Nov 18, 2014 - L (kL + tH; Ï') |θH). +pLrL + βpLEÏ (¯Vm. L (kL + tL; Ï') |θL). (PC) Ï' ... First, in any equilib
Equilibrium Collateral Constraints Cecilia Parlatore
y
New York University, Stern School of Business November 18, 2014
Abstract I study a dynamic model of optimal funding to understand the use of …nancial assets as collateral. Borrowers need to raise funds to invest in risky projects whose return is non-veri…able and, thus, noncontractible. Firms with investment opportunities tomorrow value the asset more than those without them: the asset partially solves the non-contractibility problem allowing borrowers to invest in their projects either by selling the asset or by pledging it as collateral. When investment opportunities are persistent, borrowers will value the asset more than lenders in equilibrium. This endogenous di¤erence in valuation and the non-contractibility of returns imply that collateralized debt is optimal.
1
Introduction
Collateralized debt is a widely used form of …nancing. Trillions of dollars are traded daily in debt collateralized by diverse …nancial assets, such as sale and repurchase agreements (repos) and collateralized over-the-counter derivative trades.1 Many …nancial institutions use collateralized debt to raise funds that allow them to provide intermediation services. Some of these institutions are private depository institutions, credit unions, mortgage real estate investment trusts, and security brokers and dealer. Most of these institutions are highly levered and they use the repo market as a source of …nancing. The assets that are "sold" using repos are …nancial assets which could also be sold without a repurchase agreement in …nancial markets. These …nancial assets are not productive assets and, in principle, their intrinsic value is the same independently of the assets’holder. Then, why do so many …nancial institutions choose to use these …nancial assets as collateral instead of selling them to raise funds? e-mail:
[email protected], Webpage: www.ceciliaparlatore.com. am especially indebted to Ricardo Lagos and Tom Sargent for their advice in this project. I also thank Douglas Gale
yI
for very helpful comments and suggestions. I thank Viral Acharya, Saki Bigio, Alberto Bisin, Eduardo Davila, Todd Keister, Emiliano Marambio Catan, Antoine Martin, Michal Szkup and seminar participants at NYU, Stern and Wharton for their insightful remarks. 1 See Gorton and Metrick (2012a), Gorton and Metrick (2012b), and Copeland et al. (2011 (revised 2012).
1
If we assume that the borrower values the asset more than the lender, as is the case of a family heirloom, the borrower will be better o¤ by pawning the asset than by selling it since part of the value the borrower assigns to it will not be internalized by the market. On the other hand, the lender will be happy to receive the asset as collateral since it will create incentives for the borrower to repay the debt, and in case of default the lender keeps the collateral. However, in the case of …nancial assets, there is no ad-hoc reason why borrowers would value the assets more than lenders since both borrowers and lenders can sell the assets in the same market and receive the same dividends. Yet, we observe …nancial assets being collateralized. In this paper I develop a model in which borrowers and lenders value the asset equally in autarky, but they assign di¤erent values to it in equilibrium. As a result of this endogenous di¤erence in valuations, collateralized debt contracts implement the optimal funding contract. Since the contract is an equilibrium outcome, I can also characterize the amount that can be borrowed against an asset (i.e., its debt capacity) and its determinants. The model is a discrete-time, in…nite-horizon model. There are two types of risk neutral agents, borrowers and lenders, and one durable asset which pays dividends each period. Borrowers can invest in risky projects but they need external funds to do so. The return of the projects is private information of the agent who invested in them. In order to raise funds, borrowers enter into a state contingent contract with lenders. In equilibrium, …rms value the asset more than lenders, and, therefore, choose to use collateral contracts. In my model, borrowers have the investment opportunity before the asset’s dividends are paid and, thus, cannot invest without external …nancing. This timing implies a maturity mismatch for the borrower between the need of funds to invest and the availability of the dividends. Lenders do not have access to investment opportunities. Therefore, if left in autarky, both risk neutral borrowers and lenders would value the asset by the expected discounted sum of dividends. However, once the agents are allowed to trade, the equilibrium features endogenous di¤erences in valuations and, thus, collateral contracts are optimal. The main assumptions that lead to this result are the maturity mismatch between the time in which the dividends are paid and the investment opportunity, the persistence in the role as borrowers and lenders, and the asymmetric information about the borrower’s ability to repay. Suppose that an agent in this economy will have an investment opportunity tomorrow. Holding the asset tomorrow will allow the agent to invest in the project either by selling the asset or by pledging it as collateral. Being able to raise funds against the asset in the funding market will solve the maturity mismatch problem between the timing of the dividend realization and the investment opportunity. In turn, this implies that agents with investment opportunities tomorrow will value the asset more than those without them. If the role as borrowers and lenders is persistent, as it is the case in many collateralized debt markets, it follows that agents who are borrowers today will value the asset tomorrow more than lenders will.2 This di¤erence 2 This
persistence is consistent with the observation that, in many collateralized debt markets, di¤erent types of institutions
specialize in borrowing or lending. For example, in the repo market, money market funds are usually lenders whereas hedge funds and specialty lenders are usually borrowers.
