S Tian et aL/Joumal of Banking S Finance 37(2013)2765-2778 diately as a dividend (referred to as barrier control). Val- Table 2 ues of capital holdings above c cannot be obtained because of the The impact of shock intensity on al capital holding, contagion and bailout continuous sample paths of the assumed diffusion The amounts i=0.5,l=0.3.n=0.8,x=0.1,y=098.t bank always wishes to retain a buffer of capital to reduce the ex ed cost of not meeting the regulatory capital requirement. 0.121720.121720.121720.12172 Therefore, the optimal policy is to pay dividends at as high a level as possible when C exceeds C, but otherwise to retain all earnings Condition (4)arises because control is instantaneous at the C m 0.1140260.1134920.113068 0.1405820.141071 003169200322255 boundary. Bank 1's value prior to the crisis is equal to the optimal 002778930005824490.005135380.00464646 capital level, C, when the capital is chosen optimally. Condition (5)is a consequence of an optimally selected C. Otherwise, the va- lue function could be increasing at C by a small shift of C in the direction that v<1 The impact of the anticipated government bailout probability on optimal capital holding, contagion and bailout amounts =0.5, 1=0.10, n=1, x=0.15, y=0.96. t=0.08,=0.50. roof. See Appendix E.口 In Fig. 2, we present pictures of the value function(MO), U(o) C 0.08786 and wo) to show the changes in the capital that follow different 01 0.1039640.1039190.1038 0.103826 actions(continuation, liquidation) based on the solution of the dy- 0.1039720.1039330.103893 namic models Eqs.(Al-2)and (Al-3)for U(C)and eqs.(Al-5)and 2594550002638220002683050.0027291000277646 (Al-6)for WC in Appendix A. The solid, dashed and dot-dashed Kcom000259191000263012000266922000270925000275025 lines represent VO U(O and w(C respectively. The figures in the four panels differ horizontally by parameter range and verti- cally by type of bailout Cpre( Com)is the optimal capital ratio se- when Bank 1 takes over Project G, C, with the optimal level of ex- lected by Bank 1 prior to the crisis with the anticipated preferred ante capital holding under bailout in the form of common stock stock (common stock) bailout. C is the required capital for Bank and preferred equity, Ccom and Cpre We also show the bailout I to take over Project G. The difference between C and Cpre(C amounts of common stock and preferred equity that are needed to and Com) is the minimum amount of capital that must be provided and A Shareholder's value for continuation of Project G, Kipe om, where Pre=C+[(1+n)(1-y)-nxsJl-Cpr by the government for Bank 1 to continue the project with the Km=C+((1+n)(1-y)-nxg-Coom and C is the new desired anticipated preferred stock(common stock) bailout. Panel(a) capital level C when Bank 1 takes over Project G C<akc<a and Panel (c)c<c<c<2 show the Below are the baseline parameter values case when Bank 1 has sufficient capital to take over Project G with- out bailout, while Panel(b)(t<ce <c<a) and Panel(d) 0.02 0.04 0.01 0. 02 0.05 0.05 C<Com <C<a show the case when bailout is ne Several observations can be made from Fig. 2. First, Bank 1's 5.1. The impact of shock intensity, bailout policy, and regulatory post-crisis value functions, U(C) and W(O), are non-linear functions capital requirement on contagion and bailout amounts of the capital ratio post crisis. Second, the shareholder's value U(o hen Bank 1 takes over Project G is always higher than M when Table 2 presents simulation results to illustrate the relationship between the anticipated shockintensity and Bank I's optimal capita Bank 1 liquidates Project G, no matter whether the firm receives holding, possibility of contagion, and bailout or not, or bailout takes the form of preferred nd bailout amounts to show the stock. The comparison of U(O and w(o) determines that Bank 1 economic magnitude, we assume that the total assets of Bank 1, A, will choose to continue Project G and the target level of capital are $100 billion. The capital holding required to take over Project to the crisis. Therefore, our subsequent analysis focuses on G, C, is $12. 