lecture 6 the real option approach to cost - benefit - analysis under irreversibility, risk and...

Lecture 6

The real option approach to cost - benefit - analysis under irreversibility,

risk and uncertainty

For the discussion of the approach, let us assume the evaluation of an investment in agamepark. P is the price of a trophy and I are the investment costs of the project. Therisk free rate of return r=10%. The probability of a price increase q=0.5, hence (1-q)=0.5for a price decrease as well.

A Simple Example:

- irreversible costs: I = 1600$

- immediate net-benefits: π0 = 200 $

- future net-benefits: π1H = 300$

π1L = 100$

- probability: p = 0.5

- risk – free discount rate: r = 0.10

Introduction

For the discussion of the approach, let us assume theevaluation of an investment in a gamepark. P is the price ofa trophy and I are the investment costs of the project. Therisk free rate of return r=10% . The probability of a priceincrease q=0.5, hence (1-q)=0.5 for a price decrease as well.

t=0 t=1 t=2 t =

P1= $300 P2= $300 P =300P0= $200

P1= $100 P2= $100 P =300

NPV= -1600 +

01 1.1

200t

= -1600+2200 = $600

W aiting: NPV= (0.5)[-16001.1

1600 +

01 1.1

300t

]=1.1

850 = $773

Value of flexibility: 773-600=173

g

1-g

L e t u s d e f i n e : V 0 = N P V I + I , V 1 = N P V D + I / ( 1 + r )

T h e n e t p a y o f f f r o m d e c i d i n g w h e t h e r o r n o t t o i n v e s t i m m e d i a t e l y i s :

0I,Vmax 00

T h e n e t p a y o f f , t h e o u t c o m e o f a f u t u r e o p t i m a l d e c i s i o n , c a l l e d t h e c o n t i n u a t i o n v a l u e , i s :

F 1 = m a x { V 1 – I , 0 }

T h e n e t p a y o f f t o t h e i n v e s t m e n t o p p o r t u n i t y p r e s e n t e d i n t h e f i r s t p e r i o d , o p t i m a l l y t a k e n , i s :

1000 FE

r1

1I,VmaxF

R e a l o p t i o n v a l u e : F 0 - 0

Standard Option Pricing Method

F0 = value today of the development project (what we are willing to pay tohave the option to implement the project)

F1= value of the option to develop next year (is F1 a random variable, thevalue will depend on what happens to the price)

F1 (P=300) =

01 1.1

300t- 1600= $1700

F1 (P=100) = 0.5 (

01 1.1

100t- 1600)= -500

If the price rises the option to develop will be exercised, if the price falls, theoption to develop will be unexercised.

All possible values of F1 are known. The problem is to find F0, the value ofthe option today. Note, that the option to wait has no value next year as alluncertainty has been resolved. In a more general model, uncertainty willnever be fully resolved.

Identifying F0.

1. Creating a portfolio that has two components:- the development project itself- a certain number of trophies.

2. The number of trophies will be picked in a way that the portfolio isrisk-free. The value of the portfolio is the same whether the price of atrophy goes up or down.

3. Since the portfolio will be risk-free, the rate of return from beholdingthe portfolio equals risk-free rate of return in the economy ( risk freeinterest rate).

4. Setting the portfolio equal to the risk-free rate, allows to calculate the current value of the project opportunity.

a. Consider a portfolio which contains the project and a short shell of n-trophies. The value if this portfolio is: 0 = F0 -n P0 = F0 – 200 n

b. The value of the portfolio next year is: 1 = F1 -n P1

The value of next year’s portfolio 1= F1 -n P1 depends on P1.

If P1 turns out to be high P1=300, then F1 (P1 =300)=1700 and 1 =1700-n300

If P1 turns out to be low P1=100, then F1 (P1 =100)=0 and 1 = 0-n100 Choose n in such a way that the portfolio 1 is risk-free, that is, has the same value independent of what happens to the price. P1 high P1 low 1700-n300 = -100n n = 8.5 1 = -$850 The value of the portfolio is 1=-$850 whether the price goes up or down if n is chosen to be 8.5.

