Credit Risk & Corporate Finance
Leland Endogenous Default Barrier, Debt Value, and Credit Spread Formula
Price perpetual risky corporate debt, equity-chosen endogenous bankruptcy threshold V_B, and levered firm value under Leland’s (1994) structural trade-off model.
Calculate the exact endogenous default boundary V_B = 56.25 by hand
Unlike Merton (1974), where default occurs only at fixed maturity T if assets V_T < Face, Leland (1994) allows equity holders servicing perpetual coupon C_coup to inject equity until optimal stopping threshold V_B = ((1 - shield_rate) * C_coup / r_rf) * (x_exp / (x_exp + 1)), where x_exp = 2 * r_rf / sig_A^2 (for zero asset payout) and Arrow-Debreu default price is p_B = (V / V_B)^(-x_exp).
Verify clean rational numbers: unlevered asset value V = 100 (B2), annual perpetual coupon C_coup = 6 (B3), risk-free rate r_rf = 0.06 (B4, so riskless perpetuity C / r = 100), asset volatility sig_A = 0.20 (B5, so x_exp = 0.12 / 0.04 = 3!), corporate interest shield rate = 0.25 (B6), and distress loss fraction alpha_bc = 0.40 (B7). By hand, V_B = (0.75 * 100) * (3 / 4) = 56.25! Default price is (100 / 56.25)^(-3) = (9 / 16)^3 = 729 / 4,096 = 0.1780, risky debt D_val = 100 * (1 - 729/4096) + (729/4096) * (1 - 0.40) * 56.25 = 88.2089, and yield spread = (6 / 88.2089) - 0.06 = 0.00802 (80.20 bps)!
| Leland (1994) Quantity | Closed-Form Expression | Exact Output |
|---|---|---|
| Exponent x_exp & Barrier V_B | 2*r / sig^2 = 3; V_B = ((1 - s)*C/r)*(x/(x+1)) | x = 3; V_B = 56.2500 |
| Arrow-Debreu Default Price p_B | (V / V_B)^(-x_exp) = (9 / 16)^3 | 0.1780 (729 / 4096) |
| Market Value of Debt D(V) | (C/r)*(1 - p_B) + (1 - alpha_bc)*V_B*p_B | 88.2089 |
| Levered Firm Value v(V) | V + (s*C/r)*(1 - p_B) - alpha_bc*V_B*p_B | 116.5460 |
Endogenous Bankruptcy Asset Threshold V_B Formula
Verify that higher asset volatility sig_A lowers x_exp = 2*r / sig^2 and therefore LOWERS V_B — because equity’s option to wait for an upside asset recovery becomes more valuable!
=LET(c_cp, B3, r_rf, B4, sig_a, B5, sh_rt, B6, IF(OR(c_cp <= 0, r_rf <= 0, sig_a <= 0, sh_rt < 0, sh_rt >= 1), "Invalid Leland Input", LET(x_ex, (2 * r_rf) / (sig_a^2), ROUND(((1 - sh_rt) * (c_cp / r_rf)) * (x_ex / (x_ex + 1)), 4))))Risky Debt Market Value D(V) and Credit Spread Formula
Guard immediate bankruptcy when current asset value V <= V_B (in which case bondholders recover (1 - alpha_bc) * V immediately).
=LET(v_ast, B2, c_cp, B3, r_rf, B4, sig_a, B5, sh_rt, B6, al_bc, B7, IF(OR(v_ast <= 0, c_cp <= 0, r_rf <= 0, sig_a <= 0), "Invalid Leland Input", LET(x_ex, (2 * r_rf) / (sig_a^2), v_bar, ((1 - sh_rt) * (c_cp / r_rf)) * (x_ex / (x_ex + 1)), IF(v_ast <= v_bar, ROUND((1 - al_bc) * v_ast, 4), LET(p_def, (v_ast / v_bar)^(-x_ex), ROUND((c_cp / r_rf) * (1 - p_def) + (1 - al_bc) * v_bar * p_def, 4))))))Try the example
Use the original values or change them to test your understanding.