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This skill enables Claude to apply core statistical methods to financial data, including descriptive statistics, covariance estimation, linear regression, hypothesis testing, and resampling techniques. These methods form the quantitative backbone for portfolio construction, risk measurement, and factor modeling.
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This skill enables Claude to apply core statistical methods to financial data, including descriptive statistics, covariance estimation, linear regression, hypothesis testing, and resampling techniques. These methods form the quantitative backbone for portfolio construction, risk measurement, and factor modeling.
0 — Mathematical Foundations
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Mean (Expected Value):
$$\mu = E[X] = \frac{1}{n} \sum_{i=1}^{n} x_i$$
The arithmetic average of observed values. For financial returns, this represents the central tendency of the return distribution.
Variance:
Population variance:
$$\sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2$$
Sample variance (Bessel's correction):
$$s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2$$
Standard Deviation:
$$\sigma = \sqrt{\sigma^2}$$
In finance, standard deviation of returns is commonly called volatility. Annualized volatility from monthly data: sigma_annual = sigma_monthly * sqrt(12).
Skewness:
$$\gamma = \frac{E[(X - \mu)^3]}{\sigma^3}$$
Measures asymmetry of the distribution. Negative skewness (left tail) is common in equity returns and indicates a higher probability of large losses than large gains.
Excess Kurtosis:
$$\kappa = \frac{E[(X - \mu)^4]}{\sigma^4} - 3$$
Measures tail thickness relative to the normal distribution (which has excess kurtosis of 0). Financial returns typically exhibit positive excess kurtosis (leptokurtosis), meaning fat tails and more frequent extreme events than a normal distribution would predict.
Covariance:
$$\text{Cov}(X, Y) = E[(X - \mu_X)(Y - \mu_Y)]$$
Sample covariance:
$$\hat{\text{Cov}}(X, Y) = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})$$
Covariance measures the linear co-movement between two variables. Positive covariance means they tend to move together; negative means they move inversely.
Correlation (Pearson):
$$\rho(X, Y) = \frac{\text{Cov}(X, Y)}{\sigma_X \times \sigma_Y}$$
Correlation normalizes covariance to the range [-1, +1], making it unit-free and comparable across variable pairs.
For a set of p assets with n return observations, the sample covariance matrix is:
$$\hat{\Sigma} = \frac{1}{n-1} (X - \bar{X})^T (X - \bar{X})$$
where X is the n x p matrix of returns.
The curse of dimensionality: When p (number of assets) is large relative to n (number of observations), the sample covariance matrix becomes poorly conditioned or singular, leading to unstable portfolio optimizations.
Shrinkage blends the sample covariance matrix with a structured target (e.g., the identity matrix scaled by average variance) to produce a more stable estimate:
$$\hat{\Sigma}_{shrunk} = \delta \cdot F + (1 - \delta) \cdot \hat{\Sigma}$$
where:
F = the shrinkage target (structured estimator)delta = the optimal shrinkage intensity (estimated analytically)Sigma_hat = the sample covariance matrixLedoit-Wolf determines the optimal delta that minimizes expected squared Frobenius distance to the true covariance matrix. This produces better-conditioned matrices and more stable portfolio weights.
Ordinary Least Squares estimates the linear relationship y = X * beta + epsilon by minimizing the sum of squared residuals.
Coefficient Estimate:
$$\hat{\beta} = (X^T X)^{-1} X^T y$$
Key Regression Diagnostics:
R-squared (Coefficient of Determination):
$$R^2 = 1 - \frac{SS_{res}}{SS_{tot}} = 1 - \frac{\sum(y_i - \hat{y}_i)^2}{\sum(y_i - \bar{y})^2}$$
Represents the proportion of variance in the dependent variable explained by the model.
Adjusted R-squared:
$$\bar{R}^2 = 1 - (1 - R^2) \frac{n - 1}{n - k - 1}$$
where k = number of regressors. Penalizes additional regressors that do not improve fit.
