S. Tian et al Jourmal of Banking 8 Finance 37(2013)2765-2778 2771 4. 2. Bank 1's problem Under Condition(iv), an injection of common equity dilutes exis ing shareholder interests and hence provides a stronger incentive In anticipation of crisis, Bank 1's post-crisis value functions in or Bank 1 to hold more capital to cope with the failure of other different scenarios (liquidation of the joint project, continuation banks. The term-(1-A)ovo)reflects the expected effect on Bank without bailout, or continuation with bailout)will determine Bank I's continuation value, -vO), from the shock and no government 1's choice of whether to liquidate or continue the joint asset, and bailout, which occurs with probability(1-2)odt. hence its pre-crisis value function, VO), and its target level of cap- Finally, if C< C, Bank 1s capital holding does not meet the reg- ital. C The application of standard techniques shows that the ex- ulatory requirement, and is also insufficient to take over or liqui- ante value function V satisfies Hamilton-Jacobi-Bellman(HJB date Project G Bank 1 will be liquidated henceforth) differential equation in the following general form V(C=0 pv=maxe 1+72+71-92+21-000 +A(C) 4.3. Analyti Using the dynamic stochastic programming techniques, we find where A(C) takes the following forms depending on the relation- the following analytical solutions for the value function MO)and ship between C, C, C, and C the optimal capital holding C(Core for the preferred equity bailout, and Com for the common stock bailout). p max(w(C-ED)-V(C), U(C-[(1+n)(1-y)-nxsIn v(O)I if C<C<C (i, 1i)IfC<C<C, v(C)=1 exp(mIv(C-C))+12 exp(m2v(C-o) p[U(C-[(1+n)(1-y)-nxe-v(OI if C<c<c(ii) - exp(miu( (p+ max WC-|+K2)→K2]0}-v(O 1-A)ov(O)ifC<c<c and bailout in the form of preferred equity where /R+2(p G maxmax U(C-[(1+n)(1-y)-nx+K1 R+2(p+o)a2 W(C-5+K v(C) (1-2)v(C) if C<C<C and bailout in the form of common stock G=G2(12+02(1-2+2(1-0102O12 (1-DR2 The first term 0 in brackets is the dividend payment per unit of (iii)if C<C< C and bailout in the form of preferred equity time. The next two terms [IR1+(1-D)R2-0Ve+i v(C)=2, exp(mI (C-C))+S2 exp(m2p(C-O) 02(1-D+21(1-D)o102012]Va capture the expected change in the continuation value caused by fluctuation in the bank fundamen (p+φ)2p+中 Under Condition(i)and (ii) Bank 1 can choose to liquidate or when C<C, V(C)=0. take over Project G with no government intervention. Under Condi (iv)if C<C<C and bailout in the form of common stoc tion(i), the last term, A(C) represents the expected impact, which occurs with probability ddt, on Bank 1s continuation value, MO, of V(C)=Q3 exp(mIv(C-C))+Q4 exp(m2v(C-C)) taking over or liquidating Project G, whichever is better. W is Bank Ry4925+24c Is value function if it selects to liquidate Project G, and u is Bank 1s (p+d)2 value function if it selects to take over Project G Under Condition i).the last term, 1(0), which occurs with probability odt, repre- where Ss and when C<C, V(O=0 sents the expected impact of taking over Project G on WO(the de- We can find l1, l2 in Eq.(5),521,Q2, Ce in Eq.(6).and tailed derivations of U and W functions are provided in Appendix A). 223, Q4, Com in Eq. (7). using the boundary conditions as follow Under Condition(ii)and(iv), Bank 1 expects government bail-(1)AtC, v(C)=0(2)vO)is continuous at C (3)Vd C)is continuous out if crisis occurs with an ted probability of iodt. The last at C(4)Vc=1 at c(5)V=0 at C term, A( Q), represents the expected impact on vo) trom the shock The first boundary condition states that Bank 1 will be liquidated if nd the government bailout. U(C-(1+n(1-y)-nx5Jl+ Ki)is Bank I's value function if it accepts government capital and takes the capital level is lower than the regulatory requirement, i.e Project G. wc-al+K) is Bank I's value function if it accepts C<C. Condition (2)and (3)obtains because the sample paths for the government's capital and liquidates Project G K,(K2)is the C across the C boundary are continuous. When c>C, the level amount of capital injection that maximizes Bank 1's value function and change of Bank 1's value function can be continuously adjusted if it takes over (liquidates) Project G. Under Condition(iii), the by changing dividend policy in the neighborhood of C In Condition shareholders of preferred stock will receive only the fixed dividend (4)and (5, c" is the desired long-run or target level of capitalization and will not share in the upside gain should the bank recover. at which all earnings are paid out. At C any increment to capital is4.2. Bank 1’s problem In anticipation of crisis, Bank 1’s post-crisis value functions in different scenarios (liquidation of the joint project, continuation without bailout, or continuation with bailout) will determine Bank 1’s choice of whether to liquidate or continue the joint asset, and hence its pre-crisis value function, V(C), and its target level of cap￾ital, C⁄ . The application of standard techniques shows that the ex￾ante value