Computation of sparse portfolios for financial index tracking, i.e., joint selection of a subset of the assets that compose the index and computation of their relative weights (capital allocation). The level of sparsity of the portfolios, i.e., the number of selected assets, is controlled through a regularization parameter. Different tracking measures are available, namely, the empirical tracking error (ETE), downside risk (DR), Huber empirical tracking error (HETE), and Huber downside risk (HDR). See vignette for a detailed documentation and comparison, with several illustrative examples.

The package is based on the paper:

K. Benidis, Y. Feng, and D. P. Palomar, “Sparse Portfolios for
High-Dimensional Financial Index Tracking,” *IEEE Trans. on Signal
Processing*, vol. 66, no. 1, pp. 155-170, Jan. 2018.
(https://doi.org/10.1109/TSP.2017.2762286)

The latest stable version of **sparseIndexTracking** is
available at
https://CRAN.R-project.org/package=sparseIndexTracking.

The latest development version of
**sparseIndexTracking** is available at
https://github.com/dppalomar/sparseIndexTracking.

To install the latest stable version of
**sparseIndexTracking** from CRAN, run the following
commands in R:

`install.packages("sparseIndexTracking")`

To install the development version of
**sparseIndexTracking** from GitHub, run the following
commands in R:

```
install.packages("devtools")
::install_github("dppalomar/sparseIndexTracking") devtools
```

To get help:

```
library(sparseIndexTracking)
help(package = "sparseIndexTracking")
package?sparseIndexTracking ?spIndexTrack
```

Please cite **sparseIndexTracking** in publications:

`citation("sparseIndexTracking")`

For more detailed information, please check the vignette: CRAN vignette and GitHub vignette.

`spIndexTrack()`

We start by loading the package and real data of the index S&P 500 and its underlying assets:

```
library(sparseIndexTracking)
library(xts)
data(INDEX_2010)
```

The data `INDEX_2010`

contains a list with two xts
objects:

`X`

: AT×Nxts with the daily linear returns of theNassets that were in the index during the year 2010 (totalTtrading days)`SP500`

: AT× 1 xts with the daily linear returns of the index S&P 500 during the same period.

Note that we use xts objects just for illustration purposes. The
function `spIndexTracking()`

can also be invoked passing
simple data arrays or dataframes.

Based on the above quantities we create a training window, which we will use to create our portfolios, and a testing window, which will be used to assess the performance of the designed portfolios. For simplicity, here we consider the first six (trading) months of the dataset (~126 days) as the training window, and the subsequent six months as the testing window:

```
<- INDEX_2010$X[1:126]
X_train <- INDEX_2010$X[127:252]
X_test <- INDEX_2010$SP500[1:126]
r_train <- INDEX_2010$SP500[127:252] r_test
```

Now, we use the four modes (four available tracking errors) of the
`spIndexTracking()`

algorithm to design our portfolios:

```
# ETE
<- spIndexTrack(X_train, r_train, lambda = 1e-7, u = 0.5, measure = 'ete')
w_ete cat('Number of assets used:', sum(w_ete > 1e-6))
#> Number of assets used: 45
# DR
<- spIndexTrack(X_train, r_train, lambda = 2e-8, u = 0.5, measure = 'dr')
w_dr cat('Number of assets used:', sum(w_dr > 1e-6))
#> Number of assets used: 42
# HETE
<- spIndexTrack(X_train, r_train, lambda = 8e-8, u = 0.5, measure = 'hete', hub = 0.05)
w_hete cat('Number of assets used:', sum(w_hete > 1e-6))
#> Number of assets used: 44
# HDR
<- spIndexTrack(X_train, r_train, lambda = 2e-8, u = 0.5, measure = 'hdr', hub = 0.05)
w_hdr cat('Number of assets used:', sum(w_hdr > 1e-6))
#> Number of assets used: 43
```

Finally, we plot the actual value of the index in the testing window in comparison with the values of the designed portfolios:

```
plot(cbind("PortfolioETE" = cumprod(1 + X_test %*% w_ete), cumprod(1 + r_test)),
legend.loc = "topleft", main = "Cumulative P&L")
```

```
plot(cbind("PortfolioDR" = cumprod(1 + X_test %*% w_dr), cumprod(1 + r_test)),
legend.loc = "topleft", main = "Cumulative P&L")
```

```
plot(cbind("PortfolioHETE" = cumprod(1 + X_test %*% w_hete), cumprod(1 + r_test)),
legend.loc = "topleft", main = "Cumulative P&L")
```

```
plot(cbind("PortfolioHDR" = cumprod(1 + X_test %*% w_hdr), cumprod(1 + r_test)),
legend.loc = "topleft", main = "Cumulative P&L")
```

README file: CRAN-readme and GitHub-readme.

Vignette: CRAN-html-vignette, CRAN-pdf-vignette, GitHub-html-vignette, and GitHub-pdf-vignette.