Corporate Finance - Lecture 4 PDF

University of St.Gallen (HSG) 7,107 Corporate Finance – Lecture 4: Options and Real Options Marc Arnold Objectives UNDERSTAND THAT STANDARD NPV ANALYSIS DOES NOT ACCOUNT FOR REAL OPTIONS Gain awareness of the limitations of traditional NPV analysis in evaluating real options....

University of St.Gallen (HSG) 7,107 Corporate Finance – Lecture 4: Options and Real Options Marc Arnold Objectives UNDERSTAND THAT STANDARD NPV ANALYSIS DOES NOT ACCOUNT FOR REAL OPTIONS Gain awareness of the limitations of traditional NPV analysis in evaluating real options. 1 DEVELOP A BASIC UNDERSTANDING OF OPTION PRICING Acquire knowledge of the fundamental principles behind option pricing. 2 EXPLORE THE BLACK-SCHOLES FORMULA AND BINOMIAL MODEL Gain insights into key models for pricing options, including the Black-Scholes formula and the Binomial model. 3 RECOGNIZE DIFFERENT TYPES OF REAL OPTIONS AND THEIR IMPLICATIONS Improve your ability to identify and assess various real options for better business decision-making. 4 2 Projects with Real Options COMPANY PROJECTS REAL OPTIONS can consist of the choice of whether to have a lot in common with financial options except that usually invest or not, often however, projects have options embedded they are not tradable that give managers additional choices such as shutting down a project or expanding a project (real options) FLEXIBILITY VALUATION Management is not paid to just invest and then sit back and Standard NPV analysis does not account for real options, watch the future unfold and the flexibility to react to the hence, we need tools to value these options environment has a value 3 Projects with Real Options 01 02 PROJECTS FLEXIBILITY The projects that companies often consider involve a choice of Management is not paid just to invest and then sit back to whether or not to invest. Often, projects have embedded watch how the future unfolds options that provide managers with additional opportunities, The ability to adapt to changes in the environment has value such as abandoning a project or expanding it (real options) 03 04 VALUATION REAL OPTIONS Standard NPV analysis does not consider real options; Real options can capture the value of this flexibility therefore, we need tools to evaluate these options They share many similarities with financial options, except they are typically not tradable 4 Types of Real Options OPTION TO EXPAND OPTION TO WAIT OPTION TO ABANDON OPTION TO SWITCH When launching a new product, companies often start Companies can have the option to delay an investment Companies have the with a pilot program to iron Companies have the opportunity to bail out of a out possible design problems This enables them to opportunity to switch project and to test the market potentially avoid costly production methods, input mistakes by waiting how Usually, these decisions are The company can evaluate the factors, or products markets move and then decide taken by management and not pilot and then decide whether whether to invest or not by nature to expand to full-scale production 5 How to Value Real Options NPV AND REAL OPTIONS ARE NOT MUTUALLY EXCLUSIVE In many cases it makes sense to first apply straightforward NPV and then evaluate the option on top of the no-flexibility NPV REAL OPTION VALUATION: DECISION TREES AND/OR OPTION VALUATION We will briefly look at the principles of option pricing and at examples that involve option pricing and others that do not IMPORTANT SIMILARITIES BETWEEN REAL AND FINANCIAL OPTIONS Option on a project is very similar to one on a financial asset 6 Basics of Option Pricing How To DETERMINE OPTION PRICE CALCULATION Discounting cash flows does NOT work for options We could compute cashflows in each period The risk of an option changes every time the underlying moves, hence finding one opportunity cost of capital for an option is impossible How can we compute the price of an option? Black-Scholes formula (easy to apply but not feasible for all kinds of options), particularly not for real options that often are of American type Binomial model and risk-neutral pricing (requires more work but allows to price more types of options) 7 Black-Scholes(-Merton) Formula for European Options CALL where, 𝐶0 = 𝑆0 𝑁 𝑑1 − 𝐾𝑒 −𝑟𝑇 𝑁 𝑑2 𝐶0 𝑃0 = current call and put values 𝑆0 = current price of the underlying 𝑁(𝑑) = cumulative probability that random draw from standard PUT normal distribution is smaller