Real Options Dealing with Dividends Prof. Luiz Brandão [email protected] 2009.

Real OptionsDealing with Dividends

Prof. Luiz Brandão

[email protected]

2009

IAG PUC – Rio Brandão

2

Projects that generate Cash Flows The examples we have seen up to now envolve assets or projects

that do not pay out dividends.

In practice, the main incentive a firm has to invest in a project are the cash flows the project is expected to generate for the firm and its shareholders.

As these cash flows are distributed out to the shareholders or investors, or otherwise withdrawn from the project, the value of the project decreases instantly by this amount.

The project value is then influenced and affected by the distribution of its cash flows, or dividends, which reduce its value at each period.

Note that the underlying asset is the project value, not the project cash flows.


3

Projects that generate Cash Flows If we plot the evolution of the project value in time, we shall see

that the value of the project changes instantaneously each time the cash flows are distributed.

As times passes, the value of the project increases as the expected cash flows get closer.

At each dividend payout instant, the project value suffers an instantaneous decrease

At the end of the project life, after all the dividends have been distributed, the value of the project will be zero.

Next, we will se an example of a project which is subject to dividend/cash flow distribuition


4

Projects that generate Cash Flows A project requires an investment of $1,000 and generates

a cash flow of $500 per year for five years. The WACC is 10%.

There is no residual value after the fifth year.

The evolution of the project value i:

0 455 868 1,243 1,585 Ex-Dividend PV

500 955 1,368 1,743 2,085 1,895 Pre Dividend PV

500500500500500-1000Cash Flows

543210

- 500 - 500 - 500 - 500 - 500


5

Projects that generate Cash Flows

Evolution of Project value in Time

2,085

1,743

1,368

-1000

955

0

500

455

868

1,243

1,585

1,895

-1000

0

1000

2000

3000

0 1 2 3 4 5


6

Incorporating Uncertainty In the previous example, we did not consider uncertainty in the

cash flows.

We will now assume that these cash flows are uncertain, and that the project value follows a GBM, with volatility of 18.23%. (u =1.20, d = 0.83)

We also assume that the project cash flows are a constant proportion of the project value in each period. This assures that the nodes of the binomial tree will recombine, and that the value of the project at the end of its life will be zero.

This fraction is called Dividend Rate. The Dividend Rate may be different in each period, but will be constant for all states of a particular time period.


7

Incorporating Uncertainty0 1 2 3 4 5

Cash Flows -1000 500 500 500 500 500

Project Value 1,895 2,085 1,743 1,368 955 500

Dividend Rate 0.240 0.287 0.366 0.524 1.000

1895 u = 2,274 Pre Dividend Value(545) Constant Rate

1,895 1,729 Ex-Dividend Value

1895 d = 1579(379)1201


8

Incorporating UncertaintyProject with Uncertainty

1,895

2,274

1,729

2,075

1776

1127

1352

644773

0

1,201 1233

939

536

0

1,579

1,201

310 373

0

216 259

0

180

00

1,480

1,441

1,028

782

447

1,001

543

856

714652

377

595 453

150

314

1250

1,000

2,000

3,000

0 1 2 3 4 5


9

Modeling Steps1. In the Spreadsheet, determine the dividend rate:

• In the project cash flow spreadsheet, determine the pre dividend value of the project for each year of its life.

• The dividend rate (D) is determined by the relationship Cash Flow / Project Value in each period.

• Insert this parameter in the binomial lattice model.

• Note that for projects with finite life, the dividend rate for the last period are always equal do 1.0

• The value of the project will always be the value before the distribution of dividends.


10

Modeling Steps2. Model the dividends in the binomial lattice:

• Model the project value at the end of the first period and beginning of the second according to the CRR model.

• In the branch of the tree (Get/Pay) insert the value of the cash flows for the first year. This cash flow is the result of the multiplication of the dividend rate and the end period project value.

• In the uncertainty node for the following period, deduct the value of the cash flow that was distributed, as determined in (3).

• Using the ex-dividend value obtained, determine the value of the project at the end of the next period.

• Repeat the steps above for all periodos till the last one.


11

Modeling Steps

VP0VP2 (1-D2)VP1 (1-D1) VP3 (1-D3)

VP1 (1-D1) VP2 (1-D2)

VP2 (1-D2)

VP3 (1-D3)

VP3 (1-D3)

VP3 (1-D3)


12

Example: Talion Talion Inc. owns a project that will be sold in two years and which

will generate a cash flow equivalent to 25% of its value only in year 1.

Data: (Values in €1.000) The current value of the project is €1.000

Volatility is 30% per year

WACC is 15% per year

Risk free rate is 7% per year

Model the evolution of the value of this project and determine that value of an option to expand by 40% at a cost of $200 before the sale of the project in two year.


