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

Transcription

1 O Equilibrium whe Cotiget Capital has a Market Trigger: A Correctio to Sudaresa ad Wag Joural of Fiace (2015) George Peacchi Alexei Tchistyi y March 13, 2018 Abstract This paper ideti es a error i Sudaresa ad Wag (2015), hereafter SW, that ivalidates its Theorem 1. The paper develops a model of cotiget capital (CC) with a stock price trigger that is cosistet with SW s framework ad yields closed-form solutios for stock ad CC prices. Yet the model shows that uique stock price equilibria exist for a broader rage of CC cotractual terms tha those required by SW. Speci cally, whe coversio terms bee t CC ivestors ad pealize shareholders, a uique equilibrium ca exist rather tha the multiple equilibria stated i SW. Departmet of Fiace, Uiversity of Illiois, College of Busiess, 4041 BIF, 515 East Gregory Drive, Champaig, Illiois Phoe: (217) gpeacc@illiois.edu. y Departmet of Fiace, Uiversity of Illiois, College of Busiess, 339 Wohlers Hall, 1206 S. Sixth Street, Champaig, Illiois Phoe: (217) tchistyi@illiois.edu.

2 1 Itroductio This paper otes a error i Sudaresa ad Wag (2015), hereafter SW, that ivalidates its Theorem 1. Sectio II of SW presets a cotiuous-time structural model of a bak that issues seior debt, cotiget capital (CC), ad shareholders equity. The coversio of CC from debt to equity is assumed to be triggered by the market value of the bak s shareholders equity or stock price. SW s Theorem 1 states that a uique equilibrium for the bak s stock price exists oly if a particular pricig restrictio o the CC holds both at the time of coversio ad at all times prior to coversio. SW describes coversio terms that satisfy this pricig restrictio as requirig o value trasfer betwee CC ivestors ad the bak s iitial shareholders. We show that SW s requiremet for a uique equilibrium is too severe. Rather, SW s pricig restrictio is ecessary oly at the time of coversio, ot before. Cosequetly, uique stock price equilibria ca exist for a broader rage of CC cotractual terms that pealize shareholders by trasferrig value to CC ivestors. Such a trasfer occurs whe CC coverts to a value of ew equity that exceeds the value of CC cash ows i the absece of coversio. I cocurret ad idepedet research, Glasserma ad Nouri (2016), hereafter GN, also show that uique stock price equilibria exist whe coversio trasfers value from shareholders to CC ivestors. The mai di erece betwee our paper ad GN is that we provide a speci c parametric example that yields closed-form expressios for uique equilibrium stock ad CC prices. O the other had, GN s aalysis establishes uiqueess for a more geeral class of ecoomies. We preset a special case of SW s cotiuous-time model that leads to valuatio formulas for CC ad shareholders equity. Our model s coditios for the existece ad uiqueess of equilibrium prices are broader tha those stated i SW s Theorem 1 due to a error i the theorem s proof. The impact of this error is otrivial. The abstract of SW states The o value trasfer restrictio precludes pealizig bak maagers for takig excessive risk, ad its Sectio III criticizes other research based o this coclusio. I cotrast, we show that proposals such as Calomiris ad Herrig (2013) ad Peacchi et al. (2014) that pealize a bak s iitial shareholders by trasferrig value to CC ivestors at coversio ca lead to a uique stock price equilibrium, rather tha the multiple equilibria stated i SW. To better uderstad the existece ad uiqueess of equilibrium whe CC coversio is triggered by a bak s stock price, the ext sectio presets a simpli ed model cosistet with SW s eviromet. It shows that uique equilibria for stock ad CC values exist 1

3 for coversio terms that cotradict SW s Theorem 1. Sectio 3 ideti es the error i the proof of this theorem which states that a uique equilibrium requires the CC s value to always equal its value at the time it coverts to equity. Istead, we show that the correct logic permits the CC s pre-coversio value to be less tha its coversio value, which occurs whe coversio trasfers value from shareholders to CC ivestors. Sectio 4 explais why multiple equilibria are possible i static or determiistic models but are impossible i our cotiuous-time, stochastic model. It also discusses why allowig coversio to deped o suspots does ot a ect our model s results. Sectio 5 cocludes. 2 A Couterexample This sectio develops a model that is cosistet with the cotiuous-time framework of SW yet, as will be show i our Corollary 1, cotradicts SW s Theorem 1. Our otatio follows SW s Sectio II, which we ecourage the reader to compare. All of our proofs are give i the Appedix. 2.1 Model Assumptios Let there be a risk-eutral probability space (; F; ff t ; t 2 [0; T ]g ; P ) i which the iformatio ow ff t ; t 2 [0; T ]g is geerated by the Browia motio z t. The value of a bak s assets follows the geometric Browia motio process da t = A t dt + A t dz t (1) where ad > 0 are costats. The assets geerate cash ows at the rate of a > 0, i.e., the total cash ow durig a short period dt is aa t dt. Let r > 0 be the costat risk-free rate of iterest. I risk-eutral probability measure, we should have = r a. We assume that the bak has issued a seior bod with a par value of B ad a maturity date of T. The coupo rate of the seior bod, b, is equal to the risk-free rate r. Bak regulators are assumed to close the bak whe the value of its assets falls to B so that the bak s closure (bakruptcy) date satis es = ifft 0 : A t Bg: (2) 2

4 Assumig o bakruptcy costs, the seior bod is default-free ad has a market price B t = B: (3) Besides the seior bod, the bak s capital structure cosists of shareholders equity ad CC havig a maturity date of T ad a par value of C. Prior to coversio, CC pays xed coupo iterest cotiuously at the rate c. The iitial shareholders ow shares of stock where S t is the price per share (if it exists). CC is assumed to covert automatically to m 0 additioal shares whe the equity value S t falls to the level K or lower for the rst time. Let = [0; T ] be all poits i time startig from the iitial date 0 ad icludig the CC s maturity date T > 0 durig which the equity value is compared to the trigger. The rst time a stock price is foud to be equal to or lower tha the trigger is = ifft 2 : S t Kg (4) where it is de ed that = +1 if S t > K for all t 2. The quatity m is referred to as the coversio ratio ad K as the coversio trigger. The coversio trigger is hit if the bak s per share stock price falls to K=, which is referred to as the trigger price. After cotractual coupos o the seior bod ad CC are paid, the cash ow geerated from the assets of the bak will be paid to shareholders as divideds. Therefore, before coversio, the total divided paid to shareholders durig a short period dt is (aa t rb cc)dt. After coversio ad before the seior bod s maturity or the bak s closure by regulators, the total divided paid to shareholders (icludig those ew shareholders after coversio) durig a i itesimal period dt is (aa t rb)dt. As will become clear, we also assume that A 0 > K +m + B at the iitial date 0 to rule out the case where the CC coverts immediately at the time it is rst issued. Our setup is a special case of SW s more geeral model. SW allows the bak s asset value A t to jump ad also permits a time-varyig cash ow rate a t, risk-free rate r t, volatility t, ad bod ad CC coupo rates b t ad c t. All of these parameters are costat i our model. SW also allows bakruptcy costs ad a more geeral default barrier t. We assume zero bakruptcy costs ad t = B. A major bee t of our simpli ed settig is that we are able to derive closed-form solutios for stock ad CC prices that are relatively easy to iterpret. Yet SW states o page 897 that its results hold for our case that has asset values followig a geometric Browia motio ad o bakruptcy costs. Thus, by showig that our model yields 3

