1. Real Options Dealing with Dividends Prof. Luiz Brandão brandao@iag.puc-rio.br 2009

2. 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  As these cash flows are distributed out to the shareholders or investors, or otherwise withdrawn from the project, the value of the project decreases instantly by this amount.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. IAG PUC – Rio Brandão 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. IAG PUC – Rio Brandão 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: 04558681,2431,585Ex-Dividend PV 5009551,3681,7432,0851,895Pre Dividend PV 500 -1000 Cash Flows 543210 - 500

5. IAG PUC – Rio Brandão 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 012345

6. IAG PUC – Rio Brandão 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. IAG PUC – Rio Brandão 7 Incorporating Uncertainty 012345 Cash Flows -1000500 Project Value1,8952,0851,7431,368955500 Dividend Rate 0.2400.2870.3660.5241.000 1895 u = 2,274 Pre Dividend Value (545)Constant Rate 1,8951,729Ex-Dividend Value 1895 d = 1579 (379) 1201

8. IAG PUC – Rio Brandão 8 Incorporating Uncertainty

9. IAG PUC – Rio Brandão 9 Modeling Steps 1. 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. IAG PUC – Rio Brandão 10 Modeling Steps 2. 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. IAG PUC – Rio Brandão 11 Modeling Steps

12. IAG PUC – Rio Brandão 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. IAG PUC – Rio Brandão 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. 1000 With Expansion

14. IAG PUC – Rio Brandão 14 Modeling the Underlying Asset 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. IAG PUC – Rio Brandão 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 p p

16. IAG PUC – Rio Brandão 16 Solução 1131,8 337,5 185,2 p 1-p 850,0 411,6 1713,2 p 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. IAG PUC – Rio Brandão 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

18. IAG PUC – Rio Brandão 18 Decision Tree  Step 2: Model the Binomial Lattice Dividends not yet included Without Dividends, the lattice is:

19. IAG PUC – Rio Brandão 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.

20. IAG PUC – Rio Brandão 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.

21. IAG PUC – Rio Brandão 21 Decision Tree  Step 4: Modeling the Option

22. IAG PUC – Rio Brandão 22 Decision Tree  Step 4: Modeling the Option

23. IAG PUC – Rio Brandão 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. IAG PUC – Rio Brandão 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

25. Real Options Dealing with Dividends Prof. Luiz Brandão brandao@iag.puc-rio.br 2009