CHAPTER 13
The Time Value of Money: Discounting and Net Present Values
With this chapter, we begin the third section of this book on valuation. At the start of the book, we mentioned that a CFO had three main jobs: to make good financing decisions, to make good investment decisions, and not to run out of money while doing the first two. Section one dealt with not running out of money. The second section dealt with good financing and other financial policies. This section deals with making good investment decisions. In order to do that, we must first learn the tools used in valuation. Of these, perhaps the most important is discounting and net present value (NPV). Unlike much of this book (which your authors don’t feel is included in other financial texts; otherwise we would not have written it), discounting and NPV are covered in all basic accounting and finance texts. What follows below is our take on the subject.
The Time Value of Money
The time value of money is one of the most powerful concepts in finance. It is a concept that small children understand when they say, “I want it now, not later, now!” Quite simply, the idea is that a dollar today (or anything for that matter) is worth more than a dollar tomorrow.
An easy way to start the explanation of the time value of money is to consider a bank account. If you invest $100 in a bank account at the start of the year and earn 5% annual interest on your funds during the course of the year, how much will you have at the end of the year? You will have $105, which is your original deposit of $100 plus the interest of $5 that you earned during the year ($100 * 5%). This means that with an interest rate of 5%, $100 today is equivalent to $105 in a year. Conversely, $105 in a year is worth $100 today.
Taking an amount today and computing what it is worth in the future is called compounding. Taking an amount in the future and figuring out what it is worth today is called discounting. Compounding and discounting are inverses of each other.
What if you leave the $105 in the bank for a second year, again earning 5%? How much would you have at the end of the second year? You would have $110.25. You start the second year with $105 and earn an additional $5.25, which is the interest you earned in the second year ($105 * 5%). You earned more in the second year ($5.25) than in the first ($5.00) because you began the second year with more money. In the second year, you earned $5 interest on the original $100 plus an additional $0.25 which is 5% interest on the $5 that you earned the first year. In the second year, you are earning interest on your interest. This example also means that $110.25 in two years is worth $100 today.
We can represent this visually with the following time line:
| $100 | $105.00 | |
| $ 5 | $ 5.25 | |
| $100 * 5% = $5 | $105 * 5% = $5.25 | $110.25 |
| | | | | | |
| Today | End Year 1 | End Year 2 |
We will consider today to be the present, so the $100 is our present value (or PV for short). The $105 value is a future value (FV), as is the $110.25. To differentiate between the two, we call the $105 the future value at time 1 (FV1) and the $110.25 the future value at time 2 (FV2). We can relabel our time line as follows:
| $100 * 5% = $5 | $105 * 5% = $5.25 | $110.25 |
| | | | | | |
| PV | FV1 | FV2 |
The 5% is our interest rate, denoted as r (also sometimes denoted as i). This allows us to put our time line into a formula:
We use algebraic substitution to get:
In the above example, $110.25 = $100 * (1.05)2
We can generally write that for any future period n:
Where, at an interest rate of r per period, the future value n periods from now is equal to the current value times (1 + r)n.
This compounding of a present value into a future value is a concept familiar to most, including those not involved in finance. We encounter this regularly when we put money in the bank and collect interest.
Discounting a future value to a present value is usually new to those not familiar with finance. It is, however, as noted above, the inverse of compounding. Mathematically, it works as follows:
We divide both sides of the equation FVn = PV * (1 + r)n by (1 + r)n to get PV (present value):
Using our numbers above, discounting a FV2 of $110.25 backwards for two years at an interest rate of 5% is worth $100 today.
That’s it. This is the essence of compounding (going forward), discounting (going backward), and the time value of money. We will now apply this concept to different situations.
More Compounding and Discounting
Let’s use another simple example to illustrate the above concept. Imagine you are offered a lump sum payment of $12,000 today (Option 1) or $18,000 (Option 2) at the end of four years and that the appropriate yearly interest rate is 8%.1 Which is worth more: the $12,000 today or the $18,000 in four years?
