Numeraire DCF Valuator



Where facts conquer fiction. Dig deep for facts. Dig deeper for value.





"Forecasts say much about the forecaster and little about the future." Anonymous

"Anyone might have foreseen that ... the raising of stock, of all sorts to a value above the Intrinsick, must have some fatal issue, and would fall somewhere at last so heavy as to be felt by the whole body of Trade." Observation of Daniel Defoe after the first British stock market crash of 1696.


All investing involves forecasting, even if the forecasts are implicit. The future is unknowable; thus forecasts are uncertain.

Expected growth is an incomplete forecast. An expected growth period is an incomplete forecast. An expected growth rate in combination with an expected growth horizon is a complete forecast of future growth. Past growth cannot be projected confidently into the future, and recent growth cannot be projected confidently beyond a short-term horizon.

One may have recourse to a Delphic oracle to seek knowledge of the future, but usually it is necessary to use more worldly methods to forecast future developments or quantify expectations. Forecasting can be considered from several perspectives, without placing undue emphasis on either general or specific outlooks.

For the perspective of the scientific method of induction, see Peter Medawar'a article "Is the Scientific Paper Fraudulent?", Saturday Review, August 1, 1964, and Nelson Goodman's book Fact Fiction and Forecast, 4th edition (see citation in the General Books section of the Books page at the Global Value Investing website).

For the perspective of psychology, see Dietrich Dorner’s The Logic of Failure, Ch. 5 "Time Sequences" (see citation in General Books).

Chapter 5, page 110: "Our ultimate concern in this chapter is how people form their ideas of the future. If we can identify the typical difficulties people have in dealing with time and in recognizing temporal patterns, we can suggest ways to overcome these difficulties and to improve temporal intuition."

For the perspective of financial accounting, see John Burr Williams’ The Theory of Investment Value, Ch.s XI and XII "Algebraic Budgeting" (see citation in Special Books and theory).

Chapter XI, page 128: "The formulas for the investment value of common stocks given in earlier chapters would be of little use unless some way could be found to estimate the size of the dividends whose present worth it was there proposed to take. How to estimate these future dividends is the heart of the problem, and in helping to solve this problem, the present book hopes to make a significant contribution to the art of Investment Analysis."

From the perspective of econometrics, what may be called a nave forecast assumes that the future is best forecast as an extrapolation of the past. Therefore, historical data that is deemed appropriate can be collected for purposes of data analysis. Most comprehensive statistical software packages include tools for analysis of time sequences (assumed time series). Some software will fit data to a library of standard general formulas and identify the one that best fits the data, and also optionally plot both the historical data with the fitted curve and the future projected data with the extrapolated curve for as many periods as requested.

One econometric approach is to use historical data to calculate a geometric or compounded annualized growth rate. For example, the three-year growth rate is calculated from the fourth fiscal year back to the most recently completed fiscal year. The compound annual growth rate is less stable than the regression growth rate because it depends only on the levels of the first and last data points which can be erratic, whereas the regression method uses the information from all of the data points. The geometric approach to forecasting could be called plain naive in contrast to the regression approach which could be called fancy naive.

Another econometric approach is to use historical data to calculate both a 3- and 5-year regression growth rate and their corresponding R-squared as well as a 3- and 5-year compound growth rate on Revenue, Net Income, Diluted Earnings per Share (EPS), and Dividends. The methodology for the regression growth rate calculation is as follows. The 3- and 5-year EPS regression growth rate is the compound annual growth rate in primary earnings per share. This figure is derived by the least squares method, using four data points (in the case of 3-year) or six data points (in case of 5-year) -- a base year plus 3 or 5 subsequent annual data points. Fiscal year-end figures are used for the first 3 or 5 data points, with the calculation brought up to date each interim period by time-weighting the first and last points. No calculation results for any company which does not have a positive number for the 3- or 5-year period. A 3- or 5-year growth rate is not calculated for any company which has an incomplete record of data.

Strictly speaking, statistical methods applied outside of controlled experiments do not produce forecasts or projections but rather make predictions. In addition, such predictions in the case of time sequences are outside the range of the data sample, since future time periods are never comparable to past time periods due to different conditions and ever-changing states of the world.

