From the loss distribution and its risk measures (VaR, TVaR) to the loss functions that guide estimation, pricing and decision making under uncertainty.
In actuarial work the term loss function has two complementary meanings, both central to risk management:
This page covers both: first we model the loss, then we summarize it with risk measures, and finally we use loss functions to decide optimally.
A portfolio's loss combines how many claims occur (frequency) and how much each one costs (severity).
$N$ is the frequency (Poisson, negative binomial) and the $Y_i$ are i.i.d. severities independent of $N$. $S$ is the portfolio's total loss for the period.
This splits total uncertainty into severity variability and frequency variability. For $N\sim$ Poisson$(\lambda)$ it simplifies to $\mathrm{Var}(S)=\lambda\,E[Y^2]$.
Computes the distribution of $S$ when $N$ belongs to the $(a,b,0)$ class (Poisson, binomial, negative binomial) and severity is discrete or discretized.
Large losses (catastrophes, liability) follow heavy tails. With $\alpha\le 2$ the variance is infinite; with $\alpha\le1$ not even the mean exists. Modeling the tail is critical for reinsurance and capital.
A risk measure $\rho(S)$ turns the whole loss distribution into a capital or premium figure. The choice of measure matters: VaR and TVaR can diverge sharply in the tail.
The loss level that is not exceeded with probability $\alpha$ (e.g., 99.5% under Solvency II). Simple, but not coherent: it can penalize diversification.
The average of the losses that exceed VaR. It captures tail severity and is coherent. Also called CTE or Expected Shortfall (favored by Basel and Solvency II).
A coherent measure satisfies monotonicity, positive homogeneity, translation invariance and subadditivity (diversification does not increase risk). TVaR satisfies it; VaR, in general, does not.
Variance and standard deviation principles: the premium exceeds the expected loss by a loading proportional to dispersion. The Esscher principle weights by $e^{hS}$.
When we estimate a reserve or a premium, a loss function $L(\theta,\hat\theta)$ defines which error is acceptable. The optimal estimator minimizes expected loss, and the shape of $L$ determines which estimator that is.
Under squared loss the optimal estimator is the posterior mean. It is the basis of Bühlmann credibility and of least squares regression.
Absolute loss leads to the median, which is more robust to extreme values. It generalizes to quantiles through the "pinball" loss, the basis of quantile regression.
Penalizes underestimation and overestimation differently. This matters for reserves: falling short (insolvency) costs more than overshooting, which justifies prudence margins.
In ratemaking, GLMs fit the pure premium by minimizing the deviance: Poisson for frequency, gamma for severity and Tweedie for the pure premium directly. It is the loss function of the pricing model.
Modeling the loss, measuring its risk and choosing a loss function to decide are three faces of the same problem: turning uncertainty into an actionable figure (a premium, a reserve, a capital requirement) in the least costly way under an explicit criterion.