Questions tagged [covariance]

A measure of the degree of linear association between a pair of random variables.

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51 views

Correlation between mean-variance efficient portfolios

If the covariance solution between the returns series of the minimum-variance portfolio ($A$) and any other portfolio along the efficient frontier ($B$) is $$Cov_{A, B} = \frac{1}{\mathbf{1}^T\mathbf{\...
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21 views

Is it possible to apply PCA to a time-series of covariances?

I understand that Principal Component Analysis (PCA) can be applied for cross-sectional as well as for time-series data. Nevertheless, I am trying to figure out if there is anything wrong with ...
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32 views

Update sample covariance matrix

I would like to update a covariance matrix $\mathbf{R}_T$ with a new incoming sample at time $T+1$, i.e. I would like a rank-1 update of the form $\frac{1}{T+1} [T \mathbf{R}_T + \mathbf{x}_{T+1}\...
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114 views

Covariance of mean-reverting Vasicek process?

I am dealing with a mean-reverting Vasicek process defined as: \begin{equation} S_t = S_0 e^{-at} + b(1-e^{(-at)}) + \sigma e^{(-at)} \int_{0}^{t} e^{(-as)} \ W_t \end{equation} I want to ...
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43 views

Ranking assets by covariance vs correlation

After every change in portfolio (i.e. every trade), I need to calculate a price for each asset in my portfolio. But the calculator is slow. So I want to order the sequence of price updates, from the ...
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142 views

Interpretation and units of a covariance element in portfolio risk

Given portfolio risk is $\mathbf{w}\boldsymbol{\Sigma}\mathbf{w}$ where $\boldsymbol{\Sigma}$ is the covariance matrix whose diagonal elements $\sigma^2_{n}$ are individual asset return variances and ...
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2answers
180 views

Why is portfolio optimization a convex problem if variance is concave?

Variance is concave, so portfolio risk must be too. The mean-variance model employs quadratic programming to optimize (minimize) portfolio risk. My understanding is that quadratic programming requires ...
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27 views

Covariance of Individual Return and Portfolio Return

Hi guys, Is it possible to get the covariance between the individual return and portfolio return given the correlation matrix, volatility matrix, weights matrix and return matrix? I know how to get ...
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115 views

Meaning of an identity matrix for the covariance in portfolio optimization

Instead of using a sample covariance matrix for portfolio optimization, Ledoit and Wolf use an estimator that is the weighted average of the sample covariance matrix and the identity matrix, $I$. This ...
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68 views

What do large weights above 1 in a portfolio represent?

If I have a portfolio consisting of weights -12,11,3,-2,5,-5, I know that negative weights correspond to shorting but what do these large weights represent? I thought the weights are the proportion of ...
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54 views

Can I build an efficient frontier using matrix algebra?

If i have a vector of expected returns $A$, a covariance matrix $C$ and a vector of the corresponding weights $W$ for each investment, is it possible to generate the efficient frontier with vector ...
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3answers
189 views

Simulating covariance matrices with nonzero correlation

How would you simulate a covariance matrix of 1,000 stocks where each pair has nonzero correlation? I have literally no idea how to start with this. Any suggestions?
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90 views

Variance-Covariance Matrix under $\mathbb{P}$ and $\mathbb{Q}$

I'd like to understand why $\Sigma$ is the same under both measures $\mathbb{P}$ and $\mathbb{Q}$. Is it an assumption or a general fact based on theoretical concepts?
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1answer
63 views

Covariation of Ito semimartingales

If we have two Ito semimartingales over $[0,T]$: $$d X_t^i=a^i_tdt+\sigma_t^idW_t^i,\quad i=1,2$$ What is the relationship between $$\langle X^1,X^2 \rangle_t \quad \text{and} \quad \langle W^1,W^2 \...
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45 views

Decomposition of Contribution to Variance

$C$ is a $N\times N$ covariance matrix of stock returns. Assuming $w$ is a vector of positions in each asset, the total variance of the portfolio is $$w^TCw$$ The contribution to total variance of the ...
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1answer
90 views

How do i find the covariance between two portfolios?

