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**Theory and
Practical Models**

John Burr Williams,

The Theory of Investment Value1997 reprint, Fraser Publishing, c1938, Cambridge: Harvard University Press.

**Biography | Preface
| Contents | Models**

Included here is some biographical information about John Burr Williams from his memoirs and other sources. This is followed by excerpts from his cited book: the Preface, an abbreviated Contents for Book I theory and Book II practice, and a list of practical stock valuation models.

The excerpts and quotes of this and other books by John Burr Williams on this and other pages at this web site are considered to be "fair use" as interpreted by

The Chicago Manual of Style: The Essential Guide for Writers, Editors, and Publisher, 14th edition, 1993. Chicago: The University of Chicago Press. You are encouraged to read the book.

Some biographical information about John Burr Williams appears in the book

Capital Ideasby Peter L. Bernstein who in turn draws on a booklet written by John Burr Williams entitledFifty Years of Investment Analysis(see the citations in Special Books). As Bernstein (1992:150-152) writes:

"Williams was wealthy and nearly ninety years old when he died in 1989, but he was already a highly successful investor when he began to study economics seriously in the early 1930s. As an undergraduate at Harvard, he had concentrated in mathematics and chemistry. After graduating in 1923, he went to the Harvard Business School, where he got his first taste of economic forecasting and security analysis. He then took a job as security analyst at the prestigious brokerage house ... .

Many years later, in a 1959 memoir he wrote (see the citation in General Books) for the guidance of the younger generation, Williams recalled his experience ... "How to estimate the fair value was a puzzle indeed," mused Williams. "The experience taught me a lesson. To be a good investment analyst, one needs to be an expert economist also. Hence, a few years later, I took time off to earn a Ph.D. in economics.

"At the age of thirty, in 1932, Williams enrolled in the Graduate School of Arts and Sciences at Harvard. He set out to find an economist who could explain to his satisfaction what had caused the recent debacle in the nation's economy, ... ."

"When it came time to write his doctoral dissertation, he conferred with Schumpeter about an appropriate topic. Schumpeter suggested that he choose the intrinsic value of a common stock, for which Williams's personal experience and background in economics would serve him in good stead. Williams notes that Schumpeter had a cynical, conspiratorial reason for suggesting the topic: "This choice would keep me from running afoul of the preconceptions of the other members of the Harvard faculty, none of whom ... would want to challenge my own ideas on investments.

"Williams finished his thesis in 1937, and sent it to Macmillan for publication, even though he had not yet won faculty approval for his doctorate. Macmillan returned the manuscript with the complaint that it contained algebraic symbols. McGraw-Hill made the same complaint. Finally, Harvard University Press publishedThe Theory of Investment Valuein 1938, but only after Williams had promised to pay part of the printing cost.

"Still busy as an investor, Williams waited until 1940 before taking the oral exam for this Ph.D. Schumpeter, Leontief, and Hansen were on the committee. Hansen was disturbed that Williams had published his thesis before it had been accepted by the committee; moreover, he resented Williams's skeptical view of Keynesian economics. After intense argument ... , the committee finally agreed to grant the degree."

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To outline a new sub-science that shall be known as the Theory of Investment Value and that shall comprise a coherent body of principles like the Theory of Monopoly, the Theory of Money, and the Theory of International Trade, all branches of the larger science of Economics, is the first aim of this book. To relate the abstract principles of Economics to the practical problems of investment, and to show how the theories of interest, rent, wages and profits, taxes, and money, can be applied to the evaluation of stocks and bonds, is another aim of this book. To examine certain economic consequences of the New Deal so far as the investor is concerned, and to determine the most important questions on which investment policy today must rest, is a third aim of this book. These last two aims, however, are but incidental to the primary purpose of codifying the Theory of Investment Value and making it into a department of Economics as a whole.