than d 𝑃0 = 𝐾𝑒 −𝑟𝑇 1 − 𝑁 𝑑2 − 𝑆0 (1 − 𝑁 𝑑1 ) 𝐾 = exercise price of the option 𝑟 = continuously compounded annual risk-free rate 𝑇 = time to expiration in years 𝑆0 𝜎2 𝜎 = annual standard deviation of 𝑙𝑛 + 𝑟+ 𝑇 𝐾 2 the underlying 𝑑1 = 𝑑2 = 𝑑1 − 𝜎 𝑇 𝜎 𝑇 8 Black-Scholes Option Valuation Exercise OPTION VALUATION BLACK SCHOLES Cumulative Standard Normal Distribution Table 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5239 0.5319 0.5359 We want to value a call on a stock with S0 = 100 and annual 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 volatility of 𝜎 = 0.5. The option has an exercise price of K = 95 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 and three-month expiration T = 0.25. The risk-free rate is r = 0.1 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 per year. 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 𝟏𝟎𝟎 𝟎.𝟓2 0.5 0.6915 0.9650 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 ln + 𝟎.𝟏+ 𝟎.𝟐𝟓 𝟗𝟓 2 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 d1 = ≈ 0.43 𝟎.𝟓 𝟎.𝟐𝟓 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 d2 = 0.43 − 0.5 0.25 = 0.18 0.8 0.7881 0.7610 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 N(0.43) = 0.6664 and N(0.18) = 0.5714 can be found in the 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 cumulative standard normal distribution table. 1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 𝐶0 = 100 × 0.6664 − 𝟗𝟓 × 𝑒 −𝟎.𝟏×𝟎.𝟐𝟓 × 0.5714 = 13.70 1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 9 More Flexible: Binomial Model 01 02 03 Start by constructing a tree with Determining the time steps How do we pick values for the up and potential future prices of the down movements of the underlying underlying If the time steps used to set up the in the binomial tree? tree become infinitesimally small, the binomial tree option value goes We rely on the standard deviation towards the Black-Scholes option and: value 1 + upside change = 𝑢 = 𝑒𝜎 ℎ 1 + downside 1 =𝑑= 𝑢 where 𝜎 denotes the standard deviation and h denotes the length of one-time step (in our examples, it will always be equal to one year) 10 Binominal Tree The general structure of the price path (“tree”) Example binomial tree for a stock with price of 20 today and a standard deviation of 20% 𝑆 = 20, 𝜎 = 0.2, ℎ = 1, 𝑢 = 1.22, 𝑑 = 0.82 𝑢3 𝑆0 𝑢2 𝑆0 29.84 𝑢𝑆𝑜 𝑢2 𝑑𝑆0 24.43 𝑆0 𝑢𝑑𝑆0 20 20.00 𝑑𝑆0 𝑢𝑑 2 𝑆0 16.37 𝑑 2 𝑆0 13.41 𝑑 3 𝑆0 11 Risk-Neutral Valuation To derive the value of an option, we could simply calculate its expected value Problem Easy Fix Riskiness of an option changes throughout Switch to a risk-neutral world the binominal tree, hence, discount rate Pretend investors do not care about risk, so expected would have to change as well return on everything (i.e., the discount rate) is 𝑟𝑓 , calculate expected payoff of option and discount at 𝑟𝑓 Simply use risk-neutral probabilities We can now price options in the binomial tree relatively easily Will lead to the same solution than going the hard way by using real-life probabilities and discounting future expected cash flows Note that we only need two factors to perform risk-neutral pricing: the standard deviation and the risk-free rate What we do not have to know is the actual expected price and the statistical probabilities 12 Risk-Neutral Probabilities Assuming a stock can either rise by 33% or fall by 25% and a risk-free rate of 1.5%, we can calculate the implied risk-neutral The general formula to compute the risk-neutral probability of an up movement as follows: probability of an up movement of an asset: 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑟𝑒𝑡𝑢𝑟𝑛 = 𝑟𝑓 = 𝑃𝑢𝑝 × 0.33 + (1 − 𝑃𝑢𝑝 ) × (−0.25) 𝑟𝑓 − 𝑑𝑜𝑤𝑛𝑠𝑖𝑑𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑃𝑢𝑝 = Plugging in the numbers and solving for 𝑃𝑢𝑝 yields 𝑃𝑢𝑝 = 𝑢𝑝𝑠𝑖𝑑𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 − 𝑑𝑜𝑤𝑛𝑠𝑖𝑑𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 45.69% 𝑑𝑜𝑤𝑛 and 𝑃 = 1 − 𝑃𝑢𝑝 = 54.31% 13 Risk-Neutral Valuation of a European Call Option on a Stock 1. Risk neutral valuation of a European call option for a stock with price of 20 today and a standard deviation of 20% 𝑆 = 20, 𝜎 = 0.2, ℎ = 1, 𝑢 = 1.22, 𝑑 = 0.82, 𝐾 = 23, 𝑟𝑓 = 1.5% 0.015−(−0.18) 𝑃𝑢𝑝 = = 48.75% 0.22−(−0.18) 𝑑𝑜𝑤𝑛 𝑢𝑝 𝑃 =1−𝑃 = 51.25% 2. Call option payoffs → max(0, S − K) 29.84 6.84 24.43 3.28 20.00 20.00 1.58 0.00 16.37 0.00 13.41 0.00 (6.84×0.48752 ) 3. Price of the call option: = 1.58 1.0152 14 Option Pricing via Replication