"A Theory of Financing Constraints and Firm Dynamics"

1/21 "A Theory of Financing Constraints and Firm Dynamics" G.L. Clementi and H.A. Hopenhayn (QJE, 2006) Cesar E. Tamayo Econ612- Economics - Rutgers April 30, 2012

2/21 Program I Summary I Physical environment I The contract design problem I Characterization of the optimal contract I Firm growth and survival I Contract maturity and debt limits

3/21 What they do in a nutshell I The paper develops a theory of endogenous nancing constraints.

3/21 What they do in a nutshell I The paper develops a theory of endogenous nancing constraints. I Repeated moral hazard problem

3/21 What they do in a nutshell I The paper develops a theory of endogenous nancing constraints. I Repeated moral hazard problem I The optimal contract (under asymmetric info) determines non-trivial stochastic processes for rm size, equity and debt.

4/21 Physical environment: I At t = 0 entrepreneur (E) has a project that requires initial investment I 0 > M where M in his net worth.

4/21 Physical environment: I At t = 0 entrepreneur (E) has a project that requires initial investment I 0 > M where M in his net worth. I Borrower (E) and lender (L) are risk neutral, discount future at δ. I Both agents can fully commit to a long term contract. I At each t 1, E can re-scale the project by investing additional k t. I Returns are stochastic and equal to R (k t ) if state of nature is H (with prob. p) and zero if state is L (prob. 1 p)

5/21 Physical environment: I Assumption: R () is continuous, increasing and strictly concave.

5/21 Physical environment: I Assumption: R () is continuous, increasing and strictly concave. I Assumption: R () is private information; the state of nature at t is θ t 2 Θ fh, Lg but E reports ˆθ t.

6/21 Physical environment: De nition (reporting strategy) A reporting strategy for E is ^θ = ˆθ t θ t t=1 where θt = (θ 1, θ 2,..., θ t )

6/21 Physical environment: De nition (reporting strategy) A reporting strategy for E is ^θ = ˆθ t θ t t=1 where θt = (θ 1, θ 2,..., θ t ) De nition (contract) A contract is a vector σ = α t h t 1, Q t h t 1, k t h t 1, τ t (h t ) where h t = ˆθ 1,..., ˆθ t De nition (feasible contract) A contract σ is feasible if α t 2 [0, 1], Q t 0, τ t h t 1, L 0, τ t h t 1, H R (k t ). De nition (equity and debt) Expected discounted cash ows for E is called equity, V t (σ,^θ,h t for L is called debt, B t (σ,^θ,h t 1 ) 1 ) and

7/21 The rst-best (symmetric info) I Since both are risk neutral and share δ, the optimal contract maximizes total exp. discounted pro ts of the match (E, L).

7/21 The rst-best (symmetric info) I Since both are risk neutral and share δ, the optimal contract maximizes total exp. discounted pro ts of the match (E, L). I In equilibrium L provides E with the unconst. e cient k in every t: max [pr (k) k] (1) k I So that R () strictly concave) 9 unique k 0 that solves (1). I Assume k > 0 so that per-period total surplus is: π = max [pr (k) k] = pr (k ) k k

8/21 Private information: the Pareto frontier I L designs a contract that gives her B (V ) and gives E a value V.

8/21 Private information: the Pareto frontier I L designs a contract that gives her B (V ) and gives E a value V. I The Pareto frontier of the problem is given by Gr (B(V )) and each (V, B(V )) implies a value for the match W (V ) = V + B (V ). I Begin by nding the equilibrium of the subgame that starts after L decides not to liquidate the project. I Upon continuation, the evolution of equity is given by: h V = p (R (k) τ) + δ pv H + (1 p) V Li (2) I While the evolution of debt (not in the paper): h B (V ) = pτ k + δ pb V H + (1 p) B V Li I The value of equity e ectively summarizes all the information provided by the history itself (Spear and Srivastava, 1987; Green, 1987) so it s the appropriate state variable in a recursive formulation of the repeated contracting problem.

9/21 Recursive formulation upon continuation I The optimal contract upon continuation maximizes the value for the match Ŵ (V c ), subject to LL, IC and PK constraints.

