In 1957, economist David Durand published a landmark paper in the Journal of Finance titled “Growth Stocks and the Petersburg Paradox”. It bridged a centuries-old mathematical puzzle with the wild, speculative world of equity valuation.
At its core, Durand’s paper explains a recurring headache for investors: why valuing high-growth companies is so incredibly difficult, and why the math we use to justify their prices often breaks down into absurdity.
The St. Petersburg Paradox: Infinite Value for a Few Dollars
To understand the connection, we have to start with the original paradox. Proposed by Nicolas Bernoulli in 1713, the St. Petersburg Paradox is a theoretical coin-tossing game played with the following rules:
- You toss a fair coin until it lands on “heads.”
- If it lands on heads on the first toss, you win $2.
- If it lands on heads on the second toss, you win $4.
- The payoff doubles with each successive “tails” before the first “heads” (i.e., $2, $4, $8, $16, and so on).
If you calculate the mathematical “expected value” (the average outcome if you played this game an infinite number of times), the math looks like this:
Expected Value = (1/2 * $2) + (1/4 * $4) + (1/8 * $8) + … + (1/2^n * $2^n)
Expected Value = $1 + $1 + $1 + $1 + … = Infinity
Mathematically, because the expected value is infinite, a rational person should be willing to pay any amount of money to play this game. Yet, in the real world, most people wouldn’t pay more than $10 or $20 to enter.
This discrepancy between cold mathematical expectation and human behavior is the paradox.
How It Applies to Growth Stocks?
Now, look at how we value companies. In finance, the fundamental value of a stock is the present value of all its future cash flows (often simplified using dividends). The standard formula used is the Gordon Growth Model (Dividend Discount Model):
Price = D / (r – g)
Where:
- Price is the current value of the stock.
- D is the expected dividend next year.
- r is the required rate of return (or discount rate).
- g is the constant growth rate of the dividends/earnings.
This formula works beautifully for mature, stable companies where the growth rate (g) is lower than the discount rate (r).
But what happens when we look at a red-hot growth stock where the growth rate (g) is assumed to be equal to or greater than the required rate of return (r)?
If g is greater than or equal to r, the denominator (r – g) becomes zero or negative. Mathematically, the formula breaks down, and the implied value of the stock becomes infinite. In other words, if a company can truly grow faster than its cost of capital forever, no price is too high to pay for its stock today.
This is the exact financial equivalent of the St. Petersburg Paradox.
Real-World Consequences: Speculative Bubbles
When investors temporarily suspend disbelief and behave as if high growth can continue indefinitely, market valuations skyrocket to irrational levels. This phenomenon has played out repeatedly across global business history:
1. The Dot-Com Crash (Global, 1999–2000)
During the late 1990s, early internet companies were valued not on current earnings, but on the assumption of explosive, compounding growth. Companies like Cisco Systems and Yahoo! saw their price-to-earnings (P/E) ratios climb into the hundreds or even thousands. Investors were essentially paying a massive premium to play a “game” they assumed had an infinite payoff, only for growth to sharply mean-revert, resulting in a devastating crash.
2. High-Growth Tech Giants (US, 2020–2021)
Consider the astronomical valuations of companies like Tesla during the peak of the pandemic-era stock boom. In late 2021, Tesla’s market capitalization surpassed $1 trillion, trading at a P/E ratio over 300. To mathematically justify that valuation using standard discounting models, one had to assume Tesla would maintain a massive double-digit market share of the global automotive industry and enjoy hyper-growth for decades—an assumption that faced severe reality checks as competition intensified and growth naturally slowed.
3. Chinese Tech Conglomerates (China, 2020–2021)
The rapid rise of companies like Tencent and Alibaba saw their valuations swell to unprecedented levels based on the expectation of endless domestic and international digital expansion. When regulatory changes and domestic economic shifts altered their long-term growth trajectories, investors quickly realized their “infinite growth” assumptions were flawed, causing trillions of dollars in market value to evaporate.
Resolving the Paradox: How Analysts Keep Valuations Grounded
Just as mathematicians resolved the St. Petersburg Paradox, financial analysts have to ground their valuations in reality by introducing constraints:
- The Solvency and Finite Life Constraint: In the original game, the payout doubles infinitely. In reality, the person offering the game (or the company paying dividends) has a finite amount of money and will eventually go bankrupt or cease to exist.
- Mean Reversion of Growth: No business can grow faster than the overall economy forever. Eventually, market saturation, competition, and regulatory headwinds drag a company’s growth rate (g) back down below the discount rate (r).
- Multi-Stage Valuation Models: Instead of using a simple one-stage model, analysts use two-stage or three-stage Discounted Cash Flow (DCF) models. They model high growth for a limited window (e.g., 5 to 10 years) before dropping the company into a conservative “terminal growth rate” (typically matching long-term GDP growth of 2% to 3%).
Ultimately, Durand’s application of the St. Petersburg Paradox serves as a timeless warning. Whenever a valuation model tells you that a stock’s potential is “limitless” or that “no price is too high,” remember that you are no longer investing—you are playing a theoretical coin-toss game where the house always wins.
