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Theoretically, one could have stock prices with returns $\rho_1(k)$ and $\rho_2(k)$ having mean values $\mu_1$ and $\mu_2$, but still be negatively correlated with $$ \mathbb{E}[(\rho_1(k)-\mu_1)(\rho_2(k)-\mu_2)]<0. $$ Could this kind of model have any practical applications? Could we say that pairs-trading exploits a similar model?

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It is very rare to find stocks that are reliably negatively correlated with each other. At least in absolute (as opposed to relative outperformance) terms. It can happen from time to time, but the positive correlation/beta to market usually prevails as the dominant driver of risk/returns.

This said, the classic example of the phenomenon you're asking about is stocks versus bonds. Both are (usually) priced for positive expected returns, albeit with a negative (or at least zero) correlation when they better/worse than drift. Or gold-vs-bonds, or gold-vs-stocks etc.

The practical application of this kind of thing isn't really pairs-trading. Long-short would capture the spread between two positive returns, by doubling down on the relative risks between the two assets. It's more the essence of diversification - playing for the average, while using the -ve correlation to reduce volatility.

In its purest form, the model most explicitly designed to capture this kind of effect is "risk parity". IE long of both in inverse proportion to their risk (either inverse vol, or inverse marginal contribution to portfolio risk). Albeit this is optimised for the two assets to have returns proportionate to their volatilities. The proportions invested would vary with the precise figures assumed/expected; but the concept is the same.

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