# DivRatios: Diversification Measures In FRAPO: Financial Risk Modelling and Portfolio Optimisation with R

## Description

These functions compute the diversification ratio, the volatility weighted average correlation and concentration ratio of a portfolio.

## Usage

 1 2 3 dr(weights, Sigma) cr(weights, Sigma) rhow(weights, Sigma) 

## Arguments

 weights Vector: portfolio weights. Sigma Matrix: Variance-covariance matrix of portfolio assets.

## Details

The diversification ratio of a portfolio is defined as:

DR(ω) = \frac{∑_{i = 1}^N ω_i σ_i}{√{ω' Σ ω}}

for a portfolio of N assets and ω_i signify the weight of the i-th asset and σ_i its standard deviation and Σ the variance-covariance matrix of asset returns. The diversification ratio is therefore the weighted average of the assets' volatilities divided by the portfolio volatility.

The concentration ration is defined as:

CR = \frac{∑_{i = 1}^N (ω_i σ_i)^2}{(∑_{i = 1}^N ω_i σ_i)^2}

and the volatility-weighted average correlation of the assets as:

ρ(ω) = \frac{∑_{i > j}^N (ω_i σ_i ω_j σ_j)ρ_{ij}}{∑_{i > j}^N (ω_i σ_i ω_j σ_j)}

The following equation between these measures does exist:

DR(ω) = \frac{1}{√{ρ(ω) (1 - CR(ω)) + CR(ω)}}

## Value

numeric, the value of the diversification measure.

Bernhard Pfaff

## References

Choueifaty, Y. and Coignard, Y. (2008): Toward Maximum Diversification, Journal of Portfolio Management, Vol. 34, No. 4, 40–51.

Choueifaty, Y. and Coignard, Y. and Reynier, J. (2011): Properties of the Most Diversified Portfolio, Working Paper, http://papers.ssrn.com

PMD

## Examples

 1 2 3 4 5 6 7 8 9 data(MultiAsset) Rets <- returnseries(MultiAsset, method = "discrete", trim = TRUE) w <- Weights(PMD(Rets)) V <- cov(Rets) DR <- dr(w, V) CR <- cr(w, V) RhoW <- rhow(w, V) test <- 1 / sqrt(RhoW * (1 - CR) + CR) all.equal(DR, test) 

### Example output

Loading required package: cccp
Using the GLPK callable library version 4.52
Financial Risk Modelling and Portfolio Optimisation with R (version 0.4-1)

Iteration: 0
pobj: 0.11477
dobj: -0.961072
pinf: 0
dinf: 3.76706
dgap: 1.07584

Iteration: 1
pobj: 0.113632
dobj: 0.00263219
pinf: 2.22045e-16
dinf: 0.387308
dgap: 0.111

Iteration: 2
pobj: 0.110322
dobj: 0.0805013
pinf: 2.22045e-16
dinf: 0.095544
dgap: 0.0298208

Iteration: 3
pobj: 0.107587
dobj: 0.0965035
pinf: 4.44089e-16
dinf: 0.0295032
dgap: 0.0110831

Iteration: 4
pobj: 0.105282
dobj: 0.102542
pinf: 8.88178e-16
dinf: 0.00180192
dgap: 0.00273982

Iteration: 5
pobj: 0.10454
dobj: 0.103948
pinf: 1.55431e-15
dinf: 9.80713e-05
dgap: 0.000592444

Iteration: 6
pobj: 0.104348
dobj: 0.10421
pinf: 3.10862e-15
dinf: 8.12006e-06
dgap: 0.000138232

Iteration: 7
pobj: 0.104303
dobj: 0.104281
pinf: 6.21725e-15
dinf: 3.19953e-07
dgap: 2.13694e-05

Iteration: 8
pobj: 0.104297
dobj: 0.104295
pinf: 1.17684e-14
dinf: 2.46719e-08
dgap: 2.67608e-06

Iteration: 9
pobj: 0.104297
dobj: 0.104296
pinf: 2.10942e-14
dinf: 5.00253e-09
dgap: 6.11958e-07

Iteration: 10
pobj: 0.104297
dobj: 0.104297
pinf: 3.93019e-14
dinf: 6.97906e-10
dgap: 1.06644e-07

Iteration: 11
pobj: 0.104297
dobj: 0.104297
pinf: 7.54952e-14
dinf: 5.72651e-11
dgap: 1.18903e-08

Optimal solution found.
 TRUE


FRAPO documentation built on May 2, 2019, 6:33 a.m.