# 4.3: The Discounted Cash Flow Model

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A powerful version of the Dividend Discount Models is the __Discounted Cash Flow Model__. This model states that the value of a stock is equal to the present value of all its expected future cash flow, not just its dividends. The Discounted Cash Flow Model formula is:

The cash flow for each year, CashFlow_{1}, CashFlow_{2}, CashFlow_{3}, etc., is divided by the quantity (1 + Rate)^{1}, (1 + Rate)^{2}, (1 + Rate)^{3}, etc., for as many years as we can reasonably estimate the future cash flows. Relax. **We promised you that you did not have to do any exponentiation and we are not going back on that promise.** We will do the exponentiation for the following example and then show you how we avoid the exponentiation. Your 99¢ calculator will still suffice.

Let’s say a company is going to pay their shareholders three annual dividends of $10 per share and from then on, there will be no more dividends. Let’s set our required rate of return to 7%. The calculations are:

` $10 $10 $10 $10 $10 $10`

Value **= ** —————— + ——————— + ——————— **= ** —————— + ———————— + ———————— **≅ ** $26.24

1.07 1.07^{2} 1.07^{3} 1.07 1.1449 1.2250

The model is saying the company’s stock is worth $26.24. But how often do companies pay three annual dividends and then promptly go out of business?! Plus, we keep using this term “present value.” *What does present value mean anyway?*

# What is Present Value?

__Present value__ is the value today of a lump sum principal or series of payments to be received at some future date. It is the opposite of __future value__. Present value and future value are inverse operations. We say that present value and future value are “two sides of the same coin.” Future value tells you what a single investment or series of investments made today will be worth in the future. Present value tells you what a single quantity or series of quantities that you receive in the future is worth today in the present. The present value tells us what cash flows received in the future are worth today.

There were some optional future value calculations in chapter 1. One of the problems asked you to compute the future value of a $10,000 lump sum investment in 10 years if we received an 10% average annual rate of return. We could use the exponential formula but instead, we use the future value multiplier from the __future value table__. We go across to 10% and move down to 10 years and find that the future value multiplier is 2.594.

We then multiply the $10,000 investment by the future value multiplier of 2.594 and that gives us a result of $25,940. The future value of $10,000 at a 10% average annual rate return for 10 years is $25,940. If we invest $10,000 today and receive a 10% rate of return, in 10 years, we will have $25,940.

What if we wanted to do the opposite? What if we wanted to determine how much a result that we receive in the future is worth to us today in the present? In other words, what is the present value of that payment that we will receive in the future? Again, there is an exponential formula but we promised you that you would not have to use exponents. We compute the present value using the same technique as the future value calculation except we will use the __present value table__. Let’s say we are going to receive $25,940 in 10 years. If we desire a 10% annual rate of return, what is that $25,940 worth to us today? In the present value table, go across to 10% and down to 10 years. The present value multiplier is 0.386.

We multiply the $25,940 by the present value multiplier of 0.386 and the result is $10,012.84. The result is not exactly $10,000 because the table is only using three digits of accuracy. If we had used 7 or 8 digits after the decimal point, we would receive $10,000 exactly as our result.

So, what would you rather have, $10,000 today or $25,940 in ten years? If our required rate of return is 10%, they are equivalent. The future value of $10,000 in 10 years at 10% is $25,940. The present value of $25,940 in 10 years at 10% is $10,000. They are two sides of the same coin! Calculating the future value of investments made today is called __compounding__. Calculating the present value of payments received in the future is called __discounting__. Oh, no! There’s yet another weird sounding phrase! I know. I know. “Discounting a stream of future cash flows” sounds kinda’ dumb but get used to it because those are indeed the words we use. Look, all it means is that we are going to look up numbers in the present value table and then multiply them. You Can Do It! Again, all you need is a 99¢ calculator and the __present value table__.

