An Investment Criterion Incorporating Real Options

Transcription

1 An nvestment Criterion ncorporating eal Options James Alleman, Hirofumi uto, an Paul appoport University of Colorao, Bouler, CO, UA an Columbia University, ew York, Y, UA East, okyo, Japan emple University, Philaelphia, PA, UA s: Abstract: his paper provies an investment ecisionmaking criterion uner uncertainty using real options methoology to evaluate if an investment shoul be mae immeiately, cautiously, eferre (wait-an-watch), or foregone. We evelop a ecision-making inex, which is equal to the expectation of net present value (P) normalize by its stanar eviation. Uner a lognormal assumption of the istribution of P iscounte by riskfree rate, we fin the break-even point at which the P equals the real option value (O): D.76. Using the absolute value of D, one can make sophisticate ecisions consiering opportunity losses an costs of uncertainty. his new ecision inex,, provies a criterion to make investment ecisions uner uncertainty. When making a ecision, a manager only has to observe three parameters: expectation of future cash flow, its uncertainty as measure by its stanar eviation, an the magnitue of investment. We iscuss examples using this criterion an show its value. he criterion is particularly useful when P lies near zero or uncertainty is large. Keywors: eal Options, Decision, nvestment. OPO PCE A financial option is the right to buy (a call) or sell (a put) a stock, but not the obligation, at a given price within a specific perio of time. Option pricing theory etermines the theoretical value of an option. here are several approaches to this problem, base on ifferent assumptions concerning the market, the ynamics of stock price behavior, an iniviual preferences. he most important theories are base on the no arbitrage principle, which can be applie when the ynamics of the unerlying stock take certain forms. he simplest of these theories is base on the multiplicative, binomial moel of stock price fluctuations, which is often use for moeling stock behavior.. he Binomial Moel Assume a stock traes at a price. Within one perio, the price will be either u or. Further assume we have a risk-free bon with return +r f per perio. o avoi arbitrage opportunities, we must have u > > (.) his section can be skippe by people who are familiar with option pricing theory. his interpretation is from [] f we have a stock option that allows us to buy the stock at the price K, calle the exercise price or strike price one perio later, the payoffs of the option are shown in equation (.) an Figure -. Figure. hree relate lattices Cu max( u K,) (.) C max( K,) tock u Option max(u-k, ) C Bon max(-k, ) o uplicate these two payoffs, we purchase x ollars worth of stocks an b ollars worth of the bon. One perio later, this portfolio will be worth either ux + b or x + b, epening on the path. o match the option outcomes we therefore require ux + b C u x + b C olving this equation, we have Cu C x u Cu ux uc Cu b ( u ) (.3). Combining these we fin that the value of the portfolio is Cu C x + b u uc Cu + ( u ) u Cu + C u u ow we know that the value x + b must be the value of the call option C because the payoffs of this portfolio are exactly the same as that of the stock option. (he one price principle) he portfolio mae up of the stock an the bon that uplicates the payoff of the option is often referre to as a replicating portfolio. C u Cu + C (.4) u u

