24 votes
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Why is Markowitz portfolio optimisation so popular considering it is worse than an equal weighted portfolio?

Markowitz's concepts attracted a great deal of interest from theorists (and still do), but never had much application in practice. The results from practical application were always disappointing (...
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  • 9,462
15 votes

Current industry standard for (active/passive) portfolio optimizations

I am a professor of finance who has spent his life working in the capital markets in operations, sales, compliance, and research. I would love to tell you about the existence of industry standards, ...
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  • 4,092
13 votes
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Why shrink the covariance matrix?

Have a look at this classic paper: Honey, I Shrunk the Sample Covariance Matrix by O. Ledoit and M. Wolf The abstract answers your question already: The central message of this article is that no ...
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13 votes
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cvxpy portfolio optimization with risk budgeting

The underlying problem: your ACTR constraints aren't convex The $i$th constraint on your risk contribution can be written: $$ w_i \sum_j \sigma_{ij} w_j \leq c_i s$$ And this isn't a convex ...
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  • 6,334
13 votes
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Random Portfolios vs Efficient Frontier

You seem to have two distinct problems: How to generate random portfolios How optimal portfolios are structured Ad 1) A straightforward way to simulate the weights of random portfolios is to use ...
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  • 1,933
10 votes

Maximum Sharpe portfolio (no short selling restrictions)

Let $R$ be a random vector of risky returns and let $r_f$ denote the risk free rate. Let vector of expected returns $\boldsymbol{\mu} = \operatorname{E}[R]$ and covariance matrix $\Sigma = \...
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  • 6,334
10 votes

Hedging Covid-19 and other low probability high loss risks

There's no easy answer to your question, as noob2 pointed out. You can look online for info from Universa. That fund does exactly what you are asking: https://www.universa.net/riskmitigation.html ...
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9 votes
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Application of Control Theory in Quantitative Finance

Of course, optimal control is at the core of math finance. Take few applications: Option Pricing: you have an exposure to a time dependent combination of market factors; you have some knowledge of ...
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  • 10.6k
9 votes

Why is Markowitz portfolio optimisation so popular considering it is worse than an equal weighted portfolio?

There has been a split in the community ever since Mandelbrot published his paper "On the Variation of Certain Speculative Prices." See: Mandelbrot, B. (1963). The variation of certain speculative ...
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  • 4,092
9 votes

Why is Markowitz portfolio optimisation so popular considering it is worse than an equal weighted portfolio?

It is more complicated than that: It is not the optimization per se that leads to inferior results but the data you use. Kritzman et al. makes a strong case in defense of optimization vs. 1/N in this ...
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  • 26.9k
9 votes
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Maximum Sharpe portfolio (no short selling restrictions)

There are two cases, where short sales are allowed: With riskless lending and borrowing and without. As mentioned in the comments, you just have to solve a linear system. With riskless lending and ...
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  • 2,926
9 votes
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Why techniques for portfolio optimization do not take into account the non-fractionability of stock prices?

There are a few related reasons: The optimization becomes a lot harder when only discrete values are considered. Mean variance has a closed form solution for the continuous case but the case with ...
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  • 7,652
8 votes

Why maximize expected growth rate?

The Kelly criterion is just one approach to portfolio construction (or bet sizing) that considers the risk-return tradeoff. There are many possible strategies (static or dynamic) that incorporate ...
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  • 3,250
8 votes

Why shrink the covariance matrix?

The estimation of a covariance matrix is unstable unless the number of historical observations $T$ is greater than the number of securities $N$ (5000 in your example). Consider that 10 years of data ...
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  • 3,250
8 votes

Maximum Sharpe portfolio (no short selling restrictions)

To complement @skoestimeier's answer on the shortselling-allowed case, I provide a vectorised version. Using the original notation in my post (you may change $r$ to something like $r-r_f$, but this ...
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  • 921
8 votes
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Is there a way using matrix algebra to add portfolios to a covariance matrix of assets?