While this book is addressed primarily to the intelligent investor and the professional investment analyst, it is hoped, nevertheless, that it will be of interest to the economic theorist also, because of what it has to offer on long- and short-term interest rates, liquidity preference, uncertainty and risk, the future of interest rates, the likelihood of inflation, the proper response of stock and bond prices thereto, the behavior of markets, the formation of stock prices, the liaison between speculative commodity and security prices, the incidence of various taxes, and other questions of theory.

Investment Value, defined as the present worth of future dividends, or of future coupons and principal, is of practical importance to every investor because it is thecriticalvalue above which he cannot go in buying or holding, without added risk. If a man buys a securitybelowits investment value he need never lose, even if its price should fall at once, because he can still hold for income and get a return above normal on his cost price; but if he buys it above its investment value his only hope of avoiding a loss is to sell to someone else who must in turn take the loss in the form of insufficient income. Therefore all those who do not feel able to foresee the swings of the market and do not wish to speculate on mere changes in price must have recourse to estimates of investment value to guide them in their buying and selling.

Dedicated as it is by its very title to the intrinsic or ultimate worth of investments, this book cannot help but imply as a corollary, however, a Theory of Speculation also. If marginal opinion, not intrinsic value, determines market price, as claimed in this book, and if changes in opinion, but nothing else, cause changes in price, then those who trade in the market for a living will find herein a philosophy of their work. Nevertheless, the book is not a manual of rules for speculators, nor are the estimates of investment value reached in its Case Studies intended as forecasts of the next move in stock prices.

A pioneer step in the measurement of intrinsic value was taken several years ago in a book entitledStock Growth and Discount Tablesby S. E. Guild, R. G. Wiese, Stephen Heard, T. H. Brown, and other collaborators. A further step was taken recently by G. A. D. Preinreich, in the mathematical appendix to his book,The Nature of Dividends. The present book follows the same trail still further [sic], and shows in particular how the dividends a company will pay in the future can be forecast, either by "algebraic budgets" using estimates concerning the future growth, earnings, and capital structure of a company, or by other methods.

Although the book makes frequent use of simple algebra, facility with this subject is needed only by those specialists who focus their attention on Part II; all others can read the rest of the book without knowledge of mathematics. Any symbols that are likely to trouble the non-technical reader are carefully explained when first introduced. The mathematics is not to be considered as a drawback to the analysis, however, nor as a method of reasoning which serious students can afford to neglect. Quite the contrary! The truth is that the mathematical method is a new tool of great power whose use promises to lead to notable advances in Investment Analysis. Always it has been the rule in the history of science that the invention of new tools is the key to new discoveries, and we may expect the same rule to hold true in this branch of Economics as well.

That investment analysis until now has been altogether unequal to the demands put upon it should be clear from the tremendous fluctuations in stock prices that have occurred in recent years. As will be shown in the "Post Mortems" in Book II, proper canons of evaluation, generally accepted as authoritative ,should have helped to check these price swings somewhat, and thereby reduce in some degree the violence of the business cycle, to the benefit of all the world.

BOOK I -- INVESTMENT VALUE AND MARKET PRICE

Part I Speculation in the Stock Market

I. The Difference Between Speculation and Investment

II. Does the Stock Market Predict the Future?

III. Marginal Opinion and Market Price

Part II The Pure Theory of Investment ValueIV. Does the Quantity Theory of Money Apply to Stock Prices?

V. Evaluation by the Rule of Present Worth

VI. Stocks with Growth Completed

VII. Stocks with Growth Expected

VIII. Bonds and the Price Level

IX. Stocks and the Price Level

X. Bonds with Interest Rates Changes

XI. Algebraic Budgeting

XII. Algebraic Budgeting (cont.)

XIII. Growth by Merger or Sudden Expansion

XIV. Option Warrants and Convertible Issues

XV. A Chapter for Skeptics

Part III The Economics of Interest and DividendsXVI. The Source of Interest and Dividends

