Real Option Valuation

Marking to Market Prof. Luiz Brandão

[email protected]

2009

Determining the Value of an Asset

 The value of a market asset can be obtained simply by observing the price it is being trading at.  The majority of real assets, such as projects, are not traded in the market.  In these cases, how do you determine its value?

 An important assumption in finance is that if two assets are identical they should command the same price.  One way to determine the value of a non-traded asset is to identify a traded asset that has the same risk and return characteristics.

 Since both provide the same cash flow with the same risk, the manager would be indifferent, and should command the same price. IAG PUC – Rio Brandão 2

Example

     A project has the following cash flow, with a 0.50 probability with each alternative: The following security was identified in the market, trading at $24. Observe that the market security has cash flows that are exactly 1/5 of the project’s cash flows. The value of the project is therefore five times the value of the market security, or 120. The discount rate of the asset can be determine by: 24  IAG PUC – Rio Brandão 1 

k



k

 11.7% Project ?

Market Asset $24 $200 $80 $40 $16 3

Example

 With this discount rate we can also determine the value of the project.

PV

  $120 1.117

 Note that we can use the same discount rate because both projects carry the same risk.  In this case, the market asset has a perfect correlation with the project.  Note that the NPV method (which uses the WACC) implicitly treats the company’s stock as the correlated market asset.  Efficient Markets and arbitrage concepts IAG PUC – Rio Brandão 4

Replicating Portfolio

  What if there is no single asset that replicates the cash flows and risk of the project?

In this case we can use a portfolio of assets with the same purpose. Typically we utilize an risk free security in combination with one or more risky securities.   Therefore, it is necessary to determine both the composition of the portfolio and the quantity of each of the securities within it. Example: A project and a market asset with the following cash flows, and a risk free security with a return of 5%: $20.5

$40 Project ?

$200 $80 $10 IAG PUC – Rio Brandão 5

Example

 The market security alone is not sufficient to replicate the project.   Create a replicating portfolio with a quantity A of risky security and B of the risk free security, or Φ 0 = A ($20.5)+B Determine the value of the portfolio and make it equal to the value of the project in a year: Φ 1 + = 40A + B(1+rf) = 200 Φ 0 = A ($20.5)+B Φ 1 = 10A + B(1+rf) = 80  Solving the system, we obtain

A

= 4.0 and

B

= 38.1.

 The value of the portfolio at time t=0 will be $120. Since the portfolio perfectly replicates the project, they have the same value. 6 IAG PUC – Rio Brandão

Ex: Vanatec Ltd.

 Vanatec, a traded retail company, is analyzing the decision of introducing a new product into the electronic consumer market, a wireless personal digital assistant, WPDA.  The project will take a year to complete and will require investments in production plant, equipment, and training.  The size of the market is uncertain. There is a 45% probability of a large market and 55% of a small market. Fortunately, the uncertainty will be resolved when the plant is inaugurated, but the decision to go ahead with the project or not, must be made now.  The company’s cost of Large Market

$200M

capital is 15%

($115M)

45%  The cash flow of the project is shown on the side 55% Small Market 

$100M

What would you recommend?

7 IAG PUC – Rio Brandão

Vanatec

  This project is very different from other Vanatec projects. Would the cost of capital be the same?

You observe that Wincon S.A. is a company that only makes and commercializes WPDA. Wincon’s shares are currently quoted at $22.31.    Market analysts expect that the shares will go up to $40 in the next year if the market is large or will fall to $20 if the market is small.

Wincom S.A.

What discount rate does the market use for Wincon?

Large Market 45% How can we utilize this information for the Vanatec project?

$22.31

55% Small Market

$40 $20

IAG PUC – Rio Brandão 8

Adding Flexibility

  Suppose now, that for an additional $20M we can build a flexible plant that would allow the company to utilize the plant to build PC components as well. In this case, if the WPDA market is small, the idle capacity of the plant can be redirected to the PC market, which would increase the project’s cash flow to $140M.

Large market

$200M

45%

$135M = 115+20

55% Small Market

$140M

 What is the decision now?

