# Perfect Negative Correlation - Returns and Risk

In the famous article of Sharpe "Capital Asset Prices: A theory of market equilibrium Under conditions of risk", he studies the behaviour of allocation between an asset $$i$$ and an efficient combination of assets $$g$$ in the plan where the horizontal axis represents the expected return and the vertical axis the standard deviation.

We define the portfolio $$P_\alpha$$ where we invest a proportion of $$\alpha$$ in asset $$i$$ and $$(1 - \alpha)$$ in the combination $$g$$. Playing with alpha, we obtain a curve with each point having the following coordinates $$(E_{P_\alpha}, \sigma_{P_\sigma})$$.

There is a remark that I do not understand in the fourth part: "if the correlation between $$i$$ and $$g$$ is equal to $$-1$$, the curve is not continuous." In this case, our coordinates become : $$(E_{P_\alpha}, \sigma_{P_\sigma})= (\alpha E[R_i] + (1-\alpha)E[R_g], \vert \alpha\sigma_i + (1 - \alpha)\sigma_g \vert )$$

I do not understand where is the discontinuity as both coordinates are continuous as a function of $$\alpha$$.

• I have the feeling that the author meant "differentiable" instead of "continious" – JeanGuillaume Sep 25 '19 at 15:34