Is a share of Costco Wholesale Corporation (NASDAQ:COST) worth $16.04 … or … $67.07 … or … $154.02? Actually all of these are mathematically correct according to standard valuation methods and tables used by professional analysts through to DIY investors. Of course, even though they are mathematically correct, the results are no use for anyone considering buying or selling shares in Costco. These contradictory valuation results arise because of three fundamental weaknesses with the standard intrinsic valuation methods.

The first weakness is that there are actually dozens of methods that purport to calculate the true value of a stock providing a bewildering range of outcomes. In some cases one method will show a stock is highly undervalued, others that seem just as reasonable will show that it is completely overvalued.

On top of this variation between the methods, the second weakness is that most of the methods show even more variation within their calculations. Changes of a few percentage points in the inputs give results that differ by two or three hundred percent. A common outcome is that investors and analysts unwittingly manipulate the results to support their prior opinions. While appearing to be objective, the outcome is really highly subjective.

The third weakness with the standard value formulas is that they do not have a time component. Even if we could know that a stock is 50 percent undervalued, there is a world of difference between the price moving to the intrinsic value within a year or as long as a decade. In the first case it would be a wonderful investment, in the second probably not much more than a bank rate.

Finally, all these weaknesses are made worse by what I call mathematical intimidation. This is the use of opaque mathematics, computer programs or tables (which are often quoted second-hand to begin with) without understanding the assumptions and logic behind them.

Fortunately, none of this is necessary for sensible, rational investing. In fact, in this article I will show how a careful analysis of one intrinsic value method brings us back to what we could have been doing all along, finding companies with confident growth in earnings trading at prices that will lead them to be Wealth Winners^{®}.

Over the past 15 years I have personally programmed, examined, tested and compared every method that I could find. In my recent book *The Conscious Investor* (Wiley, 2011) I closely examine and describe the most well-known of these methods.

The methods range from balance sheet methods (particularly those developed by Benjamin Graham), to discounted cash flow methods (where the outcome is usually referred to as intrinsic value), to dividend discount methods (while still an intrinsic value method, includes many variations such as return on equity, residual income valuation, and abnormal earnings growth), payback methods (that estimate the time for dividends to pay back the price of the share), and expected return methods (that estimate the expected average return of an investment). I also look at various filtering methods such as magic formula investing by Joel Greenblatt, CANSLIM by William O’Neil and factor models by Robert Haugen.

In each case I look at their assumptions followed by their strengths and weaknesses. Some of the assumptions are fairly innocent and open such as working with the entries in the balance sheet. Others require making forecasts over shortish time periods. But the most problematic assumptions require making forecasts for an infinite number of years.

## Dividend Discount Valuation

Instead of looking at the standard discounted cash flow method, consider the dividend discount method. It asserts that the true or intrinsic value is the sum of the discounted values of the dividends over the entire life of the company which is assumed to be infinite. This is the same as the discounted cash flow method except that free cash flows are replaced by dividends.

To get around the problem of forecasting dividends out to infinity, sometimes this method is recast in terms of return on equity and the payout ratio. It also uses an assumption called the clean surplus relationship which asserts that book value (equity per share) at the end of a period is equal to the book value at the start plus earnings less dividends paid.

The simplest variation assumes that ROE and the payout ratio remain constant. Next, if you know the initial book value, it is easy to calculate the earnings and dividends for the first year. Now use the clean surplus relationship to calculate the book value at the end the year. Repeating this process allows you to calculate the dividends out to infinity. Finally discount these dividends and add them up using the mathematical theory of infinite series to get the intrinsic value.

Consider Costco. (NASDAQ:COST) Over the past ten years its ROE has been as low as 10.8 percent and as high as 14.0 percent. Its average has been around 12.0 percent so we will use this as our input into the formula. (There is a difference when calculating ROE whether you use the equity at the start of the period, the end of the period, or an average of the two. For simplicity in the calculations we will suppose that the equity we are assuming is at the start of each financial year. Also, for simplicity the amounts will be written using two decimal places whereas they are carried to full accuracy.)

Similarly over the past five years the dividend payout ratio has ranged from 21 percent to 28 percent. However, in the last two years is has been 26 percent and 28 percent so we will assume that it is going to be 27 percent in the future.

Currently the book value of Costco (NASDAQ:COST) is $25.67. It is a simple matter to bundle these facts together to calculate the earnings and dividends of Costco running out into the future. As a start, since the book value is $25.67 and ROE is 12.0 percent, the earnings per share during the first year must be 12 percent of $25.67 which is $3.08. Taking the payout ratio as 27 percent means that the dividends must be 27 percent of $3.08 or $0.83.

