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)

V (n) = max

{c,z,y,e,m,n′}

• u (c) + χ V
• n′

s.t. c + z + y = n + αm,              n liquid assets c manager compensation z dividends y investment αm new equity α < 1 n′ = f (y), productive investment m = e R−1 Ω

• n′

equity rewards: e share of the firm c ≤ ψz comp linked to dividends = e ≤ m (n − γc − z) + m anti-dilution protection = γ?

6

What is shareholder value?

• Value for shareholders of the firm that takes as given manager’s behavior

Ω (n) = z (n) + R−1 [1 − e (n)] Ω

• n′

.

• 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 ψ

• (1 − α) n − αR−1γψ z′[f (y)] + αR−1f (y) − y
• (Budg Constr)
• Where ξ ≡

ψ 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

y

u [ψQ(n, y)] + χ V [f (y)]

• Guessing (and verifying later) for an interior solution The FOC is

ξ uc

• αR−1fy
• 1 − γ ψ z′

n

• − 1
• = χ fy(y) V ′

n(y)

• and envelope condition

(the time inconsistency is not here)

Vn = ξ (1 − α) uc,

• together yields the GEE (in compact form)

uc = χ (1 − α) 1 − α R−1 fy(1 − γ ψ z′

n) fy u′ c. 8

Generalized Euler Equation in Markov Perfect Equilibrium

uc = χ (1 − α) 1 − α R−1 fy(1 − γ ψ z′

n) fy u′ c.

• Today’s manager takes it into account tomorrow’s manager recklessness

via z′

n ≡ ∂z′ ∂n′ .

• The GEE collapses to a usual Euler equation when α = 0: uc = χfyu′

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 Ωt to emit.

10

Manager’s Problem - Commitment

max {ct,zt,yt,mt,et,nt+1,Ωt}

∞ t=0

∞

• t=0

χtu (ct) s.t. ct + zt + yt = nt + α mt, mt = R−1etΩt+1, et ≤ mt mt + nt − γct − zt , ct ≤ ψ zt Ωt = zt + R−1 (1 − et) Ωt+1, nt+1 = f (yt) . Euler Equation: (note the accumulated effect over time) uc,t+

t

• j=1
• −α R−1 χ−1

ψγ j uc,t−j = χ (1 − α) fy,t 1 + (1 − α) α R−1 γ ψ fy,t − α R−1 fy,t uc,t+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 W0) and the rest of the periods (with value function W ).

• The first period problem is

W0(n) = max

c,z,y,e,m,Ω,Ω′

• u (c) + χ W
• f (y), 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

W (n, Ω) = max

Ω′

u n − Ω + γc γ

• + χ

W

• f
• 1 − α − 1 + ψ − α (1 + ψγ)

ψ γ

• n + α R−1Ω′

+ 1 + ψ − α (1 + ψ γ) ψγ Ω−

• α + 1 + ψ − α (1 + ψ γ)

ψ γ

• γc
• , Ω′
• 13

Optimality Conditions of Commitment Problem

• FOC

α R−1Wn′fy + W ′

Ω = 0

• Envelope conditions

Wn = 1 γ uc + χ

• 1 − α − 1 + ψ − α (1 + ψγ)

ψγ

• W ′

nfy

WΩ = − 1 γ uc + χ1 + ψ − α (1 + ψγ) ψγ W ′

nfy

• From the envelope conditions:

Wn + WΩ = χ (1 − α) W ′

nfy

• Assuming Wn = W ′

n and WΩ = W ′ Ω in the steady state (same as before):

1 − α R−1fy = χ (1 − α) fy.

14

A Comparison of MPE and Commitment Solutions: Under-Capitalization in MPE (Steady State)

• Markov Perfect Equilibrium:

f ME

y

= 1 χ (1 − α) + α R−1 (−γ z′

n + 1)

• Commitment:

f CM

y

= 1 χ (1 − α) + α R−1

• Social Planner

f SP

y

= 1 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:

α R−1 γ χ ψ ν 0.98 0.99 0.5 0.9 1.0 0.9

• Results: z′

n = 0.036 > 0. Thus, y CM > y ME.

Commitment Equilibrium vs Markov Perfect Equilibrium y z Ω z/Ω m/Ω Commitment 0.31 0.035 0.33 0.10 0.09 Markov Perfect 0.26 0.034 0.28 0.12 0.11

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Global Solution – Markov Perfect Equilibrium

0.2 0.4 0.6 0.02 0.04 0.06 0.08 0.1 dividend (z)  = 0.5  = 1  = 1.5 0.2 0.4 0.6 0.02 0.025 0.03 0.035 0.04 0.045 0.05 consumption (c) 0.2 0.4 0.6 0.24 0.25 0.26 0.27 0.28 0.29 capital (y) 0.2 0.4 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 value of equity () 0.2 0.4 0.6 0.2 0.4 0.6 0.8 1 dividend-equity value ratio (z/) 0.2 0.4 0.6

• 0.4
• 0.3
• 0.2
• 0.1

0.1 0.2 0.3 0.4 fund raised through new equity (m)

