Real Options - Part 1 (Ch. 22)

INTRO

·        In Chapter 10, we used the decision tree approach to compute a project’s NPV taking into account the value of the real options contained in a project.

·        That’s better than ignoring the real options, but it’s only an approximate technique.

·        To properly account for the value of real options, we have to use a proper option pricing formula.

·        There are several different approaches to correctly valuing a call option.  The book discusses several of them in Chapter 21.  And then in Chapter 22, they use a different approach for each example (Black-Scholes formula in 22.1, risk-neutral valuation in 22.2 and binomial trees in 22.3)

·        We’ll stick with what we learned in Fin 300, namely the Black-Scholes formula, and apply that to all three examples

REVIEW OF BLACK-SCHOLES FORMULA

Caution: book’s version of the formula is wrong.  The correct formula is what we learned in Fin 300:

C =  P₀ N(d₁) - X e^-RT N(d₂) 

d₁ = { ln (P₀/X) + (R + 0.5 s²)* T}/{s Ö T } 

d₂ = d₁ - s Ö T  

(The formula in the book omits the R in the numerator of d₁)

C = value of call option

P₀ = value today of the underlying asset (for a stock option, this would be stock price)

X = Exercise price of the option

R = annual risk free rate over the life of the option 

T = time to maturity (in years)

s = standard deviation of annual returns on the underlying asset

·        X e^-RT is just the PV of the exercise price (discounted from the maturity date to today, assuming continuous compounding)

·        N(d₁) is the probability that a standard normal variable (Z) will have a value less than or equal to d₁.  In other words, the function N(.) is the cumulative density function for a standard normal variable.

·        Once you compute d₁ and d₂, you look up N(d₁) and N(d₂) using standard Z-tables.  These will be passed out in class (to have and to hold), and at final exam (to use and return).

Example:

Amazon’s shares are selling for $27.  A call option with an exercise price of $25 will expire in 80 days.   The yield on a 80-day Treasury bill is 1.15%.  The standard deviation of annual returns on Amazon's stock is 44%. Compute the premium for this call option.

T = 80/365 = .21918 years

d₁ =  { ln (27/25) + (.0115 + 0.5*[.44²])*.21918}/{.44 Ö .21918}

    = (.07696 + .1083*.21918)/(.20599) = 0.48885

N(d₁) = .68792

d₂ = d₁ - s Ö T    =  0.48885 - .44 Ö.21918 =  0.48885 - .20599 = 0.2829

N(d₂) = 0.61026

C =  (27 * .68792) – (25 * e^{-.0115*.21918} * .61026) = 18.574 - 15.218 = $3.36

THE OPTION TO MAKE FOLLOW-UP INVESTMENTS

·        The scenario is that if you enter the market today, you will have the opportunity to expand by making follow-up investments.  However, if you stay out of the market today, it will be too costly to enter the market later, once competitors get firmly established.

·        In such a situation, you may want to enter the market today, even if today’s project, by itself, has a negative NPV.  You don’t look just at the NPV of today’s project.  You look at the NPV of today’s project plus the value of the option to make follow-up investments.

·        Even when the NPV of today’s project is negative and the expected NPV today of tomorrow’s project is negative, the value of the option can still be positive, and make the investment worthwhile.

·        Example #1: Blitzen Computers can enter the contact lens computer market today with a Mark 1 model.  Investment would be $450 million, required return is 20% per year,  the riskfree rate is 10% per year, and the PV of future cashflows is $403.55 million.

=> The NPV of the Mark 1 project, by itself, is -$46.45 million

·        Entering the market today would position them to make a follow-up investment in three years (the Mark 2 project).  The investment at time 3 in this project will be $900 million.  The best estimate Blitzen has today is that the time 3 PV of the project’s cashflows will be $807.1 million.

=> The Mark 2 project involves double the investment, and double the expected PV.  Today’s estimate of the Mark 2 project’s future NPV is -$92.90 million.

·        However, the expected time 3 PV is the result of considerable uncertainty about the future of contact lens computers.  There’s a 50% chance the product will take off, and time 3 PV will be $1,092.1 million ($285 million higher than expected).  There’s a 50% chance the product will not do too well, and time 3 PV will be $522.1 million ($285 million less than expected).  These numbers correspond to an annual standard deviation of 35%. 

