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 $\alpha$.

  • 2
    $\begingroup$ I have the feeling that the author meant "differentiable" instead of "continious" $\endgroup$ – JeanGuillaume Sep 25 at 15:34

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