Themes. Production-Financing Interactions and Optimal Capital Structure. Outline. Background. Why Change Rate? Why Consider Financing?

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1 Production-Financing Interaction and Optimal Capital Structure John R. Birge Northwetern Univerity Theme Production deciion hould reflect: proper conideration of rik and market effect method for financing operation Financing deciion hould likewie depend on production deciion Integrated model can addre both iue 1 2 Outline Background General background Two-tage model (new vendor) Adjutment for rik Financing without bankruptcy cot Financing with bankruptcy cot Multi-period model Aumption and reviion Concluion Traditional production model Aume a fixed dicount rate No contraint on cah Why no fixed dicount rate? Rik depend on production deciion Dicount rate hould depend on rik How can financing change deciion? Order and production require payment Cot of payment depend on financing (reerve, borrowing, tretching payable, ) 3 4 Why Change Rate? Conider a 737 (capacity 100) or 747 (capacity 400) for DTW-LGA route? Suppoe demand ha mean of 200 with tandard deviation of ditribution: 100 paenger 98% of time 747 ditribution: complete demand curve Should the ame dicount rate apply to both? Load Load Why Conider Financing? Order tire for thi month production Cot $100/et Pay? Stretch until receive payment? (5%/month) Borrow (Limited? Interet to pay) Invetor? Cot of tire change depending on form of financing 6 1

2 Simplified Model: New Vendor Problem: how much to produce now for ale in future? Parameter: [no alvage value] c cot of production p elling price F ditribution of demand Production: x* that maximize over x 0 -cx + p( 0x df() + x x df()) Solution: F(x*)=(p-c)/p or x*=f ((p-c)/p) Problem: Have not dicounted future and have not conidered rik in cah flow 7 Future Payoff Function Future payoff a px* function of demand Note: Form of hare minu call x* px* Share of market x* px* Call at x* x* 8 Valuing the Future Payoff General Approache Capital aet pricing model (CAPM) Option pricing model CAPM Aume invetor can diverify (without cot) Oberve market rik premium and correlation of cah flow to market Option pricing framework (no-arbitrage, rikneutral, martingale pricing) Aume market intrument can produce rik-free return Ue probabilitie conitent with ingle rik-free rate (rik-neutral probabilitie) 9 CAPM Approach Parameter: S uncertain future cah flow r f rik-free rate r m market rate of return λ = (r m r f )/σ m2 (market price of unit of rik) E(S) expected future cah flow Cov(S, r m ) covariance of cah flow and market return Preent value (period of length one) (E(S) - λ (Cov(S,r m )))/(1+r f ) Key: Finding covariance of cah flow and market return For new vendor, explored in Singhal (1988) 10 No-Arbitrage Pricing Idea: can trade market hare (or equivalent cah flow) and firm cah flow to eliminate rik Suppoe two demand outcome, High:H (>x*) and Low: L (<x*) Firm future payoff:px* if High, pl if Low Full market hare return: ph if High, pl if Low By elling of market, can equate the payoff: p(x*- H) = p(l - L) where =(x*-l)/(h-l) High Demand Low Demand p(x* - H) p(l - L) 11 Rik-neutral Pricing The cah flow of firm minu of the market hare i rikfree Can find the p.v. of cah flow given price of market hare and rik-free bond (no-arbitrage aumption) PV = (Market hare) + Rik-free bond = (pe(s)e -r ) + (pl(h-x*)e -rf )/(H-L) Oberve: PV doe not depend on probabilitie can ue equivalent probability ditribution that provide the ame return on each invetment (rik-neutral probabilitie) Here: Prob f (H)H + (1-Prob f (H))L = (Market hare price)(e rf) Reult: no-arbitrage rik-neutral probabilitie 12 2

