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Growth Stocks and the Petersburg Paradox

In the current zero-interest rate world, where dovish Central Banks continue their accommodative monetary policy settings, future earnings become more valuable than under the previously higher interest rate regime (which is increasingly becoming a di…




In the current zero-interest rate world, where dovish Central Banks continue their accommodative monetary policy settings, future earnings become more valuable than under the previously higher interest rate regime (which is increasingly becoming a distant memory).

It’s no surprise then that many high growth stocks have been bid up to tremendous valuations, given that this often backloaded growth is ascribed a higher value when rates fall (via the discounting mechanism). However, this isn’t a new dynamic and it poses challenges for value-oriented investors in terms of what price to pay for these businesses.

In the September 1957 issue of The Journal of Finance, David Durand wrote the following:

When the growth potential of a stock becomes widely recognized, its price is expected to react favorably and to advance far ahead of stocks lacking growth appeal, so that its price-earnings ratio and dividend yield fall out of line according to conventional standards. Then the choice between growth and lack of growth is no longer obvious, and the astute investor must ask whether the market price correctly discounts the growth potential. It is possible that the market may, at times, pay too much for growth?

This idea of overpaying for growth is particularly salient in the current market we find ourselves in, where many businesses exhibiting above-normal growth have skyrocketed to seemingly expensive valuations. It brings to mind the idea of the Petersburg Paradox.

The Petersburg Paradox stems from a paper on probability that Daniel Bernoulli presented before the Imperial Academy of Sciences in Petersburg in 1798. Bernoulli puts forth the following problem.

Peter tosses a coin and continues to do so until it should land ‘heads’ on the very first throw, two ducats if he gets it on the second, four if on the third, eight if on the fourth, and so on, so that with each additional throw the number of ducats he must pay is doubled. Suppose we seek to determine the value of Paul’s expectation.

Source: The Petersburg Paradox by Durand

If playing this game, what would the expected value be? We can derive this by multiplying each probability by the payment, such that Paul’s expectation of what he will win over enough plays will be defined as ½ + 2/4 + 4/8 + 8/16 +16/32 + …

If the game is terminated after n tosses, regardless of whether a head shows or not, the series will contain n terms and its sum will be n/2. However, if the playing parties agree to continue the game until a head shows, then n is technically an infinite number, as is the sum n/2. This is what drives the paradox.

The statistical principles of expected value dictate that Paul should pay an infinite price to play this game. This is an absurdity and rational people would offer to pay much less than this to play the game.

When valuing any business, it is worth remembering that its worth is driven by the total cash flow the business is expected to generate over its lifetime, discounted back to the present at an appropriate rate. Unfortunately for investors, a discounted cash flow analysis yields a different outcome than the coin toss game in the Petersburg Paradox, and there is some price at which an investment will become value-destructive.

A key difference between investing and the hypothetical coin toss scenario is that in investing “n” is finite. All great businesses eventually get disrupted or die with a long enough timeline. Furthermore, unlike the Petersburg Paradox, the doubling sequence for each coin toss that lands on tails is not fixed, nor dependable when looking at the growth profiles of companies. Growth and returns on capital fluctuate and capitalism ensures that there are always firms looking to compete away abnormal returns. More likely than not, over a long enough period the returns of a business will be diminished (although there are of course exceptional businesses that will do a better job of sustaining their growth and return profiles).

Paying a price that requires believing growth and margin assumptions for the business that simply will never be met is a virtually guaranteed way to destroy capital in the long run. An immutable rule is that the higher the price you pay, the lower your prospective return. When investors pay rich, exorbitant multiples for high growth businesses, the implication is that a longer holding period, over which that high growth is sustained, will be required to support further gains in the stock.

This isn’t to say that over stretches of time that some businesses that look expensive, prima facie, won’t achieve growth that will render this initial valuation cheap. But for some of the valuations being ascribed to some of these high growth businesses today, the mind boggles at how they could possibly deliver the requisite growth to support these lofty valuations.

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