To answer this, we have to choose one date and then compare the value of the two options on that date (it can be any date: today, at the end of four years, or any date in between). Let’s start by finding the value of the two options at the end of four years. By definition, Option 2 is $18,000 in four years. What is the value of Option 1 in 4 years? That is, how much is $12,000 today worth in four years? From our equations above, with PV = $12,000, r = 8% and n = 4.
It is therefore better to choose the lump sum of $18,000 in four years than to take the $12,000 today and invest it at 8% for four years.
To repeat, taking the $12,000 today and computing its value in the future is called compounding. Visually:
| $12,000 | → | r = 8% | → | $18,000.00 (Option 2) versus $16,325.87 (Option 1) |
| | | | | | | | | | |
| 0 | 1 | 2 | 3 | 4 |
We can also answer the question of which option is worth more by comparing the two options today. By definition, the value of Option 1 is $12,000 today. What is the value of Option 2 today? That is, how much is $18,000 in four years, worth today? Using our equation above with FV = $18,000, r = 8%, and n = 4.
Taking the $18,000 in the future and computing its value today is called discounting.
Visually:
(Option 1) $12,000.00 versus |
||||
| (Option 2) $13,230.54 | ← | r = 8% | ← | $18,000 |
| | | | | | | | | | |
| 0 | 1 | 2 | 3 | 4 |
It is important to note that compounding Option 1 to calculate its future value and discounting Option 2 to calculate its present value both reveal that the $18,000 in four years is worth more than $12,000 today. The results are the same regardless of which direction we go (calculating future value or present value).
Thus by using compounding and/or discounting, we can take two different options and compare them at the same point in time, and we will get the same answer as to which has more value. The results are consistent: the present value of Option 2 is greater than the present value of Option 1 and the future value of Option 2 is greater than the future value of Option 1.2
The Periodic Interest Rate
When talking about interest rates and compounding, it is important to be clear about the period used. In the examples above we assumed the interest rates were yearly, meaning that they were applied once a year at the end of each year. However, it is quite common to have an interest rate compounded more than once a year. A common example in the United States is that bonds pay interest (or coupons) twice a year. This means that the rate to be compounded is the stated (or annual) rate on the bond divided by two. Let’s consider a U.S. bond with an 8% stated interest rate that is compounded twice a year. This means the bond actually pays 4% every 6 months. The 4% paid every six months is more than 8% paid once a year. Why? If the bond’s original cost is $100, then at the end of the first six months it earns $4 in interest (4% * $100). The interest for the second six months is again 4%, but this time it would be 4% of $104, which equals $4.16. If the bond matures at the end of the year, an investor would get back the original $100 cost (called the “principal”) plus combined interest of $4 and $4.16 for a total of $108.16. If the bond paid interest only once a year, to receive the same year-end amount would require an interest rate of 8.16%.
Thus, the actual interest rate over a period depends on how often the interest is compounded.
In our example above, $12,000 with an 8% stated interest rate compounded annually, gave us a total of $16,325.87 in four years. If the example is changed and the $12,000 now has a stated interest rate of 8% compounded quarterly, the computation is 2% every three months (e.g., the stated 8% rate is divided into four quarters or 2% each quarter). If we want to find how much the $12,000 is worth in four years when compounding quarterly, then r is the 2% rate and n is 16 (remember, n is the number of periods, our periods in this example are quarterly, and there are 16 quarters in four years).3
Our computation is now:
Note that FV16 at 8% compounded quarterly over four years ($12,000 * (1.02)16) gives us more than the $16,325.87 (or $12,000 * (1.08)4) calculated above where compounding was done annually.
We can use this concept to generate a once-a-year equivalent, called the annual percentage rate (APR),4 which is useful in comparing interest rates that compound over different intervals.
The formula to convert compounding more than once a year to an APR is:
APR = (1 + r/j)n*j – 1, where j is the number of times a year interest is compounded.