5 uique equilibria that violate the pricig restrictio i SW s Theorem 1, we provide a couterexample The Equilibrium Stock Price At ay time t before the CC coverts (t < ), the per-share value of commo equity is, i ratioal expectatios, h 1 S t = E t + 1 +m (A T B C)e r(t t) 1 mif;g>t + R mif;;t g aa t s rb cc e r(s t) ds (A T B)e r(t t) 1 T < + R mif;t g aa s rb i e r(s t) ds 1 <mif;t g, where E t [] deotes the expectatio coditioed o F t. The value of the CC before coversio is h C t = E t Ce r(t t) 1 mif;g>t + R mif;;t g cce r(s t) ds t + m +m (A T B)e r(t t) 1 T < + R mif;t g aa s r B e r(s t) ds 1 <mif;t g i. (5) (6) After the CC coverts to m shares ad before the seior bod matures or regulators close the bak ( t < mi f; T g), the bak s liabilities cosist of + m equity shares ad the seior bod whose value satis es equatio (3). Therefore, the per-share value of commo equity must satisfy S t = u t where the post-coversio share price is u t 1 + m A t B : (7) Note that u t i (7) is a well-de ed liear fuctio of A t for 0 t <. Give the coversio trigger K ad coversio ratio m, a pair of value fuctios, (S t ; C t ), that satisfy equatios (4), (5), (6), ad (7) is called a equilibrium. We ext cosider how stock prices must evolve if there exists oe equilibrium or multiple equilibria. Sice the equity value process is adapted to the Browia motio z t ad divideds are 1 GN allow bak assets to follow a jump-di usio process ad show that there exist uique stock price equilibria for CC cotractual terms that trasfer value from shareholders to CC ivestors. Our model where asset values follow geometric Browia motio is a special case of their geeral settig. 4

6 paid cotiuously, it ca be show that ay equilibrium stock price S t satisfyig (5) must be cotiuous over time. Lemma 1: If there are oe or more equilibria, the each equilibrium stock price is adapted to the iformatio ow ff t ; t 2 [0; T ]g ad is cotiuous i t. Moreover, its correspodig coversio time is a stoppig time with respect to ff t ; t 2 [0; T ]g. Lemma 1, whose formal proof is i the Appedix, says that there must be o jump i a equilibrium stock price, icludig o jump at a time of coversio which is F t - measurable. The ituitio for this result is that the per-share value of commo equity i (5) is the expected discouted value of the divideds per share before ad after coversio, coditioal o the iformatio geerated by the Browia motio z t. Sice this iformatio is cotiuous, a equilibrium stock price equal to this expectatio caot jump. Thus far we have ot ruled out multiple equilibria, with stock prices satisfyig (5) ad each with a di eret coversio date,. I other words, sice the coversio time (4) depeds o the stock price ad the stock price (5) depeds o the coversio time, there is a edogeeity that may ot ecessarily esure that a equilibrium is uique. However, ote that a equilibrium stock price s cotiuity requiremet implies that the prices just before coversio, S, ad just after coversio, S +, are the same ad equal to the trigger price K=. Now sice the post-coversio stock price, u t, i (7) is uique for every A t, de e A uc as the asset value such that this post-coversio stock price equals K=. From (7) this asset value is A uc = K + m + B. (8) Requirig that S t equal u t at the time of coversio ad also that S t be cotiuous has the followig implicatio: whe a equilibrium stock price exists, it must lead to coversio oly at the uique asset value A t = A uc. Coversio at ay di eret asset value implies a predictable jump i the stock s value that is icosistet with a cotiuous-time equilibrium. Ituitively, ote that cotiuity ad the de itio of coversio requires that the stock price equal the trigger just before coversio, S = K=. Coversio whe A t > A uc implies a jump up i the stock price immediately after coversio to S + = u t > K= ad is icosistet with coversio occurrig. Similarly, coversio whe A t < A uc leads to a jump dow i the stock price immediately after coversio to S + = u t < K=, which sice S t is cotiuous implies that coversio should have occurred earlier. Due to the cotiuous iformatio ff t ; t 2 [0; T ]g geerated by the 5

7 Browia motio z t, a jump i the stock price at the time of coversio is icosistet with a ratioal equilibrium. This reasoig leads to the followig propositio: Propositio 1: If there is a equilibrium stock price, the coversio happes whe A t falls to A uc for the rst time; that is, = if ft 2 [0; T ] : A t A uc g. (9) Note that this propositio rules out the possibility that coversio occurs whe A t = A uc strictly after the rst time that A t falls to A uc. The ituitio is that if S t remaied strictly above K= at the rst time that A t equals A uc, the due to the properties of a geometric Browia motio there is a strictly positive probability that A t could fall to B before it returs to A uc or the CC matures. But this bak closure evet leads to a cotradictio: at the bakruptcy date it must be that S = 0 < K=, i which case there must be a dowward jump i the stock price from above to strictly below the trigger. Propositio 1 depeds o the property that there is a strictly positive probability that bak assets ca fall su cietly far prior to maturity, which is satis ed whe > 0. Sectio 4.2 discusses the case where bak assets follow a determiistic process ( = 0) ad this property is violated. Propositio 1 implies that if there is a equilibrium, the the pre-coversio stock ad CC prices are equal to equatios (5) ad (6), respectively, where is the rst time that the asset value A t equals A uc i (8). The ext propositio shows that by settig A = A uc, oe obtais uique, closed-form solutios for these cadidate equilibrium stock ad CC prices. Propositio 2: If there is a equilibrium stock price, the the pre-coversio date t values of the stock ad CC equal S(A t ; A uc ; q) = 1 A t B C (A t ; A uc ; q) ; (10) C(A t ; A uc ; q) = c C r + e rq C c C mk c (1 F (q; A t ; A uc )) + C G(q; A t ; A uc ); r r (11) 6

