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It may be well to recognize a vital difference that has developed in the valuation of these intangible factors, when we compare earlier times with today. A generation or more ago it was the standard rule, recognized both in average stock prices and in formal or legal valuations, that intangibles were to be appraised on a more conservative basis than tangibles. A good industrial company might be required to earn between 6 per cent and 8 per cent on its tangible assets, represented typically by bonds and preferred stock; but its excess earnings, or the intangible assets they gave rise to, would be valued on, say, a 15 per cent basis. (You will find approximately these ratios in the initial offering of Woolworth preferred and common stock in 1911, and in numerous others.) But what has happened since the 1920s? Essentially the exact reverse of these relationships may now be seen. A company must now typically earn about 10 per cent on its common equity to have it sell in the average market at full book value. But its excess earnings, above 10 per cent on capital, are usually valued more liberally, or at a higher multiplier, than the base earnings required to support the book value in the market. Thus a company earning 15 per cent on the equity may well sell at 13½ times earnings, or twice its net assets. This would mean that the first 10 per cent earned on capital is valued at only 10 times, but the next 5 per cent—what used to be called the “excess”—is actually valued at 20 times.

Now there is a logical reason for this reversal in valuation procedure, which is related to the newer emphasis on growth expectations. Companies that earn a high return on capital are given these liberal appraisals not only because of the good profitability itself, and the relative stability associated with it, but perhaps even more cogently because high earnings on capital generally go hand in hand with a good growth record and prospects. Thus what is really paid for nowadays in the case of highly profitable companies is not the good will in the old and restricted sense of an established name and a profitable business, but rather their assumed superior expectations of increased profits in the future.

This brings me to one or two additional mathematical aspects of the new attitude toward common-stock valuations, which I shall touch on merely in the form of brief suggestions. If, as many tests show, the earnings multiplier tends to increase with profitability—i.e., as the rate of return on book value increases—then the arithmetical consequence of this feature is that value tends to increase directly as the square of the earnings, but inversely the book value. Thus in an important and very real sense tangible assets have become a drag on average market value rather than a source thereof. Take a far from extreme illustration. If Company A earns $4 a share on a $20 book value, and Company B also $4 a share on $100 book value, Company A is almost certain to sell at a higher multiplier, and hence at higher price than Company B—say $60 for Company A shares and $35 for Company B shares. Thus it would not be inexact to declare that the $80 per share of greater assets for Company B are responsible for the $25 per share lower market price, since the earnings per share are assumed to be equal.

But more important than the foregoing is the general relationship between mathematics and the new approach to stock values. Given the three ingredients of (a) optimistic assumptions as to the rate of earnings growth, (b) a sufficiently long projection of this growth into the future, and (c) the miraculous workings of compound interest—lo! the security analyst is supplied with a new kind of philosopher’s stone which can produce or justify any desired valuation for a really “good stock.” I have commented in a recent article in the Analysts’ Journal on the vogue of higher mathematics in bull markets, and quoted David Durand’s exposition of the striking analogy