Real Options and Decision Trees PDF

Real Options Real Options When you use discounted cash flow (DCF) methods (eg, traditional NPV) to value a project, you implicitly assume that the firm will hold the assets passively. Do managers in the real world just watch the future unfold? No! As new information arrives over time, they may: •...

Real Options Real Options When you use discounted cash flow (DCF) methods (eg, traditional NPV) to value a project, you implicitly assume that the firm will hold the assets passively. Do managers in the real world just watch the future unfold? No! As new information arrives over time, they may: • Follow up with additional investment to capitalise on initial success • Abandon or shrink a project to avoid future losses These are examples of real options: the right but not the obligation to modify a project in the future. Why Are Real Options Valuable? If the future is certain, flexibility is meaningless. • Every aspect of an investment decision (what, when, how much) can be optimised upfront. There is no need to deviate from this plan. But if the future is uncertain, flexibility is valuable. • There is opportunity to make more money/avoid losses. The more uncertain the future, the greater the value of the real options that provide this flexibility. • Valuation should account for the value of any built-in real options. Types of Real Options • Option to abandon • The project may no longer be profitable going forward. • Value comes from reducing/avoiding future losses. • Option to grow/expand • Investment may turn out to be more profitable than expected. • Value comes from being able to capitalise on additional earning opportunities. • Option to wait (timing option) • You have a positive NPV project but if implemented in the future, the NPV may be even higher. • Value comes from the ability to delay investment and learn more about market conditions. The End Decision Trees Decision Trees A decision tree is a diagram of sequential decisions and possible outcomes. Decision trees help managers determine their options by showing: • Available actions • The sequence/timing of events and actions • Possible outcomes (ie, pay-offs) A Simple Decision Tree s ces Suc Pursue project NPV = $2m Invest $200,000 Yes Fail u re Stop project NPV = 0 Test? No NPV = 0 The End Valuing Projects With Embedded Real Options Example: Coffee Shop Project Project: Build a coffee shop in London • Cost: 100K initial investment • Cash flows will last for 10 years Assume: • rf = 2.1% • βcoffee = 1 • E (Rm − rf ) = 8.4% What is the discount rate? • Use the capital asset pricing model (CAPM): E (Rcoffee ) = rf + βcoffee E (Rm − rf ) = 10.5% The Coffee Shop Project Type of Market Berkeley, CA Cleveland, OH Probability of Market Type (Idiosynchratic) 0.8 Demand Annual Cash flow (£K) High Probability of Demand (Systematic) 0.25 0.8 Med 0.50 22 0.8 Low 0.25 12 0.2 High 0.25 22 0.2 Med 0.50 12 0.2 Low 0.25 2 Two types of risk • Market type risk – idiosyncratic (eg, type of customer) • Cash flow risk – systematic (eg, economic growth) 32 Coffee Shop Project: Decision Tree Without Real Option y kele Ber Yes Build? day K to 0 0 1 st £ Inve Cle vela n 0.8) (p = d (p No NPV = 0 = 0. 2) NPV = ? NPV = ? Coffee Shop Project: Valuation Without Real Option Timeline of cash flows: CF: t −£100K £CF 0 1 £CF 2 ··· £CF £CF 9 10 Coffee Shop Project: Valuation Without Real Option Baseline valuation: no real options (ie, traditional NPV) First, determine the project NPV, given each market type. • Berkeley-type market: E (CF | Berkeley) = 0.25 × 32 + 0.50 × 22 + 0.25 × 12 = 22 NPVBerkeley = −100 + 10 X t=1 22 = 32.33 (1 + 0.105)t • Cleveland-type market: E (CF | Cleveland) = 0.25 × 22 + 0.50 × 12 + 0.25 × 2 = 12 NPVCleveland = −100 + 10 X t=1 12 = −27.82 (1 + 0.105)t Coffee Shop Project: Decision Tree Without Real Option ley ke Ber Yes Build? st Inve day £1 to 00K Cle vela n (p d (p No NPV = 0 8) = 0. = 0. 