# Risk minimization by investing in all assets with positive expected return

Suppose I have an amount $T$ to invest and $N$ available assets.

The stochastic return per invested unit of asset $i$ is $R_i$.

The variance and the expectation of $R_i$ are $\sigma^2_i$ and $\mu_i$ for $i=1,...,N$ (different across $i$).

The returns are independent across $i$.

Consider the assets with $\mu_i>s$. Let $\mathcal{N}:=\{i \text{ s.t. } \mu_i>s\}$ with cardinality $n\leq N$.

Could you give me an analytical justification (with proof) for deciding to invest in ALL assets in $\mathcal{N}$ and the associated economic intuition? In addition, I need an analytical argument that explains why I do not invest just in the asset that gives the highest expected return.

I think that what would work here is a measure of portfolio's risk that is minimised when I invest in all assets in $\mathcal{N}$. Or, in alternative, an utility function which is maximized when I invest in all assets in $\mathcal{N}$.

• What's your utility function assumption? – user2763361 Mar 4 '15 at 12:27
• The utility function is increasing in expected return and decreasign in variance. – user15526 Mar 4 '15 at 13:14

since you've assumed that all returns are independent, the covariance matrix, $C,$ is diagonal. In the comments, you are assuming that the investor is a mean-variance investor. It's a general result that every portfolio that maximizes return for a given variance is a tangent portfolio for some risk-free rate, $R.$
Let $e=(1,1,...,1).$ and let $\mu$ be the vector of expected returns.
So we have the weights $x$ satisfy $$x_i = y_i / \sum y_j$$ and $$y = C^{-1}(\mu - Re)$$ Now $C^{-1}$ is diagonal with positive entries. So you will only get negative weights if $R$ is greater than $\mu_i$ for some $i.$ However, that won't actually happen for economic reasons.