Investment

Wealth, risk, losses…and the Bernoulli hypothesis

Some families cultivate fine wines. Others build industrial dynasties. The legacy of the 18thCentury Bernoulli family of Basel, Switzerland is more subtle, involving its research into mathematical theory which laid down foundations for the modern financial world.

In his new book, Safe Haven, published by Wiley, hedge fund manager Mark Spitznagel, explains how the Bernoulli family demonstrated the importance of geometric returns, which clarify the impact of gains, and losses, on wealth over time.   

Bernoulli realised some people would consider the cost of a stake to be too high to take the risk. As he put it, a given loss “is more significant to a pauper than to a rich man.” 

Losses are more damaging than you might think, given the loss of capital involved. Spitznagel’s Universa Investments, advised by Nassim Nicholas Taleb, sets out to insure portfolios against this risk through tail risk hedging by investing in cheap put options which are out of the money. Most of the time they trundle along, not doing much, until a crash happens and their strike price rockets.

Universa, founded in 2007, generated a return of 4,000% for clients in the first quarter of 2020, when Covid-19 struck and markets tanked. Crisis-driven gains like this mean its net annualised return is three percentage points ahead of the S&P 500 index in just over a decade. 

According to Institutional Investor, Universa recommended a 3.3% weighting in tail risk hedges in 2020 to hedge a 96.7% S&P 500 index exposure.

The Bernoulli family of Basel, enriched by the spice trade, produced no less than eight renowned maths academics in the 18th Century. A descendant, Marie Curie, won a Nobel prize for her study of radioactivity in the 20th Century. 

Johann Bernoulli researched the impact of gravity on curved surfaces. His brother Jacob developed the law of large numbers paving the way for actuarial science and big data applications based on frequent sampling. 

Johann’s son Daniel became a professor of mathematics at the Academy of Science in St Petersburg. He researched planetary orbits and fluid mechanics, later inspiring research into aircraft wings.

Spitznagel is particularly impressed by the way he solved the St Petersburg paradox, where individuals are less willing than you might expect to place a wager on games that give you the chance of winning a massive prize. 

Bernoulli realised some people would consider the cost of a stake to be too high to take the risk. As he put it, a given loss “is more significant to a pauper than to a rich man.” 

The less you own, the greater the pain of loss. Equally, the more you own of a commodity in surplus the less the excitement you get from adding to the pile.

Rather than analysing arithmetic returns, Bernoulli based his research on “mean utility” taking account of whether individuals rated a given opportunity This paved the way for geometric returns, which compound up profits and losses over time to provide individuals with an accurate assessment of outcomes. 

Bernoulli refers to the potential loss of cargo by a merchant in waters infested by pirates. A merchant might look at the cost of insurance for a single voyage and decide it would erode his profits too much. But a wiser rival would reflect on the problems his business would face rebuilding its position if its capital were wiped out by a raid. 

As Spitznagel puts it: “A large loss disproportionately lowers our merchant’s geometric average return because it leaves him with a much lower stake to invest and compound on his next shipment.” 

Universa seeks to hedge its exposures by putting part of its portfolio to work in out-of-the-money put options. It costs money to keep renewing them, but the expenditure is like an insurance premium. They may lose money year on year. But averaged out, they prove their worth. 

Spitznagel has tested the outcome for an indexed portfolio with 2% hedging over 120 years. Geometric returns for an indexed portfolio weighted towards value stocks generated 9.1% a year. Those tilted towards active skill offered 9.3%. Those using tail risk insurance returned 10%.

Spitznagel says safe havens provide cover for portfolios built for speed, sometimes to meet hefty liabilities. He compares their use to Grand Prix motorists relying on a pit stop to change a damaged tyre. It costs money to maintain the facility. You lose a bit of time to get a wheel changed. But the effort keeps drivers in contention.

Spitznagel developed his career by basing his decision on a range of outcomes relating to his early ambition to play lead horn at a leading orchestra.  He realised that he was as likely to end up in an inferior position or playing in a subway.  The prospective average return was too low, so he abandoned his dream.

Anyone hoping for a detailed explanation of portfolio construction will be disappointed by this book. There is very little about Idyll Farms where Spitznagel and his wife Amy make prize-winning cheese from their flock of goats. 

But his views on the value of safe-haven investing at a time when markets are expensive and facing risks on several fronts could not be more timely. 

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