Motivating example: MCI

Transcription

1 Real Options - intro Real options concerns using option pricing like thinking in situations where one looks at investments in real assets. This is really a matter of creative thinking, playing the game of spot the option. Well known examples are The deferral option, the option to delay a project before starting it. an american call option The option to abandon projects an american put option When specified carefully, can use standard option theory to price the options, and calculate correct project present values.

2 Motivating example: MCI

3 On 11/10/97, MCI s management announced an agreement with WorldCom. Roughly, MCI shares would be exchanged for the equivalent of $51 in WorldCom stock. At the time, the deal was not a sure thing. The justice department might balk at the monopolizing. It would take about five months for the takeover to be finalized. Today s price of MCI stock is P now = The five-month T-bill rate was roughly 5% (p.a.). Before any of the takeover news hit the market, MCI was selling for $30 a share. Infer the state price probability of the deal going through.

4 We here have two possible states. In April, the price of MCI, P april is either $51 (the up state) if the deal goes through, or $30, the price the company had before the merger rumors, if everything fails

5 Given these two possible values next period, we find the possible call values. max(51 40) = max(30 40, 0) What is the implied state price probabilities here? The two-state case, where the following outcomes were possible

6 If the November price of MCI stock is $41.5, the state price probabilities implicit in the market s pricing of MCI stock is found as: P now = 1 (1 + r) 5/12 E [P april ] = /12 (pu 51 + (1 p u )30). Solving for p u : p u = As we see, using observed prices in the market, it is possible to get the market s implied probabilities. We can imply that the market charges a price of $ for an option that pays $1 in the up state (and zero otherwise). Therefore, the corresponding state-price probability, p u, equals: p u = (1.05) 5/12 =

7 Use the inferred probability to find the price of a a 4/98 call option on MCI with a $40 strike price.

8 As a matter of fact, at the close of 11/10 an MCI 4/98-$40 call was selling for 4 5/8. The discrepancy can (maybe) be explained by the fact that in addition to WorlCom, GTE had offered $40 cash per share of MCI, and there was a claim in the Wall Street Journal that GTE was considering an offer of $45. Let us plausibly posit that by January, we will know whether the Worldcom - MCI deal goes through. If the deal fails, there is still GTE, who is prepared to offer $40 $45 a share. As a reasonable estimate of the final bid from GTE, let us take the midpoint of the GTE offers, $42.5. So, by April, we have the following three possible outcomes: $51, $42.5 or $30. The first occurs for sure if it is known by January that the WorldCom takeover gets an OK. The second or third outcomes are assumed to occur randomly if the deal is called off by January.

9 Graphically, this is what we have in mind: 51 Pjanuary OK 41.5 P notok january

10 In addition to knowing the November price of MCI stock, $41 1/2, we also know the price of a January MCI call option with strike $42 1/2. This option closed on 11/10 at $1 3/4. Can we use this information to confirm the correctness of the price for the 4/98 option? At the same time, show that the current value of MCI stock is consistent with your numbers. What was the markets implied probability of the MCI-WorldCom deal going through?

11 Using this price we can find the current call price C t as C t = 1 (1.05) 5/12 E [max (P april 40, 0)] = 1 (1.05) 5/12 (pu (51 40) + (1 p u )0) = (1.05) 5/12 = 6.34.

12 As a matter of fact, at the close of 11/10 an MCI 4/98-$40 call was selling for 4 5/8. This is below the price we just estimated. Maybe we should be buying these cheap options? Before we conclude that this is a profit opportunity, we must at least consider the possibility that we have made a mistake in our pricing above. This is the complication added by the GTE offer. Let us show how we can modify the analysis to account for this. Let us plausibly posit that by January, we will know whether the deal goes through. If the deal fails, there is still GTE, who is prepared to offer $40 $45 a share. As a reasonable estimate of the final bid from GTE, let us take the midpoint of the GTE offers, $42.5. So, by April, we have the following three possible outcomes: $51, $42.5 or $30. The first occurs for sure if it is known by January that the WorldCom takeover gets an OK. The second or third outcomes are assumed to occur randomly if the deal is called off by January.