2 here is a simplifie way to view equation (.4). Defining the quantity q u (.5) an from the relation u > > assume earlier, it follows that < q <. Hence q can be viewe as a probability. his q is referre to as the risk-neutral probability. ewriting (.4) yiels (.6): Option pricing formula he value of a one-perio call option on a stock governe by a binomial lattice process is C ) ( qc + ( q ) (.6). u C Another way to obtain this risk-neutral probability is foun by solving the equation, ( qu + ( q) ). As a suggestive notation, we write (.6) as C ( ) Eˆ[ C( )] (.7) Here C() an C(-) are the option values at an -, respectively, an Ê enotes expectation with respect to the risk-neutral probabilities. We can exten this solution metho to multi-perio () options using the formula: C Eˆ[ C( )] (.8) where is the risk-free return to the time to expiration. he option price is calculate using payoffs for all cases, using the risk-neutral probability in the expectation function an iscounting with the risk-free rate.. he Continuous Aitive Moel ext, we set up the one perio continuous aitive moel, where + u (.9) :stock price at time t t u : ormal istribution with mean, variance σ : eturn of risk-free asset his moel always satisfies risk-neutral because E [ ] E[ + u] E [ ] + E[ u] i.e. E[ ] (.) f we have a call option on this stock with exercise price K at time, payoff of option C() is: C( ) [max( K,)] i.e. C ( ) if < K C( ) K if > K (.) We calculate this option price using the general option pricing formula (.8). C Eˆ[ C( )] Eˆ[ C()] () f ( ) C where f ( ) is probability ensity function of, normally istribute. ( K K ) f ( ) substituting + u, f ( ) (/ ) f ( u), u K (, + u K) f ( u) u [ ( + u K) f ( u) ] u (.) K f we introuce + u, which is normally istribute with mean, variance σ, (.) can be rewritten as a general option pricing formula for continuous outcome moel. Option pricing formula he value of a one-perio call option on a stock governe by a continuous aitive moel is K C ( K) f ( ) (.3) where : the present value of the future ranom value iscounte by risk-free rate K is the present value of exercise price, iscounte by the risk-free rate We will use this option formula in section 3.

3 Figure. Payoff an Option Price return is calle volatility. 3 When the return is efine as P[ / ], its volatility is given by ( σ / ). Payoff, Option Price Density Function of C PayoffC() Probability Although the option pricing theory has been evelope in orer to value financial options, it can be applie to the real asset or firm s project. he methoology is calle real options. n the next section, we show how it is applie. E [ ] K Exercise Price ote that when takes a lognormal istribution, this formula is equivalent to Black-choles Call Option formula []. Following the iscussion of the previous section, we exten this moel to multi-perio () options using the moel: + t t t u (.4) t where u i is the ranom variable normally istribute with mean, variance σ i.e. We can confirm risk neutral because: + u (.5) i i E[ ] E[ ] + E ui i since E [ ] for all i u i i.e. E[ ] P[ E[ ]]. An assuming u i is not correlate with any other the variance of P[ E[ ] ] is: σ [ σ ar P[ ]] ar ui i We can write this as: (.6) σ σ (.7) u, hen we can calculate one-term stanar eviation from multi-term one. he stanar eviation of the yearly j i. eal Options eal Options methoology is an approach use to evaluate alternative management strategies using traitional optionpricing theory applie to the real assets or projects. For example, when managers ecie to assess a new project, they face several choices beyon simply accepting or rejecting the investment. Other choices inclue elaying the ecision until the market is favorable, or eciing to start small an expaning later if the result seems to be superior. he traitional valuation metho, DCF analysis, fails to account for these other choices. he list of these real options is shown in able. [3]. 4 able. : Description of Options Option Description Defer o wait to etermine if a "goo" state-of-nature obtains Abanon o obtain salvage value or opportunity cost of the asset hutown & restart o wait for a "goo" state-of-nature an re-enter ime-to-buil o elay or efault on project - a compoun option Contract o reuce operations if state-of-nature is worse than expecte witch o use alternative technologies epening on input prices Expan o expan if state-of-nature is better than expecte Growth o take avantage of future, interrelate opportunities Example: alue of the Option to Defer he simplest real option alternative is the eferral option which is base on the concept of the call option, as shown in Figure. [8]. uppose you have a mining project, which is not profitable currently. A eferral option gives you the option to efer starting this project for one year to see if the price of gol rises high enough to make the investment worthwhile. We can interpret this right as a call option. he numerical example illustrates its value. able. isplays the project s present value of future cash flow. is assume to be normally istribute with mean $ million ( ) an stanar eviation $3 million ( σ). he risk-free rate is 6% (.6); the 3 A precise efinition of volatility is the stanar eviation of the return provie by the asset in one year when the return is expresse using continuous compouning. [3]. 4 Methos to valuate each kin of option are escribe in several books [4],[5],[6] an [7].

