Friday, February 11, 2011

Can low risk portfolios achieve high rewards? A statistical view

Welcome to this blog where I will try to explain some ideas regarding the implementation of portfolio strategies based on quantitative techniques (mainly Statistics and Optimization). These strategies are also based on historical data, hence their performance can be evaluated through extensive back-testing.
The ideas start with the pathbreaking work of Markowitz, designed for investors who care only about the mean and variance of a portfolio’s return. But to implement these portfolios in practice, one needs to estimate the means, volatilities and correlations of stock returns.
It is well known that next-period expected returns cannot be forecasted reasonably well from historical data. And if we try to estimate these means by fundamental or technical analysis, the forecasting error is still substantial.
For this reason, it is better to ignore these forecasts from the original framework. As Jagannathan and Ma stated: “the estimation error in the sample mean is so large that nothing much is lost in ignoring the mean altogether”.

Hence, we are going to focus on low volatility portfolios. One may argue that these portfolios will imply a low return too. But surprise: they usually perform better than portfolios that try to optimize the trade-off between risk and return.

Indeed, there exist minimum volatility indices, like the one by MSCI, to serve as relevant benchmark for financial institutions.
In subsequent posts, we will show how these low volatility portfolios continue to have low volatility in the future, and furthermore, they attain returns similar (or even better) than broad indexes like the S&P 500.
This apparent paradox is explained because the financial markets are not perfectly efficient all the time. Then, risk and return at the stock level do not need to be highly correlated in practice. Indeed, there exists evidence of stock portfolios that not always exhibit a significant positive correlation between risk and return.
This means we can increase our portfolio return per unit of risk (the well-known Sharpe ratio) by focusing only on the volatility, and ignoring the expected return.
A major concern among investors and portfolio managers is the transaction costs. In order to reduce them, it is desirable: (i) to hold a relatively small number of stocks, and (ii) to lessen the wealth traded in each period.
We propose the use of optimization techniques together with statistical analysis to select a few stocks from a broader index like the S&P 500.
And to reduce the wealth traded in each period, we propose a reasonably stable portfolio composition that does not change drastically over time. Finally, and to reduce even more the impact of transaction costs, the portfolio strategies will be rebalanced every 4 weeks.
In the next posts, we will recommend low volatility portfolios with the above properties (low volatility and good return after transaction costs), and show the corresponding back-tested results.

The back-test is as follows: at a given week, we compute the portfolio strategy, and a week later we compute the corresponding portfolio return net of transaction costs respect to the last portfolio composition. We repeat this procedure for every week in the last years obtaining the following performance measures: portfolio return mean and volatility, the Sharpe ratio, the Value-at-Risk, the correlation with the market index, etc.
Please, note that all the results reported here are back-tested results: they do not represent results of actual trading.
Finally, these ideas are inspired by the following works:
"Optimal versus Naive Diversification: How Inefficient Is the 1/N Portfolio Strategy?. Review of Financial Studies 22(5), 1915--1953 (2009).
Portfolio Selection with Robust Estimation''. Operations Research, 57(3), pp. 560-577, 2009. 


  1. The link for "Optimal vs Naive..." is instead going to a different paper by DeMiguel and Uppal.

  2. Can you please clarify specifics of the "portfolio strategy" which is being computed to generate the proposed portfolios?

  3. @quantivity

    Yes, it is just a global minimum-variance strategy.

  4. Can you please cite/define the specific algorithm used to generate the "global minimum variance" portfolio, as there are multiple ways to do so--as both "global" and "minimum variance" have multiple interpretations.

  5. @quantivity Yes, of course.

    For the optimization model I use:

    Jagannathan, R. and T. Ma (2003). Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps. The Journal of Finance 58, 1651–1684.

    For the estimation of the covariance matrix I use:

    Ledoit, O. and M. Wolf (2004). A well-conditioned estimaor for large-dimensional covariance matrices. Journal of Multivariate Analysis 88, 365–411.