13

Modeling the Underlying Asset The parameters for the

binomial approximation are:

With these parameters we can model the evolution of the project value in time.

The last column shows the value of the project if the expansion takes place.

1.350

1 0.741

tu e

d u

1000

With Expansion


14

Modeling the Underlying Asset

1.350

1 0.741

tu e

d u

1000

1349,9

740,8

750,0

411,6

1366,6

With Expansion

850,0

375,3

1713,2

337,5

1012,4

185,2

555.6

The parameters for the binomial approximation are:

With these parameters we can model the evolution of the project value in time.

The last column shows the value of the project if the expansion takes place.


15

Solution With Risk Neutral Probabilities

we can determine the value of the project with options.

We observe that the option value is €40,8.

Although for a simple problem such as this a manual solution is feasible, for more complex problems we will need a more powerful tool.

p

1-p

(1 )0.5405

rf dp

u d

p

p

1-p

1-p


16

Solução

1131,8

337,5

185,2

p

1-p

850,0

411,6

1713,2

(1 )0.5405

rf dp

u d

p

p

1-p

1-p

With Risk Neutral Probabilities we can determine the value of the project with options.

We add the cash flows in period 1 to the discounted expected values of period 2.

We observe that the option value is €131,8.

Although for a simple problem such as this a manual solution is feasible, for more complex problems we will need a more powerful tool.


17

Decision Tree Step 1: Parameters for the Underlying Asset

PV = $1.000

Vol = 30%

r = 7%

u = 1.350

d = 1/u = 0.741

p = 0.5405

u

d

p

r

VP

Vol

T1

T2


18

Decision Tree Step 2: Model the Binomial Lattice

Dividends not yet included

Without Dividends, the lattice is:

up

T2/(1+r)^2 down

T2/(1+r)^2

up

down

T2T1

1591.5 54%

[1591.5] up

873.4 46%

[873.4] down

T2

54%

[1261.6] up

873.4 54%

[873.4] up

479.4 46%

[479.4] down

T2

46%

[692.4] down

T1 [1000.0]


19

Decision Tree Step 3: Incorporate the Dividends

Insert the cash flow to be distributed at the end of year 1.

In this example, this cash flow is 25% of the value of the project in year 1.

To model the second period, the paid out dividends must be deducted from the value of the project at the end of the first period.

At the end of the second year, the project will be sold for its market value. This way, the cash flows received by the shareholders at that time will be equal to the project value at the end of year 2.

up

T2/(1+r)^2 down

T2/(1+r)^2

up

0.25*T1/(1+r) down

0.25*T1/(1+r)

T2T1

up

p PV*u

down

PV*d

T1

up

p 0.75*T1*u

down

0.75*T1*d

T2


20

Decision Tree The model of the project that reflects the payout of the dividends is shown below.

We can see that the current value of the project does not change when we include the dividend payout.

The dividend rates of T1 (0.25) and T2 (1.0) can be added as value nodes D1 and D2.

1591.5 54%

[1591.5] up

873.4 46%

[873.4] down

T2

54%

[1261.6] up

873.4 54%

[873.4] up

479.4 46%

[479.4] down

T2

46%

[692.4] down

T1 [1000.0]


21

Decision Tree Step 4: Modeling the Option

Yes

(1.4*D2*T2-200)/(1+r)^2

No

D2*T2/(1+r)^2

up

down

Expand

up

D1*T1/(1+r) down

D1*T1/(1+r)

T2T1


22

Decision Tree Step 4: Modeling the Option

1496.4

[1811.8] Yes

1193.6

[1509.0] No

Expand

54%

[1811.8] up

742.4

[1057.8] Yes

655.1

[970.5] No

Expand

46%

[1057.8] down

T2

315.4 54%

[1465.3] up

742.4

[915.5] Yes

655.1

[828.2] No

Expand

54%

[915.5] up

328.6

[501.7] Yes

359.5

[532.6] No

Expand

46%

[532.6] down

T2

173.1 46%

[739.6] down

T1 [1131.8]


23

Example: Nortak Nortak Lta. has a project that has a value of $5.000, and

generates a cash flow of: 15% of its value in the first year

25% of its value in the second year

100% of its value in the third year

The risk free rate is 8% per year.

Volatility is 25%

Determine the value of an option to expand the project at any moment by 40% at a cost of $1.500. Consider that the project may be expanded more than once.

5.2


24

Example: Adaptel Adaptel Ltd. Is analyzing a five year project as shown if the

Adaptel spreadsheet.

Determine the dividend payouts in each year.

Spreadsheet: Adaptel.xls

Real OptionsDealing with Dividends

Prof. Luiz Brandão

[email protected]

2009

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Real Options Dealing with Dividends Prof. Luiz Brandão [email protected] 2009.

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