function V satisfies Hamilton–Jacobi–Bellman (HJB henceforth) differential equation in the following general form: qV ¼maxh hþ½lR1 þ ð1lÞR2 hVc þ1 2 r2 1l 2 þr2 2ð1lÞ 2 þ2lð1lÞr1r2x12 h iVcc ( )þKðCÞ ð3Þ where K(C) takes the following forms depending on the relation￾ship between C; C; b C, and bb C : / maxfWðC nlÞ VðCÞ;UðC ½ð1 þ nÞð1 yÞ nxnlÞ VðCÞg if b C < bb C < C ðiÞ /½UðC ½ð1 þ nÞð1 yÞ nxnlÞ VðCÞ if b C < C < bb C ðiiÞ k/ maxfmaxK1>b CC ½UðC ½ð1þnÞð1yÞ nxnlþK1Þ K1; max K2> b b C C ½WðC nlþK2Þ K2;0g VðCÞ 8 < : 9 = ; ð1kÞ/VðCÞ if C < C 6 b C and bailout in the form of preferred equity ðiiiÞ and k/ max maxK1>b CC C CþK1 UðC ½ð1þnÞð1yÞ nxnlþK1Þ  h i; max K2> b b C C C CþK2 WðC nlþK2Þ h i) VðCÞ 8 >>>>< >>>>: 9 >>>>= >>>>; ð1kÞ/VðCÞ if C < C 6 b C and bailout in the form of common stock ðivÞ The first term h in brackets is the dividend payment per unit of time. The next two terms ½lR1 þ ð1 lÞR2 hVc þ 1 2 r2 1l 2 þ h r2 2ð1 lÞ 2 þ 2l1ð1 lÞr1r2x12Vcc capture the expected change in the continuation value caused by fluctuation in the bank fundamen￾tal V. Under Condition (i) and (ii) Bank 1 can choose to liquidate or take over Project G with no government intervention. Under Condi￾tion (i), the last term, K(C), represents the expected impact, which occurs with probability /dt, on Bank 1’s continuation value, V(C), of taking over or liquidating Project G, whichever is better. W is Bank 1’s value function if it selects to liquidate Project G, and U is Bank 1’s value function if it selects to take over Project G. Under Condition (ii), the last term, K(C), which occurs with probability /dt, repre￾sents the expected impact of taking over Project G on V(C) (the de￾tailed derivations of U and W functions are provided in Appendix A). Under Condition (iii) and (iv), Bank 1 expects government bail￾out if crisis occurs with an expected probability of k/dt. The last term, K(C), represents the expected impact on V(C) from the shock and the government bailout. U(C [(1 + n)(1 y) nxn]l + K1) is Bank 1’s value function if it accepts government capital and takes over Project G. W(C nl + K2) is Bank 1’s value function if it accepts the government’s capital and liquidates Project G. K1 (K2) is the amount of capital injection that maximizes Bank 1’s value function if it takes over (liquidates) Project G. Under Condition (iii), the shareholders of preferred stock will receive only the fixed dividend and will not share in the upside gain should the bank recover. Under Condition (iv), an injection of common equity dilutes exist￾ing shareholder interests and hence provides a stronger incentive for Bank 1 to hold more capital to cope with the failure of other banks. The term (1 k)/V(C) reflects the expected effect on Bank 1’s continuation value, V(C), from the shock and no government bailout, which occurs with probability (1 k)/dt. Finally, if C 6 C, Bank 1’s capital holding does not meet the reg￾ulatory requirement, and is also insufficient to take over or liqui￾date Project G. Bank 1 will be liquidated. VðCÞ ¼ 0 ð4Þ 4.3. Analytical solutions Using the dynamic stochastic programming techniques, we find the following analytical solutions for the value function V(C) and the optimal capital holding C ðC pre for the preferred equity bailout, and C com for the common stock bailout). (i, ii) If b C < bb C < C, VðCÞ ¼ P1 expðm1V ðC CÞÞ þ P2 expðm2V ðC CÞÞ /W1 1 2 r2 Vm2 1U þ RVm1U ðq þ /Þ expðm1UðC b CÞÞ /W2 1 2 r2 Vm2 2U þ RVm2U ðq þ /Þ expðm2UðC b CÞÞ ð5Þ where m1V ¼ RV þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 V þ 2ðq þ /Þr2 V q r2 V ; m2V ¼ RV ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 V þ 2ðq þ /Þr2 V q r2 V ; r2 V ¼ r2 1ðlÞ 2 þ r2 2ð1 lÞ 2 þ 2lð1 lÞr1r2x12; RV ¼ lR1 þ ð1 lÞR2 (iii) if C < C 6 b C and bailout in the form of preferred equity: VðCÞ ¼ X1 expðm1v ðC CÞÞ þ X2 expðm2v ðC CÞÞ þ RV k/ ðq þ /Þ 2 þ k/ q þ /ðC e CÞ ð6Þ when C < C; VðCÞ ¼ 0. (iv) if C < C 6 b C and bailout in the form of common stock: VðCÞ ¼ X3 expðm1V ðC CÞÞ þ X4 expðm2V ðC CÞÞ þ RV k/X5 ðq þ /Þ 2 þ k/X5 q þ /C ð7Þ where X5 ¼ ð1þnÞlR1þð1lÞR2 q C f g uþ½ð1þnÞð1yÞnxnl and when C < C; VðCÞ ¼ 0. We can find P1, P2 in Eq. (5), X1; X2; C pre in Eq. (6), and X3; X4; C com in Eq. (7), using the boundary conditions as follows: (1) At C; VðCÞ ¼ 0 (2) V(C) is continuous at b C (3) VC(C) is continuous at b C (4) VC = 1 at C⁄ (5) VCC = 0 at C⁄ . The first boundary condition states that Bank 1 will be liquidated if the capital level is lower than the regulatory requirement, i.e., C < C. Condition (2) and (3) obtains because the sample paths for C across the b C boundary are continuous. When C > C, the level and change of Bank 1’s value function can be continuously adjusted by changing dividend policy in the neighborhood of b C. In Condition (4) and (5), C⁄ is the desired long-run or target level of capitalization at which all earnings are paid out. At C⁄ any increment to capital is S. Tian et al. / Journal of Banking & Finance 37 (2013) 2765–2778 2771