Uncertainty, distributions and fat-tails

August 6, 2010

In a thoughtful article published this week in the Financial Times, PIMCO Chief Executive Mohamed El-Erian and Columbia Economics Professor Richard Clarida explore the implications of a shift in the shape of investors’ and policymakers’ expectations about the future.

“It seems that, wherever we look, the snapshot for ‘consensus expectations’ has shifted: from traditional bell-shaped curves — with a high likelihood mean and thin tails (indicating most economists have similar expectations) — to a much flatter distribution of outcomes with fatter tails (where opinion is divided and expectations vary considerably).”

They do not go quite as far as Bank of England policymaker Adam Posen, who suggested in a recent speech that the distribution of outcomes has inverted and become U-shaped. But their focus on a bell-curve with fatter tails agrees with Federal Reserve Chairman Ben Bernanke’s characterisation of the economic outlook as “unusually uncertain” at present.

El-Erian and Clarida draw five conclusions for investors:

(1) Investment strategies based on mean reversion will become less compelling. Fat-tailed distributions still have means but they will be realised less often in practice.

(2) Frequent risk-on/risk-off oscillations in sentiment will remain a persistent feature of the landscape.

(3) Hedging against tail-risks will become increasingly important.

(4) Historical benchmarks and correlations will be challenged.

(5) Less credit will be available to sustain leverage and high valuations.


With great respect, Bernanke is wrong to characterise the current outlook as “unusually uncertain.” It is no more uncertain than normal. In the last 30 years, there have been literally dozens of instances in which senior policymakers have complained about heightened or unusual uncertainty when deciding whether, when and by how much to tighten or loosen policy.

In remarks to the Fed’s Jackson Hole symposium in August 2003, Federal Reserve Chairman Alan Greenspan observed “Uncertainty is not just an important feature of the monetary policy landscape; it is the defining characteristic of that landscape.”

Uncertainty about the outlook is the policymaker’s and investors’ lament, but is in fact quite normal. At different times in the past two decades, policymakers have bemoaned uncertainty about exchange rates, the U.S. current account deficit, productivity growth, long-term bond yields, inflation, the natural rate of growth, and jobless recoveries, among many other items. The outlook always looks more certain in retrospect than it did at the time.


El-Erian, Clarida and Posen all argue the distribution of outcomes or at least expectations may have changed.

El-Erian and Clarida suggest the distribution is still bell-shaped (“Gaussian”) but with a lower peak and fatter-tails (in technical terms it has more “kurtosis”). Posen’s speech inverted the distribution completely and replaced the normal central peak with an empty centre and very fat-tails making it U-shaped.

But has the real distribution of outcomes changed, or merely our perception of them? The problem is that the actual distribution cannot be seen directly. We can only estimate it based on previous experience modified by some guesses about the future. But relying on recent experience is apt to cloud estimates about what the full distribution really looks like.

Policymakers and investors have been criticised for being fooled by the Great Moderation of the 1990s and 2000s into under-estimating tail risks. Putting too much emphasis on the stability of the recent past, they ignored what could happen during a crisis.

Now they risk making the opposite error. After being scarred by the wild gyrations of the last three years, policymakers and investors may be assigning too much weight to extreme outcomes.

Perhaps the tails of the distribution have not changed, only our estimate of them. Just as we under-estimated the tails during the 1990s and 2000s, now we may be focusing on them too much because they occurred last year, neglecting the more “average” and common returns in the middle of the distribution.

Consider Noah and the Flood. The year before the flood happened Noah probably thought an inundation was very unlikely. After all, he didn’t have an ark on hand until God told him to build one. The year after the flood, Noah probably thought the risk was much higher. But did the actual risk change? Or just Noah’s estimation of it?

Noah received a promise there would not be another flood. But after a personalised lesson in risk distributions, should Noah spend all his time and money worrying about other risks such as fire, drought and plague?


If policymakers and investors spend all their time preparing for the next 1-in-100 year cataclysm, they will be just as wrong as their predecessors who thought that it could never happen. They will take out too much “insurance” and pay a price for misjudging the distribution just as surely as their predecessors paid for not factoring in tail risk.

Investors who buy deep out-of-the money puts on equities as insurance against a crash will receive a big payoff in another market meltdown, but in the meantime lose a small amount every month as each set of puts expires worthless while they wait for the crash’s arrival. Investors who take a long position in commodities waiting for the next price spike will lose money meanwhile through the contango and cost of carry.

For these strategies to make money over the whole cycle, investors must be able to estimate the whole distribution accurately.

The same applies to policymakers. Recent disagreements within the U.S. Federal Open Market Committee and the Bank of England’s Monetary Policy Committee (MPC) are really disagreements about how much weight to assign to tail risks.

Bernanke and St Louis Fed President James Bullard have focused attention on the need to set policy to avert the risk of deflation. But it remains a tail risk rather than a central scenario.

In contrast, Dallas Fed President Richard Fisher, Kansas City Fed President Thomas Hoenig and Philadelphia Fed President Charles Plosser focus on a central scenario in which the economy continues to recovery, and have pushed for rate rises or at least no more easing for the moment. They worry the Fed will overcompensate for a small risk of deflation, in the process creating a new set of distortions and problems.

Precisely the same argument is playing out at the Bank of England, where External MPC Member Andrew Sentance has pressed for more normalisation of interest rates now the peak of the crisis has passed, while Governor Mervyn King and other members are more concerned to maintain ultra-low rates to offset lingering tail risks.

El-Erian’s and Clarida’s investment conclusions are probably correct. Recent events should give investors and policymakers a better understanding about outlying areas of the distribution. Even if the distribution itself has not changed, a better understanding of tails should make investors more cautious about piling on leverage to juice up returns on strategies that depend on mean reversion.

But we should be careful to guard against over-reacting and rushing to assume the distribution itself has changed, or start over-estimating the probability of tail risks and ignore the much higher likelihood of more moderate outcomes.

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