The Anatomy of Non-recourse Lending

Transcription

1 The Anatom of Non-recorse Lendin Janar 004 Andre Pavlov Simon Fraser Universit 8888 Universit Dr., Brnab, BC V5A 1S6, Canada Tel: Fax: and Ssan Wachter The Wharton School Universit of Pennslvania 56 Soth 37th Street Philadelphia, PA Tel: Fax: JEL Classification: G1, G13, G1

2 The Anatom of Non-recorse Lendin Lenders are freqentl accsed of mispricin the pt option imbedded in non-recorse lendin (Herrin and Wachter, 1999 and 003). In this paper we investiate the cases of market-wide mispricin and its effect on asset prices. We find two competitive eqilibria ood and nderpricin. In the ood eqilibrim, lenders accratel price the pt option and asset prices reflect their fndamental vale. In the nderpricin eqilibrim, - Competitive pressres force all lenders to nderprice the pt, - Asset prices rise above their fndamental level, - The spread of lendin over deposit rates narrows, and - Lenders have neative expected profits. Sch continin operations reslt from aenc frictions between bank shareholders and manaers and/or becase deposit insrance covers the potential losses and/or becase of limited liabilit for shareholders and manaers. We show that the transition from the ood to the nderpricin eqilibrim is onl a matter of time. This reslt holds even when all participants in both eqit and debt markets are fll rational. JEL Classification: G1, G13, G1

3 1. Introdction Investiatin the effects of the pt option imbedded in non-recorse lendin on the nderlin asset markets is not new to finance. In his 000 Presidential address to the American Finance Association, Franklin Allen discsses the role of financial instittions in asset pricin. More detailed stdies of the sbject are presented in Allen and Gale (1998 and 1999), which relate non-recorse financin to asset prices. Pavlov and Wachter (004) extend this work to show that a sinle lender ma rationall choose to nderprice the pt option imbedded in non-recorse loans. Here, we broaden this work to show the conditions nder which man lenders rationall choose to nderprice. We find that, nder certain conditions, all lenders choose to nderprice, reardless of their initial incentives or the potential ftre costs of nderpricin. This nderpricin eqilibrim is characterized b the followin: - Asset prices rise above their fndamental level, - The spread of lendin over deposit rates narrows, and - Lenders have neative expected profits. Sch continin operations reslt from aenc frictions between bank shareholders and manaers and/or becase deposit insrance covers the potential losses and/or becase of limited liabilit for shareholders and manaers. We frther analze the mechanism that leads to the econom switchin between the two eqilibria and std the economic conditions that make the nderpricin eqilibrim more likel to occr and/or more stable. We find that: - If the correctl pricin lenders cannot break-even in the ood state, all lenders switch to the nderpricin eqilibrim described above. - The probabilit of enterin the nderpricin eqilibrim increases with the vale of the pt option and the elasticit of demand for loans, bt decreases with the overall size of the debt market. - The probabilit of enterin the nderpricin eqilibrim increases with the time since the last market crash. 3

4 - Markets with loner prodct ccles experience a hiher deree of nderpricin. - The spread between lendin rates and deposit rates is neativel correlated with asset prices. - Lenders losses followin a market crash are positivel correlated with the time since the previos crash. - Lenders losses are larer for markets with loner ccles. In Section we pt forward a model of the debt markets that ltimatel shows wh rational lenders nderprice the pt option even if the correctl estimate its vale. Section.1 presents the model soltion with no nderpricin. Section. examines the manaerial incentives and sests that the are alined with maximizin shareholder vale. We present the ke reslt that a sinle lender ma choose to nderprice in section.3. Sections.4 throh.5 eneralize this reslt to the case of two or more nderpricin lenders. The most important reslt of the paper, that nder certain conditions all lenders nderprice reardless of their time horizon, is presented in Section 3. Section 4 analzes the economic environment that makes the conditions for this nderpricin eqilibrim likel to occr. Empirical implications are presented in Section 5.. Debt Market Followin Pavlov and Wachter (004), we assme banks are financial intermediaries that accept deposits and make loans to investors (borrowers) who prchase risk asset (properties) with zero eqit. All aents are risk-netral. In this settin, Pavlov and Wachter (004) show that nderpricin the pt option reslts in inflated asset prices above their fndamental vale. In what follows we investiate the circmstances nder which fll rational lenders will nderprice the pt option even if the correctl estimate its vale. We se the followin variables throhot the model: 4

