advertisement

Lecture 8: Two period corporate debt model Simon Gilchrist Boston Univerity and NBER EC 745 Fall, 2013 A two-period model with investment At time 1, the firm buys capital k, using equity issuance s and debt: the firm issues a debt with face value b, at unit price q(k, b), hence the budget constraint: χq(k, b)b + s = k The parameter χ > 1 reflects the tax shield effect - interest expenses are subsidisized. For each dollar of debt issued, the firm receives a subsidy χ − 1 . At time 2, the firm produces, and obtains a profit π = zk α , where z is an idiosyncratic shock, distributed according R ∞to a cumulative distribution function F, with mean 1 : E(z) = 0 sf (s)ds = 1. Default decision The firm will default if its profits are not large enough to repay its debt, i.e. if z < z ∗ , with z ∗ k α = b. If the firm does default, equity holders get nothing, while bondholders share the firm profits, net of bankruptcy costs. The parameter 0 < θ < 1 determines the share of profits which are recovered. Debt and Equity Value Debt value is given by a standard pricing equation, assuming risk neutrality and a discount factor β : ! Z ∞ Z z∗ zk α q(k, b) = β dF (z) . dF (z) + θ b z∗ 0 The firm equity value is Z ∞ V =β z∗ (zk α − b) dF (z). Maximization The firm chooses k, b, s, z ∗ to maximize its present discounted value Z ∞ α max∗ β (zk − b) dH(z) − k + χq(k, b)b k,b,z z∗ subject to: Z ∞ q(k, b) = β Z dF (z) + z∗ ∗ α z k z∗ 0 ! zk α dF (z) , θ b = b. The firms solves Z max β ∗ k,z + χβ ∞ k α (z − z ∗ ) dF (z) − k z∗ z ∗ kα Z ∞ Z dF (z) + z∗ 0 ! z∗ θzk α dF (z) Optimality This program can be solved by writing the two first order conditions, with respect to k and z ∗ , which determine the optimal investment and financing. First, we have (Z Z ∞ 1 = βαk α−1 ∗ (z − z ) dF (z) + χ z z∗ ∗ ∞ dF (z) + θ z∗ which can be rearranged, given that E(z) = 1, as: ( 1 = βαk α−1 Z 1 + (χ − 1) z ∗ (1 − F (z ∗ )) + (θχ − 1) z∗ zdF ( 0 Z ) z∗ zdF (z) . 0 Interpretation: note that if χ = θ = 1, we have the frictionless limit: 1 = βαk α−1 E(z), the usual MPK = user cost condition. When θ < 1, χ > 1, this is not true anymore, and the tax shield χ and the bankruptcy cost θ both tend to decrease the user cost. Optimality The second first-order condition, with respect to z ∗ , yields, after rearrangement ((1 − F (z ∗ )) (χ − 1) = χ (1 − θ) z ∗ f (z ∗ ) . This equation determines z ∗ , and hence the optimal leverage (increasing z ∗ means increasing the leverage and the probability of default). LHS: benefit of higher leverage – the tax shield conditional on no default. RHS: The cost of higher leverage is an increase in bankruptcy costs through a higher probability of default - the change in probability of default is f (z ∗ ) at the margin, and the default costs are proportional to z. zf (z) If z lognormal then z 1−F (z) is increasing, which guarantees a unique solution to this equation. Some additional intuition We can follow BGG notation. Let Z Z ∗ ∗ Γ(z ) = z dF (z) + z∗ z ∗ dF (z) 0 and ∗ z∗ Z µG(z ) = µ z∗ z ∗ dF (z) 0 denote bankruptcy costs where µ = 1 − θ. Then the equity share is 1 − Γ(z ∗ ) + (χ − 1) (Γ(z ∗ ) − µG(z ∗ )) The lender’s share is Γ(z ∗ ) − µG(z ∗ ) Firm Value Firm value is Q(z ∗ )βk α − k where Q(z ∗ ) = 1 − Γ(z ∗ ) + χ [Γ(z ∗ ) − µG(z ∗ )] zf (z) 1−F (z) is increasing implies that the term in brackets is strictly concave with an interior maximum at some z ∗∗ . So neither debt or equity claimants desire z ∗ > z ∗∗ . Optimal leverage Optimality then implies choosing leverage so that the marginal value of an additional unit of debt is equal to the marginal value of an additional unit of equity: χ Γ0 (z ∗ ) − µG0 (z ∗ ) = Γ0 (z ∗ ) This occurs at an interior value 0 < z ∗ < z ∗∗ . Since Q(0) = 1 and Q(z ∗∗ ) < 1 we have Q(z ∗ ) > 1. As long as χ > 1 issuing debt raises firm value. Otherwise it would be optimal to set z ∗ = 0 to avoid bankruptcy costs. Optimal choice of k The FOC then imply: Q(z ∗ )αk α−1 = 1/β which means that capital is above the efficient level owing to the tax subsidy on debt. We can interpret (βQ(z ∗ ))−1 as the user cost of capital. Comparative Statics The value of z ∗ is determined by the parameters χ, θ, and the distribution F. The comparative statics w.r.t. χ and θ is clear. With regard to F , an interesting comparative static is a change in the payoff