2
in valuations, together with the asymmetric information about the borrower’s ability to repay, implies that collateral contracts arise optimally in equilibrium. The extra value borrowers assign to the asset on top of the expected discounted value of dividends can be decomposed in two premia for the borrower: a liquidity premium and a collateral premium. The liquidity premium for the borrowers arises from them being able to sell the asset and use the funds to invest in the risky projects (from solving the maturity mismatch mentioned above). The collateral premium for the borrowers is the additional value they can obtain by using the asset as collateral instead of selling it. By solving the model in closed form, I am able to characterize the asset’s debt capacity completely. I show that, as one might expect, more liquid assets have a higher debt capacity. Moreover, when all borrowers can invest in the same type of projects, i.e., all projects have the same correlation with the asset’s future dividends, an increase in this correlation increases the asset’s debt capacity. Given the asymmetric information problem, the borrower has incentives to lie when the return of the projects is high. Therefore, an asset that has a higher value when the borrower has incentives to lie makes it easier to motivate the borrower to tell the truth, and, hence, is better collateral. I also …nd that when the returns of the projects are positively correlated with future dividends, a mean preserving spread of the distribution of the projects’ returns decreases the asset’s debt capacity. Finally, when borrowers can invest in di¤erent types of projects, two regimes may arise in equilibrium: one in which all borrowers use the asset as collateral and one in which some borrowers use the asset as collateral and other borrowers choose to sell the asset to raise funds. In this case, the e¤ect of changes in the correlation structure on the overall amount intermediated in the economy depends on the distribution of project types across borrowers and on the regime of the economy. There is a large literature that analyzes collateral contracts. My paper is closest to Lacker (2001) and Rampini (2005). Lacker (2001) shows that collateralized debt is the optimal …nancing contract when the borrower values the collateral good more than the lender. He shows this in a two-good, two-period model with two agents, a risk-averse borrower and a risk-neutral lender that takes the di¤erence in valuations as given. In a similar environment with non-pecuniary costs of default, Rampini (2005) studies how default varies with aggregate income when individual income is privately observed by the agents. The optimal risk sharing contract allows for default which, given the default penalties assumed, only occur when the realization of income is low. These default penalties, which are modeled as transfers of agent-speci…c goods only valued by the agent who is endowed with it, can be interpreted as collateral which is only valued by the original holder. As in these two papers, the friction that gives rise to collateral contracts in equilibrium in my model is asymmetric information between the borrower and the lender. In particular, the amount of repayment good the borrower has when he has to repay the loan is private information of the borrower. In contrast, my model is in in…nite horizon, all agents are risk neutral, and, most importantly, the di¤erence in the marginal value of the collateral good for the lender and the borrower is an equilibrium outcome: in equilibrium, the borrower values the collateral good more than the lender and, therefore, collateral contracts are optimal.
3
As Barro (1976) shows, enforcement frictions can also give rise to collateral contracts. Along these lines, Stiglitz and Weiss (1981), Chan and Kanatas (1985) and analyze collateral as a way to enforce contracts and disincentivize default. Kocherlakota (2001) analyzes optimal repayment contracts in the presence of collateral when there are enforcement frictions. Collateral contracts are optimal in Kocherlakota’s model, provided that lenders are assumed to be less willing than borrowers to substitute consumption for collateral goods. In his setup, collateral is used to force the borrower to share with the lender the returns on the project in which the borrower invested. Following Kiyotaki and Moore (1997) there is a large literature that refers to these enforcement frictions to analyze the e¤ects of collateral constraints on aggregate ‡uctuations. For example, Jermann and Quadrini (2012) …nd that a tightening of the …rms’collateral constraints contributed signi…cantly to the 2008 recession and to the downturns in 1990
2009
91 and 2001: In these papers, at the time of repayment, the lender
can recover only a fraction of the collateral value. This fraction is stochastic and depends on (unspeci…ed) market conditions. Moreover, durable assets are used not only as collateral for loans, but also as inputs for production (and, thus, selling them is not an option). In the general equilibrium literature, Geanakoplos (2003b), Geanakoplos (2003a) and Geanakoplos and Zame (2009) study the use of durable assets as collateral. They show that when collateral contracts are assumed, the set of traded assets will be determined endogenously. Araujo et al. (1994) and Araujo et al. (2005) assume collateral contracts and make collateral endogenous by allowing each seller of assets to …x the level of collateral or the bundle used as collateral. In all these papers, collateral contracts are assumed. Finally, in a search model with bilateral trading, Monnet and Narajabad (2012) show that agents prefer to conduct repurchase agreements than asset sales when they face substantial uncertainty about the value of holding the asset in the future. In contrast to the papers mentioned above, in the model presented in this paper, the asset used as collateral is not an input in production and it can be sold to raise funds. As explained above, given the persistence in the roles as borrowers and lenders, the asset is used as collateral as an optimal response to asymmetric information about the resources available to the borrower at the time of repayment. The amount that can be borrowed against the asset is determined in equilibrium, and, thus, changes in the collateral constraints faced by borrowers re‡ect changes in the economy’s fundamentals. By characterizing collateral constraints in equilibrium, my model provides a link between the amount that can be borrowed against the assets and the fundamentals in the economy beyond the value of the asset. It also allows me to identify collateral and liquidity premia which a¤ect the asset’s value and its debt capacity. The rest of the paper is organized as follows. Section 2 presents the baseline model. Section 3 de…nes and characterizes equilibrium. Section 4 extends the model to allow for a richer structure of dividends and return and for heterogeneity among borrowers. Section 5 concludes.