From the perspective of institutional dynamics, what may be called a gullible forecast is based on accepting the supposed insights of the financial industry or what has come to be known as Wall Street. Some argue that security analysts have a "sell" incentive that thwarts objective research. Evidence for such bias is the high ratio of buy to sell recommendations, often higher than 100 to 1.

The brokerage house equity research security analysts specialize in different industry groups or economic sectors, and they get bigger year-end bonuses for moving more product which is measured in shares of daily trading volume. One way to move money is by frequent upgrades and downgrades communicated to market participants by a generally compliant financial press and reinforced by carefully selected studies from the academic industry with the not so surprising result of increased share price variability. A closely-related projection of earnings per share (EPS) growth or earnings growth rate (EGR) is based on the whisper numbers circulated quasi-privately among those same analysts.

These projected numbers are called brokers’ or analysts’ consensus estimates, but they are merely simple arithmetic averages. The euphemistic term "consensus" gives the estimates the aura of greater authority of presumed expertise and the appearance of more general agreement among the analysts whose opinion is polled. One outlier in the sample of analysts' opinions can skew the consensus significantly, but publicizing an erratic forecast is not usually considered a career-enhancing move. The key is to rise above the fray and look beyond their amnesia, myopia, and manic-depressive enthusiasms and fears. Under most conditions, the consensus forecast can be considered as a practical maximum.

In an article entitled 'Speak No Evil? Analyst Turns Silent on Bank', page C1, The Wall Street Journal, 17 August 1999, Rick Brooks writes: "When a stock-research analyst is silenced, it can speak volumes about how Wall Street really works. ... Mr. Ryan, an increasingly blunt critic of First Union's top management as problems at the bank grew this spring, posed a threat to his firm's trading and investment-banking relationship with First Union. ... So, Mr. Ryan's superiors at Bear Stearns directed him not to make negative comments about first Union. ... It also reflects the pressure facing analysts in an era when many of them increasingly are viewed less as independent thinkers than as agents for a securities firm's underwriting business. ... The ruckus over Mr. Ryan also shows how easily an analyst can get into trouble despite watered-down stock-rating scales that result in nearly no "sell" recommendations from many Wall Street brokerage houses."

Security analysts are highly biased in the ratings they give the stocks they cover. The "sell" or "strong sell" ratings comprise less than 1% of all analyst ratings (buy, hold, sell), and in April 2000 are outnumbered by "buy" or "strong buy" ratings by over 70-to1. Analysts say the positive or bullish bias derives from the hesitancy to offend the companies they cover, due to fear of losing access to the company for information or fear of losing potential investment-banking business. For these reason, many analysts will not cover companies that they can't recommend to buy, and most analysts will not cover companies that are not likely to become investment banking clients.

The nave and the gullible forecasts of growth rate can be combined to create a hybrid 20/20-hindsight backward-looking and financial-industry agenda-advancing statistic that may be called the Nave-Gullible Average or NGA. When a stock is outside one's circle of competence or beyond one's experience-based informed judgment, the NGA can be used to provide a point of departure for stock valuation. Using simplifying assumptions such as a constant dividend payout ratio, a constant proportion of non-cash charges to before tax earnings, and a constant dilutive potential, it can be shown that the growth rates of earnings, earnings per share, dividends, and free cash flow to equity owners are the same. This will work for the purposes of a point of departure for in-depth analysis which necessarily entails close, detailed scrutiny of all the company financials and narrative disclosures.

As a reality check, it is useful to compare the expected growth rate of cash returns with the growth rates for individual competitors, the industry, and the economy. If a company is forecast to grow at a much higher rate than these "control" groups, then there needs to be convincing evidence to support that assertion. The primary concern is whether the management of the company can successfully translate the financial capital of stockowners entrusted to them into sales, and then, successfully translate sales into profits and ultimately dividends returned to the company stockowners.

