# Real Options in Capital Budgeting. Pricing the Option to Delay and the Option to Abandon a Project

**ABSTRACT** Traditional discounted cash-flows method for assessing projects assumes that investment decision is an irreversible one, which is not correct. Managers can and must reconsider their initial decision as the new information arises during the project life. This is managerial flexibility and it creates strategic value for a project, only if management takes advantage of the opportunities associated with an analyzed project. Real options represent a new approach in capital budgeting, using the theory of pricing financial options for investments in real assets. In this paper, we emphasize the characteristics and valuation methodologies of real options. The objective in the last section is pricing the option to delay and the option to abandon a project in construction materials field.

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Real Options in Capital Budgeting. Pricing the Option to Delay and the Option to Abandon a Project

Real Options in Capital Budgeting. Pricing the Option to Delay

and the Option to Abandon a Project

?

Nicoleta Vintilã

Candidate Ph.D. Lecturer

Academy of Economic Studies, Bucharest

Abstract. Traditional discounted cash-flows method for assessing projects assumes that investment

decision is an irreversible one, which is not correct. Managers can and must reconsider their initial

decision as the new information arises during the project life. This is managerial flexibility and it creates

strategic value for a project, only if management takes advantage of the opportunities associated with an

analyzed project. Real options represent a new approach in capital budgeting, using the theory of

pricing financial options for investments in real assets. In this paper, we emphasize the characteristics

and valuation methodologies of real options. The objective in the last section is pricing the option to

delay and the option to abandon a project in construction materials field.

Key words: capital budgeting; real options; managerial flexibility, timing options; exit options.

?

Introduction

Traditional approaches to capital budgeting, such as

discounted cash-flows (from now on DCF), cannot capture

entirely the project value, for different reasons: it is

assumed that investment decision is irreversible,

interactions between today decisions and future decisions

are not considered, and investment in assets seems to be a

passive one (management doesn’t interfere during the

life of the project).

Managerial flexibility generates supplementary value

for an investment opportunity because of managerial

capacity to respond when new information arises, while

the project is operated. Investment in real assets includes a

set of real options that management can exercise in order

to increase assets value (under favorable circumstances) or

limit loses (under unfavorable situations).

Managerial flexibility in decision-making process

introduces an asymmetry for probability distribution of

net present value (from now on NPV) for a project. An

investment opportunity value is dependent on future

uncertain events, so therefore, it will be greater than

forecasted value in the situation of passive management.

From this perspective, a project has a standard value,

determined through traditional techniques (DCF, which

does not catch adaptability and strategic value), but also

a supplementary value, coming from operational and

strategic real options held by an active management.

Decision trees are related with real option approach

by recognizing their existence, but investment value is

calculated as an average of expected cash flows (positive

or negative) weighted with probabilities associated with

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Theoretical and Applied Economics

each state. Real option appraisal eliminates potential

loses by abandoning the project when circumstances

are unfavorable (negative cash flows become zero) and

adjust discount rate to reflect the new level of risk.

Real options represent an integrated solution used

under uncertainty, by transposing the theory of financial

options to valuation of real assets, projects or even

companies, in an uncertain and dynamic environment,

where taking decisions must be a flexible process.

Specific features of real options

Unlike financial options, real options have as underlying

asset a real asset, which value is given by discounted cash-

flows plus the value of any other options associated with the

ownership of that real asset (Bruun, Bason, 2001, p. 1).

Real options are defined and valued by analogy with

financial options, but they have some specific features

(Trigeorgis,1996, pp. 127-129) that make distinction

between the two categories: not exclusively owned by

any investor (to assure the exclusive rights for a project,

someone must identify and use the competitive

advantages and raise substantial barriers to entry for other

competitors); nontradable securities (there are no

financial markets where these rights could be traded, the

only two alternatives being: exercising option or giving

up exercising); preemption for the investor who undertake

the project, despite that at the beginning the real option

was jointly held by all firms involved in that industry;

option compoundness, consisting of interdependencies

inside a project or between projects, depending on

exercising other options or taking other projects.

Black and Scholes (1973, pp. 637-657) identified the

elements affecting the theoretical price of an option in their

formula: stock price (S), exercise price (E), time to expiration

(t), variance of returns (σ) and risk free rate (r). Afterwards,

Merton (1973, pp. 141-183) completed the Black & Scholes

formula (from now on BS) with the sixth element,

dividends (δ). Models assessing financial options could be

extended for real options because of the analogy between

financial and real options, concerning the elements

influencing their value (Figure 1)(1).

Traditional methods like DCF and NPV cannot catch

the flexibility because they focus only on two

components of value creation: discounted payoffs and

investment cost. Real options capture the influence of all

six elements described before. That means a reactive

management, consisting of response from the managerial

team (through decisions they take) to the cumulated

information during operating the investment, under

uncertainty.

