In this module, we are going to discuss real options. These are options that are written on nonfinancial, more operational units. And they have optionality built into it. Meaning that we have the option of doing one particular set of operations, or another set of operations. And we want to use option theory to value an operation with built-in options. In this set of modules we're going to talk about how Option Theory can be used to evaluate non-financial investment and decision problems. Some examples of this include the valuation the lease in a gold mine. We're going to go through this in detail, valuation of an equipment upgrade option. Also in the context of gold mine, we'll go through a detailed example of how one could use Option Theory for calculating the value of such an option, valuation of a drug development process, this could include options such that you could sell your IP to another company, or options such that you it can increase the investment into a particular drug process and so on. Evaluation of a manufacturing firm that has the option of contracting its facilities to other manufacturers. The option here is to decide when to contract out your facilities and how much of it to contract out. Depending up on the market conditions you could do that. And in order to figure out what the value of such a manufacture, manufacturing firm is, one has to, accurately describe the value of such an option. And we know that in order to calculate the value of an option, we have to define an discounted expected value according to the risk neutral measure. So one can apply that technique to value. A manufacturing firm. We'll go through simple examples of an evaluation of a tolling contract at a power plant, evaluation of a gas storage facilities, and so on. The real options paradigm wants to view a project as having many sources of uncertainty. These could include market uncertainties such as price for the product, demand for the product, industry uncertainties such as mergers, mergers and acquisitions happening, innovations happening in the industry, technical uncertainties such as research and development, when a particular research breakthrough will happen, how will the development progress happen and so on? Organizational uncertainties, key personnel might leave the company and what should the company do to react to it? Political uncertainties meaning there are regulatory changes, there could be wars, there could be government changes and so on. The real options paradigm wants to manage these uncertainties by adding flexibility to a project or to a company via options. And project management in this paradigm, is all about managing flexibility. So we will explore some simple examples of real options in these modules and get back to this idea of how to introduce flexibility, how to value the flexibility, how to optimally execute this flexibility, and so on. The example we'll be using is called a Simplico Gold Mine Case. This is example 12.7 from Investment Science in Luenberger. The set of the problem is as follows. We want to evaluate the value of a ten year lease on a gold mine. The details are as follows. The price of gold, the current price of gold, is $400 per ounce, much lower than what the current prices are right now. Each year, the price increases by a factor U, which is equal to 1.2 with probability P equal to .75, or decreases by a factor D equals .9. With probably one minus B equal to 0.25. The cost of extracting gold is $200 per ounce. The maximum rate of gold extraction is 10,000 ounces per year. The interest rate R is 10%, so given the interest rate and given U and D above, we can compute the risk neutral probability Q and that turns out to be two thirds. When we're doing this risk neutral probability, we're implicitly assuming that we can buy and short sell gold. We're going to use a convention, that the cash flow, occurs at the end of the year. The revenue is the determined by the price at the beginning of the year but this revenue appears at the end of the year for bookkeeping purposes. So here are some details of the modelling. The current time, t, is equal to 0. The lease ends at t equals to 10. At the beginning of the 10th year we have to return the mine back to the owners. And therefore, the lease ends. The states in each year is given by the gold prices at the beginning of the year. Let vts denote the value in state s at the beginning of the year t. It's the value of the lease in state s at the beginning of the year t. Since the lease ends in ten years, we know that v ten s is equal to zero. Now we want to compute, vts as as a recursion. So here's my time t, I'm sitting in a particular state s, and this state s is determined by the price of gold in that particular year. From this state I can go to two possible states. U times S, or D times S. The probability of going to U times S is Q, the risk-neutral probability, and the risk-neutral probability of going to D time S is one minus Q. I already know the value here. So I know VT plus 1 US, because I am doing a recursion backwards and similarly I know VT plus 1 DS and I want to compute what is going to happen. I want to compute VTS. And the recursion is very simple. You take the revenue in year T. Plus the expected value that you would get, starting from time T plus 1 and onwards and discounted back by a factor 1 plus R. The reason the revenue is also being