5 R H = Hih paoff to the risk asset (this is 1 + retrn in the ood state) R L = Low paoff to the risk asset (in the bad state) i = Interest rate chared on the loans P = Price of the risk asset toda (R H >P>R L ) δ = Probabilit of hih retrn to the risk asset v = Vale of the pt option imbedded in the loan d = Rate paid on deposits = Nmber of loans made b each bank. (Since all projects are of the same size, is proportional to the dollar vale of the loans). c() = Total cost to the bank (i.e. pament on the deposits) p 0 (i,) = Expected profit of the bank p(i,) = Profit of the bank in the ood state. a b(e(i)) = Indstr demand for loans as a fnction of the expected interest rate, di. m = Nmber of banks (endoenosl determined) Pavlov and Wachter (004) motivate the followin cost fnction for all banks: c ( ) = v= d ( ) + v, (1) where d() is the deposit rate. The interest rate chared on the loan incorporates the vale of the pt option: v+ d i = () δ The division b δ reflects the assmption that interest is collected onl in the ood state. This specification arises from the assmption of risk-netral lenders whose onl marinal 5

6 cost is the deposit rate. Ths, the vale of the pt option becomes part of the cost of capital for the borrowers. 1 The expected profit to the bank, p, then becomes: E i v v d d v ( π0) = δ ( ) = ( + ( )) ( ) = 0 (3).1 Good Eqilibrim B eqatin the expected marinal revene to the expected marinal cost we obtain: δ i = +v (4) Sbstittin di into the expected profit condition (3), we determine that the profit of the bank in the ood state is: ( v) ( 1) v 1 π = + + = + (5) This leads s to the followin proposition that will be tilized in the rest of the paper: Proposition 1: Profits in the ood state are an increasin fnction of the optimal otpt,. To determine the optimal otpt in this base case, we areate sppl and demand: a b( + v) = m a bv = m+ b (6) 1 Tpicall, the cost of capital contains a risk premim which incorporates the risk of defalt and depends on the risk-aversion of the lenders. Or assmption of risk-netral lenders simplifies the risk premim to 6

7 The nmber of banks, m, is determined b findin the maximm m that will allow the participatin banks to break-even. In this case, m= a b( + v) (7) and the optimal otpt,, eqals one. This leads to zero expected profit. The profit in the ood state is the vale of the option, v. In this case, all lenders price the pt option correctl and, as Pavlov and Wachter (004) show, asset prices reflect the fndamental vale.. Bank Manaement Similarl to Pavlov and Wachter (004), conditional on the bank bein in bsiness, we assme the manaer s compensation to have two components: salar, S, and bons B(p) that depends on the realized bank profits. If bank manaers price the pt option correctl, the receive the salar component reardless of the state of the world. In the ood state the bank realizes positive profit and the manaers receive the bons, B(p), which is an increasin fnction of the realized profits. Pavlov and Wachter (004) motivate the assmption that nderpricin of the option is detected onl in the bad state. Ths, manaers receive S in the ood state, reardless of whether the price the option correctl or not. Frthermore, if manaers nderprice the pt option the ma be able to increase profits in the ood state. This reslts in a hiher bons in the ood state. If, however, an nderpricin is discovered, i.e. the manaer nderprices the pt and the bad state occrs, the manaer is fired and the receive zero thereafter. jst the vale of the pt option. The optimal nmber of loans (=1) in the base case is a reslt of normalization and can be set to an nmber b chanin the fixed cost in the cost fnction of the lender. 7

8 Denotin the realized profit of the bank if the manaer nderprices the pt b p, we smmarize the compensation to the manaers: Paoff Hih Low Price correctl S+B(p) S Underprice S+B(p ) 0 The above compensation scheme can arise from an one of the followin two conditions: - aenc friction between bank shareholders and manaers, or - limited liabilit for shareholders and manaers. An aenc friction that ives rise to the proposed compensation scheme cold be that bank shareholders attempt to provide performance incentives for the manaers bt do not observe nderpricin. While plasible, this jstification ltimatel assmes inefficient capital markets, and is therefore qestionable. None of the reslts of this paper reqire sch an assmption. Pavlov and Wachter (004) show, in a similar settin, that the above compensation scheme is consistent with maximizin bank eqit vale, even if shareholders perfectl observe the nderpricin behavior. These findins also hold in the absence of nderpricin. We frther assme that depositors cannot enforce correct pricin even in the absence of deposit insrance. Assmin otherwise wold reslt in the nlikel implication that depositors are more informed then shareholders. The above compensation scheme reslts in the followin condition for nderpricin: δ( B( π ) + S) + δv( T) > S + δb( π) + V( T), (8) where T denotes the time horizon of the manaer, and V(T) denotes the vale to the manaer of keepin their job. Solvin for the contination vale we obtain: 8

