How to use the package

To install the package you need to run devtools::install_github("Statistics-In-Portfolio-Theory/DOSPortfolio"), which demands that you have installed devtools. To load the package, run

library(DOSPortfolio)

The main interface to the package is DOSPortfolio::DOSPortfolio which acts as a wrapper for DOSPortfolio::wGMVOverlapping and DOSPortfolio::wGMVNonOverlapping. The big perk of using the interface is that you get input validation and more informative errors. You could use the functions directly, which would most likely not throw an error (with an exception of specific degenerate cases), though the results might not make sense. Lets simulate some data, in our simple example we assume the (log) returns are given by a t-distribution with 5 degrees of freedom.

n <- 25*2
p <- 15
data <- 5/3 * matrix(rt(n*p, df=5), ncol=p, nrow=n)

Notice that it is a matrix on long format. Each column is assumed to be the time series of an asset. We assume that the returns are ordered in time. The observation on the first row was observed before the observations on the second and so forth. To construct the DOS portfolio we need to specify the number of reallocation points and when these happen. Since data is a matrix on long format, we work with the row index. The DOS portfolio is then constructed as

reallocation_points <- c(25, 42)
# use the first subsample to estimate the relative loss
(portfolios <- DOSPortfolio(data, reallocation_points))
#> $weights
#>            [,1]       [,2]       [,3]       [,4]       [,5]       [,6]
#> [1,] 0.06928281 0.06307516 0.07207183 0.06329702 0.07025557 0.06456685
#> [2,] 0.07012946 0.06191285 0.07382107 0.06220651 0.07141704 0.06388729
#>            [,7]       [,8]       [,9]      [,10]      [,11]      [,12]
#> [1,] 0.07094686 0.06703658 0.06952505 0.06601155 0.06454546 0.05814309
#> [2,] 0.07233203 0.06715630 0.07045009 0.06579953 0.06385898 0.05538465
#>           [,13]      [,14]      [,15]
#> [1,] 0.06867475 0.06727933 0.06528809
#> [2,] 0.06932462 0.06747761 0.06484195
#> 
#> $shrinkage_type
#> [1] "non-overlapping"
#> 
#> attr(,"class")
#> [1] "DOSPortfolio"

The first parameter we configure is the number of reallocation points and when they happen. These are specified in the reallocation_points vector and specify when you recompute the weights. In this case we re-weight our portfolio on the 25th observation and 42th observation. Notice that there we do not include the last 10 observations. There are only 2 reallocation points and these do not include the last observed value. If you wanted to include a third reallocation point, just add the number of rows (or 50 in this case) to the reallocation_points vector. However, as we will see later on, this will not work in our example though there is a simple workaround.

Each row of portfolios$weights is a shrunk GMV portfolio. For each reallocation point the portfolio is computed using a convex combination using the previously estimated portfolio and on the first transition, the target portfolio. This is the “transition” which is made optimally and is specifically tailored to work in higher dimensions, when the number of assets is very large and possible close to the number of observations.

The string seen in portfolios$shrinkage_type is what type of shrinkage estimator we used to make the transitions. There are two options, "non-overlapping" and "overlapping". These can handle quite different scenarios which we can illustrate by extending the reallocation_points vector to

reallocation_points <- c(25, 42, 50)
# This will not work
try(portfolios <- DOSPortfolio(data, reallocation_points))
#> Error : Non-overlapping estimator can not handle concentration ratios above one.
#>             Consider excluding one (or more) break point(s) or provide more data.