4
2
Model
In this section I present the baseline model and derive the main result that collateral constraints arise in equilibrium as part of the optimal funding contract. Time is discrete, starts at t = 0, and goes on forever. Each period t is divided in two subperiods, morning and afternoon.3 There are two di¤erent types of non-storable, consumption goods: a morning speci…c good and an afternoon speci…c good. A unit of the morning (afternoon) speci…c good at time t that is not consumed within the morning (afternoon) at time t, perishes and disappears. There is a one in…nitely lived asset in the economy which is in …xed supply k. Holding k units of the asset yield dt k units of (afternoon) consumption good as dividend at the end of each afternoon t. In this section, I will assume that the dividend is constant, dt = d. There is a large number N of each of the two types of agents in the economy, borrowers (B) and lenders (L). All agents are risk-neutral, live forever and share the same discount factor
2 (0; 1). Lenders are
a endowed with kL;0 units of the asset at t = 0 and each period t they receive an endowment of em L and eL
units of the morning and afternoon consumption goods respectively. Borrowers start their life with kB;0 units of the asset and, every period t, they receive em B units of the morning consumption good. Borrowers receive no good endowment in the afternoon. Instead, each afternoon, each borrower has access to his own investment opportunity. Each afternoon t a borrower j can invest in his own risky, constant return, short-term project: one unit of (afternoon) consumption good invested in his project at the beginning of afternoon t yields a random payo¤ at the end of the period which is only observable by the agent who invested in the project. This payo¤ is given by
j t
2f
L; H g
where
L
a fraction pH gets a high return
< H.
H.
Each period, a fraction pL of borrowers get a low return j t
Therefore, pi = Pr
=
i
satis…es
E( ) 1
L L
whereas
for i = L; H. I assume that the project is
pro…table in expectation but it incurs losses if the low state is realized, i.e., Assumption The discount factor
L
L
1, the borrower invests all the loan amount, q, in the risky technology and gets an expected return E ( ) q. He expects to repay pH rH + pL rL in terms of consumption good and he gets dividends dkB from his asset holdings at the beginning of the afternoon. Finally, his expected continuation value (his value the following morning) depends on the contingent transfers of assets tL and tH that are part of the contract. P C states that the lender has to be at least as good participating in the contract as he would be if he didn’t participate in it. If the lender participates in the contract he gives up q units of consumption good at the beginning of the afternoon in exchanges for contingent repayments in terms of consumption good and asset. With probability pi state i = L; H is reported and the lender receives ri units of the consumption good and a ti additional units of the asset with which to enter the asset market the following morning. Assumption The lender’s endowment in the afternoon eaL satis…es
dk 1
< eaL :
L
The assumption above implies that the equilibrium loan amount is never restricted by the amount of the consumption good owned by the lenders, i.e., q < eaL . If this was not the case, a borrower with a su¢ ciently high amount of assets could issue risk free debt by pledging the dividends paid by the asset and the incentive compatibility constraints would not bind. By inspecting the constraints in the problem above, one can considerably simplify the borrower’s problem. First, in any equilibrium, the participation constraint for lenders P C will hold with equality. If it did not, the borrower could increase the loan amount and increase his expected utility without violating any of the additional constraints. Similarly, as is usual in this kind of problems, the incentive compatibility constraint will bind in the high state ICH will hold with equality. Finally, in order to maximize the size of the loan, the repayment in terms of goods in the low state, rL , will be the maximum possible, i.e., rL = The following proposition, which is proved in the appendix formalizes these arguments. Proposition 1 In the optimal lending contract the participation constraint for the lender binds E
0
VLm kL ;
0
=
q + pH rH + pH E +pL rL + pL E 9
0
0
VLm
0
VLm kL + tH ; kL + tL ;
0
j
L
j ;
H
Lq
+ dkB :
the incentive compatibility constraint for the borrower in the high state binds rL + E
0
VBm kB
0
tL ;
j
H
=
rH + E
VBm kB
0
tH ;
0
j
H
;
and the repayment in terms of consumption good in the low state is maximal rL = dkB +
L q:
From the proposition above it follows that the borrower’s problem in the funding market reduces to choosing only the two asset transfers. Corollary 1 The borrower’s problem can be rewritten as VBa (kB ; kL ; ) =
sup 2
(tH ;tL )2[0;kB ]
(E ( ) 1
L)
dkB
L
(E ( ) L) pH VLm kL + tH ; 0 + pL VLm kL + tL ; 0 E 0 VLm kL ; 1 L (E ( ) 1) + pH E 0 VBm kB tH ; 0 j H E 0 VBm kB tL ; 0 j H 1 L + pL E 0 VBm kB tL ; 0 j L + pH E 0 VBm kB tH ; 0 j H
0
+
subject to q
dkB +
=
+ q
pH E
0
0
0
0
VLm kL + tH ;
VBm kB
maxf0; pH E pL E
3
pH E
0
VLm
VBm kB
0
j
H
+ pL E
1 L tH ; 0 VBm kB tL ; 1 L 0 kL + tH ; j H + pL E tH ;
0
VBm kB
tL ;
0
0
0
0
j
VLm kL + tL ;
0
j
VLm kL ;
E
H
0
VLm kL + tL ; H
E
0
0
j
j
L 0
L
VLm kL ;
0
g
= H( ; )
Equilibrium
In this section I de…ne a recursive equilibrium of the model and characterize the unique a¢ ne equilibrium which features an optimal lending contract that can be implemented with a combination of riskless and collateralized debt. De…nition 3 A recursive equilibrium in this economy is a pair of value functions for borrowers, in the morning and in the afternoon, VBm (kB ; ) and VBa (kB ; kL ; ), a value function for lenders, in the morning and the afternoon, VLm (kB ; ) and VLa (kL ; kB ; ) ; price and quantity functions in the bilateral asset market, P (kB ; kL ; ) and k T (kB ; kL ; ), a loan contract (q (kB ; kL ; ) ; rL (kB ; kL ; ) ; rH (kB ; kL ; ) ; tL (kB ; kL ; ) ; tH (kB ; kL ; )) 10