2) NPV: 32.33 NPV: −27.82 Coffee Shop Project: Valuation Without Real Option Baseline valuation: no real options (ie, traditional NPV) Now, determine the project NPV: NPV = 0.8 × NPVBerkeley + 0.2 × NPVCleveland = 0.8 × 32.33 + 0.2 × −27.82 NPV = 20.3 NPV > 0, so you should build the coffee shop. But, can you do better? • Suppose you can hire a market research firm to immediately learn the market type. • This action creates a real option. • If Berkeley, then build; if Cleveland, then withdraw • What is the most you are willing to pay for that information? Coffee Shop Project: Decision Tree With Real Option Yes Ber Yes y (p kele Cle vela nd .8) =0 (p = Research? No Yes Build? No 0.2 ) No NPV: 0 Yes NPV: −27.82 Build? Build? NPV: 20.3 NPV: 0 NPV: 32.33 No NPV: 0 Coffee Shop Project: Valuation With Real Option Valuation with real option With the market information, we can avoid investing in the project if the market turns out to be the Cleveland type. Therefore, the expected NPV is: NPV = 0.8 × 32.33 + 0.2 × 0 = 25.86 The project with market research (NPV = 25.86) is much more valuable than without (NPV = 20.3). Therefore, you would be willing to pay up to: • (25.86 − 20.3) = £5.56K for market research Where does this extra value (the real option value) come from? • The ability to avoid the negative NPV project (Cleveland); the NPV of savings equals 0.2 × 27.82 = £5.56K Coffee Shop Project: Early Exercise, Part I We assumed that choosing to conduct market research has no impact on the timing of our project – in both cases, we can begin today (t = 0). Of course, in reality, market research takes time. Assume it takes one year to conduct this research. Choosing to undertake market research would then delay our project by one year, to t = 1. Without research, we can start immediately (t = 0). Is it worthwhile to wait? It depends! Recall option theory: • Early exercise of an American call on a non-dividend-paying stock is never optimal. • Early exercise may be optimal for a dividend-paying stock. Coffee Shop Project: Early Exercise, Part II To emphasise the cost of waiting, suppose we only have a 10-year lease on the coffee shop, starting today (t = 0). By conducting market research (ie, waiting), we give up one year of cash flow. With market research: CF: t Research −100K 0 CF 1 2 ... CF CF 9 10 CF CF 9 10 Without market research: CF: −100K CF CF t 0 1 2 ... Coffee Shop Project: Early Exercise, Part III Re-calculate the NPV with market research: NPVBerkeley,t=1 = −100 + 9 X t=1 22 = 24.22 (1 + 0.105)t The £8.11K drop in value (32.33 − 24.22) is analogous to a dividend. We miss out on the dividend by not exercising early. NPVt=0 = 0.8 × 24.22 + 0.2 × 0 = 18.98 < 20.3 1 + 0.021 Note: We discount here at the risk-free rate because we do not face any risk between t = 0 and t = 1 (we just hold cash). Takeaways • Real options are valuable. • In this case, the value came from avoiding bad projects. • By extension, actions that create real options are also valuable. • Waiting can be costly if valuable production opportunities (eg, revenues) are sacrificed. • This is similar to losing dividends when delaying the exercise of an American call option. The End The Abandonment Option Agenda 1. Temporary abandonment 2. Permanent abandonment Temporary Abandonment: Gold Mine Example, Part I • We have the rights to operate a gold mine for three years; cash flows occur