The proactive management of flexibility (Leslie,

Michaels, 1997, pp. 12-14) assumes, besides identifying

the real options embedded in a project, fairly appraising,

exercising them to seize the opportunities and taking

decisions for increasing their value, for maximizing

project value and shareholders’ wealth as well.

A prospective investor (the owner of a real option)

cannot control all the components of value creation

because of internal or external constraints (such as

technical, marketing, competition). Maximizing option

price can be obtained by focusing on those elements that

indeed can be influenced and confer competitive

advantage, keeping in mind the constraints.

Difficulties in using financial options pricing

models for real options valuation

There are certain limits for using financial options

pricing models for real options because the first category

has standard components, which could be easily identified

(part of them are clearly specified - maturity, exercise

price, or can be observed in the market - price of

underlying asset, risk free rate), while the second category

do not have such standardization and managerial

flexibility determine their pattern. Some of these potential

inconveniences, along with solutions proposed in

literature on real options are synthetically presented

further on (Bruun, Bason, 2001).

Risk neutrality

Also Cox, Ross & Rubinstein (from now on CRR)

binomial model and BS model make the assumption that

investor creates a replicating portfolio with constant

underlying asset price (S), which is risk neutral for an

instant of time. A new value for S means changing

portfolio structure because hedging ratio is not the same

anymore.

Real assets are not frequently traded in a specific market

and their price is difficult to observe and is almost

impossible to form a replicating portfolio. Under these

circumstances, the result of risk neutral valuation is not

Figure 1. Analogy between financial options and real options

Financial option (CALL or PUT) Variable Real option (investment opportunity)

Stock price

S

Discounted cash-flows for the project

Exercise price

E

Capital expenditure

Period of time that exercise decision may

be deferred

Time value of money

Time to expiration

t

Risk free rate

r

Variance of returns

σ σ

Risk for assets of the project

Lose of value by deferring investment

decision

Dividend yield

δ δ

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Real Options in Capital Budgeting. Pricing the Option to Delay and the Option to Abandon a Project

applicable anymore and it is not possible to use risk free

rate as discount rate.

Lander and Pinches (1998, pp. 537-567) proposed

either usage of discount factor corresponding to a traded

security with similar risk pattern like underlying asset, or

estimation of discount factor by applying an asset pricing

model such as CAPM.

The underlying asset

Any investor does not exclusively own this. Option

price depends on the level of competition in that

industry and on the respond of competitors to new

challenges. A study by Smit and Ankum (1993, pp. 241-

250) concluded that high growth rates for an industry

would bring new competitors, so the rate of return will

be closer to the cost of capital and the value of held

option will be smaller.

The value of underlying asset is represented by

present value of projected cash flows and is calculated

from NPV for a project (the so-called static or standard

value).

Willner (1995) built a model with discontinuous

changes in the value of underlying asset:

P(t) = P(0) × eµ×t ×

tQ×

−1

λ

λ

The model is working under certain assumptions:

the value of underlying asset increases with a

constant exponential rate, as a result of value

creation through research activity (except the

occasional up jumps owing to new discoveries and

down jumps generated by the entry of new

competitors on the market);

the up jumps will not bring the expected payoffs

because the new discoveries will attract more

competitors and some loses occur from this

situation; decrease in value can be projected from

historical data for similar projects;

the new discoveries are not correlated with market

or with entire economy, and investors are not

confronted with systematic risk;

the investment cost (exercise price) is fixed.

The model may be successfully used for assessing

multistage projects or start-up ventures.

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The exercise price

This is not fixed (it is not established from the very

beginning) and do not follow a deterministic process. It

should be found a mutual probability distribution for

value of underlying asset and exercise price to determine

the level and direction of correlation between the two

variables.

Schwartz ºi Moon (2000) developed a model with a

stochastic process for the exercise price. The level of

uncertainty is conversely proportional with investment

level, which means that uncertainty about initial cost

could be eliminated only by undertaking the project.

Pindyck (1993) identified two sources of uncertainty

about investment cost: a technological one, eliminated

only in the moment when the decision to invest is taken,

and an economic one (related with the changes in costs of

entries for the project, such as materials and labour). The

author considered a stochastic evolution for investment

cost, given by the following equation:

dK = -Idt + g(I, K)dz,

which means that exercise price (K) decreases as

investment proceeds and fluctuates along with the two

sources of uncertainty (technical and cost of entries for

the project).

Time to maturity

Exercise date for real options is not a priori specified

and it is uncertain because of exogenous factors, such as

competition or barriers to entry. Optimal exercise time

depends on various elements:

the greater the technological uncertainty, the

greater the uncertainty concerning exercise

date;

the stronger the competition and the weaker the

barriers to entry for an industry, the sooner the

exercising of the option (to take advantage on

preemption), although a delay for the project has

some benefits, because in the mean time

suplimentary information occures;

the patents or licences owned by investor protect

him against competition and permit him not to

hurry with exercising the option.