XVII. Taxes and Socialism

XVIII. Where is the Interest Rate Determined?

Part IV The Outlook for Interest Rates and the Price LevelXIX. Politics, Inflation, and Government Bonds

XX. The Future of Interest Rates

BOOK II -- CASE STUDIES IN INVESTMENT VALUEForeword to the Case Studies

Part I Current StudiesXXI. General Motors

XXII. United States Steel

XXIII. Phoenix Insurance

Part II Post MortemsXXIV. American Telephone in 1930

XXV. Consolidated Gas, and United Corporation, in 1930

XXVI. American and Foreign Power in 1930

XXVII. Conclusion

Appendix I. Bookings and Stock Prices of Unites States Steel

Appendix II. The Dividend-Paying Power of United States Steel

Index

**Models for Equity Valuation**

Chapter |
Models |
Model Types |

Formulas for the Investment Value of Common Stock | ||

V |
1 | General DCF |

VI |
2; 3 |
Declining
DDM; Constant DDM |

VII |
4, 4a,
4b, 4c; 5, 6 |
Increasing
DDM; Logistic DDM |

Formulas to Forecast Future Dividends | ||

XI |
7, 8, 9,
10, 11, 12, 13, 14; 15, 16, 17 |
Logistic
FCF Constant D/E; solve for n |

XII |
18, 19, 20 | Logistic
FCF Rising D/E (also solve for n) |

XIII |
21 | Sudden Growth (also solve for value) |

Go to graphs of Types of Growth |

The formulas for the investment value of common stocks would be of little use unless some way could be found to estimate the size of the dividends whose present worth it was proposed to take. How to estimate these future dividends is the heart of the problem.

Basic PrinciplesChapter V. Evaluation by the Rule of Present Worth, 55.

Model 1:Stocks. Equations (1a and 2), 56.

§1. Future dividends, coupons, and principle, 55. §2. Future earnings of stocks, 57. §3. Personal vs. market rate of interest, 58. §4. Compound interest at a changing rate, 59. §5. Rights and assessments, 61. §6. The formation point for income, 64. §7. The value of a right, 66. §8. Uncertainty and the premium for risk, 67. §9. Senior and junior issues of the same concern, 70. §10. The law of the conservation of investment value, 72. §11. Refunding operations, 73. §12. Marketability, 74.

Any other function that lends itself to convenient mathematical treatment may be used, so long as the general premise that stocks derive their value from their future dividends is adhered to. In order to make practical use of the foregoing mathematical analysis, the student of investments must turn to economics for this data.Economic facts, interpreted with his best judgment, must tell himwhat the probable curve of a company’s growth is, and how far along this curve the company has progressed (94).

Declining DividendsChapter VI. Stocks with Growth Complete, §1. Stocks with Declining Dividends, 76.

Model 2:Declining Dividends. Equations (1f and 2), 76. Equation (1f) is a special case of Equation (1a) wheret = nas a maximum,tis the index for years, andnis the number of years dividends are paid.

Constant DividendsChapter VI. Stocks with Growth Complete, §2. Stocks with Declining Dividends,77.

Model 3:Constant Dividends. Equation (8a), 77. Evaluation ofNet Quick Assets. Equation (8k), 84.

Growing DividendsChapter VII. Stocks with Growth Expected, §1. Dividends Increasing Forever – A Hypothetical Case, 87.

Model 4:Dividends Increasing Forever. Equations (14a, 16a, 1ac, and 15), 87-88.

Model 4a:Ifw< 1, thenV_{o}is finite. Equation (17a), 88.

Model 4b:Ifw= 1, thenV_{o}has no meaning due to division by 0. Equation (17b), 88.If

Model 4c:w> 1, thenV_{o}is infinite. Equation (17c), 89.

Whereg= annual growth of dividend-paying power,i= interest rate sought by the investor, andV_{o}= investment value per share at the start (i.e., in year 0, the present value).

Ifg<i, thenV_{o}is finite.

Ifg=i, thenV_{o}is infinite.

