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Formulas

 Price is not value, and pricing models are not valuation models. The conventional academic capital asset pricing model has one factor, the beta coefficient. Models that include beta are pricing models, not valuation models. This is not merely a matter of semantics. The difference between price and value, referred to as the margin of safety, is the raison d'etre of investment valuation independent of market pricing.

Some so-called stock valuation applications are really pricing applications with an unexplained academic beta factor lurking inside the box. That alone should be sufficient cause for concern. Pricing models are equivalent to a statistics-trained blind-folded monkey throwing darts.

The DCf Valuator model types are not mysterious secret "black boxes"; rather, they are transparent glass bowls, totally open and fully disclosed. Investment risks are not summarized by an artifact of statistical regression techniques such as the beta factor but rather as specific perils such as price-level inflation. The intrinsic value of an investment asset is what you can get out of it — generally in the form of a cash flow stream over a number of annual holding periods. The discretionary cash flow can be cash dividends paid by a joint-stock company, or the cash flow available to the common stock equity of a company that has not yet begun paying dividends. The estimation of intrinsic value involves three formulas: first, accounting for free cash flow to the common stock equity account; second, forecasting of future cash flows; and third, discounting of future cash flows to the present.

The calculation of intrinsic value of common stock in all of the models can be presented in three basic steps:

• Step 1: Calculate annual dividends or free cash flow to equity.

• Step 2: Forecast future values of net cash flows.

• Step 3: Discount the future values to the present, and total.

Word Formulas

Accounting Formula. Free cash flow to the common stock equity account, FCFE, is the cash available for distribution to the ultimate equity owners; an accounting concept that is equal to net income plus non-cash charges (depreciation, depletion and amortization) minus debt and other fixed obligations net of tax savings on interest expense minus preferred stock dividends minus fixed capital expenditures needed to maintain the company's economic productive capacity at the same level minus the increase in working capital needed to maintain the company's economic productive capacity at the same level. Such adjustments for working capital are not generally required except in unusual situations such as low-fixed capital and shifts in the cash cycle. The resulting base-period FCFE should be adjusted also to remove any non-recurring items and any cyclicality so that normal operating cash flow is measured. If a company is expected to pay cash dividends, then the future annual dividends are forecast.

Forecasting Formula. The heart of each model is the forecast of future net cash flows. The future net cash flows can be estimated individually for each future period. Alternatively, to simplify calculation, regular patterns of growth can be assumed and a formula used to estimate the future net cash flows. The assumption of exponential growth is adequate for long-term projections of five or more years.

Discounting Formula.
The time value of money is calculated according to a standard mathematical formula which has variations for different time periods, continuous or discrete time, and beginning or end of period flows. For many applications, the simplest version of the general discounting formula is adequate.

Math Formulas

Step 1: FCFE = Net Income + Non-Cash Charges (DD&A) - Long Term Debt & Fixed Obligations + Interest Expense on Long Term Debt - Income Tax Savings on Interest Expense - Preferred Stock Dividends - Fixed Capital Expenditures above Base - Working Capital Change above Base.

Step 2: FVn = FV0[(1+g)n] ,
where FV is future value,
g is the growth rate,
and n indexes values and is the number of periods.

Step 3: PVn = FVn/(1+d)n ,
where PV is present value,
FV is future value,
d is the discount rate per period,
and n indexes values and is the number of periods;
Total PV = Sum of PVn for periods 1 to N.

The stream of future values may be perpetual, in which case N equals infinity and there is no ending sale of the stock. The stream of future values may be finite, in which case N is finite and the ending sales price is the final future value term included in Total PV. The ending sales price is most conveniently expressed as a multiple of the final period FV. It is a matter of taste whether Total PV is called investment value, intrinsic value, fair value or other similar label.

Example Calculations

In Step 1, the latest quarterly dividend is multiplied by four to estimate annual dividends for the base period. Alternatively, FCFE is calculated from the most recent company external annual financial report. In the U.S. these are filed on Form 10-K documents to the Securities and Exchange Commission (SEC). See the Base FCFE help window in each investment model for an example calculation.

In Step 2, free cash flow to the common stock equity account (FCFE) can be forecast to growth from a base value in the present period of \$100 million (
FV0 = 100.00) at a compound rate of 5% per year ( g = 0.05) for a duration of five years (N = 5). The resulting projection of annual future values of FCFE is calculated as follows:

FV0 = 100.00, g = 0.05 and N = 5:

FV1 = FV0[(1+0.05)**1] = 100(1.05)**1 = 105.00
FV2 = FV0[(1+0.05)**2] = 100(1.05)**2 = 110.25
FV3 = FV0[(1+0.05)**3] = 100(1.05)**3 = 115.76
FV4 = FV0[(1+0.05)**4] = 100(1.05)**4 = 121.55
FV5 = FV0[(1+0.05)**5] = 100(1.05)**5 = 127.63

As can be seen by substituting the formula for FV into the formula for PV and simplifying, a discount rate less than the growth rate for a perpetual stream in which N equals infinity results in a PV equal to infinity. To avoid this meaningless result, known as the St. Petersburg paradox, finite growth stages are used.

A one-stage forecast has the same growth rate each year. A two-stage forecast allows for a different growth rate for a different duration for each of two stages. Only the most irregular situations can't be approximated well enough with one or two stages of growth. Take a look at the types of future
growth patterns.

