investment, capital structure, and hedging in the presence...
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Dynamic risk managementInvestment, capital structure, and hedging in the presence of �nancial
frictions
Diego Amaya, Geneviève Gauthier, and Thomas-Olivier Léautier
HEC Montréal and TSE
March 2015
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Today�s discussion
Introduction �context, objectives and contribution
Model description and parameters estimations
Main results
Extensions and future work
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Today�s discussion
Introduction �context, objectives and contributionModel description and parameters estimations
Main results
Extensions and future work
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Risk management �the fundamentals
Absent �nancial frictions, risk management does not matter: all"good" projects are �nanced, investors can diversify idiosyncratic risk,and systematic risk cannot be o oaded below cost (Modigliani andMiller, 1958 and 1963)
Asymmetric information between insiders/managers andoutsiders/investors (moral hazard, and/or adverse selection) create�nancial frictions, hence potential re�nancing constraints, hencejustify risk management (Holmström and Tirole, 2000)
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Turning these fundamentals into models of �rms�behavior
Convex incremental cost of capital: optimal heding policy function ofthe correlation between internal wealth and stochastic investmentopportunities (Froot, Sharfstein, and Stein, 1993), optimal capitalstructure (Froot and Stein, 1998)
Risk management as an inventory management program (Rochet andVilleneuve, 2011). "Cash is king": cash reserve is the state variable,bankruptcy occurs when cash runs out. Optimal/maximal cash reservelevel, optimal hedging policy: do not hedge past a certain cash reserve
Add investment and re�nancing costs to the inventory managementproblem (Bolton, Chen, and Wang, 2011)
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Objectives of this work
Builds on Léautier, Rochet, and Villeneuve, 2007
Determines a �rm�s optimal risk management policy (dividendpayments, investment level, and "hedging" policy)
inventory management programconvex cost of capitalstochastic investment opportunities
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Convex cost of capital
Re�nancing constraint progressively and continously tighter asleverage increasesMarginal cost of capital applied to the entire capital base, not simplythe incremental investment
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Stochastic investment opportunities
A portion of investment is planned (e.g., replenish depreciation)
Other investments depend on market conditions and opportunities,success of previous ventures, hence are inherently stochastic
The size of �rms is limited by the stochastic creation of opportunities,as well as searching and matching, not only by adjustment costs
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Other hypotheses
Managers maximize the value of the �rm, not only shareholders value:consistent with observed practice
The �rm can fully hedge its pro�t volatility: an oil �rm can sell all itsproduction forward, an airline can either purchase its entire oil supplyforward or index ticket prices to oil prices
The �rm re�nances itself through borrowing only: equity issuance andasset sales are not considered
The interest rate hence the cost of capital tend to in�nity as leveragetends to one
First two modeling assumptions relaxed and impact of third andfourth discussed in extensions
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Main results
Full hedging is optimal, except for very high leverage: convex capitalcost yields concave value function. Full hedging reduces volatility ofleverage, hence is optimal ... until gambling for resurrection becomesoptimal for very high leverage
Two target leverage ratios, also �xed points of the leverage dynamics.First, to the left of the cost-minimizing leverage: precautionarysavings. Second, on the right of the cost-minimizing leverage:maximum pro�table growth
Results appear robust to relaxing of hypotheses
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Today�s discussion
Introduction �context, objectives and contribution
Model description and parameters estimationsMain results
Extensions and future work
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Timing and decisions
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Pro�ts and hedging
Net Operating Pro�t less Adjusted Taxes (NOPAT):
πt+1 = x̃t+1It
Underlying source of pro�t uncertainty z̃t+1, i.i.d. and normallydistributed
Costless hedging, forward price equal to the expected spot price:
x̃t+1 = ηtE[z ] + (1� ηt )z̃t+1,
where the hedging ratio ηt satis�es:
0 � ηt � 1 (1)
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Investment and dividends payout
Investment opportunity it 2 f0, ig, i.i.d. Bernouilli trials withprobability of arrival p
Evolution of invested capital
It+1 = It + gt It � δIt = (1+ gt � δ)
where δ > 0 is the depreciation rate, and
0 � gt � δ+ it (2)
Resulting Free Cash Flow
FCFt+1 = (x̃t+1 � gt + δ)It
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Evolution of leverage