13 The usual method for attacking a problem like this is to work backward. Based on the outcomes at the terminal nodes (here April), we first find prices at time 1 (here January), and then use the time 1 prices to find the time 0 prices (now). Let us start with the case where the deal goes through, it is announced in January that the WorldCom deal will be finalized. In that case, the payoff in April is a sure thing, and we find the January price C OK january as C OK january = 1 (1 + r) 3/12 (51 40) = /12 11 =

14 If the deal does not go through, we are left with two possible outcomes at time 2, and we want to find the price C notok january. C notok january = 1 ( ( ) ) (1 + r) 3/12 p d,u january ( ) + 1 p d,u january 0, where p d,u january denotes the January state-price probability for the event that GTE takes over MCI for $42.5 per share.

15 Let us now show that market prices in November imply a $ contingent January price for a digital option that pays $1 when GTE manages to purchase MCI by April for $42 1/2. We use the recursion trick, and start with the most favorable January event: it is known that the WorldCom deal will be finalized. In that case the outcome in April is a sure thing ($51). Hence the January price of MCI, P OK january, is P OK january = 1 (1 + r) 3/12 51 = 1 51 = (1.05) 3/12 In the other January state when the WorldCom deal does not go through, the price, Pjanuary notok, would be P notok january = 1 ( ) (1 + r) 3/12 p d,u january (1 pd,u january )30

16 Going back in time, the November price of MCI should be: P now = 1 ( ) (1 + r) 2/12 p u Pjanuary OK + (1 p u )Pjanuary notok. Now consider the January option with strike 42 1/2. Using the general pricing formula its value, C is: 1 ( ( ) C = (1 + r) 2/12 p u max Pjanuary OK 42.5, 0 ( )) + (1 p u ) max Pjanuary notok 42.5, 0 = because, for sure: 1 (1 + r) 2/12 (pu ( ) + (1 p u )0), P notok january < 42.5.

17 We can thus imply p u from C = It equals Given p u = , we can imply p d,u january from P now = We get: p d,u january = Thus, we have shown that market prices in November did imply a $ contingent January price for a digital option that pays $1 when GTE manages to purchase MCI by April for $42 1/2. The digital option price is contingent upon the event that the WorldCom deal would break down. Hence, p d,u january = (1 + r)3/12 =

18 Using these prices, we find C notok january = 1 ( ( )0) = /12 Given the two possible prices at time 1 (January), we can work further backwards in time to find the time 0 (November) value of the April call: C = 1 ( ) (1 + r) 2/12 p u Cjanuary OK + (1 p u )Cjanuary notok = 1 (1 + r) 2/12 (pu (1 p u ) 1.95)

19 To find prices we need p u, the state-price probability for the January event that the WorldCom takeover goes through. We have already calculated this as p u = Given the price, we calculate the November value of the call option as C = 1 ( ( ) 1.95) = (1 + r) 2/12 That is below the market price of $4 5/8 ( $4.63), but the $0.71 difference is not necessarily sufficient to decide to write MCI April call options (one has to incorporate transaction costs and the cost of margin when writing options. In addition, the $4 5/8 price is a closing price, which could well be an ask price. The bid price relevant when one shorts may easily be 5% below the ask).

20 To illustrate the general principle that the state-price probabilities can be used to price any derivative, including the underlying security itself, we can verify the November price of MCI of $41 1/2. We will do so for the calculation where we account for the GTE offer (involving three possible states in April). We work backwards, and start with the most favorable January event: it is known that the WorldCom deal will be finalized. In that case, the January price of MCI, P OK january, is P OK january = 1 (1 + r) 3/12 51 = 1 51 = (1.05) 3/12 If the deal does not go through, one find the January price as Pjanuary notok 1 ( ( ) ) = (1 + r) 3/12 p d,u january p d,u january 30 1 = ( ( )30) 1.053/12 = 39.4.

