7.4 Returns, Volatility & Correlation

Returns, Volatility & Correlation

Pricing P&L, VaR, and portfolio risk all start from how we define returns, scale volatility, and measure co-movement. This section locks the definitions GARP expects and the traps that mix linear correlation with deeper dependence.

Simple Versus Continuously Compounded Returns

Let Pₜ be the price (or index level).

Simple (net, arithmetic) return:

Rₜ = (Pₜ − Pₜ₋₁) / Pₜ₋₁ = Pₜ / Pₜ₋₁ − 1.

Continuously compounded (log) return:

rₜ = ln(Pₜ / Pₜ₋₁) = ln(1 + Rₜ).

For small R, r ≈ R, but they differ in aggregation:

• Across time: log returns add. Multi-period log return r₁ + r₂ + … + rₕ = ln(P_{t+h}/Pₜ).
• Across assets: simple returns of a portfolio are value-weighted averages of component simple returns. Log returns of components do not average to the portfolio log return in general.

Worked example

P₀ = 100, P₁ = 110, P₂ = 100.

Simple: R₁ = 0.10, R₂ = 100/110 − 1 ≈ −0.0909. Holding-period simple return R₀→₂ = 0. But R₁ + R₂ ≈ 0.0091 ≠ R₀→₂ — simple returns do not add across time.

Log: r₁ = ln(1.1) ≈ 0.0953, r₂ = ln(100/110) ≈ −0.0953, and r₁ + r₂ = 0 = ln(P₂/P₀). Log returns telescope across time exactly.

Volatility, Variance Rate, and Implied Volatility

Volatility in risk management usually means the standard deviation of returns over a stated horizon (often annualized). If daily log-return SD is σ_d, a common annualization (≈252 trading days, i.i.d. assumption) is σ_ann = σ_d √252.

Variance is σ². The variance rate often means variance per unit time (e.g., daily variance), useful in EWMA/GARCH recursions where the model updates variance and quotes volatility as its square root.

Implied volatility is the σ plugged into an option pricing model (e.g., Black–Scholes) that matches the observed market price. It is a market forecast / quote convention, not the same object as historical realized volatility—though they are compared constantly.

Worked annualization

Daily σ_d = 0.01. Under i.i.d., σ_ann = 0.01 × √252 ≈ 0.1587 (15.9%). Variance annualizes with 252, not √252: daily variance 0.0001 → annual variance ≈ 0.0252.

Jarque–Bera Normality Test

Normal returns would have skewness S = 0 and excess kurtosis K = 0 (kurtosis 3). The Jarque–Bera statistic

JB = (n/6) (S² + (K²)/4)

(with K = excess kurtosis) is compared to χ² with 2 degrees of freedom. Large JB → reject normality.

Financial returns typically show mild negative skew and positive excess kurtosis → JB rejects in realistic sample sizes. That undermines Gaussian VaR and motivates t-distributions, mixtures, or historical/simulation methods.

Worked JB sketch

n = 500, S = −0.4, excess K = 1.5. JB = (500/6)(0.16 + (2.25)/4) = (83.333)(0.16 + 0.5625) = 83.333 × 0.7225 ≈ 60.2. χ²₂ 1% critical ≈ 9.21 → strong rejection.

Power-Law Tails

A power-law (Pareto-type) tail satisfies, for large x,

P(|X| > x) ≈ C / x^α

for some tail index α > 0. Compared with a normal (tails decay like exp(−x²/(2σ²))), power laws put far more mass on extremes when α is moderate. If α ≤ 2, variance may be infinite; if α ≤ 1, mean may be infinite—critical for whether sample variance/VaR estimators behave.

Risk practice: estimate α with tail regressions or peaks-over-threshold methods; do not assume finite-variance CLT intuition for the worst operational or market losses when tails are heavy.

Correlation Versus Dependence

Pearson correlation ρ = Cov(X,Y) / (σ_X σ_Y) measures linear co-movement only. Properties: |ρ| ≤ 1; ρ = 0 for uncorrelated variables; for jointly normal vectors, uncorrelated implies independent. Outside elliptical/Gaussian worlds, ρ = 0 need not mean independence—X² and X can be uncorrelated with X under symmetry yet dependent.

Dependence is the broader concept: any departure from independence (copulas, rank correlations, mutual information, tail dependence coefficients). Two assets can have modest Pearson ρ but explode together in crises (tail dependence)—exactly when diversification fails.

One-Factor Correlation Structures

A common equicorrelation / one-factor setup: each standardized residual Xᵢ = √ρ F + √(1−ρ) εᵢ with common factor F and idiosyncratic εᵢ, all variance 1 and independent across i, with 0 ≤ ρ ≤ 1 for the usual positive factor loading story. Then Corr(Xᵢ, Xⱼ) = ρ for i ≠ j.

Valid correlation matrix constraints matter. For an n × n equicorrelation matrix with off-diagonals ρ, positive definiteness requires

−1/(n − 1) < ρ < 1.

So ρ cannot be too negative when n is large: you cannot have every pair strongly negatively correlated simultaneously. For n = 3, ρ > −0.5; for n = 11, ρ > −0.1.

Worked PSD bound

n = 6 assets, equicorrelation ρ. Need ρ > −1/5 = −0.2. Setting all pairwise correlations to −0.5 is impossible for n = 6—the matrix would not be positive semidefinite, and a risk system would fail Cholesky / eigenvalue checks.

Factor models automatically build PSD matrices when factor and idio variances are valid—another reason risk systems prefer factor correlation structures over unconstrained sample matrices when n is large.

Dependence Measures Beyond Pearson

• Spearman’s ρ: Pearson correlation of ranks—invariant to monotone transforms.
• Kendall’s τ: probability of concordant vs discordant pairs.
• Tail dependence: lim_{u→1} P(U > u | V > u) for uniform margins—central in stress.
• Copulas: separate marginal distributions from dependence; Gaussian copula has zero tail dependence, t-copula can have positive tail dependence even with moderate ρ.

Worked dependence trap

Let Z ~ N(0,1), X = Z, Y = Z². Corr(X,Y) = 0 by symmetry, but Y is a deterministic function of X—maximal dependence in a nonlinear sense. Reporting “uncorrelated hence independent” would be false and dangerous for risk limits.

FRM Synthesis

Choose simple vs log returns for the aggregation you need; keep volatility, variance, and implied vol distinct; expect JB to reject normality; respect power-law extremes; and never confuse a single Pearson ρ with a complete dependence story—especially under one-factor PSD constraints and crisis tail dependence.