A closed-form solution to the risk-taking motivation of subordinated debtholders
AA Closed-Form Solution to the Risk-Taking
Motivation of Subordinated Debtholders ∗ Yuval HellerBar Ilan University Sharon Peleg LazarTel Aviv University Alon RavivBar Ilan UniversityFebruary 2019
Abstract
Black and Cox (1976) claim that the value of junior debt is increasing in asset riskwhen the firm’s value is low. We show, using closed-form solution, that the junior debt’svalue is hump-shaped. This has interesting implications for the market-discipline roleof banks’ subdebt.
Keywords: Risk taking, Banks, Asset risk, Leverage, Subordinated debt.JEL Classification: G21, G28, G32, G38 ∗ Yuval Heller: Bar-Ilan University, Ramat Gan, Israel, [email protected]. Peleg Lazar: Tel AvivUniversity, Tel Aviv, Israel, [email protected]. Raviv: Bar-Ilan University, Ramat Gan, Israel,[email protected]. Yuval Heller is grateful to the
European Research Council for its financial support(starting grant a r X i v : . [ ec on . T H ] J un Introduction
Over the past few decades the size and complexity of financial institutions has increased to anextent that challenges regulators’ ability to monitor them. Many proposals suggest greaterreliance on junior debt or subordinated debt (hereafter, subdebt) as a tool to disciplinebanks’ risk-taking. It is argued that the negative effect of excessive risk-taking on subdebtprice encourages subdebtholders to monitor banks closely in a way that is aligned with thedeposit insurer’s incentive (Flannery, 2001). Moreover, a change in the subdebt price cansignal a change in the bank’s risk, allowing the regulator and other market participants todiscipline the bank (Evanoff and Wall, 2001; Dewatripont and Tirole, 1993)The 2007–2009 financial crisis put into question the effectiveness of subdebt as a moni-toring tool. Calomiris and Herring (2013) claim that subdebt was ineffective in mitigatingexcessive risk due to subdebtholders’ lack of motivation to monitor banks’ asset risk. In-tuitively, when a bank is in financial distress, the only way its subdebtholders can be paidis by winning a large and risky bet; hence, the risk preferences of subdebtholders becomemore like those of shareholders. The risk preferences of the junior debt holders has beenanalyzed theoretically in Black and Cox (1976) (and further developed in Gorton and San-tomero, 1990), using the contingent claim approach for pricing corporate liabilities (Merton,1974). They show that the value of a firm’s junior debt can be expressed as a “bull spread”position, which is composed of a long call option on the firm’s asset with a strike price equalto the face value of the senior debt and a short call option with a strike equal to the totaldebt of the firm. Specifically, Black and Cox (1976, p. 369) claim that when the firm’s assetvalue is below a certain threshold (the discounted geometric mean of the face value of thedeposits and the total debt), then the value of subdebt is an increasing function of asset risk,since subdebt is then effectively the residual claimant (see also a similar claim in Gorton andSantomero, 1990, p. 122).In this note, we clarify and extend the above claim. We use a closed-form solution(as in Black and Cox, 1976) to show that when the firm’s value is below the thresholdmentioned above, the relationship between the subdebt value and the asset risk is hump-shaped (rather than increasing with asset risk), and to characterize an interior level of risk Another important tool for decreasing bank stock holders’ risk taking motivation debated heavily inrecent literature is contingent capital (CoCo). For examples of analysis of CoCo’s effect on risk takingmotivation see Martynova and Perotti (2018) and Hilscher and Raviv (2014). The revised analysis can have important implicationsfor the expected effect of subdebt on the risk-taking of financial institutions in time ofdistress. Specifically, the subdebtholders will not be motivated to increase risk as much aspossible (which is the stockholders’s motivation); rather, they will be motivated to increaserisk to some intermediate level. Thus, if the subdebtholders can affect risk-shifting, thenthey will choose a moderate level of risk and risk-shifting will be limited to the level thatmaximizes the value of the subdebt. Moreover, we show that the level of asset risk thatmaximizes the value of the subdebt is an increasing function of the firm’s leverage and ofthe proportion of the senior debt out of the firm’s total debt.Our analysis can explain the empirical results in Ashcraft (2008), which documents thatan increase in the amount of subdebt in regulatory capital has a positive effect in helpinga bank recover from financial distress. Others also document that including subdebt in abank’s capital reduced risk-shifting during the crisis period (Nguyen, 2013; John, Mehran,and Qian, 2010; Belkhir, 2013; Chen and Hasan, 2011).
