Time Value of Money - BFA.docx
Transcript
Time Value of Money In this module I'd like to talk to you about the time value of money. Have you ever wondered how finance people are able to compare cash flows that are received at different points in time? So for example, let's say we're going to get a bunch of money five years from now or 10 ye...
Time Value of Money In this module I'd like to talk to you about the time value of money. Have you ever wondered how finance people are able to compare cash flows that are received at different points in time? So for example, let's say we're going to get a bunch of money five years from now or 10 years from now, how do we evaluate what it's going to be worth today? That involves applying principles called the time value of money. And that's what we're going to discuss here in this video. In order to make those adjustments, to move the cash flows and the values between periods, we need to understand three major factors. First, how much cash are we talking about, second, what is the interest rate that's going to be applied to the cash flow over time, and third, how many years or periods away is that cash flow from today? If we knew those three things, then we can adjust to the current period to get what we call a present value of a cash flow that we received in the future, or if we have some cash we invest today, we can look at what's called the future value or what that cash will eventually be worth in the future. Those are the core principles associated with the time value of money. So, let me start out with a simple example. Let's say we have $100 today. And we want to invest this $100 at a 6% rate of interest and we want to find out in three years how much will that be worth. Well, that's a concept known as future value. And here's what we would do. First, we would draw a very simple timeline. In the first year, we'd earn 6% interest. So, $100 would turn into $106. In the second year, we would get interest on the interest, an impact called compounding. So, our $106 times 1.06 would actually get us to $112.36. In the third year, our 112.36 would grow another 6%, so we'd multiply by 1.06 again, and we'd end up with $119.10. So, the future value in three years of $100 invested at 6% annualized interest is $119.10. This illustrates another important point called compounding or compound interest, which is not only do we earn interest on our investments, but we actually earn an interest on the interest. And over time, that compounding effect can really cause our money to grow. $100 invested at 6% would grow to $320.71. Compounding is a very important and very powerful force in finance, and it must be applied when we think about the time value of money. So, let's go back to some principles that will help us understand how to move the values of these cash flows around. These are going to be called the three rules of time travel and they were developed by Berk and DeMarzo. And the three rules are as follows. Rule number one, we can only compare two values at the same point in time. Otherwise, we're comparing an apple and an orange. Rule number two, in order to move a cash flow into the future, we must compound it based on the number of periods. Rule number three, in order to take a cash flow we receive in the future and bring it back to the present or today, we must discount it. And those are going to be the three rules that we are about to apply. So, let's start out with that second rule. How do we look at the value of what will happen to our cash flow into the future? That is a concept known as future value. And you can see the formula on the screen for future value, that the future value of any cash flow is the cash flow today times one plus the expected interest rate, which is R in the formula, to the power of N, which is the number of years or number of periods that you're going to invest over time. So, let's say we wanted to know the value of $5,000 invested today at 10% interest and what that value would be in five years. So, if we put that into the formula, cash flow would become 5,000, R, the interest rate, would be 10%, and five N years. 5,000 times 1.1 to the fifth power would be the mathematical way to solve it, and that would get us an answer of $8,053. So, we're going to draw a simple cash flow timeline on the screen. Today, we represent as time zero and each period, each year is going to be one, two, three, four, and five. So, we're then going to put the cash flow that we invest today, 5,000 at time zero, and this is growing at 10% a year. So, between time zero and the first year, growing at 10%, we multiply by 1.1. So by the end of the first year, we'll have 5,500. We then reinvest the 5,500 at 10% to get to year two. So, we multiply by 1.1 again, so that gets us to 6,050 in year two, compounding the interest. To get to year three, we multiply by 1.1 a third time. So again, that turns into $6,655. To get to year four, we multiply by 1.1 a fourth time, that gets us to $7,321. And finally, to get to year five, we multiply by 1.1 a fifth time and that gets us to $8,053. So, with the timeline, we can actually see what's happening more distinctly than the formula, but the idea is that money will grow over time and we will get interest on the interest so that again, the future value of $5,000 in five years is going to be $8,053. What if we wanted to go in the other direction, take cash that we receive in the future and bring it back to today? That's a concept known as present value. So, let's do a second example with present value in mind. In this case, let's say