The next step is to apply the clean surplus relationship to calculate the book value at the end of the first year. In this case, the required book value is $25.67 + $3.08 - $0.83 or $27.92. Now repeat this step for year 2 and so on.

One more thing. We need to include a discount rate. Basically this is the rate that we would like to earn to compensate for the risks associated with investing in Costco. Assume that it is 10 percent.

The table shows the results when we do this for 10 years. I have also included years 20 and 100. People who use these methods actually implement mathematical formulas, or use tables calculated from these formulas, which calculate the result for an infinite number of years. In this case, the result is $67.07. This means that if we kept doing the calculations year after year, eventually the sum of the discounted dividends would converge to $67.07.

Notice in the table that earnings, dividends and book value are all growing at the rate of 8.76 percent per year. This is a consequence of the initial assumptions on ROE and the payout ratio.

Year | Book Value at Start of Year | Earnings per Share | Dividends per Share | Book Value at End of Year | Discounted Dividends | Running Total |

1 | $25.67 | $3.08 | $0.83 | $27.92 | $0.76 | $0.76 |

2 | $27.92 | $3.35 | $0.90 | $30.36 | $0.75 | $1.50 |

3 | $30.36 | $3.64 | $0.98 | $33.02 | $0.74 | $2.24 |

4 | $33.02 | $3.96 | $1.07 | $35.92 | $0.73 | $2.97 |

5 | $35.92 | $4.31 | $1.16 | $39.06 | $0.72 | $3.70 |

6 | $39.06 | $4.69 | $1.27 | $42.49 | $0.71 | $4.41 |

7 | $42.49 | $5.10 | $1.38 | $46.21 | $0.71 | $5.12 |

8 | $46.21 | $5.54 | $1.50 | $50.26 | $0.70 | $5.82 |

9 | $50.26 | $6.03 | $1.63 | $54.66 | $0.69 | $6.51 |

10 | $54.66 | $6.56 | $1.77 | $59.45 | $0.68 | $7.19 |

… | … | … | … | … | … | … |

20 | $126.57 | $15.19 | $4.10 | $137.66 | $0.61 | $13.61 |

… | … | … | … | … | … | … |

100 | $104,683 | $12,562 | $3,392 | $113,853 | $0.25 | $45.49 |

**Table: Return on Equity Valuation**

Often the year by year convergence to the final value is painfully slow. For example, after 10 years the result is $7.19, giving a huge error of 89 percent compared to the true value of $46.60. After 20 years the table value is $13.61, giving a sizeable error of 80 percent. Even after 100 years the table gives $45.49, so the error is still approximately 32 percent. What this shows is that the accuracy of the ROE formula for intrinsic value relies on forecasts decades, and even centuries, into the future.

Unfortunately it gets worse. Suppose you are worried about the impact of the US economy on Costco and increasing competition from other membership-based wholesale companies. You think that for the next 10 years the ROE will be 10 percent and after that it will be 8 percent. These are still excellent levels way above the majority of companies. But even with these slight changes, the intrinsic value drops to $16.04.

But wait. You believe that Costco will have increasing success overseas and that it moves into China. You reckon that the ROE will be maintained but the board will lower the payout ratio to level it was four years ago, 20 percent, to free up more money for further overseas expansion. Now the intrinsic value jumps to $154.02.

In other words, with fairly small changes in the assumptions that go into the ROE method the intrinsic value can swing from a low of $16.04 to a high of over $150. We really have no idea whether Costco is undervalued or overvalued, whether we should buy, sell, or do nothing.

## Where does price fit in?

Another confusion concerns the role of price and P/E ratios in making investment decisions. While the theory may be interesting to an academic, even if we could calculate (or approximate) intrinsic value, on its own it is useless to an investor. It needs to be compared to price to make a decision whether the stock is undervalued enough to buy or overvalued enough to sell.

But since price is involved in the final decision, it makes no logical difference whether it is included at the end or included as part of the calculations right from the start. In fact, often more insight is gained by bringing price into the calculations right at the start.

Even more surprising is the fact any decisions using the ROE method implicitly hinge on opinions whether the P/E ratio is high or low.

Consider the earlier calculations for Costco. Suppose that you accept that the intrinsic value is $67.07 and that you believe that a bargain price for Costco is a 40 percent discount or $40.24. Since the earnings are growing by 8.76 percent per year, they must have been $2.83 for the previous year. (In fact they were $2.92.) This gives a P/E ratio of 14.2. In other words, all these assumptions and calculations regarding Costco are a roundabout way of coming to a conclusion that we could have arrived at right at the outset. In round numbers, Costco is a buying opportunity if earnings grow by at least 9 percent per year, dividends are around 27 percent of earnings, and the P/E ratio is less than 14.