# What is Discounting a Stream of Future Cash Flows?

We are going to take payments that we receive in the future and calculate what those future payments are worth today. We say that “we are discounting those future payments” back to the present. The process is actually very easy but the words get in the way. Let’s use the __present value table__ and redo the example above of three annual dividends of $10 at an annual rate of 7%. We go to the present value table and find the present value multipliers for years 1, 2, and 3 at 7%.

The present value multipliers are 0.935, 0.873, and 0.816. We then multiply the future cash flows of $10 per year by each of the three present value multipliers:

Value **=** $10*0.935 + $10*0.873 + $10*0.816 **= ** $9.35 + $8.73 + $8.16 **= ** $26.24

Now that wasn’t so bad, was it? For many students, doing the calculation is often much easier than understanding what the terms present value and discounting actually mean. So please do not worry if you are still a bit confused about what the words mean. Just do the calculations … over and over and over again. The meanings will shine through soon. Another way of displaying the problem involves using a table:

Year |
Future Cash Flows |
Present Value Multipliers_{7}_{%} |
Discounted Cash Flows |

#1 | Dividend of $10 | 0.935 | $9.35 |

#2 | Dividend of $10 | 0.873 | $8.73 |

#3 | Dividend of $10 | 0.816 | $8.16 |

Total: | $26.24 |

Many find the table format is much easier to use. You multiply each year’s future cash flow by the year’s present value multiplier to get the discounted cash flow. You then compute the sum of the discounted cash flows. That gives you the present value of the future stream of cash flows.

# Examples of the Discounted Cash Flow Model

We are ready to put the Discounted Cash Flow Model into practice. Our first example did not take into account that the stock will still have worth at the end of the three years. To make the model more useful, we simply add our predicted market price of the stock at the end of the three years to the present value calculations. We treat the price of the stock at the end of the three years as a future cash flow that needs to be discounted. What if the current stock price were $125 and we predicted the stock price to be $135 at the end of three years? We add the price of the stock in the last year to the table above:

Year |
Future Cash Flows |
Present Value Multipliers_{7}_{%} |
Discounted Cash Flows |

#1 | Dividend of $10 | 0.935 | $9.35 |

#2 | Dividend of $10 | 0.873 | $8.73 |

#3 | Dividend of $10 | 0.816 | $8.16 |

#3 | Expected stock price of $135 at the end of year #3 | 0.816 | $110.16 |

Total: | $136.40 |

Notice that the present value multiplier for the expected stock price is the same as the dividend future cash flow in year #3. This is because the dividend in year #3 and the expected stock price at the end of the third year are both cash flows that we receive in the third year. We therefore use the same present value multiplier. When we sum the discounted cash flows in the last column, we compute a present value of $136.40. The current stock price is $125. The model is telling us that if we require a 7% annual rate of return, the stock is worth $136.40 while the marketplace is offering us the stock at $125. The model is saying that this stock is a potentially good investment for us if our desired rate of return is 7%.

DISCLAIMER: WARNING: REPUDIATION: DISAVOWAL: REFUTATION: ABNEGATION: RENUNCIATION: In no way should we make a final decision, either yea or nay, about whether or not we should buy or sell any stock simply based on the results of this or any other model. We are using these models to point us in the right direction. We are attempting to tilt the odds in our favor. We have a whole lot more research that we need to do before we actually decide to choose a stock as one of our investments. Got it? Good. Just wanted to make sure. Let’s continue.

Example #2: Pretzels Unlimited, symbol PU, is a stolid, imaginary company that has been making pretzels and other baked goods for almost 100 years. Their stock is currently selling for $22 per share and will pay $2.00 per share in dividends in 2023. PU expects to increase their dividends to $2.20 in 2024, $2.30 in 2025, and $2.30 in 2026. We will be selling the stock at the end of 2026 and we expect the price to be $27 per share at that time. Our required rate of return is 12%. We put the years in the first column and the future cash flows in the second column.