The return form holding this portfolio 1- 0 (payment that must be made to hold the short position) Note: Short shell is, if one sells a product today, receives the payment

today, but delivers the product at an agreed future point in time. The seller is holding a short position. The agent who buys the product holds a long position.

The expected gain on prices is zero, therefore no rational investor would be interested in the long position, if it would not make at least 10% gain. Therefore, the short sell of trophies requires a payment of 0.1* P0 to the holder of the long position, which is $20 per trophy, or 8.5*$20=$170 in total for the portfolio.

We can write the return of the holding of the portfolio: 1- 0 -170 = 1- (F0-n P0) –170

= -850- F0 + 1700 –170 = 680 - F0

We know that the return on the portfolio is risk-free and, has to be equal to the risk-free rate of return of 10% for every dollar invested: 1- 0/ 0 = r => 1- 0 = r 0

=> 680-F0 = 0.1 (F0- 1700) => 850 = 1.1 F0

=> F0 = 1.1

850=773

The irreversibility effect: some examples related to the release of herbicide tolerant sugar beets (htSB)

Assumptions:

• benefits from non-htSB 1000€/ha, infinite

• benefits from htSB 1200€/ha, infinite

• scrap value of old sugar beet planter: 500€

• price of a used new sugar beet planter 2100€

• discount rate is 10%

=> incremental benefits per hectare: 200€

=> incremental irreversible investment: 1600€

1

200200 1.1 2000

0.1

tt

t

V

Value of adopting htSB:

Irreversible investment for htSB: I =1600

NPV of for adopting htSB: V - I = 2000 - 1600 = 400

The irreversibility effect: some examples related to the release of herbicide tolerant sugar beets

(htSB)

Introducing uncertainty about future incremental benefits from htSB:

• if future looks good, incremental benefits are high, 300€/ha

• if future looks bad, incremental benefits are low, 100€/ha

• both situations are equally likely, DM is risk neutral

= > the expect value, E[V], of the project:


(htSB)

1 1

0.5 300 1.1 0.5 100 1.1 2000t t

t t

t t

E V

What is the value, V, of adopting htSB, if the future incremental benefits are low?


(htSB)

01

100 1.1 1000t

t

t

V

What is the value, V, of adopting htSB, if the future incremental benefits are high?

01

300 1.1 3000t

t

t

V

Now, we assume the farmer is flexible and can postpone his decision. Would this provide him with any additional gain?


(htSB)

In the case the gross margin:

- increases, the NPV one year from now is: 12

1600 300 1.1 1400t

t

t

NPV

or in today’s value: 0 1 /1.1 1273.NPV NPV

- decreases, the NPV one year from now is: 12

1600 100 1.1 600t

t

t

NPV

or in today’s value:

0 1 /1.1 545.NPV NPV

In the latter case the farmer would not invest. The gain from waiting is the gain from avoiding losses of 545 Euro in present value.

The economic gain from waiting can be calculated by comparing the expected value of the immediate investment with the one from waiting one year.


(htSB)

The gain from immediate investment is 400 Euro (see before).

The economic gain from waiting is the difference between the two, i.e. 236 Euro.

The gain from from postponed investment is:

02

0.5 1600 300 1.1 0.5 (0) 1.1 636t

tP

t

E NPV

At this point it is worthwhile noting the importance of the irreversibility effect. It only pays to wait when the investment costs are irreversible. This observation will be even more obvious if the incremental net-benefit would be negative in the bad case. Then the farmer would immediately stop producing htSB and move back to planting n-htSB.

If the initial investment costs were not irreversible, immediate investment would be optimal. Also, it would be optimal to invest immediately, if the investment could not be postponed due to other circumstances, such as a contract for planting htSB only offered once.