I know that the formula for covariance is But this is for two securities. How do I find the covariance between two portfolios? more specifically between the global minimum variance (GMV) and the mean-...
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101 views

Turning a covariance sum into an integral

I am reading Lorenzo's Bergomi's book Stochastic Volatility Modeling, and I have come to this passage. I just would like to understand the derivation between the first and the second equality. I ...
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1answer
48 views

$n$-day ahead forecast for asymmetric DCC-GARCH model

I am working on forecasting covariances with the use of MGARCH models. I was wondering if anyone knows how to implement a n-day ahead forecast of the aDCC (asymmetric DCC) model in R. The ...
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64 views

covariance matrix in the CAPM model

I'm running a simulation for a 5 asset portfolio, calculating the optimal weights of each asset both with the statistical model (i.e. single index) and with the CAPM. my question is: how do you ...
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1answer
156 views

Computing covariance matrix with historical data

I have been reading Active Portfolio Management by Grinold and Khan. In the chapter about risk, they mention, "The third elementary model relies on historical variances and covariances. This ...
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1answer
38 views

Decomposing Co-variance of Two Assets Terminal Prices into Forward measures

Let $X_T,Y_T$ be the terminal values of two price processes following Continuous Gaussian Motion (I.E.) let us assume no jumps. Further assume the correct forwards/futures price is given by $F^X_{t,T} ...
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58 views

Trading Bands for Tactical Asset Allocation

I am currently working through Portfolio Construction and Risk Budgeting, 4th edition, by Dr. Bernd Scherer. In chapter 12.6 the following methodology to construct trading bands around a tactical ...
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1answer
69 views

Calculating covariance from three variances

I have been asked to look to refactor some code. There is a line shown below: $\text{implied covariance} = -\frac{(\text{var}_1 - \text{var}_2 - \text{var}_3)} {2}$, where $\text{var}_1$ is the ...
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Is there a differentiable formulation of Prado's HRP algo?

I get the sense the algo is just a graph representation the dependency structure. Am wondering if there is anything written on learning the weights by optimizing some parameters rather than a forward ...
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2answers
156 views

Estimate covariance matrix using prices

We generally estimate the covariance matrix of assets using their returns instead of prices. Why is that the case? I can think of two possible reasons and would appreciate comments/feedback regarding ...
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37 views

Mean-Variance Optimization - Appropriate to use covariance of daily returns if strategy involves fixed hold of 3 months?

I'm implementing the strategy outlined Quantitative Momentum (Gray & Vogel), which basically involves investing each quarter in a set of "high-quality momentum stocks" for a fixed 3-month hold ...
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273 views

Is a more robust Covariance estimation possible?

I'm working on a mean-variance optimization problem, but instead of financial securities I'm choosing a 'portfolio' of N athletes. It is a 1-period optimization problem over one generic statistic ...
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2answers
227 views

Volatility and weights of a portfolio whose value is negative

How do you calculate the one day standard deviation (in dollars) for a portfolio that is short $30,000? How do you calculate the weightings to use? I already have the necessary covariance matrix.
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57 views

One day standard deviation of a portfolio (long/short, different scalars)

I am attempting to calculate the expected one-day standard deviation of a portfolio in dollars. In other words, I am looking for the following: "I expect my portfolio to move _______ dollars on ...
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201 views

Questions about beta, correlation, and covariance

Currently, I calculate beta, correlation, and covariance measures using daily log normal returns of Security A and Benchmark A. What would it mean if I were to use daily log normal excess returns in ...
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39 views

Covariance time frequency

I have rolling 3-year returns for an asset and a benchmark. I want to compare the covariance of the asset and benchmark, should I use the covariance of the rolling 3-year returns or the covariance ...
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1answer
73 views

Covariance matrix from GJR-GARCH?

I am implementing a AR(1)-GJR-GARCH(1,1) model to some asset returns, and I would need to have a covariance matrix but I struggle to see how I can compute one from the model I used? I know I can have ...
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1answer
217 views

Why the weight vector of 'global minimum variance' the 'eigenvector' with the minimum eigenvalue?

Question Why is it the case that the weight vector of the global minimum variance portfolio the eigenvector of the covariance matrix with the smallest eigenvalue? Question with more details I ...
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592 views

Widely accepted methods for coming up with the co-variance matrix of assets?