Annuities
An annuity is a contract with equal periodic payments. Importantly, despite the name of “annuity,” the periods do not have to be annual. Home mortgage payments are a type of annuity (a large sum is borrowed, and then equal monthly payments are made for the next 10, 20, or 30 years). Also, most debt contracts are the combination of an annuity (the periodic interest payment) as well as a final lump sum repayment.
Note, there are tables that provide the present or future value of annuities. Today, these have been replaced with computer spreadsheets where the value of an annuity can be found by simply putting each payment into a cell of the spreadsheet and discounting or compounding it.
As an example: Assume an equal payment of $300,000 is made at the end of June and December for two years, with the first payment in June 2014. Also assume an interest rate of 12% compounded semi-annually (i.e., 6% every 6 months). What is the value on January 1, 2014? The answer would look as follows:
| Date of Payment | Payment | Discount | Factor | Value |
| June 30, 2014 | $300,000 | 1/(1 + r) | 0.943396226 | $ 283,019 |
| December 31, 2014 | $300,000 | 1/(1 + r)2 | 0.889996440 | $ 266,999 |
| June 30, 2015 | $300,000 | 1/(1 + r)3 | 0.839619283 | $ 251,886 |
| December 31, 2015 | $300,000 | 1/(1 + r)4 | 0.792093663 | $ 237,628 |
| Value 1/1/2014 (r = 6%) | $1,039,532 |
Net Present Value (NPV)
In finance, investment decisions are often expressed in terms of net present value (NPV). If the NPV is positive, it is considered a good investment. If NPV is negative, it is considered a bad investment. What is an NPV? It is the present value of all the investment’s cash flows, including both inflows and outflows. Normally, investments require an initial payment (i.e., a cash outflow) followed by a series of returns (i.e., a series of cash inflows).
Let’s go back to the bank account example where $100 is deposited into the bank at an interest rate of 5% compounded annually. Now assume we withdraw our money at the end of the first year. The NPV would be calculated as follows: An initial cash outflow of $100 to the bank is followed in one year by a cash inflow of $105 if all funds are withdrawn. The NPV is the present value of these two cash flows. The present value of the $100 deposited today is $100, and since it is an outflow, it is included as –$100 in the sum. The present value of the $105 inflow in one year is $100, which is equal to the present value of $105 discounted backward by one year using a 5% interest rate. The NPV is therefore $0 (–$100 + $100).
The formula is:
Note that any individual cash flows can be positive or negative. A cash inflow is positive, a cash outflow is negative.
In our example, the value is:
A $0 NPV does not mean the investment does not earn a return. Rather, it means that the investment earns a competitive return, one that can be obtained elsewhere. If the interest rate truly is 5%, investing $100 today to receive $105 in a year is neither a good investment nor a bad investment. It is a fair return, the investor earns 5%, and has an NPV of $0. Assume, however, that instead of investing the $100 in the bank, a firm could invest the $100 in new equipment (with the same risk as the bank) and obtain $110 in a year. The NPV would be $4.76 (calculated as –$100 + $110/1.05). This is a positive NPV. If the risks of the investments are the same, the second investment is preferred to the first because it has a higher NPV.
NPV is used as an investment decision rule as follows: A positive NPV means that the future returns on the investment are greater than the risk the investment assumes. A negative NPV means that the future returns on the investment are less than the risk the investment assumes. Note that a negative NPV does not mean the total future cash flows are less than the initial investment (i.e., it does not mean, for instance, that we invest $100 and later get $90). It simply means that the future cash flows are not high enough to justify the initial investment given the required return. Thus:
Now let’s adapt our previous annuity example: Assume a series of cash flows can be generated from an initial investment of $1 million. Visually, this is:
| ($1 million) | $300,000 | $300,000 | $300,000 | $300,000 |
| | | | | | | | | | |
| 1/1/2014 | 12/31/2014 | 12/31/2015 | 12/31/2016 | 12/31/2017 |
Using a discount rate of 6%, the NPV will be $39,532 (–$1 million + $1,039,532). If you find a project with those expected cash flows and that expected interest rate, you should be willing to pay $1 million for it because it is a positive NPV project.5
Internal Rate of Return (IRR)
The internal rate of return (IRR) is sometimes advocated as an alternative investment rule to NPV. IRR is the discount rate that makes the NPV of all cash flows equal $0. The IRR is compared to a threshold required rate (also called a hurdle rate). If the IRR is above the required rate, it is a good investment; if it is lower, it is a bad investment.
