Why Separate Marginals From Dependence?
A multivariate loss model must describe two different features. Each marginal distribution describes the behavior of one risk considered on its own; the dependence structure describes how the risks move together. These are not interchangeable. The same set of marginal loss distributions can be paired with different dependence structures and therefore produce different joint probabilities.
Sklar's theorem gives a precise way to make this separation. It represents a joint distribution as a function of its marginal distributions and a copula. The result is useful when a modeler wants to retain specified marginal loss behavior while changing or assessing the dependence assumption. It also clarifies an important qualification: a copula is uniquely determined by the joint distribution when the margins are continuous, but not in general when they have jumps.
The margins specify the scales and individual loss behavior. The copula operates on the marginal cumulative probabilities, placing the risks' relative positions in their respective distributions into a common unit interval. The theorem does not say that a copula is a correlation coefficient, nor that a correlation coefficient determines a copula.
Sklar's Theorem
For continuous margins, uniqueness follows directly from the ability to invert each marginal distribution. In the bivariate case, for example, at a point \((u,v)\) choose \(x,y\) such that \(F_1(x)=u\) and \(F_2(y)=v\). Then the only possible value is \(C(u,v)=F(x,y)\). Continuity ensures that marginal cumulative probabilities cover the unit interval, including by limits at endpoints. When a margin has a jump, its cumulative distribution function skips an interval of probability values; the joint distribution does not directly specify the copula throughout that skipped region.
Why the Representation Holds
The continuous-margin argument also gives the main idea behind the theorem. If \(X_i\) has continuous distribution function \(F_i\), then \(F_i(X_i)\) is uniform on \([0,1]\). The joint distribution of these transformed variables is a copula. For any \(x_i\), the events \(X_i\leq x_i\) and \(F_i(X_i)\leq F_i(x_i)\) agree up to events of probability zero. Therefore their joint probability is both the original joint distribution evaluated at \((x_1,\ldots,x_d)\) and the copula evaluated at the corresponding marginal probabilities.
For discontinuous margins, the same existence result can be obtained by spreading the probability at each atom uniformly across the interval skipped by its marginal distribution function. The resulting transformed variables have uniform margins; their joint distribution supplies a copula representation. However, the original distribution only fixes the copula values at the marginal cumulative probabilities that actually occur. Values elsewhere can vary while leaving the original joint distribution unchanged.
The converse is also important. Starting with margins \(F_i\) and a copula \(C\), the function \(C(F_1(x_1),\ldots,F_d(x_d))\) has the specified marginal distributions: setting all arguments but one to 1 leaves that margin, because a copula's one-dimensional margins are uniform. It is a valid joint distribution because the copula assigns nonnegative probability to rectangles, and nondecreasing marginal transformations preserve that property. Thus the representation is both a description of existing joint distributions and a way to specify a joint distribution from components.
Worked Example: Recovering Dependence With Continuous Loss Margins
Suppose hypothetical annual losses \(X\) and \(Y\) have exponential marginal distributions with rates 1 and 2, respectively. Their joint model is specified by the copula \(C_\theta(u,v)=uv[1+\theta(1-u)(1-v)]\), with \(\theta=0.8\). Find the joint probability that \(X\leq\ln 2\) and \(Y\leq(\ln 2)/2\), and verify the marginal distributions and validity of the specified copula.
Method and marginal check. For \(x,y\geq0\), the specified marginal distribution functions are \(F_X(x)=1-e^{-x}\) and \(F_Y(y)=1-e^{-2y}\). The proposed dependence function has uniform margins: \(C_\theta(u,1)=u\) and \(C_\theta(1,v)=v\). Its density on the unit square is \(1+\theta(1-2u)(1-2v)\). Since \((1-2u)(1-2v)\) lies between \(-1\) and 1, for \(\theta=0.8\) this density is at least \(0.2\). It is therefore nonnegative, as required for this copula.
Evaluate the marginal probabilities. \(F_X(\ln 2)=1-e^{-\ln 2}=1-\frac12=0.5\). Also, \(F_Y((\ln 2)/2)=1-e^{-2(\ln 2)/2}=1-e^{-\ln 2}=0.5\).
Apply the copula. \(C_{0.8}(0.5,0.5)=0.5(0.5)[1+0.8(1-0.5)(1-0.5)]\) \(=0.25[1+0.8(0.25)]=0.25(1.2)=0.30\). Thus the joint probability is \(0.3000\).
Check and common mistake. Under independence the answer would be \(0.5(0.5)=0.25\), not \(0.30\). The two margins alone do not establish independence; using their product would discard the specified copula. The result is a valid probability between zero and both marginal probabilities, as it must be.
Uniqueness and Discrete Margins
With discrete losses, the copula is generally not uniquely recoverable over the whole unit cube. This is not a contradiction of Sklar's theorem: the theorem guarantees a copula representation, while uniqueness is restricted to the marginal cumulative probability values that the distributions attain. Different copulas can agree at all the points needed to reproduce the same discrete joint distribution and disagree elsewhere.
Worked Example: Two Copula Extensions for the Same Binary Losses
Consider hypothetical losses \(X,Y\in\{0,1\}\), each equal to 1 with probability \(0.5\). Their joint probabilities are \(P(X=0,Y=0)=0.4\), \(P(X=0,Y=1)=0.1\), \(P(X=1,Y=0)=0.1\), and \(P(X=1,Y=1)=0.4\). Show why these data do not uniquely identify the copula.