A third important observation is the opportunity costs of waiting. Waiting pays as the veil of uncertainty will be removed after one year, but at the same time the benefits at the end of year one are foregone. These foregone benefits of expected 200 are the opportunity costs of waiting.


(htSB)

The benefits that have been discussed, the incremental benefits, are reversible. By stopping planting htSB, incremental benefits are also foregone.

=> As there are irreversible costs there are also irreversible benefits. => These are benefits that will continue to be present even if the action that has produced them stops.

Consider, for example, a one-time subsidy of 500 Euro for planting htSB.

The E[NPV0I] increases in this case by exactly 500 Euro and the E[NPV0

I] = 900.

The E[NPV0P] from waiting in this case is:

Decision in the presence of irreversible costs and irreversible benefits

02

0.5 1600 500 300 1.1 0.5 (0) 1.1 864t

tP

t

E NPV

The E[NPV0I] > E[NPV0

P] (900 > 864) and there are no gains from waiting. The irreversible benefits reduce the irreversible cost, which leads in this case to an immediate investment.

The case of irreversible benefits is similar to the case where the adoption of transgenic crops reduces the use of pesticides harmful to human health.

Decision in the presence of irreversible costs and irreversible benefits

Another interesting question is the case where irreversible benefits decrease over time. This can be modelled by assuming where the subsidy is in the form of a loan and has to be paid back after ten years.

The E[NPV0I] is:

Decision in the presence of irreversible benefits

The E[NPV0P] is:

0 101 1

5001600 500 0.5 300 1.1 100 1.1 707

1.1

t tt tI

t t

E NPV

0 102

500 0.5 1600 500 300 1.1 0.5 (0) 1.1 776

1.1

ttP

t

E NPV

The E[NPV0I] of an immediate investment in this case is 707 Euro, which is also in this

case higher than in the case without the subsidy (636 Euro) and less than in the case with the subsidy as a grant (900 Euro). Although, in this case postponing the investment would be a better decision.

Another interesting questions related to the irreversible benefits is, whether there are gains from waiting if only irreversible benefits and no irreversible costs are present or if the net-irreversibility effect is positive. Under a positive net-irreversibility effect there will be no gains from waiting, as there are no losses that can be avoided.

The E[NPV0I] in the case of irreversible benefits only is:


The E[NPV0P] in the case of irreversible benefits only is:

01 1

500 0.5 300 1.1 0.5 100 1.1 2500t t

t tI

t t

E NPV

02 2

0.5 500 300 1.1 500 100 1.1 1.1 2273t t

t tP

t t

E NPV

The E[NPV0I] under this scenario will always be greater than the E[NPV0

P] due to the discounting effect and therefore waiting does not provide an economic gain.

The important observations about the irreversible benefits are threefold:

1. irreversible benefits reduce irreversible costs and this by the order of one. One unit of irreversible benefits compensates for one unit of irreversible costs.

2. a decrease in irreversible benefits over time, even up to a hundred percent, still has a positive impact on the value of the project.

3. a positive irreversibility effect does not provide economic gains from waiting.


•The susceptibility of pests to control agents is a non-renewable resource => the appearance of pest resistance is an irreversibility.

• Biologists and entomologists in particular argue that susceptibility to control agents, pesticides in particular, should be viewed as a renewable resource. That is, if pests become resistant to a control agent and consequently the use of the control agent stops, pest resistance breaks down after a while and pests do become susceptible again. Question: does an irreversibility effect exists?


The Special Case of Pest-Resistance

To show that an irreversibility effect does indeed exist consider the following hypothetical example for Bt-corn used against damages from the European Corn Borer (ECB): • The incremental benefits from adopting Bt-corn are assumed to be 200 at the beginning, period one, and due to price uncertainty increase to either 300 or 100 after one time period and remain at the level till the end of the fourth period. • At the end of the fourth period the ECB becomes resistant to Bt-corn and the incremental benefits decrease to zero from period five till the end of period seven.