21 Going back in time, the November price of MCI is calculated as P now = 1 ( ) (1 + r) 2/12 p u Pjanuary OK + (1 p u )Pjanuary notok = 1 (1 + r) 2/12 (pu (1 p u )39.4) = 1 ( ( )39.4) 1.052/12 = 41.5 This confirms the market price.

22 What is the imputed market probability for the WorldCom MCI deal going through? In the three-state analysis, we computed the state price probability of the up state (when WorldCom announces by January that the takeover is OKed on all fronts) to be about 22%. Does that mean that the market thinks there is only a 22% probability that the WorldCom takeover will go through? Presumably not. The true probability is colored by the fact that investors will be very rich if that state occurs. Why? The WorldCom takeover is most likely to succeed if the WorldCom stock price is high, which would only occur when there is a general increase in stock prices. Because WorldCom takes over MCI only in a state when everybody is rich anyway, the corresponding state price probability is lower than the true probability. The chance of WorldCom taking over MCI may be as high as 1/3. On 11/10/97, a one in three chance looked awfully low when compared with the stories in the newspapers. But that is what the market was signaling!

23 Real options Two characteristics Assets derive their value from the value of other assets Cash flows are contingent on the occurrence of specific events These are option-like characteristics Applying present value of expected cash flows, systematically underestimates the value of such assets.

24 Call Call option, right to buy Buying a call option. Profit 0 K S T

25 Put Put option, right to sell Buying a put option. Profit 0 K S T

26 Determinants of option value Call Put Current value of underlying + - Variance of underlying + + Dividends/payout from the underlying - + Strike price of the option - + Time to expiration + + Risk free interest rate + -

27 Option pricing models Binomial model: Replicating portfolio argument. Determinants of option value: Current price of underlying how it moves Continous time limit - Black Scholes Extension of option pricing not just simple puts and calls Caps, barrier options Compount options Rainbow option (two or more sources of uncertainty)

28 Option-pricing models: Black Scholes easier to apply, but restrictive assumptions. Binomial option pricing model: Allow for early exercise, typical for real options Allow for modelling underlying uncertainty more generally How to implement? use variance input of Black Scholes

29 The option of delaying a project Traditional investment analysis: Only accept project with higher returns than the hurdle rate. Problems: Do not consider the option built into many investment projects: Wait one period and then redo investment evaluation. Examples: Undeveloped land (real estate investor) Patent (exclusive right to develop) Natural resource company undeveloped reserves.

30 The option of delaying a project ctd Payoff of option to delay X : Initial investment. V : Value of project (NPV of future cashflows) This value is random, affected by uncertanty which will be resolved in the future. NPV decision for the firm V X Negative NPV Positive NPV

31 The option of delaying a project ctd Inputs: Value of underlying asset PV of future cashflows starting project now. Variance in the value of the asset Exercise price of the option initial investment cost Expiration of option Riskless rate (matching maturity) Cost of delay like dividend yield in option price Cost of delay = Cash flows next period/present value now If cashflows flow proportianally, 1/no years in project If these inputs are present: Charge ahead, price option

32 The calculated option price some issues Necessary assumptions to replicate the option (when pricing) trading possibilities of underlying asset. This is problematic for the kind of investments we are looking at with real option models Does this invalidate the approach? Well, think about it this way: This is how the markets should work if they applies this is still relevant as a benchmark value.

33 problems in valuing the option to delay: How well is the project approximated by the typical assumptions used in option pricing. (continous process etc) Implications in valuing the option to delay. NPV may be negative now, but may become positive. NPV may already be positive, but may still want to delay Added uncertainty may make options-like projects more valuable.