and a law of motion for ; H ( ; ), such that (1) ; (2) ; (3), and (4) are satis…ed and the law of motion for satis…es: FLm0 (k)
= pH
Z
FLa (k tH (kB ; k; d; )) dFBa (kB ) Z FLa (k tL (kB ; k; d; )) dFBa (kB )
+pL Z FBa (k) = FBm k k T (k; kL ; d; ) dFLm (kL ) Z m0 FBa (k + tH (k; kL ; d; )) dFLa (kL ) FB (k) = pH Z +pL FBa (k + tL (k; kL ; d; )) dFLa (kL ) Z FLa0 (k) = FLm0 k + k T (kB ; k; d; ) dFBm0 (kB ) :
Since the borrower has all the bargaining power in the funding market and lenders are left indi¤erent between participating in the funding market or not, in equilibrium VLa (kL ; ) = dkL + E where VLa k;
0
0
= EkB VLa k; kB ;
0
VLm kL ;
0
. Moreover, the equilibria in the asset and funding markets depend
only on the expected value of the borrowers and lenders in the following subperiod. Moreover, using (1) one gets VBm where VBa k;
0
(kB ; ) = 0
= EkL VBa k; kL ;
Z
VBa (kB ; )
P (kB ; kL ; ) dFLm (kL )
Remark 1 An equilibrium can be fully characterized by VBa (kB ; ) and VLm (kL ; ).
3.1
A¢ ne Equilibrium
For the remainder of the paper, I will focus on recursive a¢ ne equilibria, i.e., equilibria in which the value functions are a¢ ne in asset holdings. Within this class of equilibria, the equilibrium is unique.5 All proofs are in the appendix. From remark 1 an a¢ ne equilibrium is fully characterized by VBa (kB ; )
= cB ( ) kB + aB ( )
VLm (kL ; )
= cL ( ) kL + aL ( )
Using this characterization in (3) to determine the terms of trade in the asset market gives max (P0
P0 ;k0 5 The
dk0
1
cL ( ) k0 )
(cB ( ) k0
P0 )
mapping that characterizes an equilibrium is not a contraction mapping. Therefore I cannot show that the equilibrium
is unique within a more general class of value functions.
11
which implies that P0 = (1
) cB ( ) k0 + (dk0 + cL ( ) k0 )
and k T (kB ; kL ; ) = arg
max
k0 2[ kB ;kL ]
(cB ( )
d
cL ( )) k0 :
As it is usually the case with linear value functions when the terms of trade are determined through Nash bargaining, the price is an average of marginal valuations of the counterparts where the weight on an agent’s marginal valuation is his counterpart’s Nash bargaining weight. Moreover, the quantity traded maximizes surplus and, thus, trades in the asset market are e¢ cient. The linearity of the value function also implies that k T (kB ; kL ; ) 2 f kB ; kL g. Since the borrower can wait until the afternoon and sell the asset in the loan market, by setting tL = tH = kB , he will never choose to sell in the morning. Selling in the afternoon, allows the borrower to invest the proceeds of the sale in the risky projects which has a higher expected return than consuming the goods in the morning. Therefore, the terms of trade in the asset market are independent of the asset stock with which the borrower enters the market and k T (kB ; kL ; ) = kL ; for all kB ; kL , and . Given the a¢ ne speci…cation of the value functions in an a¢ ne equilibrium, the solution to the borrower’s problem in the funding market will be a corner solution. There are four possible solutions: tL = tH = 0, tL = 0 and tH = kB , tL = kB and tH = 0, and tL = tH = kB : Lemma 1 In an a¢ ne equilibrium, a borrower will choose tL tH
= kB 8 < k B = : 0
if cB ( ) < cL ( ) if cL ( ) < cB ( )
If the borrower values the asset as much as the lender, he will set tL = tH = kB and "sell" the asset to the lender. In exchange, the lender will lend the borrower cL ( ) = cB ( ) which is a fair compensation from the borrower’s perspective. However, if the borrower values the asset more than the lender, he would be getting less than his valuation if he chose to sell the asset to the lender: the lender would still pay a price cL ( ) per unit where now cL ( ) < cB ( ). In this case, the borrower will only transfer the asset if he cannot avoid it, i.e., if the low return is realized and he does not have enough resources to compensate the lender in consumption goods. Transferring the asset is costly for the borrower since he gets cL ( ) units for it while he values it cB ( ) > cL ( ). This di¤erence in valuation can be interpreted as the punishment for lying which induces truth telling and makes the state contingent contract contract incentive compatible. Proposition 2 In the unique a¢ ne equilibrium, cB ( ) = cB (d) > cL (d) = cL ( ) where cB (d) and cL (d) are a¢ ne functions of the dividend d. 12
In equilibrium, a borrower values the asset more than the lender. The extra valuation the borrower attaches to the asset arises due the persistence of the investment opportunities. Holding assets allows the borrower to take advantage of his pro…table investment opportunities in the future either by selling the assets, by pledging its dividends or by pledging the assets themselves.6 Corollary 2 In an a¢ ne equilibrium, the optimal loan contract (q ; rL ; rH ; tL ; tH ) is given by q (d) =
(d + pL cL (d) + pH cB (d)) kB 1 L (
Lq
(d) + d)
rL (d)
=
rH (d)
=
kB 1 L rL (d) + cB (d) kB
tL
=
kB , tH = 0
The optimal loan contract only depends on the aggregate state through the dividend level d. As one can see from corollary 2, borrowers are collateral constrained in the optimal loan contract: the maximum amount that the borrowers are able to borrow depends linearly on the amount of assets they have. This collateral constraint is an equilibrium outcome and it depends on how much the expected holder values the asset. With probability pL the return of the projects is low and the borrower transfers all his asset holdings to the lender whose expected discounted value of one unit of asset is cL (d). With probability pH the return of the projects is high and the borrower keeps all his assets which he values cB (d) per unit.
