The Information-Theoretic Cost of Risk Measurement

Shannon (1948) proved that the fundamental limit of communication is the channel capacity, not the message length. We prove an analogous result for risk measurement: the fundamental limit of computing coherent risk measures is the analyticity radius \(\rho\) of the loss density, not the dimension \(n\) of the underlying risk factors. For any loss random variable \(L\) whose density is analytic on its support with analyticity radius \(\rho > 1\), we show that \(N(\varepsilon) = \Theta(\log(1/\varepsilon)/\log\rho)\) spectral coefficients are both necessary and sufficient to compute every spectral risk measure to accuracy \(\varepsilon\) --- and this number is independent of \(n\). In practice, \(N = 128\) coefficients (1.04 KB) suffice for machine-precision accuracy across all tested portfolios from 5 to 10{,}000 assets. We introduce the risk entropy \(H_{\text{risk}}(L) = 1/\log\rho\) as the fundamental measure of risk-measurement complexity: fat-tailed losses have high risk entropy (hard to compress), light-tailed losses have low risk entropy (easy to compress). The framework unifies portfolio risk (lognormal sums), credit risk (CDO tranches), insurance risk (compound Poisson), and extreme-value risk (GEV/GPD) under a single information-theoretic principle. The portfolio sum case is proved constructively and formally verified in Lean 4 (Nagy, 2026a,c); the extensions are demonstrated for all loss distributions with analytic characteristic functions.