34 You are interested in aquiring the exclusive rights to market a new product that will make it easier for people to access their on the road. If you do aquire the rights to the product, you estimate it will cost you $50 mill up front to set up the infrastructure needed to provide the service. Based on your current projections, you believe that the service will generate only $10 mill in after-tax cashflows each year. In addition, you expect to operate without serious competition for the next five years. What is the present value of the project, assuming an interest rate of 15%?

35 The biggest source of uncertainty about this project is the number of people who will be interested in the product. While current market tests indicate that you will capture a relatively small number of business travellers as your customers, they also indicate the possibility that the potential market could be much larger. In fact, a simulation of the projects cash flows yield a standard deviation of 42% in the present value of the cash flows, Value this project as an option. The current risk free interest rate is 5%. Instead of using the Black Scholes formula, calculate the option price using a binomial framework, using one year for each step in the binomial tree, allowing for early exercise.

36 NPV of current project t C t NPV = 50+ ( ) ( ) ( ) ( ) 4 + =

37 Inputs to a dividend-adjusted Black Scholes calculation c = Se q(t t) N(d 1 ) Ke r(t t) N(d 2 ) d 1 = ln ( ) S K + (r q σ2 )(T t) σ T t d 2 = d 1 σ T t Value of underlying asset: (S): PV of cash flow if starting production now = Strike price initial investment needed to introduce the product 50 mill. Standard deviation of underlying 0.42 Time to expiration: 5 years Dividend yield (q): Each period loose 1 5 of the cashflows from the investment.

38 C BS (S = 33.5, K = 50, r = 0.05, q = 0.2, σ = 0.42, (T t) = 5) d 1 = N(d 1 ) = d 2 = N(d 2 ) = C BS = 1.018

39 Binomial calculation t = (T t)/n u = e σ t d = 1 u p u = e(r y) t d u d p d = 1 p u For example, the last node C 0 = e r t (p u C u + (1 p u )C d u=1.522, d=0.657

40 Price of underlying step: nodes:

41 step node... 5 S n C call value = 1.86

42 Valuing a patent Right to market and develop a product. ie. option Valuing a firm with many patents Sum of exercised options (value of commercial prediction) options (patents) currently alive potential for developing more patents (cost of developing new patents)

43 Natural Resource Options Undeveloped Reserves as options Initial value of starting production X Present value of production schedule starting now V Value max (X,V) Inputs for valuation: Available quantities of the resource, estimated value if extracted today.

44 The options to expand and abandon Similar to the investment timing option at future date, pay a cost (investment) Two step procedure First step: Low/negative NPV project Second step: Possibility to expand depending on uncertainty resolved during the first project. The second project provides the option, what one needs inputs to.

45 Valuing equity in distressed firms In distressed firms, equity is an option held by equityholders, to pay off debtholders and assume control of the firm. Value of firm Face value debt Value of equity This framework can be used to value the firm directly.

46 Exercise You are valuing the equity in a firm whose assets are currently valued at 100 million. The standard deviation in this asset value is 40%. The face value of debt is 80 million (it is zero coupon debt with 10 years left to maturity). The Treasure 10 year bond rate is 10%. Use this information to value the equity and the debt of the firm. What is the implicit interest rate on the debt? Suppose the value of the firm falls to 50. What is then the value of the equity in the firm? What about the implicit interest rate on the firm s debt?