3.2
Implementation of the Optimal Loan Contract
The optimal loan contract can be implemented using two di¤erent debt contracts simultaneously: riskless debt and collateralized debt. This implementation is not unique. Below, I describe the implementation with the maximum amount of riskless debt, which is the one that includes riskless debt at 0 interest rate. The maximum amount that can be repaid independently of the realized state, risklessly, is rL (d). Therefore, the amount of riskless debt in this implementation is rL (d). The remaining part of the loan amount q (d) is repaid in consumption goods only if the return of the project is high, whereas if it is low, the lender receives an asset transfer from the borrower. I will refer to this fraction of the loan amount as collateralized debt. In this implementation collateralized debt will be characterizes by two quantities: a loan amount qc := q (d)
rL (d) =
(pL cL (d) + pH cB (d)) kB
and an interest rate ic ic = 6 This
rH (d) rL (d) qc kb
1=
pL (cB (d) cL (d)) : (pL cL (d) + pH (cB (d)))
result holds as long as the probability of having an investment opportunitity tomorrow is higher for an agent who has
an investment opportunity today than for on who does not.
13
The loan amount qc is equal to the expected repayment the lender gets from the borrower: with probability pL the lender gets paid in assets which he values
cL (d) per unit and with probability pH he gets paid
cB (d) kB in afternoon consumption good, which is the maximum amount the borrower is willing to give up in order to keep his assets. The higher the di¤erence between the borrower’s and lender’s valuations of the asset, the higher the repayment in the high state to which the borrower can credibly promise and, therefore, the higher the interest rate on the collateralized debt. Since the implementation above is the one with the maximum amount of collateralized debt, the maximum amount that can be borrowed against one unit of the asset, its debt capacity, is given by D=
(cL (d) + pH (cB (d)
cL (d))) :
The asset’s debt capacity depends on the value of collateral for lenders when there is default (insurance) and on the extra value borrowers attach to collateral when there isn’t default (incentives). Ceteris paribus, a higher value of collateral for lenders increases the loan amount since they can recover more when there is default, while a higher (extra) value of collateral for borrowers decreases the borrowers’incentives to lie and, therefore, allows them to borrow more.
3.3
Premia
As I mentioned above, if borrowers and lenders were in autarky and could not trade with each other, they would all value the asset at its fundamental value
d 1
. However, when borrowers and lenders are allowed to
trade, their marginal valuation for the asset di¤ers from the fundamental value and between borrowers and lenders. This di¤erence between the agents’ valuation of the asset and the fundamental value of the asset can be decomposed in several premia. A borrower values the asset more than the expected discounted dividend stream for two reasons. The …rst reason is that the asset serves as a liquidity transformation device, it allows the borrower to invest when he has the investment opportunity. The extra value due to this function is captured by the private liquidity premium, which I de…ne as the di¤erence between how much a borrower would value the asset if he chose to sell it to get funds and the fundamental value of the asset. If a borrower chose to sell the asset in the afternoon, he would get d + cL (d) per unit of asset, which is the maximum amount the lender would be willing to pay for it. With this funds, the borrower would be able to invest in his risky projects and he would get the return on equity value each unit of asset
E( ) 1
L L
E( ) 1
L L
on them: Therefore, the borrower would
d + cL d , and the private liquidity premium is de…ned as E( ) 1
L
d + cL d
L
d 1
The second reason why a borrower values the asset more than the fundamental value is that he expects to use it as collateral in the following period. I de…ne the private collateral premium as the extra value a borrower gets from using the asset as collateral instead of selling it to raise funds. In an a¢ ne equilibrium, 14
a borrower who chooses to use the asset as collateral values it cB (d) per unit. Then, the private collateral premium is cB (d)
E( ) 1
L
(d + cL (d)) =
L
E( ) 1
L
pH (cB (d)
cL (d)) :
L
