Quantitative analysis forms 20% of the FRM Part 1 exam. Strong quantitative foundations are also essential for understanding risk models in Part 2.

Key Statistical Concepts

Probability Distributions

  • Normal Distribution: Bell-shaped, symmetric, fully defined by mean and variance
  • Lognormal Distribution: Used for asset prices (can't go negative)
  • Student's t-Distribution: Heavier tails than normal, used with small samples
  • Chi-Squared Distribution: Used in hypothesis testing of variance
  • Poisson Distribution: Models the frequency of events (e.g., operational risk)

Moments of a Distribution

  1. Mean: Central tendency
  2. Variance/Std Dev: Dispersion — how spread out the data is
  3. Skewness: Asymmetry — negative skew means left tail is longer
  4. Kurtosis: Tail heaviness — excess kurtosis > 0 means fatter tails than normal

Hypothesis Testing

Key Steps

  1. State null and alternative hypotheses
  2. Choose significance level (α)
  3. Calculate test statistic
  4. Compare to critical value or p-value
  5. Make decision (reject or fail to reject H₀)

Common Tests

  • t-test: Testing means with unknown population variance
  • F-test: Testing equality of variances or overall regression significance
  • Chi-squared test: Testing variance or independence

Type I and Type II Errors

  • Type I (α): Rejecting H₀ when it's true (false positive)
  • Type II (β): Failing to reject H₀ when it's false (false negative)
  • Power = 1 - β: Probability of correctly rejecting a false H₀

Regression Analysis

Linear Regression

  • Y = α + βX + ε
  • β = Cov(X,Y) / Var(X)
  • R² measures proportion of variance explained
  • Assumptions: linearity, independence, normality, homoscedasticity

Multiple Regression

  • Y = α + β₁X₁ + β₂X₂ + ... + ε
  • Adjusted R² penalizes for additional variables
  • F-test for overall significance, t-tests for individual coefficients
  • Watch for multicollinearity

Time Series Analysis

No autocorrelation

A stationary process has constant mean, variance, and autocovariance over time. Most financial time series are non-stationary.

AR, MA, and ARMA Models

  • AR(1): Y_t = φY_{t-1} + ε_t (today depends on yesterday)
  • MA(1): Y_t = ε_t + θε_{t-1} (today depends on yesterday's shock)
  • ARMA: Combines both

GARCH Models

For modeling time-varying volatility (volatility clustering):

  • σ²_t = ω + α×r²_{t-1} + β×σ²_{t-1}
  • Essential for VaR estimation and risk modeling

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