Aggregate Loss Distributions

Pricing stop-loss and measuring capital both require the full distribution of aggregate loss \$S=\sum_{i=1}^N X_i\$. Three methods dominate.

1. Panjer recursion

When the frequency \$N\$ belongs to the \$(a,b,0)\$ class (Poisson, binomial, negative binomial) and severity is discretized on \$0,1,2,\dots\$, the aggregate probabilities \$g_s=P(S=s)\$ satisfy an exact recursion:

Panjer recursion

$$g_s=\sum_{j=1}^{s}\Big(a+\frac{b\,j}{s}\Big)f_j\,g_{s-j}$$

f_j is the discretized severity probability; (a,b) parameterize the frequency class. Efficient and exact, no convolution powers required.

2. Convolution

Direct convolution sums the severity distribution \$N\$ times, weighted by \$P(N=n)\$, exact but computationally heavy for large portfolios.

3. Fourier / FFT and Monte-Carlo

The characteristic function factorizes for compound distributions, so the FFT recovers \$f_S\$ rapidly; Monte-Carlo simulation is the most flexible for complex dependency structures and reinstatements.

Compound characteristic function

$$\varphi_S(t)=P_N\big(\varphi_X(t)\big)$$

P_N is the probability generating function of frequency; phi_X the characteristic function of severity.

Aggregate Loss Distributions

1. Panjer recursion

2. Convolution

3. Fourier / FFT and Monte-Carlo

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