June 2007 Capital Adequacy Extension © Copyright 2007, CCRO. All rights reserved. Page 55 of 92 Insert 5.2 Equity Approach Example Take a company with equity at $2.0 million, equity volatility at 60% as measured by stock price standard deviation, debt at $6.0 million that will have to be repaid in one year, and the risk free rate is 5.0%. Therefore, Eo = 2.0 σE = 0.60 D = 6.0 r = 0.05 Solving equations (1) and (2) simultaneously gives us Vo = 7.69 σV = 0.16 The parameter d2 is 1.786 resulting in the probability of default, N(-d2), to be 3.70%. The market value of debt is Vo – Eo = 5.69. The present value of payment on the debt is 6.0e(-0.05*1) = 5.71. The expected loss on that debt is therefore (5.71 – 5.69) / 5.71 = 0.22%. This results in the expected recovery in the event of default (3.70 – 0.22) / 3.70 = 93.96% The above outlines an overly simplistic view that does not take into consideration certain aspects that make this method difficult to implement. Nonetheless, this example demonstrates fundamentally how this method can be utilized to calculate default probabilities, loss given defaults, and recovery rates. Below is an example of how the calculation can be set up. 93.96% Expected Recovery (ER) = (EDP - EL) / EDP 0.22% Expected Loss of 'No Default' Value (EL) = (PVD - MVP) / PVD 5.707376 Present Value of Debt (PVD) = De(-rT) 5.694613 Market Value of Debt (MVD) = Vo - Eo 3.70% Expected Probability of Default (EDP) = N(-d2) 0.962968 0.080925 1.786222 0.974191 0.060025 1.946307 0.160085 7.694613 0.05 6 1 0.600001 2.00 N(d2) N'(d2) d2 N(d1) N'(d1) d1 Asset Vol (σv) Value of Assets (Vo) RF Rate Debt (D) Time to Maturity Equity Vol (σe) Value of Equity (Eo) Vol Under- lying Strike Ito's Lemma Call Option Premium Calculated Unknowns Given 93.96% Expected Recovery (ER) = (EDP - EL) / EDP 0.22% Expected Loss of 'No Default' Value (EL) = (PVD - MVP) / PVD 5.707376 Present Value of Debt (PVD) = De ( - rT ) 5.694613 Market Value of Debt (MVD) = V o - E o 3.70% Expected Probability of Default (EDP) = N( - d 2 ) 0.962968 0.080925 1.786222 0.974191 0.060025 1.946307 0.160085 7.694613 0.05 6 1 0.600001 2.00 N(d 2 ) N'(d 2 ) d 2 N(d 1 ) N'(d 1 ) d 1 Asset Vol ( σ v ) Value of Assets (V o ) RF Rate Debt (D) Time to Maturity Equity Vol ( σ e ) Value of Equity (E o ) Vol Under - lying Strike Ito's Lemma Call Option Premium Calculated Unknowns Given