IAG PUC – Rio Brandão 9

Risk Analysis

 The substitution option eliminates part of the risk of the project. This is similar to a put option.   If the risk of the project was altered, what can we say about the discount rate of the project with the option? How can we determine the appropriate discount rate in this case?

We’ll create a market portfolio with shares of Wincom and risk free security (

rf

= 5%) that will replicate the cash flows of the project in each stage within a year. Large Market

40A + 1,05B = $200M

45% 

= 22.31A + B

55% Small Market

20A + 1,05B = $140M

IAG PUC – Rio Brandão 10

Risk Analysis

 Solving we obtain A = 3.0M and B = 76.19M. Large Market

3.0(40) + 76.19(1.05)= $200M

45%

143.12 = 3.0(22.31) +76.19

55% Small Market

3.0(20) + 76.19(1,05) = $140M

    The project is now worth $143.12

What is the discount rate of the project now? What can we conclude about the risk of a project with a substitution option?

The existence of options alters the risk of the project, and is necessary to use the replicating portfolio method to determine the appropriate discount rate. The value of the option is 143.12 – 111.54 = $31.58M

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MAD

 The replicating portfolio method allows for valuing projects at market price.  Unfortunately, with the exception of commodities, the great majority of real assets and projects are not traded in the market nor is it possible to identify a replicating portfolio for them.  In this case, there is no way to determine the market value for a project.  A solution for this problem is to assume that the Present Value of the original project, without options, is the best estimate of its market value.  This assumption is based on the same arguments used for the CAPM and is also the fundamental premise for the NPV method. IAG PUC – Rio Brandão 12

A Simple Project

 A project will have the value of $160M or 62.5M within a year, depending on the state of the economy, with a probability of 0.50

Project

0.50

$160M

0.50

$62.5M

  What is the value of this project? To answer this question, we need a market asset to determine the appropriate discount rate for the project. In the absence of such asset, we utilize the MAD. 13 IAG PUC – Rio Brandão

Utilizing the MAD

     According to the DCF method, we can determine the value of the project using its risk adjusted rate, as determined by CAPM. By doing this we implicitly assume that the twin market security for this asset is the firm´s stock.

Suppose that in this case, the rate is 11.25%. With this risk adjusted discount rate, the value of the project, without options, is $100 According to MAD, we assume that this is the market price of the project without options. We now have a market asset with the value of $100 that has expected cash flows of $160 and $62.5, as illustrated in the previous figure. 14 IAG PUC – Rio Brandão

A Simple Project

 A project will have the value of $160M or 62.5M within a year, depending on the state of the economy, with a probability of 0.50

Projeto

0.50

$160M $100M

0.50

$62.5M

  Assuming a discount rate of 11.25%, the value of this project is 100 Following MAD, we will assume that this project will be the twin market security for the project with options 15 IAG PUC – Rio Brandão

Option to Expand

  Suppose now, that the project has an option to expand capacity by 50% within year at a cost of $50M. With this option, the result of the project becomes:

Project with Option

0.50

max [160,

1.5(160)-50

]= $190M 0.50

max [

62.5

, 1.5(62.5)-50] = $62.5M

  Does this option add value to the project?

What is the value of the project now, considering the original discount rate? Is this analysis correct?

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Solution with Replicating Portfolio

  Create a replicating portfolio with A units of the project and a quantity B invested in a risk free security. The value of this portfolio today is  same value of the project in a year. = 100A +B. This will be the

Replicating Portfolio

0.50

160A+1.05B = $190M 100A+B  0.50

62.5A+1.05B = $62.5M

We find that A = 1.3077 and B = -18.315. The value of the portfolio, and also of the project, will be 100(1.3077)-18.315 =

$112.45M

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The Project’s Risk

 We can verify how the project’s risk was altered by determining the new discount rate of the project.

Project with Option

0.50

$190M $112.45M

0.50

$62.5M

  The rate is

k

 0.50(190) 0.50(62.5) greater than 11.25%. 112.45

0.1227

which is This means that the option to expand increased the risk of the project.