Year |
Future Cash Flows |
Present Value Multipliers_{12}_{%} |
Discounted Cash Flows |

2024 | $2.00 | 0.893 | $1.786 |

2025 | $2.20 | 0.797 | $1.7534 |

2026 | $2.30 | 0.712 | $1.6376 |

2026 | $2.30 + $27 = $29.30 | 0.636 | $18.6348 |

Total: | ≅ $23.81 |

Find the present value multipliers for years 1 through 4 at 12% in the __present value table__. Did you find 0.893, 0.797, 0.712, and 0.636? The present value multipliers go in the third column. With your 99¢ calculator, multiply the future cash flows by the present value multipliers to compute the discounted cash flows in the last column. Last, sum up the discounted cash flows in the last column to compute the present value. The model is saying that PU is worth $23.81 if we require a 12% rate of return. However, the price of the stock is only $22. The model says that this stock is possibly an attractive investment for us. We need to do much more research but this is a good start. PU is a stock that might just make it into our portfolio.

Notice that we added the expected stock price in the last year to the dividend in the last year. This allowed us to skip a multiplication. But more importantly, it also allows us to utilize a very powerful spreadsheet function to calculate the Internal Rate of Return.

# The Internal Rate of Return

The __Internal Rate of Return__ is a measure of what rate of return we expect to get from a series of cash flows, including positive and negative flows. In other words, we required a 12% rate of return from Pretzels Unlimited, but what do our numbers tell us will be our expected rate of return? Someday, when you take an upper-level or graduate finance or investment class, you will learn how to manually compute Internal Rate of Return. Hopefully, you will not have a sadistic professor who will require you to calculate it manually more than once! We are simply going to enter the numbers into the spreadsheet formula and press the [Enter] key, okay?

For those not familiar with an electronic spreadsheet, know that they are just like our 99¢ calculator, just a whole lot more powerful. We use Google Docs which is free to anyone who has a Google account. You may use Microsloth Excuse or maybe even LibreOffice or OpenOffice. They all work the same, kinda’ like a giant Bingo game. The spreadsheet formula is:

=IRR(**values**,**approximate-rate-of-return**) where

**values** is the block of cells containing the cash flows, both positive and negative, and

**approximate-rate-of-return** is our guess as to what the Internal Rate of Return will be

In cell C10, the formula is =IRR(C4:C8,0.12) This tells the spreadsheet to use the cash flows in cells C4, C5, C6, C7, and C8. The 0.12 (12%) is our guess of what the result will be. We can just leave it as zero and ask the spreadsheet to do its best to find the result. Notice that we must include in the initial price of the stock as a cash outflow, a negative number, in cell C4. Internal Rate of Return calculations require all cash flows, both inflows and outflows. So, what is the result from the Internal Rate of Return calculation telling us? If we pay $22 for the stock today and then we receive the expected dividends over the next four years and the stock is worth $27 at the end of the four years, our Internal Rate of Return will be 14.51%, better than our 12% desired rate of return.

Here is the __spreadsheet on the class website__. You will notice that it has two pages. The first page allows us to simply put in the cash flows and the spreadsheet will automatically calculate the present value. There is no need for us to look up the present value multipliers, multiply, and then sum the results. The second page automatically calculates the Internal Rate of Return formula calculation. Pretty handy, these electronic spreadsheets! And they are free, too, if we use Google Docs or OpenOffice or LibreOffice. Who needs to pay some company that shall remain unnamed (Micro$oft) that treats its customers like vermin, continually requiring them to endure painful and costly upgrades in the hopes that someday, somehow, their programs might actually work? (I know. I have a pathological disgust of that unnamed company. Nobody’s perfect, eh?)

