Mean-Variance Portfolios with Independent Asset Returns
Summary
The document considers how a mean-variance investor might allocate capital among assets whose returns are independent and whose expected returns exceed a threshold. With independent returns, the covariance matrix is diagonal, so the tangent portfolio weights are proportional to each asset’s expected return above a risk-free rate, divided by its variance. This explains why an optimal portfolio can include multiple assets rather than concentrating on the one with the highest expected return: allocation depends on both return and risk.
The response gives a formula for the weights and notes that negative weights arise when the risk-free rate exceeds an asset’s expected return. It does not prove that every asset above the question’s threshold should be included, or establish a specific risk measure minimized by that rule. The proposed result relies on mean-variance preferences, independence, and a suitable risk-free rate; it leaves the threshold and economic conditions for positive weights underspecified.
Key ideas
- With independent asset returns, the covariance matrix is diagonal.
- In a tangent portfolio, weights are proportional to expected excess return divided by variance.
- Portfolio allocation depends on risk as well as expected return, so the highest-return asset need not receive all capital.
- The answer does not establish that every asset above an arbitrary expected-return threshold belongs in the portfolio.
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Full text
# Risk minimization by investing in all assets with positive expected return
# 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 $\mathcal{N}$.
## Answer by Mark Joshi (score 2, accepted)
https://quant.stackexchange.com/a/16851
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.)Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.