Financial Correlation Modeling - Bottom-Up Approaches: formula sheet

Key formulas

Pearson correlation

ρ(X,Y) = Cov(X,Y) ÷ (σX × σY)

Lies between -1 and +1. Measures linear dependence only.

Covariance from correlation

Cov(X,Y) = ρ × σX × σY

Use this to build covariance matrices from volatilities and correlations.

Two-asset portfolio variance

σp² = w1²σ1² + w2²σ2² + 2 w1 w2 ρ σ1 σ2

Higher ρ means higher portfolio variance and less diversification.

Independence and correlation

Independent ⇒ ρ = 0, but ρ = 0 does not imply independent

Zero Pearson correlation can coexist with strong nonlinear dependence.

Mean-reverting correlation (discrete form)

ρ(t+1) − ρ(t) = a × (ρ_long-run − ρ(t)) + ε

a is the speed of reversion, 0 < a < 1. If ρ(t) is above the long-run level, the expected change is negative. Expected next value = ρ(t) + a × (ρ_long-run − ρ(t)).

Sample correlation

ρ = Cov(X, Y) ÷ (σX × σY)

Use it to see why a rise in covariance without a matching rise in volatilities means a rise in correlation.

Two-asset portfolio variance

σp² = w1²σ1² + w2²σ2² + 2 w1 w2 ρ σ1 σ2

Higher ρ gives higher portfolio variance, which is why rising correlation in downturns erodes diversification.

Direction summary

Recession ↑ ρ; high volatility ↑ ρ; falling equity market ↑ ρ; long run ρ → mean

These are empirical tendencies, not guarantees.

Correlation Brownian motion

dρ = σ dW

No drift, no mean reversion. Nothing keeps ρ inside [-1, +1] without a cap or transformation.

Ornstein-Uhlenbeck correlation

dρ = a(θ − ρ)dt + σ dW

θ = long-run mean, a = speed of mean reversion, σ = volatility of correlation.

OU expected correlation

E[ρ(t)] = θ + (ρ(0) − θ)e^(−a·t)

The gap to the mean shrinks exponentially at rate a.

Half-life of mean reversion

half-life = ln(2) ÷ a

Time for the expected gap to θ to halve. a is in the same time units as t.

DCC recursion

Q(t) = (1 − a − b)Q̄ + a·ε(t−1)ε(t−1)′ + b·Q(t−1)

ε are standardized residuals from GARCH. Needs a, b ≥ 0 and a + b < 1.

DCC correlation matrix

R(t) = diag(Q(t))^(−1/2) · Q(t) · diag(Q(t))^(−1/2)

Rescales Q so the diagonal equals 1. Off-diagonals are q(ij) ÷ √(q(ii)·q(jj)).

Sklar's theorem

F(x₁, x₂) = C(F₁(x₁), F₂(x₂))

C is the copula. F₁ and F₂ are the marginal CDFs. Copula and marginals are chosen separately.

Gaussian copula

C(u₁, u₂) = Φ₂(N⁻¹(u₁), N⁻¹(u₂); ρ)

Φ₂ is the bivariate standard normal CDF with correlation ρ. N⁻¹ is the inverse standard normal CDF.

Default threshold

Borrower defaults if Z_i < N⁻¹(PD_i)

Z_i is standard normal. Threshold depends only on the marginal PD.

One-factor model

Z_i = √ρ · M + √(1 − ρ) · ε_i

M and ε_i are independent standard normals. ρ is the pairwise copula correlation.

Conditional default probability

PD(M) = N[ (N⁻¹(PD) − √ρ · M) ÷ √(1 − ρ) ]

Given M, names default independently. Used to build the portfolio loss distribution.

Joint default probability

P(both default) = Φ₂(N⁻¹(PD₁), N⁻¹(PD₂); ρ)

If ρ = 0 this equals PD₁ × PD₂. Higher ρ raises it.

Tail dependence

Gaussian: 0 for ρ < 1. t copula: > 0

The key difference between the two copulas.

Joint default probability

P(A and B default) = pA × pB + ρD × √[pA(1 − pA)] × √[pB(1 − pB)]

ρD is the default correlation. If ρD = 0, defaults are independent.

Default correlation

ρD = [P(A and B) − pA × pB] ÷ {√[pA(1 − pA)] × √[pB(1 − pB)]}

Correlation of two default indicators over the same horizon.

One-factor Gaussian copula

Xi = √ρ × M + √(1 − ρ) × Zi

M and Zi are independent standard normals. ρ is the asset correlation.

Default threshold

Obligor i defaults if Xi < N⁻¹(PDi)

N⁻¹ is the inverse standard normal CDF.

Conditional PD

PD(M) = N[(N⁻¹(PD) − √ρ × M) ÷ √(1 − ρ)]

Defaults are independent once M is fixed. A low M means stress.

