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# VaR Mapping - Diversified VaR

#### Gareth

##### New Member
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

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Gareth we replicate this mapping in our XLS at https://www.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,

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#### Gareth

##### New Member
Hi David

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

##### David Harper CFA FRM
Staff member
Subscriber
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!