Sex, Lies, and Optimizing Utility

William Bernstein Victor Haghani has thought long and hard about his participation in the 1998 blowup of Long-Term Capital Management (LTCM). His big mistake, he concluded, was investing 80% of his personal assets in the firm.

This loss also cratered, in econ-speak, his “utility,” which is to say it made him miserable. The take home, he writes in The Missing Billionaires, is that he didn’t size his personal stake in LTCM properly. Had he invested only half as much, he’d have been far less unhappy. Contrariwise, had his large LTCM bet succeeded, his utility would only have been slightly higher than with a half-sized bet: There’s a huge utility gap between a net worth of $0.00 and $1 billion, and a small utility gap between $1 billion and $2 billion. The book, co-written with James White, is an exploration of this epiphany.

Haghani is a world-class financial economist who sharpened his quantitative and trading skills at Salomon Brothers during the 1980s “liar’s poker” era, where he impressed legendary bond trader John Meriwether enough to make him the youngest principal in LTCM, alongside Nobelists Myron Scholes and Robert Merton. The Missing Billionaires applies these skills to guide those who want to optimize their total lifetime utility.

First and foremost, as noted above, comes the sizing of investments. In 1956, John Kelly, a Bell Labs researcher, published a treatise on how much a gambler should bet on each draw, known thereafter as the Kelly criterion, to optimize the growth rate of one’s bankroll at any given moment. In its simplest form,

optimal bet size = p – q/b

where p is probability of a win, q is the probability of a loss, and b is the payoff. For a 60% bet that pays off 3 to 1, for example, the optimal bet size = 0.6 – 0.4/2 = 40% of your pile at that point (where the numerator means that you are returned $3 for a $1 pay in, for a profit of $2). On the other hand, if there is only a 50% chance of paying off 2 to 1, then your optimal bet size = 0.5 – 0.5/1, that is, zero. Increasing the bet size beyond the Kelly criterion decreases wealth growth, and moving much beyond twice Kelly makes a negative return highly likely.