Why Zero-Volatility Portfolios Can Have Nonzero Expected Returns
Summary
The document explains why combining two perfectly negatively correlated assets can eliminate portfolio volatility without forcing the portfolio's expected return to zero. Expected return is the weighted average of the component returns; correlation affects portfolio risk, not that weighted-return calculation. The allocation weights therefore determine both the portfolio's return and whether the opposing movements cancel in value.
An equal allocation in the special case of equal and opposite expected returns gives zero expected return. But assets can have offsetting outcomes of different magnitudes: the example of ice cream and umbrella businesses shows that a fixed mix can produce a positive payoff in either weather scenario. This is a conceptual explanation rather than an empirical study, and its certainty depends on the assumed perfect negative correlation and the stated payoff patterns. Portfolio weights must be chosen for the desired risk and return, rather than inferring returns from correlation alone.
Key ideas
- Portfolio expected return is the weighted sum of component expected returns.
- Perfect negative correlation can remove portfolio risk without making expected return zero.
- An equal allocation yields zero expected return only when the component returns offset at that allocation.
- Different payoff magnitudes can produce a nonzero fixed portfolio payoff even when asset outcomes move in opposite directions.
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# Why do two perfectly negatively correlated assets not return 0%? # Why do two perfectly negatively correlated assets not return 0%? So, per the title, why would a combination of two risky assets that have the same exact expected return and standard deviation while being perfectly negatively correlated not return 0%? Why do you just combine the weighted expected return of the two assets to get the expected return at 0 standard deviation? It seems logical that if asset A goes up by expected return and asset B goes down by the expected return, the portfolio return would be 0. But in textbooks the expected return of two perfectly negatively correlated assets is just the sum of their weighted expected returns. Why would you not subtract returns since if one goes up, the other goes down (detracting from returns)? ## Answer by Jan Sila (score 1, accepted) https://quant.stackexchange.com/a/27919 If I understand your question correctly; the (expected) return always depends on the weights that the respective factor has in the portfolio, regardless of the risk. You are trying to find the optimal portfolio (given risk with highest return, or given return with lowest risk) so in the first step you try to diversify out the risk (and get the weigths for the sought portofilo) once you get your weights, you calculate the return (that does not depend on the risk or covariances at all, just the weights and respective returns). In extreme case: 50-50 allocation between two perfectly negatively correlated assets of the same return that would result in 0 (expected) return. But say, typical example - selling ice cream and umbrellas. If it is a rainy season, umbrella stocks goes up 70\$, ice cream loses 30\$ and vice versa if it is sunny. If you invest 50-50, then regardless of the weather you get $0.5*(-30\$)+0.5*(70\$)=40$$ with certainty. Always the payoff depends on the weights (60-40) and if its a 'bad' year would give you only 10\$ etc.
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