Using the NPV formula, we set the NPV equal to zero and solve for r:
Cash Flow0 is the initial investment, Cash Flow1 is the return in 1 year and so on. Note that any individual cash flows can be positive or negative.
The decision rule with IRR is:
- If the IRR is greater than the firm’s required return, then it is a good project.
- If the IRR is less than the required return, then it is a bad project.
- If the IRR equals the required return, the investment earns a competitive return.
Intuitively, using IRR to evaluating possible investments means that if a project earns a return equal to or greater than the firm’s required return, it is a good investment. Otherwise it is not.
Let’s use our previous example of the $100 deposited in the bank at a 5% interest rate. Next assume $105 is withdrawn at the end of that first year. This investment has an IRR of 5% since the NPV using 5% (as shown above) is zero. Since the IRR of 5% equals the required return of 5%, the investment earns a competitive return.
Now let’s use the prior example of a $1 million investment today and $300,000 cash inflow once a year for four years. Also assume, as we did before, a discount rate of 6%. If we apply the IRR formula,
Solving for r we get an IRR of 7.71%. Since the IRR is above the firm’s hurdle rate of 6%, the project is considered a good investment.
In tabular form:
| Date of Payment | Payment | Discount | Factor | Present Value |
| January 1, 2014 | –$1,000,000 | 1/1 | 1.000000 | –$1,000,000 |
| June 30, 2014 | $ 300,000 | 1/(1 + r) | 0.928384 | $ 278,515 |
| December 31, 2014 | $ 300,000 | 1/(1 + r)2 | 0.861898 | $ 258,569 |
| June 30, 2015 | $ 300,000 | 1/(1 + r)3 | 0.800172 | $ 240,052 |
| December 31, 2015 | $ 300,000 | 1/(1 + r)4 | 0.742868 | $ 222,860 |
| Value 1/1/2014 (r = 15.428% semiannually or 7.714% every 6 months) | –$4* | |||
* The $4 shown above is a rounding error.
Which to Use: IRR or NPV?
The IRR and NPV rules often recommend the same decision. In the bank example above, the investment had a zero NPV, and the IRR equaled the required rate of return. This means the investment opportunity (i.e., depositing money in the bank) provides a competitive (fair) return given the risk.
In the second example above, the NPV was positive at a discount rate of 6%, and the IRR of 7.714% is also above the 6% discount rate. Thus either decision rule indicates the investment is a good one.
However, sometimes IRR and NPV give different results. This occurs for four primary reasons. They are:
First, IRR ignores the scale of a project. Assume the firm is only going to invest in one of two mutually exclusive projects (e.g., they use the same piece of land) and the hurdle rate is 12%. The two projects may have the same IRR (e.g., 15%), but one is smaller while the other is larger. For example, a project requiring $100,000 today and returning $115,000 in one year has an IRR of 15%. That is:
Project A:
Likewise, a project that requires an investment of $1 million today returning $150,000 in one year also has an IRR of 15%. That is:
Project B:
Thus the IRRs are the same and, if the discount rate is 12%, both are good investments. There is no way to judge, using IRR, which is the better investment.
However, the NPVs of the two projects are different at a discount rate of 12%.
The NPVs are:
Project A:
Project B:
The second project has a much larger NPV for the same risk and is the preferred investment if only one project is done. Thus, the IRR and the NPV rule give different answers in ranking the two projects. Under the IRR rule, the two projects are equivalent. Under the NPV rule, Project B is superior.
The second reason is that the NPV of a project changes as the discount rate changes, while the IRR does not.