This premium depends on the di¤erence in valuations for the borrower and the lender. If both agents value the asset the same, the private collateral premium is 0. In this case, the borrower would be indi¤erent between selling the asset and pledging it as collateral. When the borrower values the asset more than the lender, the private collateral premium is positive and the borrower chooses to use the asset as collateral in the optimal funding contract. Finally, a lender may value the asset more than the expected discounted sum of its dividends if he has some bargaining power in the asset market. By being able to sell the asset to agents that value the asset more than themselves, lenders can extract some of this surplus whenever their bargaining power is positive, i.e.,
< 1. This extra value is what I call a liquidity premium and it is de…ned as 0 1 d E ( ) 1 @ 1 + pH (cB (d) cL (d)) A d = (1 ) cL (d) : 1 1 L 1 + (1 ) E( ) L 1
When
L
= 1 this premium is 0 since borrowers have all the bargaining power in the asset market and keep
all the surplus from the transaction. In this sense,
is a measure of the assets liquidity. When
is high,
liquidity is low and the price at which the asset is sold in the asset market is closer to the fundamental value d 1
. A low level of
is associated with a high liquidity premium and, as one would expect it increases the
debt capacity of the asset. Proposition 3 The debt capacity of the asset increases with the assets liquidity, .i.e., @D 0 @dH jd In the model, the asset partly resolves the non-contractibility of the return of the risky projects and allows the borrower raise funds. Since in equilibrium the borrower values the asset more than the lender does, cB (dH )
cL (dL ) is the endogenous cost of defaulting on the promised amount rH and it allows the
borrower to credibly commit to reporting the projects’ return truthfully. Assets that have dividends that are more highly positively correlated with the return of the risky projects have a higher dH and a lower dL which imply a higher cost of default for borrowers. This higher cost of default allows the borrower to commit a larger amount of goods in the high state and, thus, increase the debt capacity of the asset and make it better collateral.
16
Proposition 5 If dH
dmin , where dmin < d, a mean preserving spread of the returns of the projects
decreases the debt capacity of the asset, i.e. @D @
H jE( )
d2 : Then, the expected value of the dividend conditional on the return of project j is E dt+1 j
1 t
=
H
=
E dt+1 j
2 t
=
H
=
18
d1 + (1 ) d2 ; 2pH (1 ) d1 + d2 : 2pH
Given this correlation structure, an increase in , increases (decreases) the correlation between the return of type 1 (type 2) projects and the future dividend level while keeping the unconditional probability of success, pH , and the unconditional expected dividend, d, constant. Let D =
1D
1
+
2D
2
be the average debt capacity of the asset.
Proposition 7 An increase in If
1
=
2,
has the following e¤ ects.
the economy’s average debt capacity remains unchanged, type 1 borrowers collateral con-
straint is relaxed and type 2 borrowers collateral constraint is tightened. if
1
>
2,
the economy’s average debt capacity increases, and type 1 borrowers collateral constraint is
2,
the economy’s average debt capacity decreases, and type 2 borrowers collateral constraint is
relaxed. if
1
0 is such
that the participation constraint binds. This contract still satis…es all the constraints, but it attains a value Vs0 > Vs : If rHs > rLs , ICH is the only relevant incentive compatibility constraint. For all s such that rHs > rLs and ICH is not binding, the previous argument applies and a value Vs0 > Vs can be attained. Now consider those s
0 such that rH s
rL s
qj and Vj0 > Vj : Therefore, lim Vj0
j!1
lim Vj = V
j!1
and without loss of generality one can concentrate on the sequences fVj g as de…ned above in (9), such that the loan quantities fqj g satisfy (12) with equality. Having this constraint hold with equality implies rLj =
L qj
+ dkB . Since qj
0 always, this implies that all contracts along this sequence satisfy rL > 0
which is the same as satisfying (13) with strict inequality: Suppose that for some j (14) holds with equality. This implies rHj = L qj
H qj
+ dkB and since rLj =
+ dkB this would imply that participation constraint is slack, and that the producer is giving the
non-producer all the gains from the project. From lemma 2, there exists a feasible and incentive compatible contract that attains a higher value that contract j and therefore, without loss of generality we can ignore sequences in which for some elements j, (14) holds with equality.
