VaR Mapping - Diversified VaR

[Gareth]

Hi,

I'm at a loss as to how diversified VaR is computed whe mapping linear derivatives. Undiviersified VaR is easy enough: sum(pv of cash flows x risk).

On page 67 of the official materials it says pre and post multiply by the pv of cashflows to get diversified var. But I don't get what it means.

Does anybody have any ideas please?

Thanks

 

[David Harper CFA FRM]

Hi @Gareth we replicate this mapping in our XLS at https://learn.bionicturtle.com/topic/learning-spreadsheet-jorion-chapter-6/

It's essentially similar to computing a portfolio variance in matrix version (necessary when there are many positions) where (eg) the portfolio variance, σ^2(P), is a function of the covariance matrix, let's call it COV_matrix, and the vector of portfolio weights, call it W. The portfolio variance is then given by W(T)*COV_matrix*W where W(T) is transposed W. And the covariance matrix itself is a function of correlation weights multiplied by a vector of standard deviations: COV_matrix = S(T)*corr_matrix*S; i.e., S = vector of standard deviations. Pre- and post-multiply are requirements of matrix math; but pre- and post-multiply refer to the fact that the order matters. The only difference, in this application, from my point of view, from classic portfolio variance calculation (in matrix form) is that (i) the volatiities are scaled into VaR with a 1.65/2.33 multiplier and (ii) the weights are denominated in dollars rather then %. But this are ultimately, merely differences in the units. As far as the math goes, we are effectively computing a portfolio dollar variance by assuming a correlation matrix but "tweaking up" the actual volatilities into 1.65x or 2.33 multiples. Thanks,

 

[Gareth]

Hi David

Apologies for the late reply and thank you for your rwsponse. Thats really helpful.

It's beginning tkcmake sense but there's one but that confuses me slightly still. Would S(t)corr matrixS not equal a scalar? It's a 1x3 matrix multiplied by a 3x3 matrix. This produced a 1x3 matrix. Then multiplied by a 3x1 matrix.

But this is meant to be the cov matrix. I imagine I'm getting confused somewhere.

I'm taking S = The Var% column vector as vol is scaled into that.

Thanks

 

[David Harper CFA FRM]

Hi @Gareth

For the portfolio variance, where covariance matrix is given by cov(x * x) = Σ(x*x) because it must be square with diagonal equal to variances and let's say weights/exposures are given by column vector w(x * 1), what I do in my XLS is:

1. Post-multiply: Σx = Σ(x*x) * w(x*1) = c(x*1) column vector Σx, where matrix notation is Σ(row * column)
2. Then pre-multiply x(T)*(Σx) = w(1*x)*c(x*1) = 1*1 scalar which is the portfolio variance; w(T) is transposed weight vector, so we could I think define the w as either column or vector such that portfolio variance = x(T)*Σ*x or x*Σ*x(T), but i've always post-multiplied first

On the other hand re: the covariance matrix itself, given volatility vector, σ(x * 1) which informs its own diagonal matrix (https://en.wikipedia.org/wiki/Diagonal_matrix) given by σ(x * x), and where we have a correlation matrix, ρ(x*x), the covariance matrix itself is σ(x * x) * ρ(x*x) * σ(x * x) = covariance(x*x). I can't recall if this can be solved with volatility vectors rather than the diagonal matrix equivalents. I have a very old video showing this here at

i hope that's useful, thanks!

 

[Gareth]

Got it. Thanks David!

 

[tcheung128]

Hello, I have the same question about what how to calculate the diversified VaR. What does it mean to pre and post multiply the xV (Individual VaR) at each vertex? I cannot replicate the R*xV amounts 2.330, 2.515, 2.553, 2.562, and 2.559 in the table. Thank-you,

 

[David Harper CFA FRM]

Hi @tinacheung128 pre and post-multiply simply refer to matrix (aka, dot product) multiplication; although there are probably many better illustrations of matrix multiplication, the wikipedia entry is decent at https://en.wikipedia.org/wiki/Dot_product

We have two vectors and a matrix. The pre- and post-multiplication simply refers to the fact that we perform two matrix (dot product) multiplications. The first column vector is xV = {$0.4971, $0.0541, ...} is the individual VaRs and it gets dot product multiplied by the correlation matrix which is represented by the column R*xV; i.e., correlation Matrix R multiplied by vector xV. This entails a multiplication of the {5 rows by 1 column} matrix (aka, vector) by the {5 rows by 5 rows} correlation matrix to product the {5 rows by 1 vector} column labelled R*xV. The first cell and second sells are given by the dot product operation:

• ($0.4971 * 1.000) + ($0.0541 * 0.8970) + ($0.0765 * 0.8860) + ($0.0947 * 0.8660) + ($1.9115 * 0.8550) = 2.330
• ($0.4971 * 0.8970) + ($0.0541 * 1.0000) + ($0.0765 * 0.9910) + ($0.0947 * 0.9760) + ($1.9115 * 0.9660)= 2.515
• etc

This R*xV column then gets post-multiplied by the transposed xV' vector to produce the variance because {1 row by 5 column} matrix multiplied by a {5 row by 1 column} matrix produces a {1 row by 1 column} matrix. I hope that's helpful,