The Euler theorem has found extensive applications in finance, particularly for decomposing homogeneous risk measures of degree one. However, a limitation of this approach is that it doesnâ€™t adequately isolate the true sources of risk within an investment portfolio, thus creating a gap in risk analysis.

The Minimum Torsion Bets (MTB) method offers a compelling solution to this challenge. Leveraging spectral decomposition, MTB selectively identifies uncorrelated factors that are as close as possible to the original variables among all possible matrix rotations. This approach ensures a refined understanding of the underlying risk structure.

The results of the MTB method yield a diversification distribution with several noteworthy properties:

- It is always positive, reflecting a constructive aspect of risk management;
- The distribution sums to \(1\), maintaining consistency with probabilistic principles;
- It accurately captures the true sources of risk, enabling a more precise and targeted risk management strategy;
- The distribution offers insightful interpretations, adding analytical value to the decision-making process.

In essence, the Effective Number of Minimum Torsion Bets (ENMTB) extends traditional risk measures by providing a nuanced view of risk diversification, making it a valuable tool for financial professionals seeking to optimize portfolio performance.

```
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 diversification distribution and the 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 7.6046588
```

Install 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.