CFA Level I · CFA Level I Exam

Statistical Characteristics of Asset Returns: formula sheet

Full chapter guide

Key formulas

Arithmetic mean: X̄ = (X₁ + X₂ + … + Xₙ) ÷ n; Use for the expected return in a single period. It is the average of the observations.
Geometric mean return: R_G = [(1 + R₁)(1 + R₂)…(1 + Rₙ)]^(1/n) − 1; Use for compound growth over multiple periods. Work with 1 + R, not R.
Harmonic mean: X_H = n ÷ Σ(1 ÷ Xᵢ); Use for average price paid when the same money is invested each period. Values must be positive.
Weighted mean: X̄_w = Σ(wᵢ × Xᵢ), with Σwᵢ = 1; Portfolio return is a weighted mean of asset returns.
Median position: Position = (n + 1) ÷ 2 in sorted data; For an even n, average the two middle values.
Ordering of means: Arithmetic ≥ Geometric ≥ Harmonic; Holds for positive values. Equality only if all values are equal.
Range: Range = Maximum value − Minimum value; Uses only two observations, so it is sensitive to outliers.
Mean absolute deviation (sample): MAD = Σ|Xᵢ − X̄| ÷ n; Take absolute values of deviations from the sample mean. Divide by n.
Population variance: σ² = Σ(Xᵢ − μ)² ÷ N; Use only when you have every member of the population.
Sample variance: s² = Σ(Xᵢ − X̄)² ÷ (n − 1); Use for samples. Dividing by n − 1 makes it an unbiased estimator of population variance.
Standard deviation: σ = √σ² and s = √s²; Same units as the data, so easy to compare with the mean.
Target downside deviation: √[ Σ over Xᵢ ≤ B of (Xᵢ − B)² ÷ (n − 1) ]; B is the target. Only observations at or below B contribute to the sum, but the divisor is still n − 1, where n is the total number of observations. If B equals the mean, the result is the semi-deviation, a downside deviation measured around the mean.
Coefficient of variation: CV = s ÷ X̄; Risk per unit of mean return. A lower CV means less risk per unit of return. Interpret with care when the mean is near zero or negative.
Normal distribution benchmarks: Skewness = 0; Kurtosis = 3; Mean = Median = Mode; The normal is symmetric, so every comparison is made against these values.
Excess kurtosis: Excess kurtosis = Kurtosis − 3; Many software packages and exam questions report excess kurtosis directly. Check which one a question gives you.
Positive skew ordering: Mean > Median > Mode; Long right tail. Skewness above 0.
Negative skew ordering: Mean < Median < Mode; Long left tail. Skewness below 0.
Sample skewness (idea): Skewness ≈ [1 ÷ n] × Σ (Xi − X̄)³ ÷ s³; Cubed deviations keep the sign, so a long tail on one side sets the sign. Level I rarely asks you to compute it; know its meaning.
Sample kurtosis (idea): Kurtosis ≈ [1 ÷ n] × Σ (Xi − X̄)⁴ ÷ s⁴; Fourth powers are always positive and magnify large deviations, so heavy tails raise the value.
Position of the y-th percentile: Ly = (n + 1) × y ÷ 100; n is the number of observations in the sorted data. Use y = 25, 50, 75 for quartiles. If Ly is not a whole number, interpolate.
Interpolation: Value = X(lower) + fraction × [X(upper) − X(lower)]; fraction is the decimal part of Ly. X(lower) and X(upper) are the sorted values on either side of the position.
Interquartile range: IQR = Q3 − Q1; Measures spread of the middle 50% of observations. Not affected by extreme values.
Relative frequency: Relative frequency = interval frequency ÷ total observations; Relative frequencies across all intervals sum to 1 (100%).
Quantile equivalents: Median = Q2 = D5 = P50; Q1 = P25, Q3 = P75. Each decile is 10 percentiles; each quintile is 20 percentiles.
Sample covariance: Cov(X,Y) = Σ (Xi − X̄)(Yi − Ȳ) ÷ (n − 1); Use n − 1 for a sample. Divide by n only if the question says it is a population.
Correlation coefficient: r = Cov(X,Y) ÷ (sX × sY); Unit-free. Always between −1 and +1.
Covariance from correlation: Cov(X,Y) = r × sX × sY; Rearranged form. Common when a question gives correlation and standard deviations.
Covariance of a variable with itself: Cov(X,X) = Var(X); Useful check: the diagonal of a covariance matrix holds variances.
Sample correlation in terms of deviations: r = Σ (Xi − X̄)(Yi − Ȳ) ÷ [√Σ(Xi − X̄)² × √Σ(Yi − Ȳ)²]; The n − 1 terms cancel, so you can skip them when you compute r directly.