An option can be replicated by trading in the underlying stock and the money market (risk-free asset) As the portfolio of underlying and risk-free asset always has the same payoffs as the option, both strategies must have the same price today We will have a detailed presentation with practical implications for hedging during our trading day Highly relevant in practice 15 Investment Opportunity Exercise R&D PROJECT NEW DRUG 2024 TASK INSTRUCTION Risk-free interest rate = 1%; Costs of capital = 9%; Volatility of expected income = 0.7 2024 2026 2029 Investment costs USD 26m Additional investment …yields an expected of USD 130m… income of USD 280m 56% failure 44% success 16 Investment Opportunity Exercise (continued) R&D PROJECT NEW DRUG 2024 DCF APPROACH 130 PV of the additional investment: = 127.44 1.012 280 PV of the expected profit: = 181.98 1.095 NPV = −26 + 0.44 × 181.98 − 127.44 = −2 Investment in R&D should NOT be made 17 xxx Discussion Calculate the NPV of the Project with the Real Option Approach. Please use the Excel file on Canvas to solve this exercise. 18 Investment Opportunity Exercise (continued) R&D PROJECT NEW DRUG 2024 INSIGHTS 1. Where does the value difference come from? 2. Pure DCF approach can induce wrong decisions 3. Always try to incorporate optionality in your investment decisions 4. Difficulty: Underlying parameters 19 Option to Expand Exercise MARK I TASK INSTRUCTION Investing in the Mark I is a negative NPV project, but it includes a call option on the Mark II Investment decision on the Mark II must be made in three years (maturity) Investment required in three years is USD 900m (exercise price) The underlying's value today is USD 467m (Mark II) Future value of Mark II is highly uncertain, assumption is that it would move with a standard deviation of 35%, risk-free interest rate is 5% The option to invest in the Mark II is a three-year call option on an underlying worth USD 467m with a strike price of USD 900m Feed this information into the Black-Scholes formula and obtain the option value (note that the NPV of the Mark II project is negative today!) The option to invest in the Mark II is out of the money. The option is valuable because the upside potential is big 20 Option to Expand Exercise (continued) MARK I CALCULATION Using the B/S-Formula, Mark II has a value of: What happens if the investment decision must be made in 5 years? What happens if the uncertainty (standard deviation) increases to 50%? What happens if the investment generates a cash inflow of USD 100m in one year? Negative NPV projects can be worthwhile if you get the option of a follow-up project, which is worth more than the initial NPV 21 Option to Wait Exercise COMPANY X NPV APPROACH Company X is thinking about buying a chain of computer shops and operate them for one year. The purchase price is USD 25m. Depending on how successful the computer sales will be, the cash flows at the end of the year can either be USD 32m with a probability of 0.8 or USD 22m with a probability of 0.2. Assuming a cost of capital of 5%, should Company X do this project? First, we calculate the NPV of this project and decide whether it is a good project or not: 0.8∗32+0.2∗22 NPV = −25 + = 3.571 1.05 Company X should buy the chain of computer shops 22 Option to Wait Exercise (continued) COMPANY X DECISION TREE Company X now learns about the potential to conduct a field study. The study would allow Company X to find out if the computers would be received favorably on the market. The field study, however, is going to take one year , which will delay the purchase of the shops and the cash flows by one year. Should Company X invest now, invest in one year, or not invest at all? Setting up the decision tree: NPV1 = −25 + 32 = 5.476 Invest 1.05 5.476 NPV1 = 0 NPV 0.8 Not invest today = 22 Invest NPV1 = −25 + = −4.048 1.05 4.172 0.2 0 NPV1 = 0 Not invest 23 Option to Wait Exercise (continued) COMPANY X INSIGHTS 1. The option value is equal to 4.17 − 3.57 = 𝟎. 𝟔 2. Benefit of the field study (avoid loss): If we invest today and consumers do not like the computers, we make an NPV loss of – 4.048 at t=0. The field study helps to avoid this loss. Expected benefit = 0.2 × 4.048 = 0.81 3. Cost of the field study (delay of cash flows): 32 If we invest today and consumers like the computers, we earn a positive NPV at t=0 of −25 + = 5.476. If we invest at t=1, the NPV at 1.05 5.476 t=0 is 1.05 5.476 Expected cost = 0.8 × (5.476 − ) = 0.21 1.05 4. Net benefit = 0.81 − 0.21 = 𝟎. 