Minimize portfolio variance, which should be reasonably similar to the benchmark portfolio
I am doing portfolio optimization and I would like to expand the discussion here with the following:
I have a vector of weights w_bench
that is used as a reference. I would like to optimize a portfolio weight vector w_pf
that satisfies
sum(pmin(w_bench, w_pf)) > 0.7
pmin
there is a pairwise minimum. This makes the optimized portfolio weights w_pf
look like the weights w_bench
, and the right size (0.7 in this case) determines how much they should fit. As this value gets larger, we require the portfolios to be more similar.
At first I thought I could easily do this with a package fPortfolio
(still trying). But there are still no cubes. I also think that solving this with help quadprog
would be much more intuitive, but I don't know how to incorporate this functionality into the process.
Excel implementation:
Covariance matrix:
0.003015254 -0.000235924 0.000242836
-0.000235924 0.002910845 0.000411308
0.000242836 0.000411308 0.002027183
Weight:
w_pf w_bench min
V1 0.32 0.40 0.32
V2 0.31 0.50 0.31
V3 0.38 0.10 0.10
Ss 1.00 1.00 0.72
Collapse value ( =MMULT(TRANSPOSE(H8:H10),MMULT(H3:J5,H8:H10))
) with constraints Ss(w_pf) = 1
andSs(min) > 0.7
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As you noticed, the tricky constraint is that sum(pmin(w_bench, w_pf)) > 0.7
(in fact, it is very difficult to have strict inequality, so I will do >=
instead >
, you, solve with help >= 0.7+epsilon
for a small epsilon). To approach this, we will create a new variable y_i
for each item i
in our portfolio, and we will add constraints y_i <= wpf_i
(aka wpf_i - y_i >= 0
) and y_i <= wbench_i
(aka -y_i >= -wbench_i
) where wpf_i
is the share i
in our selected portfolio (decision variable) and wbench_i
is the share i
in the control portfolio (input data). This limits y_i
no more than the minimum of these two values. Finally, we will add a constraint \sum_i y_i >= 0.7
requiring these minimum values to be at least 0.7.
All that remains is to implement it in the package quadprog
. Setting with problem data:
cov.mat <- rbind(c(0.003015254, -0.000235924, 0.000242836), c(-0.000235924, 0.002910845, 0.000411308), c(0.000242836, 0.000411308, 0.002027183))
w.bench <- c(.4, .5, .1)
n <- length(w.bench)
As we add new variables, we will place the covariance matrix (which will be placed in the optimization target) with 0 in the rows and columns corresponding to these new variables. We can do this with:
(cov.mat.exp <- cbind(rbind(cov.mat, matrix(0, n, n)), matrix(0, 2*n, n)))
# [,1] [,2] [,3] [,4] [,5] [,6]
# [1,] 0.003015254 -0.000235924 0.000242836 0 0 0
# [2,] -0.000235924 0.002910845 0.000411308 0 0 0
# [3,] 0.000242836 0.000411308 0.002027183 0 0 0
# [4,] 0.000000000 0.000000000 0.000000000 0 0 0
# [5,] 0.000000000 0.000000000 0.000000000 0 0 0
# [6,] 0.000000000 0.000000000 0.000000000 0 0 0
Now we want to create a constraint matrix for all of our constraints:
(consts <- rbind(rep(c(1, 0), c(n, n)),
rep(c(0, 1), c(n, n)),
cbind(matrix(0, n, n), -diag(n)),
cbind(diag(n), -diag(n))))
# [,1] [,2] [,3] [,4] [,5] [,6]
# [1,] 1 1 1 0 0 0
# [2,] 0 0 0 1 1 1
# [3,] 0 0 0 -1 0 0
# [4,] 0 0 0 0 -1 0
# [5,] 0 0 0 0 0 -1
# [6,] 1 0 0 -1 0 0
# [7,] 0 1 0 0 -1 0
# [8,] 0 0 1 0 0 -1
(rhs <- c(1, 0.7, -w.bench, rep(0, n)))
# [1] 1.0 0.7 -0.4 -0.5 -0.1 0.0 0.0 0.0
The first line will sum up the portfolio weights to 1, the next will be enforced \sum_i y_i >= 0.7
, the next three will be constraints -y_i >= -wbench_i
, and the last three will be constraints ypf_i-y_i >= 0
.
All that's left is to put them in the format expected by the function solve.QP
:
library(quadprog)
mod <- solve.QP(cov.mat.exp, rep(0, 2*n), t(consts), rhs, 1)
# Error in solve.QP(cov.mat.exp, rep(0, 2 * n), t(consts), rhs, 1) :
# matrix D in quadratic function is not positive definite!
library(quadprog)
mod <- solve.QP(cov.mat.exp + 1e-8*diag(2*n), rep(0, 2*n), t(consts), rhs, 1)
(w.pf <- head(mod$solution, n))
# [1] 0.3153442 0.3055084 0.3791474
(y <- tail(mod$solution, n))
# [1] 0.3 0.3 0.1
(opt.variance <- as.vector(t(w.pf) %*% cov.mat %*% w.pf))
# [1] 0.0009708365
We see that this is not a particularly interesting case, because the constraint that we worked with so much was not required. Let's increase the right side from 0.7 to 0.9 to see the constraint in action:
(rhs <- c(1, 0.9, -w.bench, rep(0, n)))
# [1] 1.0 0.9 -0.4 -0.5 -0.1 0.0 0.0 0.0
mod <- solve.QP(cov.mat.exp + 1e-8*diag(2*n), rep(0, 2*n), t(consts), rhs, 1)
(w.pf <- head(mod$solution, n))
# [1] 0.3987388 0.4012612 0.2000000
(y <- tail(mod$solution, n))
# [1] 0.3987388 0.4012612 0.1000000
(opt.variance <- as.vector(t(w.pf) %*% cov.mat %*% w.pf))
# [1] 0.00105842
In this case, the restriction was mandatory; the minimum value accepted y_1
and y_2
is taken from our new portfolio, and the minimum value accepted y_3
from the reference portfolio. We see that the variance of the optimal portfolio had a relative increase of 9.0% due to the constraint.
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