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Solutions: Fourth Group

Finance-Moeller

1. THIS EXAMPLE OF USING THE PROBABILITY THEORY METHOD TO FIND AN OPTION VALUE IS JUST A STYLIZED EXAMPLE. FOR THIS CLASS, DO NOT ATTEMPT TO VALUE OPTIONS THIS WAY UNLESS SPECIFICALLY ASKED.

a. Draw a decision tree with the relevant cash flows and information at each node.

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b. Using a discounted cash flow analysis, what is the value of just the investing in R&D phase?

NPV= -100 + (0.75*120 + 0.25*80)/1.1 = 0

c. Using a discounted cash flow analysis and assuming you must make the second investment, what is the NPV of this project?

NPV = -100 + [-100 + 0.75*120 + 0.25*80]/1.1 + [0.5625*144 + 0.375*96 + 0.0625*64]/1.1^2

NPV = 9.09

d. Now assume your company has the option to abandon the project at time 1 and not implement the new production method. Now what is the NPV using a discounted cash flow analysis.

Now you will choose to not make the second investment if you end up in the bad state of nature.

NPV = -100 + [0.75*(120-100)+0.25*80)]/1.1 + [0.75*(0.75*144+0.25*96)]/1.1^2

NPV = 13.64

e. Using the values above, what is the value of the option to abandon?

The option value is the difference between the NPV with the option (part d) and the NPV of just the first phase (part b). Option = 13.64-0 = 13.64

f. Using a binomial option pricing method, calculate the value of the option to abandon. Compare and contrast this value with the value from part (e).

This is a one period option that starts at time 0 and ends at time 1. You have the option to make the second investment. The u (1.2), d (0.8), r (0.1), t (1) and X (100) are relatively easy to find. The harder part is figuring what So should be. Since all of the cash flows are derived from the initial investment via u and d, then X equals So (100) in this case.

All of the cash inflows are So=100, Su=uSo=1.2*100=120, Sd=dSo=0.8*100=80.

The intrinsic value of the calls are Cu=120-100=20 and Cd=0 because you would not exercise if you have the right to buy the asset for 100 but it is currently worth 80.

NOTE: From this point forward, I will give the inputs into the binomial model but will not show each and every calculation in the Word document.

2. a. Assume it is a European option, what is the most you are willing to pay for this option?

The first thing you need to determine is whether the new technology represents a put or call option and whether you are long or short the option. Since you have the “right to buy” the technology, we can best categorize the real option as buying a call. Since we have an up and down state that are different (20% and 15%), we must use the binomial option pricing model to price this option. We already know the u and d are 1.2 and 0.85 (1+u and 1-d) and r is 5%. Next, we know it is a 4 year option and I choose to use a 4 period model to represent the future asset value because the variables given are all in annual terms. A good rule of thumb to use when finding the So and X for a real option is that the NPV of the project equals So-X. In this case, it is stated that the cost of the technology is $10 million (X) and today’s value of the cash inflows is $9 million. Now that we have identified all of the appropriate inputs to the binomial model, the rest is simply executing the option pricing model. Those computations are shown in the spreadsheet.

b. Assume it is an American option, what is the most you are willing to pay for this option?

To find the value of the American option, you need to pick the higher of either the call value or the exercise price at each node. For instance, at the second up state (Suu), the value of the technology is $12.96 and the value of the call (Cuu) is $4. If you would exercise at this node, the (intrinsic) value would be $2.96 (12.96-10). So at this node you would rather hold onto the call for $4 than early exercise for only $2.96. At each of the possible early exercise points (all intermediate nodes at periods 0, 1, 2 and 3), you would always choose to hold onto the call. So, the value of the American option is the same as the value of the European option.

3. a. Relative to the base case, i.e., change only one variable relative to the base case for each cell, fill in the following table.

| |Call |Put | |Call |Put |

|Base |17.64 |2.07 | | | |

|S=70 |36.09 |0.52 |S=30 |3.75 |8.18 |

|X=60 |8.19 |9.84 |X=20 |32.82 |0.04 |

|Std Dev=40% |23.51 |7.94 |Std Dev=5% |15.57 |0 |

|Rf=5% |20.14 |1.29 |Rf=0.01% |13.92 |3.9 |

|t=10 |23.07 |2.71 |t=0.01 |10.12 |0 |

Here are the computations for the base case call and put. The cumulative normal density functions are calculated on Excel by use the normsdist function.

b. The delta of the base case call is the first derivative of the option price with respect to the underlying asset value. In other words, if the value of the underlying asset goes up by $1, then the call moves by delta. Delta is found by estimating the standard normal cumulative distribution function for the computed value of d1, express as N(d1). The function in Excel to find this is normsdist(d1). These can also be easily found in cumulative normal distribution tables which can be found in the back of your textbook. From Excel, the normal distribution for the call is 0.855. So, if the value of the underlying asset goes up $1, the value of the call will increase by approximately, $0.855.

The delta of the base put is also the first derivative of the option price with respect to the underlying asset value. It is computed by finding the –N(-d1) which is -0.145. So, if the value of the underlying asset goes up $1, the value of the call will decrease by $0.145.