Again consider two possible investments, A and B. Both require an initial outlay of $1 million today. Project A returns $0 for the first three years and then $1,688,950 at the end of year four. Project B returns $357,375 at the end of each year for four years.
Project A
| ($1 million) | $1,688,950 | |||
| | | | | | | | | | |
| 1/1/2014 | 12/31/2014 | 12/31/2015 | 12/31/2016 | 12/31/2017 |
| Project B | ||||
| ($1 million) | $357,375 | $357,375 | $357,375 | $357,375 |
| | | | | | | | | | |
| 1/1/2014 | 12/31/2014 | 12/31/2015 | 12/31/2016 | 12/31/2017 |
Project A has an IRR of 14% calculated by solving for r in equation:
Project B has an IRR of 16%:
Using the IRR rule, if the discount rate is below 14%, the firm should accept the second project since it has the higher IRR.
Note, the results of the IRR evaluation are the same regardless of the firm’s hurdle rate. That is, the hurdle rate is never used in calculating the IRR.
When we use the NPV rule, the NPV changes with different discount rates. Using our NPV equation for both projects:
The NPV values for discount rates are given in the table below. At a low discount rate of 10% or 11% (actually 11% or less), the NPV of project A is greater than that of project B. At discount rates of 12% or more, the NPV of project B is greater than that of project A. Note, at a discount rate of 14% the NPV of project A becomes zero, and at a discount rate of 16%, the NPV of project B becomes zero. These are the project’s IRRs.
Thus, using the NPV rule, project A is preferred with discount rates below 12%, and project B is preferred with discount rates between 12% and 16%. Neither project should be chosen with discount rates above 16%.
In the example, the IRR of B is greater than the IRR of A. However, the NPV of B is not always greater than the NPV of A because the NPV depends on the actual discount rate used. Thus, if we are forced to choose between the two projects, using NPV may give a different answer than IRR.
Thus, the IRR rule and the NPV rule can have different answers when actual discount rates vary.
| Discount Rate | NPV of Investment A | NPV of Investment B |
| 10% | $153,582 | $132,831 |
| 11% | $112,570 | $108,737 |
| 12% | $ 73,365 | $ 85,473 |
| 13% | $ 35,871 | $ 63,002 |
| 14% | 0 | $ 41,288 |
| 15% | ($ 34,332) | $ 20,298 |
| 16% | ($ 67,202) | 0 |
| 17% | ($ 98,686) | ($ 19,636) |
| 18% | ($128,853) | ($ 38,639) |
The third reason the IRR and NPV rules may give us different answers is because the IRR rule assumes that all cash flows received are reinvested at the IRR.
That is, the IRR rule assumes any cash inflows the firm receives earn a return equal to the IRR going forward until the project ends. This is often an unrealistic assumption. In particular, if IRR is very high (which would cause a firm to normally consider the project to be a good investment), it is unlikely that cash flows from the project can be reinvested at that rate.
Using our two projects above:
| Project A | ||||
| ($1 million) | $1,688,950 | |||
| | | | | | | | | | |
| 1/1/2014 | 12/31/2014 | 12/31/2015 | 12/31/2016 | 12/31/2017 |
| Project B | ||||
| ($1 million) | $357,375 | $357,375 | $357,375 | $357,375 |
| | | | | | | | | | |
| 1/1/2014 | 12/31/2014 | 12/31/2015 | 12/31/2016 | 12/31/2017 |
Project A returns $1,688,950 at the end of year four for an IRR of 14%.
Project B returns $357,375 a year for four years for an IRR of 16%.
However, assume the payments from project B can only be reinvested to earn 11%. What is the future value on 12/31/2017 of project B? It is:
Thus, if cash flows cannot be reinvested at the IRR, using the IRR rule can lead to a suboptimal decision.
Finally, the fourth reason IRR and NPV rule may give different answers is because the IRR of a project is not necessarily unique.