8 where q T t is the CC s time util maturity, F (q; A t ; A uc ) = (x 1t (q)) + G(q; A t ; A uc ) = y 1t (q) = x 1t (q) = At A uc h t h t At A uc +z (y 1t(q)) + z 2 q p q At h t = l A uc 2 (x 2t(q)) ; (12) At A uc z (y 2t(q)) ; (13) ; y 2t (q) = h t + z 2 q p ; (14) q 2 q p ; x 2t (q) = h t + 2 q q p ; (15) q q r 2 ; = ; z = ad () is the cumulative stadard ormal distributio. 2 ; (16) Cosistet with the assumptio that coversio occurs whe A t equals A uc, it ca be veri ed from (10) ad (11) that S(A uc ; A uc ; q) = K ad C(A uc; A uc ; q) = mk. However, i order for (10) ad (11) to truly be equilibrium prices S(A t ; A uc ; q) must also exceed K the coversio trigger price,, wheever the asset value A t exceeds A uc. Otherwise, coversio would happe before A t falls to A uc, cotradictig Propositio 1. Derivig the coditios uder which S(A t ; A uc ; q) i (10) remais above K wheever A t > A uc is the basis for provig the followig theorem o the existece ad uiqueess of equilibria: 2 Theorem 1: If CC has a ite maturity ad (i) if mk maxf C; c C g, the there exists a uique equilibrium where coversio oc- r curs whe the bak s asset value declies to A uc for the rst time ad where the equilibrium stock prices before ad after coversio equal (10) ad (7), respectively; mk (ii) if (iii) if C mk < C, the there is o equilibrium stock price; CC s maturity is su cietly log. < maxfc; c C g, the there may be o equilibrium stock price if the r To uderstad the ituitio for Theorem 1, ote that coversio uambiguously pealizes shareholders whe the value of ew equity give to CC ivestors, m K, exceeds the CC s pricipal, C, ad its ucoverted perpetuity value, c C=r. Cosequetly, as the 2 GN have a somewhat di eret proof of the existece ad uiqueess of a stock price equilibrium, but their Corollary 5.1 reaches the same geeral coclusio as our Theorem 1. 7

9 bak s assets declie toward the coversio poit A uc, shareholder value declies more tha oe-for-oe with a declie i asset value esurig is strictly positive. Therefore, the cadidate equilibrium stock price (10) satis es the requiremet that it remais above the trigger price, K, prior to coversio. If, istead, coversio terms bee t shareholders by givig CC ivestors less ew equity tha the CC s pricipal or its ucoverted perpetuity value, as assets declie toward A uc the icreased likelihood of coversio ca raise iitial shareholder value ad make the slope of the cadidate stock egative. That leads to a icosistecy with equilibrium sice the cadidate stock price (10) is below the trigger price for asset values strictly greater tha A uc, implyig that coversio would ot occur at A uc. But with coversio occurrig earlier tha at A uc, the post-coversio stock price, u t, would jump above the trigger price, icosistet with equilibrium. 3 Whe coversio terms pealize shareholders so that a uique stock price equilibrium exists, the CC s market value, C t, rises with a fall i asset value. Ideed, sice coversio bee ts CC ivestors, its greater likelihood as assets declie is re ected i a rise i its market value toward its coversio value of C = m K. The followig corollary to Theorem 1 formalizes this reasoig. 4 mk Corollary 1: If > maxf C; c C g, the the uique equilibrium CC price is r C (A t ; A uc ; q) < mk= for all t strictly prior to coversio. 3 The Error i the SW Proof This sectio explais why our couterexample co icts with SW s Theorem 1 by poitig out the error i the proof of this theorem. Let us begi by recallig SW s Theorem 1, otig that our results represet a clear cotradictio. SW s proof rst derives a result aalogous to our Propositio 1, amely, that equilibrium uiqueess requires that 3 I other words, a equilibrium fails to exist because the cadidate stock price is ot a mootoically icreasig fuctio of the bak s assets for all A t A uc. Bod et al. (2010) also relate existece of equilibrium to a stock price s mootoicity with respect to a rm s assets. They aalyze a govermet s use of stock price iformatio to decide upo a itervetio that improves the rm s fudametal value ad, i tur, the value of its shareholders equity. Similarly, i our model whe coversio bee ts shareholders, a bak s cadidate stock price must both determie a itervetio (coversio) ad re ect that itervetio s improvemet i shareholders equity. Yet a di erece is that coversio does ot a ect a bak s fudametal value, but raises shareholders equity via a value trasfer from CC ivestors. 4 The Iteret Appedix illustrates the equilibrium stock ad CC prices i (10) ad (11) as a fuctio of the bak s asset value for di eret coversio terms ad realistic parameter values. 8

10 coversio occurs whe the post-coversio stock price equals the trigger price. But SW the icorrectly cocludes from this result that the CC price must always equal its coversio value, eve prior to coversio. SW s Theorem 1 is stated o page 896: THEOREM 1 (SW): For ay give trigger K t ad coversio ratio m t, a ecessary coditio for the existece of a uique equilibrium (S t ; C t ) is C t = m t K t for every t 2. I its proof, SW de es U t as the total value of shareholders equity of a bak that has idetical assets ad seior debt but has issued o CC. I our settig, U t = A t B. SW shows o pages i its Lemma A1 that this hypothetical bak s total shareholder value must equal the sum of the CC-issuig bak s total shareholder value plus CC value prior to coversio: U t = S t + C t. SW the derives the followig ecessary coditio, labeled (A10): if ft 2 : U t K t + C t g = if ft 2 : U t K t ( + m t ) =g : (A10) This coditio is equivalet to our Propositio 1, which states that if there is a equilibrium, coversio occurs the rst time A t falls to A uc. To see this, substitute U t = S t +C t ito the left-had side of (A10) (usig SW s Lemma A1) ad U t = A t B ito the righthad side of (A10), to obtai if ft 2 : S t K t g = if ft 2 : A t A uc g : (17) The step i SW s proof that we disagree with is the iferece that is draw from (A10). Speci cally, immediately after (A10) SW writes: The above equatio holds for all possible paths of U t if ad oly if K t + C t = K t ( + m t )= for all t 2, which implies m t = C t =K t for all t 2. Therefore, to have a uique equilibrium, the coversio ratio must satisfy m t = C t =K t for t 2. But while (A10) certaily implies that K t + C t = K t ( + m t )= at the istat of coversio, pre-coversio it implies oly the weaker statemet that U t > K t + C t if ad oly if U t > K t ( + m t ) =. Note that i our settig, SW Theorem 1 s requiremet for a uique equilibrium is equivalet to the CC always tradig at its par value. SW Theorem 1 s ecessary coditio is C t = m t K t = for every t 2, which whe m t ad K t are costats implies that C t = mk= is costat. Sice C T = C at maturity, C t = C = mk= ca be the oly 9

