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17 votes
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Why does the Markowitz mean-variance model require the assumption of normality?

it doesn't require normality. What it requires is that the investor's decisions are determined by mean and variance. A normal distribution is determined by mean and variance, so if you assume joint ...
Mark Joshi's user avatar
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11 votes
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What is the total correlation between assets in a portfolio?

This is indeed an interesting question. According to this website, a paper by Goldman Sachs [Tierens and Anadu (2004)] proposes three alternative methods for estimating average stock correlations: ...
vonjd's user avatar
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9 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: $$ Z_1 = w_{11} X_1 + w_{12} X_2 $$ $$ Z_2 = w_{21} X_1 + w_{22} X_2 $$ ...
Attack68's user avatar
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8 votes

What is the total correlation between assets in a portfolio?

I just want to add to vonjd's answer some info on the comparison of the 3 methods. This is too big for a comment so I'm posting as a separate answer but please upvote his answer, not mine. Do the ...
msitt's user avatar
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8 votes
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Creating a Covariance Matrix

here is how to get covariance matrix from correlations:
Valometrics.com's user avatar
8 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 ...
Quantoisseur's user avatar
8 votes
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Inverse Covariance Matrix Transformation from CAPM

This is the result of the Sherman-Morrison inversion for the sum of an invertible matrix and an outer product. You will find this (and many other helpful methods) in the Matrix Cookbook. Specifically, ...
Kermittfrog's user avatar
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8 votes
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Admissible values for non diagonal elements of correlation matrix

In your example, $$ \begin{align} V(\sum X_i) &= \sum V(x_i)+2\sum_{i=1}^n\sum_{j=i+1}^n V(X_i,X_j)\\ &=n+2\sum_i^n\sum_{j=i+1}^n\rho\\ &=n+2\rho \frac{n(n-1)}{2}\\ &=n+n(n-1)\rho \end{...
Kermittfrog's user avatar
  • 6,822
7 votes

Estimate covariance matrix using prices

If you assume that a financial asset price has a change that is a wiener process then you can view the future value of that asset as the initial value plus the sum of the independent daily changes (...
Attack68's user avatar
  • 10.8k
7 votes

Why does portfolio optimization require a positive-definite covariance matrix?

Positive definite matrix $A$ is defined as $x^TAx > 0$ for all vectors $x$. Since a term $w^T\Sigma w$ in Markowitz (and other models as well) expresses variance in returns, it is a measure of ...
Martin Vesely's user avatar
6 votes
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How can I use a more efficient volatility estimator to improve the co-variance matrix?

Let $s$ be a $N\times1$ vector of standard deviations and $C$ be an $N\times N$ correlation matrix. The covariance matrix is equal to $$\Sigma=\text{diag}(s) \ C \ \text{diag}(s)$$ where $\text{diag}...
John's user avatar
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6 votes
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Interpreting Eigenvalues of Co-variance Matrix

What you basically do here is a Principal Component Analysis (PCA). A good starting point in the financial sphere is Managing Diversification by Attilio Meucci (2010) Page 3: "The most natural ...
vonjd's user avatar
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6 votes
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Variance attribution calculation from a covariance matrix

Suppose the covariance matrix is $V$ (which is n by n) and the weights are $w$ (of length n). Then the Portfolio Variance is $V_p = w^T V w$ and the Risk Contribution (in terms of variance) of asset ...
nbbo2's user avatar
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6 votes
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Simulating covariance matrices with nonzero correlation

What does 'simulate a covariance matrix' mean? If the question means, generate an arbitrary correlation matrix for 1000 stocks, then we can choose any symmetric matrix with all 1s down the diagonal, ...
StackG's user avatar
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5 votes
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How to compute the variance of a Long-Short Equity Portfolio?

We have weights $w_A$, $w_B$ and $w_C = 1 - w_A - w_B$ that sum to $1$. With de-meaned returns $r_A$, $r_B$, and $r_C$, the portfolio variance is $$E\{[w_A r_A + w_B r_B + (1 - w_A - w_B)r_C]^2 \} = ...
RRL's user avatar
  • 3,700
5 votes
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Widely accepted methods for coming up with the co-variance matrix of assets?

