Questions tagged [covariance]
A measure of the degree of linear association between a pair of random variables.
155 questions
2
votes
1
answer
175
views
What is the optimal compromise between a sample covariance matrix and a highly structured estimator?
In Honey, I Shrunk the Sample Covariance Matrix by Ledoit & Wolf (2004), the authors mentioned:
Alternatively, one might consider an estimator with a lot of structure, like the single-factor ...
- 2,862
0
votes
0
answers
206
views
Weighted Average of Var/Covar Matrix
I have three different types of variance-covariance matrices derived from the same series of returns:
A simple variance-covariance matrix using ret.corr().
A variance-covariance matrix adjusted with ...
0
votes
0
answers
121
views
GARCH for Mean Variance Optimization
I am currently trying to carry out a mean variance optimisation, with the implementation of GARCH. I'm not sure if this is going to make complete sense as my understanding of GARCH is limited.
In the ...
1
vote
0
answers
49
views
Distribution of sample covariance times inverse covariance times sample covariance
I want to understand the distribution of the random variable:
$$S_n = \frac{1}{n^2} 1'\hat \Sigma \Sigma ^{-1} \hat \Sigma 1$$.
1 is a vector of ones of size n, and the variance is of size nxn. $\hat \...
1
vote
0
answers
107
views
Discuss how you would allocate your budget between the two assets if their correlation is 1, 0, or -1
An asset A is expected to yield a $2\%$ return with a standard deviation of $1\%$, and another asset B is expected to yield a $1\%$ return with a standard deviation of $1\%$.
Discuss how you would ...
- 111
0
votes
1
answer
105
views
Is there any relationship between the Covariance(A, B) and the variance of the synthetic asset A/B?
Let's say we have 2 pairs of currencies: EUR/USD and GBP/USD. The cross-asset (or synthetic asset) would be (EUR/USD) / (GBP/USD) = EUR/GBP.
Is there any relationship between the covariance(EUR/USD, ...
- 3
0
votes
2
answers
674
views
Why do we need the covariance when calculating portfolio VaR?
I was recently learning about value at risk and how to calculate it, and one of the steps was to calculate the covariance of the returns of the securities making up the portofolio.
This makes sense ...
0
votes
0
answers
216
views
"fix" a sample covariance matrix which is not positive semidefinite by using daily returns instead of monthly
In the portfolio optimization problem at hand, one of the constraints is that the tracking error should not be greater than $\gamma$.
The constraint is therefore:
$(\textbf{x}-\textbf{w})^\mathrm{T}\...
- 1
3
votes
1
answer
434
views
Covariance Between Two Frontier Portfolios
Based on the definitions of A, B, C, and D in "An Analytic Derivation Of The Efficient Portfolio Frontier" by Robert Merton (1972), how can I prove the following in a line-by-line derivation?...
- 31
3
votes
1
answer
166
views
Sample Variance of Portfolio
Let $w$ denote a vector of portfolio weights, $r_i$ denote the $i$th return vector, $\Sigma$ denote the Covariance matrix of $r_i$ and let $\hat{\Sigma}$ denote the sample covariance matrix of $r_i$.
...
- 175
6
votes
2
answers
241
views
Covariance of the product of log normal process and normal procces
I tried to compute the following covariance :
$$Cov(e^{\int_{t}^{T}W^1_sds},\int_{t}^{t+1}W^2_sds)$$
where $W^1_t$ and $W^2_t$ are Brownian motions such that $dW_t^1dW_t^2=\rho dt $
My idea was to ...
- 338
1
vote
0
answers
206
views
Association between a random variable and Radon-Nikodym derivative
Suppose that $X$ is a random variable and $\frac{d\mathbb{Q}}{d\mathbb{P}}$ is the Radon-Nikodym derivative. The quantity under consideration is as follows:
\begin{equation}
Cov(X, \frac{d\mathbb{Q}}{...
- 419
0
votes
1
answer
1k
views
Covariance Shrinkage - Am I getting the right variances?
I am looking into a quite simple task: shrinking the sample covariance matrix of a minor sample of monthly returns data on 5 different assets.
I am using Python to process my data and have been using ...
- 3
3
votes
2
answers
1k
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
$$
\mathbb{V}\text{ar}[I] =...
- 33
2
votes
1
answer
199
views
Help understanding the step $\sum_{j=0}^n\sum_{k=0}^ng_jg_k\text{Cov}(\epsilon_{n-1},\epsilon_{n+h-k})=\sum_{j=0}^ng_j^2+h\sigma^2$
Given is that $\epsilon_n$ is a white noise process with $\text{Var}(\epsilon_n)=\sigma^2$ and that $g_j\in\mathbb{R}$. There is a step in my lecture notes that I don't get. It says the following
$$\...
- 221
1
vote
1
answer
164
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 ...
- 700
2
votes
0
answers
117
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 ...
- 841
2
votes
1
answer
537
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 ...
- 185
1
vote
1
answer
168
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 ...
- 13
0
votes
1
answer
178
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{\...
- 3,140
1
vote
1
answer
362
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 ...
- 25
0
votes
2
answers
2k
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 ...
- 3,140
1
vote
2
answers
542
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 ...
- 3,140
0
votes
0
answers
140
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 ...
2
votes
1
answer
253
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?
- 23