Solving for the IRR rule, as seen in the equations above, is simply solving a polynomial equation for r. As such, each time the cash flows change signs (i.e., outflows being negative and inflows being positive) there is an additional IRR solution (an additional root).6 If an investment has cash flow out, then cash flows in (as our examples above do), this equals one sign change, one root, and thus one IRR solution. However, if an investment has cash flow out, then cash flow in, then cash flow out again, this equals two sign changes, two roots and two IRR solutions. Therefore, for many investments that require an initial cash outlay and then future cash flow investments (perhaps maintenance expenditures), IRR does not give a unique solution. NPV, on the other hand, gives a unique single solution each time.
Consider a project with a $1 million initial investment on January 1, 2014, a cash return of $2,100,000 at the end of the first year, then an additional $1,100,000 investment required at the end of the second year.
| ($1,000000) | $2,100,000 | ($1,100,000) |
| | | | | | |
| 1/1/2014 | 12/31/2014 | 12/31/2015 |
The NPV with any discount rate is one unique value.
As seen in the four examples, the IRR and the NPV rule sometimes give different answers as to which project is preferred. Because the NPV is more precise and considers size, changes in interest rates, reinvestment rates on incoming cash flows, and multiple roots, the NPV is the preferred investment rule.
Payback
As long as we are dealing with investment rules, we should also mention “payback.” Though not used by many firms today, in the past this was a popular technique for evaluating investment projects. It is still used by many households and individuals when considering personal investments. Payback evaluates a project by measuring the time until the initial investment in the project is repaid. In our second example above, payback is 2.8 years. The $1,000,000 initial investment has a cash flow of $357,375 a year for four years, so it is repaid after 2.8 years (i.e., $1,000,000/357,375). The rule is to choose whichever project has the shorter payback. This method may sound reasonable at first pass, and it sometimes is.
So Why Not Use Payback?
Payback does not consider the market rate of return or the time value of money. Consider a simple example with two projects, A and B, both with a $1 million investment today. Project A returns $800,000 the first year, $200,000 the second and $200,000 the third. Project B returns, $200,000 the first year, $800,000 the second and $200,000 the third. They both have identical paybacks of two years, but they do not have equal NPVs. By this point in the chapter, you should recognize that project A has a higher NPV than project B. Even though the total cash flows of the two projects are equal, project A returns four times as much in the first year. Because of discounting and or compounding, we know money in one year is worth more than money in two years. The NPV rule captures this difference in value while the payback rule doesn’t.
This does not mean that payback does not give the right answer under the right circumstances. For example, up until the early 1990s, Procter and Gamble (P&G) used payback as part of their project evaluation. This worked for them because most of their products were similar consumer goods with similar risks and cash flow profiles. P&G made an investment, advertised, and rolled out the product. Products with similar risk profiles meant the products had similar discount rates. This meant that the rank ordering of projects by payback was identical to the rank ordering of projects by NPV. Thus, P&G through experience could rank order projects and determine whether the payback period was short enough to make the investment a good one. In this instance using the payback period allowed them to essentially determine if the project had a positive NPV or not.
Projects with Unequal Lives
One frequent problem in finance is how to compare projects with different lives. This arises if a project has to be replaced at the end of its useful life. For example, imagine a homeowner has to replace a roof and can use shingles (which will last 20 years) or slate (which will last 30 years).
How do you compare the choice between shingles and slate, since they are assets with different lives? Is it still correct to simply choose the higher NPV? No. Why not? Because using shingles requires replacing them 10 years sooner (in year 20) than using slate (which would get replaced in year 30). The homeowner can’t go without a roof for years 21 through 30. The NPV of a 30-year project can’t simply be compared to the NPV of a 20-year project. We have to compare the NPVs over the same time period.
There are two ways to handle this issue:
The first is to make the time period the same for both projects. This is called “common years.” If you have a 20-year project and a 30-year project, you take both out for 60 years (i.e., find the lowest common multiple of the two different project periods). This means using the present value of three consecutive 20-year projects compared to the present value of two consecutive 30-year projects and then selecting whichever series of projects has the higher NPV.
