WITH BORROWED MONEY, THE GAMBLER WINS ONE TIME IN TWO

Picture a “finance company” in its simplest form. It starts with some money and wants to multiply it by a factor r. It can only make fair bets — bets whose average gain is zero — and it cannot risk more than it currently has. It keeps betting until it reaches its goal or goes bankrupt.

Probability theory gives a firm answer: there is no way to make a profit on average. A classic result, the optional stopping theorem, caps the chance of reaching r times your starting money before going broke at 1 in r. Want to multiply your stake by 1,000? Your odds are at most one in a thousand. This is known as the gambler’s ruin.

Lehman’s puzzle

William Kuszmaul, a computer scientist at Carnegie Mellon University, starts his short note with a real case. From 1995 to 2008, Lehman Brothers Holdings multiplied its shareholder equity by roughly 8, from about $3.4 billion to about $26 billion. With fair bets and no borrowing, that should have happened with a probability of at most about 1 in 8.

But there was a catch. When Lehman went bankrupt in 2008, it was not worth zero dollars: it was worth roughly minus $150 billion. With its roughly $26 billion, it had borrowed around $640 billion to bet with. Over the whole period, its leverage ratio — total assets divided by equity — stayed between 20 and 40. It was repeatedly risking far more than it had. The author adds that his comparison ignores many factors, such as the general growth of the stock market and new capital raised from shareholders.

The leverage game

The note turns this into a precise game. A player, Alice, starts with $1 and aims for m dollars. Every bet must be fair, but before each one she may borrow up to L − 1 times what she has, so she can lose up to L times her own money. The game ends when she reaches her goal or drops to zero or below.

The obvious move, one big bet for the whole goal, does badly. Doubling her money again and again does better. The lesson is to grow slowly, so that each win gives her more to leverage next time. The best strategy pushes this to the limit: on every bet, Alice aims to gain only a tiny fraction ε of her money, which she wins almost every time, while risking her whole stake plus the loan on the rare loss.

One chance in two of ×1,000

The author proves that this strategy succeeds with a probability that approaches

m^(−1/L)

— the target multiple raised to the power minus one over the leverage — and that no strategy can do better. The proof uses the same optional stopping theorem, applied not to Alice’s fortune but to its L-th root, a quantity that can only drift downward on average.

The numbers are striking. With leverage of 10, less than Lehman’s, the chance of multiplying your money by 1,000 without ever going bankrupt is about 50%, since 1,000^(−1/10) ≈ 0.5. More generally, with leverage L, a coin-flip chance gets you a factor of 2 to the power L. With leverage 20, that is about a million.

Who gets ruined

Back to Lehman. With leverage above 20, the toy model says an eightfold growth was not that impressive: about 8^(−1/20) ≈ 90% likely, even with bets that were merely fair. Borrowing is not free, of course; Lehman estimated it paid about 5% a year in interest. Including interest in the game does not change the conclusion, as long as the interest rate stays below 1 over the leverage.

Kuszmaul is clear that this is a toy model, written for his randomized algorithms course, and that it does not claim to capture real-world finance. Its lesson is narrower and sharper: a spectacular run of profits is not proof of good bets. With enough leverage, fair bets can build a fortune with high probability — and when the rare loss comes, it lands far below zero, on someone other than the gambler.