-of-asset-returns

Question 11Above the exam

Two portfolios have the same arithmetic mean annual return of 10% over five years, but Portfolio X has a standard deviation of returns of 5% while Portfolio Y has a standard deviation of 20%. An investor concludes their geometric mean (compound) returns must also be equal, since the arithmetic means match. This conclusion is most likely:

How sure are you?

Correct: B. Geometric mean is always less than or equal to arithmetic mean, and the GAP between them widens as return variability (variance) increases (approximately, geometric mean is approximated by arithmetic mean minus half the variance). With equal arithmetic means but Portfolio Y's much higher standard deviation (and therefore variance), Portfolio Y's actual compound (geometric) return will be meaningfully lower than Portfolio X's, even though their simple-average annual returns look identical.
A. Geometric mean is not a function of the arithmetic mean alone; return VARIABILITY also matters, which is exactly the variable that differs between the two portfolios here. Two portfolios can share an arithmetic mean and still have different geometric means.
C. The length of the measurement period does not change the underlying mathematical relationship between arithmetic mean, variance, and geometric mean; a longer or shorter period changes the specific numbers but not the fact that higher variance widens the gap between the two means.

Unit: statistical-measures-of-asset-returns