Okay then, so far we've identified how traditional NPV analysis might lead to incorrect investment decision when it fails to account for the impact of flexibility on the wealth effects of a particular investment proposal. We then spent some time defining and describing three of the most common types of real options found in practice, options to invest or defer, options to expand and options to abandon operations. It seems reasonable now to start thinking about how we might actually attempt to ascribe some value to having the right, but not the obligation, to proceed with a particular course of action in response to changing economic circumstances. So how might we value an option? Well, when we're dealing with financial options, that is, options written on financial assets such as shares, we have some fairly decent models in place. Indeed, the Black-Scholes-Merton Option Pricing Model, as detailed in these equations, yielded the Nobel Prize in economics to the two surviving researchers, Robert Merton and Myron Scholes. There are a number of challenges we face in trying to apply these financial option pricing models to settings that involve standalone projects, as is applied by the option to invest, to expand, or to abandon. Firstly, the Black-Scholes-Merton Option Pricing Model is a model for the valuation of a European-style option, that is, an option that can only be exercised at expiry. When we think about real options, however, we might note that most of them are of an American style. They can be exercised at any time up to the expiry date. Well, that's fine. As a discipline, we've developed a range of what we refer to as numerical techniques, such as binomial lattices, that can help us deal with things like the ability to exercise an option early. The problem is, though, that these models assume things like an ability to continually rebalance an investment in the underlying asset through a process known as delta hedging. And many argue that that's completely unrealistic in a real option setting. So where does that leave us? Well, the short story is that it would be really helpful to have an easier way to try to come up with the value implied by a real option, even if the approach itself provides only an approximation of the real option's value. One such approach is the decision tree approach. The decision tree approach is a discounted cash flow-based technique that allows you to assess a project that may involve a sequence of decisions through time. Let's demonstrate how decision trees work in practice. We assume that you are trying to decide whether to invest $200,000 up front in a new retail outlet with a life of five years. There's a 50% chance that the outlet will experience high demand in the first year, in which case it would remain high for the remaining four years. Each year, it would generate $150,000 per annum. If demand is low in the first year, then it will remain low for the remaining four years, generating only $50,000 each year. So where is the option? Well, the firm has the ability, at the end of the first year, to expand operations in response to high demand. Doing so will cost the firm $50,000 up front but will increase subsequent annual cash flows to $170,000 per annum. Let's assume a discount rate of 10% per annum, and further assume, for the sake of the example, that all cash flows occur at year end. Let's have a look at that decision tree. So here is the decision tree. The key with decision tree analysis is that before we can assess the decisions that we face soonest, we must first reconcile the decisions that we would make in the future depending upon the circumstances we find ourselves in. So in this decision tree we first assess the expansionary decision. Do we expand or not if demand is high in the first year? And then, having made that decision, we can go on to assess whether we should invest at all. So let's do that. Would we spend the $50,000 at the end of the first year to generate an additional $20,000 in net cash flows per annum for the remaining four years? The NPV of expanding at the end of the first year, indicated by NPV subscript 1, assuming that demand is high, is $488,877. If we don't expand in the face of high demand at the end of the first year, then we save ourself the $50,000 in expansionary cost. But the flip side is that our revenue stream is $20,000 less for each of the remaining four years. Hence, we know that if demand is high in the first year, then we will choose to expand operations. So now we can erase one arm of the decision tree which relates to a suboptimal decision given the information we have today. Furthermore, we now know the present value of the expansionary arm of the decision tree is $488,877, but it's important to recognize that this is a valuation in one year's time. Having resolved the most distant decision first, we can now roll back through the decision tree, assessing the next most distant decision, which in this case is the decision about whether we invest at all. So here we are. The wealth associated with investing in this retail outlet is calculated as follows. Firstly, we account for the initial investment of $200,000. Next we consider the low-demand arm of the tree. That's where there's a 50% chance of there being low demand in the first year, in which case low demand will continue. Across each of these five years, we generate net cash flows of only $50,000 per year. So this first expression gives us the present value of that cash flow stream. Now switching our attention to the high demand state of the world, where we know we will expand if demand is high enough in the first year, we account firstly for the fact that there's a 50% chance of finding ourselves in that high demand state, which, you will recall, generates $150,000 at the end of the first year, followed, after we expand, by $170,000 in each of the remaining four years. Fortunately, we've already calculated the present value of the remaining four payments as $488,877. So working our way through all this now, we end up with an overall NPV of the project of $185,169. Therefore, given the alternative decision, which in this case would be to not invest at all, and therefore, generate an NPV value of 0, our final decision is to invest in the project. But the question is, how does any of this assist us in providing us with an estimate of the value of the real option? Well, here's what we do. The first step is to utilize a standard decision tree to evaluate the project exactly as we did in that last example. The next step is to re-estimate the value of the project using the tree approach, but this time with the option you want to value removed from the tree. The difference between the first and second valuation gives you an approximation of the value of the embedded option in the project. So here's what the tree looks like without the option to expand. When we evaluate this tree, we only account for cash flows of $150,000 per annum in the high demand state of the world. As you can see, the calculations will be relatively straightforward. And here we go. The net present value of the project without the option to expand is equal to $179,079. We now simply compare this number with the value of the project with the embedded option. And lo and behold, the approximation of the value today of the option to expand operations in the case of high demand in the first year is equal to $6,090. Now why do I keep using the term approximation here? Well, unlike formal option pricing models, which provide very precise values for option, although they're based on a set of sometimes very restrictive assumptions, decision trees are a little bit more haphazard in that they only provide for a set of discrete outcomes. Compare the two figures at the bottom of this slide. The left-hand side figure relates to the changing value of an option as the present value of the cash flows from exercise of the option increase continuously as we move from left to right of the figure. In contrast, our decision tree only captures two discrete states of the world, high and low demand. If it was not optimal to invest, that is, to exercise the option, in either of those states of the world, the decision tree approach would suggest that the option has zero value. Whereas we know that even where options are deep, deep out of the money, provided there's some remaining term to expiry and some volatility, the option will still have a value. In summary, we've considered in this session together how to use decision trees to evaluate projects that may involve a sequence of decisions that are made over the life of the project. And truth be told, that's one of the real advantages of decision tree analysis. It gets management thinking strategically about optimal decisions to be faced in the future. We then moved on to a discussion of how we might use decision trees in practice to approximate the value of real options. Now it's been a pretty heavy module so far, right? So let's finish off by considering in the next session the extent to which Real Options Analysis is used in practice by senior financial managers.