Study guide
6.2 Hypothesis Testing
FRM Part I Financial Risk Manager Study Guide Exam
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1 Introduction & Exam Overview
1.1 Current Exam Facts 1.2 Study Strategy & Domain Weights
2 Foundations: Risk Building Blocks & Governance
2.1 Building Blocks of Risk Management 2.2 How Firms Manage Financial Risk 2.3 Governance of Risk Management 2.4 Credit Risk Transfer Mechanisms
3 Foundations: Portfolio Models, Data & ERM
3.1 MPT and the CAPM 3.2 APT and Multifactor Models 3.3 Risk Data Aggregation & Reporting 3.4 Enterprise Risk Management
4 Foundations: Disasters, Crisis & Ethics
4.1 Learning from Financial Disasters 4.2 Anatomy of the Great Financial Crisis 4.3 GARP Code of Conduct
5 Quantitative Analysis: Probability Foundations
5.1 Fundamentals of Probability 5.2 Random Variables 5.3 Common Univariate Distributions 5.4 Multivariate Random Variables
6 Quantitative Analysis: Inference & Regression
6.1 Sample Moments 6.2 Hypothesis Testing 6.3 Linear Regression 6.4 Multiple Regression
7 Quantitative Analysis: Diagnostics & Time Series
7.1 Regression Diagnostics 7.2 Stationary Time Series 7.3 Non-Stationary Time Series 7.4 Returns, Volatility & Correlation
8 Quantitative Analysis: Simulation & Machine Learning
8.1 Simulation and Bootstrapping 8.2 Machine-Learning Methods 8.3 Machine Learning and Prediction
9 Financial Markets: Institutions & Derivatives Intro
9.1 Banks 9.2 Insurance Companies and Pension Plans 9.3 Fund Management 9.4 Introduction to Derivatives
10 Financial Markets: Clearing, Futures & Hedging
10.1 Exchanges and OTC Markets 10.2 Central Clearing 10.3 Futures Markets 10.4 Using Futures for Hedging
11 Financial Markets: FX, Forwards, Commodities & Options
11.1 Foreign Exchange Markets 11.2 Pricing Financial Forwards and Futures 11.3 Commodity Forwards and Futures 11.4 Options Markets
12 Financial Markets: Option Properties, Strategies & Rates
12.1 Properties of Options 12.2 Trading Strategies 12.3 Exotic Options 12.4 Properties of Interest Rates
13 Financial Markets: Bonds, MBS, Rate Futures & Swaps
13.1 Corporate Bonds 13.2 Mortgages and Mortgage-Backed Securities 13.3 Interest Rate Futures 13.4 Swaps
14 Valuation & Risk Models: VaR, Volatility & Ratings
14.1 Measures of Financial Risk 14.2 Calculating and Applying VaR 14.3 Measuring and Monitoring Volatility 14.4 External and Internal Credit Ratings
15 Valuation & Risk Models: Credit, OpRisk & Stress
15.1 Country Risk 15.2 Measuring Credit Risk 15.3 Operational Risk 15.4 Stress Testing
16 Valuation & Risk Models: Fixed-Income Valuation
16.1 Pricing, Discounting & Arbitrage 16.2 Interest Rates 16.3 Bond Yields and Return Calculations 16.4 Duration, Convexity & DV01
17 Valuation & Risk Models: Term Structure, Options & Greeks
17.1 Non-Parallel Term Structure Shifts 17.2 Binomial Trees 17.3 The Black-Scholes-Merton Model 17.4 Option Sensitivity Measures: The Greeks
6.2 Hypothesis Testing
Key Takeaways Explain This Topic AI
H₀ is the maintained claim tested for possible rejection; H₁ is the alternative that defines one-sided versus two-sided critical regions
Type I error rejects a true H₀ (size α); Type II fails to reject a false H₀; power = 1 − β rises with effect size, n, and α
A (1 − α) confidence interval that excludes the null value is equivalent to a two-sided level-α rejection for the usual mean tests
The p-value is the smallest α at which the observed test statistic would reject H₀; it is not the probability that H₀ is true
Two-mean tests and multiple comparisons need correct variance formulas and error-rate control—unadjusted repeated testing inflates false discoveries
Last updated: August 2026
Hypothesis Testing
Risk managers and FRM candidates continually ask: is this mean return different from zero? Did two desks’ loss rates change? Is a hedge ratio equal to one? Hypothesis testing turns those questions into a controlled error framework.
Null and Alternative Hypotheses
Null hypothesis H₀ : the baseline claim (often “no effect,” “no difference,” or a regulatory threshold). You either reject H₀ or fail to reject H₀—you do not “prove” H₀ true.
Alternative H₁ (or H_a) : what you entertain when H₀ looks untenable.
Examples:
H₀: μ = 0 versus H₁: μ ≠ 0 (two-sided alpha test for mean excess return).
H₀: μ ≤ 0 versus H₁: μ > 0 (one-sided; only large positive t-stats reject).
The alternative determines the rejection region . Mixing up one-sided versus two-sided critical values is a frequent exam error.
One-Sided Versus Two-Sided Tests
Two-sided (H₁: θ ≠ θ₀): reject for very small or very large test statistics. For a normal/z test at α = 0.05, critical values are about ±1.96.
One-sided (H₁: θ > θ₀ or θ < θ₀): all of α sits in one tail. At α = 0.05, the critical z is about +1.645 (upper) or −1.645 (lower).
Use one-sided only when the research or risk question truly rules out the other direction before seeing the data. Choosing the side after peeking at the sign of the estimate inflates Type I error.