11 uique equilibrium. Istead, our Corollary 1 proves that uique equilibria ca exist whe C t < mk= for all t strictly prior to coversio. 4 Recocilig Multiple Equilibria i Other Settigs This sectio explais why multiple equilibria ca occur i settigs that are static or determiistic, but caot occur i our cotiuous-time, stochastic settig, eve if coversio is allowed to deped o ofudametal suspot iformatio. 5 As a prelimiary to comparig static ad determiistic versios of our model, let us assume that c = r ad cosider a hypothetical bak that is idetical to the CC-issuig bak but has its CC replaced with additioal ocovertible seior debt with pricipal C. Let v t be the stock price of this o-coversio bak. It is well-de ed ad give by v t = 1 A t B C (18) De e A vc as the asset level at which v t equals K=. The A vc = K + B + C (19) Note that if C < mk=, the A vc < A uc ad v t > K= u t for A t 2 (A vc ; A uc ]. 4.1 A Static Settig Cosider a static settig occurrig at the CC s maturity date, T. 6 If C < mk= ad A T 2 (A vc ; A uc ], the there are two equilibria i stock prices: ST c = u T K= is a equilibrium stock price cosistet with coversio at maturity but so is ST u = v T > K= a equilibrium cosistet with o coversio at maturity. However, this situatio at maturity could ever occur i our dyamic stochastic settig. Our Propositio 1 requires that coversio occurs at A uc, the upper boud of the iterval (A vc ; A uc ]. I other words, sice A 0 > A uc ad T > 0, it is impossible to have CC ucoverted ad A T < A uc at the same time. Moreover, the evet where the asset value equals A uc exactly at date T has zero probability ad caot a ect equilibrium prices. 5 More aalysis of the results i Sectios 4.1 ad 4.2 are give i GN Sectios 2 to SW uses a similar static example to motivate the claim of multiple equilibria i its dyamic model. 10

12 4.2 A Dyamic Determiistic Settig Two similar equilibria exist i a dyamic determiistic settig whe the followig coditios hold: = 0, A 0 > A uc, ad < 0. As a result, A t = A 0 e t is decreasig over i time. I additio, the CC s maturity date must be such that T 2 l(avc=a0 ) ; l(auc=a 0), so that A T 2 (A vc ; A uc ]. I oe equilibrium, coversio occurs whe the asset level falls to A uc for the rst time, which is at time = l(auc=a 0). This equilibrium s stock price is S c t = ( v t 1 mk C e r for t 2 [0; ] u t for t 2 [; T ] (20) I the secod equilibrium CC ever coverts, ad the stock price is S u t = v t for t 2 [0; T ]. However, it is importat to poit out that sice v t < K= whe A t < A vc, the secod equilibrium critically depeds o the assumptio that the asset level caot drop below A vc prior to the maturity date T. I the stochastic settig with > 0, oly the rst equilibrium exists. The secod equilibrium is ruled out by Propositio 1 that requires coversio to happe whe the asset level drops to A uc for the rst time. As discussed i Sectio 2.2 ad i the proof of Propositio 1, if the asset level is below A uc ad CC is ot coverted, there is a strictly positive probability that A t could fall to B prior to maturity. But the S = 0, leadig to a cotradictio. As a result, the secod equilibrium i the determiistic settig does ot exist whe > A Settig with Suspot Iformatio Our basic model assumes all price processes are adapted to the iformatio ow geerated by the Browia motio, z t, determiig the bak s fudametal asset value, A t. Sice this iformatio ow is cotiuous, a equilibrium stock price caot jump. Now geeralize this iformatio ow to iclude a ofudametal Poisso suspot process, N t, that is urelated to z t. Oe might cojecture that there are multiple suspot equilibria, sice this discotiuous suspot iformatio ca geerate jumps i the bak s pre-coversio stock price(s) by a ectig the time of coversio. 7 Propositio 3.3 of GN also proves that if it is impossible for A t to reach A vc prior to maturity, the S t = v t is a equilibrium. They describe this o-coversio equilibrium as of limited practical iterest ad models that violate this trigger accessibility coditio as rather cotrived. 11

13 However, as detailed i the Iteret Appedix, additioal suspot equilibria ca be ruled out. We rst ote that the post-coversio bak s stock price, u t, cotiues to be give by (7) sice the post-coversio bak s capital structure is ua ected by the time of coversio. The rest of the proof is based o the martigale represetatio theorem that allows the pre-coversio stock price to jump oly whe the Poisso suspot iformatio arrives. But if a suspot-geerated jump i the price causes coversio whe A t > A uc, it must be followed by a secod o-suspot, upward jump to the post-coversio stock price that violates the martigale represetatio theorem. The martigale represetatio theorem is also violated if A t falls to A uc without triggerig coversio: there is a strictly positive probability that A t declies to B where bakruptcy occurs ad S = 0 before there is a suspot jump or A t returs back to A uc or the CC matures. But this bakruptcy evet implies a o-suspot, dowward jump. Hece, suspots caot geerate additioal coversio times or equilibrium stock prices so that Propositios 1 ad 2 cotiue to hold ad the statemet of our Theorem 1 remais valid. 5 Coclusio Our model of a bak that issues CC with a stock price trigger is cosistet with SW s cotiuous-time framework ad yields closed-form solutios for stock ad CC prices. We prove that uique equilibria exist wheever CC coversio terms pealize (bee t) iitial shareholders (CC ivestors), leadig to a CC s pre-coversio price that is below its coversio value. This result is a couterexample to SW s coclusio that a uique equilibrium requires o value trasfer coversio terms ad that a CC s market value always equals its coversio value. We explai the co ictig results by idetifyig a error i the proof of SW s Theorem 1. This correctio has implicatios for CC proposals that seek to reduce a bak s riskshiftig ad debt overhag icetives. CC ca be desiged to pealize bak shareholders for takig excessive risk without ecessarily geeratig a multiplicity or absece of stock price equilibria. 12