The second way to handle this is by computing what is called the “equivalent annual cost” (sometimes called the “equivalent annual revenue”). This method is somewhat analogous to computing the IRR.
Equivalent annual cost calculates the annual cash flows in our NPV equation necessary to make the NPV = $0 (given the initial cash investment and the market discount rate). Note, the IRR method of evaluating investments solves the same equation for the discount rate that makes the NPV = $0 (in this case given the initial cash investment and the cash flows instead). Instead of solving for the IRR that makes the NPV = $0, we solve for the cash flows that make the NPV = $0. The formula to do this is:
where r is given and you solve for the cash flows.
For example, imagine the following comparison:
- Project one: capital outlay of $100 million, 10-year life, and a 10% discount rate.
- Project two: capital outlay of $160 million, 15-year life, and a 10% discount rate.
Solving both projects for NPV = $0:
Thus, the equivalent annual cost is $16.3 million for project one.
Thus, the equivalent annual cost is $21.0 million for project two.
The investment rule in this case is to select the project with the lowest equivalent annual cost. In this case, it would be project one with an annual cost of $16.3 million versus an annual cost of $21.0 million for project two.
One way to think about this is that the payments for the initial cash outlay are being converted into an equivalent annual payment (like an annual rental payment). In fact, the two methods described above (making the time periods the same and equivalent annual costs) are identical. When comparing projects with different lives, the one with the lower required annual cash outflow per year is the one with the higher NPV.
Perpetuities
When doing valuation, we often use perpetuity formulas. In concept, valuation takes cash flows out until they end and discounts them back to the present. In reality, the period over which we are comfortable projecting out cash flows may be far less than the period over which cash flows are expected to exist. To give an example, a firm like Apple might roll out a new product/service like Apple Pay. The actual life of the project could be 50 years or more. However, analysts are only comfortable projecting cash flows out 5–10 years. The question then is: How should we handle the valuation of cash flows after the period over which we feel comfortable projecting?
The perpetuity formula is the answer. It values a set of periodic payments that last forever. The formula for the present value of equal cash flows that last forever is:
PV = cash flow/r
Let’s go back to our example of the bank and assume we earn $5 a year interest forever. This is a perpetuity, and the present value of $5 a year forever at a discount rate of 5% is ($5 / 0.05) = $100. Think of it this way: If we start with a deposit of $100 in the bank and leave it there forever at 5%, each year it will generate $5 a year in interest. Thus, $5 a year in interest forever at 5% is worth $100 today.
A Perpetuity with Growth
Now suppose the periodic payment does not stay the same, but grows over time at a constant rate g. The formula for the present value of cash flows with constant growth g, discounted at r, is:
Consider our example of Apple Pay. Imagine Apple expects the cash flows to be $100 million the first year and then growing with inflation of 2% a year. Further assume Apple’s discount rate for this project is 12%. This means the perpetuity value (i.e., the present value of the cash flows in perpetuity) is $100 million/(12% – 2%) = $1 billion.
Perpetuities are often used in firm valuation. They are particularly useful when calculating terminal values, a topic we will explain in more detail in the coming chapters on valuation.
Summary
This chapter introduces a number of tools used in making investment decisions. These tools are dependent on the concepts of the time value of money, compounding and discounting. In particular, we discussed net present value, internal rate of return, and payback. Although often consistent with each other, we demonstrated the superiority of NPV in analyzing the merits of an investment and deciding between alternative investments. The NPV investment rule is quite simple: Discount back all the cash flows at a required rate of return, and if the NPV of the cash flows is positive, do the investment. If it is negative, don’t. Note, projects with a zero NPV provide a competitive rate of return equal to the required rate of return.
Coming Attractions
The next two chapters apply the tools just introduced to the task of making good investment decisions. We begin with generating the free cash flows (FCF) to the firm using pro formas. This is followed by how to calculate the cost of capital, which is used to discount the cash flows, and how to determine a terminal value. Together, this determines the value of a project or firm.