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Our whitepaper “The Optimization Machine: A General Framework for Portfolio Choice” presented a logical framework for thinking about optimal methods of portfolio formation given specific assumptions regarding expected relationships between risk and return. We explored the fundamental roots of common portfolio weighting mechanisms, such as market cap and equal weighting, and discussed the rationale for several risk-based optimizations, including Minimum Variance, Maximum Diversification, and Risk Parity.

For each approach to portfolio choice we examined the conditions that would render the choice mean-variance optimal. For example, market cap weighting is mean-variance optimal if returns are completely explained by CAPM beta, or in other words, if all investments have the same expected Treynor ratios. Minimum variance weighted portfolios are optimal if all investments have the same expected return, while Maximum Diversification weighted portfolios are optimal if investments have the same Sharpe ratios.

The Portfolio Optimization Machine framework prompts questions about how well academic theories about the relationships between risk and return explain what we observe in real life. While academics would have investors believe investments that exhibit higher risk should produce higher returns, we do not observe this relationship universally.

For instance, we show that both the Security Market Line, which expresses a relationship between return and stock beta, and the Capital Market Line, which plots returns against volatility, are either flat or inverted for both U.S. and international stocks over the historical sample. In other words, stock returns are either independent of, or inversely related to risk.

We also examined the returns to major asset classes, including global stocks, bonds, and commodities. For asset classes, there appears to be a positive relationship between risk and return, at least when returns are analyzed across different macroeconomic regimes. Normalized for inflation and growth environments, stocks and bonds appear to have equal Sharpe ratios in the historical sample.

The Sharpe ratio of diversified commodities has been about half of the Sharpe ratio observed for stocks and bonds since 1970 when conditioned on regime. However, we highlight that our analysis may produce bias against commodities, given that there were few regimes that would have been favorable to commodities in our historical sample. With such a small sample size, we believe it is premature to reject the hypothesis that commodity risk should be compensated at the same rate as risk from stocks and bonds.

Our whitepaper presented a great deal of theory, and offered guidance from history about the nature of the relationship between risk and return. Armed with this guidance, we can invoke the Optimization Machine decision tree to make an educated guess about optimal portfolio choice for different investment universes.