Questions tagged [covariance]
A measure of the degree of linear association between a pair of random variables.
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questions
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34 views
Covariance between integral of brownian motion and brownian motion
Let
$$
I = \int_0^1W_tdt,
$$
where $W_t$ is a Brownian motion.
From Integral of Brownian motion w.r.t. time we have that
$$
\mathbb{E}[I]=0,
$$
by Fubini's theorem. And that
$$
\mathbb{V}\text{ar}[I] =...
0
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0answers
16 views
rotational invariant estimator
An estimator $\Xi(E)$ an estimator of $C$ given $E$ is said to be rotationally invariant if and only if $\Xi (OEO' ) = O\Xi (E)O'$ where $O$ is any rotation matrix.
Question: why $\Xi(E)$ can be ...
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1answer
44 views
Show that $\text{Cov}[X_r,X_s]=\text{Cov}[X_{r+h},X_{s+h}]$ for $X_t=a+bZ_t+cZ_{t-2}.$
Problem: Let $\{Zt\}$ be a sequence of independent normal random variables, each with mean $0$ and variance $\sigma^2$, and let $a$,
$b$, and $c$ be constants. Is $X_t=a+bZ_t+cZ_{t-2}$ a (weakly)
...
2
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1answer
53 views
Help understanding the step $\sum_{j=0}^n\sum_{k=0}^ng_jg_k\text{Cov}(\epsilon_{n-1},\epsilon_{n+h-k})=\sum_{j=0}^ng_j^2+h\sigma^2$
Given is that $\epsilon_n$ is a white noise process with $\text{Var}(\epsilon_n)=\sigma^2$ and that $g_j\in\mathbb{R}$. There is a step in my lecture notes that I don't get. It says the following
$$\...
1
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1answer
81 views
Show that $\text{Cov}[Z_t,Z_{t+h}]=\text{Cov}[Z_s,Z_{s+h}].$
Problem: If $X\sim\text{WN}(\mu,\sigma^2).$ Let then $Z$ be the process defined by \begin{equation}
Z_t=\sum_{i=0}^na_iX_{t-i} \end{equation} for some coefficients $a_1,...,a_n\in\mathbb{R}$ with ...
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21 views
open-close intraday demeaned log return calculation
open-close return is basically what I feed into the realized kernel volatility and recently I noticed the realized kernel covariance/variance is generating negative value so I had to retrace my ...
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1answer
29 views
Regression of stochastic integral on Wiener process
This question is a follow-up from the following: conditional expectation of stochastic integral
so I won't repeat myself regarding assumptions and notation.
Using Brownian bridge approach, we know ...
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0answers
36 views
Disjoint covariance matrix estimation
I have always estimated correlations and variances disjointly and later combine them to construct covariance matrices. Specifically, variances are estimated in a univariate setting (only using the ...
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0answers
39 views
How to reduce a covariance matrix after clustering?
I have an N = 100 covariance matrix. I am clustering the covariance matrix say into 5 clusters.
How can I compute the reduced ...
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0answers
63 views
What is the difference between np.cov(array) and array.cov()?
I'm trying to find a covariance matrix, so when i use returns.cov() on my returns variable, I get a good result. Unfortunately, when i want to use ...
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vote
1answer
268 views
Is there a way using matrix algebra to add portfolios to a covariance matrix of assets?
What I want to do is the following:
Let's say I have two assets 1 and 2, and have a 2x2 covariance matrix.
Then I have two portfolios A and B made of weights from assets 1 and 2.
What I would like to ...
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0answers
15 views
Interpreting factor coefficients when correlation flips
I am looking at mainly value and growth factor coefficients of a fund during the recent Covid market “crisis”.
I have found that said fund had a negative coefficient to value at the start of 2020 (let’...
1
vote
1answer
61 views
Help Setting a Monte Carlo Simulation
I am trying to replicate the steps of the Barras, Scaillet, Wermer(2010) paper for a Monte-Carlo Simulation. More specifically the steps in Appendix B.1 (Attached image).
I have so far done the ...
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0answers
50 views
Do you need multi-period ahead covariance forecast, in order to construct portfolios with weekly/monthly rebalancing?
Suppose I want to rebalance my portfolio each week. Do I then need weekly covariance forecasts, from some multivariate volatility model to do this? Ie. Insert the weekly covariance forecast $\Sigma_{t+...
1
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1answer
72 views
Covariance AR(2) Process [closed]
I am not sure what the formula is for the covariance of an AR(2) process, described by
$X_t - \mu = \phi_1(X_{t-1} - \mu) + \phi_2(X_{t-2} -\mu ) + \epsilon_t$
where $\mu$ denoted the process mean ...
0
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1answer
62 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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0answers
25 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 ...
0
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1answer
44 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}\...
1
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1answer
129 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 ...
0
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0answers
49 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 ...
2
votes
3answers
155 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 ...
0
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2answers
404 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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0answers
31 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 ...
0
votes
2answers
165 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 ...
0
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2answers
87 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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0answers
58 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
249 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?
2
votes
1answer
100 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?
1
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1answer
73 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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208 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 ...
1
vote
1answer
192 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-...
3
votes
1answer
103 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 ...
1
vote
1answer
68 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 ...
1
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0answers
152 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 ...
0
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1answer
305 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 ...
0
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1answer
55 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} ...
2
votes
1answer
73 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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vote
0answers
33 views
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 ...
2
votes
2answers
174 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 ...
4
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6answers
330 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 ...
2
votes
2answers
433 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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0answers
71 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 ...
5
votes
1answer
209 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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0answers
42 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 ...
0
votes
1answer
83 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 ...
4
votes
1answer
303 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 ...
3
votes
2answers
903 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 ...
1
vote
1answer
64 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 ...
4
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0answers
100 views
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
214 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)\...