18 IAG PUC – Rio Brandão

Risk of the Option to Expand

 The option to expand is a call option with the following values: 0.50

Option Only

$30 $12.45M

0.50

$0  What is the corresponding discount rate?

k

 0.50(30) 12.45

0.204

 This shows that a call option is riskier than the underlying asset IAG PUC – Rio Brandão 19

Project with Abandonment Option

  Suppose now that the project has an option to abandon for $92.5M in a year. With this option, the project payoffs are:

Project with Option

0.50

max(

160

,92.5) = $160M 0.50

max(62.5,

92.5

) = $92.5M

  The option will only be exercised if the worst scenario occurs.

To value the project with the option, we utilize the replicating portfolio. IAG PUC – Rio Brandão 20

Replicating Portfolio

  Create a portfolio with A units of the project and a quantity B invested in a riskless asset. The value of this portfolio today is project in each stage.  = 100A +B. Next, we compare the value of the portfolio in a year to the value of the

Replicating Portfolio

160A+1.05B = $160M 100A+B 62.5A+1.05B = $92.5M

 This results in A = 0.6923 and B = 46.8865. The value of the portfolio, and the project, will therefore be 100(0.6923)+46.8865 =

$116.12M

21 IAG PUC – Rio Brandão

The Project’s Risk

 What is the risk of the project with the option to abandon?

Project with Option

0.50

$160M $116.12M

0.50

$92.5M

   

k

116.12

which is less than the original rate.

  This indicates that the abandonment option reduces the risk of the project. 22 IAG PUC – Rio Brandão

Risk of the Option to Abandon

 The option to abandon is a Put Option with the following results:

Option Only

0.50

$0 $16.12M

0.50

$30    Note that the expected value of $15 is less than the value of the option today. The corresponding discount rate is -6.9%, given that

k

 0.50(30)    0.069

16.12

This indicates that the Put Option is less risky than the project.

23 IAG PUC – Rio Brandão

Comments

       Both the call and put options pay $30M each. Both has the same probability (0.50) of occurrence. The call option pays $30M in high state and has a price of $12.45M

The put option pays $30M in low state and has a price of $16.11M.

Why are the prices different?

The answer has to do with the fact that investors are risk averse.

For that reason, they value cash flow received in bad times more highly than the same cash flow received in good times, when the investor has the greatest wealth. 24 IAG PUC – Rio Brandão

Option to Expand - Call

    The option to expand pays when the market is good, it pays nothing when the market is bad and the project generates reduced/limited cash flow. This increases the volatility of the cash flows of the project. Therefore, it implies that the option to expand has a positive correlation with the market. Like the investors with a decreasing marginal utility, an asset that generates cash flows in times of high wealth is less desirable and will have a lower value, which implies a higher rate of return and greater risk 25 IAG PUC – Rio Brandão

Abandonment Option - Put

    The abandonment option doesn’t generate cash flow when the market and project are doing well, but generate positive cash flow when the project is doing poorly. This reduces the volatility of the project’s cash flows. This means that the option to abandon has a negatively correlation with the market. Because this asset generates cash flows in time of low wealth it is more desirable and therefore will command a higher price, which implies a lower rate of return and risk. 26 IAG PUC – Rio Brandão

Computing Option Values

Example

    Initech obtained a concession that allows it to invest in a project in two years. Data:(Values in $1,000)    The value of the project today is $1,000 In one year, the value will be $1,350 or $741, depending on the market conditions. In two years, the value will be $1,821, $1,000 or $549.

  WACC is 15% The risk free discount rate is 7% Initech can opt to extend the project by 30% at a cost of $250 it in two years if it decides to build it. What is the value of this option? 28 IAG PUC – Rio Brandão

Underlying Asset Modeling

  We will use a binomial lattice to model the project. The values are added to the binomial lattice.

1000   With this data we can model the evolution of the project’s value over time. The last column shows the value of the project at each stage with the expansion. IAG PUC – Rio Brandão With Expansion 29

Underlying Asset Modeling

  We will use a binomial lattice to model the project. The values are added to the binomial lattice.

