Building Joint Loss Models from Marginals and Copulas

From Marginal Loss Models to Joint Probabilities

An actuarial model may specify the distribution of each line of business separately, but questions about the combined portfolio depend on how the losses occur together. A copula provides a way to combine marginal loss distributions into a joint model: the marginals describe each loss on its own, while the copula specifies how their distribution functions move together. Once the joint model is defined, it can answer questions such as the probability that both losses exceed thresholds or the probability of a large loss in one line conditional on a loss in another.

This tutorial focuses on using a specified copula and specified marginals. It does not require estimating the copula or deriving a particular copula family. The calculations are aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Definition: Let $X$ and $Y$ be two loss random variables with cumulative distribution functions $F_X$ and $F_Y$. A copula $C$ is a joint distribution function on the unit square whose two marginal distributions are uniform on $[0,1]$. For continuous marginals, the joint cumulative distribution function is $$ F_{X,Y}(x,y)=P(X\le x,Y\le y)=C(F_X(x),F_Y(y)). $$ The copula inputs are marginal probabilities, not loss amounts.

For continuous marginals, each marginal probability $F_X(x)$ and $F_Y(y)$ lies in $[0,1]$, and the copula combines those probability-scale values. This separation is useful: changing a marginal loss model changes the probability assigned to a threshold, while the copula continues to describe how the marginal probabilities are associated. The same copula can therefore be paired with different marginal distributions.

The formula directly gives the probability that both losses are at or below their thresholds. Other event probabilities follow from it using complements and differences. For instance, subtracting the joint lower-tail probability from the appropriate marginal probability gives a mixed event, while inclusion-exclusion gives a joint upper-tail probability.

Useful Probability Formulas

Write $u=F_X(x)$ and $v=F_Y(y)$. The joint lower-tail probability is $C(u,v)$. The other common events are:

\begin{aligned} P(X\le x,Y>y)&=u-C(u,v),\\ P(X>x,Y\le y)&=v-C(u,v),\\ P(X>x,Y>y)&=1-u-v+C(u,v). \end{aligned}

The last equation is inclusion-exclusion. Start with the probability one, subtract the events $X\le x$ and $Y\le y$, then add their intersection back because it was subtracted twice. This explains why an upper-tail probability cannot usually be found by taking $1-C(u,v)$: that complement includes both mixed regions as well as the joint upper tail.

For a rectangle, take differences of joint CDF values at its corners. If $x_1<x_2$ and $y_1<y_2$, then

P(x_1<X\le x_2,\ y_1<Y\le y_2) =C(F_X(x_2),F_Y(y_2))-C(F_X(x_1),F_Y(y_2)) -C(F_X(x_2),F_Y(y_1))+C(F_X(x_1),F_Y(y_1)).

The signs follow by subtracting the lower and left regions from the joint lower-left probability at the upper-right corner, then adding back the overlap subtracted twice. For continuous losses, whether a threshold is written with $<$ or $\le$ does not affect these probabilities. For distributions with point masses, endpoint conventions matter.

A conditional probability is a ratio: the numerator is the probability of the joint event, and the denominator is the probability of the conditioning event. For example, provided $F_Y(y)<1$,

P(X>x\mid Y>y) =\frac{1-F_X(x)-F_Y(y)+C(F_X(x),F_Y(y))} {1-F_Y(y)}.

A conditional probability is undefined if its conditioning event has probability zero. Always check the denominator and ensure that the numerator represents the event being conditioned on.

Worked Applications

Worked Example: Joint Exceedance for Two Exponential Losses

A hypothetical portfolio has annual losses $X$ and $Y$, measured in thousands of dollars. Assume $X$ is exponential with mean 100 and $Y$ is exponential with mean 200. The specified copula is $C_\theta(u,v)=uv+\theta uv(1-u)(1-v)$, with $\theta=0.6$. Find the probability that both losses exceed their respective medians, then find the probability that $X$ exceeds its median given that $Y$ exceeds its median. Treat losses as continuous.