9 (1 δ) VT ( ) < δ( B( π ) B( π)) (1 δ) S (9) Given that V is an increasin fnction of T, and B is an increasin fnction of p, we can write the followin proposition: Proposition : If nderpricin increases profits in the ood state, p >p, there exists a time horizon T* sch that manaers with time horizons shorter then T* will nderprice the pt option. Frthermore, T*(p - p) is an increasin fnction of the additional profits obtained in the ood state if the pt option is nderpriced. The intition behind Proposition is based on the tradeoff between increased profits in the ood state and discover of nderpricin in the bad state. Lon-term manaers have a reat deal to lose if the nderprice and are discovered. Ths, lon-term manaers wold not nderprice. Short-term manaers, however, have relativel little to lose if their nderpricin is discovered. For them the benefit of increased profits in the ood state is sfficient to nderprice and risk discover. Frthermore, the above compensation scheme is consistent with maximizin shareholder vale. Ths, shareholders with limited liabilit ma provide incentives for the manaers to nderprice. This possibilit is stronest for shareholders who have little eqit compared to the paoff from nderpricin in the ood state. Therefore, Proposition holds even if there is no aenc friction or informational asmmetr between manaers and shareholders. Frthermore, we can phrase Proposition in terms of shareholders eqit: Proposition a: If nderpricin increases profits in the ood state, p >p, there exists a bank eqit level, E*, sch that shareholders of banks with lower eqit prefer nderpricin. Frthermore, E*(p - p) is an increasin fnction of the additional profits obtained in the ood state if the pt option is nderpriced. 9

10 To be consistent with previos research, we focs on bank manaement and se the first form of Proposition in the remainder of this paper. This also allows s to inclde a friction between manaers and shareholders as a possible case for nderpricin. Nonetheless, if sch a friction does not exist, all of or reslts can be restated in terms of shareholder eqit b tilizin Proposition a..3 Incentives for a Sinle Lender to Underprice the Pt Option We adopt the followin additional notation: = nmber of loans made b the bank that nderprices the pt option. = nmber of loans made b the banks that do not nderprice the option (ood banks), when there is one bank that nderprices. p = profit of the bank that nderprices the pt π = profit of the banks that price correctl in the presence of a bank that nderprices We formlate the followin proposition: Proposition 3: In the ood state, the bank that nderprices the pt receives hiher profits and the other banks receive lower profits relative to the base case, i.e., π > π > π (10) Proof: The marinal revene for the bank that nderprices is. Ths, the interest rate chared on the loans is: δ i = + v = (11) 10

11 Eqatin areate demand and sppl, we obtain the followin eqation for the otpt of the lenders: a b( ) ( 1) = m + (1) Sbstittin the expression for from eqation (11) into eqation (1), leads to the followin expression for the optimal otpt of the bank that nderprices: a( + v) v(1 + b( + v)) = ( a bv) (13) Eqation (7) implies that a bv = m +b > 1. It is then easil verified that >1=. Invokin Proposition 1, we conclde that the profits of the nderpricin bank are hiher relative to the base case, i.e., π > π. Usin eqation (11) we obtain an expression for the otpt of the banks that do not nderprice in the presence of one nderpricer: v = 1 ( a bv ) (14) Ths, the optimal otpt of the correctl pricin banks is below the otpt in the base case, < 1 =. Invokin Proposition 1, we conclde that the profits of the correctl pricin banks are lowered b the presence of an nderpricer, i.e., π > π. Q.E.D. Combinin Propositions and 3 directl ield the followin reslt: Reslt 1: There exists a time horizon T 1 * sch that a manaer with shorter time horizon, T<T 1 *, chooses to nderprice the pt option. Sch a choice leads to 11