Portfolio expected loss

EL = Σ (PDi × LGDi × EADi)

Expected loss does not depend on correlation. Unexpected loss does.

Conditional default probability given another default

P(B | A) = P(A and B) ÷ P(A)

Useful for contagion-style questions.

Tranche width

Tranche width = Detachment point − Attachment point

Example: a 3%-7% tranche has width 4%. Thin tranches are more sensitive to the loss distribution.

Tranche loss

Tranche loss = min(max(L − A, 0), D − A), where L is pool loss, A attachment, D detachment

Loss is zero below A and capped at D − A. Express as a percentage of tranche width by dividing by (D − A).

Expected pool loss

E[L] = PD × (1 − R), as a fraction of pool notional

Assumes homogeneous names: equal notionals and a common PD and recovery rate R. Independent of correlation. Correlation changes only the distribution around this mean.

Correlation sensitivity rules

Equity: value ↑ as ρ ↑. Senior: value ↓ as ρ ↑. Mezzanine: mixed.

Equity is long correlation; senior is short correlation.

Base correlation decomposition

Tranche [A, D] = Equity(0, D) − Equity(0, A)

Each equity piece is priced with its own base correlation, taken from the calibrated base correlation curve.

Gaussian copula one-factor model

Ai = √ρ × M + √(1 − ρ) × Zi

M is the common factor, Zi the idiosyncratic factor. Name i defaults if Ai falls below a threshold set by its PD.

Quick revision

• Correlation measures co-movement; it is not causation and a single number cannot describe full dependence.
• Pearson correlation captures linear dependence only; rank measures capture monotonic dependence.
• Correlations tend to rise in market stress, which reduces diversification when you need it most.
• Correlation lies between -1 and +1, and its estimate depends on the horizon and data frequency used.
• A copula links marginal distributions to a joint distribution and separates dependence from the marginals.
• The Gaussian copula has no tail dependence, so it understates joint extreme events.
• Higher default correlation leaves the expected number of defaults unchanged but raises the variance of the loss distribution, making both very few and very many defaults more likely (a fatter right tail and more mass near zero).
• For the holder of the tranche (long the tranche, such as a protection seller), equity tranche value generally rises with higher default correlation and senior tranche value generally falls. A protection buyer on the same tranche has the opposite sign.
• Mezzanine tranche sensitivity to correlation is mixed and depends on attachment and detachment points.
• Bottom-up models build portfolio loss from single names; they depend heavily on the correlation input.
• Correlation estimated from calm periods can badly understate stress dependence.
• Always state the direction of the effect and the reason in one sentence before choosing an option.

Common mistakes

• Saying zero correlation means independence. Fix: Remember that Pearson captures only linear dependence. Independence implies zero correlation, not the reverse, except for jointly normal variables.
• Treating Pearson correlation and financial correlation as identical. Fix: Financial correlation is the broader concept covering any dependence between financial variables, including default correlation and implied correlation.
• Saying correlation is constant over time. Fix: Remember that estimates change with the sample period and market state. Fixed inputs are a modelling simplification.
• Treating mean reversion as immediate or certain. Fix: Reversion is gradual and governed by a speed parameter. Correlation can stay away from the mean for a long time.
• Treating correlation Brownian motion as mean-reverting. Fix: Brownian motion has no drift term. Only OU (and DCC through Q̄) pulls toward a long-run level.
• Saying the OU process guarantees correlation stays within -1 and +1. Fix: A Gaussian OU process is unbounded. Bounds need to be imposed through capping or a transformation.
• Saying the Gaussian copula has tail dependence Fix: For ρ < 1 the Gaussian copula has zero tail dependence. Only copulas such as the t copula have positive tail dependence.
• Treating copula correlation as the correlation between default events Fix: Copula correlation is between latent normal variables. Default-event correlation is smaller for low PDs.
• Treating asset correlation as default correlation. Fix: Check the wording. Asset correlation goes into the copula. Default correlation goes into the Bernoulli joint formula. Default correlation is usually lower.
• Using σ = p(1 − p) without the square root. Fix: Variance is p(1 − p). Standard deviation is √[p(1 − p)]. The formula needs standard deviations.

Exam tips

• Expect applied scenarios where a desk assumes constant correlation. The answer usually points to dynamic behaviour and correlation breakdown in stress.
• Know the word-level distinctions: linear versus nonlinear, static versus dynamic, observed versus implied.
• In calculations, check ρ's effect on variance before computing, so you can spot a wrong option quickly.
• For correlation risk, identify whether the position gains or loses when correlation rises.
• Expect scenario questions: a stress period is described and you must say what happens to correlation and diversification.
• Watch for absolute words like always or never in options. They are usually wrong.
• For mean reversion numbers, compute the gap to the long-run level first, then apply the speed.
• Link every answer to a risk consequence: understated VaR, weaker diversification or underestimated stress losses.