• At the end of period seven, the ECB becomes susceptible again to Bt-corn.

• To keep the example simple, we assume that the incremental benefits increase to 200 Euro until infinity as the ECB will also be susceptible till infinity. The costs of pest resistance in present value terms are 1600 Euro. These are extra costs beyond the lost incremental benefits of period five, six and seven.



ECB bt resistantECB bt susceptible ECB bt susceptible

1 2 3 4 5 6 7 8 9 10 …

Figure: Example for appearance and breakdown of ECB resistance to Bt-toxin.

100

200

300

0



The value of Bt-corn from immediate adoption is:



4 4

10

1 1 8

1600 200 1.1 0.5 300 1.1 0.5 100 1.1 200 1.10 60t t t

t t tI

t t t

E NPV

The result for a postponed adoption is:

5

02 9

0.5 1600 300 1.1 200 1.1 171t t

t tP

t t

E NPV

The example illustrates that even though pest resistance can be reversible from a biological point of view, from an economic point of view an irreversibility effect may exist.

Scope

ReversibilityPrivate External

Reversible

Quadrant 1

Private Reversible Benefits (PRB)Private Reversible Costs (PRC)

Quadrant 2

External Reversible Benefits (ERB)External Reversible Costs (ERC)

Irreversible

Quadrant 3

Private Irreversible Benefits (PIB)Private Irreversible Costs (PIC)

Quadrant 4

External Irreversible Benefits (EIB)External Irreversible Costs (EIC)

Private and Public Irreversibilities

Figure. The Two Dimensions of an Ex-Ante Social Benefit-Cost Analysis including irreversibilities

Private and Public Irreversibilities

An example:

The decision rule to release htSB is formulated as, to release htSB if the net reversible social benefits W, the sum of quadrant 1 and quadrant 2 in figure 2, are greater than the net irreversible costs, the sum of quadrant 3 and quadrant 4, multiplied by a factor greater than one, the so-called hurdle rate :

( )W I R

As the social irreversible costs, I=PIC + EIC, and benefits, R=PIB+EIB, of transgenic crops are highly uncertain, instead of identifying the net reversible social benefits W required to release transgenic crops in the environment, the maximum tolerable social irreversible costs I* under given net social reversible benefits W and social irreversible benefits R are identified:

*I R W

Member State W (€/ha) R (€/ha) Hurdle Rate I* (€/ha) Total I* (€)Austria 251 3.36 2.88 91 1,842,164Belgium & Luxembourg 168 2.09 1.26 135 5,852,023Denmark 178 2.06 1.73 105 2,864,870Finland 251 0.74 3.69 69 976,108France 179 1.05 1.25 145 24,964,742Germany 179 1.57 1.36 134 27,846,376Greece 264 7.97b 3.12 93 1,771,502Ireland 116 -0.96b 2.29 50 691,951Italy 330 2.32 1.82 183 22,682,730The Netherlands 121 0.83 1.31 94 4,630,433Portugal 354 -0.65b 1.67c 212 615,218Spain 252 0.53 2.10 121 7,258,219Sweden 150 0.18 3.01 50 1,226,127UK 127 1.78 1.76 74 5,135,522EU 199 1.59 1.67a 121 102,628,681

Table. Hurdle Rates and Annual Net Private Reversible Benefits (W), Social Irreversible Benefits (R), and Maximum Tolerable Social Irreversible Costs (I*) per

Hectare Transgenic Sugar Beet

a sugar beet area-weighted average of the individual Member States’ hurdle rates.b The extreme estimates for Greece, Ireland and Portugal are probably due to data inconsistencies. These countries only cover 4% of total EU sugar beet area, almost not affecting the EU average.c No data on margins has been found for Portugal. We use the EU area-weighted average.

lecture 6 the real option approach to cost - benefit - analysis under irreversibility, risk and...

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