9/21 Recursive formulation upon continuation I The optimal contract upon continuation maximizes the value for the match Ŵ (V c ), subject to LL, IC and PK constraints. I In recursive form, the program to be solved upon continuation is: h Ŵ (V ) = max pr (k) k + δ pw V H + (1 p) W V Li k,τ,v H,V L s.t. h V = p (R (k) τ) + δ pv H + (1 p) V Li (PK) τ δ V H V L (ICC) τ R (k), V H 0, V L 0 (LL)

10/21 Recursive formulation before liquidation decision I If project is liquidated, E receives Q while L receives S project is not liquidated, they get V c, B (V c ). Q. If

10/21 Recursive formulation before liquidation decision I If project is liquidated, E receives Q while L receives S project is not liquidated, they get V c, B (V c ). I Pure strategies may not be optimal for some values of V so α 2 [0, 1] and L o ers a "lottery" to E. Q. If I Thus, in recursive form, the program to be solved prior to liquidation: W (V ) = max α2[0,1],q,v c αs + (1 α) Ŵ (Vc ) s.t. : αq + (1 α) V c = V (PK) : V c 0, Q 0 (LL)

11/21 Regions for V (Propositions 1 & 2) The domain of V can be partitioned in three regions: I Region I: When 0 V V r, liquidation is posible and randomizing is optimal with α (V ) = (V r V ) /V r

11/21 Regions for V (Propositions 1 & 2) The domain of V can be partitioned in three regions: I Region I: When 0 V V r, liquidation is posible and randomizing is optimal with α (V ) = (V r V ) /V r I Sketch of argument: α = 1 ) W (V ) = S while α = 0 ) W (V ) = Ŵ (V ). Now W > S ) 9! V r and α 2 (0, 1) s.t.v V r implies that αs + (1 α) Ŵ (V r ) > max S, Ŵ (V ). I Intuition: As V! Vr expected value Ŵ (V ) rises above S and L liquidates with low probability (draw graph). I Region III: When V V = pr (k ) / (1 δ) the total surplus is the same as under symmetric information ( rst-best), i.e., W (V ) = W.

12/21 The borrowing constraint region (Propositions 1 & 2 cont.) I Region II: When V r V < V :

12/21 The borrowing constraint region (Propositions 1 & 2 cont.) I Region II: When V r V < V : (a) There is no liquidation in current period and V 7! W (V ) is strictly increasing and,

12/21 The borrowing constraint region (Propositions 1 & 2 cont.) I Region II: When V r V < V : (a) There is no liquidation in current period and V 7! W (V ) is strictly increasing and, (b) The optimal capital advancement policy is single-valued and s.t. k (V ) < k (the rms is debt-constrained) I Sketch of argument: suppose that the optimal repayment policy for region II was τ = R (k) implying that the ICC binds (see below the proofs for both of these results). Then: R (k) = δ(v H V L ) which implies that increasing k is only incentive compatible if V H V L also increases. But W (V ) concave implies that doing so is costly! (draw graph)

13/21 Optimal repayment policy (proposition 3) I The optimal repayment function satis es τ = R (k) for V < V and τ = 0 for V V.

13/21 Optimal repayment policy (proposition 3) I The optimal repayment function satis es τ = R (k) for V < V and τ = 0 for V V. I Intuition:

13/21 Optimal repayment policy (proposition 3) I The optimal repayment function satis es τ = R (k) for V < V and τ = 0 for V V. I Intuition: I We know that maxv 2V W (V ) = W and from props 1&2 we know that W (V ) = W.

13/21 Optimal repayment policy (proposition 3) I The optimal repayment function satis es τ = R (k) for V < V and τ = 0 for V V. I Intuition: I I We know that maxv 2V W (V ) = W and from props 1&2 we know that W (V ) = W. Now, at given t, L delivers promised utility Vt either by allowing τ < R (k) or by promising higher future value.

13/21 Optimal repayment policy (proposition 3) I The optimal repayment function satis es τ = R (k) for V < V and τ = 0 for V V. I Intuition: I I I We know that maxv 2V W (V ) = W and from props 1&2 we know that W (V ) = W. Now, at given t, L delivers promised utility Vt either by allowing τ < R (k) or by promising higher future value. Risk neutrality and common δ ) V! V in the shortest time possible is optimal.