25
6.2 6.2.1
Uniqueness Asset Market
Given the a¢ ne structure of the equilibrium, using the results for prices and quantities in the asset market, one gets that the value functions for lenders and borrowers in the morning before entering the asset market are, respectively, VLm (kL ; ) = P (kB ; kL ; ) = (1
) cB ( ) +
and VBm (kB ; ) =
cB ( )
d+ E
Z
0
cL
0j
0
d + E 0 cL
kL
kdFLm (k) + cB ( ) kB :
Using the guessed functional form for the value functions gives cL ( ) = (1
) cB ( ) +
d + E 0 cL
and E 0 cL Therefore,
"
cL ( ) = (1 6.2.2
0
=
(1
) cB ( ) +
) E 0 cB 1 (1
d+
0
+ d
) E 0 cB 1
0
: 0
+ d
!#
:
Funding Market
Applying the results in proposition 1 implies that, in an a¢ ne equilibrium, the value of a borrower who enters the loan market with kB units of the asset and is matched with a lender with kL units of the asset can be written as
VBa (kB ; kL ; )
=
max
tH ;tL 2[0;kB ]2
(E ( )
1) q + dkB +
+ pH E 0 (cB ( ) (kB + E
0
cB
0
tH ) j
L)
d + cL
(pH cL ( ) tH + pH cL ( ) tL )
+ pL E 0 (cB ( ) (kB Z 0 0 kdFLm (k)
tL ) j
H)
s.t. q q
=
dkB + max
(pH cL ( ) tH + pL cL ( ) tL ) pH cB ( ) (tH tL ) 1 L 0 pH E 0 cL j H tH + pL E 0 cL 0 j L tL + pL E
rH
= q
pH E
0
cL
0
j
H
tH
pL E
0
cL
0
j
L
tL + p L E
rL
= q
pH E
0
cL
0
j
H
tH
pL E
0
cL
0
j
L
tL + p H E
0
0
(16) 0
0
cB
j
cB
0
j
L
cB
0
j
H
L
(tH
(tH (tH
tL ) ; 0(17) tL ) tL )
Given the a¢ ne speci…cation of the utility functions, the solution to the borrower’s problem in the afternoon will be in corner solution. If the constraint (17) is ignored, there are four possible solutions: 26
tL = 0 = tH , tL = 0 and tH = kB , tL = kB and tH = 0, and tL = kB = tH : If tL
tH then the constraints
on q are satis…ed. If tL < tH , (17) might bind. In the appendix I show that (17) can’t bind in equilibrium. Therefore, ignoring the constraints on q ; (17) ;and using the guessed functional form for VPa (kB ; kL ; ) ; one can match coe¢ cients and get cB (d)
(E ( ) L) d+ 1 L (E ( ) 1) pH E 1 L
=
+ pH E aB ( )
=
E
0
cB
0
j
0
cB
0
0
cL j
d0 + E
@tH + pL E @kB @tL @kB
H
0
cL
0
+ pL E 0 cL 0 j L 1 Z 0 00 kdFLm (k) :
cB
00
j
@tH @kB
H
@tH @kB
1
H
0
cB
0
0
pH E
j
@tL @kB
L
@tL @kB
(18)
tL :
(19)
The following four lemma’s help characterize the optimal funding contract. Lemma 5 (17) can’t bind in equilibrium. Proof. Suppose (17) binds, then, dkB +
pH E
L
0
cL
0
j H tH + pL E cB (dH )
0
0
cL
j
L
tL
= tH
If rH = 0 binds, q =
pH E
0
cL
0
j
H
tH
pL E
0
0
cL
j
L
tL (1
L)
+ pL dkB
and using the de…nition of q together with (11) this implies pL dkB +
q =
pH E
0
0
cL
j
H
tH + pL E 1 L
0
cL
0
j
L
tL (1
pH
L)
:
Putting these last two equations together gives pH E which implies pH E
0
0
cL
0
cL
0
j
j
H
H
tH + pL E tH + pL E
0
0
0
cL cL
0
j
j L
L
tL (2
pH
L) L
=
L pL dkB
tL < 0; a contradiction.
Proposition 8 The distributions of assets converges to a degenerate distribution m m a a lim FB;t (k) ; FL;t (k) ; FB;t (k) ; FL;t (k) = (1; )
t!1
Proof. If asset transfers are not contingent, H ( ) = . Suppose that the borrowers transfer the asset only if the low state is realizes. Then, the probability that a borrower has 0 assets in the afternoon of time t + 1 is a Pr kB;t+1 =0 =
a Pr kB;t =0 +
1 X
a pL Pr kB;t =s
s=1
27
!
a Pr kB;t = 0 + pH
1 X
n=1
a Pr kB;t =n
!
where the …rst term represents the fraction of agents who transferred s units of the asset in the afternoon of time t and who had 0 assets at the beginning of the afternoon. The second term represents the probability meeting a lender in the asset market who did not hold any assets: the lenders who were matched with borrowers with no assets the previous afternoon or those who were matched with borrowers who did not transfer assets. Then, a Pr kB;t+1 =0
a pH Pr kB;t = 0 + pL
=
a = pH pL Pr kB;t =0
2
a pL Pr kB;t = 0 + pH
a + p2L + p2H Pr kB;t = 0 + pH pL
a for all t > 0 and Pr kB;0 = 0 = 0. Let pH = p: De…ne the operator T as follows,
T x = x + p (1
2
p) (1 + x)
