Transcript A Cursory Introduction to Real Options

A Cursory Introduction to Real
Options
Andrew Brown
5/2/02
Background
• Real Options Analysis (ROA) was developed as a
method to find the value of projects and assets
more accurately then earlier methods such as
Discounted Cash Flows (DCF) and Decision Tree
Analysis (DTA).
• ROA is based on financial option pricing, as
developed by Black and Scholes (1973) and
Merton.
Traditional Valuation Methods
• The DCF method can be done various, equivalent
ways:
– Adjust all future cash flows to account for the risk in
the project and then discount the cash flows to present
values using the risk-free interest rate.
– Adjust the discount rate to take into account the risk of
the project and discount all future cash flows to present
values using the risk-adjusted interest rates. This is
what is commonly thought of as Net Present Value
(NPV).
• Problem- DCF does not take into account that the
project can be changed once started. This leads to
DCF undervaluing projects.
DCF Example
T=1
$170-$115
T=0
0.5
0$
0.5
$65-$115
The risk-adjusted discount rate is 17.5%. The
expected value of this project is:
NPV = 0.5(170-115)/1.175 + 0.5(65-115)/1.175
= -$6.48
• The problem with the net present value in
the previous example is that it doesn’t take
into account the ability for the project to be
managed. For example, it may be possible
that the project can be abandoned if the low
inflow is going to occur.
• DTA tries to put decision making into the
valuation calculation. The next slide shows
the same example using DTA assuming that
the cost of the project can be avoided if the
low outcome is to occur.
DTA Example
T=1
MAX[$170-$115,0]
T=0
0.5
0$
0.5
MAX[$65-$115,0]
NPV = 0.5*MAX[170-115,0]/1.175
+ 0.5*MAX[65-115,0]/1.175
= $23.40
DTA Problems
• Problem- allowing decisions to be made during
the project changes how risky the project is.
• This implies that the risk-adjusted discount rate of
17.5% is no longer valid.
• Because the risk of the project has been lowered
the risk-adjusted interest rate should be lower as
well…
• Therefore, the DTA undervalued the project.
Options- A Different Perspective
• Another way to look at previous example is
as an option.
• At time zero a contract is entered into that
gives the firm the right, but not the
obligation, to pay $115 for a cash flow
worth either $170 or $65, each with 50%
probability.
• What is the value of this contract? DCF
said -$6.48 and DTA said $23.40.
Formulating ROA
• To use ROA an assumptions must be made:
– Changes in the value of the project are spanned
by existing assets in the economy.
• Spanning implies 2 things:
– A dynamic portfolio of other investments exists
whose value is perfectly correlated with the
value of the project.
– Decisions made about the project will not affect
the options available to investors.
Contingent Claims Analysis
• The spanning assumption gives a better way to
value the example project.
–
–
–
–
–
Let the risk-free interest rate be r = 10%
F0 is the value of the opportunity at time 0.
F1 is the value of the opportunity at time 1.
P1 is the payoff of the project at time 1.
Let n be a number of shares in the twin portfolio. Each
share is worth the expected value of the payoffs from
the portfolio, i.e. $115.
• Set up a portfolio containing the project and short
selling n shares of the twin portfolio.
The value of this portfolio at time zero is:
V0 = F0 - 115*n
At time one the value is:
V1 = F1 – n* P1
= (170-65) – 170n OR
= (0) – 65n
The return will be risk free if V1 is always the
same… (170-65) – 170n = 0 - 65n
Solving for n gives: n = 0.524
This gives the value of V1 to be: -$34.08
The portfolio is risk free and should return the risk
free interest rate, minus the cost of the short
position:
Return = V1 - V0 - .10*115*n
= (-34.08) – (F0 - 115*0.524) – 1.15*0.524
= 25.58 - F0
The return must be equal to 10% of the initial cost,
i.e.,
25.58 - F0 = 0.1(F0 - 115*0.524)
F0 = $28.73
So, ROA gives an even greater value to the project
then DCF (-$6.48) and DTA ($23.40)
Real Options Analysis
• There are three ways to evaluate real
options:
– Black Scholes
– Binomial Trees
– Monte Carlo Analysis
Black-Scholes
• To use the Black-Scholes equation to price a
financial option, 5 values are needed. Listed
below are the 5 financial values needed and their
ROA equivalents:
Financial Value
Current Stock Price
Exercise Price
Time to Expiration
Stock Value Uncertainty
Risk-free Interest Rate
Real Value
PV of Exp. Cash Flows
Investment Cost
Time Opportunity Lasts
Project Value Uncertainty
Risk-free Interest Rate
When Black Scholes Works
–
–
–
–
–
Good for “one shot” projects.
Lognormal distribution assumed
No interaction between various options
Only one source of uncertainty
Only one maturity date
Some of these limitations can be avoided by rederiving or modifying the Black Scholes
equation.
Monte Carlo Simulation
• Repeat the following a bunch of times:
– Draw random numbers for each random variable.
– Decide what choice would be made based on those
number.
– Calculate the NPV
• Average the NPV over all of the runs to get an
approximation of the value of the project.
• DCF would be the same as always making the same choice
above, regardless of the drawn random numbers.
• The value of the real option is the difference between the
DCF and the simulation as the number of runs goes to
infinity, i.e., Average NPV = DCF + RealOption
When to use Monte Carlo
• There are multiple sources of uncertainty.
• Can use distributions besides lognormal.
• Monte Carlo simulation is slow and
computationally expensive.
Binomial Trees

p is the risk neutral probability that the
random value goes up in a period
ue
d  1/ u
1  rf  d
p
ud
T/N
u is the multiplier the value goes up by.
d is the multiplied the value goes down by
Rf- per period risk free rate
T/N – number of periods
V*u
p
p
1-p
p
V
V*u*d=V*d*u=V
1-p
V*d
V*u*u
1-p
V*d*d
Binomial Trees
• Can handle multiple options
• Assumes lognormal distribution
• Multiple dates for the different options to
mature.
• Handles only one source of uncertainty.
• Is the same as Black Scholes as the number
of intervals, N, goes to infinity.