Re: [R] 12th Root of a Square (Transition) Matrix

Duncan Murdoch Fri, 18 Jun 2010 03:19:58 -0700

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 belowshould 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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and provide commented, minimal, self-contained, reproducible code.

Re: [R] 12th Root of a Square (Transition) Matrix

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