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Risk free rate

The discount rate is the rate of return for a riskless

security, with the same maturity as the real option. But

exercise date for the option is uncertain, so is quite

difficult to identify the appropriate discount factor. More

than that, rate of return is not constant over time. The

solution is similar with previous cases: rather considering

a stochastic evolution for discount rate than a

deterministic one.

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Theoretical and Applied Economics

The volatility of the underlying asset

Volatility is the only element not directly observable

for financial options and it has to be estimated for the

entire period until maturity. If the underlying asset is

traded, the historical variance is extrapolated for future

periods. If the underlying asset is not traded (the case of

real options), volatility is difficult to appraise. Luehrman

(1998) suggested some alternative solutions. First, the

observed risk for a financial market index could be used

as a proxy for the project risk (adjusted with the level of

individual, specific risk). Second, the volatility could be

estimated from historical data regarding similar projects

from related industries. Third, volatility could be obtained

from probability distribution of projected cash flows,

when applying Monte Carlo simulation.

Another problem is that volatility is not constant over

long periods, but this inconvenient could be eliminated

by applying a GARCH model, if possible.

The dividend yield

It is represented by lose of cash flows until maturity of

the real option (for example, by deferring the project).

Pricing real options in capital budgeting - the

option to delay and the option to abandon a

project

A various range of real options are identified and used

as powerful instruments in capital budgeting: timing

options assume that investor may postpone the investment

decision until specific information arise and help him to

understand, even partly, the uncertainty connected with

the analyzed project; staging options are very useful for

assessing multistage projects, when uncertainty is not

resolved over time and investor must undertake the project

even in small increments, in order to learn about cost pattern

and profitability of the project; exit (abandon) options

allow investor to avoid or at least reduce loses if bad

circumstances appear, by turning negative cash-flows into

null payoffs; operating options enable the firm to organize

operations for adjusting its processes to business

environment and react to economic changes by scaling up

to enhance earnings or scaling down to reduce damages,

depending on given circumstances; flexibility options

consist of purchasing or building a flexible production

capacity or asset, so that it has two or more different uses,

depending on market conditions; growth options are

usually associated with strategic investments, which

sometimes have negative NPV, but are indispensable for

implementing following projects with substantial positive

NPV, greater (in module) than loses from the initial project.

The objective of this section is to explain in detail the

working mechanism and asses two types of real options

from the categories discussed above, namely the option

to delay and the option to abandon a project, with wide

applications in capital budgeting.

Pricing the option to delay a project

On the one hand, the possibility of deferring a project

is significantly valuable because investor needs more time

to learn about uncertain variables of the project. On the

other hand, this value is diminished with the lost cash

flows for period that investment was deferred. So the

option is exercised only if potential earnings from delay

exceed loses in such a situation.

Projects with negative NPV (initially rejected from

the point of view of traditional investment analysis) could

turn to positive NPV in the future. Decision to postpone a

project is equivalent to holding a CALL option, which

provides the right, but not the obligation, to undertake

the project sometimes in the future, when the holder

decides to do so. Even profitable projects from the very

beginning (with positive NPV) acquire a plus of value by

delay of the project, if conditions related to competition,

barriers to entry or exclusive rights for a product or

technology allow that.

Pricing an option to delay a project requires

identification of the variables (Damodaran, 2002) in the

model:

underlying asset - is represented by the project

itself; its value (S) is calculated with DCF method;

exercise price - is the cost of implementing the

project; the model is working under assumption

that this is constant in real terms and it is affected

only by inflation; its evolution is deterministic,

but not stochastic;

time to maturity - is established as the period of

time that investor enjoy of exclusivity for the

analyzed project or at least has an important

competitive advantage which allow him deferring

the project without risking its achievement by

another firm;

risk free rate - is represented by the expected rate

of return for a riskless security (treasury bill or

treasury bond), with the same maturity with real

option;

volatility of the underlying asset - it appears

because of the errors associated with estimation of

the financial cash-flows and the value of

underlying asset and it is the most difficult element

to appraise because the underlying asset is not

traded. Monte Carlo simulation is used for

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Real Options in Capital Budgeting. Pricing the Option to Delay and the Option to Abandon a Project

assessing variance, because the distribution of

probability for the components of cash flows (size

of the market, market share) is defined and random

numbers (normally distributed, for example) are

generated for the specification errors of the

variables in the model. Different scenarios for

expected DCF result from here. Rolling a great

number of simulations (it is recommended to do at

least 3,000 iterations), the standard deviation of

DCF is obtained, and then it is used as a proxy for

volatility of underlying asset in the option pricing

model.

dividend yield - delay of investment generates lose

of cash-flows for each year. There are two

situations:

–if annual cash-flows are evenly distributed in

the period until maturity of the option (n years),

the lost value for the first year is 1/n and it

increases with time (1/(n-1) in the second year,

1/(n-2) in the third year and so on);

– if annual cash-flows are not evenly distributed,

the cost of postpone the project with one year

is given by the following formula:

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Cost of delay = (PV of CF in the future –

– PV of CF at the current moment) / PV of CF at the current moment

After identifying all these six elements, all we need is

to apply a pricing option model (BS or CRR), but also

keep in mind the difficulties emphasized in the previous

section.

