On Discounting Cashflows
After compound interest, discounting cashflows is the most important skill that every financially fit person should know, but doesn’t know.
Oddly, discounting cashflows never, ever, gets taught in schools. The math is junior high–level math, so I would think high school and college students could be required to master the use of this powerful skill.
This ain’t calculus. It’s way easier. And it’s way more useful in your life.
Why Is This Not Taught?
I’ve thought for a long time about what could be the reasons for never teaching discounted cashflows to high school students.
Is it because Super Evil Mega Corp Conglomerate (the one that secretly controls everything, including the company you bought this book from) does not want people to understand how investing actually works, so they hide this information in a giant conspiracy of ignorance intended to keep the rich wealthy and the poor as impoverished sheeple? Maybe, yes?
I mean, no, not really.
But if it’s helpful to think of the silence around discounted cashflows as a secret conspiracy that you can discover and undo by reading this chapter, then go ahead and use that as motivation.
Is it because high school teachers—and the vast majority of college professors—have zero idea how investing works? In part, probably, yes.
Perhaps the cone of silence exists because discounting cashflows is just too complicated for the average educated person to grasp? NO!
That’s the point of this chapter. This is not too complicated. Be optimistic that you can understand this stuff. Be skeptical of the way the investment world presents its seemingly sophisticated face, when underlying it all is this pretty simple math.
Finally, is it because maybe I’m wrong, and discounting cashflows is not an essential skill? No. I am not wrong.
This is the fundamental skill used to invest money. Now, you may not want to invest money for a living, and frankly I don’t blame you in the least for that choice. But in that case you will need to make certain financial choices that involve someone else investing for you, such as your bank, your government, your insurance company, and your broker. And you want to know the whats, whys, and hows with your money. Your ability to understand the skill behind investing can open up a whole world of understanding about what it is that the person you hired is actually doing.
Learning the Math
It’s not hard, and if you’ve learned compound interest, then it’s kind of a snap.
But it does involve math and poking around with a spreadsheet.
Just like the last chapter on compound interest, you should definitely open up a spreadsheet as you read through this chapter, as it’s the only way to actually learn how to discount cashflows yourself.
So What Is It?
Discounting cashflows—in the simplest mathematical sense—is just the opposite action to compound interest.
I mean opposite in the way that subtraction is the opposite action of addition and division is the opposite action of multiplication. And—for that matter—square roots are the opposite of raised-to-the-power-of exponents.
In that same sense, discounting cashflows is the opposite of compound interest.
And just like the compound interest formula showed us how the mathematical bridge of interest rates and time connects today’s money to future money, the discounted cashflows formula showed us the same bridge, only in reverse. We connect future money to a value of money today.
Calculating compound interest, you recall from the last chapter, tells us precisely how money (or kittens, or snowballs) grows from a certain known amount today (PV) into a known amount in the future (FV), through the intervention of an interest rate (Y) and multiple compounding periods (N).
The formula we learned in the previous chapter on compound interest is FV = PV ∗ (1 + Y)^N.
Again, that’s how a known amount of feral kittens today (PV) becomes a predictably known amount of kittens in the future (FV) through a growth rate (Y) and a number of compounding periods (N). (And a lot of kitty litter along the way.)
Discounting cashflows moves in the opposite direction. We move backward in time, from the future back to the present.
Specifically, the discounting cashflows formula tells us how a certain known amount of money in the future (FV) can be “discounted” back to a certain known amount in the present (FV) through the intervention of an interest rate (Y) and multiple compounding periods (N).
Notice that we use the exact same variables in both formulas. Notice, also, that the only difference mathematically is that we’re solving for a different number.
The discounted cashflow formula simply reverses the algebra of the compound interest formula.
The discounted cashflow formula solves for present value, so that:
PV = FV / (1 + Y)^N
If you remember your algebra skills, you will see that this is the exact same formula as compound interest, except that instead of “solving for” or isolating FV, we’ve “solved for” and isolated PV.
So why do we care about discounting cashflows?
Discounting cashflows is the basis for all investing. Full stop. Punto. End of story.
It allows us to estimate the present value, or in plainer language—how much we would pay today for a future amount of money.
And notice that’s exactly what we’re doing when we invest money. We pay for, or invest, some amount of money today, in order to get a larger amount of money returned, some time in the future. The discounted cashflows formula tells us how much we should pay for that future money. How much is that future money worth to me today? That’s what I’ll pay. That’s investing.