3 Rik-neutral Pricing If market i arbitrage free, then can find a rikneutral equivalent meaure, F f, for new vendor Reult: F f (x*)=(p-e rf c)/p or x*=f f ((p-e rf c)/p) How to find F f? If demand ditribution i log-normal, can treat a if market hare future trade a a geometric Brownian motion with rate r, rik premium δ=r-r f Reult: F(e δ x*)= (p- e rf c)/p or x*=e -δ F ((p-e rf c)/p) Note: time-value and rik adjutment Equivalent to Singhal CAPM reult 13 What about Financing? Have rik and time-value adjutment but aume allequity financing for cx* Suppoe k available in equity and mut borrow D to meet cah requirement Debt holder receive D(e rd )if demand i greater than b, p o.w. ( b = D(e rd )/p) Reult: D e rf = 0 b p df f () + D e rd b df f () D(e rd ) Debt Holder Payoff b 14 New Vendor with Borrowing (Perfect Market) Optimal production x* hould now maximize over x 0 -ce rf x + p( 0x df() + x x df()) ubject to: c x k + e -rf ( 0 b p df f () + D e rd b df f ()) Reult: If contraint lack, ame x*; If contraint tight, eparate to obtain ame x*. Without taxe or cot of bankruptcy, production deciion independent of financing (Form of Miller-Modigliani irrelevance of capital tructure) 15 New Vendor with Corporate Tax Suppoe all-equity and corporate tax Suppoe corporate tax rate τ o that equity holder ha return a follow (no lo carryover): Y E (x) = px- τ(p-c)x if x p - τ(p-cx) if x > * p(1-τ) p if * > p Aume rik neutral pricing: find x*.t. F f (x*) = [p(1-τ)-c(e rf -τ(1-f f (*))]/(p(1-τ)) < (p-ce rf )/p Tax optimal production point * x Payoff with Tax 16 Debt and Equity Financing Suppoe financing by debt and equity Interet payment produce tax hield If bankruptcy (return cannot cover Equity payoff debt payment), fixed recovery cot of B p(1-τ) Equity holder payoff (now p include tax hield) Y E (x) = px- τ((p-c)x-(e rd ) D) D(e rd ) if x * x b p - τ(p-cx -(e rd ) D) D(e rd ) if x > * p D(e rd ) if * > b 17 -B Debt Holder Cah Flow Debt Holder Payoff Y D (x) = D (e r D) if b p Β if b > Rik-neutral pricing of debt De r f = b 0 (p-b) df f () + D e r D b df f () D(e rd ) b 18 3

4 Overall Value Function Find x* and D* to maximize over x 0 and D cx V(x,D) = x [px e rd D-τ (px-cx-(e rd )D)]dF f () + *x [p e rd D-τ (p-cx-(e rd )D)]dF f () + b * (p- e rd D) df f () + D e rd b df f () + 0 b (p-b) df f () - c x (e rf ) Auming no boundary value, oberve V/ x = p(1-τ) x df f () + cτ * df f () c e rf i.e., marginal after-tax revenue plu marginal tax benefit equal marginal cot Debt influence through * (breakeven with debt): V/ D = τ(1-f f (*))(r D + D( r D / D))-B F( b )/ D i.e., expected bankruptcy cot equal expected tax hield. 19 Obervation on Leveraged Firm With no bankruptcy cot (B=0), Solution approache D = cx (all debt) Production decreae a a function of financial leverage (i.e., Optimal x decreae with D, in particular, for all-equity optimum, x e, x* < x e ) Suppoe bet leverage for production of x e i D e, then optimal leverage, D* > D e General: Time value, rik, corporate tax, and financial leverage all lead to lower production than ideal cae. Extenion: Agency effect, parameter effect. 20 Multiple Period and General Production Requirement One-period implification Debt mut be re-paid in full No deciion on bankruptcy (abandonment) or continuation No obervation period or deciion on delay Production/pricing implification No general production capacity tructure No pricing deciion 21 Multiperiod Cae General t-period future value of ale x t : c t x t Suppoe no contraint on production, aume market rate r, preent value: e -rt c t x t Contraint limit production to K, then price a hare minu call, ue rik-neutral pricing e -rf t [ 0 c t min(,k) df f ()] Period t Payoff c t K K 22 Alternative Computation Condition Lognormal demand ditribution Future on period-t demand follow geometric Brownian motion F f (A) = F( e (r-r f )(t) Α) for any A Reult for ale lot (demand above K): Call at K = e -r f t c t (-K) + df f () = e -(r-r f )(t) e -r f t e (r-r f )(t) c t (-K) + df f () = e -rt c t (-e (r-r f )(t) K) + df() PV of production up to K: PV = e -rt (c t c t (-e (r-r f )(t) K) + )df() or max e -rt c t x()df().t x(), x() e δ t K a.. 23 General Period-t Contraint Aume rik neutral meaure, general period t: max e -rft c t x t ()df f ().t x t (), A t x t () b a.. Aume rik premium δ, lognormal ditribution Subtitution for d.f. F: x t ( )=e δt x t () e δt = A t e -δt x t ( )= A t x t () b t where ha d.f. F Equivalent form: max e -rt c t x t ()df().t x t (), A t x t () e δt b a

