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In this module, we'll discuss in detail one approach to valuation of real options,

namely decision trees.

Valuation of options on financial assets, is well developed and typically,

one of two approaches is used.

These two approaches are referred to as the binomial model and

the Black-Scholes formula.

The use of these two approaches for

evaluation of real assets is somewhat complex and difficult to understand.

Moreover, these two approaches require the estimation of the standard deviation

sigma, of continuously compounded annualized return from the project.

On the other hand, a decision tree approach to value real options is easy to

understand, although this approach requires the estimation of

several probabilities, as we will see next.

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Let's consider a simple example motivated by the case

Cypress Telecommunications Authority that we talked about in the previous module.

When an organization is considering upgrading its information system,

the amount of investment required now has been estimated to be said 225 in

thousands of the unit of currency.

It's not clear how well this upgradation will be used.

If the upgradation is a success, it's estimated that the incremental cash flow

during the first year, attributable to this expansion of the information system,

will be 80 in thousands.

The probability of success estimated as 0.55.

Note that although the incremental cash flows occur during the year, we assume,

for simplicity, that all the cash flows occur at the end of year one.

Alternatively, the amount 80 should be interpreted as the future

that is at the end of year one.

Value of incremented cash flows occurring during year one.

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Moreover, the upgradation is successful at the end of year one, or equivalently,

the beginning of year two, some uncertainty in the subsequent incremental

cash flows attributable to this upgradation is resolved, but

not completely.

In this case, it is estimated that the sum of the incremental cash

flows from year two onwards till the life of the project,

when the entire information system will be changed completely.

Discounted to the end of year two is expected to be 300 with a probability

0.6 or 150 with probability 0.4.

This uncertainty whether it will 300 or

150 is not resolved at the beginning of year two.

On the other hand, if the upgradation of the information system is not a success,

then incremental cash flows attributable to this upgradation during the first

year will be only ten in thousands with probability 1- 0.55 = 0.45.

In this situation, it is estimated that the sum of the incremental cash flows from

year two onwards till the life of the project, discounted to the end of year two

would be 200 with probability of 0.6 or 100 with probability 0.4.

Again, this uncertainty whether it will be 200 or

100 is not resolved the beginning of year 2.

The yearly discount rate for this project is assumed to be 10%.

The first project is to upgrade the information system now.

The data are represented in a diagram as shown.

This diagram is an effort as to the tree.

Using the data represented in the tree,

the decision problem is whether to accept the project or not.

A typical approach to solving this decision problem

is to use the Net Present Value Rule, which states that,

accept the project if NPV is greater than zero.

Otherwise, reject the project.

We'll illustrate the application of NPV for this problem.

We'll start at the end node.

In this example, we have two end nodes, namely denoted as 1a and 1b.

We must consider each end node in turn, and

work backwards still we reach the starting node denoted as b.

The calculations are done as follows.

At node 1a, the value of the expected incremental future cash flows of 300 and

150 are multiplied by the respective probabilities of 0.6 and 0.4, and

added to arrive at the total.

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Similar calculations are done for node 1b in the same diagram.

Thus, at node 1a and node 1b, we have the expected incremental

cash flows as 298.18 and 155.45, respectively.

Now, working backwards to node b, we first multiply the values 298.18 and

155.45 calculated for nodes 1a and 1b by the respective

probabilities of 0.55 and 0.45 and add it to get the total.

This total is discounted to get the present value that is at node B,

and subtract the investment cost of 225 to get the NPV.

This gives an NPV of -12.32, as shown in the diagram.

So the project upgrade information system is not worth undertaking by itself.

Now, as in the case of Cyprus Telecommunication Authority,

suppose upgrading the information system is necessary for

the organization to be able to expand its operations one year from now.

So, we will next consider the project to upgrade the information system and

expand operations.

As shown in the diagram,

if upgrading the information system is successful, the expected incremental cash

flows of the end of the first year would be 80 as before.

But with the expansion of operations, the sum of the incremental cash flows

from year 2 onwards till the life of the project, discounted to the end of year

2 is expected to be 500 with probability of 0.6 or 250 with probability 0.4.

As before, this uncertainty of whether it'll be 500 or

250 is not resolved at the beginning of year two.

On the other hand, if the upgradation for the information system is not a success,

the incremental cash flows attributable to its upgradation during the first year will

be only ten and thousands with probability 1- 0.55 equal to 0.45 as before.

In this situation, it is estimated that the sum of the incremental cash flows from

the year two onwards till the life of the project, discounted to the end of

year two, would be 210 with probability 0.6 or 105 with probability 0.4.