Example 1: An Inheritance
A simplified example should help to get us started, then later in the chapter—and in the Appendix—we’ll tackle some further examples to solidify our understanding.
How much would an inheritance coming in 5 years be worth, today?
If I know for example I’m set to inherit $5,000 from my uncle 5 years from now, what would that be worth to me today, like, right now? I know we don’t “pay” for an inheritance, but you can think of it as “How much would I pay today to get $5,000 in 5 years?”
If I assume a certain interest rate or yield—also known in this context as a “discount rate” but mathematically it represents the same exact thing—I can calculate the worth to me today of the future $5,000.
By the way, again, if you want to start to learn this formula for real, open up a spreadsheet right now and let’s calculate the answer. Simply reading the words here will not suffice. Nor for that matter will “cheating” and using a financial calculator, because that’s not a flexible enough tool—in my opinion—for solving any discounted cashflow situation you could encounter in your real life. You have to actually follow along with the mathematics using a spreadsheet. Just trust me on this one.
OK, back to our inheritance. Are you excited? I’m excited for this.
We know the FV is $5,000, the amount we expect in the future.
We know the N is 5, for 5 years’ worth of discounting.
PV, the present value of the future inheritance, is what we’re solving for with the formula.
So what’s Y?
Let’s assume for the moment a 4% Y, or discount rate. (More on that assumed Y in a moment, just below the math solution.)
If Y is 4%, then the math involves just plugging in numbers for FV, N, and Y into the discounted cashflows formula, which we remember from above is
PV = FV / (1+Y)^N
(I implore you—once more—to open up a spreadsheet here and create a simple spreadsheet formula to solve this math. Doing it by hand, or even with a financial calculator, just isn’t efficient or robust enough.)
So, PV (what we’re trying to solve for) is equal to $5,000 / (1 + 0.04)^5
That’s easy-peasy math for your spreadsheet formula.
The present value is $5,000 / 1.216653.
Or $4,110 rounded to the nearest dollar.
In investment terms, that means we should be willing to pay $4,110 today in order to receive $5,000 in inheritance, 5 years from now. Again, we don’t really pay for an inheritance, but the example is meant to show what that future inheritance is worth today, 5 years in advance of payment.
By the Way, How Did I Come Up with 4%?
Frankly and honestly, I made up the 4% for the example.
I don’t just say I made it up to be flippant. I mean to emphasize that I made it up because making up Y, or the proper yield or discount rate (remember, those mean the same thing!) is a key to effectively using the discounted cashflows formula.
In fact, any time you discount cashflows, you have to “make up,” or assume, a certain Y or discount rate, and the Y assumption you use is as much art as science.
Is that 4% Y I assumed correct?
I don’t know, but it’s reasonable, and that’s usually the most we can say about any assumed Y. How do we come up with a reasonable Y number?
Y as an interest rate or discount rate (remember: same thing!) reflects a combination of:
Only some of these things can be known at any time, so only some of our Y is scientifically knowable. The rest has to be assumed according to best estimates. That’s why we can reasonably say that sometimes this Y assumption is as much art as science.
The rest of this chapter—after the following caveat—will provide examples of how we use discounted cashflows in real-life investing.
Do You Have to Learn This?
Can you do just fine without learning how to discount cashflows? Yes.
Do 999 out of 1,000 people know how to do this? No.
And it turns out many of those 999 people out of 1,000 do just fine in life. They have friends. They seem normal and well adjusted. Some of them even grow wealthy. They remain ignorant of the importance and usefulness of discounting cashflows, but they get by OK in life. All true.
But I want to mention three reasons to learn this skill.
First, most generally, getting wealthy may involve learning things that others haven’t bothered to learn or haven’t been able to learn. Being different is to your advantage here.
Second, all investing, properly understood, is based fundamentally on discounting cashflows. Seriously, all investing!
That statement probably sounds weird because, as I mentioned already, 999 out of 1,000 people don’t know how to do this. Yet many more people than 1 in a 1,000 believe they are “investing,” without knowing this technique. Which means they “invest”—in an important sense—blind.
Or they are doing something different than investing, such as guessing, or speculating, or fingers-crossing, or something else. I don’t know what.
Third, even if you never invest yourself—even if you hire a fund manager or investment adviser to do this for you—you should understand discounting cashflows. Knowing the math of discounting cashflows is going to help you judge the extent to which the people you hired know what they are doing.