In Step 3, assume the investor's opportunity cost of capital is 6% (r=0.06) based on the current yield on 30-year U.S. government bonds and the terminal or ending selling price (
FVe) is 10 times the ending FCFE (Price-to-FCFE ratio= 10). The resulting projection of annual present values of FCFE and Price is calculated as follows:

PV1 = FV1/[(1+0.06)**1] = 105.00/(1.06)**1 = 99.06
PV2 = FV2/[(1+0.06)**2] = 110.25/(1.06)**2 = 98.12
PV3 = FV3/[(1+0.06)**3] = 115.76/(1.06)**3 = 97.19
PV4 = FV4/[(1+0.06)**4] = 121.55/(1.06)**4 = 96.28
PV5 = FV5/[(1+0.06)**5] = 127.63/(1.06)**5 = 95.37

FVe = FV5(10) = (127.63)(10) = 1276.30

PVe = FVe/[(1+0.06)**5] = 1276.30/(1.06)**5 = 953.70

Further, assume there are 100 million common shares outstanding (Shares=100.00). The resulting intrinsic value per share (Value) is calculated as follows:

Total PV = PV1 + PV2 + PV3 + PV4 + PV5 + PVe
Total PV = 99.06 + 98.12 + 97.19 + 96.28 + 95.37 + 953.70
Total PV = 1439.72

Value = Total PV/Shares = 1439.72/100.00 = 14.40

Once the intrinsic value is estimated, we can calculate the maximum safe price. Assume the investor's required margin of safety is 25% (Margin=0.25) for this particular stock at this time. The resulting safety price is calculated as follows:

Safety Price = Value (1-Margin)
Safety Price = 14.40 (1-0.25) = 14.40 (0.75) = 10.80

We can compare current market price with the safety price, or alternatively, calculate the spread between intrinsic value and current market price and compare it to the margin of safety. Assume the current market price is \$10 per share (Market Price=10.00). The spread represents the potential for price appreciation (capital gain or loss) of the stock.

Market Price of \$10.00 is less than the Safety Price of \$10.80

Spread = Value - Market Price = \$14.40 - \$10.00 = + \$4.40
Spread % of +31% is greater than the Safety Margin % of +25%

If all other facets of valuation are acceptable, and all other selection criteria of the investor are met, then this could be a stock to buy.

The Total Present Value in the DCF Valuator value models is identical to the familiar Net Present Value (NPV) without the investment cash outflow netted out. The rate of return in the DCF Valuator return models is identical to the familiar Internal Rate of Return (IRR), a concept closely related to the NPV. The IRR is the rate at which the NPV is equal to zero, or equivalently, the rate at which the Total Present Value is equal to the investment cash outflow. The IRR in the DCF Valuator return models assumes that the investment cash outflow is equal to the Total Present Value and that all intermediate cash flows between the investment cash outflow and the ending cash inflow are reinvested to yield the same rate as the IRR.

As John Burr Williams says in his classic book entitled The Theory of Investment Value on pages 58-59, "In applying the foregoing formulas, each investor should use his own personal rate of interest. If one investor demands 10 per cent and another 2 per cent as minimum wages of abstinence, then the same stock or bond will be accorded a lower value by the one than by the other. The only case in which the market rate of interest should be applied is when the analyst is speaking not for himself but for investors in general. Then he should use the pure interest rate as it is expected to be found in the open market in the years to come."

Using the standard discounting formula, an example calculation of the expected rate of return on an investment in common stock using the General-Purpose Discounted Cash Flow (DCF) model is presented in

Desirable Dilemma

Ideally, an investor can identified several desirable stocks that are currently priced below their estimated investment value at his or her current personal opportunity cost discount rate. The dilemma is to select the stock or stocks to buy. The following is a hypothetical list of such stocks:

 Common Stock Fair Value Market Price Spread Amount Spread Percent Alpha 14.40 10.00 + 4.40 + 31% Bravo 25.00 20.00 + 5.00 + 20% Charlie 100.00 90.00 + 10.00 + 10% Delta 200.00 100.00 + 100.00 + 50% Echo 500.00 300.00 + 200.00 + 40%

To avoid manipulative situations, assume these five stocks each meets a \$5.00 minimum stock price at latest close and a 10,000 share minimum average daily trading volume for the past six months. Assume an investor personally requires a minimum 25 per cent and \$5.00 absolute margin of safety for all these stocks and has accumulated \$10,000 to invest in stocks.

Which stocks should be bought? Eliminate Alpha due to insufficient absolute margin of safety and Bravo and Charlie due to insufficient percentage margin of safety. Rank the remaining stocks by spread, not absolute but percent spread, from highest to lowest. This results in Delta (50%) first and Echo (40%) second.

How much money should be invested in each? John Burr Williams recognizes the advantages of diversification that results in buying different stocks over time, but at any particular buying opportunity, he recommends buying "the best at the price" to achieve the highest long-term returns. Thus, buy one round lot or 100 shares of Delta common at \$100 per share for a total of \$10,000 trade.

The Best at the Price

The Theory of Investment Value, 1997, John Burr Williams
Chapter I. The Difference between Speculation and Investment

1. Real Worth and Market Price. Separate and distinct things not to be confused, as every thoughtful investor knows, are real worth and market price. No buyer considers all securities equally attractive at their present market prices whatever these prices happen to be; on the contrary, he seeks "the best at the price." He picks and chooses among all the stocks and bonds in the market until he finds the cheapest issues. Even then he may not buy at all, for fear that everything is too high and nothing will give him his money's worth. If he does buy, and buy as an investor, he holds for income; if as a speculator, for profit. But speculators as a class can profit only by trading with investors, to whom they can sell only for income; therefore in the end all prices depend on someone's estimate of future income. Of investment value in this sense some men will make one estimate, others another, and of all the estimates only one will coincide with the actual price, and only one with the true worth.