Inversion of a real matrix and the solution of a set of linear equations using Crout's method, calling Fortran code by Alan Miller. Compile with
gfortran kind.f90 linear_solve.f90 xmatrix_inverse.f90.
A correlation matrix of dimension n where all the off-diagonal elements have value r has an inverse matrix whose diagonal elements equal
(1 + (n-2)*r) / ((1-r) * (1 + (n-1)*r))
and off-diagonal elements equal
r / ((r-1) * (1 + (n-1)*r))
Given the expected returns mu and covariances cov of assets, when taking positions w the expected portfolio return is mu' * w, the variance is w' * cov * w, and optimal portfolio weights are proportional to cov^(-1) * mu. The program below specifies expected returns for 3 assets and studies how the optimal portfolio weights depend on the average correlation.
module kind_mod
implicit none
private
public :: dp
integer, parameter :: dp = selected_real_kind(15, 307)
end module kind_mod
module linear_solve
use kind_mod, only: dp
implicit none
public :: equicorr, inverse_equicorr_off_diag, inverse_equicorr
contains
pure function equicorr(n, r) result(xmat)
! return a correlation matrix with equal off-diagonal elements
integer , intent(in) :: n ! dimension of correlation matrix
real(kind=dp), intent(in) :: r ! off-diagonal correlation
real(kind=dp), allocatable :: xmat(:,:)
integer :: i
allocate (xmat(n, n), source = r)
do i=1,n
xmat(i,i) = 1.0_dp
end do
end function equicorr
pure function inverse_equicorr_off_diag(n, r) result(y)
! value of off-diagonal elements of the inverse of an equicorrelation matrix
integer , intent(in) :: n
real(kind=dp), intent(in) :: r
real(kind=dp) :: y
y = r / ((r-1) * (1 + (n-1)*r))
end function inverse_equicorr_off_diag
pure function inverse_equicorr_diag(n, r) result(y)
! value of diagonal elements of the inverse of an equicorrelation matrix
integer , intent(in) :: n
real(kind=dp), intent(in) :: r
real(kind=dp) :: y
y = (1 + (n-2)*r) / ((1-r) * (1 + (n-1)*r))
end function inverse_equicorr_diag
pure function inverse_equicorr(n, r) result(xmat)
! return the inverse of a correlation matrix with equal off-diagonal elements
integer , intent(in) :: n ! dimension of correlation matrix
real(kind=dp), intent(in) :: r ! off-diagonal correlation
real(kind=dp), allocatable :: xmat(:,:)
integer :: i
real(kind=dp) :: ydiag
ydiag = inverse_equicorr_diag(n, r)
allocate (xmat(n, n), source = inverse_equicorr_off_diag(n, r))
do i=1,n
xmat(i,i) = ydiag
end do
end function inverse_equicorr
end module linear_solve
program xequicorr_port
! find optimal portfolio as a function of correlation, given expected
! returns and assuming all volatilities (standard deviations) are 1,
! so that the covariance matrix equals the correlation matrix
use kind_mod , only: dp
use linear_solve, only: equicorr, inverse_equicorr
implicit none
integer :: ir
integer, parameter :: n = 3, nr = 10
real(kind=dp) :: r, xmat(n,n), xinv(n,n), mu(n), w(n), expected_ret, &
sd_ret
mu = 1.0_dp
mu(1) = 2.0_dp ! stock 1 has higher expected return
print "('expected asset returns:',*(f12.3))", mu
print "(/,*(a12))", "corr", "w1", "w2", "w3", "mean_ret", "sd_ret", "mean/sd"
do ir=1,nr
r = (ir-1) * 0.1_dp
xmat = equicorr(n, r)
xinv = inverse_equicorr(n, r)
w = matmul(xinv, mu)
w = w / sum(abs(w))
expected_ret = sum(w*mu)
sd_ret = sqrt(sum(w * matmul(xmat,w)))
print "(*(f12.3))", r, w, expected_ret, sd_ret, expected_ret/sd_ret
end do
end program xequicorr_portOutput:
expected asset returns: 2.000 1.000 1.000
corr w1 w2 w3 mean_ret sd_ret mean/sd
0.000 0.500 0.250 0.250 1.500 0.612 2.449
0.100 0.556 0.222 0.222 1.556 0.683 2.277
0.200 0.625 0.187 0.187 1.625 0.754 2.155
0.300 0.714 0.143 0.143 1.714 0.828 2.070
0.400 0.833 0.083 0.083 1.833 0.908 2.018
0.500 1.000 0.000 0.000 2.000 1.000 2.000
0.600 0.833 -0.083 -0.083 1.500 0.742 2.023
0.700 0.714 -0.143 -0.143 1.143 0.542 2.108
0.800 0.625 -0.188 -0.188 0.875 0.377 2.320
0.900 0.556 -0.222 -0.222 0.667 0.228 2.928
For the case of uncorrelated assets, twice as much weight is given to asset 1 than the other 2 assets, since its expected return is twice as high. As correlation increases, the weights given to assets 2 and 3 decrease, since they have lower returns and are less effective at diversifying asset 1. For correlation of 0.5 asset 1 gets weight 1.0 and the other assets weight 0.0. As correlation increases still further, the optimal portfolio goes long asset 1 but now bets against assets 2 and 3 (shorts them), since doing so hedges the position in asset 1 and reduces risk. One see that this boosts the Sharpe ratio (the last column, labeled mean/sd). What hedge funds are supposed to do is go long and short assets to maximize the Sharpe ratio.
An individual investor may be unable or unwilling to take short positions and may impose a long-only constraint. This results in a quadratic programming problem that can be solved by the quadprog package of @loiseaujc.