Note that if x 2 [0; 1], then T x 2 [0; 1]. Moreover, a a Pr kB;t+1 = 0 = T Pr kB;t =0
and a a Pr kB;t+1 = 0 = T t+1 Pr kB;0 = 0 = T t+1 0
Note that jx T t0
T xj
T t+1 0
2
= p (1
p) (1 + x)
= p (1
p) 1 + T t 0
2
0 and therefore contradicts tL = 0. Lemma 1 In an a¢ ne equilibrium, a borrower will choose tL
=
tH
=
kB 8 < k B : 0
if cB ( )
cL ( )
if cL ( )
cB ( )
Proof. The proof of this proposition follows from the FOC of the borrower’s problem in the funding market using the two lemmas above. Lemma 7 tL = kB = tH is not an equilibrium. Proof. Suppose tL = kB = tH in equilibrium. In this case, cB ( ) cL ( )
= =
(E ( ) 1 (1
L)
=
0
and
(d + cL ( ))
L
) cB (d) + 29
d+
(1
) cB ( ) + d 1
Therefore, cL ( ) =
(1
) cB ( ) + d ; 1
and
(E( ) L ) 1 L d + (E(1 ) L L )
cB ( ) = 1
(1
)
which implies + cL ( ) = 1 and cB ( ) =
(E ( ) 1
Thus, cL ( )
L) L
0
@ (E ( ) 1
cB ( ) =
(E( ) 1
0
1) @
(1
) d
L)
(1
L
+
@d +
L
(E( ) 1
+
0
L) L
(E( ) 1
1
) L)
L
+
(E( ) 1
+
(E( ) 1
d+ 1
+
(1 L) L
L) L
(E( ) 1
1
) d
A:
(1
)
(1
)
L) L
(1
)
11
dAA < 0
what would imply that the derivative of the objective function is decreasing in tH and, thus, tH = 0; which is a contradiction. Proposition 2 In the only a¢ ne equilibrium, tL = kB and tH = 0 . Proof. From the previous lemmas in this section, the only candidate left for equilibrium is tL = kB and tH = 0: Assume tL = kB and tH = 0. Then, (18) becomes cB ( ) E
0
0
cB
j
= =
H
(E ( ) 1 (E( ) 1
L) L L)
L
d+
E (dj
H)
L
0
+ pL E
pL E
L) @ d+
0
j 0
cB
+ pH E
L
L)
j
L
cB
0
0
j
H
; and
:
L
0
cB
0
0
cB
0
pH (E(1 )
1
(E ( ) cB ( ) = 1
pL E
j
L
+ pH (E(1 )
pH (E(1 )
1
L) L
dH
L) L
1
A:
(20)
To have tL = kB and tH = 0 be chosen by the producer, the objective function must be increasing in tL and decreasing in tH ; i.e., E( ) 1
L L
pL E
0
cL
0
j
pL E
L
0
0
cB
j
+
L
E( ) 1
1 L
pH E
0
cB
0
j
H
>0
(21)
and E( ) 1
L L
E
0
cL
0
j
E
H
0
0
cB
j
H
< 0:
(22)
Let di = E (dj i ) and d = E (d). Using (20) evaluated at d = dL and d = dH the derivative of the objective function with respect to tL becomes 0 E( ) L p E 0 cL 0 j L (1 ) 1 L @ L 1 pH (E(1 ) L L ) + (E(1 )
dL + L) L
30
pH (1
d (1
)
pH dH +
pL )dH + pL dL (1 )
d (1
)
1 A
Using that (1
pL ) dH + pL dL = (1
+ pH ) dH + pL dL = (1
) dH + d
the expression above becomes E( ) 1
L
(1
L
)
pH (E(1 )
1 since E
cL
0
pL E
L)
0
0
cL
j
d
dL +
L
(1
(E ( ) 1
+
)
1)
pH
L
diH +
d (1
)
>0
L
0
j
dL +
L
d (1
0: Therefore, the derivative of the objective function is positive
)
under this guess which in turn implies tL > 0: The sign of the derivative of the objective function with respect to tH depends on the sign of the di¤erence in marginal valuations between the borrower (borrower) and the lender. Using the de…nition of cL ( ) and cB ( ) together with the law of iterated expectations and that
0
converges, one gets that cL ( ) = cL (d)
and cB ( ) = cB (d). Using this in the equation above, this di¤erence can be rewritten as " !# (1 ) cB d + d cL (d) cB (d) = (1 ) cB (d) + d+ cB (d) 1 ! (1 ) cB d + d = d+ cB (d) 1 (E ( ) 1 (E ( ) 1
=
1)
(1 1 L 1) (1 d+ 1 L
)
d+
)
(E ( ) 1 (E ( ) d L) 1 1 L (1
L)
)
d
+
1
L
cB d +
(E ( ) 1
L) L
cB d
But cB d pL cL (dL ) + pH (E ( ) pH 1 (E ( ) pH 1
since
(E( ) 1
L) L
(E ( ) 1
L)
dH +
L L) L
dH +
L)
dH
L
d 1 d 1
(E ( ) d L) 1 1 L d (E ( ) > 1 pH 1 1 d > pL cL (dL ) 1 d > pH d H + 1 d d = pL dL + 1 1 >
L) L
d 1
pL cL (dL )
> 1:
Then, cL (dH )
cB (dH )
(E ( ) 1
1) L
dH +
(1 1
)
(E ( ) 1
L) L
d 1
cB d
< 0;
and this implies tH = 0: Therefore, if the contract implied by tL = kB and tH = 0 is a solution to the borrower’s problem in the project market.
31
d
6.2.3
Value function coe¢ cients with one project
When dt = d for all t, the coe¢ cients for the marginal valuations are given by ) (E(1 )
(1 cL (d)
=
cB (d)
=
(1 0 @
+
pH (E(1 )
) 1
(1
L) L
pH (E(1 )
1 L)
(E( ) L ) (1 1 L (E( ) L ) pH 1 L
) 1
) pL (E(1 )
(1
L
L) L
d
L) L
pH ) ) pL (E(1 )
(1
L) L
1
Ad
Proposition 3 The debt capacity of the asset increases with the assets liquidity, .i.e., @D