discounted by a factor 1 plus R is because we assume that the revenue occurs at the end of the year and therefore it has to be discounted back by the annual interest rate R. The revenue in ERT can be written very simply to be the maximum of s minus c and 0 times the maximum rate at which you can produce gold, which is g. What does this mean? If the current price of gold is greater than the cost of extracting gold, so s minus c is strictly positive, then you will extract gold at the maximum possible rate. If s minus c is equal to 0, or less than 0, then you will not operate the mine. So implicitly we are assuming that there is no cost for shutting down operations in any given year. We can shut it down in one year, and reopen the mine the next year, and we don't pay any cost for that. Since we're interested in pricing this lease, all the expectations with, must be with respect to the risk-neutral measure. Or the risk-neutral probability. And that's why we are highlighting the fact that here the expectation is with respect to the risk-neutral probability. Written out in detail this is Q times the value at the up state. That's 1 minus Q times the value of the down state. We know the value at time ten. We can compute it backwards using lattice to compute V zero of S, which is the value of the lease at time zero. In the next module, I'm going to work through this in an Excel spreadsheet and show you exactly how this works out. Now we want to take this example a little bit further. If you think about this example for a moment, it is actually a sequence of option. We don't do it in that way, we don't compute it explicitly in that manner. But it is a sequence of options. At the beginning of every year, I can look at the price of gold and decide I have the option to either run the gold mine, or not run the gold mine. So, in every year, I have the option of running the gold mine or not running the gold mine. And that option, has been implicitly calculated here. It's a maximum of S minus C and zero. I can choose which one I want to opt for at the beginning of every year. In the next example we will explicitly include another option and then try to value it. So, this is an example with an equipment enhancement option. The details of the equipment upgrade are as follows. The cost of the upgrade is $4 million. It's a one time fixed cost. If you upgrade the equipment, the new rate of production goes up to 12,500 ounces per year instead of 10,000 ounces per year. But at the same time. The cost of production goes up to $240 per ounce instead of $200 per ounce that had existed before. This upgrade is an option, in that it can be exercised at any time over the lease period, but once the upgrade is in place, it applies for all future years. When the lease ends they upgrade the equipment in the worst part for the miners, and the question is, what is the value of this upgrade option? So we're going to solve this in stages. First thing we're going to define, is a, value function with the in place. So V up will denote a value function. The value of the mine in state S at time T with the equipment upgrade already in place. We can compute this using the same computations as we did for VS except that now we have to uses these new parameters, G up and C up. In order to calculate what V happens. Now, this is the value that you would get if the upgrade was already in place. But now you have to pay for the upgrade, and how does one model it? The upgrade is an American option that pays V up S minus $4 million if exercised in state S on AT. We haven't upgraded equipment yet. We are sitting on a situation where the mine is operating, but the equipment upgrade has not happened. If I decide to upgrade, I'll move to a different mine operating conditions at that particular time. So, here's my states s if I decide to upgrade. My value that I will get will be v up ts now, in this particular state. But in order to upgrade I will have to pay 4 million dollars so you end up getting 4 million dollars as The cost that you pay for this upgrade. Once exercised, the equipment is upgraded and the mine is operating according to policy corresponding to v up t, meaning that if we decide to operate a mine in v up t, we operate the mine once the equipment is placed. If we decide in a particular [INAUDIBLE] the cost. Of extracting gold is too high, we don't operate the mine,and so on. so let UTS define the value instead as in [UNKNOWN] with an upgrade option. This time, the equipment upgrade has not happened, we only have the option of upgrading the equipment. So what do you do? In every state. And every time you have two actions available to you. Either you exercise the option or you don't exercise the option and you continue. If you exercise the option, the process stops. You move to the lattice that corresponds to the upgraded mine. You pay 4 million dollars for it and you manage the mine as if the equipment upgrade was already in place. So that's the exercising option. If you continue, then you have not upgraded your cost of producing gold still remains at 200. Your maximum rate at which you can produce gold still remains as 10,000. And this recursion is just as if you were continuing with old equipment. And the decision now is which of these two actions to choose, whether to upgrade or to continue. And we will show you in an Excel spreadsheet how these calculations are done.