14 6 Appedix 6.1 Proof of Lemma 1 Let be some radom time de ed o the probability space (; F; P ). If coversio happes at time, the the discouted per share realized payo to shareholders at date T plus divideds paid from date 0 to date T is equal to Y = 1 e rt (A T B C) 1T <mif;g + R T e rs aa 0 s rb cc 1 s<mif;g ds + 1 e rt (A +m T B) 1T < + e rr T e r(s ) aa 0 s rb 1 s< ds (A.1) Sice Y is a realized stream of cash ows discouted as of date 0, it is ot a stochastic process but a radom variable. De e X t E t Y where E t [] deotes the expectatio coditioal o F t, i.e., iformatio geerated by the Browia motio z t. By the de itio of coditioal expectatio, X t is measurable with respect to F t. Moreover, by the law of iterated expectatios, X t is a martigale adapted to ff t ; t 2 [0; T ]g, the ltratio geerated by Browia motio z t. Hece, X t is a cotiuous process sice all martigales adapted to a Browia ltratio are cotiuous. To make explicit that our proof of Lemma 1 allows for multiple equilibria, let the superscript i idicate that a variable correspods to the i th equilibrium. 8 The i each equilibrium i, there is a equilibrium stock price h St i 1 = E t + 1 +m e r(t t) (A T B C) 1T <mif i ;g + R T e r(s t) aa t s rb cc 1 s<mif i ;gds e r(t t) (A T B) 1 i T < + R T e r(s t) aa t s rb i 1 i s<ds, (A.2) ad a correspodig coversio time i = if t 2 : S i t K (A.3) where i = 1 if S i t > K 8t 2. We ote that process S i t is adapted to the iformatio ow ff t ; t 2 [0; T ]g, sice S i t is a coditioal expectatio give F t. 8 Our argumet does ot deped o the umber of di eret equilibria, which could be ucoutable. 13

15 Let X i t deote X t whe = i. Comparig X i t X i t = Z t 0 to S i t i (A.2) yields e rs 1 aa s r B c C 1 s<mif i ;g + 1 +m aa s r B 1 i s< ds + e rt S i t; (A.4) which ca be rewritte as Z t St i = e rt X i t e rs 1 aa s rb cc 1 s<mif i ;g + 1 aa +m s rb 1 i s< ds : Sice X i t 0 (A.5) ad the time itegral i (A.5) are cotiuous i t, St i must be cotiuous i t. Cosequetly, each equilibrium stock price, St, i must be cotiuous. Moreover, the coversio time i i (A.3) that correspods to the stock price St i ca be classi ed as a hittig time. Sice St i is a cotiuous process, i is a stoppig time with respect to ff t ; t 2 [0; T ]g Proof of Propositio 1 Sice the post-coversio price u t i (7) is strictly mootoe i A t, coversio at ay asset level other tha A uc would lead to a jump i the stock price, which caot be a equilibrium accordig to Lemma 1. Thus, coversio ca occur oly whe A t = A uc. To ish the proof we eed to show that coversio must happe whe A t falls to A uc for the rst time. Our argumet is that if it did ot, it would lead to a cotradictio. So, suppose that coversio did ot happe the rst time A t = A uc. Now de e the stoppig time " = if ft 2 [0; T ] : A t A uc "g (A.6) for some " > 0. Due to the properties of a geometric Browia motio, at time " there is a strictly positive probability that A t will declie to B before it returs back to A uc or the CC matures. I other words, there is a strictly positive probability that the bakruptcy date,, ca occur before the coversio date or maturity date. 10 Yet at bakruptcy shareholders lose all claims o the bak s assets, ecessitatig that S = 0. However, i this case coversio must occur whe A t is strictly below A uc, which 9 See Karatzas ad Shreve (1991) pages 6-7. That i is a stoppig time implies that oe ca decide based o F t whether or ot coversio has occurred: the coversio evet is F t -measurable. 10 The bak is closed by regulators whe A t = B. This assumptio is speci c to our model, yet ay sesible model of a levered rm leads to bakruptcy whe assets hit some critical lower boud. 14

16 leads to a cotradictio. Cosequetly, i equilibrium coversio must happe before time " for ay " > 0. Takig the limit "! 0 yields that coversio must happe at = if ft 2 [0; T ] : A t A uc g. 6.3 Proof of Propositio 2 Our model has similarities to the default-risky, ite maturity debt model of Lelad ad Toft (1996). Let f(s; A t ; A uc ) be the risk-eutral probability desity of the rst passage time of A t to A uc at date t + s, ad let F (s; A t ; A uc ) be correspodig cumulative distributio fuctio. The the oly possible cadidate equilibrium CC value at date t is C(A t ; A uc ; q) = Z q + 0Z q e rs c C (1 F (s; A t ; A uc )) ds + e rq C (1 F (q; At ; A uc )) 0 e rs mk f(s; A t; A uc )ds: (A.7) The rst term is the discouted risk-eutral expected value of the coupo ow, which is paid s periods i the future with probability (1 F (s; A t ; A uc )). The secod term is the risk-eutral expected discouted value of repaymet of pricipal, ad the third term is the risk-eutral expected discouted value of the shares give to CC ivestors at coversio if coversio occurs. Itegratig the rst term by parts yields C(A t ; A uc ; q) = c C r + e rq C where G(q; A t ; A uc ) c C mk (1 F (q; A t ; A uc )) + r Z q 0 e rs f(s; A t ; A uc )ds: cc G(q; A t ; A uc ); r (A.8) Harriso (1990) ad Rubistei ad Reier (1991) show that F ad G equal (12)-(16). (A.9) Fially, the pre-coversio per-share stock value must be equal to the asset value mius the value of the seior debt ad CC: S(A t ; A uc ; q) = 1 A t B C(A t ; A uc ; q) : (A.10) 15

