Corporate Finance & Banking Macro
Holmström-Tirole Pledgeable Income & Leverage Multiplier
Calculate unit pledgeable income rho_0 = p_H*(R - b/dp), the Holmström-Tirole (1997) equity multiplier 1/(1 - rho_0/gamma), and maximum borrowing capacity I_max - A.
Why positive-NPV firms still hit hard net-worth leverage caps: pledgeable income vs total return!
Jean Tirole (2006) and Bengt Holmström & Jean Tirole (1997) unified corporate finance around one fundamental distinction: Total Expected Return rho_1 = p_H * R vs Pledgeable Income rho_0 = p_H * (R - b / dp)! Suppose a borrower with liquid net worth ("skin in the game") A > 0 can scale a project of size I at gross investor required rate gamma >= 1. Diligent management achieves success probability p_H (yielding R * I), whereas shirking lowers success by dp = p_H - p_L > 0 while yielding private benefit b * I. Incentive compatibility requires paying the entrepreneur at least (b / dp) * I upon success, leaving AT MOST rho_0 * I = p_H * (R - b / dp) * I pledgeable to outside lenders — capping total investment at I_max = A / (1 - (rho_0 / gamma))!
Verify by hand on Net_Worth_A = 40 (B2), Success_Prob_pH = 0.80 (B3), Effort_Delta_dp = 0.30 (B4), Gross_Project_Return_R = 1.75 (B5), Unit_Private_Benefit_b = 0.30 (B6), and Investor_Gross_Rate_gamma = 1.00 (B7): Total Expected Return is 0.80 * 1.75 = 1.40 > 1.00 (40% positive NPV!), yet moral-hazard agency wedge b / dp = 0.30 / 0.30 = 1.00 leaves Unit Pledgeable Income rho_0 = 0.80 * (1.75 - 1.00) = 0.6000! Consequently, Equity Multiplier is 1 / (1 - 0.60) = 2.5000, Max Project Scale is 2.50 * 40 = 100.00, and Max External Debt is 100 - 40 = 60.00!
| Holmström-Tirole Metric (A=40, p_H=0.80, dp=0.30, R=1.75, b=0.30) | Formula | Verified Output |
|---|---|---|
| Total Social Return per $1 (rho_1) | p_H * R | 1.4000 (+40% NPV) |
| Unit Pledgeable Income (rho_0) | p_H * (R - b / dp) | 0.6000 (< gamma = 1.00) |
| Equity Investment Multiplier (k_mult) | 1 / (1 - rho_0 / gamma) | 2.5000x Net Worth |
| Max Project Scale I_max (Debt = I_max - A) | k_mult * A | 100.00 (Max Debt = 60.00) |
Unit Pledgeable Income rho_0 & Equity Multiplier Formula
Guard both NPV viability (p_H * R > gamma) and finite leverage (0 <= rho_0 < gamma; if rho_0 >= gamma, the project is 100% self-financing without net worth!).
=LET(a_nw, B2, p_hi, B3, d_pr, B4, r_un, B5, b_pv, B6, gm_r, B7, IF(OR(a_nw <= 0, p_hi <= 0, p_hi > 1, d_pr <= 0, d_pr > p_hi, r_un <= 0, b_pv < 0, gm_r <= 0), "Invalid Pledgeability Input", IF(p_hi * r_un <= gm_r, "Negative NPV Project", LET(rh_0, p_hi * (r_un - (b_pv / d_pr)), IF(rh_0 >= gm_r, "Self-Financing (No Net-Worth Cap)", IF(rh_0 <= 0, "Zero External Borrowing Capacity", ROUND(1 / (1 - (rh_0 / gm_r)), 4)))))))Maximum Project Scale I_max & Credit-Crunch Sensitivity Formula
Test raising Investor_Gross_Rate_gamma from 1.00 to 1.20 during a monetary tightening: Equity Multiplier contracts from 2.50x to 2.00x, shrinking I_max from 100.00 to 80.00!
=IF(ISTEXT(E3), IF(E3 = "Zero External Borrowing Capacity", B2, E3), ROUND(E3 * B2, 4))Try the example
Use the original values or change them to test your understanding.