On Equilibrium When Contingent Capital Has a Market Trigger: A Correction to Sundaresan and Wang Journal of Finance (2015)

• DOI: http://doi.org/10.1111/jofi.12762
• Author: GEORGE PENNACCHI,ALEXEI TCHISTYI
• Date: 01 June 2019
• Published date: 01 June 2019

THE JOURNAL OF FINANCE •VOL. LXXIV, NO. 3 •JUNE 2019

On Equilibrium When Contingent Capital Has a

Market Trigger: A Correction to Sundaresan and

Wan g Journal of Finance (2015)

GEORGE PENNACCHI and ALEXEI TCHISTYI∗

ABSTRACT

This paper identifies an error in Sundaresan and Wang (2015, hereafter SW) that

invalidates its Theorem 1. The paper develops a model of contingent capital (CC)

with a stock price trigger that is consistent with SW’s framework and yields closed-

form solutions for stock and CC prices. Yet, the model shows that unique stock price

equilibria exist for a broader range of CC contractual terms than those required by SW.

Specifically, when conversion terms benefit CC investors and penalize shareholders,

a unique equilibrium can exist rather than the multiple equilibria stated in SW.

THIS PAPER NOTES AN ERROR IN SUNDARESAN AND WANG (2015, hereafter SW)

that invalidates its Theorem 1. Section II of SW presents a continuous-time

structural model of a bank that issues senior debt, contingent capital (CC),

and shareholders’ equity. The conversion of CC from debt to equity is assumed

to be triggered by the market value of the bank’s equity or stock price. SW’s

Theorem 1states that a unique equilibrium for the bank’s stock price exists

only if a particular pricing restriction on the CC holds both at the time of

conversion and at all times prior to conversion. SW describes conversion terms

that satisfy this pricing restriction as requiring “no value transfer” between

CC investors and the bank’s initial shareholders.

We show that SW’s requirement for a unique equilibrium is too severe. In

particular, SW’s pricing restriction is necessary only at the time of conversion,

not before. Consequently, unique stock price equilibria can exist for a broader

range of CC contractual terms that penalize shareholders by transferring value

to CC investors. Such a transfer occurs when CC converts to a value of new

equity that exceeds the value of CC cash flows in the absence of conversion. In

concurrent and independent research, Glasserman and Nouri (2016, hereafter

GN) also show that unique stock price equilibria exist when conversion trans-

fers value from shareholders to CC investors. The main difference between

our paper and GN is that we provide a specific parametric example that yields

∗George Pennacchi and Alexei Tchistyi are with the Department of Finance at the University

of Illinois. The authors thank the Editor, Philip Bond; an anonymous Associate Editor; and three

anonymous referees for valuable comments. Wehave not received external financial support for this

research and have no conflicts of interest as identified in the Journal of Finance’s disclosure policy.

DOI: 10.1111/jofi.12762

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1560 The Journal of Finance R

closed-form expressions for unique equilibrium stock and CC prices, while GN’s

analysis establishes uniqueness for a more general class of economies.

We present a special case of SW’s continuous-time model that leads to valu-

ation formulas for CC and shareholders’ equity. Our model’s conditions for the

existence and uniqueness of equilibrium prices are broader than those stated

in SW’s Theorem 1due to an error in the theorem’s proof. The impact of this

error is nontrivial. The abstract of SW states that “The ‘no value transfer’ re-

striction precludes penalizing bank managers for taking excessive risk,” and

its Section III criticizes other research based on this conclusion. In contrast,

we show that proposals such as those in Calomiris and Herring (2013)and

Pennacchi, Vermaelen, and Wolff (2014) that penalize a bank’s initial share-

holders by transferring value to CC investors at conversion can lead to a unique

stock price equilibrium, rather than the multiple equilibria claimed in SW.

To better understand the existence and uniqueness of equilibrium when CC

conversion is triggered by a bank’s stock price, in the next section, we present a

simplified model consistent with SW’s environment. We show that unique equi-

libria for stock and CC values exist for conversion terms that contradict SW’s

Theorem 1. In Section II we identify the error in the proof of this theorem, which

states that a unique equilibrium requires the CC’s value to always equal its

value at the time it converts to equity. We further show that the correct logic

permits the preconversion value of CC to be less than its conversion value,

which occurs when conversion transfers value from shareholders to CC in-

vestors. In Section III, we explain why multiple equilibria are possible in static

or deterministic models but are impossible in our continuous-time, stochastic

model. We also discuss why allowing conversion to depend on “sunspots” does

not affect our model’s results. Finally, in Section IV we conclude.

I. A Counterexample

In this section, we develop a model that is consistent with the continuous-

time framework of SW but, as shown in our Corollary 1, contradicts SW’s

Theorem 1. Our notation follows SW’s Section II, which we encourage the

reader to review for comparison. All of our proofs are given in the Appendix.

A. Model Assumptions

Let there be a risk-neutral probability space (,F,{Ft,t∈[0,T]},P)in

which the information flow {Ft,t∈[0,T]}is generated by the Brownian

motion zt. The value of a bank’s assets follows the geometric Brownian motion

process

dA

t=μAtdt +σAtdzt,(1)

where μand σ>0 are constants. The assets generate cash flows at the rate of

a>0, that is, the total cash flow during a short period dt is aA

tdt.Letr>0be