17
votes
Accepted
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 ...
- 6,853
11
votes
Accepted
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:
...
- 27.3k
9
votes
Accepted
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:
$$ Z_1 = w_{11} X_1 + w_{12} X_2 $$
$$ Z_2 = w_{21} X_1 + w_{22} X_2 $$
...
- 9,215
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 ...
- 741
8
votes
Accepted
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, ...
- 6,425
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 (...
- 9,215
7
votes
Accepted
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 ...
- 750
6
votes
Accepted
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}...
- 5,391
6
votes
Accepted
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 ...
- 27.3k
6
votes
Accepted
6
votes
Accepted
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 ...
- 10.9k
6
votes
Accepted
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, ...
- 2,996
6
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 ...
- 781
5
votes
Accepted
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 \} = ...
- 3,595
5
votes
Accepted
Portfolio with lots of subportfolios
One way to this is the following (you can code all these constraints if you use the right software, I am doing such things using mathematica)
You define $w_{i,j}$ which is the weight of asset $j$ in ...
- 13.6k
5
votes
Accepted
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 ...
- 2,980
5
votes
Accepted
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 ...
- 2,880
5
votes
Accepted
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:
...
- 4,186
4
votes
Accepted
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 ...
- 9,332
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 ...
- 10.9k
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 ...
- 2,915
4
votes
Accepted
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 ...
- 9,332
4
votes
Accepted
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 ...
- 8,032
4
votes
Accepted
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 ...
- 227
4
votes
Accepted
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,...
3
votes
Accepted
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 ...
- 9,332
3
votes
Accepted
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 ...
- 1,456
3
votes
Generating a random covariance matrix with variances in range
A useful decomposition is, in R's matrix notation, V = S %*% C %*% S, in which S is
a matrix with the standard deviations on the ...
- 3,186
3
votes
Ledoit-Wolf Shrinkage estimator not giving positive definite covariance matrix
In theory, the Ledoit and Wolf shrinkage estimator is supposed to guarantee a positive-definite matrix, given that it adds a positive-definite matrix (the target) to a semi-positive one (the sample ...
- 141
3
votes
Semi-variance/Downside Risk, what about the rest of the covariance matrix?
one solution that works is set up the usual correlation matrix and pre- and post multiply by a diagonal matrix with semi standard deviations down the diagonal taking care that they are not zero
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