Aggregate Loss Models

Aggregate loss models describe the total dollar loss from a portfolio over a period as a random sum of an uncertain number of claims, each of uncertain size. This "collective risk model" underlies stop-loss pricing, aggregate deductible rating, and capital modeling.

The collective risk model

Let $N$ be the (random) claim count and $X_1, X_2, \ldots$ be i.i.d. severities independent of $N$. The aggregate loss is

$S = \sum_{i=1}^{N} X_i, \qquad S = 0 \text{ if } N = 0.$

$S$ is a compound distribution. Its moments follow from conditioning on $N$ (the law of total expectation/variance):

$E[S] = E[N]\,E[X], \qquad \text{Var}(S) = E[N]\,\text{Var}(X) + \text{Var}(N)\,E[X]^2.$

When $N$ is Poisson with mean $\lambda$, $\text{Var}(N) = E[N] = \lambda$, so $\text{Var}(S) = \lambda\,E[X^2]$ — a clean result exploited throughout ratemaking and capital work. The moment generating function factors as $M_S(t) = M_N(\ln M_X(t))$, which underlies both the recursive and transform-based computation methods below.

The (a, b, 0) class and Panjer's recursion

Many standard counting distributions — Poisson, negative binomial, binomial — satisfy a recursion on their probabilities $p_k = P(N=k)$:

$\frac{p_k}{p_{k-1}} = a + \frac{b}{k}, \qquad k = 1, 2, \ldots$

For Poisson($\lambda$): $a=0, b=\lambda$. For negative binomial$(r,\beta)$: $a = \beta/(1+\beta)$, $b=(r-1)a$. For binomial$(m,q)$: $a = -q/(1-q)$, $b=(m+1)a$.

If severity is discretized on integers $0,1,2,\ldots$ with pmf $f_X(j)$, Panjer's recursion computes $f_S(s) = P(S=s)$ recursively:

$f_S(0) = \begin{cases} P_N(f_X(0)) & \text{if } f_X(0) > 0 \\ p_0 & \text{if } f_X(0)=0 \end{cases}, \qquad f_S(s) = \frac{1}{1-a f_X(0)}\sum_{j=1}^{s}\left(a+\frac{bj}{s}\right) f_X(j)\, f_S(s-j).$

This avoids simulating or convolving the full distribution of $N$ and is exact (up to the severity discretization) for the entire $(a,b,0)$ family.

Worked example: Poisson frequency, three-point severity

Let $N \sim \text{Poisson}(\lambda = 2)$ so $a=0, b=2$, and let severity be discrete with $f_X(1) = 0.5$, $f_X(2) = 0.3$, $f_X(3) = 0.2$ (no mass at 0).

Step 0: $f_S(0) = P_N(f_X(0)) = P_N(0) = e^{-\lambda} = e^{-2} = 0.135335$.

Step $s=1$: only $j=1$ contributes ($f_X(j)=0$ for $j>1$ when $s-j<0$ is excluded, and here $j\le s=1$):

$f_S(1) = \left(a + \frac{b\cdot 1}{1}\right) f_X(1) f_S(0) = (0+2)(0.5)(0.135335) = 0.135335.$

Step $s=2$: sum over $j=1,2$:

$f_S(2) = \left(2\cdot\tfrac{1}{2}\right) f_X(1) f_S(1) + \left(2\cdot\tfrac{2}{2}\right) f_X(2) f_S(0) = (1)(0.5)(0.135335) + (2)(0.3)(0.135335)$ $= 0.067668 + 0.081201 = 0.148869.$

Step $s=3$: sum over $j=1,2,3$:

$f_S(3) = \left(2\cdot\tfrac{1}{3}\right)(0.5)f_S(2) + \left(2\cdot\tfrac{2}{3}\right)(0.3)f_S(1) + \left(2\cdot\tfrac{3}{3}\right)(0.2)f_S(0)$ $= (0.6667)(0.5)(0.148869) + (1.3333)(0.3)(0.135335) + (2)(0.2)(0.135335)$ $= 0.049623 + 0.054134 + 0.054134 = 0.157891.$

Continuing this recursion out far enough and summing $1 - \sum f_S(s)$ gives the survival function needed to price a stop-loss layer or aggregate deductible directly, without ever simulating a single claim.

FFT and simulation alternatives

The Fast Fourier Transform method computes the compound distribution by (1) discretizing severity onto a fine grid, (2) computing the discrete Fourier transform of $f_X$, (3) applying the frequency distribution's pgf/mgf in transform space — for Poisson, the transform of $S$ is $\exp(\lambda(\hat f_X(t)-1))$ — and (4) inverting via inverse FFT. This is $O(n\log n)$ versus Panjer's $O(n^2)$ and is preferred for large supports or heavy severity tails, though it requires care with aliasing (wrap-around) if the grid is too short relative to the tail.

Monte Carlo simulation — simulate $N$, then simulate $N$ severities and sum, repeat many times — is the most flexible approach (handles frequency-severity dependence, mixed/non-$(a,b,0)$ frequency, copula structures) at the cost of simulation noise in the tail, which stop-loss and high-layer XoL pricing is especially sensitive to; variance reduction (importance sampling, control variates) is often needed for rare, high layers.

Applications: aggregate deductibles and stop-loss

An aggregate stop-loss with deductible $d$ pays $E[(S-d)_+] = \sum_{s>d}(s-d) f_S(s)$, the stop-loss net premium, directly computable once $f_S$ is known from Panjer/FFT/simulation. An aggregate deductible (cedent retains the first $d$ of aggregate losses) is the mirror image, with expected retained loss $E[\min(S,d)]$. These aggregate features appear throughout reinsurance (aggregate stop-loss treaties), retro rating plans, and large-account loss-sensitive programs.

Pitfalls

• Applying Panjer's recursion to a frequency distribution outside the $(a,b,0)$ class (e.g., a mixed Poisson without deriving the correct $a,b$) without verifying the recursion holds.
• Discretizing severity too coarsely relative to the deductible/limit, biasing the stop-loss premium.
• Ignoring frequency-severity dependence (e.g., larger events driving both more claims and higher severity) when the compound-independence assumption is not appropriate — climate/cat perils often violate it.
• Truncating the FFT grid too short, causing tail probability to alias back into the body of the distribution.

Exam relevance

Aggregate distributions, Panjer recursion, and stop-loss moments are central to CAS Exam 8 and SOA Exam STAM/ASTAM.

Further reading

• Klugman, Panjer & Willmot, Loss Models: From Data to Decisions
• Panjer (1981), Recursive Evaluation of a Family of Compound Distributions, ASTIN Bulletin