4.10.4 Aggregate Loss & the Loss Distribution Approach (LDA)

Basic Prerequisites: Probability & Measure Theory and 03 · Frequency–Severity.

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1. Intuition & Practical Objective

The objective: fuse the frequency and severity halves into one object - the distribution of annual total loss $S$ - and read the capital number off its tail. This is the Loss Distribution Approach (LDA), the quantitative core of operational-risk capital. Given a frequency law for $N$ and a severity law for $X$, LDA computes the aggregate-loss distribution $F_S$, from which:

• Expected loss $\text{EL}=\mathbb{E}[S]=\lambda \mathbb{E}[X]$ (the cheap average);
• Operational VaR ${\text{VaR}}_{99.9}=F_S^{-1}(0.999)$ (the one-year, 99.9% tail);
• Unexpected loss ${\text{UL}}_{99.9}={\text{VaR}}_{99.9}-\text{EL}$,

and the Basel II AMA capital is $\text{EL}+\text{UL}$ at the 99.9% / one-year soundness standard (¶667).

Three “aha”s:

1. $S$ is a random sum of a random number of losses - a compound Poisson process. Its distribution is not a scaled-up severity; it is a convolution of severity with frequency, and it is heavier-tailed than either alone.
2. The 99.9% quantile cannot be “seen” in the data - no bank has 1,000 years of history. So $F_S$ must be built (by convolution, recursion, or Monte Carlo), not observed.
3. Everything hinges on the severity tail. At equal expected severity, a Pareto tail gives a 99.9% loss ~5.8× the lognormal’s. LDA is a machine for turning a severity-tail choice into a capital number - the machine is correct, the input is everything.

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2. Mathematical Ground Truth & Derivations

The compound Poisson aggregate

With $N\sim \text{Poisson}(\lambda )$ and $X_i\sim F_X$ iid independent of $N$,

$$S=\sum _{i=1}^N{X}_i(S=0\text{when }N=0).$$

Expectation (Wald’s identity) and variance:

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

Moment-generating function (the cleanest expression of the convolution):

$$M_S(t)=\mathbb{E}[e^{tS}]=\mathbb{E}[\mathbb{E}[e^{tS}\mid N]]=\sum _{n=0}^{\infty }\frac{e^{-\lambda }{\lambda }^n}{n!}[M_X(t){]}^n=\exp [\lambda (M_X(t)-1)].$$

This is the compound-Poisson signature: the aggregate MGF is the exponential of the severity MGF, encoding how Poisson frequency and severity convolve.

LDA capital

$${\text{VaR}}_{\alpha }=F_S^{-1}(\alpha ),{\text{UL}}_{\alpha }={\text{VaR}}_{\alpha }-\mathbb{E}[S],{\text{Capital}}_{\text{AMA}}=\text{EL}+{\text{UL}}_{0.999}.$$

Analytic computation: Panjer recursion

When severity is discretized to integer units with probabilities $f_j=\mathbb{P}(X=j)$, the aggregate probability mass $g_k=\mathbb{P}(S=k)$ obeys the Panjer recursion (valid for the Poisson, $\mathbb{P}(N=n)=\frac{{\lambda }^n}{n!}{e}^{-\lambda }$, which is of the form $p_n=(a+\frac{b}{n})p_{n-1}$ with $a=0,b=\lambda$):

$$g_k=\frac{1}{1-a{f}_0}\sum _{j=1}^k(a+\frac{bj}{k})f_j{g}_{k-j},$$

(Here $g_0=\mathbb{P}(S=0)={\sum }_n{p}_n{f}_0^n=e^{-\lambda (1-f_0)}$.) The recursion gives $F_S$ exactly (up to discretization error) in $O(K^2)$ - the classical alternative to Monte Carlo used throughout Panjer (2006). Both routes converge to the same aggregate distribution.

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3. Computational Implementation - Monte Carlo LDA with two severity tails

Simulate the compound Poisson directly, read $\text{EL},{\text{VaR}}_{99.9},\text{UL}$ from the empirical aggregate, and compare a lognormal severity against an equal-mean Pareto severity to expose the tail’s dominance. Stdlib only.

Read the last line: the MC expected loss matches the exact $\lambda \mathbb{E}[X]$ (Wald), and both severity choices cost the same “average” - yet the Pareto’s 99.9% loss is 5.8× the lognormal’s (€7.88M vs €1.35M), and its VaR/EL ratio jumps from 2.24 to 13.1. The aggregate tail inherits the severity tail almost wholesale.

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4. Failure Modes & First-Principles Breakdowns

1. The tail is an input, not an output. LDA correctly propagates whatever severity tail you feed it - which is why a lognormal “fit” produces €1.35M while a defensible Pareto produces €7.88M. Choosing the wrong family is the dominant error and is invisible in the EL.
2. MC tail error. The 99.9% quantile is estimated from $0.001M$ of $M$ paths - at $M=200,000$ only ~200 paths define it, so ${\text{VaR}}_{99.9}$ carries sampling noise that MC alone cannot shrink quickly (Glasserman Ch 1: $O(M^{-1/2})$). Use Panjer recursion or variance reduction when the tail must be tight (06 · Advanced Extensions).
3. Independence is built into the formula. $\mathbb{E}[S]=\lambda \mathbb{E}[X]$ and the MGF factorization assume independent events. Stress-time clustering and cross-event-type dependence break the compound-Poisson convolution, and LDA then under-states the true aggregate (the dependence failure mode, 05 · Failure Modes).
4. Capital addition across cells. Basel allows correlations across business-line/event-type estimates (AMA ¶669(d)); naively summing cell VaRs (perfect-correlation assumption) overstates total capital, while assuming independence understates it. Either way the choice is a model assumption to validate, not a freebie.

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5. References

• Panjer, Operational Risk: Modeling Analytics (2006)
• McNeil, Frey & Embrechts, Quantitative Risk Management (2015)
• Glasserman, Monte Carlo Methods in Financial Engineering (2004, Springer)
• BCBS, Basel II (2006), ¶667 (99.9%/one-year AMA soundness standard) and ¶669(d) (correlations across estimates)
• Shevchenko, Modelling Operational Risk Using Bayesian Inference (2011)

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6. Connected Graph Bridges

• Base: Probability & Measure Theory · 03 · Frequency–Severity
• Back: 02 · Loss Event Types
• Forward: 05 · Failure Modes · 06 · Advanced Extensions
• Sibling: VaR & Expected Shortfall (the risk-measure it computes) · EVT & Fat Tails (the tail it needs)