We use the expanded contingent claims valuation model derived by Black and Cox (1976)to encompass the case of multiple debt claims (junior and senior). Our model and notationsclosely follow Gorton and Santomero (1990).
We begin by describing the firm’s liability structure and expressing the values of the differentclaims.A firm is funded by stock with market value S , and two types of debt with differentpriorities as claimants, such that at debt maturity the junior debt is repaid only after thesenior debt is repaid in full. Both loans are zero-coupon loans maturing at time T . The senior We present a closed-form solution to the sensitivity of a bull spread option strategy to asset risk inPeleg Lazar and Raviv (2017) and Peleg Lazar and Raviv (2019). However, the focus in that paper is on thesensitivity of the bank’s stock to asset risk when bank assets are a risky debt claim. The question whether and to what extent subdebtholders can affect the choice of asset risk is a separatequestion that is beyond the scope of this paper. In Heller, Peleg Lazar, and Raviv (2019) we apply agame-theoretic bargaining analysis to study the equilibrium risk induced by joint control of the bank byshareholders and subdebtholders. F S and its market value is B S and the junior debt’s face value is F J and itsmarket value is B J . The value of the firm’s assets, V , follows a geometric Brownian motionunder the risk-neutral measure according to the following equation: dV t = r · V t · dt + σ · V t · dW ,where r is the instantaneous risk-free rate of return, σ is the instantaneous volatility of assetvalue, and dW is a standard Wiener process under the risk-neutral probability measure.If at debt maturity the value of the firm’s assets is greater than the sum of the facevalues of all its debt, F S + F J , then both debtholders are repaid in full and equityholdersreceive the residual. By contrast, if the firm’s asset value at maturity is below the sum ofthe face values of all its debt, but above the face value of the senior debt, F S < V < F S + F J ,then the senior debt is repaid in full and the junior debtholders are the residual claimants(equityholders do not receive any payoff). Otherwise, if the firm’s asset value is below theface value of the senior debt, V < F S , the senior debtholders receive all the asset value andthe junior debtholders and equityholdesr are not repaid at all.Hence, the senior debtholders’ payoff at maturity is the minimum between the firm’sasset value and the face value of their debt: B S,T = min { V T , F S } . This expression can berearranged and expressed as B S,T = F S − max { F S − V T , } . As discussed in Merton (1974),this payoff is equivalent to the payoff of a risk-free debt with face value F S and a shortposition in a European put option. Therefore, the present value of the senior debt is B S,t = F S · e − r ( T − t ) − P ut t ( V t , F S , σ, T − t, r ) , where P ut t ( V t , F S , σ, T − t, r ) is the value of a European put option on the firm’s asset valueat time t , with a strike price equal to the face value of senior debt F S , asset risk σ , and timeto maturity T − t . Under the above described geometric Brownian motion, the value of theoption can be found using the Black and Scholes (1973) equation.The junior debt’s payoff at maturity is the minimum between the value of assets left afterrepaying the senior debt, if any, and the face value of the junior debt min { V T − F S , F J } , aslong as the payoff is nonnegative, i.e., B J,T = max { min { V T − F S , F J } , } . This payoff canbe rearranged and expressed as B J,T = max { V T − F S , } − max { V T − ( F S + F J ) , } , which isequivalent to a long position in a European call option with a strike price equal to the facevalue of the senior debt, F S , and a short position in a European call option with a strikeprice equal to the sum of the face values of all of the firm’s debt, F S + F J . Therefore, the4alue of the junior debt prior to maturity is B J,t = Call t ( V t , F S , σ, T − t, r ) − Call t ( V t , F S + F J , σ, T − t, r ) , (1)where Call t ( V t , · , σ, T − t, r ) is the value of a European call option according to the Blackand Scholes (1973) equation.The stock’s payoff at debt maturity is S T = max { V T − ( F S + F J ) , } . This payoff can bereplicated by a European call option on the value of the firm’s assets, with a strike price equalto the sum of the face values of all the firm’s debt (Galai and Masulis, 1976). Therefore, thevalue of stock at time t is S t = Call t ( V t , F S + F J , σ, T − t, r ) . The value of the firm’s assets and the payoff to each of its claimholders at debt