5 Extreme Cae All production contraint lack: max e -rt c t x t ()df().t x t (), A t x t () e δt b a.. equivalent to: e -rt c t df() (market hare correct by definition) All production contraint tight: A t x t ()=e δt b a.. with ome bai B t, o that x B t ()=B t (e δt b) for all, equivalent to: e -rt c t B t (e δt b) df() = e -rt e (r-rf)t c t B t (b) = e -rft c t B t (b) (rik-free dicounting correct by definition) Reult: method interpolate between value given by market 25 Multitage Model Private equity model (dividend either poitive or negative in each period) One-period debt paid in each period Option to pay debt or abandon in each period Production require cah in each period Corporate tax IPO or Sale at end of horizon General framework: Maximize rik-neutral (or tranformed market rik) dicounted cah flow ubject to ale demand, ale inventory plu production, production capacity, cahin dividend interet =cahout + productioncot +capacitycot + taxe, capacityforward = 0 if cahout Goal in Multi-period Model Conitent model for valuation Expandable for marketing, pricing deciion Obervation objective: Effect of rik premium, demand volatility on production, capital tructure Relationhip of abandonment timing to parameter Equity cah flow change under varying parameter Effect of IPO/Sale timing on production, capital tructure Aumption Price exogenou (fixed) Set price to ditributor (e.g., electricity) Can relax (competitive or monopoly) For monopoly, nonconvexity in optimization Rik-neutral pricing (contructing a rik-free hedge) How to contruct a hedge? If NPV>0, inconitency Proce: Trade option and aet to create rik-free ecurity Creating Bet Hedge Reult of Reidual Rik Underlying aet: Max potential ale in market Option: Plant with given capacity Other marketable ecuritie: Competitor hare Overall all ecuritie min reidual volatility Due to incompletene, ome volatility remain (otherwie, NPV=0) In binomial model, ale can be H+v 1 or L+v 2 where v 1 and v 2 in ome range Hedge ratio i a function of v 1 and v 2 ( (v 1,v 2 )) Effective dicount rate (and price of option) i in a range determined by v 1 and v 2 Analogy to general cae: can reduce to ome range on the rik premium ue other criteria (competitor, expanded utility) to chooe

6 Capacity Implication Can adjut capacity limit by varying dicount factor with rik neutral aumption on forecat Can vary contraint multiplier with original forecat ditribution All optimal policie for the given range are conitent (i.e., cannot be beaten all the time) Repone on Incompletene Ue equilibrium and utility function approache Beware of model complexity Critical factor: range of outcome conidered Other challenge: Effect of pricing deciion Effect of competitor Ditribution change from deciion Operational and Financial Hedging in Global Market Objective: Determine capacity level in different market, production in each market, ditribution acro market, and ue of financial hedging intrument to maximize total global value Challenge: Demand and exchange rate may change Correlation among demand and exchange What i enough capacity? What i the optimal mix of financial and operational hedge? Situation Market in US, Europe, and Aia Can add capacity in each place Have hitory of exchange rate and demand in each market Can tranport acro market Know capacity cot Goal Find the bet capacity number in each market Decide what meaure to ue How well can you do? Concluion Production deciion depend on: proper conideration of rik and market effect method for financing operation Financing deciion depend on production deciion Integrated model addre both iue Relaxing aumption lead to range of outcome Financial and operational hedge can be compared with comprehenive view 35 6