From the perspective of intrinsic value calculations, you may "back in" to a solution. Do a valuation using the model for deterministic single-point value with goal-seeking. Use realistic assumptions about the levels of the input variables, and use the NGA as a first approximation for the growth rate if the company is outside your circle of competence. The goal-seek level of a variable is calculated while the levels of all other variables are kept the same. Consider the resulting goal-seek quantity for each of the input variables that is necessary to justify on one hand the market price, and on the other hand the price adjusted for a margin of safety. With this information, ask the question, What degree of confidence do I have that this quantity for this variable will be realized?

Goal Seeking

The goal-seek analysis uses a function called root. Root finds the value of x such that f(x) = 0. Root must be supplied with either a single point guess for the value of x or both a lower-limit and upper-limit. Root uses different methods to find the value of x depending on which of these starting methods is used. If it is provided a value for x1 only, then the function must converge in a "downhill" direction from the x1 value. If it is provided a value for both x1 and x2, the values f(x1) and f(x2) must be of different signs. If this condition is met, root will always find a solution. Root is an approximated function and is subject to the tolerance set by precision.

Double-precision floating point numbers using the standard 8-byte IEEE numeric format can hold values as large as +/-1.79x10e308 (that is shorthand notation for 1 followed by 308 zeros). You can specify extremely large integer literals, but you may lose precision in the trailing digits. All integers between -2e53 and 2e53 can be represented exactly. However, most integer manipulations are performed on 32-bit integers, which range from -2e31 to 2e31. The upper limit of the range of numbers that can be stored in 16-byte size is trillions of times larger than the GDP of the entire planet Earth represented in the smallest currency unit. A so-called "underflow" error or "overflow" error is caused by a floating point calculation that is out of the computer's machine precision range. For example, 0.1**500 is not zero, but it is smaller than any number that can be represented by 16-byte numbers. As long as both lower and upper limits are supplied for each goal-seek variable, these two types of error should not occur.

Lower and upper limits for realistic solutions are set on each of the goal-seek variables: Years Forecast, FCFE Growth Rate, Discount Rate, Price-to-FCFE Ratio, and Base Year-Zero FCFE. A goal-seeking level is calculated for each of these variables with the levels of all other variables held constant. The goal-seek level for each variable is calculated for two targets: market price, and safety margin price derived from the market price.

For any given goal-seek variable, the condition of different signs for the two limits may not be met in every valuation case. Even if met, not a single root but rather multiple roots may exist within the designated limits. If the signs are not different for any particular goal-seek variable, a change in the value of one or more of any of the other input variables may cause the signs of the limits to be different for this goal-seek variable and thus result in a goal-seek solution.

If any one of the ten goal-seek calculations (five input variables with the market price target, and the same five input variables for the safety margin target) does not meet the opposite-signs test, there will be returned a message "NA" for that calculation which means that a single root does not exist between the specified bounds which are very broad practical limits. If a goal-seek variable has no answer for the market price target, then it is likely to have no answer for the safety margin target also.


In the book entitled "Value Investing : A Balanced Approach" by Martin J. Whitman (see citation in Special Books), Chapter 4 "Problems Faced by Research Departments", a section called "Analyses That Focus on Base Case Forecasts Rather Than Alternative Scenarios", pages 92-93, it is written:

"An investor putting $100 million or more of his or her own funds into an equity situation in which the funds will be tied up on a permanent or semi-permanent basis usually does not rely as heavily on base forecasts as do [broker-dealer] research department analysts [sell side] and conventional money managers [buy side]. This is understandable because the latter groups enjoy marketability and can, at least theoretically, undo investments rapidly. One consequence of this, though, is that there is far less need and desire for research department analysts and conventional money managers to conduct in-depth investigations: Given high portfolio turnover, it becomes terribly unproductive to do in-depth investigations. Furthermore, it becomes impossible for even the largest organization to conduct in-depth investigations when the number of securities in a portfolio runs to hundreds of issues.

"By contrast, large investors holding equities on a permanent or semi-permanent basis tend to investigate relatively thoroughly on two bases: the base case and the reasonable worst case. Most also put the optimistic case into the investigation mix."

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