Quantitative MethodsModule 3 of 11

Statistical Measures of Asset Returns

Concepts

Formulas

Decisions

Quiz Questions

Key Concepts

7 concepts covered in this module.

Measures of Central Tendency

Mean (arithmetic, geometric, harmonic), median (middle value, robust to outliers), mode (most frequent).

Quantiles

Quartiles divide data into 4 parts, quintiles into 5, deciles into 10, percentiles into 100.

Variance & Standard Deviation

Variance = average squared deviation from mean. SD = √Variance. Population uses N; sample uses N-1.

Coefficient of Variation (CV)

CV = s/X̄. Measures relative dispersion — useful for comparing risk across assets with different means.

Skewness

Positive skew: right tail longer (mean > median > mode). Negative skew: left tail longer (mean < median < mode).

Kurtosis

Leptokurtic (excess kurtosis > 0): fatter tails, more peaked. Platykurtic (< 0): thinner tails. Mesokurtic = normal.

Correlation

Standardized covariance: ρ = Cov(X,Y) / (σ_Xσ_Y). Range: -1 to +1. Measures linear association only.

Formulas

6 essential formulas for this module.

Population Variance

σ² = Σ(X_i - μ)² / N

Where: μ = population mean, N = population size

Sample Variance

s² = Σ(X_i - X̄)² / (n - 1)

Where: X̄ = sample mean, n-1 for Bessel correction

Coefficient of Variation

CV = s / X̄

Where: s = standard deviation, X̄ = mean

Covariance

Cov(X,Y) = Σ(X_i - X̄)(Y_i - Ȳ) / (n - 1)

Where: For sample covariance

Correlation

ρ_XY = Cov(X,Y) / (σ_X × σ_Y)

Where: Range: -1 ≤ ρ ≤ +1

Downside Deviation

DD = √[Σ min(R_i - R_target, 0)² / (n - 1)]

Where: Only considers returns below target

Decision Frameworks

2 decision frameworks to guide your analysis.

When to use Median vs Mean?

Median: when data is skewed or has outliers
Mean: when data is symmetric and approximately normal

When to worry about kurtosis?

Risk management: leptokurtic returns mean more extreme events than normal distribution predicts
VaR calculations: fat tails underestimate tail risk if assuming normality

Mind Map

Visual overview of how concepts connect in this module.

Statistical Measures

Central Tendency

Arithmetic Mean

Median

Mode

Geometric Mean

Harmonic Mean

Dispersion

Range

MAD

Variance

Standard Deviation

Downside Deviation

Shape

Skewness (+/-/0)

Kurtosis (lepto/platy/meso)

Normal: skew=0, excess kurt=0

Association

Covariance

Correlation coefficient

Scatter plots

Spurious correlation

Study Statistical Measures of Asset Returns

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Time Value of Money in Finance

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