·        This means that with a probability of 0.5, the time 3 NPV of the Mark 2 project will turn out to be 1,092.1 – 900 = $192.1 million.  With a probability of 0.5, the time 3 NPV of the Mark 2 project will turn out to be 522.1 – 900 = -$377.90 million.

·        What is relevant at time 3 is not today’s expected NPV of -$92.90 million, but the actual NPV of $192.1 million or - $377.90 million.  If the actual NPV turns out to be positive, then they will go ahead and make the follow-up investment.  If the actual NPV turns out to be negative, they will not make the follow-up investment. 

·        From our earlier discussion of decision trees, we can recognize that the opportunity to make this future investment is valuable (even though the expected NPV today of the time 3 investment is negative)

·        Before we turn to the option-pricing-computation, let’s use the decision tree approach.  (Among other things, we’ll be able to see the size of the approximation error when we use decision trees in place of the option pricing approach.)

The decision tree will look as follows:

                                INVEST IN      time 3 NPV = 192.10

                                                                                 MARK 2

                                    MARK 2 PV HIGH 

                                         prob 0.5                                         

                              DON'T INVEST  time 3 NPV = 0

                                                                           IN MARK 2  

            INVEST IN          

              MARK 1                                                    INVEST IN        time 3 NPV < 0

                                                                                  MARK 2

                                           MARK 2 PV LOW                                            

                                                prob 0.5

                              DON'T INVEST  time 3 NPV = 0     

                                                                                IN MARK 2  

The Mark 2 project will only be taken if the PV turns out to be high.

The expected time 3 value from the Mark 2 project is then 0.5*192.1 = 96.05 million

The total time 0 NPV of the project = time 0 NPV of the Mark 1 project + the time 0 PV of the 96.05 million expected at time3 from the Mark 2 project = -46.45 + 96.05/(1.2^3) = -46.45 + 55.58 = $9.13 million

·        Now consider the option pricing approach.  The total time 0 NPV of the project = time 0 NPV of the Mark 1 project + the time 0 value of the option to invest at time 3 in the Mark 2 project

·        The Mark 2 project is an option to spend $900 million at time 3 to “buy” the project’s future cashflows.  If the time 3 PV turns out to be high, Blitzen will exercise the option and invest in the project.  If the time 3 PV turns out to be low, Blitzen will just let the option expire without exercising it.

o       Maturity = 3 years

o       Annual standard deviation = 35%

o       R_f = 10% per year

o       X = $900m

o       P₀  = the value today (three years before maturity) of the project

= PV today of the expected time 3 value of the cashflows

                = 807.1/(1.2^3) = $467.07 million

d₁ = { ln (P₀/X) + (R + 0.5 s²)* T}/{s Ö T } 

    = { ln (467.07/900) + (.10 + .5*.35^2)*3}/{0.35*3^0.5} = -0.284

d₂ = d₁ - s Ö T  = -0.284 - 0.35*3^0.5 = -0.890

N(d₁) = 0.38974

N(d₂) = 0.18673

C =  P₀ N(d₁) - X e^-RT N(d₂)  = 467.07*0.38974 -  (900 * e^-.1*3 *0.18673)

   = 182.0365 – 124.4997 = 57.54

Total NPV of project = -46.45 + 57.54 = 11.09 million

 (Decision tree approach gave $9.13.  In this instance, the decision tree NPV was an underestimate.  It

 doesn't always work like that.)

·        Example #2 (in-class exercise): 

      Your company can enter a new market today with a first generation product.  The project requires an investment of $200 million and has a required return of 14% per year.  It will generate cashflows that will start next year with $16.25 million and grow at 5% per year forever.  The riskfree rate is 8% per year.

·        Entering the market today would position you to make a follow-up investment in two years (with a second-generation product).  The time 2 investment in this project will be $500 million.  It will have the same required return as the first generation project.  Cashflows are expected to start at time 3 with $38 million, and grow at an expected rate of 6% per year forever.

·        However, there is considerable uncertainty about the time 2 PV of the cashflows from the second-generation project.  With a probability of 0.5 the PV could be $155 million more than expected, and with a probability of 0.5 the PV could be $155 million less than expected.  These numbers correspond to an annual standard deviation of 30%. 

·        What is the project’s NPV using the decision tree approach?  What is the correct NPV, using the Black-Scholes formula?

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