maturity ispresented in Figure ?? . As pointed out by Gorton and Santomero (1990, p. 122), “If the promised payment ofthe senior debt is close to the value of the firm, then junior debt is effectively the residualclaimant and will behave like an equity claim. If, however, the value of the firm is significantlyhigher than the promised payment on the senior debt, then the junior debt will behave likedebt.” More precisely, the sensitivity of the value of junior debt to the level of asset risk isdivided into two segments defined by the threshold (Black and Cox, 1976, Eq. 10; Gortonand Santomero, 1990, Eq. 7):ˆ V ≡ e − (cid:16) r + σ (cid:17) ( T − t ) (cid:112) F S · ( F S + F J ) , (2)which is a function of the geometric mean of the face value of senior debt and the sum of theface values of all of the firm’s debt. Black and Cox (1976, p. 360) conclude that “Analysisof the function shows J [the bond’s value] is an increasing (decreasing) function of σ for V less than (greater than) ˆ V .”We clarify this statement by noting that when the value of the borrower’s assets is belowthis threshold, the relationship between the market value of the subdebt claim and asset risk5s hump-shaped , and the maximum value of the subdebt claim is reached at the level of assetrisk defined as follows (the proof is presented in the Appendix): σ max ≡ arg max σ B J,t ( σ ) = (cid:115) T − t ln (cid:18) F S · ( F S + F J ) V t (cid:19) − r, (3)if V ≤ V ∗ ≡ e − r · ( T − t ) (cid:112) F S · ( F S + F J ); otherwise there is no internal solution and σ max = 0.The level of asset risk that maximizes the value of the junior debt, σ max , is an increasingfunction of the firm’s leverage ( F S + F J to V ) and the ratio of senior debt to asset value ( F S to V ).The following result summarizes the above analysis of the value of the bank stock as afunction of the asset’s risk. Proposition 1.
The value of the junior debt is (1) decreasing with asset risk if V t > V ∗ and (2) hump-shaped (unimodal) in asset risk if V t < V ∗ , and, in this case, its maximum isobtained for risk level σ max . Moreover, the risk level that maximizes the value of junior debtis higher than the initial risk (i.e., σ max > σ ) if and only if V t < ˆ V . The proposition is demonstrated in Figure 1. In panel (a) the value of assets is aboveboth ˆ V and V ∗ and the value of junior debt is decreasing with asset risk. By contrast, inpanel (b) the value of junior debt is hump-shaped with respect to asset risk and achieves itsmaximum value when asset risk equals σ max = 26 . The effect of the proportion of junior debt
We find that both the range of assetvalues where risk-shifting takes place and the level of asset risk that maximizes the value ofjunior debt decrease with the proportion of junior debt. To see this, note that both ˆ V (Eq.2) and σ max (Eq. 3) decrease when the face value of junior debt F J increases while the sumof the face values of the total debt F S + F J remains unchanged (where an increase in F J whilekeeping F S + F J unchanged implies a decrease in F S ). This result is demonstrated in Figure2, where the sum of the face values of the total debt is constant and equal to F S + F J = 100while the face value of the junior debt changes. Under the assumption of continuous dividends, qV t dt , we find that the both the threshold under whichrisk shifting takes place and the optimal level of risk when risk shifting takes place are slightly higher andequal ˆ V ≡ e − (cid:16) r − q + σ (cid:17) ( T − t ) (cid:112) F S · ( F S + F J ) and σ max ≡ (cid:114) T − t ln (cid:16) F S · ( F S + F J ) V t (cid:17) − r + 2 q . a) Asset value: V = 100 (b) Asset value: V = 62 Figure 1: The value of junior debt as a function of the level of asset risk.
The face value of thesenior debt is F S = 60 and the face value of the junior debt is F J = 10. In addition, the time to maturity isone year and the risk-free rate is r = 1%. Given these values, ˆ V = 63 . igure 2: The level of asset risk that maximizes the value of junior debt for different propor-tions of junior debt. The figure depicts the equilibrium level of asset risk chosen by the junior debtholders.The different lines (ranging from 10% to 30%) correspond to different proportions of junior debt out of thetotal debt. The sum of the face values of both the senior debt and the junior debt is fixed at F S + F J = 100.The initial level of asset risk is σ = 10%. In addition, the time to maturity is one year and the risk-freerate is r = 1%. eferences Ashcraft, A. B. (2008): “Does the market discipline banks? New evidence from regulatorycapital mix,”
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