Standard Errors:
$$SE(\hat{\beta}) = \sqrt{\hat{\sigma}^2 \cdot \text{diag}((X^T X)^{-1})}$$
where sigma_hat^2 = SS_res / (n - k - 1).
t-statistic:
$$t = \frac{\hat{\beta}_j}{SE(\hat{\beta}_j)}$$
Tests whether each coefficient is significantly different from zero.
In finance, the single-factor regression R_i - R_f = alpha + beta * (R_m - R_f) + epsilon is the CAPM regression, where alpha is the risk-adjusted excess return and beta is market sensitivity.
Normal Distribution: Symmetric, fully characterized by mean and variance. Used as a baseline model for returns, though real returns deviate from normality.
Log-Normal Distribution: If ln(X) is normal, then X is log-normal. Asset prices (not returns) are often modeled as log-normal, ensuring prices cannot go negative.
Student-t Distribution: Has heavier tails than the normal. Parameterized by degrees of freedom nu; lower nu means fatter tails. Commonly used to model financial returns more realistically. As nu -> infinity, converges to the normal.
Chi-Squared Distribution: The distribution of a sum of squared standard normal variables. Used in variance tests and as the sampling distribution of (n-1)*s^2 / sigma^2.
Non-parametric resampling technique for estimating the sampling distribution of a statistic.
Algorithm:
n, draw B bootstrap samples, each of size n, with replacement.B bootstrap statistics to estimate confidence intervals, standard errors, or bias.Confidence Interval (Percentile Method):
The (1 - alpha) confidence interval is given by the alpha/2 and 1 - alpha/2 percentiles of the bootstrap distribution.
Bootstrap is especially useful in finance when:
t-test (mean): Tests whether a sample mean differs significantly from a hypothesized value.
$$t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$$
with n - 1 degrees of freedom.
F-test (joint significance): Tests whether a group of regression coefficients are jointly zero. Used in multi-factor models.
$$F = \frac{(SS_{restricted} - SS_{unrestricted}) / q}{SS_{unrestricted} / (n - k - 1)}$$
where q = number of restrictions.
Jarque-Bera Test (normality): Tests whether sample skewness and kurtosis are consistent with a normal distribution.
$$JB = \frac{n}{6} \left(\gamma^2 + \frac{\kappa^2}{4}\right)$$
where gamma = sample skewness and kappa = sample excess kurtosis. Under the null of normality, JB follows a chi-squared distribution with 2 degrees of freedom. Financial return series almost always reject normality due to fat tails and skewness.
| Formula | Expression | Use Case |
|---|---|---|
| Sample Mean | x_bar = (1/n) * sum(x_i) | Central tendency |
| Sample Variance | s^2 = (1/(n-1)) * sum((x_i - x_bar)^2) | Dispersion |
| Annualized Volatility | sigma_annual = sigma_period * sqrt(periods_per_year) | Risk standardization |
| Skewness | gamma = E[(X-mu)^3] / sigma^3 | Asymmetry |
| Excess Kurtosis | kappa = E[(X-mu)^4] / sigma^4 - 3 | Tail thickness |
| Covariance | Cov(X,Y) = E[(X-mu_X)(Y-mu_Y)] | Co-movement |
| Correlation | rho = Cov(X,Y) / (sigma_X * sigma_Y) | Standardized co-movement |
| Shrinkage Estimator | Sigma_shrunk = delta*F + (1-delta)*Sigma_hat | Stable covariance matrix |
| OLS Coefficients | beta_hat = (X'X)^(-1) X'y | Linear regression |
| R-squared | 1 - SS_res / SS_tot | Model explanatory power |
| t-statistic | t = beta_hat_j / SE(beta_hat_j) | Coefficient significance |
| Jarque-Bera | JB = (n/6) * (gamma^2 + kappa^2/4) | Normality test |
Given: Monthly returns (in %) for a fund over 12 months:
[2.1, -0.5, 1.8, -3.2, 4.5, 0.3, -1.1, 2.7, -0.8, 3.4, 1.2, -0.6]
Calculate: Mean, volatility, skewness, excess kurtosis, and Jarque-Bera test statistic.