Users prefer BS model because it consists in a simple

formula:

CALL option price =

××

=

)()(

21

dNeEdNeS

trt

××−

×−× −δ

PUT option price =

×−

=

)()(

21

dNeEdNeS

trt

−××+−×

×−×−δ

where:

t

tr

σ

ES

d

×

××+−+

=

σδ

]21)[()ln(

2

1

and

tdd

probability of normal distribution.

×−=

σ

12

and N(d) represents the cumulated

Binomial model requires laborious calculus because

valuation proceeds iteratively backwards, from the last

period to the current time moment (the more periods until

option maturity, the more complex the determination of

option price).

More than that, binomial model considers only a

finite number of periods (it is a model in discreet time).

It leads, at limit, to BS model (in continuous time), which

unlike the first one, assumes that time to maturity is

divided into an infinite number of periods. That is why

the value derived from BS formula is always smaller

than value provided by binomial model (but close to it).

To apply CRR model, when the values of the six

variables for BS model are known, Damodaran proposes

two formulas(2):

u (the “up” factor) =

dtrdt

e

×−+×

) 2/(

2

σσ

d (the “down” factor) =

where dt = 1/number of periods from a year, until maturity.

dtrdt

e

×−+×−

) 2/(

2

σσ

We stressed before that option value is reducing as

time passes (the loss in value measured by dividend yield

is enhancing). Repeating the calculus for delay option

price, an equilibrium point can be determined, where

strategic NPV (including option premium) decreased and

became equal to standard NPV. This is the moment for

exercising the option (by investing), only if, of course,

NPV is positive. Beyond this point, deferring does not

create value anymore, but destroy part of it. Therefore,

the real duration of delay option is always shorter than

the initially projected one (because of competition,

appearance of similar products).

Pricing the option to abandon a project

The possibility to renounce to a project under

unfavourable circumstances represents, in fact, another

real option for investor. The option to abandon (a PUT

option this time) will be exercised if DCF generated

subsequently by the project are even negative or positive,

but inferior to salvage value gained after project

abandonment.

Value of the underlying asset is represented again by

estimated DCF and volatility could be assessing with the

same tools as before (similar traded companies from

industry, simulations). Option maturity is the period for

adopting abandonment decision and it may coincide with

time remaining for operating the project.

Exercise price is the salvage value obtained from

abandonment of investment, estimated at the current

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Theoretical and Applied Economics

moment of time. Pricing model is applied under the

hypothesis that real assets do not depreciate (salvage value

is fixed over time). Lose of value is quantified through

dividend yield and it is determined(3) as: δ = 1/period

remaining from the project life.

The actual value from abandonment may be inferior

to the estimated one, from different reasons: there is not

demand for such a technology or an organized

second-hand market is not operational.

Until now, we implicitly supposed that liquidation

value is positive. There are situations when from

abandonment does not result something or worse, there is

a cost to cover for. In this case, the investor will renounce

to the project only if loses from still operating the project

are bigger (in absolute value) than the costs implied by

liquidation procedure.

The price of option to abandon a project increases if

investor builds a flexible operational structure from the

very beginning, which allows him to take easier the

decision concerning abandonment. This objective can

be accomplished by employing labor for limited time, by

renting or taking in leasing the fixed assets, by choosing

a more expensive, but flexible technology.

Case study: pricing of real options - option

to delay and option to abandon a project

Investment presentation and assumptions(4)

A construction company from Bucharest analyses

the opportunity to invest in a project for producing

metallic tiles for roofs (a green field project). There will

be obtained two types of products (A and B), with

different technical features. The firm will install an

annual production capacity of about 1,500 thousands

pieces (type A) and 500 thousands pieces (type B). The

life of the proposed project is ten years (2007 - 2016).

The cost required by implementing the project is 3,670

thousands USD, consisting of land, buildings and

technology, and it will be entirely financed with equity

capital. Salvage value, estimated for the end of

exploitation period of investment (including tax

shields), is 1,485 thousands USD.