172 billion. It is apparent that Com is always higher than case of continuation of the project. third the pre-crisis target or equal to Cpre. Contagion occurs for a wider range of values of o in 1, which is endogenously determined by anticipation of preferred stock bailout than common stock bailout solving Eq (3), depends on payoffs under these different scenarios. (we use numbers in bold to indicate contagion). For example, when p=0., Com is $.1071 billion, exceeding the required capital ra- 5. Bank optimal capital holding, interbank contagion, and tio to take over Project G, while Chre is s113068 billion, lower than C government bailout That is, Bank 1 is willing to set aside s2. 8 billion more for an antici Ited common stock bailout. Intuitively, if Bank 1 views a shock and Since we cannot obtain a closed-form solution for the optimal a common stock bailout as likely, it will keep more capital in order to apital holding for Bank 1, we use simulations to examine the im- Table 4 pact of a number of parameters on Bank 1s optimal capital he The impact of the regulatory capital ratio on optimal capital holding, contagion and and whether interbank contagion will emerge. These factors in- bailout amountsφ=0.5,=0.5,l=0.1,n=2,x=0.15.y=095,=050. clude exogenous variables and factors related to Bank 1's exposure 0.11 0.12 to Project G. For each case, we compare the required capital level for interested readers to investigate other cases. Please also refer our working paper ersion for more detailed simulation results and discussionspaid immediately as a dividend (referred to as barrier control). Val￾ues of capital holdings above C⁄ cannot be obtained because of the continuous sample paths of the assumed diffusion process. The bank always wishes to retain a buffer of capital to reduce the ex￾pected cost of not meeting the regulatory capital requirement. Therefore, the optimal policy is to pay dividends at as high a level as possible when C exceeds C⁄ , but otherwise to retain all earnings. Condition (4) arises because control is instantaneous at the C⁄ boundary. Bank 1’s value prior to the crisis is equal to the optimal capital level, C⁄ , when the capital is chosen optimally. Condition (5) is a consequence of an optimally selected C⁄ . Otherwise, the va￾lue function could be increasing at C⁄ by a small shift of C⁄ in the direction that Vc < 1. Proof. See Appendix B. h In Fig. 2, we present pictures of the value function (V(C), U(C) and W(C)) to show the changes in the capital that follow different actions (continuation, liquidation) based on the solution of the dy￾namic models Eqs. (A1-2) and (A1-3) for U(C) and Eqs. (A1-5) and (A1-6) for W(C) in Appendix A. The solid, dashed and dot-dashed lines represent V(C), U(C) and W(C) respectively. The figures in the four panels differ horizontally by parameter range and verti￾cally by type of bailout. C pre C com   is the optimal capital ratio se￾lected by Bank 1 prior to the crisis with the anticipated preferred stock (common stock) bailout. b C is the required capital for Bank 1 to take over Project G. The difference between b C and C preðb C and C com) is the minimum amount of capital that must be provided by the government for Bank 1 to continue the project with the anticipated preferred stock (common stock) bailout. Panel (a) C < b C < C pre < bb C  and Panel (c) C < b C < C com < bb C  show the case when Bank 1 has sufficient capital to take over Project G with￾out bailout, while Panel (b) C < C pre < b C < bb C  and Panel (d) C < C com < b C < bb C  show the case when bailout is necessary. Several observations can be made from Fig. 2. First, Bank 1’s post-crisis value functions, U(C) and W(C), are non-linear functions of the capital ratio post crisis. Second, the shareholder’s value U(C) when Bank 1 takes over Project G is always higher than W(C) when Bank 1 liquidates Project G, no matter whether the firm receives bailout or not, or bailout takes the form of preferred or common stock. The comparison of U(C) and W(C) determines that Bank 1 will choose to continue Project G and the target level of capital prior to the crisis. Therefore, our subsequent analysis focuses on the case of continuation of the project. Third, the pre-crisis target level of capital, C pre C com  , which is endogenously determined by solving Eq. (3), depends on payoffs under these different scenarios. 