Method. Convert each median threshold to its marginal CDF value, evaluate the copula, and use inclusion-exclusion for the joint exceedance. This avoids treating the lower-tail copula probability as an upper-tail probability.

For an exponential distribution with mean $m$, $F(t)=1-e^{-t/m}$ for $t\ge 0$. Its median $t$ satisfies $1-e^{-t/m}=0.5$, so $t=m\ln 2$. The median thresholds are therefore $100\ln 2$ and $200\ln 2$ thousand dollars, and each has marginal CDF value 0.5.

Evaluate the copula at these probability values:

C_{0.6}(0.5,0.5) =(0.5)(0.5)+0.6(0.5)(0.5)(0.5)(0.5) =0.25+0.0375=0.2875.

Thus, $P(X\le 100\ln 2,\ Y\le 200\ln 2)=0.2875$. Inclusion-exclusion gives the joint exceedance:

P(X>100\ln 2,Y>200\ln 2) =1-0.5-0.5+0.2875=0.2875.

The conditioning event $Y>200\ln 2$ has probability $1-0.5=0.5$. Hence $P(X>100\ln 2\mid Y>200\ln 2)=0.2875/0.5=0.5750$. The joint probability is no greater than either marginal exceedance probability, and the conditional probability is between zero and one, as required.

Worked Example: One Loss in an Interval and the Other Above a Threshold

Assume a hypothetical annual loss $X$ is uniform on $[0,100]$ thousand dollars, and a second loss $Y$ is exponential with mean 50 thousand dollars. Use the specified copula $C_{0.5}(u,v)=uv+0.5uv(1-u)(1-v)$. Find $P(20<X\le70,\ Y>50\ln 2)$, then find the probability of that event conditional on $20<X\le70$.

Method. For each endpoint of the $X$ interval, calculate $P(X\le x,Y>y)=F_X(x)-C(F_X(x),F_Y(y))$. Subtract the lower-endpoint result from the upper-endpoint result. This method handles the interval and upper-tail event without incorrectly using the copula value alone.

The marginal CDF values at the relevant thresholds are $F_X(20)=0.2$, $F_X(70)=0.7$, and $F_Y(50\ln 2)=1-e^{-(50\ln 2)/50}=0.5$. For the upper endpoint,

C_{0.5}(0.7,0.5) =(0.7)(0.5)+0.5(0.7)(0.5)(0.3)(0.5) =0.35+0.02625=0.37625.

Therefore $P(X\le70,Y>50\ln 2)=0.7-0.37625=0.32375$. At the lower endpoint,

C_{0.5}(0.2,0.5) =(0.2)(0.5)+0.5(0.2)(0.5)(0.8)(0.5) =0.10+0.02=0.12,

so $P(X\le20,Y>50\ln 2)=0.2-0.12=0.08$. Subtracting gives $P(20<X\le70,Y>50\ln 2)=0.32375-0.08=0.24375$. The conditioning event has probability $P(20<X\le70)=(70-20)/100=0.5$. The conditional probability is therefore $0.24375/0.5=0.4875$. The joint event probability is less than the probability of the interval being conditioned on, and the conditional result is valid.

Worked Example: Conditional Exceedance Given a Loss Interval

Assume $X$ is exponential with mean 100 thousand dollars and $Y$ is exponential with mean 200 thousand dollars. Their specified copula is $C_{-0.8}(u,v)=uv-0.8uv(1-u)(1-v)$. Find the probability that $Y$ exceeds its median, conditional on $X$ being above its median but no greater than its 75th percentile.

Method. Express both endpoints of the conditioning interval as marginal probabilities. Calculate the upper-tail probability for $Y$ at each endpoint of $X$, then subtract. This is a conditional interval calculation, not a conditional probability at a single loss value.

