(như thể hiện trong hình Nếu equityholders quyết định ngừng theo dõi, sự năng động của giá trị tài sản trở thành , nhưng họ tiết kiệm chi phí giám sát r γ. Trốn tránh việc trở nên tối ưu cho equityholders bất cứ khi nào sự mất mát tức thời của E giá trị vốn chủ sở hữu ít hơn chi phí này theo dõi. | REBALANCING THE THREE PILLARS OF BASEL II 277 As for the total value of the bank equation the second term is an option value that is maximized when Al Ae -O- D y . a 1 At this threshold the value of the bank s equity has a horizontal tangent as represented in figure E Ae 0. If equityholders decide to stop monitoring the dynamics of asset value become dA . . _ p - Ap dt a dW .a but they save the monitoring cost ry. Shirking becomes optimal for equityholders whenever the instantaneous loss of equity value E A A p is less than this monitoring cost. Because E Ae 0 see equation this condition is always satisfied in the neighborhood of the liquidation point. However we have to check that this incentive constraint binds after the bank becomes insolvent. This is true whenever ÀAe D or a D y a 1 D which is equivalent to the condition of proposition namely D -1 This ends the proof of proposition . Minimum Capital Ratio Suppose bank regulators impose a closure threshold Ar D y if the bank s asset value hits Ar the bank is liquidated and shareholders receive nothing. By an immediate adaptation of equation shareholders value becomes E A a - y - D D y - Ar AK. Ar7 The condition for eliminating shirking is va Ar E A A p yr. 2 78 CHAPTER 9 Using equation we see that this is equivalent to A A Mr a - a D Y - Ar AV Yr ArJ Ap Provided that AR Y D this will be checked ex post the left-hand side of equation is increasing in A therefore equation is equivalent to AR a 2 - a D y Ap or Ar Ai . a D y Yr Ap. a 2 aR represents the minimum asset value that preserves the incentives of the banker. The associated capital ratio is R _ At - D _ Y a r Ap - D p aR a D y Yr Ap. Subordinated Debt Consider now that the bank issues a volume B of subordinated bonds paying a coupon cB per unit of time and randomly renewed with frequency m. The market value of these bonds B A as a function of the bank s asset value .