Here’s a scary yet true statement: the vast majority of investment advisers—certainly a vast majority of financial salespeople masquerading as investment advisers—do not know how to discount cashflows. Which means you should consider firing them and hiring yourself!1
Or something.
I mean, don’t really fire them right away, because even these financial salespeople may be able to help you in other ways, as we’ll discuss in Chapter 15. Just not, probably, as experts on investing.
OK, what I really mean is this: discounting cashflows is the basis for all investing. If you mastered the technique, and then realized how little this fundamental skill is understood by the people you pay to help you, you will:
Application to Business Owners
This may seem like a tangent, and more related to the section of Chapter 21 about entrepreneurship, but the ability to calculate discounted cashflows does more than just make you literate in investing. That skill is also the financial basis of good decision making when it comes to business growth, business investment, and merger activities.
You see, the decision to pay for something today in order to receive cashflows in the future is what an entrepreneur or CEO has to make.
Should I buy this company? Well, a key piece of information is whether the future cashflows, discounted back to the present, justify the cost today.
Should I build out this new product line? Well, what are the future cashflows you can estimate based on that product? When you discount them back to the present, are they more or less than what you need to pay today?
In the Appendix for this chapter I discuss the use of discounted cashflow analysis for deciding which price to accept when selling a business.
When our elected officials agree to pay long-term public pensions to retirees, calculating discounted cashflows is how we figure out the cost in today’s dollars. When a retiree elects to receive a $50,000 pension for the next 24 years, we can know precisely what that is worth, if we apply a discount rate to each of those $50,000 payments.
OK, now, if you open up a fresh spreadsheet, you really can master the discounted cashflows formula using this chapter, and the Chapter 5 Appendix, with a little patience and practice.
Example 2: Single Lump Sum Like an Annuity Payout
Let’s start simple, spreadsheet open.
A builder’s insurance company offers you a $25,000 lump sum payment to compensate you for the pain and hardship of an injured pet hit by an errant beam that fell from his construction site.
Picture a big piece of wood that hurt the dog’s paw. The dog will likely make a full recovery, but the developer/builder offered you this settlement to avoid a costly lawsuit with bad public relations potential.
Importantly, however, the settlement will be paid out 10 years from now. Note, by the way, that this is common practice in injury-settlement cases.
Lump sums get offered far into the future. This is partly because such agreements incentivize the victim/beneficiary to comply with the terms of the settlement for the longest period of time. But also importantly, as we will see, it’s much cheaper for the insurance company to make payments deep into the future.
Now, back to the math.
Let’s assume the insurance company is a very safe, stable company, and we expect moderate inflation, so the proper Y, or discount rate for the next 10 years, is around 3%.
How much is that settlement worth to us today?
Let’s go to the spreadsheet.
We set up our formula in a spreadsheet that the value today, or present value (PV), is equal to FV / (1 + Y)^N.
We know the future payout, FV, is $25,000.
We know how many years we have to wait, so N is 10.
We’ve assumed a Y of 3%.
The present value will be equal to $25,000 / (1 + 3%)^10.
This is easy-peasy math for your spreadsheet, which tells us the present value is $18,602, rounded to the nearest dollar.
What does this mean in practice? We’re not going to invest $18,602 in this future $25,000 insurance payout, but it can be very helpful for us to understand that the future $25,000 payment really only costs the insurance company about 75% of what it first appears to cost.
Incidentally, using discounted cashflows can give us extraordinary insight into how an insurance company operates, and how it makes money. In a related story—as we’ll discuss in Chapter 16, “On Insurance”—you don’t want to buy too much insurance.
If you’d like to do more discounted cashflows practice before moving on to Chapter 6, I recommend turning to the following Appendix to get further practice on this essential investing technique.
Appendix
This Appendix makes the most sense after reading Chapter 5 on discounting cashflows and working through that math.
If you’d enjoy a deeper dive using video and watching me walk through some discounted cashflows examples, I encourage you to visit my website:
http://www.bankers-anonymous.com/blog/discounted-cashflows-deeper-dive
This section gets us closer to how discounting cashflows math gets used in real life, by actual banks, insurance companies, entrepreneurs, and bond and stock investors. I want to warn you upfront that there’s some math here. Many of you may be turned off by math. The first thing I’ll say about your aversion is that you can absolutely do great in life without using this math, and Chapters 6 through 21 do not require you to be able to do this on your own.
The second thing I’ll say is that I passionately believe that many more people can and should learn this, as it can change your financial life. Third, it drives me crazy that personal finance books for a general audience never bother to teach this the right way.