at the beginning of each year (ie, the first cash flow occurs immediately ) • The mine produces 50K ounces each year • Costs of extraction: $230/ounce • Gold is currently selling for $220/ounce • The risk-free rate is 5% • The price of gold has a CAPM beta of 0 • Therefore, discount rate = rf = 5% Temporary Abandonment: Gold Mine Example, Part II Growth in the price of gold has two equally likely outcomes: The Price of Gold, +20%, -10% 317 264 220 0.5 0.5 0.5 238 238 0.5 198 0.5 0.5 178 E0 (P1Gold ) = 0.5 × 264 + 0.5 × 198 = $231 E0 (P2Gold ) = 0.25 × 317 + 0.5 × 238 + 0.25 × 178 = $242.75 Temporary Abandonment: Gold Mine Example, Part III Traditional NPV calculation (no abandonment option): NPVGold Mine = 50K × (220 − 230) 50K × (231 − 230) + 1.05 50K × (242.75 − 230) + = $126K (1.05)2 Temporary Abandonment: Gold Mine Example, Part IV Here is a closer look at the net cash flows (in $’000s) in each scenario: Cashflows 4350 =(317-230)(50) 1700 -500 0.5 0.5 0.5 400 400 0.5 -1600 =(238-230)(50) =(238-230)(50) 0.5 0.5 -2600 =(178-230)(50) Temporary Abandonment: Gold Mine Example, Part V Suppose you can temporarily abandon production if the gold price is too low. • You can avoid scenarios that would produce negative net cash flows. • Insert zeros at nodes where the CF is negative and recalculate NPV. Cashflows 1700 0 0.5 0.5 0 4350 =(317-230)(50) 400 400 =(238-230)(50) 0.5 0.5 0.5 0.5 0 =(238-230)(50) Temporary Abandonment: Gold Mine Example, Part VI There are four distinct scenarios for the CFs at t = 1 and 2: 1. Price goes up and up, with probability (w.p.) 0.5 × 0.5 = 0.25 → CFs = (1700,4350) 2. Price goes up and then down, w.p. 0.5 × 0.5 = 0.25 → CFs = (1700,400) 3. Price goes down and then up, w.p. 0.5 × 0.5 = 0.25 → CFs = (0,400) 4. Price goes down and down, w.p. 0.5 × 0.5 = 0.25 → CFs = (0,0)  NPVGold Mine  1700 4350 = 0 + 0.25 + 1.05 (1.05)2   1700 400 + 0.25 + 1.05 (1.05)2   400 + 0.25 0 + + 0.25 × 0 = $1978K (1.05)2 The project is much more valuable with the option! Permanent Abandonment You receive a non-retractable offer to buy your company for $150 million at any time within the next year. Given the following tree of projected cash flows ($ millions), what is the value of the offer? • Use a discount rate of 10%. 0.6 CF2 = 120 CF1 = 100 0.6 0.4 CF0 = 0 0.6 0.4 CF2 = 90 CF2 = 70 CF1 = 50 0.4 CF2 = 40 Permanent Abandonment: Naive Approach • Value of the cash flows at node 100 (t = 1): 100 + (120 × 0.6 + 90 × 0.4) = 198 > 150 1 + 0.1 • At node 50 (t = 1): 50 + (70 × 0.6 + 40 × 0.4) = 102.72 < 150 1 + 0.1 • At year 0: (198 × 0.6 + 102.72 × 0.4) = 145.4 < 150 1 + 0.1 Does this mean you should sell the company today? • No! We have ignored the value of the abandonment option. • Use backwards induction → given optimal abandonment decision at t = 1, what is the t = 0 value? Permanent Abandonment With the option, cash flows would look like this: 0.6 0.6 CF1 = 100 0.4 CF0 = 0 0.4 CF2 = 120 CF2 = 90 CF1 = 150 Therefore, the value of the company today with the option is: (198 × 0.6 + 150 × 0.4) = 162 > 150 1 + 0.1 Don’t sell today! • Option value = 162 − 145 = 17 Permanent Abandonment (cont.) What is the source of the option value? • The abandonment option lets you sell the firm for $150 million at t = 1 when the continuation value is only $102.72 million. • This event occurs with probability = 40%. • Expected PV of the gain is therefore (150−102.72)×0.4 1.1 = 17 • This is the same as the option value calculated in the previous slide! The End The Growth Option Option to Expand You are deciding on an executive flying service. Passenger demand forecasts are as follows: • First year demand • Pr(High) = 60% • Pr(Low) = 