Solution:
Mean:
x_bar = (2.1 + (-0.5) + 1.8 + (-3.2) + 4.5 + 0.3 + (-1.1) + 2.7 + (-0.8) + 3.4 + 1.2 + (-0.6)) / 12
x_bar = 9.8 / 12
x_bar = 0.8167% per month
Annualized return (approximate): 0.8167% * 12 = 9.8%
Sample Standard Deviation:
Deviations from mean: [1.283, -1.317, 0.983, -4.017, 3.683, -0.517, -1.917, 1.883, -1.617, 2.583, 0.383, -1.417]
Squared deviations: [1.646, 1.734, 0.967, 16.133, 13.566, 0.267, 3.674, 3.547, 2.614, 6.674, 0.147, 2.007]
Sum of squared deviations = 52.977
s^2 = 52.977 / 11 = 4.816
s = sqrt(4.816) = 2.195% per month
Annualized volatility: 2.195% * sqrt(12) = 7.60%
Skewness:
Sum of cubed standardized deviations:
gamma = (1/n) * sum[((x_i - x_bar)/s)^3] (using adjusted formula for sample)
gamma approx 0.075 (slightly positive, near symmetric)
Excess Kurtosis:
kappa = (1/n) * sum[((x_i - x_bar)/s)^4] - 3
kappa approx -0.42 (platykurtic, lighter tails than normal)
Jarque-Bera Test:
JB = (12/6) * (0.075^2 + (-0.42)^2 / 4)
JB = 2 * (0.00563 + 0.04410)
JB = 2 * 0.04973
JB = 0.099
The JB critical value at 5% significance (chi-squared, df=2) is 5.99. Since 0.099 < 5.99, we fail to reject the null hypothesis of normality. With only 12 observations, however, the test has low power, and we should not conclude the data is truly normal.
Given: 24 monthly observations:
R_i - R_f): mean = 0.8%, std = 4.2%R_m - R_f): mean = 0.6%, std = 3.8%Calculate: CAPM alpha and beta, R-squared, and assess statistical significance.
Solution:
Beta:
beta = Cov(R_i, R_m) / Var(R_m)
= rho * sigma_i * sigma_m / sigma_m^2
= rho * sigma_i / sigma_m
= 0.85 * 4.2 / 3.8
= 0.939
Alpha:
alpha = mean(R_i - R_f) - beta * mean(R_m - R_f)
= 0.8% - 0.939 * 0.6%
= 0.8% - 0.564%
= 0.236% per month (approximately 2.84% annualized)
R-squared:
R^2 = rho^2 = 0.85^2 = 0.7225
72.25% of the fund's return variance is explained by the market factor.
Standard Error and t-statistic for alpha:
Residual std = sigma_i * sqrt(1 - R^2) = 4.2% * sqrt(1 - 0.7225) = 4.2% * 0.5268 = 2.213%
SE(alpha) = residual_std / sqrt(n) = 2.213% / sqrt(24) = 0.452%
t(alpha) = 0.236 / 0.452 = 0.522
With 22 degrees of freedom (n - 2), the critical t-value at 5% significance (two-tailed) is approximately 2.074. Since |0.522| < 2.074, the alpha is not statistically significant. Despite the positive point estimate, we cannot conclude the fund generates genuine risk-adjusted outperformance with this sample size.
Standard Error and t-statistic for beta:
SE(beta) = residual_std / (sigma_m * sqrt(n-1)) = 2.213% / (3.8% * sqrt(23)) = 2.213% / 18.226% = 0.121
t(beta) = 0.939 / 0.121 = 7.76
Since |7.76| >> 2.074, the beta is highly statistically significant, confirming the fund has meaningful market exposure.
n - 1 (Bessel's correction) in the denominator when estimating variance from a sample. Using n underestimates the true variance.p approaches or exceeds the number of observations n, the sample covariance matrix becomes singular or poorly conditioned. Apply Ledoit-Wolf shrinkage or factor-based covariance models to obtain stable, invertible matrices for portfolio optimization.See scripts/statistics_fundamentals.py for computational helpers.