| Title: | Dynamic Optimal Shrinkage Portfolio |
| Version: | 0.1.0 |
| Maintainer: | Erik Thorsén <erik.thorsen@math.su.se> |
| Author: | Taras Bodnar 
[aut],
Nestor Parolya
[aut],
Erik Thorsén
[aut, cre] |
| Description: | Constructs dynamic optimal shrinkage estimators for the weights of the global minimum variance portfolio which are reconstructed at given reallocation points as derived in Bodnar, Parolya, and Thorsén (2021) (<doi:10.48550/arXiv.2106.02131>). Two dynamic shrinkage estimators are available in this package. One using overlapping samples while the other use nonoverlapping samples. |
| License: | GPL-3 |
| URL: | https://github.com/Statistics-In-Portfolio-Theory/DOSportfolio |
| Encoding: | UTF-8 |
| RdMacros: | Rdpack |
| RoxygenNote: | 7.1.1 |
| Depends: | R (≥ 3.5.0) |
| Suggests: | knitr, rmarkdown, testthat (≥ 3.0.0), HDShOP |
| Config/testthat/edition: | 3 |
| Imports: | Rdpack (≥ 0.7) |
| VignetteBuilder: | knitr |
| NeedsCompilation: | no |
| Packaged: | 2021-09-09 08:27:08 UTC; ethor |
| Repository: | CRAN |
| Date/Publication: | 2021-09-13 07:30:08 UTC |
A set of tools for constructing Dynamic Optimal Shrinkage estimator of the global minimum variance portfolio.
Description
The DOSPortfolio package consists of two shrinkage estimators for the Global
Minimum Variance (GMV) portfolio. These are implemented in the packages main
interface: DOSPortfolio. The shrinkage is performed at fixed
reallocation points in a dynamic manner where the sample estimator of the
GMV weights is shrunk towards the holding portfolio. The reallocation points
are specified by a deterministic sequence given a priori. The estimation is
performed such that the shrinkage coefficients are optimal for large
portfolios, e.g. there are many assets in comparison to observations. The
main interface validates the main assumptions of the theory used to derive
these methods.
Methods
DOSPortfolio: (Bodnar et al. 2021)
References
Bodnar T, Parolya N, Thorsén E (2021). “Dynamic Shrinkage Estimation of the High-Dimensional Minimum-Variance Portfolio.” arXiv preprint arXiv:2106.02131. https://arxiv.org/abs/2106.02131.
A helper function for computing beta coefficients used in the case of the overlapping sample (Bodnar et al. 2021)
Description
The function computes the beta coefficients from Eq. (2.20) in Bodnar et al. (2021), which are used in the recursive computation of the dynamic shrinkage estimator of the GMV portfolio weights in the case of overlapping samples.
Usage
ComputeBeta(i, j, Psi)
Arguments
i |
an integer greater than one. |
j |
an integer greater than one. |
Psi |
vector, the vector of the optimal shrinkage intensities computed in the previous step of the recursion. |
Value
a number
References
Bodnar T, Parolya N, Thorsén E (2021). “Dynamic Shrinkage Estimation of the High-Dimensional Minimum-Variance Portfolio.” arXiv preprint arXiv:2106.02131. https://arxiv.org/abs/2106.02131.
A helper function for computing D-coefficients used in the case of the overlapping sample (Bodnar et al. 2021)
Description
The function computes the D_{j,i} coefficients from Eq. (2.21) in Bodnar et al. (2021),
which are used in the recursive computation of the dynamic shrinkage estimator of the GMV portfolio weights in the case
of overlapping samples.
Usage
ComputeD(Ci, Cj)
Arguments
Ci |
a number equal to the concentration ratio of period i |
Cj |
a number equal to the concentration ratio of period j |
Value
a number
Computes a convex combination between two vectors.
Description
Computes a convex combination between two vectors.
Usage
ConvexCombination(x1, x2, lambda)
Arguments
x1 |
a vector. |
x2 |
a vector. |
lambda |
a numeric value between 0 and 1. |
Value
a vector
The Dynamic Optimal Shrinkage Portfolio interface.
Description
This is the main function to compute the weights of the global minimum variance portfolio by using the dynamic optimal shrinkage estimators presented in Eq. (2.11) and Eq. (2.23) of Bodnar et al. (2021). It implements two different estimators for the shrinkage coefficients, one using overlapping samples (see, Eq. (2.23) of Bodnar et al. (2021)) and one using non-overlapping samples (see, Eq. (2.11) of Bodnar et al. (2021)).