4 Defer Option Present value of a project s future cash flow Call Option tock price Fig.. ExP (Expane P). M nvestment to acquire the project assets K Exercise price 6.4 M Length of time the ecision may be eferre ime to expiration ime-value-of-money r isk-free rate of return iskiness of the project assets σ ariance of returns on stock exercise price one year later is $ million. With these assumptions, the project s present value is $3.8 or $/ M P + O ExP Conventional alue of Project Flexibility value to efer able. A mining project Defer Option ariable Present value of operating future cash flow $ million nvestment in equipment K $3.8 million Length of time the ecision may be eferre year isk-free rate r f.6 iskiness σ $3 million Conventional P is given by -K million. his project woul have been rejecte uner P criterion. However, applying call option formula in the pricing equation (.3), the value of eferring the project one year is calculate as the efer O (eal Option alue): ( K O C K) f ( ) ( K) exp K π σ 3.8 ( ) ( ) ( 3.8) exp π 3. (.) he flexibility that allows us to efer this project is value at. million. Aing P an O gives positive value 6.4 or his is calle Expane P or ExP. ExP represents the value of this project incluing future flexibility [4]. Consequently, your optimum ecision now is efer, i.e. wait an watch the gol market! 3. HE EW DECO MAKG CEO Decision uner conitions of uncertainty shoul be mae on the basis of the current state of information available to ecision makers. f the expectation of the P were negative for the investment, the conventional approach woul be to reject the investment. However, if one has the ability to elay this investment ecision an wait for aitional information, the option to invest later has value. his implies that the investment shoul not be unertaken at the present time. t leaves open the possibility of investing in future perios. For the purpose of analyzing the relationship between P an the option value associate with the single investment, we assume that the ranom variable of interest is the present value of future cash flow, which is assume to be normally istribute ~ ( m', ). he investment cost is assume to be a constant. n the conventional metho, P is expresse as: P E [ - ] E[] - m' - (3.) We examine the two cases: ( a, a) A are efine as Act : ( a ) o not invest now when P <, an Act : ( a ) invest now when P >. 3. Case : Act o not invest as P < Here, following Herath & Park [], we introuce a loss function. When we o not invest, the cash flow is equal to. But imagine the situation that >, where the opportunity loss is recognize as -. herefore the loss function of Act is: L ( a, ) if < - if > (3.)

5 he expecte opportunity loss can be calculate as: [ L( a, )] L( a E, ) f ( ) ( ) f ( ) (3.3) his function is the payoff of a call option, (.) using the pricing formula of a call option iscusse in the previous section, (.3). Assuming we can efer this investment to obtain new information, we know this value is the same as the efer option for the investment. Moreover, the value is also equal to the expecte value of perfect information (EP) for this investment opportunity []. O (eal Option alue) ( ) f ( ) (3.4) We can see the similar relationship in Figure 3. as we saw before in Figure.. Figure 3. Opportunity Loss function an O (P<) Loss function, O Density Function of O L(a, ) nvestment m' When the terminal istribution of is normal, the real option value can be calculate using the unit normal linear loss integral: ( D L ( D) D) f ( ) (3.5) where f ( ) is the stanar normal ensity function O ( D) (3.6) 5 L where m' D When the manager makes the ecision to invest, her optimal ecision is not to invest if P m'- <. hen she may compare the P an the efer option value. f she fins that the option value is larger than the absolute value of P ( m'- ), she has the option to efer an watch for positive changes in the investment opportunity. f the option value is too small to compensate the P (< ), she will abanon this investment proposal. 5 his expression is only for the case P < though general expression is possible. Probability Decision Criterion : (Case of P < ) O > P Wait an watch the opportunity carefully O < P Do not invest We can solve the equation, O P (3.7) for m' D. From (3.) an (3.6), (3.7) is expresse in ( D) m' (3.8) Divie by L, we have: (because of > ) olving this equation, D is: L D D ( D) D m' L ( D) i.e. L ( D) D (3.9) ( D) f ( ) D f ( ) D f ( ) D exp D DΦ π D where Φ ( ) ( D ) D a a f ( x) x D.76 (3.) An also the left han sie of equation (3.9) is ecreasing as D increases because: ( L ( D) D) < for all D > (3.) D From (3.) an (3.), we can confirm that former Decision Criterion can be written in; Decision Criterion ': (Case of P <) D < D Wait an watch the opportunity carefully D > D Do not invest 3. Case : Act invest as P > imilarly, in the opposite case we can iscuss loss function as []: L( a, ) if < (3.) if >

