The proposed analytic solution in this article confirms that in terms of risk minimization (i.e., higher Sharpe ratios), minimum variance portfolios are superior to risk parity and maximum diversification portfolios. The low number of securities in the minimum variance portfolio proves that risk can be minimized by including fewer, less correlated stocks rather than just by adding more stocks.

## Whatâ€™s Inside?

The historical fact that low-volatility stocks tend to outperform the market combined with investors’ move toward risk management in the wake of the financial crisis has led to the increased interest in equity portfolios with the lowest possible variance. With an objective function of maximizing the Sharpe ratio of a portfolio, the authors provide an analytic solution for a long-only, constrained, maximum diversification portfolio—a tangent portfolio on the efficient frontier. The analytic solution also tries to provide justification for the reasons behind the selection of an investable asset class for a risk-based portfolio and the reasons behind the particular weight assigned to it.

## How Is This Research Useful to Practitioners?

A lot of research has been done already on such subjects as risk parity since the original
concept was introduced in the 1990s by Bridgewater Associates. Risk parity is a
mean–variance objective function. A risk parity portfolio generally lies inside the
efficient frontier, and it is a very difficult task to construct the portfolio with a large
number of investable sets. Similarly, a maximum diversification portfolio is an objective
function that tries to maximize a portfolio’s Sharpe ratio. It is a tangent portfolio
on the efficient frontier. The objective function of the minimum variance portfolio is
constructed in such a way that the portfolio lies on the left-most tip of the *ex
ante* efficient frontier.

Empirical results indicate that the relatively low number of stocks in the maximum diversification and minimum variance portfolios is the result of an objective function in which the risk can be reduced by selecting fewer, less correlated, and less risky stocks. These stocks are predominantly driven by a long-only threshold correlation or threshold beta. According to the analytic solution provided by the authors, individual assets are included in the portfolio if their correlation or beta is lower than the threshold limit. In their previous research, they proposed an analytic solution for optimal stock weights of a long-only, constrained, minimum variance portfolio. They empirically proved that systematic risk is a decision-making factor for a negative weight and that idiosyncratic risk can lower the optimal weight but cannot be used to exclude the security from the portfolio.

When used to look at the performance of 1,000 U.S. stocks during 1968–2012, the analytic solution illustrates that the explicit objective of low risk is achieved best by the minimum variance portfolio, followed by the risk parity portfolio and then the maximum diversification portfolio.

## How Did the Authors Conduct This Research?

With an objective of comparing risk-based, long-only portfolios, the authors simulate various risk-based portfolios, such as a long-only minimum variance portfolio, maximum diversification portfolio, and risk parity portfolio. Using the CRSP database to gather data from 1968 to 2012, they construct these portfolios out of the investable set of 1,000 large-cap stocks. Back testing is carried out on this set of stocks by using a single index model (single risk factor model) with no other portfolio constraints, such as maximum exposure limit.

Although numerous research papers and back-test results have been published on minimum variance and maximum diversification portfolios, the authors are the first to empirically study an example of a large risk parity portfolio composed of 1,000 stocks. The striking feature of their analytic solution is that it allows portfolio managers to quickly ascertain exposure weights for any investable set.

When the single risk factor model is used as a foundation, all assets have only one source
of common risk. The authors postulate numerical equations to arrive at individual asset
weights in the risk-based portfolios. The same equations are used to construct risk-based
and benchmark portfolios at the end of each month from 1968 to 2012 for the set of 1,000
common stocks. Based on the results of 60 months of observations, the authors use a
shrinkage factor of one-third toward the mean for long idiosyncratic risk instead of
one-half, which was used for predicting *ex ante* beta values.

Although the maximum diversification portfolio generates average excess returns at par compared with the minimum variance portfolio, on the risk front, it disappoints managers with portfolios that have the highest standard deviations. One of the reasons for this result could be errors in the estimations in the correlations matrix, which can be better estimated by using a multifactor risk model instead of single-factor risk model.

## Abstractorâ€™s Viewpoint

Inconsistent with prevalent academic views of a direct relationship between risk and return, research shows that the top three quintiles of portfolios have an inverse relationship between risk and return. Similarly, research shows that a concentrated portfolio with a few selective, less correlated stocks can outperform the market portfolio consistently.