- Setup H₀ example H₁ α = 0.05 critical z (σ known)
- Two-sided μ = 0 μ ≠ 0 ± ±1.96
- Upper one-sided μ ≤ 0 μ > 0 reject if z ≥ 1.645
- Lower one-sided μ ≥ 0 μ < 0 reject if z ≤ −1.645
- Type I, Type II, Size, and Power
Type I error : reject H₀ when H₀ is true. Size (significance level) α = P(Type I).
Type II error : fail to reject H₀ when H₁ is true. β = P(Type II) depends on the specific alternative, n, and α.
Power = 1 − β = P(reject H₀ | specific H₁ is true).
Trade-off: lowering α (fewer false alarms) typically raises β (more misses) unless you increase sample size. Larger |μ − μ₀|, larger n, and smaller σ all raise power.
Worked example: size versus power intuition
Test H₀: μ = 0 vs H₁: μ > 0 with z = X̄ / (σ/√n), σ = 10, n = 25, α = 0.05. Critical value: X̄ ≥ 1.645 × (10/5) = 3.29.
If the true mean is μ = 5, then X̄ ~ approx N(5, 2²). Power ≈ P(X̄ ≥ 3.29 | μ = 5) = P(Z ≥ (3.29 − 5)/2) = P(Z ≥ −0.855) ≈ 0.80. If instead true μ = 2, power falls because 2 is closer to the null—same α, harder to detect.
Confidence Intervals and Tests
For the usual two-sided test of H₀: μ = μ₀ with known σ (z-test):
(1 − α) CI: x̄ ± z_{α/2} · (σ/√n)
Equivalence : reject H₀ at level α ⟺ μ₀ lies outside the (1 − α) CI.
If a 95% CI for mean daily P&L is [0.4, 1.2], then H₀: μ = 0 is rejected at 5% two-sided; H₀: μ = 0.7 is not. This link is a high-yield FRM fact for mean tests and coefficient tests in regression.
Worked calculation
n = 36, x̄ = 2.0, σ = 6, α = 0.05 two-sided.
SE = 6/6 = 1. CI: 2 ± 1.96 × 1 = [0.04, 3.96].
z for H₀: μ = 0 is (2 − 0)/1 = 2.0 > 1.96 → reject. Indeed 0 is outside the CI. For H₀: μ = 1, z = 1.0 < 1.96 → do not reject; 1 ∈ [0.04, 3.96].
p-Values
The p-value is the probability, computed under H₀, of obtaining a test statistic at least as extreme (toward H₁) as the one observed.
Small p-value → data are surprising under H₀ → reject at common α levels if p ≤ α.
p-value is not P(H₀ true | data).
p-value is not the probability of Type I error for this specific experiment after seeing the data; α is chosen before the test as the long-run Type I rate under H₀.
For two-sided z = 2.0, p ≈ 2 × P(Z ≥ 2) ≈ 2 × 0.0228 = 0.0456 < 0.05 → reject at 5%. For one-sided upper test with the same z = 2.0, p ≈ 0.0228.
Tests for Two Means
Independent samples , common known variances (or large-n with estimated SEs):
H₀: μ_X − μ_Y = 0, z = (x̄ − ȳ) / √(σ_X²/n_X + σ_Y²/n_Y)
If variances unknown but assumed equal, use a pooled-variance t-statistic; if unequal, use Welch-style SE. FRM questions often give SEs or variances explicitly—plug in carefully.
Paired samples (same firms before/after, matched books): work with differences D_i = X_i − Y_i and run a one-sample test on μ_D. Pairing removes shared noise and can raise power.
Worked example: two independent means
Desk A: n = 40, x̄ = 1.5, s = 2. Desk B: n = 40, ȳ = 0.5, s = 2. Treat s as σ for a large-sample z.
SE(diff) = √(4/40 + 4/40) = √0.2 ≈ 0.447.
z = (1.5 − 0.5)/0.447 ≈ 2.24.
Two-sided p ≈ 0.025 → reject equal means at 5%. A 95% CI for μ_A − μ_B: 1.0 ± 1.96 × 0.447 ≈ [0.12, 1.88], which excludes 0—same decision.
Multiple Testing Bias
If you run m independent tests each at α = 0.05, and all nulls are true, the probability of at least one false rejection is 1 − (1 − α)^m ≈ mα for small α (union bound / Bonferroni intuition).
- m tests (true nulls) Approx. P(≥1 false reject) at α = 0.05
- 1 0.05
- 10 ~0.40
- 20 ~0.64
- 100 ~0.994
Multiple testing bias (data mining / p-hacking): scanning many desks, factors, or cutoffs and reporting only “significant” hits produces a flood of Type I errors. Mitigations include Bonferroni (use α/m), FDR control, pre-registration of hypotheses, and out-of-sample validation—central in quant research and model validation culture.
Worked sketch
A researcher tests 40 factor premia at α = 0.05 with no adjustment. Expected false positives under global null ≈ 40 × 0.05 = 2. Finding “two significant factors” is unsurprising noise. Bonferroni would require p ≤ 0.05/40 = 0.00125.
Practical FRM Checklist
State H₀/H₁ and whether the test is one- or two-sided before computing the statistic.
Choose α (size) knowing the Type I / Type II trade-off.
Compute the statistic and p-value; cross-check with the CI when testing a point null.
For two groups, pick independent vs paired formulas deliberately.
If many hypotheses were tried, discount unadjusted significance.
Hypothesis testing does not replace economic judgment: a statistically significant 0.1 bp mean difference may be irrelevant, and a noisy but large risk difference may fail to reject with short samples. Still, size, power, p-values, and CI equivalence are the grammar of quantitative FRM inference.