5. Bank optimal capital holding, interbank contagion, and government bailout Since we cannot obtain a closed-form solution for the optimal capital holding for Bank 1, we use simulations to examine the im￾pact of a number of parameters on Bank 1’s optimal capital holding and whether interbank contagion will emerge.19 These factors in￾clude exogenous variables and factors related to Bank 1’s exposure to Project G. For each case, we compare the required capital level when Bank 1 takes over Project G, b C, with the optimal level of ex￾ante capital holding under bailout in the form of common stock and preferred equity, C com and C pre. We also show the bailout amounts of common stock and preferred equity that are needed to maximize shareholder’s value for continuation of Project G, K pre and K com, where K pre ¼ C u þ ½ð1 þ nÞð1 yÞ nxnl C pre; K com ¼ C u þ ½ð1 þ nÞð1 yÞ nxnl C com and C u is the new desired capital level C⁄ when Bank 1 takes over Project G. Below are the baseline parameter values: r1 R1 r2 R2 x12 q A 0.02 0.04 0.01 0.02 0.05 0.05 1 5.1. The impact of shock intensity, bailout policy, and regulatory capital requirement on contagion and bailout amounts Table 2 presents simulation results to illustrate the relationship between the anticipated shock intensity and Bank 1’s optimal capital holding, possibility of contagion, and bailout amounts. To show the economic magnitude, we assume that the total assets of Bank 1, A, are $100 billion. The capital holding required to take over Project G, b C, is $12.172 billion. It is apparent that C com is always higher than or equal to C pre. Contagion occurs for a wider range of values of / in anticipation of preferred stock bailout than common stock bailout (we use numbers in bold to indicate contagion). For example, when / ¼ 0:9; C com is $14.1071 billion, exceeding the required capital ra￾tio to take over Project G, while C pre is $11.3068 billion, lower than b C. That is, Bank 1 is willing to set aside $2.8 billion more for an antici￾pated common stock bailout. Intuitively, if Bank 1 views a shock and a common stock bailout as likely, it will keep more capital in order to Table 3 The impact of the anticipated government bailout probability on optimal capital holding, contagion and bailout amounts / = 0.5, l = 0.10, n = 1, x = 0.15, y = 0.96, s = 0.08, n = 0.50. k 0.1 0.3 0.5 0.7 0.9 b C 0.08786 0.08786 0.08786 0.08786 0.08786 C pre 0.104008 0.103964 0.103919 0.103873 0.103826 C com 0.10401 0.103972 0.103933 0.103893 0.103852 K pre 0.00259455 0.00263822 0.00268305 0.0027291 0.00277646 K com 0.00259191 0.00263012 0.00266922 0.00270925 0.00275025 Table 2 The impact of shock intensity on optimal capital holding, contagion and bailout amounts k = 0.5, l = 0.3, n = 0.8, x = 0.1, y = 0.98, s = 0.10, n = 0.50. / 0.1 0.3 0.5 0.7 0.9 b C 0.12172 0.12172 0.12172 0.12172 0.12172 C pre 0.115931 0.114752 0.114026 0.113492 0.113068 C com 0.11694 0.117928 0.139893 0.140582 0.141071 K pre 0.0297867 0.0309655 0.031692 0.0322255 0.03265 K com 0.0287782 0.0277893 0.00582449 0.00513538 0.00464646 Table 4 The impact of the regulatory capital ratio on optimal capital holding, contagion and bailout amounts / = 0.5, k = 0.5, l = 0.1, n = 2, x = 0.15, y = 0.95, n = 0.50. s 0.08 0.09 0.10 0.11 0.12 b C 0.0948 0.10665 0.1185 0.13035 0.1422 C pre 0.11154 0.104859 0.114848 0.124838 0.134828 C com 0.111553 0.1234 0.117999 0.126985 0.136416 K pre 0.0034052 0.0219367 0.023797 0.0256574 0.0275179 K com 0.0033923 0.00339517 0.0206462 0.0235102 0.0259295 19 We used Mathematica to generate simulation results. Due to space constraint, we only report a subset of simulation results. However, the code is available upon request for interested readers to investigate other cases. Please also refer our working paper version for more detailed simulation results and discussions. 2772 S. Tian et al. / Journal of Banking & Finance 37 (2013) 2765–2778