Find the copula values required by the joint distribution. Both marginal distribution functions equal \(0\) below 0, \(0.5\) from 0 up to but not including 1, and \(1\) at and above 1. At the only nontrivial pair of attained marginal values, \(C(0.5,0.5)=P(X\leq0,Y\leq0)=0.4\). The margins and boundary conditions fix the other required values. In particular, copula values with either coordinate 0 are 0, and \(C(u,1)=u,\ C(1,v)=v\).
Construct two different valid extensions. Let \(M(u,v)=\min(u,v)\), \(\Pi(u,v)=uv\), and define \(D(u,v)=uv+\frac{1}{8\pi^2}\sin(2\pi u)\sin(2\pi v)\). The mixed derivative of \(D\) is \(1+\frac12\cos(2\pi u)\cos(2\pi v)\), which is at least \(0.5\); its boundary values give uniform margins. Thus \(D\) is a copula. Both \(C_1=0.6M+0.4\Pi\) and \(C_2=0.6M+0.4D\) are also copulas, as mixtures of copulas.
At \((0.5,0.5)\), \(M=0.5\), \(\Pi=0.25\), and \(D=0.25\), so \(C_1(0.5,0.5)=C_2(0.5,0.5)=0.6(0.5)+0.4(0.25)=0.4\). They agree at all required boundary points as well. Yet at \((0.25,0.25)\), \(C_1=0.6(0.25)+0.4(0.0625)=0.175\), whereas \(C_2=0.175+\frac{1}{20\pi^2}\), which differs from \(0.175\).
Conclusion and common mistake. Both copulas reproduce the stated binary joint distribution because the margins only attain 0, 0.5, and 1. Their disagreement at \((0.25,0.25)\) does not change that distribution. It would be incorrect to claim that the joint binary data identify one unique copula across the full unit square. A particular extension may be selected for a model, but the data in this example do not uniquely determine it.
Using the Copula for Joint Events
Once the joint distribution is represented by its margins and copula, probabilities of rectangles can be calculated by inclusion-exclusion. For continuous losses with joint distribution \(F(x,y)=C(F_X(x),F_Y(y))\), the probability that both losses fall in specified intervals is the alternating sum of the joint cumulative probabilities at the four corners. This is a consequence of subtracting the portions below the lower boundaries from the probability below the upper boundaries.
Worked Example: A Joint Loss Rectangle Under Dependence
Use the exponential margins and the copula \(C_{0.8}\) from the first example. Find the probability that \(X\) is between the points where its marginal cumulative probabilities are 0.2 and 0.6, and \(Y\) is between the points where its marginal cumulative probabilities are 0.2 and 0.6. Boundaries are included at the upper ends and excluded at the lower ends.
Method. The four corner values use \(u,v\in\{0.2,0.6\}\). Direct substitution gives \(C_{0.8}(0.6,0.6)=0.36[1+0.8(0.4)(0.4)]=0.40608\). Likewise, \(C_{0.8}(0.2,0.6)=0.12[1+0.8(0.8)(0.4)]=0.15072\), and by symmetry \(C_{0.8}(0.6,0.2)=0.15072\). Finally, \(C_{0.8}(0.2,0.2)=0.04[1+0.8(0.8)(0.8)]=0.06048\).
Apply inclusion-exclusion. The requested probability is \(0.40608-0.15072-0.15072+0.06048=0.16512\). To four decimal places, the probability is \(0.1651\).
Check and common mistake. If the risks were independent, the probability would be \((0.6-0.2)(0.6-0.2)=0.16\). The specified dependence changes it to \(0.16512\). A common error is to use \(C(0.6,0.6)\) alone; that is the probability below both upper bounds, not the probability of the rectangle. The four-corner calculation correctly removes the regions below either lower bound and adds back their overlap.
What the Separation Does—and Does Not—Tell You
The marginal distributions answer questions about each risk separately. The copula captures the remaining joint structure needed to calculate multivariate probabilities. If a marginal distribution is changed while the copula is held fixed, the dependence description on the probability scale is retained, but the losses' joint distribution on the original dollar scales changes. Conversely, holding the margins fixed and changing the copula changes joint behavior without changing either single-risk distribution.
This representation is especially important in risk aggregation: a model of individual loss distributions does not, by itself, determine the distribution of the combined losses. The copula is one way to specify the dependence needed for that joint model. Its selection and calibration are separate modeling questions; Sklar's theorem establishes the representation, not which dependence model is appropriate.
Check Your Understanding
For each problem, identify which information comes from the margins and which comes from the copula.
- A continuous bivariate loss model has \(F_X(a)=0.3\), \(F_Y(b)=0.6\), and \(C(0.3,0.6)=0.22\). Find \(P(X\leq a,Y\leq b)\). (Answer: 0.2200)
- Two continuous margins are paired with an independence copula. If \(F_X(x)=0.4\) and \(F_Y(y)=0.7\), calculate \(P(X\leq x,Y\leq y)\). (Answer: 0.2800)
- A joint distribution has continuous margins. Two proposed copulas both represent it but differ at a point in the unit square. Is this possible? State the relevant condition. (Answer: No; with continuous margins, the copula is unique on the unit square.)
- For continuous margins, suppose \(C(0.7,0.5)=0.38\), \(C(0.2,0.5)=0.12\), \(C(0.7,0.1)=0.08\), and \(C(0.2,0.1)=0.03\). Find the probability that \(X\) lies between marginal probabilities 0.2 and 0.7 and \(Y\) between 0.1 and 0.5. (Answer: 0.2100)
- Two binary losses each have probability 0.5 of being positive. Their joint distribution fixes the copula at the attained marginal probability pairs, but two copula extensions disagree elsewhere. Does that disagreement imply different binary joint distributions? (Answer: No; off-range copula values do not affect the joint distribution at these discrete margins.)