Many finance books that try to teach the fundamentals of discounting cashflows and bond investing make use of a discounted cashflows table, to which readers can refer, to figure out the present value of future bond payments. But these tables are a terrible idea. First, following the widespread use of computers, nobody in the real investment world after say, 1982, would ever refer to a discounted cashflows table to value a bond. And yet, that method keeps showing up in books.
At this point, with spreadsheets available to all, tables in books should be banned as a method for discounting cashflows. Anyway, that’s a pet peeve of mine. So I want you to learn this the right way.
One more throat-clearing note before we plunge into the math. The examples to follow are how an entrepreneur selling a business, a bond investor, and a stock investor would use discounting cashflows to determine financial value. To work up to this, you should already master the Examples 1 and 2 from this chapter. If you take your time with those examples, you will be ready to tackle these examples.
The examples below apply specifically as follows:
Example 3: Entrepreneur
Example 4: Bond Investor
Example 5: Stock Investor
If you’d like to learn this for real, just like for Examples 1 and 2, I recommend taking your time, with a spreadsheet open, to follow along each step. The math does not go beyond early high school skills, but is not “intuitive” without literally working through the examples on the spreadsheet. If you do master these skills, I optimistically propose you’ll have a better intuition about how the financial world around us works.
Example 3: Multiple Annual Future Payments—Entrepreneur Selling a Business
Let’s add a little complexity over the earlier Chapter 5 examples by adding in multiple future payments for which we need to calculate a present value.
Let’s assume you are ready to retire, and you plan to sell a business you own. The business purchasers agree to pay you $300,000 for your business. But instead of offering all that money now, they plan to pay you in annual installments of $30,000, each and every year for the next 10 years.
You’re tempted, but you want to know, what is this payout really worth today?
We can even think about a scenario in which you have a competing offer of $200,000 for your business, with all the cash paid to you upfront, this year.
To calculate the worth of 10 future payments, we need to discount those payments to the present day, which we can then compare to the $200,000 all-cash offer.
Let’s assume the business you plan to sell is somewhat risky, making those 10 future pension payments also somewhat risky. The right discount rate (or Y) is something like 12% annually.
[About that 12% Y assumption: please know that the 12% discount rate (Y) is really an assumption based on as much art as science. If my business is somewhat risky, I might base my discount rate on comparable “junk bond” interest rates for other risky companies. I could also take into account prevailing interest rates of the economy, as well as inflation expectations. Is 12% the right discount rate? I don’t know, but I can start with that and change it later. If you program the formula into your spreadsheet correctly, changing the Y just takes a moment.]
Now we can calculate each of the annual payments separately.
Let’s start with the furthest-in-the-future payment of $30,000.
The payment 10 years from now will have a future value (FV) = $30,000, a discount rate (Y) of 12%, and the number of years (N) is 10. Plugging those numbers into our present value formula of PV = FV / (1 + Y)^N, we get $30,000 / (1 + 12%)^10, which my spreadsheet tells me is equal to $9,659.
Now how about the next-furthest-away payment?
The payment due 9 years from now will have a FV of $30,000, a Y of 12%, and an N of 9.
Plugging that into the formula PV = FV / (1 + Y)^N we get $30,000 / (1 + 12%)^9. My spreadsheet produces a present value for this payment of $10,818.
I follow this process for all 10 separate annual payments—in fact this all takes about 8 seconds if you can do an auto-fill on your spreadsheet—and I add up all 10 annual payments to arrive at a value of $169,507, rounded to the nearest dollar.
This is a lot less than the $300,000 headline offer I thought I was getting at first glance.
Is That a Good Deal or Not?
That depends.2
At least now we have a way of comparing the 10-year installment payout offer versus another offer that might be for the entire company in one single payment.
If somebody offers $200,000 today, versus the $300,000 in 10 annual installments, a good case could be made that $200,000 today is worth more.
If you’ve got the discounting formulas set up on your spreadsheet, you can see that the comparison depends on what discount rate, or Y, you assume those 10 future payments deserve.
When I play around with assumptions on my spreadsheet, I see that an 8% discount rate makes the 10-year payout of $300,000 roughly equal to $200,000 today. If the payments are not very risky, or inflation stays low, 8% might be a fair discount rate. Again, this is a judgment call involving art as well as science. But the formula helps us look at those different scenarios, comparing money today versus multiple future payments.
Intuition Around Y, and Using Y to Help Your Intuition
At this point, using this example, it may be worth pausing to reflect again on the concept of Y, or discount rate.