distribution F. If F becomes more risky, at least for most distributions, we will have less debt and less capital. Intuitively, the debt contract becomes more inefficient, making the cost of the debt higher. (Note: this is ex-ante higher risk, not ex-post: this is different from risk-shifting) Financial Frictions Now assume that equity is fixed at some predetermined value n (Net worth) and set χ = 1. The firm borrows some initial amount qb = k − n We can write the problem as: Z ∞ α max∗ β (zk − b) dH(z) − k + n + q(k, b)b k,b,z z∗ subject to: Z ∞ q(k, b) = β dF (z) + z∗ ∗ α z k Z = b. q(k, b)b = k − n 0 z∗ ! zk α θ dF (z) , b Maximization Max problem Z max β ∗ k,z + λ β ∞ k α (z − z ∗ ) dF (z) − k z∗ ∗ α Z ∞ z k Z dF (z) + z∗ ! z∗ α θzk dF (z) ! − (k − n) . 0 We can write this as maxk,z ∗ {βQ(z ∗ )k α − λ(k − n)} where Q(z ∗ ) = 1 − Γ(z ∗ ) + λ (Γ(z ∗ ) − µG(z ∗ )) Optimality FOC: Qβαk α−1 = λ λ Γ0 (z ∗ ) − µG0 (z ∗ ) = Γ0 (z ∗ ) β (Γ(z ∗ ) − µG(z ∗ )) k α = k − n Entrepreneur’s Investment Opportunity Entrepreneur starts period with net worth N . Entrepreneur borrows: B = QK − N, Q = price of capital (exogenous) Project payoff: ωRk QK α , 0<α<1 Rk = aggregate (gross) rate of return on capital (exogenous) ω = idiosyncratic shock to project’s return 2 2 Assume: ln ω ∼ N ( −σ 2 , σ ) ⇒ E[ω] = 1 Information Structure No asymmetric information ex ante: Rk is known to both lender and entrepreneur before investment decision. ω is realized after investment decision. Asymmetric information ex post: ω is freely observed by entrepreneur. To observe ω, lender must pay µωRk QK α Parameter 0 ≤ µ < 1 measures the cost of monitoring and hence the magnitude of credit market frictions. Debt Contract Entrepreneur and lender agree to a standard debt contract (SDC) that pays lender an amount D as long as bankruptcy does not occur. If ωRk QK α ≥ D: Entrepreneur pays D to lender and keeps residual profits. If ωRk QK α < D: Entrepreneur declares bankruptcy and gets nothing. Lender pays bankruptcy cost to monitor entrepreneur and keeps profits net of bankruptcy cost. Payoffs to Entrepreneur and Lender Bankruptcy occurs if ω ≤ ω: ω≡ D Rk QK α Expected payoff to entrepreneur: Z ∞ Z k α ωR QK dΦ(ω) − ω ω ∞ Rk QK α dΦ(ω) ω Expected payoff to lender: Z (1 − µ) ω k α Z ∞ ωR QK dΦ(ω) + ω 0 Rk QK α dΦ(ω) ω Competitive loan market: Lender must earn an expected (gross) rate of return R on the loan amount B. Payoffs as a Share of Expected Profits (Rk QK) Define: ω Z Γ(ω) ≡ Z ∞ ωdΦ(ω) + ω 0 dΦ(ω) ω Z µG (ω) ≡ µ ω ωdΦ(ω) 0 Entrepreneur’s expected share of profits: 1 − Γ(ω) Lender’s expected share of profits: Γ(ω) − µG(ω) Optimal Contract Choose K and ω to solve: max [1 − Γ(ω)] Rk QK α K,ω subject to the lender’s participation constraint: [Γ(ω) − µG(ω)] Rk QK α = R(QK − N ) Lagrangean: n o max [(1 − Γ(ω)) + λ (Γ(ω) − µG(ω))] Rk QK α − λR(QK − N ) K,ω λ = Lagrange multiplier on the lender’s participation constraint and hence measures the shadow value of an extra unit of net worth to the entrepreneur. The term in brackets reflects total firm value when valued using the shadow price of external funds. Optimality Conditions FOC w.r.t. ω̄: λ= Γ0 (ω) ≥1 [Γ0 (ω) − µG0 (ω)] FOC w.r.t. K: α [(1 − Γ(ω)) + λ (Γ(ω) − µG(ω))] Rk QK α−1 = λRQ FOC w.r.t. λ: [Γ(ω) − µG(ω)] Rk QK α = R(QK − N ) External Finance Premium FOCs imply: αRk QK α−1 = ρ(ω̄)RQ ρ(ω̄) = h λ [1−Γ(ω)]+λ[Γ(ω)−µG(ω)] i ≥1 ρ(ω̄) = external finance premium (EFP) EFP is increasing in the default barrier ω: ρ0 (ω̄) > 1 Leverage and Default: The default barrier ω̄ is increasing in leverage: QK ψ(ω̄) = N 1 − (1 − α)ψ(ω̄) where ψ(ω̄) = λ [Γ(ω) − µG(ω)] 1+ ≥1 1 − Γ(ω) ψ 0 (ω̄) > 0 Intuition: An increase in leverage requires a higher default barrier to increase the payoff to the lender relative to the entrepreneur. The increase in the default barrier also implies a higher shadow value of external funds λ. An increase in net worth reduces the default barrier and lowers the premium on external funds. Example: Constant Returns to Scale (α = 1) The default barrier is determined by the rate of return on capital relative to the risk-free rate of return: Rk = ρ(ω̄) R Given ω̄, capital expenditures are determined by available net worth: QK = ψ(ω̄) N Combining these, we obtain a positive relationship between the premium on external funds and leverage: Rk QK =s , s0 > 0 R N