Usage
DOSPortfolio(
data,
reallocation_points,
relative_loss = NULL,
target_portfolio = rep(1, ncol(data))/ncol(data),
shrinkage_type = "non-overlapping"
)
Arguments
data |
an n by p matrix of asset returns. Columns represent different assets rows are observations, where n>p, containing, for instance, log-returns. |
reallocation_points |
a vector of reallocation points. The reallocation points determine when the holding portfolio should be reconstructed and its weights should be recomputed. |
relative_loss |
possibly a numeric or NULL. The initial value of the
relative loss in the variance of the target portfolio. If its NULL, then it
will be initialized with the first subsample and the function
|
target_portfolio |
a vector which determines the weights of the target portfolio used when the shrinkage estimator of the global minimum variance portfolio is constructed for the first time. |
shrinkage_type |
the type of shrinkage estimator to use. The two implemented approaches are "non-overlapping" and "overlapping". |
Value
An S3 class which contains the matrix of the constructed weights of the dynamic shrinkage estimator of the global minimum variance portfolio and the type of the shrinkage estimator (i.e., "overlapping" or "non-overlapping") that was used in its construction. Each row of the weight matrix corresponds to the reallocation point and the column corresponds to the asset.
References
Bodnar T, Parolya N, Thorsén E (2021). “Dynamic Shrinkage Estimation of the High-Dimensional Minimum-Variance Portfolio.” arXiv preprint arXiv:2106.02131. https://arxiv.org/abs/2106.02131.
See Also
Section 2.1 and 2.2 of (Bodnar et al. 2021)
Examples
n <- 250*2
p <- 80
c <- p/n
reallocation_points <- c(120, 240)
data <- sqrt(5/3) * matrix(rt(n*p, df=5) , ncol=p, nrow=n)
weights <- DOSPortfolio(data, reallocation_points, 1)
Constructor for the DOSPortfolio class
Description
Constructor for the DOSPortfolio class
Usage
new_DOSPortfolio(
data,
reallocation_points,
target_portfolio,
relative_loss,
shrinkage_type
)
Arguments
data |
an n by p matrix of asset returns. Columns represent different assets rows are observations, where n>p, containing, for instance, log-returns. |
reallocation_points |
a vector of reallocation points. The reallocation points determine when the holding portfolio should be reconstructed and its weights should be recomputed. |
target_portfolio |
a vector which determines the weights of the target portfolio used when the shrinkage estimator of the global minimum variance portfolio is constructed for the first time. |
relative_loss |
possibly a numeric or NULL. The initial value of the
relative loss in the variance of the target portfolio. If its NULL, then it
will be initialized with the first subsample and the function
|
shrinkage_type |
the type of shrinkage estimator to use. The two implemented approaches are "non-overlapping" and "overlapping". |
Value
a DOSPortfolio class.
Computes the relative loss of the target portfolio used
Description
The function computes the initial value of the relative loss in the variance of the target portfolio as given in Eq. (2.10) of Bodnar et al. (2021).
Usage
r0Strategy(data, target_portfolio, c)
Arguments
data |
an n by p matrix of asset returns. Columns represent different assets rows are observations, where n>p, containing, for instance, log-returns. |
target_portfolio |
a vector which determines the weights of the target portfolio used when the shrinkage estimator of the global minimum variance portfolio is constructed for the first time. |
c |
a numeric which is the concentration ratio. |
Value
vector
References
Bodnar T, Parolya N, Thorsén E (2021). “Dynamic Shrinkage Estimation of the High-Dimensional Minimum-Variance Portfolio.” arXiv preprint arXiv:2106.02131. https://arxiv.org/abs/2106.02131.
Examples
n <- 200*2
p <- 80
data <- 5/3 * matrix(rt(n*p, df=5), ncol=p, nrow=n)
# set a target portfolio, such as equally weighted
b <- rep(1,p)/p
r0Strategy(data, b, p/n)
The recursive estimation for updating the relative loss in the variance of the holding portfolio.
Description
This function implements equation (2.19), the recursive estimation for updating the relative loss in the variance of the holding portfolio in the overlapping scheme.
Usage
rUpdateOverlapping(Psi, c, prev_R, K)
Arguments
Psi |
a vector of shrinkage coefficients. |
c |
a numeric between 0 and 1. It is the concentration ration. |
prev_R |
a numeric greater than 0, the previous value of the relative loss |
K |
a numeric parameter. |
Value
a number
Validates input to the DOSPortfolio function.
Description
This function validates the assumptions made to derive the analytic formulas implemented in the different functions of the package. It is called for its side-effects.