Remember that the assumed Y takes into account a combination of three factors: prevailing interest rates, inflation expectations, and the risk of the future payments themselves. Some of this is not knowable, meaning Y must always be a “best guess” combination of art and science.
If you’ve got 10 future payments from our example programmed into your spreadsheet, all linked to the same discount rate (your Y value), you can use your spreadsheet to give you intuition around the effect of changing risk assumptions.
I mean by that the following. If your business is very risky, we would apply a very high discount rate to all future payments from the business.
Maybe 20%? Your spreadsheet will spit out a value of $125,774 for the sum of 10 years’ worth of $30,000 payments, discounted at a 20% rate.
Maybe 25%? Your spreadsheet will spit out a value of $107,115 for the sum of 10 years’ worth of $30,000 payments discounted at a 25% rate.
If you know your business is highly risky as you look to sell, you will intuitively know to take the $200,000 cash upfront, rather than wait a long time for $300,000 in cash. The discounted cashflows formula helps you put a mathematical value on that intuition, using a numerical representation of that risk. The numerical representation of that risk, again, is the variable Y.
To go in the other direction, maybe the future cashflows of your business, by contrast, are very safe? Maybe you only apply a 5% discount rate? Well then, the sum of all those future payments are worth $231,652 today, according to my spreadsheet. Take the long payout, because that’s worth more than the $200,000 all-cash offer. With a safe series of cashflows, you’ll have intuition that a long payout is OK. With the discounted cashflows formula, again, you can put a mathematical value today on that intuition. You begin to think and act like a fundamental investor.
More Complex Examples. The next few examples add yet another layer of complexity, but also get us even closer to how the discounted cashflows formula gets used in practice by investors, insurance companies, and banks.
Example 4: Multiple Future Cashflows, at Semiannual Intervals, Like a Bond
People who buy bonds, bond investors, are putting up money today in order to receive a series of future payments. Determining how much money a bond investor would pay for those future payments is the number one skill of a professional bond investor.
Bond investors use the discounted cashflows formula as the fundamental tool of their trade. With it, bond investors take each individual cashflow of a bond, discount it back to its present value, and add up all the different present values. The sum of all the present values matters because it’s how much any investor would pay for a bond.
Wall Street wizards in all their glorious mathematical ingenuity have come up with an infinite variety of bond types. In the interest of simplicity, however, we are not going to describe this delightful variety. I’m going to describe below the most boring animal in the bond zoo: a semiannual, fixed rate, “bullet” bond. It does not amortize, that is to say, pay off principal early. It does not have a floating interest rate. It pays all principal upon maturity. This is what we’d call a “plain-vanilla” bond, for those of you who enjoy ice cream metaphors.
Cashflows That Come More Frequently Than Annually
A typical bond pays investors semiannually, meaning two payments per year. Those semiannual payments will be one-half of the stated fixed interest rate—or coupon—of the bond.
So if we look to evaluate a 4% bond, then we would expect two payments per year, each consisting of 2% of the bond principal. I get to 2% because 2% is half of the stated 4% interest rate, but you knew that already.
If the stated fixed interest rate were 12%, we would expect two semiannual payments of 6%. If the stated fixed interest rate were 7%, we would expect two semiannual payments of 3.5%. By now most likely you are seeing the pattern.
You may be wondering, quite rightly, how much money gets paid by a bond, when stated in percent terms. I wrote “2% semiannual payment,” but you’re possibly thinking “2% of what?”
Good question.
The 2% (or whatever percent payment) is a percentage of the original principal of the bond.
Oh yeah, sorry, I forgot to mention: bonds (the plain-vanilla ones, anyway) always have a fixed amount of principal. This is also known as the face amount of the bond. This amount always has to be repaid at the end of the life of the bond, also known as the “maturity” of the bond.
The semiannual coupon is calculated as a percentage of the face amount.
So, for example, on my $1,000,000 bond, a 2% semiannual coupon will be $20,000, because that’s 2% of $1,000,000. Here are more examples so that the math sinks in. A 5% coupon on a $5,000,000 bond would be $250,000, because that’s 5% of $5,000,000. A 7% coupon on a $100,000 bond would be $7,000, because that’s 7% of $100,000. A 2.75% coupon on a $50,000 bond would be $1,375, because that’s 2.75% of $50,000. Again, by now you are most likely seeing the pattern.