1349,9 1822,1 With Expansion 2118,8 1000   With this data we can model the evolution of the project’s value over time. The last column shows the value of the project at each stage with the expansion. IAG PUC – Rio Brandão 740,82 1000 548,8 1050,0 463,5 30

Underlying Asset Modeling

  We will use a binomial lattice to model the project. The values are added to the binomial lattice.

1349.9

1822.1

With Expansion 2118.8

1000   With this data we can model the evolution of the project’s value over time. The last column shows the value of the project at each stage with the expansion. IAG PUC – Rio Brandão 740.82

1000 548.8

1050.0

463.5

31

Options Modeling

 Value of the Project with the Option PV0 IAG PUC – Rio Brandão PVA PVB     With the option, the value of the project in two years changes in some cases.

This alters the risk of the project, which prevents the use of WACC to determine the expected value of the project. In this case, we use the replicating portfolio method to determine the value of the project PVA, PVB and PV0. This will be done separately for each one. 32

Options Modeling

 Value of the Project with the Option 2118,8 VPA 1050,0 VP0 VPB 548,8     With the option, the value of the project in two years changes in some cases.

This alters the risk of the project, which prevents the use of WACC to determine the expected value of the project. In this case, we use the replicating portfolio method to determine the value of the project PVA, PVB and PV0. This will be done separately for each one. 33 IAG PUC – Rio Brandão

Solution by Replicating Portfolio

 VPA Calculations: Underlying Asset: Project w/o Options 1349,9 1822,1 1000 Project with Option VPA 2118,8 1050,0    Create a replicating portfolio adopting the MAD assumption. Since this portfolio has the same results and risk as PVA, it will necessarily have to have the same value. Determining the values of A and B, we obtain PVA. 34 IAG PUC – Rio Brandão

Solution by Replicating Portfolio

 VPA Calculations: Underlying Asset: Project w/o Options 1349,9 1822,1 1000 Project with Option VPA 2118,8 1050,0    Create a replicating portfolio adopting the MAD assumption. Since this portfolio has the same results and risk as PVA, it will necessarily have the same value. Determining the values of A and B, we obtain PVA.

1349,9 A + B 1822,12 A + 1,07B = 2118,8 1000 A + 1,07B = 1050,0 A = 1,30 B = - 233,64

VPA = 1521,2

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Solution by Replicating Portfolio

 VPB Calculations: Underlying Asset: Project w/o Options 740,8 1000 548,8 Projeto com Opção VPB 1050,0 548,8    Create a replicating portfolio adopting the MAD assumption. Since this portfolio has the same results and risk as PVA, it will necessarily have the same value. Determining the values of A and B, we obtain PVA.

36 IAG PUC – Rio Brandão

Solution by Replicating Portfolio

 VPB Calculations: Underlying Asset: Project w/o Options 740,8 1000 548,8 Projeto com Opção VPB 1050,0 548,8    Create a replicating portfolio adopting the MAD assumption. Since this portfolio has the same results and risk as PVA, it will necessarily have the same value. Determining the values of A and B, we obtain PVA.

740,8 A + B 1000 A + 1,07B = 1050,0 548,8 A + 1,07B = 548,8 A = 1,11 B = - 56,84

VPB = 766,1

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Solution by Replicating Portfolio

 VP0 Calculations: 1000 Underlying Asset: Project w/o Options 1349,9 740,8 Project with Options VP0    Create a replicating portfolio adopting the MAD assumption. Since this portfolio has the same results and risk as PVA, it will necessarily have the same value. Determining the values of A and B, we obtain PVA.

38 IAG PUC – Rio Brandão

Solution by Replicating Portfolio

 VP0 Calculations: Underlying Asset: Project w/o Options 1000 1349,9 740,8 Projeto com Opção VP0 1521,2 766,1    Create a replicating portfolio adopting the MAD assumption. Since this portfolio has the same results and risk as PVA, it will necessarily have the same value. Determining the values of A and B, we obtain PVA.

1349,9 A + 1,07B = 1521,2 1000 A + B A = 1,24 B = - 142,43

VP0 = 1097,4

740,8 A + 1,07B = 766,1 39 IAG PUC – Rio Brandão

Real Option Valuation

Marking to Market Prof. Luiz Brandão

[email protected]

2009