we're going to receive $20,000 in five years and our expected rate of interest is 15% annually. Well, the formula for present value is we take the cash flow and we divide it by one plus the interest rate to the number of years as a power. So in this case, we take the cash flow of $20,000, divided by one plus 0.15 or 1.15 for the interest rate to the fifth power because we're receiving that cash flow in five years. If we were to look at that on a timeline, we could put on year five $20,000 and we want to come back today or time zero. In either case, we're going to end up with an answer of $9,944. So, let's walk through the timeline process. If we take $20,000 and we divide by 1.15 to get from year five to year four, we end up with a year four value of 17,389. If we divide by 1.15 again, we get a year three value of 15,121. If we divide by 1.15 again, we get a year two value of 13,149. If we divide by 1.15 again, we get a year one value of 11,434. And if we divide by 1.15 a final time, we get $9,944. So, again, the present value of $20,000 received in five years at a 15% interest rate is $9,944 today. That's what it's worth to us today. We could have gone in the other direction, which says if we take $9,944 and we invest it at 15% for five years, it will grow to become $20,000. And so, we could be indifferent between the two, $20,000 in five years is worth a little under $10,000 today. In finance, we often think in terms of present value because it's much easier to think in terms of current dollars than future dollars. So, you will often see most cash flows converted into a present value format. The nice thing is once the cash flows are in the present value format, they can be added or subtracted against each other regardless of the time period in which they're earned. A little terminology. I've been using the concept of interest rate. Internally to the business, people might also refer to this as something called the hurdle rate or the expected rate of return. Externally, it might also be called the cost of capital. And you might hear people refer to it as the discount rate. These terms are fairly interchangeable in finance when it comes to time value of money. It's the R in the equation or again, the risk-based expected rate of return that we are either going to invest the money or the rate we use to bring the money back to today. Another way of thinking about this, to kind of visualize what's happening in the formula, is to create what we call a discount factor table. Using the concept of present value, we're going to actually create a table, applying the formula based in a dollar of cash flow, and we're going to use interest rates across the columns, and we're going to use number of periods or years down the rows. So, what that will allow us to do is to see the value today of a dollar and years in the future, down the rows, at whatever interest rate we want to apply. So for example, let's say that we're going to get a dollar 10 years from today and our expected rate of interest is 7%. That means, applying the formula, and you can see this on the table, a dollar in 10 years at 7% is worth 51 cents today. And in 10 years, if the interest rate were to change to 10%, that same dollar would only be worth 39 cents. And if we were looking at a really risky project and that interest rate or hurdle rate were 25%, a dollar received in 10 years would only be worth 11 cents today. So obviously, time matters to the present value as does the expected rate of interest. And as we go longer and as it takes us a higher level of risk, then you can see the cash flows are worth less and less to us today. By the way, the opposite is true if you're spending money. Meaning, if you defer money into the future that you spend, it doesn't cost you as much in present value terms to spend money in the future. That's the concept again of time value of money. So, let's close out by going back to our original example of present value and use the discount factor table to get to the same solution. In this case, $20,000 that we receive five years from now at a 15% interest rate or discount factor. This table uses a discount factor that has been expanded to 5 decimal places to illustrate this example. So in this case, we look down five years, we look across to 15%, and we see 0.49718. That's called the discount factor. So, what we do is we multiply the 20,000 by 0.4978 and we get a result, - $9,944. Present value is also related to another concept known as net present value or NPV. When we have cash flows that have all been adjusted to the same time. We can add them up. The sum of all of those cash flows at the same time is known as the net present value of a project. Look at the example that we've provided on the screen. It's a very simple project that lists cash going out and cash coming in over a five year period of time. We can net these cash flows in each of the years Because they all occur at the same period of time. But to bring them all back to today we would then have to discount the cash flows to bring them back to a present value. We then add them all up. If a project has a net present value of 0 it means after we've taken out the expected rate of return are cash outflows exactly are equal are cash inflows and no value would have been created. A positive net present value means that after we've taken out all of the expected rates of return we have generated more cash than we have spent and the amount of the NPV is the value that we are creating. If a project has a negative NPV that means that after all of the returns have been