Show what copula functions are for and how they are used in finance.
The idea
A copula is a function that joins separate marginal distributions into one joint distribution while carrying all of the dependence between the variables. Call the marginal cumulative probabilities of the two variables u and v, each between 0 and 1, and let C be the copula. Sklar's theorem says the joint cumulative probability can always be written as follows. Joint = C(u, v). This lets a risk manager calibrate each asset's own distribution, for example a default time or a fat-tailed return, and then choose the dependence structure separately. Bottom-up credit models use copulas to turn individual default probabilities into joint default probabilities for CDO tranches and portfolio loss. Useful anchors follow. Independence gives a joint probability equal to u × v. Perfect positive dependence gives the smaller of u and v. The joint probability always lies between max(u + v − 1, 0) and min(u, v). Correlation alone does not fix a joint distribution unless the variables are jointly normal, which is why copulas are needed.
Formulas
Joint distribution by Sklar's theorem
F(x₁, x₂) = C(F₁(x₁), F₂(x₂))
- C
- the copula, carrying all the dependence
- F₁
- the marginal cumulative distribution of the first variable
- F₂
- the marginal cumulative distribution of the second variable
- x₁
- the value of the first variable
- x₂
- the value of the second variable
Any joint distribution can be written this way. The copula takes marginal probabilities between 0 and 1, not raw values, and the marginals stay as separately modelled.
A practice question
What is the main purpose of a copula function in financial modelling?
- To join separately specified marginals into a joint distribution with a chosen dependence. - correct
- To estimate the volatility of each asset from a long window of historical return data.Why not: Confuses a copula with a marginal volatility estimator; a copula contains no information about each variable's own distribution.
- To replace the marginal distributions of all the variables with a single standard normal distribution.Why not: Treats the copula as a normalising transformation of the marginals; the marginals are kept and only linked.
- To measure the average linear correlation between two assets over a fixed window.Why not: Reduces a copula to a Pearson correlation estimate; a copula describes the whole dependence structure, not one coefficient.
By Sklar's theorem a joint distribution equals a copula applied to the marginal cumulative distributions. The marginals stay as modelled and the copula supplies the dependence between them.
A harder one
The one-year default probabilities of two borrowers are 3% and 5%. Whatever copula links them, the highest joint default probability consistent with these marginals is:
- 5.0%.Why not: Takes the larger marginal; the upper bound min(u, v) uses the smaller one, because both cannot default more often than the rarer one.
- 0.15%.Why not: Multiplies 3% and 5%, which is the independence copula, not the upper bound.
- 3.0%. - correct
- 8.0%.Why not: Adds the two probabilities; that is an upper bound on the chance at least one defaults, not on both.
The Frechet upper bound is min(u, v) = min(3%, 5%) = 3%, reached by perfect positive dependence. The lower bound is max(u + v - 1, 0) = 0, and independence gives 0.15%.
The hardest tier
Two obligors have one-year default probabilities of 5% and 10%. A copula gives a joint default probability C(0.05, 0.10) = 0.02. The probability that neither defaults within the year is closest to:
- 85.5%.Why not: Multiplies the survival probabilities (0.95 x 0.90), which assumes independence and ignores the copula's joint default probability.
- 87.0%. - correct
- 85.0%.Why not: Computes 1 - 0.05 - 0.10 and forgets to add back the joint default probability, so the both-default case is subtracted twice.
- 98.0%.Why not: Computes 1 - 0.02, which excludes only the case where both default, not the cases where one defaults.
P(neither) = 1 - P(A) - P(B) + P(both) = 1 - 0.05 - 0.10 + 0.02 = 0.87. Inclusion-exclusion: the both-default case was removed twice, so it is added back once.
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