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Copyright © 2003 Pearson Education, Inc. Slide 17-2
Real options
•The application of derivatives theory to the operation and valuation of real investment projects
•A call option is the right to pay a strike price to receive the present value of a stream of future cash flows
•An investment project is the right to pay an investment cost to receive the present value of a future cash flow stream
Investment Project Call Option
Investment Cost = Strike Price
Present Value of Project = Price of Underlying Asset
Copyright © 2003 Pearson Education, Inc. Slide 17-3
Investment and the NPV rule
• NPV rule:– Compute NPV by discounting expected cashflows at the
opportunity cost of capital
– Accept a project if and only if its NPV is positive and it exceeds the NPV of all mutually exclusive alternative projects.
• Example under certainty: – Invest in a $10 machine, that will produce one widget/year forever
at a cost of $0.90/widget. The widget will sell at $0.55 next year and the price will increase at 4% per year. Risk-free rate is 5% per year. The investment can occur anytime. Should it? If yes, when?
Copyright © 2003 Pearson Education, Inc. Slide 17-4
Investment and the NPV rule (cont.)
• Example under certainty: (cont.)
– Static NPV:
NPVInvest today = $27
NPVWait five years = $30.49
NPVWait 23.82 years = $35.03
=$ .0 551
105
104
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105
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105
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3 2 3
.
.
.
.
.$0.
. . .$10 $27
Copyright © 2003 Pearson Education, Inc. Slide 17-5
Investment and the NPV rule (cont.)
• The project as an option:– The decision to invest is analogous to the decision to exercise an
American option early• Exercise price ~ investment cost• Underlying asset ~ value of the project
– Tradeoff between three factors: • Dividends foregone by not exercising: cashflows from selling widgets • Interest saved by deferring the payment of exercise price• Value of the insurance lost by by exercising (the implicit put option): since
no uncertainty, there is no insurance value
– S = $55 (=$0.55/(0.05-0.04)), K = $28 (=$10 + $0.90/0.05), r = 0.04879 (=ln1.05), = 0, = 0.0095669 (=ln1.05 – ln1.05) C = $35.03
Copyright © 2003 Pearson Education, Inc. Slide 17-6
Investment under uncertainty
• Example:– A project requires an intial investment of $100, and is expected to
generate a perpetual cashflow stream, with a first cash flow $18 in one year, expected to grow at 3% annually. Assume a discount rate of 15%. If the project can be delayed for up to two years, should the project be accepted? If yes, when?
– Static NPV: $18/(0.15 – 0.03) – $100 = $50
– Trade-off between three factors:• Foregone initial cashflow: $18
• Interest savings: $7 (7% x $100)
• Value of preserved insurance: Is it more than the loss due to delay?
Loss of $11 if delayed
Copyright © 2003 Pearson Education, Inc. Slide 17-7
Investment under uncertainty (cont.)
• Example: (cont.)
– What is the value of the insurance? Need to know the volatility of cashflows:
Copyright © 2003 Pearson Education, Inc. Slide 17-8
Investment under uncertainty (cont.)
• Example: (cont.)
– S = $150, K = $100, r = 6.766% (=ln1.07), = 0.5, t = 2 years, = 0.1133 (=ln(1+($18/$50))), p* = 0.335 C = $55.80 > $50
Copyright © 2003 Pearson Education, Inc. Slide 17-9
Real options in practice
• The decision about whether and when to invest in a project ~ call option
• The ability to shut down, restart, and permanently abandon a project ~ project + put option
• Strategic options: The ability to invest in projects that may give rise to future options ~ compound option
• Flexibility options: Ability to switch between inputs, outputs, or technologies ~ rainbow option
Copyright © 2003 Pearson Education, Inc. Slide 17-10
Real options in practice (cont.)
• Peak-load electricity generation– Plant idle when price of electricity is less than the cost of fuel
– Plant online when electricity price spikes or fuel price declines
– Similar to owning a strip of call options on electricity expiring daily, with a strike price of cost of variable inputs
– Spark spread: Selec – H x Sgas, where H is plant efficiency measure
– Profit = max(Selec – H x Sgas, 0), a European exchange option
–
– By rewriting using put-call parity:
Value of Plant ) BSCall F H F r t rE t G t ti
n
ii i i( , , , , ,, ,
1
e F H F BSPut F H F r t rrt
E t G ti
n
E t G t ti
n
ii
i i i i i( ) ( , , , , ,, , , ,
1 1 )
Static NPV Option Not to Operate
Copyright © 2003 Pearson Education, Inc. Slide 17-11
Real options in practice (cont.)
• Research and development– Costs incurred to acquire technology to be used in future projects
– Projects in the future only undertaken if they have positive NPV
Copyright © 2003 Pearson Education, Inc. Slide 17-13
Commodity extraction as an option
• Incur the extraction costs to realize the value of the resource: defer investment and stop and start production
• Example: Single-barrel extraction under certainty– A plot of land contains one barrel of oil that can be extracted by
paying $13.60. Currently, oil sells for $15/barrel, effective annual lease rate is 4%, and effective annual risk-free rate is 5%
– How much is the land worth? • $1.40 (= $15 – $13.60) [no real options, immediate extraction]
• $1.796 [considering real options, trigger price $16.918 at t = 12.575]
• If the lease rate of an extractive commodity is zero, it is best to leave the commodity underground forever
Copyright © 2003 Pearson Education, Inc. Slide 17-14
Commodity extraction as an option (cont.)
• Example: Single-barrel extraction under uncertainty– The same example as before, but oil price volatility is 15%
– How much is the land worth? • $3.7856 [considering real options, trigger price $25.3388]
• When uncertainty is introduced, the optimal exercise time is later, and the project value, and trigger price is higher
Copyright © 2003 Pearson Education, Inc. Slide 17-16
Commodity extraction as an option (cont.)
• Example: Valuing an infinite oil reserve– Example 17.1: Suppose S0 = $15, r = 5%, = 4%, c = $8, and the
investment cost I is $180• The value of the producing well is $15/0.04 – $8/0.05 = $215.
• Per barrel extraction cost is (c/r + I) = $13.60
• CallPerpetual [$15/0.04, $8/0.05 + 180, 0.000001, ln(1.05), ln(1.04)]
= [$44.914, $422.956]
• The value of the well is $422.956
• Extraction occurs when S = 0.04 x $422.956 = $16.918
– Example 17.2: Assumptions above + oil price volatility = 0.15• CallPerpetual [$15/0.04, $8/0.05 + 180, 0.15, ln(1.05), ln(1.04)]
= [$94.639, $633.469]
Copyright © 2003 Pearson Education, Inc. Slide 17-17
Commodity extraction with the option to shut down and restart production
Copyright © 2003 Pearson Education, Inc. Slide 17-18
Commodity extraction with the option to shut down and restart production (cont.)