Practice: Statistical Measures of Asset Returns

Quantitative Methods. 13 question(s) in this unit's pool (2 above the exam). Free up to ten a day; the coach picks which ones based on what you have already answered and when each is next due.

Quantitative MethodsStatistical Measures of Asset Returns
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Today's practice

Pick an answer, say how sure you are, then reveal. Every wrong choice gets its own explanation. Questions you have already answered correctly and confidently stay out of the way until they are due for review again.

Question 1Exam level

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

How sure are you?

Correct: 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.
A. Averaging 10, -5, 20 as simple arithmetic and rounding. Arithmetic mean of 8.33% does not account for compounding. The -5% year reduces the base for the next year's gain.
C. Arithmetic mean calculation (10-5+20)/3 = 25/3 = 8.33%. Arithmetic mean overstates compound return when there is variance in returns. It ignores the sequence-of-returns effect.

Unit: statistical-measures-of-asset-returns

Question 2Exam level

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:

How sure are you?

Correct: B. Sample variance = sum of squared deviations / (n-1) = 0.0045 / (10-1) = 0.0045 / 9 = 0.000500. Recalculating: 0.0045/9 = 0.0005. Answer C = 0.000500 = 0.0005. Sample variance = 0.0045/(10-1) = 0.0005. Correct answer is C. Sample standard deviation = sqrt(0.0005) = 2.24%.
A. Dividing sum of squared deviations by n (10) instead of n-1 (9): 0.0045/10 = 0.000045. Wait: 0.0045/10 = 0.000045 (if deviations already in decimal form this would be population variance). This is the population variance formula (divide by N). For a sample, we divide by N-1 to obtain an unbiased estimate of population variance.
C. You might be tempted to choose C because it matches the sum of squared deviations divided by the sample size, but the correct formula for sample variance uses n-1 in the denominator, not n, making C incorrect.

Unit: statistical-measures-of-asset-returns

Question 3Exam 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).

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Question 4Exam 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 5Exam 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 6Exam 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 (-).

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Question 7Exam 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 8Exam 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 9Exam 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 10Exam 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 11Exam 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 12Above 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 13Above 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