La situación actual de los directivos

Similar to the definition in Chapter 2, the liquidation time τd is the first-passage time to a

specified asset-liability ratiod∗,

τd = in f{t≥0 : ¯vt ≤d

∗

}. (4.14)

In the case that regulators allow the firm to fail without enforcing conversion (τd < ξ), the

barrier is the level DEF0 = d∗L. This case is only possible for NSI financial institutions.

In the event that conversion is enforced prior to liquidation (τd > ξ), the barrier is the level

DEF1= d∗(LD+LS). This case is possible for both NSI and TBTF financial institutions

Being consistent with Chapter 2, we assume the financial institution keeps operating until liquidation happens. LetEx[·] be the expectation with the asset value diffusion process starting

atV0 = x. Given the fixed imposed loss 100βCCB% to the notional value of CCB at conversion,

6_{We don’t have a formal mathematical proof that the intensity function in (4.13) is such that conversion to} precede liquidation with probability one, but the numerical evidence (in Table 4.10) indicates that this probability is extremely close (if not exactly equal to) one.

the fair coupon rates (par yields) are solved from the following equation system LD= cDLD r 1−Ex[e −rτd_]+_R DLDEx[e−rτd], (4.15) LS = cSLS r 1−Ex[e −rτd_]+_R SLSEx[e−rτd], (4.16) LJ = cCCBLJ r 1−Ex h e−rτdi+₍₁₋β CCB)LJEx h e−rξ1{ξ≤τd} i +RJLJEx h e−rτd1 {τd<ξ} i , (4.17) whereτd = min{τd, ξ}and the recovery rates at liquidation for deposits, the senior bond and

CCB are RD, RS and RJ respectively. Only whenτd < ξ (i.e. regulators never enforce con-

version) is there a recovery payment to the CCB investor. To respect seniority, we assume

RD >RS >RJ.

To estimate the credit spreads, we need to calculate the Laplace transform of the liquidation time and conversion time in the equation systems. When the conversion trigger is objective then the Laplace transform of the conversion time can be expressed in terms of Laplace transforms of hitting times of affine geometric Brownian motion to fixed levels, which is available in closed form (Metzler [9]) and happens to involve the confluent hypergeometric function. However, in this chapter, the conversion timeξis no longer the hitting time of asset value to a fixed level. Thus, the analysis is more complicated and we require the transform of the “killing time”ξas well as two conditional transforms. Although we are not able to compute these transforms in closed form, we are able to characterize them as solutions to second-order ordinary differential equations. By numerically solving the ordinary differential equations, we obtain the value for the transforms.

In the following, we state the theorems of relevant differential equations and the cor- responding boundary conditions. Theoretical proofs are presented in Appendix D. For the reader’s convenience, we summarize the results in Table 4.1 at the end of this section.

Theorem 4.2.1 Let U(x)=Ex[e−αξ1{ξ≤τd}]. Then U(x)is the solution of the following second-

order ordinary differential equation

GU(x)−(k(x)+α)U(x)+k(x)=0, x>DEF0, (4.18)

where the operatorG= 1₂σ2(x)_dxd22 +µ(x,C0)

d dx.

Different boundary conditions are applied for different categories of intensity functions. Under the NSI intensity function, the boundary conditions areU(DEF0)= 0 and limx→+∞U(x)= 0. The first condition makes sense because as the asset value approaches to the level DEF0,

it is almost sure that the NSI institution will fail without the conversion of contingent capital (i.e., ξ > τd). In other words, as x → DEF0, the value of the indicator 1{ξ≤τ_d} approaches to zero. As the asset value approaches infinity, we have the boundary condition limx→+∞U(x)=0 according to the assumption (4.9). Under the TBTF intensity function, regulators will always enforce conversion prior to liquidation, leading to the boundary conditionU(DEF0)= 1. This

is because as the asset value approaches to the levelDEF0, contingent debt is more likely to be

converted to common shares. As a result, we haveξ →0 asx→ DEF0. The second boundary

4.2. Model 83

Theorem 4.2.2 Let Z(x)=Ex[e−ατd1{τd<ξ}]. Then Z(x)is the solution of the following second-

order ordinary differential equation

GZ(x)−k(x)Z(x)−αZ(x)= 0, x> DEF0, (4.19)

where the operatorG= 1₂σ2(x)_dxd22 +µ(x,C0)

d dx.

