Transcript From financial options to real options

From financial options to real options
3. Real option valuations
Prof. André Farber
Solvay Business School
ESCP March 10,2000
Valuing real options
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Back to Portlandia Ale
• Portlandia Ale had 2 different options:
– the option to launch (a 2-year European call option)
• value can be calculated with BS
– the option to abandon (a 2-year American option)
• How to value this American option?
– No closed form solution
– Numerical method: use recursive model based on binomial
evolution of value
– At each node, check whether to exercice or not.
– Option value = Max(Option exercised, option alive)
Valuing real options
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Valuing a compound option (step 1)
• Each quaterly payment (€0.5 m) is a call option on the option to launch
the product. This is a compound option.
• To value this compound option::
• 1. Build the binomial tree for the value of the company
0
1
2
3
4
5
6
7
14.46
17.66
11.83
21.56
14.46
9.69
26.34
17.66
11.83
7.93
32.17
21.56
14.46
9.69
6.50
39.29
26.34
17.66
11.83
7.93
5.32
47.99
32.17
21.56
14.46
9.69
6.50
4.35
58.62
39.29
26.34
17.66
11.83
7.93
5.32
3.56
up
u=1.22, d=0.25
down
8
71.60
47.99
32.17
21.56
14.46
9.69
6.50
4.35
2.92
•
Valuing real options
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Valuing a compound option (step 2)
• 2. Value the option to launch at maturity
• 3. Move back in the tree. Option value at a node is:
Max{0,[pVu +(1-p)Vd]/(1+rt)-0.5}
0
1
1.74 4.24
0.42
2
7.88
1.95
0.00
3
12.71
4.57
0.53
0.00
4
18.79
8.35
2.16
0.00
0.00
5
26.25
13.30
4.93
0.62
0.00
0.00
6
35.29
19.47
8.87
2.36
0.00
0.00
7
46.27
26.94
13.99
5.30
0.67
0.00
8
59.60
35.99
20.17
9.56
2.46
0.00
0.00
0.00
0.00
0.00
0.00
0.00
p = 0.48, 1/(1+rt)=0.9876
=(0.482.46+0.520.00)  0.9876
Valuing real options
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When to invest?
• Traditional NPV rule: invest if NPV>0.
Is it always valid?
• Suppose that you have the following project:
– Cost I = 100
– Present value of future cash flows V = 120
– Volatility of V = 69.31%
– Possibility to mothball the project
• Should you start the project?
• If you choose to invest, the value of the project is:
• Traditional NPV = 120 - 100 = 20 >0
• What if you wait?
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To mothball or not to mothball
• Let analyse this using a binomial tree with 1 step per year.
• As volatility = .6931, u=2, d=0.5. Also, suppose r = .10 => p=0.40
• Consider waiting one year..
V=240 =>invest NPV=140
V=120
V= 60 =>do not invest NPV=0
• Value of project if started in 1 year = 0.40 x 140 / 1.10 = 51
• This is greater than the value of the project if done now (20
• Wait..
• NB: you now have an American option
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Waiting how long to invest?
• What if opportunity to mothball the project for 2 years?
V = 480 C = 380
V=240 C = 180
V=120 C = 85
V = 120 C = 20
V= 60 C = 9
V = 30 C = 0
85>51 => wait 2 years
• This leads us to a general result: it is never optimal to exercise an
American call option on a non dividend paying stock before maturity.
• Why? 2 reasons
– better paying later than now
– keep the insurance value implicit in the put alive (avoid regrets)
Valuing real options
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Why invest then?
• Up to know, we have ignored the fact that by delaying the investment,
we do not receive the cash flows that the project might generate.
• In option’s parlance, we have a call option on a dividend paying stock.
• Suppose cash flow is a constant percentage per annum  of the value
of the underlying asset.
• We can still use the binomial tree recursive valuation with:
p = [(1+rt)/(1+t)-d]/(u-d)
• A (very) brief explanation: In a risk neutral world, the expected return r
(say 6%) is sum of capital gains + cash payments
• So:1+r t = pu(1+ t) +(1-p)d(1+ t)
Valuing real options
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American option: an example
•
•
•
•
•
Cost of investment I= 100
Present value of future cash flows V = 120
Cash flow yield  = 6% per year
Interest rate r = 4% per year
Volatility of V = 30%
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•
•
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Option’s maturity = 10 years 
Binomial model with 1 step per year
Immediate investment : NPV = 20
Value of option to invest: 35
WAIT
Valuing real options
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Optimal investment policy
•
Value of future cash flows
(partial binomial tree)
0
1
2
3
120.0 162.0 218.7 295.2
88.9 120.0 162.0
65.9 88.9
48.8
4
398.4
218.7
120.0
65.9
36.1
5
537.8
295.2
162.0
88.9
48.8
26.8
•
•
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•
•
Investment will be delayed.
It takes place in year 2 if no down
in year 4 if 1 down
Early investment is due to the loss
of cash flows if investment
delayed.
Notice the large NPV required in
order to invest
Valuing real options
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A more general model
• In previous example, investment opportunity limited to 10 years.
• What happened if their no time frame for the investment?
• McDonald and Siegel 1986 (see Dixit Pindyck 1994 Chap 5)
• Value of project follows a geometric Brownian motion in risk neutral
world:
• dV = (r-  ) V dt +  V dz
• dz : Wiener process : random variable i.i.d. N(0,dt)
• Investment opportunity :PERPETUAL AMERICAN CALL OPTION
Valuing real options
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Optimal investment rule
•
•
•
•
•
Rule: Invest when present value
reaches a critical value V*
If V<V* : wait
Value of project f(V) =
aV if V<V*
V-I
if V V*
f (V )  aV 
1
2
 
V* 
r 
2
 (
r 
2
1
2r
 )2  2
2

I
 1
V*  I
a  *
V
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Optimal investment rule: numerical example
•
•
•
•
Cost of investment I = 100
Cash flow yield  = 6%
Risk-free interest rate r = 4%
Volatility = 30%
Sensitivity analysis

V*
2%
341
4%
200
6%
158
• Critical value V*= 210
• For V = 120, value of investment opportunity f(V) = 27
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Value of investment opportunity for different volatilities
250.0
200.0
f(V)
150.0
100.0
50.0
0.0
0
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300
V
Sigma = 0
Sigma = 0.2
Sigma = 0.3
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