Vineer Bhansali, managing director and portfolio manager at Pimco, says there has been a shift in asset allocation modelling away from reliance on a single mean point, towards multiple points of equilibrium.
Unimodal versus Bimodal Distributions
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.
|Optimal Allocations: For those who are mathematically inclined, the computation of the optimal allocation to risky assets begins by specifying a very standard power utility function, W^(1-a)/(1-a), which represents the preferences of a typical investor with constant relative risk aversion and W is the initial wealth of the investor. We used a typical parameter of a=5. The investor has a choice to allocate between the risky asset (stocks) and keep money in cash at a return rate of 25 basis points annually. (The curious reader is encouraged to experiment with other assumptions).|
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.
None of these observations should seem surprising if one realizes that a mixture of two normal regimes can yield a result that is abnormal. Fat tails and negative skewness can arise from even the mere possibility of multiple equilibria, even though both equilibria in themselves are normal. This practice of generating very complex distributions from mixtures of simple, normal distributions is well known among statisticians and has applications in many fields of practical import: medicine, astronomy and casino gambling to name a few. In the present context, the two “normals” are the mixture of the old normal and the New Normal that PIMCO has been talking about for a few years now.