A rigorous and detailed analysis of the project

(financial analysis of the company, investment cost,

European and Romanian market of metallic tiles and

perspectives for growth rates, market share, sale prices,

cost components, forecasted inflation rate, fiscal

implications, discount rate) was performed before and it

doesn’t make the object of this study. That’s why the

hypothesis for assessing free cash-flows are concisely

presented, to create the basis for identifying and pricing

real options associated with this investment. Estimation

of free cash flows is done in American dollars (USD)

because some cost elements are specified in USD.

Value of sales for the two categories of products, on

internal market, is of approximately 4,000 thousands

USD. The annual growth rate, in real terms, is estimated

to 20-22% for the first two years, 10% for the next four

years, respectively 1-2% for the last four years. This

foreseeing represents the combined results of estimations

for market size (as quantity) and unit sale prices (in real

terms) for the two categories of products, for the entire

life of the project. The firm wants to achieve a market

share of 20-30% in the first two years, following an

increasing of this percent with ten percentage points

every year, until production capacity is completely

utilized (according to forecasting, this event will occur

in the sixth year of operating).

The expenditures were appraised under subsequent

assumptions: unit variable cost represents 60% of turnover

(in the first two years), then it enhances annually with

two percentage points until it reaches 70%, which is kept

constant afterwards; fixed costs (others than amortization)

are established to a level of 80 thousands USD (in real

terms) and amortization is about 166 thousands USD (in

nominal terms).

Projected inflation rate is 2% per year (for USD) and

nominal discount rate is 18.5%. There are also used in

calculations a corporate income tax rate of 16% and an

average of 30 days for assessing working capital investment.

Investment ratios under certainty

Keeping in mind the assumptions described in the

previous subsection, we have estimated annual turnover

and operational expenses (in nominal terms), then financial

margins. It was also taken into account the tax shields

provided by the recover of financial loses from the first

year in the following years (when profits are obtained).

Finally, nominal free cash flows were appraised and

discounted with nominal cost of capital (in fact, a cost of

equity capital). The result is an NPV of 1,297 thousands

USD. Profitability index is greater than one (1.34),

indicating that the investor recovered all expenses and

gained a net discounted profit of 34% from invested

capital. Internal rate of return is 25% and exceeds the

cost of capital (18.5%). The discounted payback period

is about 4.5 years.

All these calculations lead to conclusion that investor

must accept the analyzed project and they are exhibited

in Table 1.

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Real Options in Capital Budgeting. Pricing the Option to Delay and the Option to Abandon a Project

Pricing the option to delay the project

The sensitivity analysis presumes that there are certain

intervals for possible values taken by variables from free

cash-flows model, and thus investment analysis is

transposed in an uncertain environment. Strong arguments

(related with projected trend for construction and

construction materials market, competition level,

evolution of imports and currency exchange rate, using

similar products in consumption, fiscal policy regarding

corporate income tax, firm policies for marketing,

employment and wage system, and evolution of utilities

costs) help us to establish the inferior and the superior

limits for these intervals. The new values for NPV of the

project are pictured in Table 2 and they are graphically

illustrated in Figure 2.

Sensitivity analysis – NVP

Table 2

- thousands USD -

30%

Variable

Market size

Market share

Unit sale price (real terms)

Unit variable cost (real terms)

Fixed expences (real terms)

Days for working capital

-50%

-30%

-20%

-10%

921.4 1,122.7 1,297.3 1,440.8 1,570.9 1,700.2

921.4 1,122.7 1,297.3 1,440.8 1,570.9 1,700.2 1,829.5 2,084.3

–177.4

1,792.4 1,297.3

1,362.1 1,329.7 1,313.5 1,297.3 1,281.0 1,264.8 1,248.6 1,232.4

1,434.6 1,379.7 1,352.2 1,324.7 1,311.0 1,297.3 1,283.5 1,269.8 1,256.0 1,242.3 1,214.8 1,159.9

-5%

0%

5%

10%

15%

20%

50%

–48.1

451.5

561.1 1,297.3 2,033.4

802.1

307.0 –189.7

Investment ratios for the project under certainty

Table 1

- thousands USD -

2015

2016

YEAR

2006

2007

2008

2009

2010

2011

2012

2013

2014

2017

FCF=NOPAT+Amo–∆Imo–∆WorkCap

–3,777.6

446.8

575.6

779.5 1,059.7 1,366.0 1,860.9 1,761.3 1,735.7 1,695.0 3,681.5

Cost of capital (nominal)

Discount factor

18.5% 18.5% 18.5% 18.5% 18.5% 18.5% 18.5% 18.5% 18.5% 18.5% 18.5%

118.5% 140.4% 166.4% 197.2% 233.7% 276.9% 328.1% 388.8% 460.7% 546.0% 647.0%

DCF

5,074.84

377.1

409.9

468.4

537.4

584.6

672.1

536.8

446.4

367.9

674.3

NPV 1,297.3

IRR 25.0%

Profitability index 1.34

Discounted payback period 4.5 years

PV of future CF (delay of the project)

4,563.55

420.9

373.3

445.4

510.5

579.9

573,4

446,4

367.9

304.5

541.3

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Theoretical and Applied Economics

It is easy to notice that the project is very sensitive to

little changes for unit sale price, with a great variance of

the results. A decrease for price of only 9% means negative

NPV. This situation is possible to achieve, because the

value is placed inside the interval of variation, which is

(-10%, +5%). A negative NPV is also obtained for a fall of

30% in internal market size or in market share of the new

firm. Treated independently, the two variables induce

the same effect, because their product represents the

quantity sold by the company. The value of 30% is even

the inferior limit for market share interval (-30%, +30%),

but is outside the interval for market size (-10%, +15%).

