Derive the discretized version of a continuous distribution using the method of rounding and local moment matching. | Free Exam ASTAM Lesson

Panjer recursion and FFT both demand a discrete severity on an arithmetic grid. You discretize because the algorithms cannot consume a continuous density. Two standard recipes, two different priorities.

Why discretize. Aggregate loss $S = X_{1} + \dots + X_{N}$ requires evaluating a compound distribution. Panjer recursion needs $f_{X}$ at points $0, h, 2 h, \dots$ on an arithmetic grid. FFT convolution likewise needs a discrete PMF, ideally one whose length is a power of two. A continuous severity has to be projected onto that grid, and the projection introduces error. Your job is to control the error by choosing both the span and the method that match the aggregate quantity you actually care about.

Two discretization philosophies. Both methods produce a PMF ${f_{0}, f_{1}, f_{2}, \dots}$ on the grid $0, h, 2 h, \dots$ for a non-negative severity $X$ . They differ in WHAT they preserve.

Method of rounding preserves probability mass on each rounding interval. Simple, fast, but only first-order accurate for the mean.
Local moment matching preserves a chosen number of moments exactly on each interval. More algebra, but the discrete mean (or higher moments) of the aggregate is right by construction.

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Common mistakes

Treating the leftmost cell as $F (h)$ instead of $F (h /2)$ . The rounding interval at zero is half-width, so $f_{0} = F (h /2)$ .
Forgetting the $- x_{k} m_{0}^{k}$ term in the local moment matching numerator. Without it, $b_{k}$ is wrong and the discrete mean drifts.
Mis-labeling $p$ . Two-point matching is $p = 1$ (matching the 0th and 1st moments). Three-point matching is $p = 2$ .

Bottom line

Method of rounding (mass dispersal): $f_{0} = F (h /2)$ and $f_{j} = F (j h + h /2) - F (j h - h /2)$ . Total probability preserved.
Local moment matching with $p$ moments: each interval of length $p h$ gets $p + 1$ masses so the first $p$ moments of $X$ match.
Most common variant matches 0th and 1st moments ( $p = 1$ ), preserving probability and mean exactly.
Rounding biases the mean, while local moment matching at $p = 1$ forces the discrete mean to equal $E [X]$ exactly.

Exam shortcut

For exponential $(θ)$ with span $h$ , rounding gives $f_{0} = 1 - e^{- h / (2 θ)}$ and $f_{j} = e^{- (j h - h /2) / θ} - e^{- (j h + h /2) / θ}$ . Memorize this exact form. For local moment matching on any severity, set up two equations per interval, solve for $a_{k}, b_{k}$ , then add overlapping contributions at each grid point. Don't try to derive a closed form per family. Asked which method preserves the mean, the answer is local moment matching with $p = 1$ . Asked which preserves cell-by-cell probability, the answer is rounding.

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