So instead of having a method that we believe avoids such things as price, P/E ratios and dividend yields, in reality the use of formulas or tables has simply obscured their use.

## Discounted Cash Flow Methods

The same extreme variations apply to discounted cash flow methods. Small variations in the input variables give outputs that vary by many multiples. It is impossible to tell whether a company is undervalued or overvalued.

Another flaw of all these methods involving infinite sums is that it is impossible to verify the accuracy of the input variables. For example, if one person says that the company will grow by 10 percent in its final stage and another sys that it will grow by 11 percent, then it is impossible to determine who is more correct. Yet, the difference between 10 percent and 11 percent may give a value that differs by 50 to 100 percent.

## What to do?

Applying mathematics formulas to valuation methods can be fun, but not if it leads to overconfidence and misleading results. The problem is that hidden inside the calculations are absurd assumptions and impractical requirements. Unsuspectingly we have been led astray by mathematical intimidation. I have been through it all: Writing high level code for the hedging operations of international companies through to designing new foreign exchange option strategies.

But the more I see, the simpler I want any method I use. Warren Buffett, the record-breaking chairman and CEO of Berkshire Hathaway, wrote in 1997:

“Your goal as an investor should simply be to purchase, at a rational price, a part interest in an easily-understandable business whose earnings are virtually certain to be materially higher five, ten and twenty years from now.”

This seems like a good starting point: find companies with virtually certain significant growth in their earnings. Buffett explains the result of finding such companies:

“Put together a portfolio of companies whose aggregate earnings march upward over the years, and so also will the portfolio's market value.”

Finally he tells us this is what he does:

“Though it's seldom recognized, this is the exact approach that has produced gains for Berkshire shareholders.”

It’s hard to get anything clearer than this. To see how it works in practice, consider Expeditors International and FacstSet Research Systems. The table shows that over the past ten years, the earnings and price of Expeditors have both grown by approximately the same amount, around 16 percent per year. Similarly, the earnings and price of FactSet have also grown by approximately the same amount, approximately 19 percent per year. This reinforces Buffett’s injunction to seek out companies that will grow their earnings because that will drive the price.

10 Years Ago | Today | Growth Ratio | Growth Per Year | ||

ExpeditorsInternational | EPS | $0.38 | $1.46 | 3.84 | 16.13% |

Price | $14.00 | $53 | 3.79 | 15.94% | |

FactSetResearch Systems | EPS | $0.64 | $3.27 | 5.11 | 19.87% |

Price | $19.00 | $96.10 | 5.06 | 19.73% |

Of course, not all companies will work out as close as this. Consider Wal-Mart over the past ten years. Its earnings have grown by around 12 percent per year but the price is much the same as it was ten years ago. This is because the P/E ratio was around 40 but now it is approximately 14.

This means that we need to add one more requirement for earnings. It is important not to pay too much for them. If we can be confident about the future growth of earnings and we do not pay too much for these earnings, we are going to do well. If we can top this up with a healthy stream of dividends, all the better.

Confidence in the growth of earnings comes from a proper analysis of areas such as debt, the stability of earnings and sales growth, and the company’s economic moat. It also comes from knowing that management is honest, rational and acting in the best interests of shareholders. Knowing what constitutes a reasonable price to pay for the earnings comes from applying six strict rules. These ideas are explained in detail in Chapter 11 of

*The Conscious Investor*.

Finally, no discussion of finding real value in the share market would be complete without talking about of margins of safety. Before making any investment decision we need to stress test or apply practical margins of safety to assumptions or forecasts we make in three areas: future business performance, market opinion and board dividend policy. Even better, as discussed in my book, is to do this in an automated way to remove any prejudicial biases. Putting these steps together is the best way that I know to achieve consistent success in the stock market.

Dr John Price has published over 60 papers in mathematics, theoretical physics and finance.

He has personally programmed, tested, and compared over 30 different stock valuation methods in his search for the best approaches. This led to the development of investment software called Conscious Investor

^{®}. It also led to his recent book

*The Conscious Investor: Profiting from the Timeless Value Approach*(Wiley, 2011) in which he describes the assumptions along with the strengths and weaknesses of these methods.

Alex Morris- 5 years agoSpot on Dr. Price. Thank you for a wonderful article that should be read by all investors.