Ifg>i, thenV_{o}is infinite.

The foregoing analysis shows that if the rate of growth (of dividend-paying power) is less than the rate of interest (used for discounting dividends), the stock has a finite value, even though growth continue without limit.

Rapid then Slower Dividend GrowthChapter VII. Stocks with Growth Expected, §2. Dividends Increasing Rapidly, Then Slowly, 89

An initial dividend-growing period followed by a dividend-maturing period. The process of evaluating a stock whose dividends increase in this way may be divided into two parts corresponding to the two periods of growth.

Model 5:Logistic Growth. Equation (27a), 94.

The logistic curve employed above is not the only one that may be used to represent the course of dividend payments on the stock of a company passing through stages of rapid then slower expansion.

Chapter VII. Stocks with Growth Expected, §3. Special Case Wherew= 1, 94.

Model 6:Special Case Wherew= 1. Equation (22c) and Equation (27b), 95.

A Growing Company with Constant LeverageChapter XI. Algebraic Budgeting, §1-6, 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.¶ The solution to be proposed is straightforward; it consists in making a budget showing the company's growth in assets, debt, earnings, and dividends during the years to come. This budget is not drawn up with debit and credit, however, using a journal and ledger as an accountant would do, but is put in algebraic form in a way that is altogether new to the accountant's art. ¶ The simplifying assumptions can be set down mathematically in the form of seven constant characteristics of such a company: (i) The return on invested assets stays the same; (ii) The ratio of stocks to bonds in its capitalization stays the same; (iii) The rate of interest paid on bonded debt and other senior securities stays the same; (iv) The rate of growth of invested asses stays the same; (v) We prove that for a company with these properties until the point of slackening growth arrives, the rate of net dividend, expressed as a ratio to book value, stays the same; (vi) Since all terms on the right-hand side of these last equations are constant, according to characteristics (i) to (iii) set forth above, it follows that the rate of earnings onbookvalue of common is also constant under these conditions. We prove the reinvestment rate, expressed as a ratio tobookvalue, constant like the rate of earnings onbookvalue of common; (vii) Since the growth in assets in percent per year is constant by hypothesis it follows that the reinvestment rate is also constant. Having proved that both the rate of net dividend and the reinvestment rate are constant, we can now see that the pure dividend rate, expressed as percent ofbookvalue, is also constant. We can now show that the dividends themselves follow a compound interest law till the point of inflection is reached. ¶ Since dividends thus increase according to a compound-interest law up to the point of inflection in the company's growth, the value of its stock may be found by one of the formulas already worked out in Chapter VII, §2, which is Equation (22b).

Model 7:Logistic Growth. Equation (67a), 140. Derived from Equation (22b). Its dividends beyond the point of inflection.

Model 8:Book Value at the Point of Inflection. Equation (75a), 143.

Model 9:Price-Earnings Ratio at the Point of Inflection. Equation (79a), 144.

Model 10:Yield at the Point of Inflection. Equation (90a), 146.

Chapter XI. Algebraic Budgeting, §7. Special Case Wherew= 1, 147.

Model 11:Logistic Growth. Equation (67b), 148. Derived from Equation (67a).

Model 12:Book Value at the End. Equation (75b, 75c, or 75d), 148. Derived from Equation (75a).

Model 13:Price-Earnings Ratio at the End. Equation (79b or 79c), 148. Derived from Equation (79a).

Model 14:Yield at the End. Equation (90b), 148. Derived from Equation (90a).

These formulas are very convenient for the hasty appraisal of growing companies. In appraising such companies, it is often enlightening topose the question,How far ahead is the market discountinga progressive increase in dividends? To determine the market's implied consensus horizon, the equations forV_{o }can be rearranged into equivalent equations forn, the number of years dividends are paid or cash flow is received, and then solved.

Model 15:Book Value at the End. Equation (75e or 75f), 150. To answer this question, an inverted form of Equation (75c) may be used to solve forn.

