So let's get started with some project evaluation. And as we learned in the first course of this finance specialization, this kind of casual analysis, forms the basis for most evaluation model in modern finance theory. So. So let's begin that analysis of this kind of casual analysis with a question. And the question is very simple. What would you prefer $1,000 now or the promise of $1,000 a year from now? And why? Well, of course, most of us would prefer the $1,000 in our hand now. And there are at least three reasons for this. Firstly, in the presence of inflation, that is a general increase in prices over time, the $1,000 now can buy more goods and services than the anticipated $1,000 a year from now. Secondly, the $1,000 in our hand now is inherently less risky. In fact, it has no risk. You're holding it. As compared with the promised $1,000 a year from now. And thirdly, even in the absence of risk, or inflation, we still prepare catchphrase, earlier rather than later. The reason for that is we can invest them, and generate a turnover in the intervening period. So the net result of those three contributors to what we call the time value of money. The net result is that we can't directly compare cash flows that occurred different points in time. And that's important for project evaluation purposes, because when we evaluate a project, we want to aggregate the cash flows that we expect the project to produce. Those cash flows, if they occur at different points in time, can't simply be added together, we need to make an adjustment. So how do we do that? Let's have a look. The way that we account for the time value of money when dealing with cash flows that occur at different points in time is through the discounting technique. The formula here provides us with that illustration. The present value of a cash flow that occurs sometime in the future, time N, needs to be discounted back at a rate of R percent per period over N periods. Which looks fairly confusing, and we'll highlight this with some examples. But essentially what we're doing is applying a penalty rate of R percent per period, either N periods against a cash flow that occurs in N periods time. Sounds very confusing doesn't it, but as soon as we go through a calculation you'll see exactly what's going on. The point to be made though is that as we convert future expected cash flows into their present values today, well now those cash flows are comparable. Now, we can add them together. Now, we can evaluate projects that have formed from portfolios of cash flows that occurred at different points in time. Let's have a look. So our example here involves calculating the present value of $800,000 expected in a years time, given a discount that reflects risk, opportunity cost, and expected inflation of 10% per year. So the nominal cash flows had $100,000, expected to be received in a years time, and that's $800,000. In its present value today Is $727,273. So what's the intuition here? As an investor, I'm indifferent between $727,273 today or the promise of $800,000 in a year's time. Will that promised future cash flow can be validly discounted at a rate of 10% per annum. Now, we can look at this in another way, as well. If we would only invest $727,273 today, in the bank at a rate of 10% per annum, then in a years time we would expect to have $800,00. So what happens if the cash flow was expected in two years time, rather than in a single year? Well, now the present value declines, as we have to wait yet another year to receive the cash flow. Indeed, it declines to $661,157. Now, intuitively once again, I'm indifferent between $661,157 today or this promised cash flow of $800,000 in two years time. With a risk, opportunity costs an expected inflation over the intervening period is accurately captured by the 10% per annum discount rate. Once again, if I were to invest $661,157 today at 10% per annum for two years that at the end of that period I'd have $800,000 dollars. So just to go over that once more, the present value of a cash flow that occurs in end periods time is equal to that cash flow, discounted by R percent per period. To the power of N periods. Very simple, very straight forward. So obviously, the riskier a cash flow, the higher our expectations of inflation, or the greater the opportunity cost associated with the investment, the higher the discount rate that we're going to use to discount those future cash flows. That's exactly right. So let's go back to our example, and let's assume that it's a much riskier cash flow than at first we assumed. So let's use a discount rate of 20% per annum. We're still just waiting a single year for this cash flow. Instead of this, the present value, being $727,000, present value of the cash flow now is $666,667. Intuition, well intuitively what's going on here, if i was to invest $666,667 at 20% per annum for a single year, that would accumulate, at the end of that year, to $800,000, just like a bank deposit, if you like. So this graph illustrates very neatly and simply the relationship between the discount rate, present value and the nominal value of the cash flow. Now, if money didn't have a time value, that is, if we didn't have to account for the impact of risks, opportunity costs and expected inflation, then there wouldn't be a difference between nominal and present values. And the present value of the cash flow would be it's nominal value of $800,000. As the time value of money becomes more important, as risk opportunity cost or expected inflation increase, the discount rate increases and the present value of the expected feature cash flow declines. So in this example, we can see that a discount rate of 10% per annum, the present value of $800,000 expected in a years time is equal to, well $727,273. If a discount rate was higher, G2 for example, the cash flow being riskier, well the present value of the $800,00 expected cash flow would be even lower. So for example, given a discount rate of 20% per annum, rather than only 10%, the present value of that cash flow is only $666,667, rather than $727,273. So as you can see, we have this inverse relationship between discount rates and present values, as indicated by this downward sloping curve. So that's Discounted Cash Flow analysis also known as DCF analysis. In summary, we started with the motivation that we can't directly compare cash flows that are expected to occur at different points in time. And if you can't compare them, well you can't aggregate them for project evaluation purposes. The reason they're not comparable is because money has a time value. Money has a time value for at least three reasons. Risk, opportunity cost and expected inflation. And what we do is we incorporate those three contributors to the time value of money into a single discount rate that we then discount future expected cash flows with. Back to their present value today. That then enables us to aggregate those cash flows, and that's essentially the ideal for project evaluation purposes. What you'll see in the next couple of lecture, is the discussion of the variety of techniques that employ this method.