6 he expectation of loss is given by: E[ L( a, )] L ( a, ) f ( ) ( ) f ( ) (3.3) hese equations are similar to the payoff an price of the put option (Figure 3.). Assuming we can efer this investment, the option value is the same as the expectation of the loss. O ( ) f ( ) (3.4) By symmetry, (3.4) can be written as: O ( D) (3.5) L where m' D Figure 3. Loss Function an O (P>) Loss Function an O L(a, ) Density Function of m' O nvest carefully if < < D Wait an watch if D < < Do not invest if < D m' where, D.76 Consequently, we know that only observing three parameters, m ', an gives us sufficient information to make more sophisticate ecisions uner uncertainty, expressing them in form of the new ecision-making inex,. 3.3 What an D mean? What o an D mean? First, can be seen as P ivie by its stanar eviation. n other wors, is the ratio of P to its uncertainty. Because oes not epen on the size of the project, it can be calle uncertainty-ajuste P or risk-normalize P. We can easily compare several risky projects of which the sizes are ifferent. ext, when - D, option value to efer is equal to expecte loss of Figure 3.3 ummary of the Criterion P P < O P > O P < O < -D -D < < Decision not nvest wait an watch n this case, optimal ecision is invest as P m'- >. Comparing the P with the value of this option, which is same as the cost of uncertainty, we arrive at a similar criterion: Decision Criterion : (Case of P > ) O > P nvest carefully O < P nvest ewriting this as: Decision Criterion ': (Case of P > ) O E[ P]m'- O E[P]m'- P P > O P < O P > O < < D D < Decision nvest carefully nvest D < D nvest carefully D > D nvest Here we introuce ecision-making inex ( m' ), which is given from eliminating the absolute value sign from D. We can combine the two ecision criteria as: O E[P]m'- O E[P]m'- Combine Decision Criterion P, namely, D is the break-even point of expectation of P an its option value, or the point where ExP. nvest if D <

7 Let s calculate the probability that the payoff of this efer option is positive if -D at time. he probability is calculate as follows, P [ > ] P > σ m + m P > σ m m P > σ σ P > [ ] [ D ] P >.39 where is stanar normal istribution. Because is normally istribute with mean m an stanar eviation σ, (-m )/σ is normally istribute mean, stanar eviation. herefore, the probability that the payoff of the efer option is positive is 39%. Does it seem to be a high probability to abanon this option? Yes, it oes! he criterion < -D means, Do not invest now but oes not mean Abanon the efer option. he efer option itself has value though the expectation of P is eeply negative. f holing the option oes not require any cost, we o not have to throw it away! Just wait an watch what happens in the next perio. On the other han, if D, the probability that the project will be out of the money is also 39%, by symmetry. When the manager makes her ecision to invest as D, there is still 39% probability of losing money. f the manager wante a positive P with probability 9%, shoul be higher than.8. t might be the case that the manager coul set a higher for the ecision criterion if she woul not care about opportunity losses. he traeoffs between opportunity loss an cost of uncertainty are shown in Figure 3.4 n the next section, we will show an example of this criterion. Figure 3.4 raeoffs of Losses Loss Expecte Loss Probability Payoff > Expecte Opportunity Loss - - -D D Probability 4. AMPLE CAE ix nepenent Projects f we have projects shown in able 4., how can we make ecision using our new criterion? We have assets that have current value, time to expiration, exercise price K at time, volatility σ, an risk-free rate r f. o calculate, we have to solve m, an σ. Assuming the value of at time is normally istribute with mean (+ r f ), m because m is expresse in present value. After setting P(K) K/(+ r f ), σ σ, we can calculate ( m' ) /. herefore, we fin for each project an make ecision to invest as shown in able 4.. Luehrman [8], [9] efine option space having two axes value-to-cost ( / P ( K ) ) an volatility ( σ ) an showe that ecision criterion epens on the region in the option space. Using his example, we fin similar results with our criterion. Our new ecision criterion can is an integrate, simplifie version of Luhrman s metho. Furthermore, project E in able 4. is what we illustrate before in section, the Mining Project, an the new criterion gives the same ecision, wait an watch! ariable A B C D E F $. $. $. $. $. $. K $9. $9. $. $. $. $ σ 3% 3% 3% % 3% 4% rf 6% 6% 6% 6% 6% 6% $. $4.43 $. $4.4 $3. $ P(K) $. $9.9 -$. -$6.84 -$3.77 $. +infinite.469 -infinite Exercise ecision invest invest o not invest Current asset value K Exercise Price (at time ) ime to expiration (year) σ tanar eviation of return (per year) isk-free rate of return (% per year) r f σ t.5 tanar eviation of -P(K) Conventional P (m'-) uncertainty ajuste P 5. COCLUO o not wait an invest watch invest carefully Applying real option valuation methoology, we have shown that the new ecision inex the uncertainty ajuste P an D.76 the break-even point of