# But What If a Company is Not Paying Any Dividends?

When we reviewed the problems with the initial Dividend Discount Models, we found that they simply did not work if the company was not paying any dividends. The present value of nothing received in the future is zero. This is not the case with the Discounted Cash Flow Model. Unlike the other Dividend Discount Models, the Discounted Cash Flow Model can still be used if there are no dividends. We simply treat the expected future price of the stock as a single future cash flow. Very cool!

Example 3: Genes ’R’ Us, symbol GRUS, an exciting, dynamic, make believe San Diego-based biotechnology company, is currently selling for $21 per share. It pays no dividends and is currently losing money. They are working on a drug that will cure baldness. We believe that GRUS will sell for around $50 per share in five years. Our required rate of return is 13%. How can we determine if this is a potentially good investment?

Let’s construct the the cash flow table:

Year |
Future Cash Flows |
Present Value Multipliers_{12}_{%} |
Discounted Cash Flows |

2023 | $0 | 0.885 | $0 |

2024 | $0 | 0.783 | $0 |

2025 | $0 | 0.693 | $0 |

2026 | $0 | 0.613 | $0 |

2027 | $0 + $50 = $50 | 0.543 | $27.15 |

Total: | $27.15 |

Actually, we did not need to construct the entire cash flow table. All we really needed was 2027, year #5, since the present value of zero dividends is zero. We could have simply multiplied $50 by the present value multiplier for 13% for 5 years, 0.543. $50 future value times 0.543 equals $27.15. The model is telling us that we believe GRUS is worth $27.15 while it is selling for only $21 per share. Once again, the model is pointing us in the direction of the company as a potentially worthwhile investment.

Unlike the table above, when we use an electronic spreadsheet to calculate the Internal Rate of Return, we are forced to include all the years, even those with no cash flow:

Year |
Cash Flows |
Internal Rate of Return |

($21.00) | Initial cash outflow is negative $21.00 | |

2023 | $0 | There are no dividends |

2024 | $0 | |

2025 | $0 | |

2026 | $0 | |

2027 | $50.00 | Expected price of stock in 5 years |

18.95% | =IRR(B2:B7,0.13) |

If Genes ‘R’ Us does reach $50 in five years, then we will have achieved almost a 19% rate of return. Pretty awesome!

*Wait a minute!* Hopefully, by now, you can now look at both Pretzels Unlimited and Genes ‘R’ Us and make some simple observations. Which company is the safer alternative? Which company is offering their investors cash each year and growing that stream of income? If you answered, Pretzels Unlimited, you have been paying close attention. Which company is the more risky investment? Which company offers the potential for great reward but also could fall down, crack open, and dissolve into a pool of tears? If you answered, Genes ‘R’ Us, give yourself a gold star for today’s very important lesson.

Pretzels Unlimited could easily be one of the blue-chip or income-oriented companies, big, stodgy, growing at a very slow pace or simply not growing at all and throwing off gobs of cash to their investors because they just don’t need the money to reinvest in the company anymore. Genes ‘R’ Us, on the other hand, could easily turn our $500 or $1,000 investment into $50,000 if they hit the big time because their drug works and is approved by the authorities as safe and effective and everyone who is bald is going to fork over gobs of money to buy it. However, Genes ‘R’ Us could also easily turn our $500 or $1,000 into 50¢ when it turns out the drug doesn’t work or turns people’s livers into pate. If the truth be told, a half million years ago, Your Humble Author was a sucker for these small, biotechnology startups based here in San Diego. I would go to the annual meetings and talk to the employees. The technology was so cool and was going to change the world and, uh, well, it didn’t always work out the way it was supposed to. Luckily for me and my wife, I only put a tiny percentage of our investments into these very speculative ventures. I called it our “Vegas Fund” … and it lived down to its name. Now I concentrate on companies like Pretzels Unlimited and call the account our “Benjamin Graham Fund.”

