VERY PRELIMINARY Motivation and Content Many firms are not publicly - PowerPoint PPT Presentation

Equity Issuance in Non-Traded Firms: A Tale of Time Inconsistency José-Víctor Ríos-Rull Tamon Takamura Yaz Terajima Penn, CAERP Bank of Canada Bank of Canada November 12, 2019 Wharton Macro Lunch VERY PRELIMINARY

Motivation and Content • Many firms are not publicly traded. - Manager Compensation cannot depend on Firm Value . Not traded. - It depends on dividends. - Manager captures value • Trade-off between investment, dividends and new equity issuance. • We pose an environment where there is a firm featuring a manager who makes the decisions 1. Issues outside equity and dividends ( time inconsistency problem); 2. Is impatient ; and faces moral hazard through limited liability. 2

Main Results • Under-capitalization due to time-inconsistency problem. Time inconsistency problems exist because of: - Reoptimization of dividend payment, and dilution of existing equity. - When interpreting as banks with limited liability additional moral hazard problem • Some regulation may fix it. 3

Outline 1. Simple model to highlight the time-inconsistency issue i Time-consistent solution in Markov Perfect Equilibrium ii Commitment solution iii Insufficient Investment 2. A Model with loan, deposit and default (firms are banks) i Default and equity valuation ii Higher leverage together with under-capitalization 4

The Environment • Consider a manager with preferences u ( c ) and discount rate χ that issues equity to fund investment. • Issuance is costly. • The manager itself has no equity. • Its compensation is linked to dividends • It has no commitment to the amount of new equity issuance 5

Simple Model: Manager’s Problem (Assuming in Markov Perfect Equilibrium) � � n ′ �� V ( n ) = max u ( c ) + χ V s.t. { c , z , y , e , m , n ′ }  liquid assets n      c manager compensation  c + z + y = n + α m , dividends z    y investment    α m new equity α < 1 n ′ = f ( y ) , productive investment � n ′ � m = e R − 1 Ω equity rewards: e share of the firm c ≤ ψ z comp linked to dividends = m e ≤ anti-dilution protection = ( n − γ c − z ) + m γ ? 6

What is shareholder value? • Value for shareholders of the firm that takes as given manager’s behavior � n ′ � Ω ( n ) = z ( n ) + R − 1 [ 1 − e ( n )] Ω . • Substituting we get R − 1 Ω( n ′ ) = n + m − ( 1 + γ ψ ) z (Accounting Rule) Ω( n ) = n − ψ γ z ( n ) (Value of firm) � � z = Q ( n , y ) = ξ ( 1 − α ) n − α R − 1 γψ z ′ [ f ( y )] + α R − 1 f ( y ) − y ψ (Budg Constr) ψ • Where ξ ≡ 1 + ψ − α ( 1 + ψγ ) • Tomorrow dividends are written as a direct choice of tomorrow’s manager. This is where the time inconsistency is. 7

Yielding a compact Manager’s Problem with a clean characterization V ( n ) = max u [ ψ Q ( n , y )] + χ V [ f ( y )] y • Guessing (and verifying later) for an interior solution The FOC is � � � � α R − 1 f y 1 − γ ψ z ′ = χ f y ( y ) V ′ ξ u c − 1 n ( y ) n • and envelope condition (the time inconsistency is not here) V n = ξ ( 1 − α ) u c , • together yields the GEE (in compact form) χ ( 1 − α ) n ) f y u ′ u c = c . 1 − α R − 1 f y ( 1 − γ ψ z ′ 8

Generalized Euler Equation in Markov Perfect Equilibrium χ ( 1 − α ) n ) f y u ′ u c = c . 1 − α R − 1 f y ( 1 − γ ψ z ′ • Today’s manager takes it into account tomorrow’s manager recklessness n ≡ ∂ z ′ via z ′ ∂ n ′ . • The GEE collapses to a usual Euler equation when α = 0: u c = χ f y u ′ c . • The manager considers the “cost” of increasing y through Ω [ f ( y )] = − ψ γ z [ f ( y )] + f ( y ) . 9

Comparison with Commitment • To see how much of an issue this is (when α > 0) we pose the problem under commitment. • We proceed by posing the model as of time zero and looking at the implied Euler equation after in a generic future period. • The manager still optimally chooses how much equity Ω t to emit. 10

Manager’s Problem - Commitment � ∞ χ t u ( c t ) max s.t. ∞ { c t , z t , y t , m t , e t , n t + 1 , Ω t } t = 0 t = 0 c t + z t + y t = n t + α m t , m t = R − 1 e t Ω t + 1 , m t e t ≤ m t + n t − γ c t − z t , c t ≤ ψ z t Ω t = z t + R − 1 ( 1 − e t ) Ω t + 1 , n t + 1 = f ( y t ) . Euler Equation: (note the accumulated effect over time) � � j � t χ ( 1 − α ) f y , t − α R − 1 χ − 1 � u c , t + ψγ u c , t − j = u c , t + 1 . 1 + ( 1 − α ) α R − 1 γ � ψ f y , t − α R − 1 f y , t j = 1 11

Writing the Commitment Problem Recusively: The first Period • The problem for the manager under commitment can be recursively written by separating the first period problem (with value function W 0 ) and the rest of the periods (with value function W ). • The first period problem is � f ( y ) , Ω ′ � � � W 0 ( n ) = max u ( c ) + χ W . c , z , y , e , m , Ω , Ω ′ s.t. all relevant constraints. 12

