6.1 Sample Moments

Sample Moments

FRM quantitative analysis leans on sample moments: statistics computed from data that estimate unknown population features. Markets give you one path of returns; risk models need statements about means, variances, tails, and co-movement. This section builds the vocabulary and formulas you will reuse in hypothesis tests and regression.

Estimator Versus Estimate

An estimator is a function of the sample—a random variable before you see the data. An estimate is the realized number once the sample is fixed.

• Estimator: X̄ = (1/n) Σ X_i (a random rule).
• Estimate: for returns {0.01, −0.02, 0.03}, x̄ = 0.0067 (one number).

Properties such as bias, variance, and consistency belong to the estimator. Saying “this estimate is biased” is loose language; exam answers should attach bias to the procedure.

Sample Mean, Variance, and Standard Deviation

For i.i.d. draws X_1, …, X_n from a population with mean μ and variance σ²:

Sample mean

x̄ = (1/n) Σ_{i=1}^{n} x_i

Sample variance (unbiased for σ² under i.i.d. normality / classical assumptions)

s² = [1/(n − 1)] Σ_{i=1}^{n} (x_i − x̄)²

Sample standard deviation

s = √s²

Why n − 1? Using the sample mean instead of μ “uses up” one degree of freedom. The divisor n − 1 makes E[s²] = σ² (unbiasedness for variance). Dividing by n yields a downward-biased variance estimator—sometimes used in maximum-likelihood contexts, but not the default FRM “unbiased sample variance.”

Worked example: mean and variance

Daily P&L (in $000): 2, −1, 4, 0, 5.

n = 5, Σ x_i = 10, so x̄ = 10/5 = 2.

Deviations from 2: 0, −3, 2, −2, 3. Squared: 0, 9, 4, 4, 9. Sum of squares = 26.

s² = 26/(5 − 1) = 6.5, so s ≈ 2.55.

If someone divides by n = 5, they get 5.2—systematically smaller. On FRM, default to the unbiased (n − 1) form unless the question specifies MLE.

Bias

Bias of an estimator θ̂ for parameter θ is:

Bias(θ̂) = E[θ̂] − θ

• Unbiased: E[θ̂] = θ for all θ (bias = 0).
• Biased but useful: a small bias with much lower variance can beat an unbiased estimator on MSE = Variance + Bias².

Classic facts:

• X̄ is unbiased for μ when E[X_i] = μ.
• s² with (n − 1) is unbiased for σ² under standard i.i.d. assumptions.
• The MLE variance with divisor n is biased low for finite n.

Bias does not mean “wrong for this sample.” A fair coin’s sample proportion of heads can be 0.9 in ten flips; the estimator can still be unbiased.

BLUE: Best Linear Unbiased Estimator

Under the Gauss–Markov setting (linear model, errors with mean zero, homoskedastic, uncorrelated), the OLS estimator is BLUE: among all linear and unbiased estimators of the coefficient, it has the smallest variance.

For estimating μ with i.i.d. observations and constant variance, the sample mean is BLUE among linear unbiased estimators of μ: no other weighted average with weights summing to 1 has lower variance while staying unbiased.

BLUE is narrower than “best estimator in the world.” Nonlinear estimators (or biased ones) might have lower MSE. Exam trap: equating BLUE with “always optimal under every criterion.”

Consistency

Consistency: θ̂_n → θ in probability as n → ∞. For every ε > 0,

P(|θ̂_n − θ| > ε) → 0 as n → ∞.

Unbiasedness is about the center of the sampling distribution at fixed n. Consistency is about the distribution collapsing onto the truth as the sample grows. An estimator can be biased for every finite n yet consistent (bias shrinks to zero). Conversely, an unbiased estimator that does not concentrate can fail to be useful.

Law of Large Numbers and Central Limit Theorem

Law of Large Numbers (LLN). For i.i.d. X_i with finite mean μ, the sample mean X̄_n converges in probability to μ. Sample averages become reliable location measures in large samples.

Central Limit Theorem (CLT). For i.i.d. X_i with mean μ and finite variance σ²,

√n (X̄_n − μ) / σ → N(0, 1) in distribution as n → ∞.

Practically: even if returns are not normal, the distribution of the sample mean is approximately normal for large n. That underwrites z-tests, many confidence intervals, and asymptotic regression inference.

Worked example: CLT scaling

Suppose daily returns have μ = 0.0004 and σ = 0.01. For n = 100 trading days,

SE(X̄) = σ/√n = 0.01/10 = 0.001.