Increasing the variable costs with at least 13% determines

a negative NPV, but the value is situated outside the

interval (-5%, +10%), even it is close enough to its

superior limit.

To gather supplementary information about unknown

variables (sold quantity, market share, unit sale price), the

company considers the possibility to delay the investment

decision, which means that investor owns a CALL real

option, appraised in this section. The first step is to asses

the elements from BS and binomial option pricing models.

The value of the underlying asset is represented by

discounted cash flows of 5,074.84 thousands USD and

exercise price is the cost of implementing the project of

3,670 thousands USD (constant in real terms).

Time to maturity is strictly related with the period

while investor can maintain the advantage of the first

Romanian producer in this market consisting entirely of

imports. As we mentioned before, the company does not

exclusively own this option, because other investors

could be equally interested of such a project. As

construction industry is growing, is expected that

construction materials industry follow the same trend, so

it is a great probability that new competitors entry on this

expansive market. Using of top technology and high level

of investment needed are significant barriers to entry in

this industry, offering a supplementary advantage for

investor. Therefore, under given circumstances, we

appreciate that deferring the project with more than 3

years could drive to lose of preemption right.

Risk free rate is assimilated to the rate of return for

government bonds with maturity in 3 years. Because the

Public Finance Ministry in Romania did not issue any

more US dollars denominated bonds, we use as a proxy

the rate of return for treasury bills with maturity at 3 years,

issued by United States Treasury (4.7% at the end of year

2006) plus the country risk premium for Romania.

International rating agencies noted Romania, for

government bonds denominated in a foreign currency

with BBB (Fitch Ratings(4), last modified in august 2006),

Baa3 (Moody’s(5), improved in October 2006, from Ba1)

and BBB- (Standard & Poor’s(6)). USA rating is Aaa or

AAA that is the greatest note for all agencies. Country

default spread is zero for government bonds noted with

Aaa and 2% (or 200 points) for Baa3(7). In conclusion, for

the binomial model we use an interest rate (a so-called

risk free rate, composed of the two elements) of 6.7% (in

discreet time), corresponding to a rate of 6.486% for BS

model (in continuous time).

Volatility of the underlying asset is estimated from

the distribution of probability of DCF from Monte Carlo

Figure 2. Sensitivity analysis - Change in NPV

-8.000.000

-6.000.000

-4.000.000

-2.000.000

0

15%

2.000.000

4.000.000

6.000.000

8.000.000

10.000.000

-50%

-30%

-20%

-10%

-5%

0%

5%

10%

20%

30%

Change

50%

NPV

Market size

Market Share

Unit sale price

Unit variable cost

Fixed expences

days for working capital

Page 9

55

Real Options in Capital Budgeting. Pricing the Option to Delay and the Option to Abandon a Project

simulation (4,000 scenarios were included). For every

rolling of simulation is generated the distribution of

probability for NPV and the present value of free cash-

flows PV(FCF). Rolling subsequently the simulation for

many times, the average for natural logarithm of

discounted free cash-flows - E(ln(PV(FCF))) is situated

between 15.4 and 15.5, and standard deviation

ó(ln(PV(FCF))) falls between 14.56 and 15.54. We could

be tempted to use the maximum value for ó (the maximum

level of risk), but we know that the greater the volatility

of the underlying asset, the greater the value of CALL

option. That is why we decide to use an average of the

two values, which is σ = 15%, corresponding to a

dispersion σ2 = 0.0225.

Because the financial flows generated by investment

are not evenly distributed during the life of project,

dividend yield is calculated with the following formula:

Cost of delay = (PV of CF in the future - PV of CF at the current

moment) / PV of CF at the current moment.

Present value of cash-flows (in the future) in the

situation of deferring the project with one year is

determined under some assumptions: the forecast for

exogenous variables is extended with one year to 2017

(market size and unit sale prices remain the same in 2007,

even the company doesn’t invest, projected inflation rate

is still 2%, and corporate income tax is unchanged 16%);

for endogenous variables, strictly related to firm features

and policies, the foreseeing is lagged with one year, the

life of the project falls now between 2008 and 2017 (for

instance, market share for the first year is again 25%, but

it is applied now to another level for market size, so the

quantity sold in the first year is different from the initial

scenario). The variable expenses have the same weight,

but they are computed from other values for turnover.

