If two assets are perfectly negatively correlated, the portfolio variance can be which of the following as you mix them in equal weights?

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Multiple Choice

If two assets are perfectly negatively correlated, the portfolio variance can be which of the following as you mix them in equal weights?

Diversification works best when assets move in opposite directions. When two assets are perfectly negatively correlated, their returns offset each other precisely when you combine them. With equal weights, the portfolio variance is 0.25 Var1 + 0.25 Var2 + 0.5 Cov12. If the correlation is -1, Cov12 = -√(Var1 Var2). Plugging that in gives portfolio variance = 0.25(Var1 + Var2 - 2√(Var1 Var2)) = 0.25(√Var1 − √Var2)². This value is always nonnegative and becomes zero only when Var1 equals Var2. So the variance can be reduced toward zero due to the perfect negative relationship, and it hits zero exactly if the two assets have the same volatility. If the volatilities differ, the variance is still smaller than the individual variances but not necessarily zero. That's why the best choice is that it can be reduced toward zero. The other options don’t fit: zero regardless would require equal risk, becoming the average variance isn’t how equal-weight diversification behaves, and diversification under perfect negative correlation reduces risk rather than increases it.

Diversification works best when assets move in opposite directions. When two assets are perfectly negatively correlated, their returns offset each other precisely when you combine them.

With equal weights, the portfolio variance is 0.25 Var1 + 0.25 Var2 + 0.5 Cov12. If the correlation is -1, Cov12 = -√(Var1 Var2). Plugging that in gives portfolio variance = 0.25(Var1 + Var2 - 2√(Var1 Var2)) = 0.25(√Var1 − √Var2)². This value is always nonnegative and becomes zero only when Var1 equals Var2.

So the variance can be reduced toward zero due to the perfect negative relationship, and it hits zero exactly if the two assets have the same volatility. If the volatilities differ, the variance is still smaller than the individual variances but not necessarily zero. That's why the best choice is that it can be reduced toward zero. The other options don’t fit: zero regardless would require equal risk, becoming the average variance isn’t how equal-weight diversification behaves, and diversification under perfect negative correlation reduces risk rather than increases it.

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