We are not saying that you should never choose a Genes ‘R’ Us as one of your investments. We are not saying that you should only choose companies like Pretzels Unlimited. However, for the vast majority of us individual investors without the benefit of global research teams based all around the world speaking dozens of languages, the Pretzels Unlimited’s of the world are more likely to help us successfully build prudent, long-term wealth. How ‘bout this strategy? For every one Genes ‘R’ Us you find, choose four or five Pretzels Unlimited’s. Would that work for you?

What is most important from this discussion is that you learn to identify the risks inherent in the companies you research. We want you to have your eyes wide open. If you do choose a speculative issue such as Genes ‘R’ Us, have the courage of your conviction and we wish you the best of luck and success. But realize that you are assuming a large risk. You are taking a big gamble. Be prepared for volatility. (Translation from personal experience: “I bought it at $11.88 and I sold it at 30¢.” That was Alliance Pharmaceuticals. They were working on artificial blood! No more blood banks or pleas for people to donate blood! Uh, don’t bother looking for them. Alliance Pharmaceuticals is gone, not the blood banks. The SEC officially revoked their securities in 2013 but they were long gone way before then. That was one of my speculative issues. Remember that speculation is our industry euphemism for, “Aye! I lost a lotta’ money!”)

You know what is next, right? We once again revisit Mr. Benjamin Graham’s definition of an investment:

“An investment operation is one which, upon thorough analysis promises safety of principal and an adequate return. Operations not meeting these requirements are speculative.” ‒

The Intelligent Investor, Benjamin Graham

We also revisit our original definition of an investment:

“An investment is any vehicle into which resources can be placed with the expectation that it will generate positive income, or that its value will be preserved or increased, or both."

So now when you approach a potential investment, you will look at it with fresh eyes. You will investigate what type of cash flows the investment will return to you in the form of income or capital gains or both. You also now have the tools to value those future cash flows, the Dividend Discount Models, including the very powerful Discounted Cash Flow Model, and the equally powerful Internal Rate of Return calculation, given to you courtesy of an electronic spreadsheet. You will also be able to estimate the relative risk of the investment. Congratulations, Dear Students, you are now official Investment Gurus!

# You Must Learn How to Discount a Future Stream of Cash Flows!

We know that for some individuals, as soon as they see numbers and formulas and symbols and calculations, their mouths become dry, their eyes gloss over, and they vow to completely ignore whatever they see in front of them. Don’t do it! Don’t Give Up! Never Give Up! Go back and read the above sections again. Listen to or watch the accompanying presentation. Practice the calculations and the worksheets. There are answer keys and commentaries. Do them. If they don’t make sense and you are confused and you get the wrong results, go play volleyball or walk the dog or ride your bike. Then come back and do it all over again. You Can Do It! The calculations are very easy once you do them a few times. Remember that all you need is a 99¢ calculator and the present value table. Learn to use the free electronic spreadsheets and you don’t even need the calculator or present value table. You can’t leave BUS-123, Introduction to Investments, without knowing how to discount a future stream of cash flows. These calculations are going to be on exam #2, exam #3, exam #4 and the final exam so you may as well learn how to do them now. *You must learn how to discount a future stream of cash flows!* (Please. Remember it is really bad for my self-esteem if you don’t.)

# Other Valuation Models

As mentioned, there are __numerous valuation models__. We have concentrated on the Dividend Discount Models. One of my favorite aspects of investing is that a person will never, ever learn all that there is to know about investments. You have the rest of your life to explore the various models. Please contact me when you find a model that is as good or better than the Discounted Cash Flow Model, okay?

You may be thinking, “Okay, Mr. Know-It-All, this is all great, but just where are we supposed to get all this historical information, anyway? And just who decides what next year’s earnings or dividends per share, the dividend growth rate, etc. are going to be, let alone the expected price of a stock in 3 to 5 years?!”

Before the Internet (BI?), this information was not readily available. Normally, you would ask your broker for it or you would use one of the securities industry’s trusted information sources. Traditionally, the most respected source was *The Value Line*, the subject of our next section.