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I have the following risk function:

$\mathbf{Risk}(x):=\mathbb{E}[R(x)]+\delta\mathbb{E}[|R(x)-\mathbb{E}[R(x)]|]$

where $R(x)$ is the portfolio return and $\delta$ is any positive scalar. My textbook assumes this function is convex without proving it. Any ideas how it could be shown it is convex?

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Since $R(x)$ is linear in $x$, we just need to prove $\mathbb{E}(x) + \delta \mathbb{E}(\mathbb{|x-\mathbb{E}(x)|})$ is convex. Now $\mathbb{E}(x)$ is linear and $\delta$ is a positive constant, so we only need to prove $f(x):=\mathbb{E}(|x-\mathbb{E}(x)|)$ is convex in $x$. Then it is straightforward to prove by definition: for $0\leq t <1$,

$f(tx+(1-t)y) = \mathbb{E}(|tx + (1-t)y -\mathbb{E}(tx + (1-t)y)|) = \mathbb{E}(|t(x-\mathbb{E}(x)) + (1-t)(y-\mathbb{E}(y))|) \leq \mathbb{E}(|t(x-\mathbb{E}(x))| + |((1-t)(y-\mathbb{E}(y))|) = tf(x) + (1-t)f(y)$,

as desired. The inequality used is simply the triangle inequality.

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