17 6.4 Proof of Theorem 1 (i) The CC s coversio value exceeds its pricipal ad its coupo value i perpetuity: mk maxf C; c C r g; that is, coversio always bee ts CC ivestors. Whe A t > A uc = K +m + B, substitutig (11) ito (10) yields S(A t ; A uc ; q) = 1 (A t B C(A t ; A uc ; q)) > 1 A uc B C(A t ; A uc ; q) = K + 1 mk C(A t ; A uc ; q) = K + 1 mk c C (1 G(q; A t ; A uc )) e rq C r (A.11) c C (1 F (q; A t ; A uc )) : r Note that the cumulative distributio fuctio F (q; A t ; A uc ) 1. I additio, G(q; A t ; A uc ) F (q; A t ; A uc ); (A.12) sice G(q; A t ; A uc ) is give by (A.9) with r > 0. Because of mk maxf C; c C g ad (A.12), the term i square brackets i the last lie r of (A.11) is oegative. The implicatio is that S(A t ; A uc ; q) > K for ay q 0 ad ay A t > A uc ; that is, the stock price remais above the coversio trigger as log as the asset level remais above A uc. Thus, S(A t ; A uc ; q) is the uique equilibrium price prior to coversio. (ii) The CC s coversio value is less tha its pricipal: mk < C; that is, CC ivestors receive less tha the pricipal value at coversio. Note that a equilibrium stock price must be equal to K whe A t = A uc ad must be greater tha K for all A t > A uc. This requires that the stock price is icreasig i A t ear A uc. However, we ow show that this is ot the case whe mk < C. 16

18 Takig the derivative of S(A t ; A uc ; q) i (10) with respect to A t t ; A uc ; t = t ; A uc ; q) 1 t At=Auc = e rq C c (q) t At=A mk uc c t : (A.13) At=Auc If A t = A uc, the h t = 0, (x 2t (q)) = 1 (x 1t (q)) = p q, ad (y 2t (q)) = 1 (y 1t (q)) = z p q. As a result, we t t = At=Auc 1 A uc 1 A uc 2( p q) + 2 (p q) p q ( z) + 2z(z p q) + 2 (zp q) p q ; (A.14) ; (A.15) where () deotes the stadard ormal desity fuctio. Importatly, ote that as q! t! At=Auc t! 1: At=Auc Now equatio (A.13) ca be rewritte as t; A uc ; t = 1 + C (q) t At=Auc 1 e rq C (q) t cc (q) t At=A uc t : At=Auc (A.16) As q! 0, the secod term coverges to 1 whe mk < C. The last two terms i (q) coverge to zero t t are of the order of magitude of pq 1, uc At=A uc while (1 e rq ) rq << q At=A t uc At=A (z2 2 )q! 0. Thus, uc 2 p 2 p q 17

19 if mk < C, the lim Auc ; t = 1. (A.17) At=Auc Therefore whe mk < C, S is decliig i the bak s assets at a time su cietly close to maturity whe the asset level is ear A uc. This meas that the cadidate stock price falls below the trigger before the asset level drops to A uc. Accordig to Propositio 1, it caot be a equilibrium stock price. Cosequetly, a equilibrium stock price does ot exist for this case. (iii) The CC s coversio value exceeds its pricipal but is less tha its coupo value i perpetuity: C mk < c C r : Whe mk < c C there are parametric coditios such that a equilibrium stock price r will ot exist if the CC s maturity is su cietly log. This follows because as q! 1, the model is equivalet to the perpetual maturity CC model aalyzed i Peacchi ad Tchistyi (2015) where uder our assumptios the cadidate stock price equals S(A t ; A uc ; q = 1) = 1 q where A uc ; q = t A t B cc r 1 At A uc! m K At r 2 > 0. They show that = 1 cc At=Auc 1 r ( + m) B K m K A uc! # (A.18), (A.19) which is egative whe mk < c C ad K < cc B = ( + (1 + ) m), implyig that r r the cadidate stock price falls below the trigger before the asset level declies to A uc. From this result it ca be deduced that if CC has a ite but su cietly log maturity, there will be similar parametric coditios that lead to o equilibrium stock price. 18

20 6.5 Proof of Corollary 1 Corollary 1 cosiders the situatio where mk strictly exceeds maxf C; c C g ad coversio r has ot yet occurred so that A t > A uc. 11 Equatio (11) ca be rearraged as: C(A t ; A uc ; q) = c C r + e rq C = mk + C c C r < mk mk + = mk mk < mk mk = mk mk c C mk c (1 F (q; A t ; A uc )) + C G(q; A t ; A uc ) r r mk e rq c (1 F (q; A t ; A uc )) C (1 G(q; A t ; A uc )) r cc mk e rq c (1 F (q; A t ; A uc )) C (1 G(q; A t ; A uc )) r r (1 G(q; At ; A uc)) e rq (1 F (q; A t ; A uc)) c C r cc r cc r (1 F (q; At ; A uc)) e rq (1 F (q; A t ; A uc)) 1 e rq (1 F (q; A t ; A uc )) < mk (A.20) The rst iequality follows from the fact that mk > C. The secod iequality follows from the fact that G(q; A t ; A uc ) < F (q; A t ; A uc ) whe A t > A uc ad q > 0. The last iequality follows from the fact that F (q; A t ; A uc ) < 1 ad mk > c C r We also assume q > 0 so that maturity has ot yet occured. If q = 0, the trivially C (A t ; A uc ; 0) = C < mk= by assumptio. 12 Iequality G(q; A t ; A uc ) < F (q; A t ; A uc ) < 1 is explaied i the proofs of Propositio 2 ad Theorem 1. 19

21 Refereces Bod, P., Goldstei, I. ad Prescott, E.: 2010, Market-based corrective actios, Review of Fiacial Studies 23, Calomiris, C. ad Herrig, R.: 2013, How to desig a cotiget covertible debt requiremet that helps solve our too-big-to-fail problem, Joural of Applied Corporate Fiace 25, Glasserma, P. ad Nouri, B.: 2016, Market-triggered chages i capital structure: Equilibrium price dyamics, Ecoometrica 84, Harriso, J. M.: 1990, Browia Motio ad Stochastic Flow Systems, rst ed, Robert E. Krieger, Malabar, Florida. Karatzas, I. ad Shreve, S. E.: 1991, Browia Motio ad Stochastic Calculus, secod ed, Spriger-Verlag, New York. Lelad, H. E. ad Toft, K. B.: 1996, Optimal capital structure, edogeous bakruptcy, ad the term structure of credit spreads, Joural of Fiace 51, Peacchi, G. ad Tchistyi, A.: 2015, Cotiget covertibles with stock price triggers: The case of perpetuities. Uiversity of Illiois workig paper. Peacchi, G., Vermaele, T. ad Wol, C.: 2014, Cotiget capital: The case of CO- ERCs, Joural of Fiacial ad Quatitative Aalysis 49, Rubistei, M. ad Reier, E.: 1991, Breakig dow the barriers, Risk Magazie 4, Sudaresa, S. ad Wag, Z.: 2015, O the desig of cotiget capital with a market trigger, Joural of Fiace 70,

