Vineer Bhansali of PIMCO writes this superb note on financial portfolios.

In finance, we assume a normal distribution of any asset price series with one peak (unimodal) and tail risks really limited and seen as a black swan event. The reality is financial time series have fatter tails implying risks have a bigger probability of happening and hitting portfolios. Moreover, in today’s times you could actually have two peaks (called bimodal) highlighting probability of multiple equilibrium (one can look at both Eurozone crisis and US recovery for instance).

In such a scenario, building asset portfolios based on unimodal is nots just risky but dangerous as well as one might be taking higher exposures to risky assets than required:

*When constructing the “normal” returns chart we used the long-term history of the S&P 500 Index as a proxy to approximate the stock market (1951 through 2010) and assumed a normal distribution: 10% average annual return and 20% volatility, as measured by standard deviation. For the bimodal distribution, we assumed that there were two regimes: the first is the one shown in our normal distribution (10% average return and 20% volatility), but the second “bad” regime is one where equities go down 50%, and then become trapped in that new scenario. A group of Deutsche Bank analysts led by Vinay Pande has been writing for a few years that equity market returns realized in the recent past are bimodal. Indeed, in one of our client meetings, my colleague Marc Seidner raised the possibility that the future looks a lot more bimodal than the past ever did, based on this evidence and similar data.*

*To illustrate this, we assumed for our example that there was only a 10% chance of the second regime happening, but once it happens the environment is a sticky, local equilibrium – a “hole that is hard to climb out of.” The interested reader can make up an infinite number of plausible scenarios such as these, and is encouraged to question accepted lore of asset allocation and portfolio construction under such multimodal distributions. In this note we will attempt to do exactly such an exercise.*

*For the bimodal distribution that results from combining the normal and bad regimes, the average return is 4% and the volatility is 26% (versus a 10% average return and 20% volatility for the unimodal normal distribution). This is simply because the bad regime has sufficient weight to reduce the overall returns. There is also negative skewness (of -0.58) in the bimodal curve versus zero skewness for the normal distribution, and excess kurtosis (a measure of whether data are peaked or flat) of 0.19 over the normal distribution, reflecting the magnitude of unlikely outcomes, or how fat the tails are (under “old normal” circumstances they are rather flat). All of these statistics are not too far from what one would glean from looking at the implied distributions from current option prices in broad equity indices; but with the important difference that traditional option pricing models get their fat tails and skewness from building in the skew ex-post on top of a unimodal distribution.*

*If we start with an assumption that we would allocate 50% of the portfolio to equities in the unimodal case, what would the optimal allocation be in the bimodal case, assuming our risk preferences are unchanged? By following a very traditional portfolio optimization exercise which involves a little bit of math, the answer turns out to be that the optimal allocation would be only 10%! In other words, one would have to de-risk by almost 80% from the current optimal allocation to arrive at the mathematically optimal result (see disclosures at the end of this article for a more detailed explanation of our computations). The prospect of being trapped in a low return, low probability event requires us to, as Mohamed El-Erian would say, “generally play defense and selectively play offense.”*

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