On 18/06/2010 2:01 AM, Corey Gallon wrote:> Dear R-tisans,
>
> I am trying to calculate the 12th root of a transition (square) matrix, but
can't seem to obtain an accurate result. I realize that this post is laced
with intimations of quantitative finance, but the question is both R-related and
broadly mathematical. That said, I'm happy to post this to R-SIG-Finance if
I've erred in posting this to the general list.
>
> I've pulled down an annual transition matrix from the latest
Moody's Corporate Default Study, and I'm using this (with the default
row added manually) as the basis for this calculation. (I've pasted the
dput of the resulting matrix below.) According to Hull, Appendix E [1], an
arbitrary root of a square matrix (A) can be calculated by multiplying the
inverse matrix of eigenvectors (X-inv) by the nth-root of diagonalized matrix of
eigenvalues (Lambda-star) by the matrix of eigenvectors (X) -- all of these
eigenvectors(values) being calculated from the matrix for which one wishes to
calculate the nth root. The equation is as follows:
>
> A = X-inv %*% Lambda-star %*% X
>
This is wrong: you've swapped X and X-inv. So your final line below
should be
nth_root <- X %*% L_star %*% X_inv
Duncan Murdoch> I've written the code below to implement this, but the result
doesn't seem to be correct. (I can't raise the resulting matrix to the
12th power to calculate the original matrix.) I believe that the reason for
this is the order in which R returns the eigenvalues (i.e. a vector in
descending order) and the order in which I've created the matrix of
eigenvectors, but I may be wrong in this suspicion.
>
> I defer to the collective wisdom of the community, and hope that minds
greater than mine may provide insight.
>
>
>
> Cheers,
>
> Corey
>
>
>
>> dput(trans_matrix)
>>
>
> structure(c(0.9426, 0.0047, 0, 4e-04, 0, 5e-04, 0, 0, 0, 0, 0,
> 0, 9e-04, 0, 0, 0, 0, 0, 0.0308, 0.8205, 0.0254, 4e-04, 9e-04,
> 0, 0, 0, 0, 4e-04, 0.0016, 0, 0, 0, 0, 0, 0, 0, 0.021, 0.1291,
> 0.7978, 0.034, 0.0043, 0.0025, 0.0011, 3e-04, 9e-04, 4e-04, 0,
> 0, 0, 0, 0, 0, 0, 0, 0.0056, 0.0394, 0.1174, 0.8366, 0.0509,
> 0.0094, 0.0023, 0.0022, 0.0014, 0.0017, 0, 0, 0, 0, 0, 0, 0,
> 0, 0, 0.0016, 0.0448, 0.0944, 0.8253, 0.0569, 0.011, 0.0051,
> 6e-04, 0.0021, 0.0016, 0, 0, 0, 0, 0, 0, 0, 0, 0.0016, 0.0067,
> 0.024, 0.0873, 0.8108, 0.0677, 0.0105, 0.004, 0.0017, 0, 9e-04,
> 0, 0, 0.0039, 0.0045, 0, 0, 0, 0.0016, 0.0024, 0.0064, 0.0179,
> 0.0833, 0.7838, 0.0758, 0.012, 0.0021, 0.0024, 0, 9e-04, 0, 0.0013,
> 0.0015, 0, 0, 0, 0, 0.0024, 0.0014, 0.0068, 0.0202, 0.0908, 0.7694,
> 0.0744, 0.0136, 0.0079, 0.0045, 9e-04, 0.0024, 0, 0, 0.0024,
> 0, 0, 0.0016, 0, 0.0014, 0.004, 0.0089, 0.0175, 0.0983, 0.8066,
> 0.1032, 0.0267, 0.0063, 0.0054, 0.0024, 0, 0.003, 0, 0, 0, 0,
> 0.0024, 0, 0.0022, 0.002, 0.0079, 0.0178, 0.0632, 0.7605, 0.1422,
> 0.0308, 0.0099, 0.006, 0.0013, 0.003, 0, 0, 0, 0, 0, 0, 0, 0.0017,
> 0.0014, 0.0086, 0.0117, 0.0574, 0.6787, 0.1014, 0.0425, 0.006,
> 0.0039, 0.003, 0.0047, 0, 0, 0, 0, 0, 3e-04, 0.0012, 0.0014,
> 0.0022, 0.0109, 0.0227, 0.0589, 0.6814, 0.1058, 0.041, 0.0078,
> 0.0045, 0.0024, 0, 0, 0, 6e-04, 0, 0, 2e-04, 0.0096, 0.0038,
> 0.0034, 0.012, 0.0212, 0.0661, 0.6609, 0.159, 0.0362, 0.012,
> 0.0024, 0, 0, 0, 0, 0, 0, 2e-04, 3e-04, 0.0016, 0.0029, 0.0058,
> 0.0118, 0.029, 0.0597, 0.6205, 0.1332, 0.0331, 0.0047, 0, 0,
> 0, 0, 0, 0, 0.001, 0.0011, 3e-04, 0.002, 0.0074, 0.0157, 0.0244,
> 0.0515, 0.0916, 0.6546, 0.1203, 0.0353, 0, 0, 0, 0, 4e-04, 0,
> 0, 3e-04, 0, 6e-04, 8e-04, 0.0047, 0.0199, 0.0172, 0.0325, 0.0699,
> 0.7098, 0.1318, 0, 0, 0, 0, 0, 0, 5e-04, 0.0017, 0.0022, 0.004,
> 0.0021, 0.0102, 0.0217, 0.0244, 0.0241, 0.044, 0.0737, 0.5271,
> 0, 0, 0, 0, 7e-04, 0, 7e-04, 0.0023, 0.0019, 0.0014, 0.0062,
> 0.0165, 0.0136, 0.0199, 0.0145, 0.044, 0.0316, 0.2894, 1), .Dim = c(18L,
> 18L), .Dimnames = list(c("1", "2", "3",
"4", "5", "6", "7", "8",
> "9", "10", "11", "12",
"13", "14", "15", "16", "17",
"18"), c("AAA",
> "AAp", "AA", "AAm", "Ap",
"A", "Am", "BBBp", "BBB",
"BBBm", "BBp",
> "BB", "BBm", "Bp", "B",
"Bm", "CCC.to.C", "D")))
>
>
> ------ BEGIN PASTE ------
>
> # create a matrix of eigenvectors of the transition matrix
> X <- eigen(trans_matrix)$vectors
>
> # create a diagonalized matrix of the eigenvalues of the transition matrix
> L <- diag(eigen(trans_matrix)$values)
>
> # calculate inverse of matrix of eigenvectors of the transition matrix
> X_inv <- solve(X)
>
> # calculate the 12th root of the eigenvalues in the diagonal matrix
> L_star <- L ^ (1/12)
>
> # calculate the 12th root of the transition matrix
> nth_root <- X_inv %*% L_star %*% X
>
> ------ END PASTE ------
>
>
> References:
>
> [1] Hull, John. Risk Management and Financial Institutions. Prentice Hall,
2007.
>
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