Approx. 95% CI for μ: 0.0004 ± 1.96 × 0.001 ≈ [−0.00156, 0.00236] if we knew σ and used the true μ only for illustration—in practice replace μ with x̄. The key FRM takeaway is the 1/√n shrinkage of mean uncertainty.

Skewness and Kurtosis

Population skewness (third standardized moment) measures asymmetry:

Skew = E[(X − μ)³] / σ³

• Positive skew: long right tail (common for some credit losses flipped in sign, or lottery-like payoffs).
• Negative skew: long left tail (typical concern for equity index returns and many trading books).

Population kurtosis (fourth standardized moment):

Kurt = E[(X − μ)⁴] / σ⁴

Normal distribution: kurtosis = 3. Excess kurtosis = Kurt − 3. Fat tails mean excess kurtosis > 0: extreme moves more often than a normal with the same σ would predict. Risk managers care because VaR/ES calibrated under normality understate tail loss when kurtosis is high.

Sample versions replace expectations with averages of cubed/fourth powers of standardized residuals (software may apply small-sample corrections). FRM questions usually test interpretation more than obscure correction constants.

Quantiles

The p-quantile (percentile) q_p satisfies P(X ≤ q_p) = p (with care for discrete distributions). The median is the 0.5-quantile. Sample quantiles sort the data and interpolate as needed.

Quantiles are robust to outliers relative to the mean: one crash day moves x̄ a lot but may change the median less. VaR is literally a quantile of the loss distribution. Interquartile range (q_0.75 − q_0.25) is a scale measure that ignores extremes.

Worked sketch

Ordered losses: 1, 2, 2, 5, 8, 9, 20 (n = 7). Median = 5. Sample mean ≈ 6.7, pulled by 20. A 90% historical VaR-style reading near the upper order statistics would focus on the right tail of losses—not on the mean alone.

Covariance and Correlation

For paired observations (X_i, Y_i):

Sample covariance

cov̂(X, Y) = [1/(n − 1)] Σ (x_i − x̄)(y_i − ȳ)

Sample correlation

r = cov̂(X, Y) / (s_X s_Y), with −1 ≤ r ≤ 1.

Correlation is scale-free linear association. Covariance keeps units (product of the two series’ units). Correlation zero does not imply independence outside special cases (e.g., jointly normal): nonlinear dependence can remain.

Worked example

Two-asset excess returns over four days:

X: 1, 2, 0, 1 → x̄ = 1, s_X = √[(0+1+1+0)/3] = √(2/3) ≈ 0.816 Y: 2, 3, 1, 2 → ȳ = 2, deviations mirror X’s pattern

Products of deviations: 0, 1, 1, 0; sum = 2; cov̂ = 2/3 ≈ 0.667; r = 0.667/(0.816×0.816) ≈ 1.0 (perfect linear lock in this toy sample).

Coskewness and Cokurtosis

Higher co-moments matter when portfolios care about crash co-movement, not just variance.

Coskewness involves third-order products such as E[(X − μ_X)² (Y − μ_Y)] / (σ_X² σ_Y) (definitions vary by normalization). Positive coskewness with the market can mean an asset tends to deliver upside when the market is already up—or, depending on sign convention and which variable is squared, encodes asymmetric co-movement. FRM takeaway: coskewness captures how skewness is shared across assets.

Cokurtosis involves fourth-order products such as E[(X − μ_X)³ (Y − μ_Y)] / (σ_X³ σ_Y) or related forms. It describes tail co-movement: whether assets crash together beyond what correlation implies.

In risk budgeting and alternative risk premia, two portfolios with identical volatilities and correlations can differ sharply in coskewness/cokurtosis with the market—changing downside diversification in stress.

Putting Sample Moments to Work

A risk dashboard that reports only mean and volatility is incomplete. Add skewness and kurtosis for shape; quantiles for VaR-style levels; correlation for linear hedges; and, when the mandate is crash diversification, co-moment diagnostics. Estimators are random; larger n helps via LLN/CLT for means, but dependence, structural breaks, and fat tails still break naive i.i.d. formulas—topics that later FRM readings extend.

Master the distinction between estimator and estimate, the (n − 1) variance, bias versus consistency, BLUE’s precise claim, and the interpretation of higher moments and co-moments. Those ideas feed directly into hypothesis testing and regression inference.