As I mentioned above, the typical bond has a set principal payment that gets paid at the end, or maturity, of a bond. A typical 3-year bond, therefore, makes six semiannual coupon payments and then a final principal repayment at the end of 3 years.
To continue with our hypothetical $1,000,000 bond with a 4% interest rate, the payments would look like this:
6 months—$20,000
12 months—$20,000
18 months—$20,000
24 months—$20,000
30 months—$20,000
36 months—$20,000 + $1,000,000
To value a bond—to fully calculate its present value—we need to add up all of the separate present values of the interim coupon payments, plus the present value of the final maturity payment.
That means we take each cashflow made at different future time intervals, apply a discount specific to that time interval, and then add up all the present values to get a single value for the bond. This may sound complex when written out in words, but the use of a spreadsheet makes all this math pretty straightforward, as explained in the following section.
Calculating Six Present Values
Let’s take these bond cashflows one at a time, starting with the first $20,000 cashflow, due in 6 months. This is the point where you absolutely have to open up a spreadsheet for the following bit to make any sense at all. Otherwise you’ll just be reading my word salad jibberish. OK, deal?
We know the future value (FV) of the payment is $20,000.
For six regular payments at regular intervals, we can assign an N of one through six to each of the six future payments. For the first $20,000 due in 6 months, we can use an N of 1.
The discount rate (Y) requires us to mix a little artistic judgment to our financial science. Let’s say I expect the bond to be quite safe, and inflation to be low, so I will apply a 3% annual discount rate to the bond payments.
Really you can, and should, try applying different discount rates (Y) to these future payments. The best way to understand discounting cashflows is to model up future bond payments and insert different discount rates, and see how the present values change. This, again, is the fundamental task of professional bond investing. On this particular bond, if you use a 4% discount rate on all of the future bond cashflows, the value will be exactly $1,000,000. If you use a 5% discount rate, the present value will be less than $1,000,000. This is how some bonds can be worth less than their original principal amount (applying a high discount rate, or yield), while other bonds can be worth more than their original principal amount (applying a low discount rate, or yield).
Adjusting Y for Semiannual Payments
Up until this example, we have used Y as an annual discount rate.
In this example, however, before we calculate the present value of our first payment, we have to make an additional adjustment for the discount rate of the payment, because it’s not made at an annual interval.
Because we usually discuss discount rates in annual terms, we can accurately say we will apply a 3% annual discount rate.
For a payment made in 6 months, however, we have to apply a discount rate that’s been adjusted for semiannual payments.
We always do this by dividing our discount rate (Y) by the number of payments being made in a year, or the variable P.
In this case, for example, two semiannual payments require us to divide our 3% Y by 2. If we worked with quarterly payments we would have to divide our annual-rate Y by 4.
A few more examples may help set the pattern. If we analyzed payments made monthly, we would divide Y by 12. If we analyzed payments made six times per year, we would divide Y by 6. By now you are probably getting the pattern.
Applying a 3% annual discount rate to my bond cashflows, therefore, requires me to input 1.5% as my adjusted Y in my spreadsheet.
For payments made more frequently than annually, you can elect to follow the pattern I’ve set above, which is adjust Y by dividing your annual Y by the number of payments per year. Or, to get the same result, you can introduce a variable p in your discounted cashflows formula, in which p represents the number of payments made per year, and the formula could be updated to PV = FV / (1 + Y/p)^N.
Back to the First Coupon Payment
All right, so, now we can input in our spreadsheet the proper values to figure out the present value of our first coupon.
We know present value of the first coupon (PV) = FV / (1 + Y/p)^N, so I program my spreadsheet to tell me the dollar value today of $20,000 / (1 + 1.5%)^1.
My handy spreadsheet tells me that’s worth $19,704, rounded to the nearest dollar.
The Rest of the Payments
Next, I program my spreadsheet to tell me the dollar value today (PV) of the second coupon. Notice that as we’ve moved away from annual discounting to semiannual discounting, the variable N is no longer number of years, but rather number of discounting periods.
The N of my second time period is 2, so my PV formula must be $20,000 / (1 + 1.5%)^2. My handy spreadsheet spits out a value for me of $19,413, rounded to the nearest dollar.
So far, so good. Same method, third coupon, $20,000 / (1 + 1.5%)^3, gets me a present value of $19,126, rounded to the nearest dollar.
