So discounting cash flows, what does it entail? Why do we consider it to be the gold standard in finance? Let's take a closer look. We start with a simple example. I've got two propositions for you, which one would you prefer? I either give you $100 now or I promise to give you $100 in a year's time. Which one do you prefer? Wouldn't be surprised if you said that you very much preferred the $100 right now, thank you very much. And that is indeed what most people would intuitively reply. They prefer money in hand rather than to wait for the same amount of money at some point in the future. But why is that? Why is the $100 today somehow valued differently from $100 in the future? Well, we have at least three good reasons why you might want to prefer the $100 right now, expected inflation. We've seen over many, many years that dollars, over time reduce in value due to price increases. Inflation erodes the value of money over time. So what you can buy with $100 today is going to be worth probably more than what you can buy with the $100 in a year's time. Second reason why you might want to prefer the $100 now is that you might not get it. I might walk away and you'll end up empty-handed in one year's time. Risk, will the cash flow actually occur? If you get the money now, there is no risk. But one year from today, who knows what can happen in that time period. And lastly, opportunity costs. We'll discuss the opportunity costs in a lot more detail in the remainder of this module. The key point is that we cannot directly compare dollars, cash flows, that occur at different points in time. We need to consider the actual date that a cash flow eventuates. So, we need to consider the value of those cash flows at different points in time, somehow accounting for expected inflation, for risk, and for opportunity costs, alternative investment opportunities for investors. The way we do that is as follows. So, we account for what we labeled the time value of money, how a dollar changes in value from one time period to the next time period by systematically discounting future cash flows. The formula here tells you how we do that. We take the cash flow and we divide through by 1 plus a discount rate, r, to the power of n. And n indicates the number of years you have to wait until you're entitled to the cash flow, where the cash flow is indicated for the year that it will occur, cash flow at n. If we divide the cash flow through by 1 plus the discount rate, we divide through by something that is larger than 1, weâ€™ll end up with a value which is going to be smaller than the cash flow. That is the present value. The present value in the presence of expected inflation risks and opportunity costs will be less than the cash flow entitlement, and years down the track. It allows us to express future cash flows into present dollars, equivalent present dollars. We label those equivalent present dollars, the present value of a future cash flow. That, then, will allow us to bring all future cash flows, whether they occur in one year's time, two years' time or ten years' time, we can bring them back to the present. And once we've got them all valued at the same decision period, today, now, we've got comparable cash flows that allow us to make an informed investment decision. So, let me give you an example. What is the present value of that $100 that I promised you earlier, but I will only give it to you in one year's time? And an appropriate discount rate, we'll discuss the choice of that discount rate a little later, and appropriate discount rate being 5% per annum. Take the cash flow, $100 in one year's time, divide by 1 plus that discount rate of 5%, 1.05 to the power, the number of the years, 1 in this case, and that tells us that the present value in this example is $95.24. So, $100 in one year's time is valued today at a discount rate of 5% at only $95.24. So, what's the intuition here? Or, consider it another way, take that $95.24, and take that right now, borrow $95.24, invest it at the discount rate. Let's assume a bank is offering you a 5% interest rate per annum. Invest the $95.24 at that 5% and what do you get? Exactly the $100 in one year's time. So that suggests a neat link between an opportunity to invest at 5% over a one-year period, linking a future cash flow to a present value of $95.24. Now what would happen if that cash flow didn't occur in one year's time, but in two years' time instead? No problem, same formula. Take that cash flow. In two years, n equals 2 of $100, but now discount by a slightly higher discount factor, and that is going to be 1 plus the same discount rate of 5%, but now to the power of n equals 2 years. And it will tell us that the present value of a $100 cash flow that you will be entitled to in two years' time, today is only worth the present value of $90.70, so you've lost the further $5, almost $5, in terms of present value. The further away the cash flow entitlement, the smaller the present value. So the intuition here is the same. Borrow the $90.70 today. Go to the bank, invest it now for two years. And what will you get after two years? Exactly the $100. So there is a very clear link between future values of cash flows and present values of cash flows. And we can in fect move either way. So rather than discounting future cash flows to find their equivalent present value, we can also move cash flows into the future, take a present value, take a present cash flow, today's cash flow, and see what that is worth at some point after some number of years in the future. We call that compounding, rather than discounting. So, we discount future cash flows to the present, we compound present cash flows to the future. So how does that work