𝟔 5. Note that 0.6 is also the maximum amount Company X should be willing to pay for the field study! 24 Option to Wait Exercise (continued) COMPANY X INSIGHTS Assume, the two states differ in a more extreme way and the shops' cash flows are either USD 35m or only USD 10m. How much is the option worth now? Invest 35 NPV1 = −25 + = 8.333 1.05 8.333 0.8 NPV1 = 0 Not invest 6.349 Invest 10 NPV1 = −25 + = −15.476 0.2 1.05 0 Not invest NPV1 = 0 1. Note that the NPV if we invest at t=0 is still 3.5714 (mean is unchanged!) 2. Option value = 6.349 − 3.571 = 2.778 → Why is it larger? 25 Option to Wait Exercise (continued) COMPANY X INSIGHTS 1. The NPV from investing today is positive, yet it is an optimal strategy to not invest because the option to wait has a positive value 2. The NPV from investing today can be interpreted as if we disregard the field study 3. An investment opportunity with an option to wait can be compared to a call option in which the exercise price is the investment, and the underlying is the value of the project → If we invest, we exercise the option (and forgo the remaining option value) 4. The option to postpone an investment is more valuable the higher the volatility of the cash flows, and it is less valuable the higher the early expected cash-flows from the project 5. Note the similarity to American call options If the underlying pays no dividends, the option is worth more alive than exercised and should never be exercised early If the underlying does pay dividends, these payments reduce the ex-dividend price and possible payoffs at maturity, so early exercise might be rational Dividends do not always prompt early exercise but if they are large enough, call option holders capture them by exercising before the ex- dividend date 26 Option to Wait and Risk-Neutral Pricing (BMA, Malted Herring) Exercise BMA, MALTED HERRING TASK INSTRUCTION Investment now or in one year, investment cost of 180 The project's value (= discounted future cash flows) is 200 now and either 160 or 250 in one year from now, depending on how the market for the product develops → If the market is positive, revenues will be 25, if the market is negative, revenues will be 16 The NPV of the project at time 0 is 200−180=20 250 (𝟐𝟓𝟎 − 𝟏𝟖𝟎 = 𝟕𝟎) 200 𝑁𝑃𝑉 = 20 (?) 160 (𝟎) What is the value of an option that pays out either 70 or 0 in one period from now? 27 Option to Wait and Risk-Neutral Pricing Exercise (continued) BMA, MALTED HERRING CALCULATIONS Risk-neutral pricing (assuming a risk-free rate of 5%): 50+25 Return in the good state of the world = = 37.5% 200 −40+16 Return in the bad state of the world = = −12% 200 Risk-neutral probability of an up movement: 𝑟𝑓 − 𝑑𝑜𝑤𝑛𝑠𝑖𝑑𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 0.05 − (−0.12) 𝑃𝑢𝑝 = = = 0.343 𝑢𝑝𝑠𝑖𝑑𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 − 𝑑𝑜𝑤𝑛𝑠𝑖𝑑𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 0.375 − (−0.12) Value of a call option that pays out either 0 or 70 in one year from now: 0.343×70+0.657×0 = 22.9 1.05 The option is worth USD 20m if exercised immediately and USD 22.9m if exercised in one year Wait and then invest next year only if demand turns out to be high 28 xxx Discussion Derive what happens if the revenues are 50 when the market is positive (note that everything else is fixed, so the project’s value today must still be 200). Please also discuss the intuition behind your solution. 29 Examination Task We can use risk-free interest rates and probabilities to price a real option because a) the price of an option does not depend on the risk preferences but on a no arbitrage argument b) this approach gives us a lower bound for the value of the option c) this approach gives us an upper bound for the value of the option d) the discount rate changes at each node in the tree e) the uncertainty is resolved at each node in the tree 30 Key Take-Aways REAL OPTIONS ARE INHERENT IN MANY BUSINESS DECISIONS 1 STARTUP FIRMS OFTEN REPRESENT A REAL OPTION, HENCE VALUATED THROUGH REAL OPTION VALUATION 2 RECOGNIZING VALUABLE REAL OPTIONS IS CRUCIAL FOR MAKING DECISIONS THAT INCREASE VALUE 3 TRADITIONAL METHOD IGNORES REAL OPTIONS, POTENTIALLY LEADING TO POOR INVESTMENT DECISIONS 4 31 Institute of Accounting, Control and Audit University of St.Gallen (ACA-HSG) Prof. Dr. Marc Arnold Tigerbergstrasse 9 CH-9000 St.Gallen [email protected] Tel.: +41 71 224 74 13 www.aca.unisg.ch