Remaining payments on my spreadsheet look like this:
Payment 4: =$20,000 / (1 + 1.5%)^ 4= $18,844
Payment 5: =$20,000 / (1 + 1.5%)^5 = $18,565
Payment 6: = $1,020,000 / (1 + 1.5%)^6 = $932,833
Remember to notice that Payment 6 is a combination of the sixth coupon payment and the return of the original principal of $1,000,000, as is typical of a plain-vanilla bond.
When you add those all up, you get a value of $1,028,486, rounded to the nearest dollar.
In plain language, a bond investor seeking a 3% yield from this bond would be willing to pay $1,028,486. The price a bond buyer is willing to pay flows directly from the discount rate he or she applies to all future bond cashflows. In the bond investing world yield, or discount rate, is the most important choice an investor makes. The price, or present value, flows mathematically from the choice about yield.
A Bit More on Bond Pricing, Probably More Than You Really Want to Know
In the bond world, professionals quote bond prices as a percentage of 100%, so they might refer to this bond above as a bond priced at 102.8486 or—to further confuse outsiders3—the professionals might quote this in fractional terms, such as 102 and 27/32nds, which is a roughly, but not precisely, accurate price.
Now, if that same bond investor sought to earn a 5% annual yield, the bond investor would need to update this annual Y to 5, and an adjusted Y/p of 2.5% applied to each semiannual cashflow in the spreadsheet. If you’ve built your spreadsheet formulas already, you can see how switching Y/p to 2.5% changes all of the present values immediately.
The new value of a 4% coupon bond shifts to $972,459, or 97.2459, or maybe just 97¼ for short.
Example 5: Discounting Multiple Future Cashflows at Quarterly Intervals, Like a Stock
Investing in stocks, for the fundamental investor, is really just the purchase in today’s dollars of future cashflows, in the form of profits, if any, and dividends, if any. In that sense, the skill of fundamental stock investing is to discount the value of all future cashflows into present values. The sum of all these cashflows then becomes the price at which you should pay for a stock.
The Example of a Risky Stock
Let’s say we were looking at buying stock in a particularly risky company. It’s so risky, in fact, that we should apply a 35% discount rate to all future cashflows. With our limited amount of funds, we can purchase one millionth of the company, which coincidentally (to keep our math kind of simple) produces precisely 4 million dollars in profits a year, or 1 million dollars per quarter of a year.
As a purchaser of one millionth of the company, we would be buying $4 of profit per year, or $1 in profit per quarter.
OK, that’s the setup. How much is our stock worth?
Admittedly I’ve created a simple scenario, but one I hope will illustrate how fundamental stock investing can work. Because given all the information I’ve provided, we can know the fundamental value of this stock precisely!
In one quarter of a year from now, with our stock purchase, we will in essence “own” a $1 cashflow, in the form of our one millionth ownership of this risky company’s profits.
How much is that $1 worth, discounted by 35%, for one quarter? Easy-peasy, with our discounted cashflow formula:
We have future cashflow (PV) of $1.
We have a discount rate (Y) of 35%.
We have a number of periods in the year (p) of 4, since we’re counting quarterly profits.
And we are looking at the first period (N), which is one quarter from now.
I calculate this cashflow as $1 / (1 + 35% / 4)^1.
I plug that into my spreadsheet and find out that it equals $0.92.
I can then repeat the process for the next few quarters and then the next few years’ worth of $1 cashflow profits, which in my spreadsheet is the matter of about 8 seconds’ worth of work using the auto-fill function.
Quarter 2 profits will be $1 / (1 + 35% / 4)^2—which is equal to $0.85
Quarter 3 profits will be $1 / (1 + 35% / 4)^3—which is equal to $0.78
Quarter 4 profits will be $1 / (1 + 35% / 4)^4—which is equal to $0.71
I can continue to calculate each future cashflow separately, and then add them all up in my spreadsheet.
Adding up the next 20 quarters, or 5 years’ worth of profits, I get a value of $9.29. If I extend my spreadsheet out to say, 50 years’ worth of future $1 quarterly profits, the total value of all the discounted cashflows is $11.43.
When I do that, I can say that the stock is fundamentally worth $11.43.
This is super-useful to know, because if the stock trades at something less than $11.43, I could reasonably argue that on a fundamental basis, the stock is cheap. If the stock trades at something higher than $11.43, I could reasonably argue that the stock is expensive. This, right here, is exactly what all fundamental stock investing attempts to do. You model up all the future cashflows of a business, then discount cashflows to the present day, and add them all up. The sum of all the cashflows is the fundamental value, today, of that business.
