In their latest Investment Insights article titled – Rational Numbers, Lindsell Train discuss why perpetual growth stocks don’t actually exist in real life. Here’s an excerpt from the letter:
The St. Petersburg Paradox as laid out by Bernoulli in the eighteenth century, debates the appropriate wager for a coin toss game whereby the payout doubles for every successive tail that’s thrown. As the theoretical number of headless flips trends out to eternity (even with their decreasing likelihood) the series’ expected value actually sums to infinity (i.e., P=1×1/2+2×1/4+…etc). By analogue to our growth stocks, the game’s probability halving as the payout doubles has the same effect as a company’s growth rate equalling the discount rate.
The paradox in both cases stands that although a participant is in practice guaranteed to win/earn a finite amount, their theoretically fair entry fee is infinite. Many investors have referenced to this specific problem of valuing growth stocks, including John Burr Williams (widely credited with deriving DCFs in the first place) and Benjamin Graham, who included a whole chapter on them in his 1962 (fourth and final author-updated) edition of Security Analysis.
At this point though, most sensible practitioners will pause to remind us that this isn’t really a problem, as such perpetual growth stocks don’t actually exist in real life. (As the saying goes, ‘anyone who believes exponential growth can go on forever in a finite world is either a madman or an economist’). Some moderation is clearly called for – but how, and how much? Growth rates do not need to be that high before we run into trouble. If the market’s fair rate of return is, say 8-9% nominal then any double-digit perpetuity growth rate will blow up the equation. So, should we dial back the growth, or rationalise our time horizon?
Arbitrarily constraining growth forecasts seems peremptory, but if left to run – even over finite periods – high rates eventually stop being credible. $1,000 compounded at 10% pa over 250 years will match the current gross domestic product of the US economy. Give it another 15 years and it will exceed today’s entire global output. So, whilst bounding the problem resolves the infinity paradox, it can still yield unrealistic figures . What company could possibly grow at such a rate for so long?
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