Correlating Subportfolios with Their Combined Portfolio

# Calculate correlation between two sub portfolios and the combined portfolio


# Calculate correlation between two sub portfolios and the combined portfolio












I have two sub portfolios (lets call them portfolio a & portfolio b - a portfolio is just a vector of weights that sum to 1) that combine to create a total portfolio. I also have a 2 x 2 covariance matrix.

```
 | Variance a     Covariance ab |   
 | Covariance ab  Variance b    |
```

By taking the square root of the diagonals of this covariance matrix and I able to create a 2 x 2 correlation matrix so I can get the correlation between portfolio a and b.

What I would like to know is the correlation of portfolio a with the total portfolio (that is portfolio a and portfolio b combined) & the correlation of portfolio b with the total portfolio?

Edit

I should mention that portfolios a & b have the same investable universe. So let says our investable universe is 2000 stocks. portfolio a is a 2000 x 1 vector as is portfolio b. Row i in both vectors correspond to the same company which is linked to another matrix.

```
  Company             Portfolio a        portfolio b
  ABC                 0                  0.25
  DEF                 0.1                0.05
  GHJ                 0.05               0.05
  IJK                 0.12               0
  ...                 ...                ...
  2000th Company      0.03               0.02

  Total               1                  1
```

The 2 x 2 covariance matrix was calculated by the following,

```
   P' * COV * P

   P is a 2000 x 2 matrix column 1 is portfolio a, 2nd column is portfolio b
   COV is a 2000 x 2000 matrix of all the stock variance & covariances.
```

Edit In reference to user12348 post. Another Post provides alternate solution, while Marco's solution also works.

## Answer by Matt Wolf (score 3, accepted)

https://quant.stackexchange.com/a/11203

You can obtain the covariance between 2 portfolios by multiplying the row vector, containing the weights of portfolio A with the variance-covariance matrix of the assets and then multiplying with the column vector, containing the weights of assets in portfolio B.

Equally you can set up a new portfolio A+B by creating a new column vector that contains the combined weights of column vectors A and B and perform same as above to obtain your covariance between A and A+B. Same follows for B and A+B.

Hope this helps.

## Answer by Marco Breitig (score 8)

https://quant.stackexchange.com/a/11207

To clarify notation, you have an universe of $n=2000 \space$ stocks and two portfolio vectors $\mathbf{a},\mathbf{b}\in\mathbb{R}^{n}$ with $\left\|\mathbf{a}\right\|_{1}=\left\|\mathbf{b}\right\|_{1}=1$. Further, you have Estimators for the true Variance $\operatorname{Var}\left[\mathbf{a}\right]$ resp. $\operatorname{Var}\left[\mathbf{b}\right]$ and the Covariance $\operatorname{Cov}\left[\mathbf{a},\mathbf{b}\right]$.

Normally you'd multiply the portfolio weights with the random return vector $\boldsymbol{\mu}\in\mathbb{R}^{n}$ to get the portfolio returns $p_{\mathbf{a}} := \mathbf{a}^{T}\boldsymbol{\mu}$. This portfolio return now is a random variable and it makes sense to speak of its Variance etc. In slight abuse of Notation you can call the portfolio value $a$, which is now a real number. The same holds for the portfolio value $b$.

For a new portfolio $\alpha{a}+\beta{b}=:{c}\in\mathbb{R}\space$ with weights $\alpha+\beta=1$ you want to know $\operatorname{Var}\left[{c}\right]$.

For two correlated random vectors the following holds (http://en.wikipedia.org/wiki/Covariance#Properties): $$\operatorname{Var}\left[{c}\right] = \operatorname{Var}\left[\alpha{a}+\beta{b}\right] = \alpha^{2}\operatorname{Var}\left[{a}\right] + \beta^{2}\operatorname{Var}\left[{b}\right] + 2\alpha\beta\operatorname{Cov}\left[{a},{b}\right]$$

You can also see this if you write down the quadratic form of your Variance-Covariance-Matrix of portfolios ${a}, {b}$ with the corresponding weighting vector that forms portfolio ${c}$. For that, note that $$ {c} = \begin{pmatrix} \alpha\\\beta \end{pmatrix}^{T} \begin{pmatrix} {a} \\ {b} \end{pmatrix} = \alpha{a}+\beta{b} $$ holds. Therefore you can write $$\operatorname{Var}\left[{c}\right] = \operatorname{Var}\left[ \begin{pmatrix} \alpha\\\beta \end{pmatrix}^{T} \begin{pmatrix} {a} \\ {b} \end{pmatrix}\right] \overset{(1)}{=} \begin{pmatrix} \alpha\\\beta \end{pmatrix}^{T}\operatorname{Var}\left[ \begin{pmatrix} {a} \\ {b} \end{pmatrix}\right] \begin{pmatrix} \alpha\\\beta \end{pmatrix}$$ $$ = \begin{pmatrix} \alpha\\\beta \end{pmatrix}^{T} \begin{pmatrix} \operatorname{Var}\left[{a}\right] & \operatorname{Cov}\left[{a},{b}\right] \\ \operatorname{Cov}\left[{a},{b}\right] & \operatorname{Var}\left[{b}\right] \end{pmatrix} \begin{pmatrix} \alpha\\\beta \end{pmatrix}$$ $$ = \alpha^{2}\operatorname{Var}\left[{a}\right] + \beta^{2}\operatorname{Var}\left[{b}\right] + 2\alpha\beta\operatorname{Cov}\left[{a},{b}\right]\textrm{,} $$ with (1) the matrix-wise principle on how to get a constant out of the Variance, see (http://en.wikipedia.org/wiki/Covariance#A_more_general_identity_for_covariance_matrices).

