Concentration Risk in Rural Loan Books: A Merton–Vasicek Derivation of the Basel Retail Capital Function and a Two-Factor Monte Carlo for District-Concentrated Agricultural Lending

The Basel IRB capital function rests on the asymptotic single-risk-factor (ASRF) model: a single systematic factor drives defaults and the portfolio is infinitely granular. I derive the function from a Merton firm-value model, proving the default-threshold result with Itô's lemma and the Vasicek limiting loss distribution with a conditional law of large numbers. I verify the 99.9% quantile against Monte Carlo and test the ASRF assumptions for a loan book concentrated in a few districts, as is typical of an Indian regional rural bank. At PD = 5% and LGD = 45%, the Basel "other retail" function (correlation 0.053; formula as published by the BCBS) gives K = 5.31% of exposure. Finite granularity matters little above about 1,000 borrowers. District concentration matters a great deal. With asset correlation 0.15 within a district and 0.05 across districts, unexpected loss at the 99.9% level is 2.23 times K for a single-district book, 1.42 times for three districts and 1.11 times for ten. PD, LGD and correlation values are scenarios, not estimates for any named bank. As context, the RBI reports a 5.1% gross NPA ratio for agricultural loans of scheduled commercial banks in March 2026.