Real Options Dealing with Dividends Prof. Luiz Brandão [email protected] 2009.
-
Upload
earl-oliver -
Category
Documents
-
view
217 -
download
2
Transcript of Real Options Dealing 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As these cash flows are distributed out to the shareholders or investors, or otherwise withdrawn from the project, the value of the project decreases instantly by this amount.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.
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
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:
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
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
0 1 2 3 4 5
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.
IAG PUC – Rio Brandão
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
IAG PUC – Rio Brandão
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
IAG PUC – Rio Brandão
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.
IAG PUC – Rio Brandão
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.
IAG PUC – Rio Brandão
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)
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.
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.
1.350
1 0.741
tu e
d u
1000
With Expansion
IAG PUC – Rio Brandão
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.
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
(1 )0.5405
rf dp
u d
p
p
1-p
1-p
IAG PUC – Rio Brandão
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.
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
u
d
p
r
VP
Vol
T1
T2
IAG PUC – Rio Brandão
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]
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.
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
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.
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]
IAG PUC – Rio Brandão
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
IAG PUC – Rio Brandão
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]
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
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