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1

The variance of the portfolio is $$ V_p=\sum_i \sum_j w_i w_j Cov(r_i,r_j)$$ because of the properties of $Cov(\cdot,\cdot)$ we can rewrite this as $$ V_p=\sum_i w_i Cov(r_i,\underbrace{\sum_j w_j r_j}_{R_P})$$ where $R_p$ is the return on the portfolio. So we have $$ V_p=\sum_i w_i Cov(r_i,R_p)$$ QED (And it is true of every portfolio, not just the ...


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Let us start with some underlying math. First, $\sigma=\sqrt{\sigma^2}$, but the minimum variance unbiased estimator (MVUE) for standard deviation is not the square root of the MVUE of the variance, $\hat{\sigma}\ne\sqrt{\hat{\sigma^2}}.$ Taking the square root of the unbiased sample estimator of the variance introduces bias because it is a non-linear ...


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The Markowitz mean-variance model takes in some target expected portfolio return $\mu_T$ as an input and returns optimal portfolio weights $\boldsymbol\omega$ that minimize risk for that return. Repeating this for a series of target returns, $\boldsymbol\mu_T$, manifests two different efficient frontier curves (series of efficient portfolios) depending on ...


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The maximum attainable return is unbounded as in the model you can borrow without limit. However, what matters in that model is what is the maximum sharpe ratio you can attain. That has bounds, which are given by the Hansen-Jagannathan distance. Let me show you what the JH distance looks like. From the law of one price: \begin{equation} 1 = E [R_{i,t+1} ...


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Do you have correctly formulated the problem for the solver ? If you want to maximise a function (the sharpe ratio) $f$, it is equivalent to minimise $-f$. This kind of confusion (minimising instead of maximising) would basically lead to a similar outcome as yours.


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