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

{c,z,y,ℓ,d,e,m}

• u (c) + χ
• η∗(d,y)

V

• f
• d, y, η′

; Ω

• G
• dη′
• continuation value of survival

+ χ V G

• η∗′ (d, y)
• utside option
• s.t

c + z + y = n + αm Budget Constraint ℓ = d + y Balance Sheet m = e R−1

• η∗(d,y)

Ω

• f
• d, y, η′

dG

• η′

Equity Issuance c ≤ ψz Compensation Constraint e ≤ m (n − γc − z) + m Dilution Constraint f (ℓ, y, η′) = Rℓ · ℓ1−νη′ −

• Rd + 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∗. f

• ℓ, y, η′

= max

• Rℓ−γη′ − (Rd + ζ (ℓ − y))
• ℓ + (Rd + ζ (ℓ − y)) y, n
• Q (ℓ, y, n) = ξ

ψ

• (1 − α) n+

αR−1

• η′∗(ℓ,y)
• f
• ℓ, y, η′

− ψγ z′ f (ℓ, y, η′)

• dG
• η′

− y

• η′∗ (ℓ, y) =

A +

• Rd + ζ(ℓ − y)
• · (ℓ − y)

Rℓ · ℓ1−ν

21

The problem of the manager then simplifies to...

V (n) = max

ℓ,y

• u [ψ Q (ℓ, y, n)] + χ
• η′∗(ℓ,y)

V

• f
• ℓ, y, η′

G(dη′) + χ G

• η′∗ (ℓ, y)
• V
• With Optimality conditions (again under interiority)

[y] ψ uc Qy = −χ

• η∗(ℓ,y) V ′

n fy G(dη′)

[ℓ] ψ uc Qℓ = −χ

• η∗(ℓ,y) V ′

n fℓ G(dη′)

[envelope] Vn = uc ψ Qn

22

Generalized Euler Equations

[y] uc

• αR−1
• η′∗(ℓ,y)
• fy
• ℓ, y, η′

1 − ψγ z′

n

• f (ℓ, y, η′)
• dG
• η′

+ η∗

y γψz(A)g(η∗)

• − 1
• = χ
• (α − 1)
• η∗ u′

c fy G(dη′) + g(η∗) η∗ y (ℓ, y) 1

ξ [V (A) − V )]

• [ℓ]

uc

• αR−1
• η′∗(ℓ,y)
• fℓ
• ℓ, y, η′

1 − ψγ z′

n

• f (ℓ, y, η′)
• dG
• η′

+ gη∗

ℓ γψz(A)g(η∗)

• = χ
• (α − 1)
• η∗ u′

c fℓ G(dη′) + g(η∗) η∗ ℓ (ℓ, y) 1

ξ [V (A) − V )]

• 23

GEEs continued

[y] 0 = ψ uc ·

1 1+ψ−α(1+γψ)

• −1

direct loss in consumption from y↑ +αR−1

η∗(d,y) fyG(dη′)

gain in equity valuation from more loans −αR−1

η∗(d,y) γψz′ nfyG(dη′)

loss in equity valuation from higher z′ −αR−1γψz(A)g(η∗)(−η∗y)

• loss in equity valuation from lower default

+χ

• η∗(d,y)

ψ(1−α) 1+ψ−α(1+γψ) u′ cfyG(dη′)

gain in future (consumption) from more loans + χ

ξ [V (A) − V ] g(η∗)(−η∗y)

gain in continuation value from lower default

24

Two State Example

• Suppose η′ ∈ {0, 1} and ζ(d) = κ · d. Let p1 = Pr (η′ = 1) .
• Long-surviving bankers’ leverage and capital can be orderd as

leverageME > leverageCM > leverageSP, y ME < y CM < y SP.

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Proof (Step 1)

• Compare a Markovian manager and a committed manager.
• Regardless of com. tech., ℓ (y) is the same under moral hazard.

Dℓf (ℓ (y) , y, 1) = 0.

• DRS implies dℓ (y) /dy < 1 and leverage is decreasing in y.
• Again, y ME < y CM due to time inconsistency. Hence,

leverageME > leverageCM.

26

Proof (Step 2)

• Compare a committed banker and the social planner.
• Only the banker is protected by limited liability.
• Solutions to d and ℓ satisfy dCM > dSP and ℓCM < ℓSP:

dCM = p−1

1

• χ (1 − α) + αR−1−1 − Rd

2κ > R−1 − Rd 2κ = dSP, ℓCM =

• χ (1 − α) + αR−1

p1 (1 − γ) R 1/γ <

• R−1p1 (1 − γ) R

1/γ = ℓSP

• This implies that y CM < y SP. As a result,

leverageCM > leverageSP,

27

Conclusion

• Time inconsistency problem arises in firms where manager’s compensation

is linked to profits not value (effective problem in non-traded firms).

• It generates excessive dividends and equity issuance
• When added to environments with moral hazard such as banking or other
• nes where there is an external cost in bankruptcy the problem leads to

too much borrowing and thus excessive leverage of banks.

• It may also lead to insufficient investment, although this depends on the

details.

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