taken out we are not generating enough cash to pay back the original investment and the project will not create value. NPV is a simple way to quantify the overall value of a project that takes into account the time value of money. NPV is expressed in dollars. Because of this it also helps us understand the magnitude of the value of projects. It is often used in ranking projects when we have choices to make across many options. A concept that is related two net present value is something called the internal rate of return or the IRR of a project. IRR represents the average rate of return that we earn on our cash over the life of a project. To calculate the IRR we put all of the cash flows on a timeline. But rather than discounting the cash flows we use excel or some other program to solve for the R which is the interest rate that will cause these cash flows to equal NPV 0. By doing so this will give us a rate of return for the project. Once we know the average return on investment over time for a project we compare it with our hurdle rate or cost of capital to make a choice as to whether to proceed for the project. IRR tells us the return and investment of our project. NPV tells us the magnitude of the value created or destroyed. For example, I might have a project that has a 40% internal rate of return but it only generates $1,000,000 worth of MPV. I might have another project that generates a 20% IRR but creates $500 million of NPV. Even though the 20% project has a lower rate of return than the 40% project an organization might prefer it just because of the size of the value that it is creating. If a 3rd project has an 8% IRR and the cost of capital is 10% and that project has a $5m negative NPV – that project would likely be rejected since it would destroy value. Either of the other projects would be preferred. The other core concept that we use in finance is something called break even. Break even is the point at which we start earning a positive cash flow on a project or investment. Accounts look at cost as we've defined under generally accepted accounting principles as either direct cost of goods sold or indirect overhead. Finance people look at cost differently. We care much more about cost behavior. We think about what's going to happen to cost when volume changes. So we classify costs as either variable cost or fixed cost. Variable costs change with volume. Fixed costs do not change with volume. A good example of a variable cost is a toll road. If you use the toll road you pay a toll. If you don't use the toll road you don't pay any money. The more you use the toll road the more tolls you pay. This is a variable cost. A fixed cost might be spending on a building. Regardless of whether there's any people in that building we still have to pay the cost of the building. It's not going to change. Depreciation is a good example of a fixed cost. Regardless of the volume the amount we spend on annual depreciation will not change. The one thing that is often confusing to people is the per unit cost impact of fixed and variable. Generally even though variable costs change with volume the amount we spend per unit often stays about the same. Going back to the toll road. If I use a toll road more I'm still paying the same toll every time I use that road. That's a variable cost. From a depreciation standpoint if I produce more units then the cost for every unit will go down as I am spreading that fixed cost over more units. If I produce fewer units than the cost is going to up because I am spreading the same cost over fewer units. To estimate a breakeven point I start with a price and subtract my variable cost. What is left is something known as contribution margin. This margin contributes to my covering my fixed costs and eventually to making a profit. I then take all of my fixed costs and I divide by the contribution margin I make when I sell a unit. That answer is going to tell me how many units I need to break even and reach cash flow 0. Any volume above that amount I'm making positive cash flow. Any volume less than that amount I'm making a negative cash flow. Let's say that I'm Nike and I want to introduce a new shoe. I'm going price a pair at $100 and the cost to manufacture the shoes and get it to the customer which are my variable costs equals $30 per pair. If I take my $100 price and I subtract my $30 variable cost then I would have $70 of contribution margin every time I sold a pair of shoes. Now let's say that I spend $20 million to develop the shoes and all of the other expenses associated with getting it to the market. These become my fixed costs. How many pairs would I need to sell to break even. Well if I took my $20 million and I divided it by $70 that would tell me that I would need to sell - 285,714 pairs of shoes to breakeven. Any sales above that amount would now be generating positive cashflow for my business. Any sales less than that amount would be losing money. This is also why fixed costs can be risky. Let’s say I can develop the shoes for $10m rather than $20m. But as a result my variable cost goes up to $40 per pair of shoes rather than $30 per pair I was spending. What happens to my breakeven? Getting $100 per pair as a price doesn’t change. But because I am paying $40 for the shoes my contribution margin is now $60. If I divide $10,000,000 by $60 I see that I now have to sell 166,667 pairs to breakeven. This is a much lower sales target. A breakeven analysis is not only useful for telling me when my cash flow turns positive – it can also help me understand the risk of my project.