Amortization has the same value in nominal terms, but

the amortization process is also lagged with one year. We

do not forget about loses recover procedure (needed for

financial lose in the first year of operating) and investment

in working capital when we recalculate the discounted

cash flows. The result is PV of CF in the future = 4,563.55

thousands USD and dividend yield = -10.075% (the sign

indicates that this is a cost).

BS model for pricing the CALL option to delay this

project leads to following results: d1 = 0.96296 and N(d1)

= 0.83222, d2 = 0.70315 and N(d2) = 0.75902, so CALL

option price is 828.66 thousands USD.

We can also use binomial model to asses the option,

but value of the underlying asset changes this time only

once per year, so dt = 1, the “up” factor u = 1.2284 and the

“down” factor d = 0.9101. Investor may exercise the option

in each year until maturity (three years), which means

that this is an American CALL option. Dividend yield

diminishes the price of the underlying asset in each point

of the binomial tree. For an example, a value of the

underlying asset of 5,074.84 thousands USD become, after

one year, S (u - ä) = 5,722.88 or S (d - ä) = 4,107.09

and so on (Figure 3).

7,277.77

6,453.67

5,722.885,222.99

5,074.84

4,631.55

4,107.093,748.34

3,323.89

2,690.04

Figure 3. The binomial tree for the value of the underlying

asset

Risk neutral probabilities are p =

du

dr

−

−+

1

= 49.29%

and 1-p = 50.71%.

Option pricing is done for every point of the binomial

tree, starting with the third year and going backwards to

the current moment of time. As an example, we picture

the binomial tree for discreet time value of option to delay

this project, in Figure 4.

3,607.77

2,404.74

1,469.58 1,552.99

852.77

754.68

365.84 78.34

36.19

0.00

Figure 4. The binomial tree for the CALL option to delay the

project

CALL option value from the binomial model is 852.77

thousands USD, bigger than the value obtained from BS

model (828.66 thousands USD), because we divided the

time until maturity in a very small number of periods

(only three). We also built a tree with 12 periods (changing

Page 10

56

Theoretical and Applied Economics

the value of the underlying asset every trimester).

Trimestrial rate of return is 1.68%, u = 1.0726 and

d = 0.9232, risk neutral probabilities are p = 0.6262 and

1 - p = 0.3738, and trimestrial dividend yield is 2.51875%.

The value of CALL option is now 847.57 thousands USD,

smaller than in the situation of annual change for the

underlying asset, but still not so close to the value from

BS model. Despite of this, unlike other securities traded

continuously, the underlying asset for a real option is

always represented by discreet value of DCF (usually,

estimated once a year).

For pricing financial options, binomial tree is a

reliable tool only if year is divided in a large number

of periods. However, for real options, a binomial tree

with annual change for price of the underlying asset is

more appropriate than BS model. For this reason, we

say that CALL option to delay the project has a

maximum value of 852.77 thousands USD

(representing 23.2% of capital investment), but could

not be smaller than 828.66 thousands USD (which

means 22.6% of project value).

Pricing the option to abandon the project

We can imagine scenarios for correlated evolution of

variables from the free cash-flows model with negative

NPV for the project. The possibility to abandon the project

if one of these pessimistic scenarios occurs becomes

valuable. Therefore, the investment has attached a PUT

option to abandon.

The price of the underlying asset is again the present

value of projected cash flows and that is 5,074.84

thousands USD. Exercise price is represented by the

salvage value from the abandonment, estimated at the

current moment of time by cumulating market values for

land, buildings and technology (the value is fixed over

time, in real terms). Experts of real estate industry appraise

the market value of land to 360 thousands USD.

Buildings have a long life period (legal amortization

period is 40 years) and we can take their production cost

(440 thousands USD) as a proxy for the market value after

10 years (no matter that the book value decrease with

every passed year). Because the company constructed

the buildings itself, the production cost is, for sure,

smaller than their market value. Experts estimate that

market value is at least 660 thousands USD, with 50%

bigger than accounting value.

Market value for equipments and installations is hard

to asses, because it is an inflexible and specialized

technology, and there is not an active second-hand market

for this kind of equipment. That is why we consider

(subjectively, of course) that our old technology worth at

least 1.600 thousands, approximately half of the price for

a similar new technology. Doing so, we do not risk

overestimating the salvage value, because for a PUT

option, the bigger the exercise price, the bigger the price

of the option.

The time to maturity is thought to be 10 years, because

investor may decide to abandon the project anytime during

the exploitation period.

