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Enrico Schumann
Enrico Schumann
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Additional note: The example covariance-matrix hides some fairly-strong dependencies between the asset returns, which may not be obvious from looking at the data:

library("NMOF")
R <- randomReturns(na = 4, ns = 100,
                   sd = sqrt(diag(S)),
                   rho = cov2cor(S), exact = TRUE)
pairs(R)

scatterplot of correlated returns

Computations based on such a matrix (such as a marginal-risk calculation) will typically be sensitive and react strongly to small changes ("perturbations") of inputs. In the example, you can get around this by increasing the iterations:

w <- riskParityPortfolio(S)$w
FRAPO::mrc(w, S)
## [1] -194.71   -8.58  151.97  151.32

w <- riskParityPortfolio(S, maxiter = 1e4)$w
FRAPO::mrc(w, S)
## [1] -67.3  11.2  78.2  77.9

w <- riskParityPortfolio(S, maxiter = 1e5)$w
FRAPO::mrc(w, S)
## [1] 23.1 24.7 26.1 26.1

w <- riskParityPortfolio(S, maxiter = 1e6)$w
FRAPO::mrc(w, S)
## [1] 25 25 25 25

(I use the marginal-risk function mrc from Bernhard Pfaff's FRAPO package.) But of course, this is all the result of an empirical problem (not a computational one): the assets are highly-correlated, and so an algorithm will always have trouble differentiating between linear combinations of those assets.

Additional note: The example covariance-matrix hides some fairly-strong dependencies between the asset returns, which may not be obvious from looking at the data:

library("NMOF")
R <- randomReturns(na = 4, ns = 100,
                   sd = sqrt(diag(S)),
                   rho = cov2cor(S), exact = TRUE)
pairs(R)

scatterplot of correlated returns

Computations based on such a matrix (such as a marginal-risk calculation) will typically be sensitive and react strongly to small changes ("perturbations") of inputs. In the example, you can get around this by increasing the iterations:

w <- riskParityPortfolio(S)$w
FRAPO::mrc(w, S)
## [1] -194.71   -8.58  151.97  151.32

w <- riskParityPortfolio(S, maxiter = 1e4)$w
FRAPO::mrc(w, S)
## [1] -67.3  11.2  78.2  77.9

w <- riskParityPortfolio(S, maxiter = 1e5)$w
FRAPO::mrc(w, S)
## [1] 23.1 24.7 26.1 26.1

w <- riskParityPortfolio(S, maxiter = 1e6)$w
FRAPO::mrc(w, S)
## [1] 25 25 25 25

(I use the marginal-risk function mrc from Bernhard Pfaff's FRAPO package.) But of course, this is all the result of an empirical problem (not a computational one): the assets are highly-correlated, and so an algorithm will always have trouble differentiating between linear combinations of those assets.

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Enrico Schumann
Enrico Schumann
  • 3.8k
  • 12
  • 8

Many risk-parity implementations simply use the inverse-vol rule (i.e. weights are proportional to 1 over vol), and then all weights are (strictly) positive by construction.

A number of implementations that do the full optimization indeed set non-negativity as an explicit constraint. But in general and without such constraints, negative weights may occur when you equalize risk-contributions (though negative weights may be unlikely).

An example in R for a 4-by-4 covariance matrix, which is of rank 4 and positive definite:

S <- structure(c(0.000366309632978563, -0.000105943107569848,
                 -0.000119667135542588, 0.000169622848883779,
                 -0.000105943107569848, 0.000187730511335559,
                 -6.49497066288097e-05, -0.000167162505503921,
                 -0.000119667135542588, -6.49497066288097e-05,
                 0.000102201079006482, 1.88829418811814e-05,
                 0.000169622848883779, -0.000167162505503921,
                 1.88829418811814e-05, 0.000208993337170307),
               dim = c(4L, 4L))

The implied correlation matrix:

cov2cor(S)
##        [,1]   [,2]   [,3]   [,4]
## [1,]  1.000 -0.404 -0.618  0.613
## [2,] -0.404  1.000 -0.469 -0.844
## [3,] -0.618 -0.469  1.000  0.129
## [4,]  0.613 -0.844  0.129  1.000

I do not use the riskParityPortfolio package, but it seems to support negative weights, though the default is to not allow them:

library("riskParityPortfolio")
riskParityPortfolio(S)
## $w
## [1] 0.2313 0.2990 0.4588 0.0109
## 
## $relative_risk_contribution
## [1] -1.9471 -0.0858  1.5197  1.5132
## ....


riskParityPortfolio(S, w_lb = -1)
## $risk_concentration
## [1] 1.93e-14
## 
## $w
## [1] -0.0829  0.4391  0.2167  0.4271
## 
## $relative_risk_contribution
## [1] 0.25 0.25 0.25 0.25
## ....