12 - neative expected profit for the nderpricin bank, - hiher profits for the nderpricin bank in the ood state, - maximm expected manaement compensation for the nderpricin manaer, - lower profits in the ood state for the banks that price the pt option correctl (the still have zero expected profits over all states), - lower compensation for the manaers of the banks that price correctl, and - lower lendin rates. The most important implication of Reslt 1 is that the presence of an nderpricin manaer leads to lower lendin rates. Interest rates no loner fll reflect the vale of the pt option imbedded in the loans. We will come back to this point below..4 Incentives for Two Lenders to Underprice the Pt Option Conditional on one lender nderpricin the pt option, we now examine the incentives for a second lender to nderprice. We introdce the followin additional notation: = nmber of loans made b each of the two banks that nderprice the pt option. = nmber of loans made b the banks that do not nderprice the option (ood banks), when there are two banks that nderprice. p = profit of the banks that nderprice the pt π = profit of the banks that price correctl in the presence of two banks that nderprice We formlate the followin proposition: 1

13 Proposition 4: In the presence of one nderpricer, a bank can increase the ood state profits b nderpricin. Frthermore, the other banks who price correctl in the presence of two nderpricers receive lower profits relative to the case of one nderpricer, i.e., > > > (15) π π π π Proof: Analoosl to the proof of Proposition 3, we eqate the demand and sppl: a b( ) = ( m ) + (16) This leads to the followin expression for the optimal otpt of a bank that nderprices: a( + v) v( + b( + v)) = ( a bv) (17) Comparin expression (17) to (13) shows that < which verifies the first part of Proposition 4. Comparin expression (17) with that for iven b (14), verifies that proves the second ineqalit of Proposition 4. Eqatin the marinal costs of the nderpricin and correctl pricin banks, we obtain that >, which v = (18) It is easil verified that Q.E.D. < which proves the third ineqalit of Proposition 4. 13

14 Combinin Propositions and 4 we obtain the followin reslt: Reslt : In the presence of one nderpricer, there exists a time horizon T * sch that a manaer with shorter time horizon, T<T *, chooses to switch from pricin correctl to nderpricin. Sch a choice leads to - neative expected profit for both nderpricin banks, - lower profits for the nderpricin banks in the ood state, - lower profits in the ood state for the banks that price the pt option correctl (the still have zero expected profits over all states), - lower compensation for the manaers of the banks that price correctl, and - lower interest rates..5 Incentives for K Lenders to Underprice the Pt Option We now examine the case when K lenders nderprice the pt option. The central qestion is whether there is still incentive for et another lender to start nderpricin. We adopt the followin notation: k = nmber of loans made b each of the banks that nderprice the pt option. k = nmber of loans made b the banks that DO NOT nderprice the option (ood banks), when there are K banks that nderprice. p k = profit of the banks that nderprice the pt π k = profits of the banks that price correctl in the presence of K banks that nderprice Eqatin demand and sppl with K nderpricers, we obtain the followin eqilibrim condition: a b( ) ( ) k = m K k + Kk (19) 14

15 Analoosl to Propositions 3 and 4, this leads to the followin expressions for k and k : k a( + v) v( K + b( + v)) = ( a bv) (0) and k Kv = 1 ( a bv ) (1) To determine the nderpicin incentive we compare the otpt of K nderpricers with the otpt of a ood bank in the presence of K-1 nderpricers: k v 1 k 1 = 1 > 0 () ( a bv) Combined with Propositions 1 and, this leads to the followin reslt: Reslt 3: In the presence of K nderpricers, there exists a time horizon T k * sch that a manaer with a shorter time horizon, T<T k *, chooses to switch from pricin correctl to nderpricin. Sch a switch leads to - neative expected profits for all nderpricin banks, - lower profits for all nderpricin banks in the ood state, - lower profits in the ood state for the banks that price the pt option correctl (the still have zero expected profits over all states), - lower compensation for the manaers of the banks that price correctl, and - lower interest rates. 15