Interestingly, with a 35% discount rate, I only need to extend my calculations to about 15 years before the value of each quarterly $1 future profit shrinks to an infinitesimally small size, a size that no longer affects the present value of the company. In plain terms, for a highly risky company with a 35% discount rate applied to its profits, the far-out future (beyond 15 years) doesn’t matter all that much. For safer companies, with lower discount rates, far-out future years matter more to fundamental value.
A Few Observations of Fundamental Investing
Although I’ve kept the example extremely simple, I hope this bit of math above explains a few phenomena of fundamental stock investing. For example:
This discounting of future profits of a stock explained above is the basis for what’s known as fundamental or “value” investing. The godfather of this approach is Benjamin Graham (and his more-famous disciple, Warren Buffett), and I recommend Graham’s book The Intelligent Investor for more on the use of discounted cashflows to invest this way.
Buffett famously rejected investing in volatile technology-based businesses, presumably because he found it too difficult to accurately discount future, unknowable, cashflows.
And Now, for the Unfortunate Reality of Stock Investing
In real-life investing, unfortunately, all the information is not as clear and unambiguous as I’ve presented it in the above example.
We generally wouldn’t really know, for example, that a company will produce profits, like clockwork, of $1 per quarter. Profits could go down or profits could go up, complicating our calculations. The imprecise nature of modeling future profits means fundamental investing is not as mathematically precise as it at first appears.
Similarly, we don’t really know if 35% is the right discount rate to apply to future quarterly profits.
If the company for some reason became less risky, and I could reasonably apply a more generous discount rate like 25% to reflect that decline in risk, suddenly the sum of present values of future cashflows jumps to $16, and we need to consider up to 25 years, or even more, worth of future quarterly profits in our calculation.
I have two other, seemingly contradictory thoughts, on this idea. I hope that the fact that they are contradictory is not too off-putting. The world, including the investment world, is a complex, sometime contradictory, place. The following two contradictory truths are one example of this complexity.
First, anyone who is NOT discounting cashflows like this is—in some fundamental way—not actually investing. They are gambling, or guessing, or counting on the “greater fool” theory, or praying, or punting, or whatever. The greater fool theory incidentally is the idea of buying something in the hopes that it will go up in price, purchased by a greater fool than me, so I can make a profit.
Second—and here’s the contradictory idea—applying the correct discount rate and correctly assuming future cashflows in fact requires a tremendous amount of guesswork about fundamentally unknowable future events.
So even if you’re doing everything “correctly” the way Benjamin Graham and Warren Buffett would advocate, you’re still in a way gambling or guessing or praying or punting or whatever. How can anyone really know what the profits of any company will look like in 5 years? You can’t know. So frustrating.
I don’t say this to endorse gambling and praying and punting with your money, but rather to introduce an essential modesty about how much anyone can actually know about future unknowable things.
What I mean is this: the more certain the rocket-scientist financial modeler seems—even one properly discounting cashflows—the more skeptical we need to be to counteract that certainty.
This ends the Appendix on discounting future cashflows, which I hope has given some insight into how this skill is the fundamental basis for all legitimate investing activity. You don’t have to understand this to be financially savvy, but I hope you will be left with the following ideas.
Be skeptical of complicated-seeming investment presentations. At base, all of it comes down to variations on this chapter’s math.
Be optimistic that with a little bit of practice and some time, you could do all the math required for fundamental investing.
Understanding how much time and effort it could take to actually come up with the correct estimate of future cashflows, plus their proper discount rate, you might be left feeling modest about your own ability to do this better than someone else.
These are all healthy responses to learning about discounting cashflows.
1If you have an investment adviser, or if you are considering hiring one, try out the test above on your person. Say, “I’m going to inherit $5,000 in exactly 5 years. Assuming a 4% discount rate, what is the precise present value of that inheritance, today?” I wonder how many of your candidates will know to plug in $5,000 / (1 + 0.04)^5 to a spreadsheet or their calculator to get $4,110? Possibly very few.
2I’m skipping the issue of whether there are tax differences between being paid money upfront and receiving installment payments. Taxes are important—although as I emphasize in Chapter 10, “On Taxes,” not too important. For the moment, we’re trying to simplify for the sake of understanding the math, so we disregard the taxes to focus on the math theory.
3It’s not really to confuse outsiders, but rather is a historical convention of the bond market to use fractions for pricing rather than decimals. The larger the size of the bond and the more efficient the bond market pricing, the more likely the price would be decimalized rather than fractionalized. But, whatever.