22 Iteret Appedix for O Equilibrium whe Cotiget Capital has a Market Trigger: A Correctio to Sudaresa ad Wag Joural of Fiace (2015) George Peacchi Alexei Tchistyi y March 13, 2018 Abstract This Appedix shows that the paper s results o the existece ad uiqueess of equilibria are robust to the possibility of suspot equilibria. The appedix also provides graphical illustratios of the equilibrium stock ad cotiget capital (CC) prices usig realistic parameter values. It also graphs the relatioship betwee the post-coversio bak s equity value ad the pre-coversio CC ad trigger values that is the basis of the error i Sudaresa ad Wag (2015). Departmet of Fiace, Uiversity of Illiois, College of Busiess, 4041 BIF, 515 East Gregory Drive, Champaig, Illiois Phoe: (217) gpeacc@illiois.edu. y Departmet of Fiace, Uiversity of Illiois, College of Busiess, 461 Wohlers Hall, 1206 S. Sixth Street, Champaig, Illiois Phoe: (217) tchistyi@illiois.edu.

23 This appedix shows that our basic model s results o the existece ad uiqueess of equilibria are robust to the possibility of suspot equilibria. I additio, the appedix graphs the model s equilibrium stock ad cotiget capital (CC) prices usig realistic parameter values. It also illustrates the relatioship betwee the post-coversio bak s equity value ad the pre-coversio CC ad trigger values that is the source of the error i Sudaresa ad Wag (2015), hereafter SW. 1 Suspots I our basic model, all price processes are adapted to the fudametal iformatio ow geeratig chages i the bak s asset value. As a robustess check, we ow cosider the possibility of suspot equilibria ad d that this extesio does ot chage the basic model s results. 1 The bak s fudametal asset value process cotiues to be give by equatio (1) i the text: da t = A t dt + A t dz t. (1) I additio, cosider a suspot process, modeled as a Poisso process, N t, that is urelated to this fudametal asset value process. Moreover, allow for the possibility that the i th equilibrium stock price before coversio, S i t, might deped o N t because the coversio time give by equatio (A.3) i the text, i = ifft 2 : S i t Kg, (A.3) could deped o N t. However sice the post-coversio bak s capital structure does ot deped o the time of coversio, its stock price, u t, caot be a ected by the suspot process. Its stock price cotiues to be give by equatio (9) i the text: S t = u t 1 + m A t B. (9) With this Poisso suspot process, a pre-coversio equilibrium stock price might jump, but oly whe N t jumps. To show this, slightly modify the proof of Lemma 1 by rede ig F t as the iformatio geerated by the joit history of the fudametal Browia motio z t ad the suspot Poisso process N t. As before, let Y be the discouted, 1 Suspot equilibria are ot cosidered by SW or Glasserma ad Nouri (2016). 1

24 per-share realized stream of cash ows paid to shareholders whe coversio happes at date : Y = 1 e rt (A T B C) 1T <mif;g + R T e rs aa 0 s rb cc 1 s<mif;g ds + 1 e rt (A +m T B) 1T < + e rr T e r(s ) aa 0 s rb 1 s< ds, (A.1) The X t E t Y is a martigale adapted to ff t ; t 2 [0; T ]g. Accordig to the martigale represetatio theorem (as i Propositio 4 of Appedix I i Du e (2001) or Lemma 4.24 of Jacod ad Shiryaev (2003)), X t ca jump oly whe the process N t jumps. De ig X i t as X t whe = i, the the stock price give by equatio (A.5), S i t = e rt X i t Z t 0 e rs 1 aa s rb cc 1 s<mif i ;g + 1 aa +m s rb 1 i s< ds (A.5) must be cotiuous except whe N t jumps. Now cosider whether there could be a equilibrium stock price, St, i where coversio does ot occur whe the fudametal asset value, A t, rst equals A uc. First, for reasos similar to those outlied i the proof of Propositio 1, we ca rule out the possibility that coversio occurs strictly after the rst time that the asset level, A t, falls to A uc. Suppose that coversio has ot yet occurred at time ", de ed by equatio (A.6) as " = if ft 2 [0; T ] : A t A uc "g (A.6) where " > 0. The due to the properties of geometric Browia motio, there is a positive probability that A t will declie to B before there is a jump i N t or A t returs back to A uc or the CC matures. Cosequetly, there is a strictly positive probability that the bakruptcy date,, occurs before the coversio date or maturity date. However, S i = 0 because regulators close the bak whe A = B ad shareholders lose all claims o the bak s assets. This positive probability evet requires a dowward jump i St i from above the trigger that is ot geerated by N t. Such a jump violates the martigale represetatio theorem ad is icosistet with equilibrium. Secod, we ca also rule out the possibility that coversio happes whe A t > A uc. Suppose that a jump i N t triggers coversio by causig the stock price to jump dow to or below the coversio trigger K= whe A t > A uc. The immediately after coversio, say +, the equilibrium stock price must equal the post-coversio stock price (9). But with A + > A uc it must be that S + = u + > K=. So if a suspot led to a dowward, 2