Question What are the widely accepted ways for coming up with co-variance matrix of assets after the Markowitz's modern portfolio theory? Question explained in more detail After Modern portfolio ...
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1answer
61 views

Generate scenarios of multiple related parameters

Assume I have three industry datasets: interest rates, inflation and unemployment. Data contains information of last ten years and it's monthly. Now, I would like to create N possible scenarios of ...
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Why are thousand-ish-factor vendor risk models not extremely overfit and inaccurate?

Many vendor risk models have many hundreds, or even thousands of factors (many of which are highly correlated with each other). Underlying all these risk models is some sort of covariance matrix in ...
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2answers
159 views

Prove ρ(X,Z) = ρ

The covariance of two random variables $X$ and $Y$ is defined by: $$\mathrm{Cov}(X,Y)= \operatorname{E}(X-\operatorname{E}(X))(Y-\operatorname{E}(Y))=\operatorname{E}(XY)-\operatorname{E}(X)\...
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1answer
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Co-variance of Portfolio A with Portfolio B

I'm trying to calculate the correlation between two separate portfolios. I've used A*COV(AB)*B to calculate the co-variance of each portfolio where: A = Array ...
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Is Ledoit-Wolf Shrinkage with a Constant Correlation Prior Reasonable for a Stock/Bond Mix?

I've been looking into Ledoit-Wolf shrinkage but I've found the papers concentrate on large numbers of assets that tend to all be highly correlated. Often a universe of large cap stocks. I'm ...
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1answer
99 views

Covariance and Beta: can anyone explain this calculation?

Let us consider a simple equity portfolio that has exposures to only two factors: 0.5 exposure to value and 0.8 exposure to momentum. Let us assume that the volatilities of the two factors are 3% for ...
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1answer
453 views

How to calculate the covariance between two stochastic integrals?

How to calculate the covariance between the integral of a Brownian motion at different times: $$\text{Cov}\left(\int^{t_1}_0\sigma(t)dW_t,\int^{t_2}_0\sigma(t)dW_t\right)\ ?$$ I know the answer is: $$\...
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1answer
294 views

target market correlation for long / short equity portfolio

Given a long / short equity portfolio, I want to have some net long market exposure. My portfolio volatility is fixed to a target, so I don't think it makes sense to target market beta. I think I ...
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1answer
121 views

What are the units of the variance of returns?

I am a little confused about the units of the variance of returns. One way to compute that would be to look at the units of returns- $$r=\frac{1}{\Delta t}\ln\frac{P(t+\Delta t)}{P(t)}=\text{...
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447 views

Filtering smallest eigenvalues

In Risk Budgeting and Diversification Based on Optimized Uncorrelated Factors [1], which introduces minimum torsion bets, Meucci gives an example involving the computation of covariance matrices on ...
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3answers
2k views

Negative variance?

Using the formula w*Cov*t(w) I can generate a negative portfolio variance. What are the implications of a negative variance? Should I just assume it's zero? A negative variance is troublesome ...
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216 views

Variance covariance matrix - number of periods required

Hi I am reviewing the example of Barra risk model in the following document page 23 there is the statement: "Estimating a covariance matrix for, say, 3,000 stocks requires data for at least 3,...
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CAPM model as a regression

The CAPM model states that the returns of a stock are- $r_s=r_f+\beta (r_m-r_f)+\varepsilon_s$ The $\beta$ defined above is then calculated as $\frac{cov(r_s,r_m)}{var(r_m)}$. My question is ...
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147 views

Finding a minimum variance portfolio when using a regulariser?

I am aware that the minimum variance portfolio of a market with $n$ securities can be shown to be: \begin{equation} w^* = (1^T_n\Sigma^{-1}1_n)^{-1}\Sigma^{-1}1_n, \\ s.t. \ \ 1^T_nw = 1 \end{...
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1answer
31 views

Factor Models: uncorrelated errors don't impact covariances of assets

This question stems from time series factor models (e.g., CAPM, Fama-French, etc.), but is a broader idea. I am trying to comprehend how adding noise to a time series (e.g., error/residual from a ...
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219 views

Parametric VaR of a portfolio of a stock and an option on that stock

I understand how to calculate the parametric VaR of a stock and an option separately. But I don't understand how one can calculate the VaR of a portfolio of a stock and an option on that stock using ...