However, this will work

(portfolios <- DOSPortfolio(data, reallocation_points, shrinkage_type = "overlapping"))
#> $weights
#>            [,1]       [,2]       [,3]       [,4]       [,5]       [,6]
#> [1,] 0.06928281 0.06307516 0.07207183 0.06329702 0.07025557 0.06456685
#> [2,] 0.07465137 0.05570507 0.08316368 0.05638221 0.07762032 0.06025783
#> [3,] 0.07651752 0.05314318 0.08701928 0.05397857 0.08018036 0.05875999
#>            [,7]       [,8]       [,9]      [,10]      [,11]      [,12]
#> [1,] 0.07094686 0.06703658 0.06952505 0.06601155 0.06454546 0.05814309
#> [2,] 0.07973017 0.06779568 0.07539070 0.06466719 0.06019256 0.04065199
#> [3,] 0.08278331 0.06805955 0.07742963 0.06419988 0.05867947 0.03457197
#>           [,13]      [,14]      [,15]
#> [1,] 0.06867475 0.06727933 0.06528809
#> [2,] 0.07279552 0.06853658 0.06245913
#> [3,] 0.07422793 0.06897360 0.06147576
#> 
#> $shrinkage_type
#> [1] "overlapping"
#> 
#> attr(,"class")
#> [1] "DOSPortfolio"

The issue concerns what is called the “concentration ratio” which we denote \(c_i=p/n_i\), where \(n_i\) is the size of each subsample. In our case these are 25, 17, 8. When the methods are derived for the “non-overlapping” shrinkage estimator it is assumed that \(c_i \in (0,1)\). This is due to the assumption that the sample covariance matrix needs to be non-singular. For the “non-overlapping” estimator we do not need to make such a rigid assumption, we only need that the first value \(c_1 \in (0,1)\) and the rest can be \(p>n_j\) for \(j=2,...,T\). One methods moves in windows of size \(n_i\) and the other extends the data to include all observations. We use \(N_I=\sum_i^I n_i\) data to estimate each portfolio weight.

The last two parameters that are available to configure is the target portfolio, which we will denote \(\mathbf{b}\), and an intial seed/estimate for the relative loss parameter. The target portfolio can be any portfolio whose values sums to one. The default value is the EW portfolio previously mentioned but it can be any other portfolio of your choice.

reallocation_points <- c(25, 42)
new_target <- runif(p, -1, 1)
new_target<- new_target/sum(new_target)
(portfolios <- DOSPortfolio(data, reallocation_points, target_portfolio = new_target))
#> $weights
#>            [,1]       [,2]      [,3]       [,4]       [,5]       [,6]      [,7]
#> [1,] 0.08190014 0.01485708 0.1260101 0.01418530 0.09871228 0.04108800 0.1151250
#> [2,] 0.08435591 0.02628987 0.1177301 0.02676723 0.09673967 0.04551436 0.1078659
#>            [,8]       [,9]      [,10]      [,11]        [,12]      [,13]
#> [1,] 0.07385225 0.09630610 0.07742194 0.03881509 -0.011751581 0.08697632
#> [2,] 0.07178860 0.09276249 0.06982581 0.04425078 -0.005959531 0.08473256
#>           [,14]      [,15]
#> [1,] 0.07816182 0.06834013
#> [2,] 0.07493839 0.06239796
#> 
#> $shrinkage_type
#> [1] "non-overlapping"
#> 
#> attr(,"class")
#> [1] "DOSPortfolio"

The initial estimate for the relative loss parameter demands some more detail. The relative_loss refers to how the following quantity \[ r_0 = \mathbf{1}'\boldsymbol{\Sigma}^{-1}\mathbf{1} \mathbf{b}'\boldsymbol{\Sigma}\mathbf{b} -1 \] which describes how the variance of our target portfolio \(\mathbf{b}'\boldsymbol{\Sigma}\mathbf{b}\) relates to the variance of the GMV portfolio \(1/\mathbf{1}'\boldsymbol{\Sigma}^{-1}\mathbf{1}\). A value of \(0\) would indicate that we know the variance of the portfolio and it is the target portfolio. The argument is equal to NULL per default which implies that we use a simple estimate of the quantity. We use the sample covariance matrix \(\mathbf{S}\) from the first subsample to estimate \(\boldsymbol{\Sigma}\) and its consistent inverse estimate \((1-c)\mathbf{S}^{-1}\). The factor \(1-c\) comes from the fact that the inverse sample covariance matrix is biased when \(p\) is allowed to grow together with \(n\), that is when \(p\rightarrow \infty\) s.t. \(n>p\). We can use any other estimate, though we need to make sure that the estimate is consistent, e.g. the estimate \(\hat{r}_0\) converges to \(r_0\) in probability.