Angulo, M. T.; Saavedra, S.

Biodiversity is commonly summarized by macroecological mean patterns, most prominently the species-area relationship (SAR) linking habitat area to expected species richness. Yet conservation, policy, and economic decisions increasingly require risk metrics: probabilities of rare but consequential biodiversity shortfalls, including local collapse. Such tail risks are central in finance and insurance but remain difficult to quantify in ecology because the data needed to estimate full richness distributions are rarely available at decision scales. Here we provide a mechanistic route from species-area relationships to biodiversity risk metrics. We show that when regional species abundances are well approximated by Fisher's log-series, a minimal immigration-extinction mechanism yields a closed-form stationary distribution for local richness whose structure tightly couples the mean SAR to richness variability and lower-tail probabilities. This coupling implies exact fluctuation-response identities and an explicit integral transform that reconstructs collapse probabilities and other tail risk measures directly from the mean SAR. These results define ecological analogues of financial risk metrics---such as collapse probability and lower-tail quantiles---without requiring direct estimation of the full richness distribution. Using high-resolution ForestGEO tree censuses spanning tropical, subtropical, and temperate forests, we find empirical support for these predictions across spatial scales. Together, our results show how widely measurable species-area relationships can be elevated from descriptive averages to operational tools for biodiversity risk assessment and reliability-based conservation planning.