Under the NSI intensity function, we have limx→∞Z(x) = 0 because as the asset value approaches to infinity, the hitting timeτd →+∞. The other boundary condition isZ(DEF0)=

1 due to the reason that as x → DEF0, we have τd → 0 and the value of the indicator 1{τd<ξ} approaches to one. Under the TBTF intensity function, the boundary conditions areZ(DEF0)=

0 and limx→∞Z(x) =0. As x → DEF0, it is more likely that regulator will enforce conversion

(ξ→ 0) before the failure of the institution leading to the indicator1{τd<ξ}approaches to zero.

Theorem 4.2.3 Let W(x) = Ex[e−ατd]. Then W(x)solves the following second-order ordinary

differential equation

GW(x)−(k(x)+α)W(x)+k(x)W1(x)= 0, x> DEF0, (4.20)

where the operator G = 1₂σ2(x)_dxd22 +µ(x,C0)dxd and W1(x) = Ex[e

−ατd_{], which is calculated}

based on the post-conversion diffusion process, i.e., the fixed coupon payment for the diffusion process is C1.

Based on the assumption (4.12), limx→+∞W(x)=0 holds for both TBTF and NSI intensity functions. Under the NSI intensity function, the other boundary condition is W(DEF0) = 1.

This is because asx→ DEF0, we haveτd →0. In the TBTF case, asx→ DEF0,τdconverges

to the hitting time of an affine geometric Brownian motion beginning at the levelDEF0to the

level DEF1. Therefore,W(DEF0) can be expressed in terms of the known Laplace transform

for such hitting times, i.e.W(DEF0)=EDEF0[e

−ατd_{] (see Theorem B.1.6 in Appendix B).} Table 4.1 gives a summary of the relevant differential equations and the corresponding boundary conditions.

Table 4.1: ODEs and Boundary Conditions

Transforms ODEs and Boundary Conditions

U(x)= Ex[e−αξ1{ξ≤τd}]

GU(x)−(k(x)+α)U(x)+k(x)=0, x> DEF0

TBTF U(DEF0)=1, limx→+∞U(x)= 0 NSI U(DEF0)=0, limx→+∞U(x)= 0

Z(x)=Ex[e−ατd1{τd<ξ}]

GZ(x)−k(x)Z(x)−αZ(x)= 0, x> DEF0

TBTF Z(DEF0)=0, limx→+∞Z(x)=0 NSI Z(DEF0)=1, limx→+∞Z(x)=0

W(x)= Ex[e−ατd]

GW(x)−(k(x)+α)W(x)+k(x)W1(x)= 0, x> DEF0

TBTF W(DEF0)=EDEF0[e

−ατd_{], lim}

x→+∞W(x)=0

NSI W(DEF0)=1, limx→+∞W(x)=0 So far we derived the ordinary differential equations and their boundary conditions cor- responding to the transforms needed in the equation system. We do not need to derive the

differential equation for the termEx[e−rτd] in (4.17) since it equals to the summation ofU(x)

andZ(x). By solving the differential equations, we can estimate the par yields under the con- version term of fixed imposed loss.

Similar to our discussions under the objective conversion trigger in Chapter 2, we also investigate the interval for reasonable conversion terms when the conversion trigger depends on regulatory discretion. Recall Figure 1.4 in Chapter 1, a reasonable conversion term must guarantee the liability seniority in the capital structure and the existing shareholders are not rewarded for the poor performance of the firm at the same time.