40% • Second year demand • • • • Pr(High | First-year High) = 80% Pr(Low | First-year High) = 20% Pr(High | First-year Low) = 40% Pr(Low | First-year Low) = 60% One strategy is to purchase a Turboprop plane for $550K that will generate the following cash flows (in $’000s). Use a 10% discount rate. Option to Expand 0.6 0.8 CF2,H = 960 0.2 CF2,L = 220 0.4 CF2,H = 930 0.6 CF2,L = 140 CF1,H = 150 CF0 = −550 0.4 CF1,L = 30 E(CF1 ) = 0.6 × 150 + 0.4 × 30 = 102 E(CF2 | First year High) = 0.8 × 960 + 0.2 × 220 = 812 E(CF2 | First year Low) = 0.4 × 930 + 0.6 × 140 = 456 ∴ E(CF2 ) = 0.6 × 812 + 0.4 × 456 = 669.6 NPVTurbo = −550 + 102 669.6 + = 96.12 1 + 0.1 (1 + 0.1)2 The project is positive NPV. But can we do better? Option to Expand An alternative strategy is to purchase a Piston-engine (ie, smaller) plane for $250K today and another for $150K if demand in the first-year is high. 0.8 CF1,H = 100 Buy? − Yes 1 tt = 150 a No 0.6 CF2,H = 800 0.2 CF2,L = 100 0.8 CF2,H = 410 0.2 CF2,L = 180 CF0 = −250 0.4 0.4 CF2,H = 220 0.6 CF2,L = 100 CF1,L = 50 Option to Expand 0.8 CF1,H = 100 Buy? Yes t =1 0 at 5 1 − No 0.6 CF2,H = 800 0.2 CF2,L = 100 0.8 CF2,H = 410 0.2 CF2,L = 180 CF0 = −250 0.4 0.4 CF2,H = 220 0.6 CF2,L = 100 CF1,L = 50 First decide if you should buy the second plane at t = 1 when demand is high. NPVBuy,t=1 = −150 + NPVDon’t,t=1 = (0.8 × 800 + 0.2 × 100) = 450 Buy! 1 + 0.1 (0.8 × 410 + 0.2 × 180) = 331 1 + 0.1 Option to Expand CF1,H = 100 Buy? Yes 1 −150 at t = 0.8 CF2,H = 800 0.2 CF2,L = 100 0.6 CF0 = −250 0.4 0.4 CF2,H = 220 0.6 CF2,L = 100 CF1,L = 50 Then, determine the NPV of the resulting Piston-engine strategy. E(CF1 ) = 0.6 × (100 − 150) + 0.4 × 50 = −10 E(CF2 | First year High) = 0.8 × 800 + 0.2 × 100 = 660 E(CF2 | First year Low) = 0.4 × 220 + 0.6 × 100 = 148 ∴ E(CF2 ) = 0.6 × 660 + 0.4 × 148 = 455.2 NPVPiston = −250 + −10 455.2 + = 117.11 1 + 0.1 (1 + 0.1)2 NPVPiston = $117K > NPVTurbo = $96K (staged implementation is usually better). Option to Expand Suppose you ignore the option to expand in the Piston-engine strategy. Then, 0.8 CF2,H = 410 0.2 CF2,L = 180 0.4 CF2,H = 220 0.6 CF2,L = 100 CF1,H = 100 0.6 CF0 = −250 0.4 CF1,L = 50 E(CF1 ) = 0.6 × (100) + 0.4 × 50 = 80 E(CF2 | First year High) = 0.8 × 410 + 0.2 × 180 = 364 E(CF2 | First year Low) = 0.4 × 220 + 0.6 × 100 = 148 ∴ E(CF2 ) = 0.6 × 364 + 0.4 × 148 = 277.6 NPVPiston-NE = −250 + 80 277.6 + = 52.15 1 + 0.1 (1 + 0.1)2 Therefore, the value of the option to expand is equal to NPVPiston − NPVPiston-NE = $65K Option to Expand What is the source of the option value? • The growth/expansion option gives you the ability to invest $150K at t = 1 for a second Piston plane and get an incremental CF at t = 2 of: • 800 − 410 = $390K if demand is High w.p. 0.8 • 100 − 180 = −$80K if demand is Low w.p. 0.2 • The NPV of this incremental investment at t = 1 is: −150 + (0.8 × 390 + 0.2 × −80) = $119.1K 1.1 • This opportunity exists with probability 0.6 • The present value (t = 0) of this investment is therefore: 0.6 × 119.1 = $65K 1.1 • The same as the previous option value! The End The Timing Option Option to Wait The option to wait has value: Even if a project has a positive NPV now, it may be even more valuable if delayed until a more opportune time. • Option value = Intrinsic value + Time premium • Intrinsic value: our profit (ie, NPV) if we exercise right now • Time premium: value of being able to wait Option Price Option Value Intrinsic Value Stock Price Option to Wait (cont.) But remember the coffee shop example! There is a trade-off if waiting means we lose revenues or incur additional costs (akin to foregone dividends). 1 1 Source: Berk and DeMarzo The End