13/21 Optimal repayment policy (proposition 3) I The optimal repayment function satis es τ = R (k) for V < V and τ = 0 for V V. I Intuition: I I I I We know that maxv 2V W (V ) = W and from props 1&2 we know that W (V ) = W. Now, at given t, L delivers promised utility Vt either by allowing τ < R (k) or by promising higher future value. Risk neutrality and common δ ) V! V in the shortest time possible is optimal. Limited liability then implies τ = R (k) until V = V

14/21 Evolution of equity when V < V I From prop. 3 we know that τ = R (k) for V r V < V is optimal.

14/21 Evolution of equity when V < V I From prop. 3 we know that τ = R (k) for V r V < V is optimal. I Next, notice that τ = R (k) implies that the ICC binds (lemma 2).

14/21 Evolution of equity when V < V I From prop. 3 we know that τ = R (k) for V r V < V is optimal. I Next, notice that τ = R (k) implies that the ICC binds (lemma 2). I To see this, recall that from prop. 2, V < V ) k (V ) < k. Thus, suppose that the optimal k is s.t. the ICC is slack: τ = R (k) < δ V H V L. Then one could increase k thereby increasing the total surplus without violating the ICC, contradicting optimality. I Next, ICC binding ) R (k) = δ(v H V L ). Recall that from prop. 2 V r V ) α (V ) = 0. Summarizing: V = αq + (1 α) V c = V c = δ hpv H + (1 p) V Li

15/21 Evolution of equity when V < V I The authors show that V L (V ), V H (V ) are nondecreasing.

15/21 Evolution of equity when V < V I The authors show that V L (V ), V H (V ) are nondecreasing. I Moreover, starting from any equity value V 0 2 [V r, V ) after a nite sequence of good shocks V 0! V. I Likewise, after a nite sequence of bad shocks V 0! V r or below, triggering randomized liquidation.

Dynamics of equity 16/21

17/21 Dynamics of equity I Finally, it is easy to see that fv t g is a submartingale with two absorving sets: V t < V r and V t > V 8 t

17/21 Dynamics of equity I Finally, it is easy to see that fv t g is a submartingale with two absorving sets: V t < V r and V t > V 8 t I Simulations (R (k) = k 2/5, p = 0.5, δ = 0.99, S = 1.5):

18/21 Optimal k advancement policy I We ve seen that V 0 2 [V r, V ) ) k (V ) < k (the rms is debt-constrained).

18/21 Optimal k advancement policy I We ve seen that V 0 2 [V r, V ) ) k (V ) < k (the rms is debt-constrained). I Now, it is di cult to characterize V 7! k (V ) in general (it is nonmonotonic).

18/21 Optimal k advancement policy I We ve seen that V 0 2 [V r, V ) ) k (V ) < k (the rms is debt-constrained). I Now, it is di cult to characterize V 7! k (V ) in general (it is nonmonotonic). I However, simulations show that conditional on success, capital grows at a positve rate, i.e. k(v H )/k (V ) > 1 while k(v L )/k (V ) < 1. I This contributes to the "cash- ow coe cient" debate; if Tobin s q is a su cient statistic for investment, then cash ows should not matter.

19/21 Firm growth and survival I Firm size is captured by k

19/21 Firm growth and survival I Firm size is captured by k I Starting frm the same V 0 2 [V r, V ) simulate many di erent shock paths.

19/21 Firm growth and survival I Firm size is captured by k I Starting frm the same V 0 2 [V r, V ) simulate many di erent shock paths. I As seen before, rms eventually either exit or reach the unconstrained optimum size. I Since k < k w/e V t < V surviving rms grow with age; size and age are positively correlated in accordance with empirical evidence. I Mean and variance of equity growth decrease sistematically. I Conditional probability of survival increases with V. Given that V increases over time (for surviving rms), survival rates are positively correlated with age and size.

Firm growth and survival 20/21

21/21 Beyond rm dynamics (not presented here) I The authors show that the optimal (long term) contract can be replicated by a sequence of one-period contracts i it is renegotiation-proof.

21/21 Beyond rm dynamics (not presented here) I The authors show that the optimal (long term) contract can be replicated by a sequence of one-period contracts i it is renegotiation-proof. I The contract is renegotiation-proof i collateral S is greater than the maximum sustainable debt.