You are interested in $$ \operatorname{Cov}\left[{a},{c}\right] = \operatorname{Cov}\left[{a},\alpha{a}+\beta{b}\right]$$ $$ \overset{(2)}{=} \alpha\operatorname{Cov}\left[a,a\right]+\beta\operatorname{Cov}\left[a,b\right] = \alpha\operatorname{Var}\left[a\right]+\beta\operatorname{Cov}\left[a,b\right] \textrm{.} $$ The equality (2) follows from the definition of Covariance and some manipulations: $$ \operatorname{Cov}\left[{a},{c}\right] = \operatorname{Cov}\left[{a},\alpha{a}+\beta{b}\right]$$ $$= \mathbb{E}\left[\left(a-\mathbb{E}\left[a\right]\right)\left(\alpha{a}+\beta{b} - \mathbb{E}\left[\alpha{a}+\beta{b}\right]\right)\right]$$ $$ = \mathbb{E}\left[\left(\alpha{a^{2}}-2\alpha{a}\mathbb{E}\left[a\right]+\alpha\mathbb{E}\left[a\right]^{2}\right) + \left(\beta{ab}-\beta{a\mathbb{E}\left[b\right]}-\beta{b}\mathbb{E}\left[a\right]+\beta\mathbb{E}\left[a\right]\mathbb{E}\left[b\right]\right)\right]$$ $$ = \mathbb{E}\left[\alpha\left(a-\mathbb{E}\left[a\right]\right)^{2} + \beta\left(a-\mathbb{E}\left[a\right]\right)\left(b-\mathbb{E}\left[b\right]\right)\right] = \alpha\operatorname{Var}\left[{a}\right] + \beta\operatorname{Cov}\left[a,b\right] $$ You can get the Correlation from the Covariance in the obvious way, devide the Covariance by the square root of the product of the Variances of both random variables.

Hope I didn't miscalculate and that is the answer you were looking for.

## Answer by user12348 (score 2)

https://quant.stackexchange.com/a/11269

Let there be n stocks, 2 portfolio a and b. c is a combined portfolio of portfolio a and portfolio b. $\Sigma $ is variance-covariance matrix of the n assets. Weight vectors for portfolios a and b are $$w_{pa},w_{pb}\in\mathbb{R}^{n} ,$$ $$\left\|w_{pa}\right\|_{1}=\left\|w_{pb}\right\|_{1}=1 $$

then $$ Var(a)= w_{pa}' \Sigma w_{pa} $$ $$Var(b)= w_{pb}' \Sigma w_{pb}$$ $$Cov(a,b)=w_{pa}' \Sigma w_{pb} $$

Let us define combined portfolio c such that its weight vector $$w_{pc}\in\mathbb{R}^{n},$$ $$\left\|w_{pc}\right\|_{1}=1 $$ $$\alpha w_{pa}+\beta w_{pb}=:w_{pc}\in\mathbb{R}\space$$ with weights $\alpha+\beta=1$

Then we have in matrix form:

$$\begin{pmatrix} Cov(a,c) \\ Cov(b,c) \end{pmatrix}=\begin{pmatrix} Var(a) & Cov(a,b) \\ Cov(a,b)) & Var(b) \end{pmatrix} \begin{pmatrix} \alpha \\ \beta \end{pmatrix}$$ It can also be expressed in terms of the asset weights in the sub portfolios as: $$\begin{pmatrix} Cov(a,c) \\ Cov(b,c) \end{pmatrix}=\begin{pmatrix} \alpha w_{pa}' \Sigma w_{pa} + \beta w_{pa}' \Sigma w_{pb} \\ \beta w_{pb}' \Sigma w_{pb} + \alpha w_{pa}' \Sigma w_{pb} \end{pmatrix}$$

This reconciles the matrix calculations in solution to this question and what @MarcoBietig has posted here. Here is multivariate portfolio solution.

## Answer by chetan (score -1)

https://quant.stackexchange.com/a/39094

Use Ledoit - Wolf estimator instead of normal cov matrix