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Say I have a portfolio of expected return $10\%$ and volatility $20\%$. If I have another asset that is either one of:

  1. Negatively correlated
  2. Positively correlated
  3. Uncorrelated

With negative expected return $\mu < 0$ and volatility $\sigma$. From intuition, I think that if we are allowed to use leverage, we should be adding this to portfolio under scenarios 1 and 3 to reduce risk (and apply leverage to achieve desire rate of return). Is this true? How would I size this position if I want to target $10\%$?

Is this scenario similar to the case of shorting one asset and buying another that are positively correlated to each other? In both instances (long/short positively correlated or long/long negatively.. or zero correlated), they should be risk reducing. And if we're allowed to use leverage we should be ale to achieve target return at lower risk? Though this also depends on the bounds of expected return and correlation?

Basically, is it ever smart to add something with negative expected value to a portfolio depending on its correlation to the portfolio?

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If you're not in negative correlation territory you will not increase expected return over volatility with your negative return asset. Even at zero correlation it will add volatility (in dollar terms) but you need to decrease it to compensate for the decrease in return. Then again if you are talking about total expected return rather than the return minus the risk free rate, then a negative return would be fine in so far the risk free rate itself would be even lower (and you could use it to leverage your portfolio).

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