Model 16:Price-Earnings Ratio at the End. Equation gives (79d), 150. A similar transformation of equation (79c) may be used to solve forn.

Model 17:Yield at the End. Equation (22e), 150. Likewise, instead of transforming (90b), it is usually better to transform (22c) and solve forn.

A Growing Company with Increasing LeverageChapter XII. Algebraic Budgeting (continued), 151.

Model 18:Book Value at the End. Equation (93), 156.

Model 19:Price-Earnings Ratio at the End. Equation (99), 159.

Model 20:Yield at the End. Equation (99), 159.

The foregoing equations for the valueVof a company’s stockcan also be used to find thenumber of yearsthat it would take for the company to reshape its capitalization to any specified leverage. We have simply to treatnV_{o}as a known quantity, setting it equal to the present market priceM_{o}, and solve the equation fornby a method of successive approximations.

Growth by Merger or Sudden ExpansionChapter XIII. Growth by Merger or Sudden Expansion, 161.

When companies grow by merger or sudden expansion, rather than by slow development as has been discussed heretofore, then their dividends may be affected by a change in several factors at once, for their invested assets, their return on investment, their ratio of stocks to bonds, and so forth, may all undergo a change at the same time. Such a multitudinous change in characteristics would be in contrast to the cases discussed so far, which have all had the following properties: (i) the rate of return on invested assets,a= a constant; (ii) the rate of interest on bonds and preferred,b= a constant; (iii)u= 1+g= a constant, whereg= the growth in assets in percent per year; and (iv) either leverage,l=A/C= a constant, where A = net investment per share of common stock =B+Cby definition, where B = senior securities per share of common at end of the year andC =common stock's book value per share at end of the year, or the reinvestment rate, expressed as a ratio tobookvalue,r= zero.

¶ In describing such a company's condition at the start, its development during the period of growth, and its condition at the end, one would need to list eighteen characteristics in all, of which any eleven might be taken as known (or assumed for the sake of argument) and seven might be treated as unknown. A convenient way to group these characteristics into knowns and unknowns is as follows: Knowns --A_{0}= assets per share at start;C_{0}= book value of common at start;M_{0}= market price of common at start;K= actual dividends paid during growth (assuming continuance of present dividend rate);n= time involved in period of growth (assumed);l_{n}= leverage to be attained at end (assumed);c_{0}= rate of earnings on common at start;c_{n'}= rate of earnings on common at end (assumed);e_{n}= price-earnings ratio at end (assumed);i= interest rate;s_{n*}= amount of an annuity of $1 at the ratei; Unknowns --A_{n}= assets per share at end;C_{n}= book value of common at end;M_{n}= market price at end;S= subscriptions to new stock during growth;R= reinvested earnings during growth;G= earnings on common during growth;z= increase in size of company, in percent.

¶ Since sudden growth such as we are now considering usually takes place in a short period of from two to five years, the interest on any dividends received from or assessments paid to the company during the period in question is a relatively unimportant item. It is permissible to simplify the problem by assuming that the net dividend is the same in each of thenyears of rapid growth.

¶ Of the seven unknowns listed above, one in particular,z, is always of interest, because it represents the amount of growth in the specified number of yearsnrequired to justify the present price M.

Model 21:Formula for z. Equation (121), 170. Formula for V. Equation (123), 171

Chapter XXVII. Conclusion. The foregoing case studies are intended to be suggestive rather than exhaustive. They are designed to show how the buyer of stocks and bonds should go about it to find out if he is getting his money's worth.The last word on the true worth of any security will never be said by anyone, but men who have devoted their whole lives to a particular industry should be able to make a better appraisal of its securities than any outsider can. And with the coming of better professional appraisals of the leading issues on the Stock Exchange should come fairer, steadier prices for the investing public.

Note: The continuous S-shaped LOGIT growth curve can be approximated by a multi-stage growth curve, typically two or three successive stages, each with a of constant rate of growth.

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