1. Binnenlandse Francqui Leerstoel VUB 2004-2005 Real Options
André Farber
Solvay Business School
University of Brussels

2. Valuation: the traditional approach
Standard approach: V = PV(Expected Free Cash Flows)
Free Cash Flow = CF from operation + CF from investment
Valuation technique: discounting using a risk-adjusted discount rate
Setting the discount rate: Capital Asset Pricing Model
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3. Making Investment Decisions: NPV rule
NPV : a measure of value creation
NPV = V-I with V=PV(Additional free cash flows)
NPV rule: invest whenever NPV>0
Underlying assumptions:
one time choice that cannot be delayed
single roll of the dices on cash flows
But:
delaying the investment might be an option
what about flexibility ?
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4. Capital Budgeting: What techniques are used?
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5. A simple example: the widget factory
Cost of factory I: $4,000
Produces one widget per year forever with zero operating cost
Price of widget uncertain:
t = 0
t = 1
t = 2
P1 = $300
P2 = $300
Proba = 0.50
P0 = $200
Proba = 0.50
P1 = $100
P2 = $100
Risk-free interest rate = 5% β = 0
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6. Traditional NPV calculation
Expected price = 0.5 × $ × $100 = $200
Expected cash flows
t = 0
t =1
t = 2
t =3
Operating CF
$200
Investment CF
-$4,000
What would you decide?
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7. What if we wait one year? P1 = $300
→ NPV = -4, /0.05 = $2,300
DECISION: INVEST
Invest →NPV = $200
Wait →
DECISION: WAIT
P1 = $100
→ NPV = -4, /0.05 = -$1,900
DECISION:DO NOT INVEST
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8. Real options take a number of forms
If an initial investment works out well, then management can exercise the option to expand its commitment to the strategy.
For example, a company that enters a new geographic market may build a distribution center that it can expand easily if market demand materializes.
An initial investment can serve as a platform to extend a company's scope into related market opportunities.
For example, Amazon.com's substantial investment to develop its customer base, brand name and information infrastructure for its core book business created a portfolio of real options to extend its operations into a variety of new businesses.
Management may begin with a relatively small trial investment and create an option to abandon the project if results are unsatisfactory.
Research and development spending is a good example. A company's future investment in product development often depends on specific performance targets achieved in the lab. The option to abandon research projects is valuable because the company can make investments in stages rather than all up-front.
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9. Portlandia Ale: an example
(based on Amram & Kulatilaka Chap 10 Valuing a Start-up)
New microbrewery
Business plan:
€4 million needed for product development (€0.5/quarter for 2 years)
€12 million to launch the product 2 years later
Expected sales € 6 million per year
Value of established firm: €22 million
(based on market value-to-sales ratio of 3.66)
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10. DCF value calculation
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11. But there no obligation to launch the product..
The decision to launch the product is like a call option
By spending on product development, Portlandia Ale acquires
a right (not an obligation)
to launch the product in 2 years
They will launch if, in 2 years, the value of the company is greater than the amount to spend to launch the product (€12 m)
They have some flexibility.
How much is it worth?
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12. Valuing the option to launch
Let use the Black-Scholes formula (for European options)
At this stage, view it as a black box (sorry..)
5 inputs needed:
Call option on a stock Option to launch
Stock price Current value of established firm
Exercise price Cost of launch
Exercise date Launch date
Risk-free interest rate Risk-free interest rate
Standard deviation of Volatility of value
return on the stock
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13. Volatility
Volatility of value means that the value of the established firm in 2 years might be very different from the expected value
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14. Using Black Scholes Current value of established firm = 14.46
Cost of launch =
Launch date = 2 years
Risk-free interest rate = 5%
Volatility of value = 40% 
and…magic, magic…..value of option = € 4.96
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15. The value of Portlandia Ale
(all numbers in € millions)
Traditional NPV calculation
PV(Investment before launch)
PV(Launch)
PV(Terminal value)
Traditional NPV
Real option calculation
Value of option to launch
Real option NPV
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16. Let us add an additional option
Each quarter, Portlandia can abandon the project
This is an American option (can be exercised at any time..)
Valuation using numerical methods (more on this later..)
Traditional NPV calculation
Traditional NPV
Real option calculation
PV(Investment before launch)
Value of options to launch and
to abandon
Real option NPV
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17. Real option vs DCF NPV Where does the additional value come from?
Flexibility
changes of the investment schedule in response to market uncertainty
option to launch
option to continue development
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18. 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)
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19. Binomial option pricing model: review
Used to value derivative securities: PV=f(S)
Evolution of underlying asset: binomial model
u and d capture the volatility of the underlying asset
Replicating portfolio: Delta × S + M
Law of one price: f = Delta × S + M uS fu S dS fd Δt
M is the cash position M>0 for investment M<0 for borrowing
Consider the following example. You want to value a call option with three months to maturity. The strike price of the option is 20 and the spot price of the underlying asset is 20. The risk-free interest rate (with continuous compounding) is 5%. The volatility of the underlying asset is 30%.
You want to use the binomial option pricing model with Δt=3/12=0.25
The up and down factors are: u = and d = The two possible stock prices three months later are uS = and dS = 17.21
The values of the call option at the end of the period are: fu = Max(0, )=3.24 and fd = Max(0, )=0
The composition of the replicating portfolio is: Delta = and M = You should buy shares and borrow
As a consequence, the value of the call option is: f = ×20 – = 1.61.
As an exercise, you can verify that similar calculations for a put option would lead to the following results:
fu = 0, fd = 2.79
Delta = , M =
f = 1.36
r is the risk-free interest rate with continuous compounding
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20. Risk neutral pricing
The value of a derivative security is equal to risk-neutral expected value discounted at the risk-free interest rate
p is the risk-neutral probability of an up movement
Risk-neutral pricing is a very powerful technique to price derivative securities. The difficulty is to be able to calculate the risk neutral probabilities. This calculation is straightforward in the binomial model. It requires more advanced mathematical techniques in more general model.
To continue the example in the previous slide, the risk neutral probability of an up movement is:
p = (e.05×0.25 – 0.861)/( ) = 0.504
The value of the call and the put options are:
Call f = (0.504 × 3.24)/ e.05×0.25 = 1.61
Put f = (0.504 × 2.79)/ e.05×0.25 = 1.36
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21. 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
u=1.22, d=0.25 up
down
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22. 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}
0.00
p = /(1+rt)=0.9876
=(0.48 0.00) 
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23. When to invest?
Traditional NPV rule: invest if NPV> 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 = = 20 >0
What if you wait?
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24. To mothball or not to mothball
Let analyze 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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25. 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 = V = 120 C = 20
V= 60 C = 9
V = C = 0
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)
85>51 => wait 2 years
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26. 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)
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27. 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%
Option’s maturity = 10 years 
Binomial model with 1 step per year
Immediate investment : NPV = 20
Value of option to invest: WAIT
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28. Optimal investment policy
Value of future cash flows
(partial binomial tree)
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
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29. 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 (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
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30. 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*
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31. Optimal investment rule: numerical example
Cost of investment I = 100
Cash flow yield  = 6%
Risk-free interest rate r = 4%
Volatility = 30%
Critical value V*= 210
For V = 120, value of investment opportunity f(V) = 27
Sensitivity analysis
 V*
2% 341
4% 200
6% 158
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32. Value of investment opportunity for different volatilities
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33. Some Applications of Real Options
Valuing a Start-Up
Exploring for Oil
Developing a Drug
Investing in Infrastructure
Valuing Vacant Land
Buying Flexibility
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