Study guide
5.4 Multivariate Random Variables
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
5.4 Multivariate Random Variables
Key Takeaways Explain This Topic AI
Joint discrete laws are probability matrices; marginals sum out other variables; conditionals renormalize a slice of the joint
Covariance measures co-movement; correlation standardizes covariance to [−1, 1]; uncorrelated does not imply independent except in special cases (e.g., jointly normal)
For Y = aX + b, means and covariances transform linearly; portfolio variance needs all pairwise covariances, not just variances
Var(w₁X₁ + w₂X₂) = w₁²σ₁² + w₂²σ₂² + 2 w₁ w₂ Cov(X₁, X₂)—the core diversification formula
Conditional expectation E[Y|X=x] is the minimum-MSE predictor of Y given X; i.i.d. sums scale means by n and variances by n (std. by √n)
Last updated: August 2026
Multivariate Random Variables
Market, credit, and operational risks move together. QA–4 develops joint distributions, dependence measures, and portfolio variance—the bridge from single-name math to book-level risk.
Joint Probability Matrices
For discrete RVs (X, Y), the joint PMF p(x, y) = P(X = x, Y = y) can be written as a matrix with rows indexed by x and columns by y. Entries are nonnegative and sum to 1.
Worked joint matrix
Let X = 1 if equity book loses money today, else 0. Let Y = 1 if credit book loses money, else 0.
- Y=0 Y=1 Row sum (marginal X)
- X=0 0.45 0.15 0.60
- X=1 0.20 0.20 0.40
- Col sum 0.65 0.35 1.00
So P(X=1, Y=1) = 0.20. Both books lose money on 20% of days in this toy history.
Marginals and Conditionals
Marginal of X: p_X(x) = Σ_y p(x, y). From the table, P(X=1) = 0.40. Marginal of Y: P(Y=1) = 0.35.
Conditional of Y given X=x: p(y|x) = p(x, y) / p_X(x). Given equity lost money (X=1):
P(Y=1|X=1) = 0.20 / 0.40 = 0.50,
P(Y=0|X=1) = 0.20 / 0.40 = 0.50.
Unconditionally P(Y=1) = 0.35, so equity losses raise the chance of credit losses from 35% to 50%—dependence is visible without computing covariance yet.
For continuous vectors, joint PDFs f(x, y) replace matrices; marginals integrate; conditionals are f(x, y)/f_X(x).
Covariance, Correlation, and Independence
Covariance: Cov(X, Y) = E[(X − μ_X)(Y − μ_Y)] = E[XY] − μ_X μ_Y.
Correlation: ρ(X, Y) = Cov(X, Y) / (σ_X σ_Y), defined when σ_X, σ_Y > 0. Always −1 ≤ ρ ≤ 1. Correlation is invariant to positive scale changes of each variable separately; Cov(aX, bY) = a b Cov(X, Y).
Independence means the joint factors: p(x, y) = p_X(x)p_Y(y) for all x, y (or f(x, y) = f_X f_Y). Independence ⇒ Cov = 0 (when variances exist), but zero covariance does not imply independence in general. Classic counterexamples exist with nonlinear dependence. Special case: if (X, Y) are bivariate normal , then uncorrelated ⇔ independent.
Worked covariance on the 0–1 matrix
E[X] = 0.40, E[Y] = 0.35, E[XY] = P(X=1, Y=1) = 0.20.
Cov(X, Y) = 0.20 − 0.40·0.35 = 0.20 − 0.14 = 0.06.
Var(X) = 0.40·0.60 = 0.24, Var(Y) = 0.35·0.65 = 0.2275.
ρ = 0.06 / √(0.24·0.2275) ≈ 0.06 / 0.2337 ≈ 0.257.
Positive correlation matches the conditional probability increase computed earlier.
- Concept Definition / test Common trap
- Independence Joint = product of marginals Equating to “ρ = 0” always
- Cov = 0 Uncorrelated Claiming independence
- ρ = ±1 Perfect linear relation Claiming identical outcomes if scales differ
- Bivariate normal Uncorrelated ⇔ independent Extending that equivalence to all laws
- Linear Transforms of Vectors
- If Z = aX + bY + c, then E[Z] = aE[X] + bE[Y] + c and
Var(Z) = a²Var(X) + b²Var(Y) + 2ab Cov(X, Y).
- For a portfolio return R_p = Σᵢ wᵢ Rᵢ,
- Var(R_p) = Σᵢ wᵢ² σᵢ² + Σᵢ≠ⱼ wᵢ wⱼ Cov(Rᵢ, Rⱼ) = wᵀ Σ w
in matrix form, where Σ is the covariance matrix. Diversification works when covariances are small or negative; in stress, correlations often rise and Var(R_p) jumps even if weights are unchanged.
Worked two-asset variance
σ₁ = 20%, σ₂ = 30%, ρ = 0.4, w₁ = w₂ = 0.5.
- Var(R_p) = (0.5)²(0.20)² + (0.5)²(0.30)² + 2(0.5)(0.5)(0.4)(0.20)(0.30)
- = 0.25·0.04 + 0.25·0.09 + 0.5·0.4·0.06
= 0.010 + 0.0225 + 0.012 = 0.0445.
σ_p = √0.0445 ≈ 21.1%.
Equal-weight average of standalone vols would be 25%; diversification cuts vol to ~21% here. If ρ → 1, σ_p → 0.5·20% + 0.5·30% = 25%. If ρ → 0, σ_p → √(0.010 + 0.0225) ≈ 18.0%.
Conditional Expectation
The conditional expectation E[Y|X = x] is the mean of the conditional distribution of Y given X = x. As a random variable, E[Y|X] is a function of X. It is the minimum mean-squared-error predictor of Y among functions of X. Tower property: E[E[Y|X]] = E[Y].
In the binary example, E[Y|X=1] = 0.50 and E[Y|X=0] = P(Y=1|X=0) = 0.15/0.60 = 0.25. Knowing X updates the expected credit-loss indicator.
Risk uses: forecasting losses given macro factors, expected exposure given market moves, and regression as linear conditional-mean modeling (later QA readings).
i.i.d. Sums
If X₁, …, Xₙ are i.i.d. with mean μ and variance σ², the sum Sₙ = Σᵢ Xᵢ satisfies E[Sₙ] = nμ and Var(Sₙ) = nσ², so Std(Sₙ) = σ√n. The sample mean X̄ = Sₙ/n has E[X̄] = μ and Std(X̄) = σ/√n—the √n risk-reduction law for independent risks.
Independence is essential for Var(Sₙ) = nσ². With pairwise correlation ρ among every pair (equal-correlation model),
Var(Sₙ) = nσ² + n(n−1)ρ σ² = nσ²[1 + (n−1)ρ],
which grows like n²ρ σ² when ρ > 0 is fixed—systematic risk that diversification cannot kill. That formula is why “many small independent loans” and “many loans tied to one factor” produce different capital answers.
Worked i.i.d. versus correlated
One-period losses on n = 100 exposures, each with σ = $1m.
Independent: Std(S) = 1·√100 = $10m.
Equal ρ = 0.2: Std(S) = √{100[1 + 99·0.2]} = √{100·20.8} = √2080 ≈ $45.6m.
Correlation multiplies portfolio risk far more than adding names reduces it.
Synthesis for Exam Day
Read joint tables carefully (marginals vs joints). Do not equate ρ = 0 with independence unless normality (or another listed special case) is given. For portfolio questions, write wᵀΣw or the two-asset expansion explicitly. Use conditional probabilities and E[Y|X] when information is revealed. For aggregation, ask whether risks are i.i.d. or factor-linked before applying √n rules.