16 The essence of Reslt 3 is that reardless of the nmber of nderpricers on the market, there is alwas a potential incentive for a ood manaer, i.e., manaer who prices correctl, to bein nderpricin. 3. Underpricin Eqilibrim The manaement compensation scheme described above assmes that the bank is in bsiness. A necessar condition for stain in bsiness is that the bank makes positive profits at least in the ood state: ( ) k k vk 1 0 π = + > (3) In other words, k v+ 4 + v = > 0. (4) Sbstittin (1) into (4) reslts in the followin condition for K: ( a bv)(+ v 4+ v K < Kmax = (5) v where K max denotes the maximm nmber of nderpricers that will allow for the ood banks to make positive profit in the ood state. This maximm will be bindin onl if K max exceeds the total nmber of banks, m. Usin expression (7) for m, we obtain the followin condition for K < m: max a > bv 4 + v 4+ v (6) 16

17 which simpl states that the total debt market has to be lare relative to the vale of the pt option. If a bank cannot trn positive profit even in the ood state, i.e. K>K max, the manaer faces certain dismissal. If, however, the manaer nderprices the pt, the manaer receives salar pls bons in the ood state and onl looses his or her job if the bad state occrs. It is ths clear that all remainin manaers will nderprice the pt reardless of their time horizon. This allows s to formlate the followin reslt: Reslt 4: For a sfficientl lare market (condition (6)), if the nmber of nderpricin banks exceeds K max all remainin manaers will switch to nderpricin reardless of their time horizon. The new eqilibrim has the followin characteristics: 1. The interest rate on the loans reflects onl the deposit rate and does not inclde the pt option at all. The new interest rate is lower relative to the base case 3. All banks make the same nmber of loans 4. All banks make positive profit in the ood state althoh this profit is neliible in a lare market 5. All banks have neative expected profit 6. The deposit rate is hiher relative to the base case, althoh this difference is neliible in a lare market 7. More loans are made relative to the base case, althoh this increase is neliible in a lare market Proof: Let * denote the optimal nmber of loans when all banks nderprice. The marinal cost to the bank onl reflects the cost of obtainin new deposits, *, and does not inclde the pt option. Eqatin areate sppl and demand provides the followin eqation for the optimal otpt: 17

18 a b = m (7) Sbstittin (7) into (7) provides the followin expression for the optimal otpt: a * = a bv (8) This is reater then the optimal otpt in the base case (=1). For a lare debt market, i.e., a>>bv, * approaches the otpt in the base case. The profit in the ood state is iven b: a * 1 1 π = = a bv (9) Expression (9) is reater then zero, bt approaches zero for a lare debt market. In other words, the profits in the ood state are small and approach zero, while the losses in the bad state remain nchaned. This leads to neative expected profits for the lenders. Usin Eqation (1), we obtain the followin expression for the deposit rate in the new eqilibrim: a a bv d( *) = + a bv a (30) Since a bv = m +b > 1, a is reater than 1, and it is easil verified that d(*) is reater than, which is the deposit rate in the base case. Expression (30) also sests that for a lare market, i.e. a>>bv, d(*) approaches the deposit rate in the base case. 18

19 Finall, we compare the interest rates chared on loans in the base case and the nderpricin eqilibrim. The base case interest rate is iven b (4), while the new interest rate, i*, is a i* = a bv. (31) It is easil verified that the new interest rate is lower then the interest rate in the base case, i*<i. Q.E.D. 4. Determinants of K max In what follows we examine the inflence of the economic environment on the critical nmber of nderpricin banks, K max. We smmarize the reslts in the followin statement: Reslt 5: The followin relationships hold: Kmax Kmax Kmax < 0, > 0, and < 0 v a b Proof: Based on eqation (5), we compte the followin derivatives: Q.E.D. 4 v a 1 + bv 1 K * 4+ v 4+ v = < 0 v v Kmax + v 4+ v = > 0 a v Kmax = + v + v < b ( 4 ) 0 (3) 19