47 The inputs to the Black Scholes model to value this equity S = 100, K = 80, (T t) = 10 years, r = 0.10, σ = 0.4. The continously compounded interest you need in the Black Scholes calculation: r = ln( ) = 0.095% C BS (S = 100, K = 80, r = , σ = 0.4, (T t) = 10) d 1 = N(d 1 ) = d 2 = N(d 2 ) = C BS = E 75.05

48 E This is the value of equity. The value of the bond is the difference between this value and 100, B = = The implicit interest rate: = 100 (1 + r) 10 r = 14.89%

49 If the value of the firm falls to 50, equity is valued at C BS (S = 50, K = 80, r = , σ = 0.4, (T t) = 10) d 1 = N(d 1 ) = d 2 = N(d 2 ) = C BS = The bonds are then valued at B = = which implies an interest rate of B = = r = 17.4% 100 (1 + r) 1 0

50 Implications of viewing equity as an option: equity may have value even if value of the firm is currently below bond promised payment. Note also that this can be used to figure out properties of the bonds of the firm, such as implicit probabilities interest rates, or alternatively, implicit probabilities of default.

51 Another perspective on real options

52 A call option is the right to pay a strike price to receive the present value of a stream of future cash flows (represented by the price of the underlying asset). An investment project is the right to pay an investment cost to receive the present value of a stream of future cash flows (represented by the present value of the project Note the similarities: Investment project Call option Investment cost Strike price Present value of project Price of underlying asset

53 The correct use of NPV 1. Compute NPV by discounting expected cash flows at the opportunity cost of capital. 2. Accept a project if and only if its NPV is positive and it exceeds the NPV of all mutually exclusive alternative projects

54 Option-like features in investment decisions. Decision of whether and when to invest. [Call option] Ability to shut down, restart, abandon projects. [Put option] Ability to be flexible about choice of inputs, outputs, production technologies [Flexibility options] Ability to invest in projects that may give rise to new options. [Strategic options]

55 Suppose you own a tract of land, and you find oil in the ground. The current oil price is 15/barrel. Suppose this is a small find, with a small number of barrels of oil. What is the nature of the decision problem? Specifically, suppose that the find is one barrel of oil. This barrel can be extracted by paying X = The effective annual risk free rate is r = 5%. The oil forward curve is such that the effective annual lease rate, δ, is 4% (constant over time). What can you currently sell the land for? What uncertainties will affect the decision problem? Suppose this is a large find, you expect to be able to extract oil from it in the indefinite future. What is the nature of the decision problem?

56 For a small find, the decision problem is when to pay the production cost, extract the oil, and sell it. This will depend on the spot price of oil. The higher the oil price, the better.

57 Take the specific case. Here there is no uncertainty, the forward price for oil specification ( ) 1 + r T F 0,T = S δ implies that the spot rate grows at = %. If we were to extract the oil immediately, get = To find the correct answer we don t need to solve an option problem, this is a case of deterministic optimization: Choose T to maximize NPV = S T K (1+r) T. The solution: Defer extraction as long as the lease rate (loss) of 4% is less than the cost savings on delaying one more year.

58 Let r d and δ d be daily rates ( 1 S 1 + r ) d K > S K 1 + r d 1 + δ d I e defer investment as long as S K < rd 1+r d δ d 1+r d Want to invest when S = X r d/(1 + r d ) δ d /(1 + r d ) = i.e. wait until the oil price has reached (from 15) which is to wait for t = years.

59 How much to pay for the land? ( ) NPV = =

60 What is the option equivalent: Deciding when to exercise a perpetual American call What are the uncertainties we have to think about? Cost of extraction may grow exercise earlier, benefit from waiting is less. For example, if extraction costs grow at 0.5% p.a. exercise (ie extract) after 1.32 years. Price of oil is uncertain. Intuitively, what changes: With uncertainty, option part of waiting becomes greater More time to wait for the oil price to get larger. Will always wait longer. Easiest seen in terms of a trigger price Can be found using the formula for a perpetual call Increasing volatility, higher option value

61 Suppose we have an infinite number of barrels in the ground. What is the nature of the decision problem? When to start producing? Should you stop producing once started? Should you restart production if stopped? Main determinant: Initial cost of developing Cost of shutting down Cost of restarting Oil price

62 Structure of the solution Three oil prices initial investment trigger Production starts when oil price hits barrier Shutdown trigger Production is stopped if the oil price goes below this barrier. Restart trigger Production is started again if oil price goes above this

