# [R] The best solver for non-smooth functions?

Cren oscar.soppelsa at bancaakros.it
Wed Jul 18 22:00:19 CEST 2012

```# Hi all,

# consider the following code (please, run it:
# it's fully working and requires just few minutes
# to finish):

require(CreditMetrics)
require(clusterGeneration)
install.packages("Rdonlp2", repos= c("http://R-Forge.R-project.org",
getOption("repos")))
install.packages("Rsolnp2", repos= c("http://R-Forge.R-project.org",
getOption("repos")))
require(Rdonlp2)
require(Rsolnp)
require(Rsolnp2)

N <- 3
n <- 100000
r <- 0.0025
rc <- c("AAA", "AA", "A", "BBB", "BB", "B", "CCC", "D")
lgd <- 0.99
rating <- c("BB", "BB", "BBB")
firmnames <- c("firm 1", "firm 2", "firm 3")
alpha <- 0.99

# One year empirical migration matrix from Standard & Poor's website

rc <- c("AAA", "AA", "A", "BBB", "BB", "B", "CCC", "D")
M <- matrix(c(90.81,  8.33,  0.68,  0.06,  0.08,  0.02,  0.01,   0.01,
0.70, 90.65,  7.79,  0.64,  0.06,  0.13,  0.02,   0.01,
0.09,  2.27, 91.05,  5.52,  0.74,  0.26,  0.01,   0.06,
0.02,  0.33,  5.95, 85.93,  5.30,  1.17,  1.12,   0.18,
0.03,  0.14,  0.67,  7.73, 80.53,  8.84,  1.00,   1.06,
0.01,  0.11,  0.24,  0.43,  6.48, 83.46,  4.07,   5.20,
0.21,     0,  0.22,  1.30,  2.38, 11.24, 64.86,  19.79,
0,     0,     0,     0,     0,     0,     0, 100
)/100, 8, 8, dimnames = list(rc, rc), byrow = TRUE)

# Correlation matrix

rho <- rcorrmatrix(N) ; dimnames(rho) = list(firmnames, firmnames)

# Credit Value at Risk

cm.CVaR(M, lgd, ead, N, n, r, rho, alpha, rating)

# Risk neutral yield rates

Y <- cm.cs(M, lgd)
y <- c(Y[match(rating[1],rc)], Y[match(rating[2],rc)],
Y[match(rating[3],rc)]) ; y

# The function to be minimized

sharpe <- function(w) {
- (t(w) %*% y) / cm.CVaR(M, lgd, ead, N, n, r, rho, alpha, rating)
}

# The linear constraints

constr <- function(w) {
sum(w)
}

# Results' matrix (it's empty by now)

Results <- matrix(NA, nrow = 3, ncol = 4)
rownames(Results) <- list('donlp2', 'solnp', 'solnp2')
colnames(Results) <- list('w_1', 'w_2', 'w_3', 'Sharpe')

# See the differences between different solvers

rho
Results[1,1:3] <- round(donlp2(fn = sharpe, par = rep(1/N,N), par.lower =
rep(0,N), par.upper = rep(1,N), A = t(rep(1,N)), lin.lower = 1, lin.upper =
1)\$par, 2)
Results[2,1:3] <- round(solnp(pars = rep(1/N,N), fun = sharpe, eqfun =
constr, eqB = 1, LB = rep(0,N), UB = rep(1,N))\$pars, 2)
Results[3,1:3] <- round(solnp2(par = rep(1/N,N), fun = sharpe, eqfun =
constr, eqB = 1, LB = rep(0,N), UB = rep(1,N))\$pars, 2)
for(i in 1:3) {
Results[i,4] <- abs(sharpe(Results[i,1:3]))
}
Results

# In fact the "sharpe" function I previously defined
# is not smooth because of the cm.CVaR function.
# If you change correlation matrix, ratings or yields
# you see how different solvers produce different
# parameters estimation.

# Then the main issue is: how may I know which is the
# best solver at all to deal with non-smooth functions
# such as this one?

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```