20 The most important claim of Reslt 5 is that the larer the option vale, the smaller the critical nmber of nderpricers. In other words, hihl volatile markets are sbstantiall more likel to enter the eqilibrim in which all lenders completel inore the pt option in the loans the are providin. Followin a low realization all nderpricin manaers are discovered and replaced. Let G 0 (T) denote the cmlative distribtion fnction of the time horizons of the manaers followin a low realization. Notice that G 0 (T 1 ) = 0, i.e., there are no manaers with time horizons shorter then T 1, defined in Reslt 1 as the ctoff point for a sinle manaer to nderprice. Each time period, the time horizon for all manaers oes down b one. Ths, the cmlative distribtion fnction t intervals after the last low realization, G t, can be expressed as follows: G ( T) = G ( T + 1) = G ( T + t). (33) t t 1 0 The nmber of nderpricin manaers, K t, at time t is the soltion to the followin eqation: K = mg ( T( K )) = mg ( T( K ) + t). (34) t t t 0 t Proposition 5: The nmber of nderpricin manaers, K t, is an increasin fnction of time since the last low realization. Proof: Notice that the critical time horizon that indces nderpricin, T(K t ), is a fnction of the nmber of nderpricin manaers, K t. Let Dp k = πk k 1 denote the chane in the ood state profits and D k = k k 1 denote the chane in optimal otpt for a bank that beins to nderprice in the presence of K-1 nderpricers. Eqation () shows that D k is independent of K. Since the profit is a qadratic fnction of the optimal otpt (Eqation (5)), the chane in profits is an increasin fnction of the optimal otpt: π 0

21 π = + ( + v) k k k k 1 (35) Eqation (1) shows that k 1 is a decreasin fnction of K. Since D k is independent of K, π k is a decreasin fnction of K. Invokin Proposition, we find that T K < 0. (36) Differentiatin eqation (34) with respect to time ives: K t T K = m( T ( Kt + t)) + 1, (37) K t where denotes the probabilit densit fnction of T. Rearranin (37) we obtain: T K 1 m( T ( Kt + t)) = m( T ( Kt + t)) (38) K t The riht hand side of Eqation (38) is positive. Eqation (36) sests that T 1 m( T ( Kt + t)) >0 (39) K Therefore, K t > 0 (40) Q.E.D. 1

22 Since the nmber of nderpricin manaers is an increasin fnction of time since the last low realization, it is onl a matter of time ntil K t reaches the critical point K max at which all manaers switch to nderpricin reardless of their time horizon. We smmarize the above reasonin in the followin reslt: Reslt 6: The nmber of nderpricin manaers increases with time since the last market crash (i.e. low realization). Given enoh time withot a crash, the critical level of nderpricin, K max, is achieved and all manaers switch to nderpricin. Reslt 6 immediatel leads to the followin important conclsion: Reslt 7: Asset Markets with loner ccles sffer, on averae, a hiher deree of nderpricin. 5. Empirical Implications The immediate empirical implication of Reslt 4 is that nderpricin of the pt option reslts in inflated asset prices. This is not directl testable becase otsiders do not observe the vale of the pt option or the fndamental price of the asset. We se the spread of the loan rate over the deposit rate as a prox for the nderpricin. This spread compensates the lenders for providin the pt option imbedded in their loans. Drin periods of widespread nderpricin, lenders reqire little or no compensation for the pt option, and the spread of lendin over deposit rates is redced. As shown above, this is reflected in hiher market price of the nderlin asset. Frthermore, periods of widespread nderpricin are associated with increased lendin activit. In order for lenders to raise enoh capital to spport this increased lendin activit the increase deposit rates. This leads to the testable implication that deposit rates are positivel correlated with asset prices. We smmarize the above reasonin in the followin empirical prediction:

23 Reslt 8: The spread between lendin rates and deposit rates is neativel correlated with asset prices. Deposit rates are positivel correlated with asset prices. Reslts 7 and 8 also sest the followin empirical implication: Reslt 9: Lenders losses followin a market crash are positivel correlated with the time since the previos crash. Frthermore, lenders losses are larer for markets with loner ccles. Note that all empirical implications of or model relate one or more smptoms of nderpricin to observed asset prices. Therefore, none of them impl casalit nor is the mechanism we propose the onl one that can enerate sch conclsions. 6. Conclsion We pt forward a model of efficient asset markets that demonstrates how nderpricin of the pt option imbedded in non-recorse loans reslts in inflated prices for assets in fixed sppl. Frthermore, we examine the conditions nder which fll rational lenders choose to nderprice this pt option. In particlar, the presence of short-term plaers in the debt market creates competitive pressres that force all lenders to nderprice the option reardless of their time horizon. Once this happens, retrnin to pricin the pt option correctl can onl happen after a market crash. Since the vale of the pt option is nobservable b otsiders, the above conclsions cannot be tested directl. The proposed model provides for some axiliar empirical implications. 3

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