# optimalPortfolio: Optimital portfolio In ExpectedReturns: Computation of implied expected returns

## Description

Function wich computes the portfolio's weights

## Usage

 1 optimalPortfolio(Sigma, mu = NULL, semiDev = NULL, control = list())

## Arguments

 Sigma a (N x N) covariance matrix. mu a (N x 1) vector of expected returns. Default: mu = NULL. semiDev a vector (N x 1) of semideviation. Default: semiDev = NULL. control control parameters (see *Details*).

## Details

The argument control is a list that can supply any of the following components:

type

method used to compute the optimal portfolio, among 'mv', 'minvol', 'erc', 'maxdiv' and 'riskeff' where: 'mv' is used to compute the weights of the mean-variance portfolio. The weights are computed following this equation:

w = 1 / γ Σ^{-1} μ

'minvol' is used to compute the weights of the minimum variance portfolio. 'erc' is used to compute the weights of the equal-risk-contribution portfolio. For a portfolio w, the percentage volatility risk contribution of the i-th asset in the portfolio is given by:

% RC_i = w_i[Σ w]_i / (w' Σ w)

Then we compute the optimal portfolio by solving the following optimization problem:

argmin { ∑_{i=1}^{N} (% RC_i - 1/N)^2}

'maxdiv' is used to compute the weights of the maximum diversification portfolio where:

DR(w) = (w' σ)/(√(w' Σ w)) ≥ 1

is used in the optimization problem. 'riskeff' is used to compute the weights of the risk-efficient portfolio:

w = argmax ( w'J ξ)√(w'Σ w)

where J is a (N x 10) matrix of zeros whose (i,j)-th element is one if the semi-deviation of stock i belongs to decile j,ξ = (ξ_1,…,ξ_{10})'. Default: type = 'mv'

constraint

constraint used for the optimization, among 'none', 'lo' and 'gross' where :

'none' is used to compute the unconstraint portfolio

'lo' is the long-only constraint.

'gross' is the gross constraint.

Default: constraint = 'none'.

gross.c

Gross exposure constraint. Default: gross.c = 1.6.

gamma

Risk aversion parameter. Default: gamma = 0.89.

## Value

A (N x 1) vector of optimal portfolio weights.

## Author(s)

David Ardia <[email protected]> and Jean-Philippe Gagnon Fleury.

## References

Amenc, N., Goltz, F., Martellini, L., Retowsky, P. (2011). Efficient indexation: An alternatice to cap-weightes indices. Journal of Investment Management 9(4), pp.1–23.

Ardia, D., Boudt, K. (2013). Implied expected returns and the choice of a mean-variance efficient portfolio proxy. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2215042.

Choueifaty, Y., Coignard, Y., (2008). Toward maximum diversification. Journal of Portfolio Management 35 (1), pp.40–51.

Choueifaty, Y., Froidure, T., Reynier, J., (2011). Properties of the most diversified portfolio. Working paper.

Das, S., Markowitz, H., Scheid, J., Statman, M., (2010). Portfolio optimization with mental accounts. Journal of Financial and Quantitative Analysis 45, pp.311–334.

DeMiguel, V., Garlappi, L., Uppal, R., (2009). Optimal versus naive diversification: How inefficient is the 1/n portfolio strategy. The Review of Financial Studies 22(5), pp.1915–1953.

Fan, J., Zhang, J., Yu, K., March (2009). Asset allocation and risk assessment with gross exposure constraints for vast portfolios.

Maillard, S., Roncalli, T., Teiletche, J., (2010). The properties of equally weighted risk contribution portfolios. Journal of Portfolio Management 36(4), pp.60–70.

Martellini, L., (2008). Towards the design of better equity benchmarks. Journal of Portfolio Management 34, Summer,pp.34–41.