25 jump that triggered coversio, the there must be a secod upward jump i the stock price ot caused by N t. This secod jump violates the martigale represetatio theorem so that it could ot occur i equilibrium. Thus, coversio caot occur whe A t > A uc. To coclude, coversio ca occur oly whe A t = A uc, implyig that the suspot process has o e ect o the equilibrium coversio time. Therefore, Propositio 1 holds eve whe the iformatio ow F t icludes the joit history of the fudametal Browia motio z t ad the suspot Poisso process N t. With Propositio 1 uchaged, Propositio 2 gives the same uique cadidate stock ad CC prices as i the basic model sice the equilibrium coversio time is uchaged. Moreover, Theorem 1 cotiues to determie the coditios for which these cadidate stock ad CC prices are cosistet with a uique equilibrium. Thus, all results are ua ected by the presece of the suspot process. 2 Illustratios This sectio provides graphic illustratios of equilibrium stock ad CC prices as a fuctio of the bak s asset value. It also graphs the relatioship betwee the post-coversio bak s equity value ad the pre-coversio CC ad trigger values which is the basis of the error i SW. 2.1 Equilibrium Stock ad CC Prices Our Theorem 1 states that uique stock ad CC price equilibria exist wheever mk maxfc; c C g. Here we provide graphs of equilibrium stock ad CC prices for CC coversio r terms that satisfy this criterio. The followig bechmark parameter values are assumed. Parameter Value Parameter Value Seior Debt Pricipal, B 96 Coversio Trigger, K 8 Seior Debt Coupo Rate b 3.0% Iitial Equity Holder Shares, 1 CC Pricipal, C 5 Risk-eutral Cash ow Growth, 0.0% CC Coupo Rate, c 3.0% Volatility of Asset Returs, 4.0% CC Maturity 5 years Risk-free Iterest Rate, r 3.0% These parameters imply that the value of seior debt equals bb=r = B = 96, which is also the value of assets at which regulators would close the bak. The volatility of asset returs, = 4%, equals what Peacchi et al. (2014) estimate to be the average asset 3

26 retur volatility for Bak of America, Citigroup, ad JPMorga Chase over the period 2003 to The risk-eutral cash ow growth rate of = 0 implies that divideds paid to equity holders declie to equal zero at the time that the bak is closed by regulators. Note that sice C = cc=r = 5, K = 8, ad = 1, the umber of shares that provide CC ivestors with a value of equity exactly equal to the CC s par value is m = C=K = 5 = 0:625. Whe m ad K are costats, SW states that this is the oly case for which 8 a uique equilibrium exists. I cotrast, our Theorem 1 shows that uique equilibria exist for all m C=K = 0:625 ad that there is o equilibrium stock price for m < C=K = 0:625. Pael 1 of Figure 1 illustrates stock price equilibria based o equatios (10) ad (11) i the text for values of m = 0:625, m = 1, ad m = 1:5, which are equivalet to the values of equity received by CC ivestors of mk= = C, mk= = 1:6 C, ad mk= = 2:4 C, respectively. The gure graphs the stock prices for each bak asset value ad shows that as coversio terms are more (less) favorable to CC ivestors (iitial shareholders), the bak asset at which coversio occurs, A uc, rises. 3 Ituitively, for a give value of bak assets, less favorable shareholder coversio terms reduce the stock s pre-coversio value, leadig to earlier coversio. Pael 2 of Figure 1 graphs the correspodig CC prices i equatio (11) for the same values of m. It shows that oly whe m = 0:625 does C t = mk= = C. Cosistet with Corollary 1, whe m = 1 or m = 1:5, the C t < mk= strictly prior to coversio ad maturity. The examples i Figure 1 varied the shares give to CC ivestors, m, while keepig the CC s coupo rate of c = 3% xed. As a result, the CC s value was higher for larger m. Now cosider a di eret comparative static where the CC s coupo rate is adjusted for di eret m so that the CC s market value remais the same for a give bak asset value. Speci cally, suppose that CC is issued at date 0 ad its coupo rate is set to make its market value equal its par value: C 0 = C = 5. Assume that the bak s total asset value at this issuace date is A 0 = 120, so that sice B 0 = B = 96, the S 0 = A 0 B 0 C 0 = 19. Figure 2 illustrates this case for three di eret examples, each of which imply C (A 0 = 120) = C = 5: m = 0:625 ad c = 3:00%; m = 0:6875 ad 2 The model i Peacchi et al. (2014) assumes assets follow a jump-di usio process. Their estimate of a total retur volatility of 4% is broke dow betwee a di usio compoet volatility of 3% ad a jump compoet volatility of 1%. The asset retur volatility estimates for idividual baks are 4.2%, 4.4.%, ad 3.3% for Bak of America, Citigroup, ad JPMorga Chase, respectively. 3 From equatio (8) i the text oe sees that for m = 0:625, 1, ad 1:5, the asset value at which coversio occurs is A uc = 109, 112, ad 116, respectively. 4

27 c = 2:255%; m = 0:75 ad c = 1:357%. Pael A of Figure 2 shows that as the umber of shares give to CC ivestors icreases from m = 0:625, 0:6875, ad 0:75 the the bak asset level at which coversio occurs icreases from A uc = 109, 109:5, ad 110, respectively. Pael B co rms that C (A 0 = 120) = 5 for m = 0:625, 0:6875, or 0:75, but that the CC prices are always less tha or equal to their correspodig coversio values of 5, 5:5, or 6, respectively. Agai, Corollary 1 is co rmed. 2.2 Post-Coversio Equity versus CC ad Trigger Values SW s proof of its Theorem 1 correctly shows i its equatio (A10) that a uique equilibrium requires that whe the post-coversio bak s total equity value U t = ( + m) u t rst equals K ( + m) =, it must also be the rst time that U t equals K + C t. 4 However, it the icorrectly cocludes that this coditio implies K ( + m) = = K + C t for all t 2. I other words, C t must always equal m K so that m K = C is the oly coversio terms that yield a uique equilibrium. Figures 3 illustrates that this coclusio is icorrect by graphig U t (blue dotted lie) ad K + C t (red solid lie) for m K = 1:6C, so that m = 1. This is the same coversio terms ad parameter values used i Figure 1 s secod example where coversio terms are strictly favorable (ufavorable) to CC ivestors (iitial shareholders) ad coversio occurs whe A uc = 112. Sice K = 8 ad m = = 1, the K( + m)= = 16. Cosistet with SW s equatio (A10) ad our Propositio 1, the gure shows that U t rst equals K +C t at the same (coversio) time that U t rst equals K(+m)=. Yet the gure clearly demostrates that equatio (A10) ca still hold with K +C t < K ( + m) = strictly prior to coversio, which explais why there is a broader rage of coversio terms that pealize iitial shareholders ad permit a uique stock price equilibrium. Figure 4 presets oe additioal example where m K = 1:2C, so that m = 0:75. It also assumes that the CC s coupo rate is c = 1:357% so that the CC s market price equals its par value of C = 5 whe the bak s assets equal 120. This was the third example illustrated i Figure 2. Agai, oe sees that U t = K ( + m) = = K + C t at the level of assets where coversio occurs (A uc = 110), but that K + C t < K ( + m) = strictly prior to coversio. 4 As discussed i the text, this coditio is equivalet to our Propositio 1 which requires that coversio occur whe the post-coversio stock price, u t, rst equals the trigger price, K=. 5