The Euler theorem has been widely used in finance as a way to decompose homogeneous risk measures of degree one. Unfortunately, this decomposition does not isolate the true sources of risk.
The Minimum Torsion Bets (MTB) offers a solution to this problem: it uses the spectral decomposition to “pick” the uncorrelated factors that are as close as possible to the original variables, among all living matrix rotations.
The output is a diversification distribution with the following properties: it’s always positive, sums to 1, capture the true sources risk and have an insightful interpretation. The above characteristics put the Effective Number of Minimum Torsion Bets as a generalization of the Marginal Contribution to Risk (MCR).
library(uncorbets)
# prepare data
returns <- diff(log(EuStockMarkets))
covariance <- cov(returns)
# Minimum Torsion Matrix
torsion_mat <- torsion(covariance)
# Prior Allocation (equal weights, for example)
w <- rep(1 / ncol(returns), ncol(returns))
# Compute allocation and diversification level
effective_bets(b = w, sigma = covariance, t = torsion_mat)
#> $p
#> [,1]
#> DAX 0.2673005
#> SMI 0.2404370
#> CAC 0.2776369
#> FTSE 0.2146256
#>
#> $enb
#> [1] 3.980549
# maximize the effective number of bets (enb)
max_effective_bets(x0 = w, sigma = covariance, t = torsion_mat)
#> $weights
#> [1] 0.2227163 0.2603372 0.2114589 0.3054876
#>
#> $enb
#> [1] 4
#>
#> $counts
#> nfval ngval
#> [1,] 47 9
#>
#> $lambda_lb
#> [,1]
#> DAX 0
#> SMI 0
#> CAC 0
#> FTSE 0
#>
#> $lambda_ub
#> [,1]
#> DAX 0
#> SMI 0
#> CAC 0
#> FTSE 0
#>
#> $lambda_eq
#> [1] 1.162481e-06
#>
#> $gradient
#> [,1]
#> DAX 1.966953e-06
#> SMI -4.768372e-06
#> CAC 8.940697e-06
#> FTSE -3.337860e-06
#>
#> $hessian
#> DAX SMI CAC FTSE
#> DAX 5.3149468 -1.4603802 -1.4893686 -0.5169268
#> SMI -1.4603802 5.2212615 -3.6528175 -0.9075766
#> CAC -1.4893686 -3.6528175 6.3451210 -0.5218244
#> FTSE -0.5169268 -0.9075766 -0.5218244 7.6046588Install the released version from CRAN with:
Install the development version of uncorbets from github with:
Meucci, Attilio and Santangelo, Alberto and Deguest, Romain, Risk Budgeting and Diversification Based on Optimized Uncorrelated Factors (November 10, 2015). Available at SSRN: https://www.ssrn.com/abstract=2276632 or http://dx.doi.org/10.2139/ssrn.2276632
Attilio Meucci (2021). Portfolio Diversification Based on Optimized Uncorrelated Factors (https://www.mathworks.com/matlabcentral/fileexchange/43245-portfolio-diversi-cation-based-on-optimized-uncorrelated-factors), MATLAB Central File Exchange. Retrieved July 8, 2021.