Multivariate volatility models for replacing the sample covariance matrix with in the mean-variance portfolio selection model: RiskMetrics 1996 EWMA (Exponentially weighted moving average) covariance ...
develarist's user avatar
  • 3,040
5 votes
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Double objective in portfolio optimization

There is nothing wrong mathematically (nor ethically) with this objective function. However, this objective is weird in a couple of ways. First, there is no weighting on these which implies you prefer ...
kurtosis's user avatar
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5 votes
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For portfolio variance, why doesn't $Var(X w) = w^\top \Sigma w$?

I'm not a Python programmer, however, reading the reference manual of np.var, you're using the "biased" version of the variance estimator. Instead use the unbiased variance estimator: ...
Pleb's user avatar
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4 votes
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Average Correlation

He is forced to use some tricks because Excel can only take average of a rectangular area, but he wants the avg of upper non-diagonal elements of the matrix only. So he subtracts $\frac{1}{n}$ (the ...
Alex C's user avatar
  • 9,382
4 votes

Hierarchical Risk Parity with allocation constraints?

EDITED You are right. We have to look town to the "leaves" in each iteration. I would do it the following way: If $L_i^{(j)}$ is the set of indices in the $j$ branch ($j \in \{1,2\}$), then we ...
vanguard2k's user avatar
  • 2,925
4 votes

Variance Matrix with 'nan' values

This is a common problem in covariance matrix estimation, with several possible solutions. One of the simplest involves two steps: (1) You compute each element of the covariance matrix on a 'best ...
nbbo2's user avatar
  • 11.5k
4 votes
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Why annualizing sampled covariance matrix changes stock weight vector?

Q1. Calculating the GMVP involves three operations: Inverting the covariance matrix $\Sigma$ Multiplying the inverse by a column vector of 1's on the right: $x=\Sigma^{-1} \mathbf{1}$ Normalizing ...
Alex C's user avatar
  • 9,382
4 votes
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Variance-Covariance Matrix under $\mathbb{P}$ and $\mathbb{Q}$

Just to expand on Alex answer. Empirically it is simply not true. Focusing on the diagonal of the variance-covariance matrix, we know that there is a large variance risk premium. Take a look at table ...
phdstudent's user avatar
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4 votes

Meaning of an identity matrix for the covariance in portfolio optimization

You can think of it in Bayesian terms. To start with, knowing nothing at all about stocks, you might assume that stock returns are i.i.d with unit variance. This would be your prior. It is very simple ...
nbbo2's user avatar
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4 votes
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Meaning of an identity matrix for the covariance in portfolio optimization

OK, so think of it this way... Your standard (Markowitz) covariance matrix is a sample observation. That may or not be close to the population sigmas and correlations of your sampled markets. Even if ...
demully's user avatar
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4 votes
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Why does Hierarchical Risk Parity ignore the clusters generated?

This turns out to be a general drawback of the HRP algorithm, as pointed out by Pfitzinger, J., & Katzke, N. (2019) (my highlights): As shown in Figure 2.3, the naive bisection rule can violate ...
Doggie52's user avatar
  • 227
4 votes
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Sample Variance of Portfolio

Yes, indeed. It's a simple Linear Algebra and Expectation result: Given: $Var(w'r) = \mathbb{E}[(w'r)^2] = \mathbb{E}[(w'rr'w)]$ With $w$ and $r$ the vectors of weights and returns. As $w$ is constant,...
André Bittencourt's user avatar
3 votes
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Filtering smallest eigenvalues

After writing an email to Meucci directly, I posted the question in his LinkedIn Group ARPM - Advanced Risk and Portfolio Management. Below are his answer and the answer of other group members, which ...
Hans-Peter Schrei's user avatar
3 votes
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Portfolio Optimisation/Covariance Estimation on a large scale

Broadly speaking, as you probably already know, there are 2 approaches to estimating large covariance matrices: 1) Shrinkage Methods like Ledoit-Wolf that try to reduce the noise in a large matrix (N ...
Alex C's user avatar
  • 9,382
3 votes
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Transform raw forecasts into orthogonal forecasts

I do not have access to this book but I suppose the decomposition is the cholesky decomposition (if you use R, simply generate it with chol(cov(g)) where g is a ...
Stefan Voigt's user avatar
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