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 $\mathcal{N}$.

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

(See my book "introduction to mathematical portfolio" for more details.)

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    $\begingroup$ How did you derive y and x_i? $\endgroup$ – user15526 Mar 6 '15 at 17:28
  • $\begingroup$ well it's a standard result, you can proceed via Lagrange multipliers to maximize return for a given variance. $\endgroup$ – Mark Joshi Mar 6 '15 at 19:55

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