As quantitative asset managers, a key tenant of our investment philosophy is what we call Quantitative Integrity™. At the heart of Quantitative Integrity is a belief that success is achieved through a balance of quantitative analytics and qualitative insight. Data can never trump insight, but theories must always be supported by data.

A key part of Quantitative Integrity is understanding not just the math behind a model, but intuitively grasping how the model will behave when applied to financial data. Simple toy examples can be a useful tool in beginning to develop this understanding.

Post-2008, there has been huge mainstream interest in more effective approaches to diversification (see risk parity, all-weather, etc.). Attilio Meucci has done some extremely interesting work on this front^{1}. Like Markowitz, Meucci believes an investor should seek the best risk/return trade-off. Meucci keeps this basic framework, but utilizes a non-traditional risk measure.

Meucci’s risk metric is the effective number of independent bets contained in the portfolio. In non-financial applications, this risk metric is pretty straightforward to understand. Suppose a sports fan wants to place $100 worth of bets this weekend. He decides to bet on the Ravens over the Lions, the Spurs over the Jazz, the Blackhawks over the Kings and Manchester United over Aston Villa. We can be pretty confident that the outcomes of these four bets are independent and so the effective number of independent bets in this sports betting portfolio is four.

This type of analysis gets a little bit more complicated in finance, where asset returns are generally not independent. Principal component analysis is a tool that can be utilized to express a portfolio of non-independent assets as a portfolio of independent sub-portfolios, which then allows us to calculate the effective number of independent bets.

Let’s compare the results for Meucci’s approach to the results of the more traditional mean-variance optimization approach through a simple example. Consider a simple CAPM world with 30 individual stocks. Each stock’s return follows the model below:

We assume that the risk-free rate is 2%, the expected market excess return is 6%, the beta for every stock is 1, the standard deviation of the market excess return is 15% and the standard deviation of the idiosyncratic returns are 15%. The expected return and variance of an individual stock i are computed as follows:

The covariance and correlation between stock i and stock j are:

Since all 30 stocks have the same expected return, the expected return of any fully invested portfolio will be 8%. As a result, the mean-variance optimal portfolio will be the same as the minimum volatility portfolio. The minimum volatility portfolio will invest 1/30^{th} of its capital in each of the 30 stocks. This portfolio has a volatility of 15.2% and a Sharpe Ratio of 0.39. We are able to diversify away almost all idiosyncratic risk.

The results using Meucci’s approach are quite different. By removing the idiosyncratic risk, the mean-variance optimal portfolio essentially has one bet, the market. The mean-variance portfolio is actually the least diversified portfolio in terms of the independent number of bets. The portfolio with the most independent bets will hold only one stock. This one stock portfolio will have two independent bets, the market and undiluted exposure to the idiosyncratic exposure of that stock.

What’s going on here? Are these models contradictory? Is one right and one wrong? Here is where understanding a model intuitively is important. Based on the Meucci methodology, the equally weighted portfolio effectively has one independent bet. This is true mathematically. However, in the financial world we must consider that the equally weighted portfolio actually maximizes the number of independent bets taken (each 1/30^{th} allocation to a stock results in a 1/60^{th} bet on the market and a 1/60^{th} bet on the idiosyncratic risk of that stock, so overall there is a ½ bet on the market and a 1/60^{th} bet on each of the stock’s idiosyncratic risks); it’s just that the effects of the idiosyncratic bets will on average cancel each other out. The market is not the only bet that the portfolio takes, but it is the only bet that cannot be diversified away and thus is the only bet that matters in terms of the portfolio’s end result.

In terms of more practical applications, the takeaway is that it does make sense to maximize the number of independent bets but only when those bets cannot be diversified away through complimentary exposures.

^{1} See http://ieor.columbia.edu/files/seasdepts/industrial-engineering-operations-research/pdf-files/IEOR_Oxford_Meucci.pdf and http://www.risk.net/data/risk/pdf/technical/2009/risk_0509_technical_im.pdf

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