Financing �ow
FFt+1 = r (λt )Dt � (Dt+1 �Dt ) + dt Itwhere Dt book value of debt, λt =
DtItleverage, r (λt ) interest rate
Free Cash Flow = Financing Flow
(x̃t+1 � gt + δ)It = r (λt )Dt � (Dt+1 �Dt ) + dt ItResulting motion equation
Λt+1 = 1�1+ x̃t+1 � µ (λt )� dt
1+ gt � δ
where µ (λt ) = λt (1+ r (λt ))
λt+1 =
8<:0 if Λt+1 < 0
Λt+1 if 0 � Λt+1 � 11 if 1 < Λt+1
(3)
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Value of the �rm
Vt = Et
8>>>><>>>>:∞
∑s=t+1
(xs � gs�1 + δ)Is�1s�1∏k=t
(1+ w(λk ))
9>>>>=>>>>;()
vt = Et
8>>>><>>>>:∞
∑s=t+1
xs � gs�1 + δs�1∏k=t
(1+ w(λk ))
s�2∏u=t(1+ gu � δ)
9>>>>=>>>>;where vt = Vt
Itis the relative value of the �rm (average Tobin�s Q)
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Bellman equations
Jt (λt , it ) the value of the �rm for the optimal controls is recursivelyde�ned by:
Jt (λt , it ) = maxgt ,ηt ,dt
s .t . (1), (2), (3)
11+ w(λt )
�E[z ]� gt + δ+
(1+ gt � δ)Et fJt+1g
�
Terminal value:
JT+1 (λT+1, iT+1) =E[z ]� g + δ
1+ w (λT+1)
∞
∑s=T+2
�1+ g � δ
1+ w(λT+1)
�s�2�T=
E[z ]� (g � δ)
w(λT+1)� (g � δ)
where (g � δ) is the long-term net growth rate
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First-order derivatives
De�ne
Θt =1
1+ w(λt )fE[z ]� gt + δ+ (1+ gt � δ)Et fJt+1gg
Then:∂Θt
∂dt=
11+ w(λt )
Et
�∂Jt+1∂λt+1
�∂Θt
∂gt=
11+ w(λt )
��1+Et fJt+1g+Et
�∂Jt+1∂λt+1
�1� λ̃t+1
���∂Θt
∂ηt=
11+ w(λt )
Et
�∂Jt+1∂λt+1
(z̃t+1 �E [z ])�
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Estimation of the parameters
Statistical analysis of annual data for a 20 year panel of 854 industrial �rms
g � δ long-term net growth rate 2.1%δ depreciation 12%i investment opportunity 14.7%p prob. of arrival of investment opp. 21.2%E [z ] expected ROIC 8.4%σz std. dev. of ROIC 5.8%
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Estimated cost of capital and (after tax) interest rate
λ̄ � 31.7% minimizes the cost of capital
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Today�s discussion
Introduction �context, objectives and contribution
Model description and parameters estimations
Main resultsExtensions and future work
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(Stationary) value function
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Optimal hedging policy
For λt+1 � bt+1, Jt+1 is concave, hence ηt = 1: λt+1 is deterministicAmaya, Gauthier, Léautier (TSE) Dynamic risk management 03/15 23 / 34
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Next period (expected) leverage
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Optimal dividend and investment policies
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Impact of investment opportunity in current period
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Leverage dynamics
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Sketch of the proof
Backwards induction from (T + 1)As long as there is no truncation at the optimum:
λt+1 =µ (λt ) + gt � δ+ dt �E [z ]
1+ gt � δ+(1� ηt ) (z̃t+1 �E [z ])
1+ gt � δ
= yt+1 + (1� ηt ) ε̃t+1
De�neϕt+1 (yt+1) = Et fJt+1 (λt+1, ı̃t+1)g
If E [z ]� w�λ̄�> 0, ϕT+1 (yT+1) satis�es
ϕT+1 (yT+1) concave, unique maximum λ̄T+1 2 [aT+1, bT+1] � [0, 1]
9!λ̂T+1 2�λ̄T+1, bT+1
�/�
ϕT+1 (x)� 1+ (1� x) ϕ0T+1 (x) � 0, x � λ̂T+1
�(P)
Suppose ϕt+1 (yt+1) satis�es property (P). Then, we (1) determinethe optimal controls at date t, and (2) prove that ϕt (yt ) satis�esproperty (P)
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Larger investment opportunity
i = 20% instead of 14.6%
Same solution structure
Higher precautionary savings: low target leverage 30% instead of 31%
Higher value: maximum value 3% higher
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No hedging restriction: same solution structureExpected next period leverage
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Impact of hedging restriction
Higher precautionary savings: low target leverage 29.3% instead of31%. "Hedging is tax advantaged equity"
Lower value: maximum value 8% lower when hedging restricted.Consistent with some empirical estimates (Allayanis and Weston,2001)
Suggests structure of solution unchanged if we add a non-hedgeablerisk
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Today�s discussion
Introduction �context, objectives and contribution
Model description and parameters estimations
Main results
Extensions and future work
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Maximizing shareholder value
Shareholder value St = 11+w (λt )
(dt It + St+1)Bellman equations for shareholder value per unit of equitySt = St
(1�λt )It
(1� λt ) St =1
1+ w (λt )maxdt ,gt ,ηt
�dt+
(1+ gt � δ)Et f(1� λt+1) St+1g
�Terminal value
ST+1 (λT+1) =1
(1� λT+1)
�JT+1 (λT+1)� λT+1
1+ r (0)1+ r (λT+1)
�Preliminary analysis suggest the same solution structure
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Other next steps
Equity issuance: �xed and variable costs, relevant only for highleverage
Asset sales: (1) reserve price (decreasing with leverage), (3)bargaining between buyer and seller, but also (2) stochastic saleopportunity
Correlation between pro�ts and investment opportunities, seriallycorrelated pro�ts, non-normal distributions
"Large risks": no truncation assumptions violated
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