If your two assets are denoted by random variables $X_1$, $X_2$, with 2x2 covariance matrix $\mathbf{Q}$ and the portfolios: $$ Z_1 = w_{11} X_1 + w_{12} X_2 $$ $$ Z_2 = w_{21} X_1 + w_{22} X_2 $$ ...
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  • 8,037
8 votes
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How can I measure returns such that the average is useful?

Take the log return between days.
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7 votes

Why shrink the covariance matrix?

Transaction costs - even for banks, funds etc, every trade has an associated cost, so if you would be buying a small number of shares, it's probably cheaper to carry the risk and not make those small ...
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  • 3,589
7 votes
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On learning the bayesian approach to portfolio optimization

An introductory presentation by Michael Brandt from a seminar of Inquire Europe is Bayesian Portfolio Construction. His review Portfolio Choice Problems has a section on decision theory which could ...
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  • 866
7 votes
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Is it possible to deal with non-normal distribution in Black-Litterman model?

Well there are two main things to consider here. Many implementation of Black-Litterman use the market portfolio and the ex post volatility and correlation structure to back out implied returns to ...
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  • 2,874
7 votes
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Replicate a Portfolio with Given Payoff

Consider the case where we are interested in decomposing a continuous and piece-wise linear European payoff function $V \left( S_T \right)$ over $n$ intervals with $n + 1$ node points $S_i$ for $i = 0,...
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7 votes

Portfolio Optimization and Global Minimum Variance Portfolio (GMV)

1) To be honest, any horizon is problematic in this respect. Simple sampling statistics 101 will tell you that the standard error around any estimate of true mean returns is the root time * variance. ...
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  • 4,896
7 votes
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Can a capital market line have a negative slope?

Two separate cases were identified by R.C. Merton in 1972: In the economically more relevant case, where $r_f < b/c$, efficient portfolios are combinations of a long position in [the tangency] ...
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  • 9,462
7 votes
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Why does portfolio optimization require a positive-definite covariance matrix?

To supplement the other answer, yes there are optimization reasons for the covariance matrix being symmetric positive definite (SPD). All positive definite matrices are invertible and its inverse is ...
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7 votes
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Which portfolio is more "diversified": the $\frac{1}{N}$, the MDP or the max decorrelation?

First of all, I am not sure what you mean by the ratio in your second point. However, I will try to give you a partial answer at least. There is a very comprehensive overview of these by EDHEC, page 4....
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  • 2,874
7 votes

Closed-form analytical solution for the variance of the minimum-variance portfolio?

A few more steps beyond your last equation gives the answer. With $C = \mathbf{1}^T\mathbf{\Sigma}^{-1}\mathbf{1}$, we have $$\sigma_P^2 = [C^{-1} \mathbf{\Sigma}^{-1}\mathbf{1}]^T \mathbf{\Sigma} [C^{...
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  • 3,250
7 votes

What's the point of resampling?

The "estimation problem" in Portfolio Optimization is a serious one. The parameters (returns and covariances) are known very imprecisely. For example the covariance between stocks and bonds ...
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  • 9,462
6 votes
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Utility Theory and portfolio optimization - Proof of a lemma

This is the well known Euler's equation for optimality. The trick here is to setup the budget constraint correctly. Your initial wealth $W_0$ is irrelevant. The terminal (risky) wealth is, $$W = W_0( ...
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  • 515
6 votes
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Mean Variance Portfolio theory and real-world problem?

Mean-variance (MV) is a framework rather than a prescription. This framework allows one to make, discuss, and defend his investment decision. In practice, there are many ways to make adjustments to ...
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6 votes
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Maximum Certainty Equivalent Portfolio with Transaction Costs

Seems like a small mistake in the last equation. It should read $\Delta^* = A^{-1} \left[\mu-\gamma \Sigma \omega_c - \frac{1}{\iota'A^{-1}\iota} \iota' A^{-1}(\mu-\gamma \Sigma \omega_c )\iota\...
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