Banking & Financial Intermediation
Diamond Delegated Monitoring & Diversification Formula
Calculate Douglas Diamond’s (1984) delegated-monitoring cost advantage of a diversified bank over duplicated direct saver monitoring (m*K) and unmonitored debt.
See how pooling independent loans collapses the cost of monitoring the monitor!
Douglas Diamond (1984) answered why financial intermediaries exist: if each borrower requires capital from m = 100 small savers (B2) and monitoring costs K_mon = 0.02 (2.00% of principal, B3), direct saver monitoring wastes m * K_mon = 2.0000 (200% duplicated cost), forcing direct markets into unmonitored debt with deadweight default loss D_unmon = 0.0800 (8.00%, B4)! Delegating monitoring to a single bank cuts physical audit cost to 1 * K_mon = 0.0200 per loan — PROVIDED depositors do not have to monitor the bank itself! Instead, the bank issues fixed-rate deposits requiring Expected Gross Return R_dep = 1.05 (B8) backed by a non-pecuniary bankruptcy penalty equal to any deposit shortfall!
Verify Diamond’s exact two-loan diversification proof on independent projects succeeding with probability p = 0.80 (B5) for R_H = 1.70 (B6) or failing (1 - p = 0.20) for R_L = 0.50 (B7 -> E[R] = 1.46): (1) Undiversified 1-Loan Bank (N = 1) must promise deposit face F_1 = (1.05 - 0.20 * 0.50) / 0.80 = 1.1875, incurring Expected Delegation Penalty D_1 = 0.20 * (1.1875 - 0.50) = 0.1375 and Total Cost K_mon + D_1 = 0.1575 (15.75% — worse than unmonitored debt 8.00%!); (2) Diversified 2-Loan Bank (N = 2) earns mixed-state average (1.70 + 0.50)/2 = 1.10 >= R_dep = 1.05 whenever at least ONE loan succeeds, defaulting ONLY when BOTH fail ((1 - p)^2 = 0.04) at face F_2 = (1.05 - 0.04 * 0.50) / 0.96 = 1.0729 <= 1.10! Delegation Cost plunges 83.3% from 0.1375 to D_2 = 0.04 * (1.0729 - 0.50) = 0.0229, cutting Total Bank Cost per loan to 0.0200 + 0.0229 = 0.0429 (4.29% < 8.00% unmonitored debt, and converging to K_mon = 0.0200 as N -> infinity)!
| Financing Architecture (m=100, K=0.02, p=0.80, R_H=1.70, R_L=0.50) | Formula | Total Cost / Loan |
|---|---|---|
| Direct Duplicated Monitoring (m Savers) | m * K_mon | 2.0000 (200.00%) |
| Undiversified 1-Loan Intermediary (N = 1) | K_mon + (1 - p)*(F_1 - R_L) | 0.1575 (15.75% > 8% Direct) |
| Diversified 2-Loan Intermediary (N = 2) | K_mon + (1 - p)^2 * (F_2 - R_L) | 0.0429 (4.29% Beats Direct!) |
| Asymptotic Diversified Bank (N -> inf) | K_mon + 0 (LLN Limit) | 0.0200 (2.00% First-Best Monitor) |
Undiversified (N = 1) vs Two-Loan Diversified (N = 2) Bank Cost Formula
Check piecewise mixed-state solvency: if F_2_low <= (R_H + R_L)/2, the 2-loan bank survives single-loan defaults; otherwise compute the high-face regime where default also occurs in the mixed state!
=LET(m_sv, B2, k_mn, B3, p_ok, B5, r_hi, B6, r_lo, B7, r_dp, B8, e_rt, p_ok * r_hi + (1 - p_ok) * r_lo, IF(OR(m_sv < 1, k_mn < 0, p_ok <= 0, p_ok >= 1, r_lo < 0, r_dp <= r_lo, e_rt <= r_dp), "Invalid Diamond Input", LET(p_bb, (1 - p_ok) ^ 2, r_md, (r_hi + r_lo) / 2, f_lw, (r_dp - p_bb * r_lo) / (1 - p_bb), d_tw, IF(f_lw <= r_md, p_bb * (f_lw - r_lo), LET(p_mx, 2 * p_ok * (1 - p_ok), f_hi, (r_dp - p_mx * r_md - p_bb * r_lo) / (p_ok ^ 2), p_mx * (f_hi - r_md) + p_bb * (f_hi - r_lo))), ROUND(k_mn + d_tw, 4))))Net Intermediation Advantage Over Unmonitored Debt Formula
Positive spread D_unmon - Total_Bank_Cost confirms financial intermediation strictly dominates direct unmonitored bond markets!
=IF(ISTEXT(E4), E4, ROUND(B4 - E4, 4))Try the example
Use the original values or change them to test your understanding.