8 P an O (real option value) gives a clear solution to make a ecision uner uncertainty. When making ecision, managers have to observe only three parameters: expectation of future cash flow, its uncertainty, an the amount of investment to acquire the project. An also we have iscusse some examples using our new criterion an shown its usefulness. EFEECE [] Davi G. Luenberger, nvestment cience, Oxfor University Press, 998 [] Hemantha Herath an Chan Park, eal Options aluation an ts elationship to Bayesian Decision- Making Methos, he Engineering Economist, ol. 46, o. [3] James Alleman an Paul appoport, Moeling egulatory Distortions with eal Options, he Engineering Economist, volume 47, number 4,, pp [4] Lenos rigeorgis, eal Options: Managerial flexibility an strategy in resource allocation, M Press, 996 [5] Martha Amram, alin Kulatilaka, eal Options, HB Press, 999 [6] om Copelan, laimir Antikarov, eal Options: A Practitioner s Guie, exere LLC, [7] om Copelan, im Koller, an Jack Murrin, aluation: Measuring an Managing the alue of Companies, McKinsey & Company nc [8] imothy Luehrman, nvestment Opportunities as eal Options: Harvar Business eview, 5-67, July- August 998 [9] imothy Luehrman, trategy as a Portfolio of eal Options: Harvar Business eview, 89-99, eptember-october 998 [] James mith an obert au, aluing isky Projects: Option Pricing heory an Decision Analysis, Management cience ol. 4, o.5, May 995 [] James Alleman an Eli oam (es.), he ew nvestment heory of eal Options an ts mplication for elecommunications Economics, Kluwer Acaemic, 999 [] Avinash Dixit an obert Pinyck, nvestment Uner Uncertainty, Princeton University Press, 994 [3] John Hull, Options, Futures an other Derivatives, Prentice-Hall, [4] obert chlaifer, ntrouction to tatistics for Business Decisions, McGraw Hill, 96 [5] Peter Boer, he eal Options olution: Fining otal alue in a High-risk Worl, John Wiley & ons, nc. [6] Aswath Damoaran, Dark ie of aluation, Prentice-Hall, [7] Michael Mauboussin, Get eal, Creit uisse Equity esearch, June 999 [8] Harriet embhar, Leyuan hi, Chan Park, eal Option Moels For Managing Manufacturing ystem Changes in he ew Economy, he Engineering Economist, ol.45, o.3 [9] Daisuke Yamamoto, ntrouction to eal Options, oyo-keizai, (in Japanese)