Quick revision

Arithmetic mean = Σx ÷ n. Geometric mean of returns = [Π(1 + R)]^(1/n) − 1.
For returns that vary, the geometric mean is lower than the arithmetic mean. They are equal only when all returns are identical.
Harmonic mean = n ÷ Σ(1/x), used for averaging prices paid with equal money amounts per period.
Population variance divides by N. Sample variance divides by n − 1.
Standard deviation = √variance and is in the same units as the returns.
Coefficient of variation = standard deviation ÷ mean, a measure of risk per unit of return. The mean must be nonzero (and positive for the usual interpretation).
For a typical unimodal distribution, positive skew gives mean > median > mode and negative skew gives mean < median < mode.
Excess kurtosis = kurtosis − 3. A positive value means fatter tails than a normal distribution (leptokurtic).
Quartiles split ordered data into four parts, and the interquartile range is Q3 − Q1.
Covariance = Σ(X − mean X)(Y − mean Y) ÷ (n − 1) for a sample. Its size depends on units.
Correlation = Cov(X, Y) ÷ (σX × σY) and always lies from −1 to +1.
Correlation shows linear association only. It does not prove causation, and a zero correlation can hide a non-linear link.

Common mistakes

Taking the geometric mean of returns instead of 1 + returns. Fix: Always convert to 1 + R, multiply, take the root, then subtract 1.
Using the arithmetic mean to describe multi-year growth. Fix: Use the geometric mean for compound growth. The arithmetic mean overstates it when returns vary.
Dividing by n when the data is a sample. Fix: Treat data as a sample unless the question says it is the entire population. Use Sx, not σx, on the calculator.
Reporting variance when the question asks for standard deviation, or vice versa. Fix: Underline the requested statistic in the stem. Take the square root as a deliberate last step.
Calling a distribution with a long left tail positively skewed because the peak sits on the right. Fix: Name the skew by the tail. Long left tail is negative skew, with mean < median < mode.
Treating a kurtosis of 3 as fat-tailed. Fix: Compare total kurtosis with 3, or compute excess kurtosis = kurtosis − 3 and compare with 0.
Using n × y ÷ 100 instead of (n + 1) × y ÷ 100. Fix: Use (n + 1) × y ÷ 100 as in the CFA curriculum.
Not sorting the data before finding the position. Fix: Always rewrite the data from smallest to largest first.
Dividing by n instead of n − 1 for sample covariance Fix: Unless the data is stated to be a population, use n − 1. Look for the word sample.
Interpreting covariance size as strength of relationship Fix: Covariance depends on units and volatility. Use correlation to judge strength.

Exam tips

Options are listed smallest to largest, so use the ordering arithmetic ≥ geometric ≥ harmonic to discard the wrong ones fast.
Words like 'compound', 'cumulative' and 'over the period' point to geometric. 'Average cost per share' with equal money points to harmonic.
When you see a trap option equal to the arithmetic mean in a geometric or harmonic question, it is almost always wrong unless all values are equal.
For median, always sort first and check whether n is odd or even.
Check your calculator is in the right mode and clear memory registers before multi-step root calculations.
Look for the words 'sample' and 'population' first. They decide the divisor and often decide which distractor is wrong.
When a question asks which asset is riskier per unit of return, think CV, not standard deviation.
For target downside deviation, count all observations in the divisor and use n − 1 as in the curriculum formula.