4. Essentially this project has three components. The first component is the base case NPV of the original $100 million investment and $90 million cash inflows where the NPV=-100+90=-10. The second component is the option to expand the investment anytime in the next three years which costs $45 million and increases cash inflows by 40%. This is most similar to buying an American call option. The third component is the option to liquidate the investment anytime in the next three years for $65 million. This is most similar to buying an American put option.

There are two ways to solve the value: Add up the value of the three components or use a binomial decision tree with the best possible cash flows (base NPV, expansion or liquidation) in each node. From a managerial perspective, the individual components method allows you to see the value of each of the three decisions which makes managing the process easier.

I’ll detail how to find the value via the three components and the alternate method is only shown on the spreadsheet.

The NPV of the project without the options is -10, -100+90.

The put option is the right to liquidate (sell) the company for $65 million. The European version of this option is easily computed by using a three period binomial with the follow values: So=90, X=65, u=1.65, d=1.65, r=5%. The risk neutral probability of an up move in this problem is 0.5385 (p).

5. Market and cash flow information: The risk free rate is 5%, the appropriate discount rate for the investment is 10%, assume that for each investment the last reported year’s cash flows continue forever with a growth rate of 2% and the standard deviation of the cash flows from the investment is 50%. NOTE: The FCF number excludes capital expenditures.

a) You begin to explain to them the concept of real options and they seem willing to listen. Draw and label a decision tree which characterizes the Indian investment opportunity.

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b) What is the value of the joint venture without the strategic issues?

The NPV of the project without strategic issues is just the NPV of the initial investment.

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c) What is the value of the joint venture with the strategic issues?

To value the JV with the strategic issues we need to add the three components: NPV of the JV, the option to open the sales office and the option to buy out the JV partner. There is one major wrinkle in the options. The buyout option is actually nested in the sales office option so you will value the buyout option as if you make a joint decision at time 1, to open the sales office and buy the buyout option. Both of these options are call options and we can use the Black Scholes model to value both of these options. For both options, the r is 5% and the standard deviation is 50%. The time to maturity of the sales office call option is one year and the buyout option is two years (t=1 to t=3). Now we need to estimate the So and X for both projects. First lets identify X for both options. Remember we want to find what is equivalent to the “initial capital spending” for the project. Looking at the Capex line for the Sales office it is clear that the initial investment is the 70 because Capex returns to a “maintenance” level in year 2 of 1. For the JV buyout the answer is still as obvious, it is clear the first two years of expenditures are unusually high and Capex returns to a “maintenance” level. Clearly we can not just add the 120 and 130 so need to recognize that the $130 is in different year dollars than the 120. So X=120+130/(1+r). That hard part is, what is r? Two choices, the appropriate risky discount rate or the risk free rate. Though I think both are viable choices, I am going to choose the risky rate, 10%, because that discounting is consistent with the DCF only solution (which also uses the discount rate). So X is $238.20 for the buyout option.

Now we need to find the beginning S for the two options. The buyout option begins in year 1, so I am going to first compute the cash inflows in year 3 (the first year of the project) then I’m going to bring those flows back to time 1 (my choice, I just want to see the NPV). Alternatively, you could immediately bring the flows back to time 1.

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Putting these X and S values into the BS model, along with the standard deviation of 0.5, t of 2 and Rf of 0.05, the call option value is $88.59.

Next we will compute the S of the sales office option which includes the nested buyout option. Since this option starts with the JV, this option begins at time 0 so we need So.

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Putting these X and S values into the BS model, along with the standard deviation of 0.5, t of 1 and Rf of 0.05, the call option value is $83.25.

So the overall value of the joint venture which includes the sale office call option and the nested call option of the JV buy out is $58.87 (-24.38+83.25).

d) You have an elevator ride to convince the CEO that the strategic issues should not have a value of zero. Be concise.

The option to open a sales office then eventually buy out the JV has real strategic value ($83.25 to be concise) because of the possible positive cash flows projected from these two actions. Whether we eventually choose to exercise them or not is irrelevant because the valuation method (real options) already takes this into account.

6. Note that the exercise decision (maturity, T) on parts a-c occur at time 1. (These images are from Campbell Harvey’s website and p1 represents p, the risk neutral probability).

a. You are renovating your plant and you need to make a final decision whether to add another production line.

The option to expand is a call option because you have the right to “buy” (invest) in the new production line.

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b. Your company produces personal computers. Because the speed of technology changes is unpredictable, you manage your inventory and production process so you can stop production quickly.

The option to abandon is a put option because you have the right to “sell” (stop) the production process.

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c. Instead of buying additional warehouse space you have leased so you can easily decrease your amount of storage.

By leasing the storage space you easily have the ability to contract your business so this is a put option. Again you basically have the right to “sell” a portion of your storage capacity.

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d. You have purchased a production process which is very flexible. It allows you to leave the plant open for successive years, shut down then reopen a year later or abandon the process all together.

This is a series of calls and puts. Assuming staying open is the base case, then you also have the option to close in the down state at time 1 (put option). Plus you have two additional options at time 2, either reopen (call option) or abandon (put option).

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7 & 8) We will cover these in class and a spreadsheet with the answers will be online.

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