How Perfectly Correlated Stocks Can Form a Zero-Volatility Portfolio

Summary

The document asks whether a portfolio of two stocks can have both lower volatility and a higher return than either stock, and whether adding more stocks changes the answer. The response gives a mathematical edge case: if two assets have identical volatility and perfectly correlated movements, taking equal amounts long and short cancels their fluctuations. If their returns differ, that spread has a nonzero return despite the zero volatility under the stated assumptions. Scaling both positions scales the return while leaving the volatility at zero.

This construction demonstrates a possibility, not a practical investment recipe. It assumes a perfect, stable correlation and equal volatility, permits short positions, and abstracts from financing, borrow costs, transaction costs, liquidity constraints, and changes in the relationship. The claimed ability to scale return without bound follows from the idealized setup; real portfolios cannot rely on those assumptions. The answer does not separately work through the case of more than two assets, though the same logic can extend when a zero-risk linear combination exists.

Key ideas

• With equal volatility and perfectly correlated movements, equal long and short positions can cancel fluctuations.
• If the two assets have different returns, the spread can retain a nonzero return while its volatility is zero in this idealized case.
• Scaling the positions scales the spread return without changing its zero volatility under the assumptions.
• The result relies on exact, stable correlation and omits practical costs, constraints, and market changes.
• A similar construction with more assets requires a suitable zero-risk linear combination.

Tags

# Puzzler on construction of a 2-stock portfolio


# Puzzler on construction of a 2-stock portfolio












Lets say you get to choose any 2 stocks.

Is it possible that a portfolio can be built from the two stocks (long or short either, any weighting) that has a lower volatility (standard deviation of movements), and simultaneously higher percent return than either of the two underlying securities alone.

I want to say yes, but I think the answer is no. In the event of a correlation of -1 between A and B:

A: a stock with high stddev, and small positive return

B: a stock with same stddev, and small negative return

Long A short B = much higher returns, much higher stddev

Short A Long B = 0 return, 0 stddev

I currently have an infinite loop randomly building portfolios to test this at home. Bonus: Is it possible with a portfolio with more than 2 stocks in the portfolio?

## Answer by M. Jeunesse (score 1, accepted)

https://quant.stackexchange.com/a/29921

$X$ and $Y$ perfectly correlated with same vol. So $X-Y$ has no volatility at all and for any $n$, $n(X-Y)$ has still no volatility.

If $r_X > r_Y$ are returns of $X$ and $Y$, then $n(X-Y)$ has return $n(r_X-r_Y)$ which can be as big as you want with $n$ big enough.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.