For $X$, the median is $100\ln 2$, with CDF value 0.5. Its 75th percentile is $100\ln 4$, since $1-e^{-\ln 4}=0.75$. For $Y$, the median is $200\ln 2$, with CDF value 0.5. Evaluate the copula at the upper endpoint of the $X$ interval:

C_{-0.8}(0.75,0.5) =(0.75)(0.5)-0.8(0.75)(0.5)(0.25)(0.5) =0.375-0.0375=0.3375.

Thus $P(X\le100\ln 4,Y>200\ln 2)=0.75-0.3375=0.4125$. At the lower endpoint,

C_{-0.8}(0.5,0.5) =(0.5)(0.5)-0.8(0.5)(0.5)(0.5)(0.5) =0.25-0.05=0.20.

It follows that $P(X\le100\ln 2,Y>200\ln 2)=0.5-0.20=0.30$. Subtracting the lower-endpoint probability from the upper-endpoint probability gives the joint event: $P(100\ln 2<X\le100\ln 4,Y>200\ln 2)=0.4125-0.30=0.1125$. The conditioning interval has probability $0.75-0.5=0.25$, so the required conditional probability is $0.1125/0.25=0.4500$. The numerator is smaller than the conditioning probability, and the result lies between zero and one.

Checks, Assumptions, and Common Traps

A calculation is only as sound as the inputs and events it represents. Before evaluating a joint or conditional probability, check the following:

Use probability-scale inputs. Evaluate $F_X(x)$ and $F_Y(y)$ first. Do not insert a loss amount directly into a copula that takes inputs in $[0,1]$.
Identify the event represented by $C(u,v)$. It is $P(X\le x,Y\le y)$, not a joint exceedance probability. Use complements and inclusion-exclusion for upper tails.
Take differences for intervals. A probability involving $x_1<X\le x_2$ requires the relevant CDF probabilities at both endpoints.
Check conditioning denominators. Divide by the probability of the conditioning event, not by the marginal probability of the event being predicted.
Check the marginal assumptions. The displayed formulas use the specified marginal CDFs. A mistaken scale, time unit, or threshold produces incorrect copula inputs even if the copula calculation is correct.

For continuous marginals, Sklar's representation gives the joint CDF through the marginal CDFs and the copula. With discrete marginals, a copula representation still exists, but the copula is generally not unique at points outside the ranges of the marginal CDFs. The joint CDF values determine probabilities, but formulas for exact point events or interval endpoints must account for jumps. In particular, do not assume that changing $<$ to $\le$ is harmless when a loss distribution has atoms.

Key takeaway: Map loss thresholds through their marginal CDFs, evaluate the copula to obtain the joint lower-tail probability, and then use complements, differences, or conditional-probability ratios to match the event asked for.

Check Your Understanding

For each problem, use the stated marginal distributions and copula; report probabilities to four decimal places.

Let $X$ be uniform on $[0,80]$ thousand dollars and $Y$ be exponential with mean 40 thousand dollars. Assume the losses are independent. Find $P(X>60,Y>40)$. (Answer: 0.0920)
Let $X$ be uniform on $[0,100]$ and let $Y$ be exponential with mean 50, both measured in thousands of dollars. Use $C_{0.4}(u,v)=uv+0.4uv(1-u)(1-v)$. Find $P(X\le30,Y>50\ln2)$. (Answer: 0.1290)
Use $C_{0.8}(u,v)=uv+0.8uv(1-u)(1-v)$. Find the probability that both losses exceed their medians when both marginal distributions are continuous. (Answer: 0.3000)
Let $X$ be uniform on $[0,100]$ and $Y$ exponential with mean 100, in thousands of dollars. Use $C_{-0.5}(u,v)=uv-0.5uv(1-u)(1-v)$. Find $P(20<X\le80,\ Y\le100\ln2)$. (Answer: 0.3000)
Let $X$ and $Y$ be continuous losses with median thresholds $x_m$ and $y_m$. Use $C_{-0.6}(u,v)=uv-0.6uv(1-u)(1-v)$. Find $P(X>x_m\mid Y>y_m)$. (Answer: 0.4250)

What You'll Learn