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There is no generic solution. However, the KKT conditions are of the forms \begin{align*} \begin{cases} Qy + \lambda_1 \mu +\lambda_2 Py = 0,\\ \mu^T y = 1,\\ y^TPy \leq k^2 \sigma^2,\\ \lambda_2 \big( y^TPy - k^2 \sigma^2\big) = 0. \end{cases} \end{align*} Here, the condition $$\lambda_2 \big( y^TPy - k^2 \sigma^2\big) = 0$$ means that two cases need to ...

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I just pushed Python implementations of some common portfolio optimizers to my GitHub. It uses the CVXOPT library to solve the resulting quadratic programs. It supports the construction of Markowitz portfolios, minimum variance portfolios and tangency portfolios (both long-only or long/short).

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With a dedicated portfolio all interest rate risk has been eliminated, since you hold ZCB's that deliver exactly the cash you need when you need it. So it is a perfect hedge against i.r. risk. With an immunized portfolio you have reduced i.r. risk (by matching duration and convexity) but you still have a probability distribution of outcomes. Yields at ...

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