for our example? Well let's assume that you opted for the $100 today, you could go to the bank, invest the $100 today for two years, n equals 2 at a discount rate and an interest rate of 5% per annum. The future value, FV, after two years, n equals 2, would be that cash flow today, the present value of $100, multiplied by 1 plus the discount rate, the interest rate of 5%, to the power n, two years. And that tells you that the future value of $100 today invested for two years at 5% is worth $110.25. Now why would we want to do that? Why would we want to move money forwards in time, into the future? Well consider the case where you have a decision date which isn't today. The example I gave you before that an investor has to make up their mind about investing in Kellogg's today, at the share market price, as it is quoted today, that would require us to work out present values of future cash flows entitled by ownership in Kellogg's. Consider the scenario where the decision point is not today but some time in the future. For example, where you want to work out what you need to do at the time of retirement, when you are entitled to the pension, in that scenario your decision point is not today, but you would be investing in your pension fund today. But you would want to know the future value of that pension fund sometime in the future when you decide to retire. So there is a need to move money to the future as there is a need to move money to the present. In either scenario what we want to achieve is that we bring all cash flow entitlements to a fixed point in time. Whether that is the future or whether that is the present makes little difference. The principle is the same. It will allow us to compare cash flows to add cash flows together, which in fact happen at different points in time. That is the core of discounting cash flow valuation. So, how do you choose the discount rate? Well here's a few pointers. We typically start with the risk-free rate of interest, the 5% that I just mentioned to you before, guaranteed by the bank. Then you might want to consider the fact that the cash flows that you're entitled to, if you invest it in Kellogg's shares, are, in fact, not risk-free. So there is some likelihood that some of the cash flows might not actually occur. To allow for that scenario, and to embed that in the discount rate, you might want to look for rates of return on similar risky investments, similar to investing in shares of Kellogg's. Or, maybe you're able to actually work out the exact risk premia as they are attached to future cash flows that Kellogg's is promising you. Then, you might want to consider the other opportunity costs. So what are the other investments that are available to you? And how would that have a bearing on how you value a future cash flow entitlement in Kellogg's? And lastly on this list, expected inflation. You would need to form a position on what you think inflation is going to be over your investment horizon in shares of Kellogg's. The bigger the impact of any of these factors is going to be, the higher the discount rate will be as it is applied to the future cash flows, and therefore the smaller the present value will be. The higher the risk, the lower the present value. So continuing our example here, our expectation of $100 in a year's time, let's boost the risk of this investment. We're now really unsure of whether we will actually be receiving the cash flow of $100 in a year's time. So, rather than use a discount rate of 5%, we've increased the discount rate by a further 10% to 15% per annum. Take that same cash flow, divide now by 1.15, and we find that the present value of that same $100 has dropped from $95 to $87 A significant drop from the $95.24 that we saw with a discount rate of 5%. So the higher the risk, the higher the discount factor, the lower the present value, the more we discount future cash flow entitlements. Just to illustrate that with a bit more detail, for a series of cash flow entitlements, let's assume that Kellogg's was actually promising cash flows of $100 for the next ten years. If we discount these $100 cash flow entitlements, as they occur at the end of each year at 5%, look at the red bars, you see that they steadily decrease in present value. So whereas the first entitlement would be worth $95.24 after one year, then after ten years, the cash flow that Kellogg promises in ten years' time is going to be worth only slightly over $60 in present value. If we increase the discount rate to reflect the riskiness of those cash flows to 15%, look at the green bars, you can see that there is an immediate drop off in the present value of future cash flows. And you can see that the cash flow that is promised after ten years is now only worth about $17, in present value. So, given its importance in valuation, given its gold standard in finance, let's summarise what we've just seen. Cash flows expected to occur at different points in time are not directly comparable. Do not make the mistake of adding up cash flows that occur after one year, two years, until ten years and assume that that is what you should be willing to pay for those cash flows. We've seen that the further in time, the lower the present value of those cash flows is going to be. So causes for the discounting for the time value of money, the deterioration of future cash flow entitlements: risk, opportunity cost, and expected inflation. We've seen that with discounted cash flow valuation, we've got a very easy technique, easy to implement, to bring future cash flows back to their present value, and that makes them comparable and solves the problem. Discounted cash flow is used for many project evaluations and, in fact, firm valuations.