63 ial estment ger start ger utdown ger Start Stop Restart

64 ial estment ger start ger utdown ger Start Stop Restart

65 A project costs 100 to initiate. The project produces an infinite stream of cashflows starting 1 year after investment. The cashflows are expected to increase by 3% thereafter. The risk free rate is 7%, the project beta is 1.33 and the market risk premium is 6%. 1. Suppose the project cashflow next year is expected to be 18. Value the project. 2. Suppose the project does not have to be initiated immediately, it can be delayed for one or two years, but if it is not initiated by year 2 it is no longer possible to start it. Also suppose that there is uncertainty about the initial cash flows from the project, they are lognormally distributed with volatility 50%. Value the project.

66 Static NPV calculation PV = E[C 1] r g = 18 r g Use CAPM to find cost of capital: r = r f + β(e[r m ] r f ) = = 15% PV = = = 150 NPV = = 50

67 Uncertainty about first year s cashflow. What is different? Learning about possible cash flow valuable, but loose one or two years of cash flows. Using the usual CRR binomial approximation E[CF 2 ] = ue[cf 1 ] = e σ E[CF 1 ] E[CF 1 ] E[CF 2 ] = de[cf 1 ] = e σ E[CF 1 ]

68 Plugging in numbers for the next two periods

69 With this info, can calulate tree of project values if project is started at any point in the tree = = = = = = How to value this

70 Natural: Binomial option framework But: be aware that this is not a case where the project is traded. We calculate the fair price, what another security with the same cash flows will be worth. Risk neutral probability, need to adjust for dividend yield y: q = er y d u d Why: Forfeit one year of project cashflows, or 18. Inn annual percentage terms, forfeit = 12% Translate to continously compounded interest: r = ln(1 + r f ) = ln( ) = 6.776% y = ln( ) = 11.33% q = er y d u d = e = 0.335

71 Will then work our way backwards in the tree using the terminal values V uu = = V ud = = 50 V dd = 0 Now consider period 1. Take upstate first. If we don t start the project, have a project worth V u = e r (qv uu +(1 q)v ud ) = e ( ( )50)

72 Here it is better to start the project immediately, valued at = In the down state V d = e r (qv ud +(1 q)v dd ) = e ( ( )0) = V

73 V At time 0, exercising has a value of 50. What is the value of keeping the project alive V 0 = e r (qv u +(1 q)v d ) = e ( ( )15.65 The option should be kept alive Increase in value relative to static NPV: = 5.80.

74 Consider a gold mine with an estimated inventory of one million ounces and a capacity output rate of 30,000 ounces per year. The price of gold is expected to grow 3% a year. The firm owns the right to this mine for the next 20 years. The cost of opening the mine is $100 million, and the average production cost is $250 per ounce. Once initiated, the production cost is expected to grow 5% a year. The standard deviation in gold price is 20%, and the current price of gold is $375 per ounce. The riskless rate is 6%.

75 The inputs to the model are as follows 1. Value of the underlying asset: Present value of expected gold sales (50,000 ounces a year) ) ( ) (50, ) ( , = mill mill = 47.24mill 2. Exercise price: Cost of opening mine: 100 mill. 3. Variance in ln( gold price ) = Time to expiration of the option: 20 years. 5. Riskless interest rate: 6% 6. Dividend yield = Loss in production for each year of delay 1 20 = 5%.

76 Now, this is an option which can either be valued as an European call option, in which case we will use the Black Scholes formula, and calculate the Black Scholes value of the call: 3.19 million. However, this is more realistically an american option which can be exercised early, and hence the more correct price is calculated using an binomial approximation to an American call: Value = 3.70 million

77 Summary Real Options Analysis: Using tools from option pricing in real investment decision. Look for: Contingent choice Ask: What is the observable contractible important exogenous factor relevant for profitability of investment project