Again, this uncertainty of whether it will be 210 and

105 is not resolved at the beginning of year two.

The investment calls for an expansion of operations at the end of year 1 is 75.

The yearly discount rate for this combined project of upgradation followed by

expansions of evaluations one year later is assumed to be 10% as before.

This data as shown in the diagram below.

Project 2 is the composite project, upgrade the information system now and

expand operations one year from now.

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We again start at the end node.

In this example also, we have two end nodes, namely nodes denoted as 1b and 1d.

We must consider each end node in turn.

Work backwards till we reach the starting node denoted as b.

The calculations are done as follows.

At node 1b, the value of the expected incremental future cash flows are 500 and

250, multiplied by the respective probabilities of 0.6 and

0.4, and added to get the total.

This total is discounted to the end of year one, and

the investment cost of expansion is deducted

to obtain the net incremental cash flow attributable to expense and of operations.

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We then add the incremental cash flow of 80 during the first year to the value

calculated for node 1b to get the value of 368.64 for node 1a.

This calculation for node 1a shown in the next diagram.

Similar calculations are done for 1b and node 1c in the next diagram.

Thus, at node 1a and 1c we have the expected incremental cash flows

as 368.14 and 87.73, respectively.

Now working backwards to node b, we first multiply the values 368.84 and

87.73, calculated for nodes 1a and 1c by the respective

probabilities of 0.5 and 0.45, and then add them to get the total.

This total is discounted as the present value, that is to node b, and

we subtract the investment cost of 225 to get

the net present value of -4.79 as shown on the diagram.

So the composite project of upgrade information system and

expand operations is also not worth undertaking.

A little reflection about the composite project will suggest that the follow-on

project of expanding the operations need not necessarily

be undertaken at the end of the first year.

Although at current time,

we may anticipate the upgradation of the information system would be a success.

And the follow-on project expansion of operations would be undertaken at the end

of the first year.

There is no need to decide now, that is, at the current time, to undertake

the follow-on project of expansion of operations at the end of the first year.

It should be kept in mind that the organization has the option to expand

operations at the end of the first year, but not the obligation to do so.

The decision to expand operation or not may be made at the end of the first year,

when uncertainty pertaining to the success of the project,

upgrade the information system would be resolved.

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For the sake of completions, we look at the possibility of expansion or

no expansion of operations at the end of the first year for

both outcomes of the success our failure of the first project,

which is upgrade the information system.

The different possibilities are shown in the diagram below.

Project 3, upgrade the information system now

with an option to expand operations at the end of the first year.

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Now at nodes 1a and 1d, we must decide whether to expand operations or not.

Let us first consider node 1a.

If we decide to expand, the expected value of incremental cash flows,

net of the investment cost of operation, as calculated before,

is equal to 500 times 0.6 plus 250 times 0.4.

That sum divided by 1.1, and

to which we add -75 + 80 to get 368.64.

On the other hand, if at node 1A, we decide not to expand operations,

the expected value of incremental cash flows is 300 times 0.6

plus 150 times 0.4 then the sum divided by 1.1 plus 80 equals 298.18.

So the decision at node 1a is to expand the operations, and

we have the expected incremental cash flow as 368.64.

Next we consider node 1b to decide whether we should expand operations or

not at that node.

If we decide to expand, the expected value of incremental cash flows,

net of the investment cost of expansion,

as calculated before is 210 into 0.6 plus 105 into 0.4,

the sum divided by 1.1, And to which we add -75 + 10 = 87.73.

On the other hand, if at node 1d we decide not to expand operations,

the expected value of incremental cash flows is 200 times 0.6 + 100 times 0.4,

the sum divided by 1.1 + 10 = 155.45.

So the decision at 1d is not to expand operations.

And we have the expected incremental cash flows as 155.45.

It does have the following diagram.

Finally, we backtrack to node b, with the values at nodes 1a and

1d as calculated above.

Now the expected net incremental cash flow at node b is 368.64

multiplied by 0.55 plus 155.45 multiplied by 0.45.

And the sum is divided by 1.1, and we subtract the investment

cost of -225, and get a net present value of 22.91.

So, we accept the project.

Note that the NPV of project 3 equals NPV of project 1 plus

value of the option to expand at the end of year 1.

So the value for the option is NPV of project 3 minus NPV of project 1.

Which is equal to 22.91 minus -12.32, which is equal to 35.23.

This concludes the module on valuation of real options, decision trees.

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