Risk free rate is equal to rate of return for treasury

bonds with 10 years maturity (issued by US Treasury),

which was 4.67% at the end of year 2006, plus country

risk premium for Romania, estimated to 2%. The result is

a discount rate of 6.67% for the binomial model (in discreet

time) and 6.457% for BS model (in continuous time).

Volatility of the underlying asset is the same with

dispersion obtained in Monte Carlo simulation σ2 =

0.0225. The dividend yield is δ = 1/the remained period

from the life of the project = 10%.

Pricing the PUT option to abandon the project with

BS model generates the following results: d1 = 0.88401

and N(-d1) = 0.18835, d2 = 0.40966 and N(-d2) = 0.34103,

and PUT option value is 116.82 thousands USD.

We also built a binomial tree for the underlying asset

for 10 years, with dt = 1 (annual change for the discounted

cash-flows of the project). The multiplying factors are u =

1.2281 and d = 0.9098. This PUT option is also an

American option and it may be exercised anytime in the

period of 10 years until maturity. Risk neutral

probabilities are p = 49.3% and 1-p = 50.7%. Option

pricing is done by calculating its value in every single

point of the tree, backwards from the tenth year to the

current moment of time.

PUT option value from the binomial model is 142.5

thousands USD, bigger than the value obtained from BS

model (116.82 thousands USD), for the same reasons

reported before, for the option to delay the project. So we

consider that PUT option to abandon the project has a

maximum value of 142.5 thousands USD (representing

3.9% of capital investment), but could not be smaller

than 116.82 thousands USD (which means 3.2% of project

value).

Page 11

57

Real Options in Capital Budgeting. Pricing the Option to Delay and the Option to Abandon a Project

Conclusions

The first conclusion is obvious: pricing real options

associated with the investment of producing metallic tiles

conduct the investor to recommendation of adopting the

project. NPV was positive even before identifying and

pricing the real options. Option to delay and option to

abandon bring a supplementary value of 26% of invested

capital, and represents an enhancement of approximately

75% for NPV, in comparison with situation when real

options were ignored.

The option to delay the project has a significant value,

because uncertainty associated with this investment is

greater than investor thought it would be. Sensitivity

analysis reveals that the project is extremely sensitive to

changes in unit sale price and unit variable cost. On one

hand, to postpone the investment decision is a valuable

opportunity because new information arises while time

passes. On the other hand, deferring the project exposes

the company to potential competition. For such

investments with multiple sources of uncertainty, the only

solution is to undertake the project.

Option to abandon the project has a small price,

because of the influence coming from the reduced value

for volatility (ó is only 15%) and from undervalued

exercise price (these two elements diminish the price of a

PUT option). If abandonment appears in the first two or

three years, this value is certainly bigger.

The value for the underlying asset is taken from project

appraisal in a certain environment, using the same

discount rate of 18.5%. But real options embedded in a

project reduce the risk of investment, and we must use a

smaller discount rate, so the value for the underlying asset

increases, which implies that the price of option to delay

enhances, while the price of option to abandon decreases.

Real options are reliable tools for capital budgeting

only if they are used for complex strategic projects. Real

options are used rather to conceptualize projects than to

appraise them, to change organizational structure in a

company, needed to gather the maximum potential from

a project.

In conclusion, using real options is more related with

company management than with valuation methodology.

Applying real options successfully assumes training the

managerial team in real options spirit. Managers must

detach from decisions adopted only with DCF or NPV

and to be ready to renounce to a project if it is proven to

be more efficiently that way.

Notes

(1)

Source: Mauboussin M., 1999, pp. 8 ºi Bruun S., P. Bason,

2001, „Essay One: What Are Real Options?”, pp. 5 (abordare

preluatã ºi adaptatã din Timothy A. Luehrman, 1998, „In-

vestment Opportunities as Real Options: Getting Started on

the Numbers”, Harvard Business Review, July – August,

pp. 51 – 58.

Damodaran A., 2002, Chapter 5: „Option Pricing Theory and

Models” ºi Capitolul 28: „The Option to Delay and Valuation

Implications”. A similar formula also in Richard A. Brealey

and Stewart C. Myers, 1996, „Principles of Corporate Finance”,

(2)

Fifth Edition, The McGraw–Hill Companies Inc., New York,

pp. 598, where u =

e

ºi d =

See Damodaran A., 2002, Chapter 29: „The Option to

Expand and Abandon: Valuation Implications”

Source: www.fitchratings.com, Fitch Ratings Ltd., New York

Source: www.moodys.com, Moody’s Investors Service

Source: www.standardandpoors.com, Standard & Poor’s,

The McGraw–Hill Companies, New York

Source: www.sjsu.edu/faculty/watkins/countryrisk, according

the data given by Moody’s

dt

×σ

dt

e

× −σ

, and d = 1/u.

(3)

(4)

(5)

(6)

(7)

Page 12

58

Theoretical and Applied Economics

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