Usage
validate_input(
data,
reallocation_points,
target_portfolio,
relative_loss,
shrinkage_type
)
Arguments
data |
an n by p matrix of asset returns. Columns represent different assets rows are observations, where n>p, containing, for instance, log-returns. |
reallocation_points |
a vector of reallocation points. The reallocation points determine when the holding portfolio should be reconstructed and its weights should be recomputed. |
target_portfolio |
a vector which determines the weights of the target portfolio used when the shrinkage estimator of the global minimum variance portfolio is constructed for the first time. |
relative_loss |
possibly a numeric or NULL. The initial value of the
relative loss in the variance of the target portfolio. If its NULL, then it
will be initialized with the first subsample and the function
|
shrinkage_type |
the type of shrinkage estimator to use. The two implemented approaches are "non-overlapping" and "overlapping". |
Value
NULL, only called for its side effects
Sample estimator of the weights of the global minimum variance portfolio
Description
The functions computes the sample estimate of the weights of the global minimum variance portfolio (see, e.g., Eq. (1.4) of Bodnar et al. (2021))).
Usage
wGMV(data)
Arguments
data |
an n by p matrix of asset returns. Columns represent different assets rows are observations, where n>p, containing, for instance, log-returns. |
Value
a vector, which is the Global Minimum Variance Portfolio.
Examples
n <- 200
p <- 80
data <- 3/5 * matrix(rt(n*p, df=5), ncol=p, nrow=n)
weights <- wGMV(data)
# since the covariance matrix is the identity-matrix the estimated weights
# should be close to the equally weighted portfolio.
mean(abs(wGMV(data) - 1/p))
Dynamic optimal shrinkage estimator of the weights of the global minimum variance portfolio when non-overlapping samples are used.
Description
The function implements the dynamic shrinkage estimator of the weights of the global minimum-variance portfolio when the overlapping samples are used as given in Eq. (2.11) of Bodnar et al. (2021) .
Usage
wGMVNonOverlapping(data, reallocation_points, target_portfolio, relative_loss)
Arguments
data |
an n by p matrix of asset returns. Columns represent different assets rows are observations, where n>p, containing, for instance, log-returns. |
reallocation_points |
a vector of reallocation points. The reallocation points determine when the holding portfolio should be reconstructed and it is weights should be recomputed. |
target_portfolio |
a vector which determines the weights of the target portfolio used when the shrinkage estimator of the global minimum variance portfolio is constructed for the first time. |
relative_loss |
possibly a numeric or NULL. The initial value of the
relative loss in the variance of the target portfolio. If it is NULL, then it
will be initialized with the first subsample and the function
|
Value
a matrix of the constructed weights at each reallocation point of the dynamic shrinkage estimator of the global minimum variance portfolio when non-overlapping samples are used.
References
Bodnar T, Parolya N, Thorsén E (2021). “Dynamic Shrinkage Estimation of the High-Dimensional Minimum-Variance Portfolio.” arXiv preprint arXiv:2106.02131. https://arxiv.org/abs/2106.02131.
See Also
section 2.1 (Bodnar et al. 2021)
Examples
n <- 200*2
p <- 80
reallocation_point <- c(199)
data <- 3/5 * matrix(rt(n*p, df=5), ncol=p, nrow=n)
target_portfolio <- as.vector(rep(1,p))/p
wGMVNonOverlapping(data, reallocation_point, target_portfolio, 1)
Dynamic optimal shrinkage estimator of the weights of the global minimum variance portfolio when overlapping samples are used.
Description
The function implements the dynamic shrinkage estimator of the weights of the global minimum-variance portfolio when the overlapping samples are used as given in Eq. (2.23) of Bodnar et al. (2021).
Usage
wGMVOverlapping(data, reallocation_points, target_portfolio, relative_loss)
Arguments
data |
an n by p matrix of asset returns. Columns represent different assets rows are observations, where n>p, containing, for instance, log-returns. |
reallocation_points |
a vector of reallocation points. The reallocation points determine when the holding portfolio should be reconstructed and it is weights should be recomputed. |
target_portfolio |
a vector which determines the weights of the target portfolio used when the shrinkage estimator of the global minimum variance portfolio is constructed for the first time. |
relative_loss |
possibly a numeric or NULL. The initial value of the
relative loss in the variance of the target portfolio. If it is NULL, then it
will be initialized with the first subsample and the function
|
Value
a matrix of the constructed weights at each reallocation point of the dynamic shrinkage estimator of the global minimum variance portfolio when overlapping samples are used.
References
Bodnar T, Parolya N, Thorsén E (2021). “Dynamic Shrinkage Estimation of the High-Dimensional Minimum-Variance Portfolio.” arXiv preprint arXiv:2106.02131. https://arxiv.org/abs/2106.02131.
See Also
Examples
n <- 200*2
p <- 80
reallocation_points <- c(199)
data <- matrix(rt(n*p, df=5), ncol=p, nrow=n)
target_portfolio <- as.vector(rep(1,p))/p
wGMVOverlapping(data, reallocation_points, target_portfolio, 1)