Writing the Commitment Problem Recusively: Period 2 Onwards • Substituting the constraints into the problem • The commitment is to a value of the shares � n − Ω + γ c � W ( n , Ω) = max u + χ γ Ω ′ � �� � 1 − α − 1 + ψ − α ( 1 + ψγ ) n + α R − 1 Ω ′ W f ψ γ + 1 + ψ − α ( 1 + ψ γ ) Ω − ψγ � � � � α + 1 + ψ − α ( 1 + ψ γ ) , Ω ′ γ c ψ γ 13

Optimality Conditions of Commitment Problem • FOC α R − 1 Wn ′ f y + W ′ Ω = 0 • Envelope conditions � � 1 1 − α − 1 + ψ − α ( 1 + ψγ ) W ′ W n = γ u c + χ n f y ψγ − 1 γ u c + χ 1 + ψ − α ( 1 + ψγ ) W ′ W Ω = n f y ψγ • From the envelope conditions: W n + W Ω = χ ( 1 − α ) W ′ n f y • Assuming W n = W ′ n and W Ω = W ′ Ω in the steady state (same as before) : 1 − α R − 1 f y = χ ( 1 − α ) f y . 14

A Comparison of MPE and Commitment Solutions: Under-Capitalization in MPE (Steady State) • Markov Perfect Equilibrium: 1 f ME = χ ( 1 − α ) + α R − 1 ( − γ z ′ y n + 1 ) • Commitment: 1 f CM = y χ ( 1 − α ) + α R − 1 • Social Planner 1 f SP = y R − 1 • Insufficient capitalization if z ′ n > 0 . y SP > y CM > y ME . 15

Numerical Results (Steady State) • Functional forms: u ( c ) = log ( c ) , f ( y ) = y ν . • Parameter values: R − 1 α γ χ ψ ν 0 . 98 0 . 99 0 . 5 0 . 9 1 . 0 0 . 9 n = 0 . 036 > 0. Thus, y CM > y ME . • Results: z ′ Commitment Equilibrium vs Markov Perfect Equilibrium Ω z / Ω m / Ω y z Commitment 0 . 31 0 . 035 0 . 33 0 . 10 0 . 09 Markov Perfect 0 . 26 0 . 034 0 . 28 0 . 12 0 . 11 16

Global Solution – Markov Perfect Equilibrium dividend (z) consumption (c) capital (y) 0.1 0.05 0.29  = 0.5  = 1 0.045 0.08 0.28  = 1.5 0.04 0.06 0.27 0.035 0.04 0.26 0.03 0.02 0.25 0.025 0 0.02 0.24 0.2 0.4 0.6 0.2 0.4 0.6 0.2 0.4 0.6 value of equity (  ) dividend-equity value ratio (z/  ) fund raised through new equity (m) 0.8 1 0.4 0.7 0.3 0.8 0.6 0.2 0.5 0.1 0.6 0.4 0 0.4 0.3 -0.1 0.2 -0.2 0.2 0.1 -0.3 0 0 -0.4 0.2 0.4 0.6 0.2 0.4 0.6 0.2 0.4 0.6

An Extension with Default, Borrowing and Investment

Firms are Banks Not Normal Firms • Here investment is in risky loans • Crucially there is Borrowing • Unsecured debt • Government Insured (i.e. deposits) (se we do not have to worry about interest rate penalties for excessive risk). These deposits may be increasing in costs of acquisition ζ ( d ) • Brings an obvious concern which implies regulation 18

Model with Loan, Deposit and Default � V ( n ; Ω) = max u ( c ) + χ { c , z , y ,ℓ, d , e , m } � � � � � � d , y , η ′ � � � d η ′ � η ∗ ′ ( d , y ) V f ; Ω G + χ V G s.t η ∗ ( d , y ) � �� � � �� � outside option continuation value of survival c + z + y = n + α m Budget Constraint ℓ = d + y Balance Sheet � � � d , y , η ′ �� � η ′ � m = e R − 1 Ω f dG Equity Issuance η ∗ ( d , y ) Compensation Constraint c ≤ ψ z m e ≤ Dilution Constraint ( n − γ c − z ) + m � � f ( ℓ, y , η ′ ) = R ℓ · ℓ 1 − ν η ′ − R d + ζ ( ℓ − y ) · ( ℓ − y ) Loan Returns η ∗ ( ℓ, y ) given by f ( ℓ, y , η ′ = η ∗ ) = A Default Threshold set by regulator 19

Excessive Leverage due to Moral Hazard • We can separate the problem in stages • Without loss of generality • Imagine the choice of loans conditional on equity issuance. • It displays excessive lending (relative to what the regulator wishes due to the moral hazard of not paying for deposits) 20

Using some functions to reduce clutter • Define auxiliary variables for cash-flow f , dividends Q , and default threshold η ∗ . � ℓ, y , η ′ � �� � � R ℓ − γ η ′ − ( R d + ζ ( ℓ − y )) f = max ℓ + ( R d + ζ ( ℓ − y )) y , n � Q ( ℓ, y , n ) = ξ ( 1 − α ) n + ψ � � � � ℓ, y , η ′ � − ψγ z ′ � �� � η ′ � α R − 1 f ( ℓ, y , η ′ ) f dG − y η ′∗ ( ℓ, y ) � � R d + ζ ( ℓ − y ) A + · ( ℓ − y ) η ′∗ ( ℓ, y ) = R ℓ · ℓ 1 − ν 21