## Examples

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 # For the examples, we simply generate a 100 x 25 random matrix. set.seed(3214) T = 100 N = 25 rets = matrix(rnorm(T*N), nrow = T, ncol = N) mu = meanEstimation(rets) Sigma = covEstimation(rets) semiDev = semidevEstimation(rets) #Computes the mean-variance portfolio without constraint and gamma = 0.89. optimalPortfolio(mu = mu, Sigma = Sigma) #Computes the mean-variance portfolio without constraint and gamma = 1. optimalPortfolio(mu = mu, Sigma = Sigma, control = list(gamma = 1)) #Computes the mean-variance portfolio with without and gamma = 0.89. optimalPortfolio(mu = mu, Sigma = Sigma, control = list(type = "mv")) #Computes the mean-variance portfolio without constraint and gamma = 0.89. optimalPortfolio(mu = mu, Sigma = Sigma, control = list(type = "mv", constraint = "none")) #Computes the mean-variance portfolio with the long only constraint and gamma = 0.89. optimalPortfolio(mu = mu, Sigma = Sigma, control = list(type = "mv", constraint = "lo")) #Computes the mean-variance portfolio with the gross constraint, gross constraint parameter = 1.6 and gamma = 0.89. optimalPortfolio(mu = mu, Sigma = Sigma, control = list(type = "mv", constraint = "gross")) #Computes the mean-variance portfolio with the gross constraint, gross constraint parameter = 1.2 and gamma = 0.89. optimalPortfolio(mu = mu, Sigma = Sigma, control = list(type = "mv", constraint = "gross", gross.c = 1.2)) #Computes the minimum volatility portfolio without constraint. optimalPortfolio(Sigma = Sigma, control = list(type = "minvol")) #Computes the minimum volatility portfolio without constraint. optimalPortfolio(Sigma = Sigma, control = list(type = "minvol", constraint = "none")) #Computes the minimim volatility portfolio with the long only constraint. optimalPortfolio(Sigma = Sigma, control = list(type = "minvol", constraint = "lo")) #Computes the minimum volatility portfolio with the gross constraint and the gross constraint parameter = 1.6. optimalPortfolio(Sigma = Sigma, control = list(type = "minvol", constraint = "gross")) #Computes the minimum volatility portfolio with the gross constraint and the gross parameter = 1.2. optimalPortfolio(Sigma = Sigma, control = list(type = "minvol", constraint = "gross", gross.c = 1.2)) #Computes the equal-risk-contribution portfolio without constraint. optimalPortfolio(Sigma = Sigma, control = list(type = "erc")) #Computes the equal-risk-contribution portfolio without constraint. optimalPortfolio(Sigma = Sigma, control = list(type = "erc", constraint = "none")) #Computes the equal-risk-contribution portfolio with the long only constraint. optimalPortfolio(Sigma = Sigma, control = list(type = "erc", constraint = "lo")) #Computes the equal-risk-contribution portfolio with the gross constraint and the gross parameter = 1.6. optimalPortfolio(Sigma = Sigma, control = list(type = "erc", constraint = "gross")) #Computes the equal-risk-contribution portfolio with the gross constraint and the gross parameter = 1.2. optimalPortfolio(Sigma = Sigma, control = list(type = "erc", constraint = "gross", gross.c = 1.2)) #Computes the maximum diversification portfolio without constraint. optimalPortfolio(Sigma = Sigma, control = list(type = "maxdiv")) #Computes the maximum diversification portfolio without constraint. optimalPortfolio(Sigma = Sigma, control = list(type = "maxdiv", constraint = "none")) #Compute the maximum diversification portoflio with the long only constraint. optimalPortfolio(Sigma = Sigma, control = list(type = "maxdiv", constraint = "lo")) #Computes the maximum diversification portfolio with the gross constraint and the gross parameter = 1.6. optimalPortfolio(Sigma = Sigma, control = list(type = "maxdiv", constraint = "gross")) #Computes the maximum diversification portfolio with the gross constraint and the gross parameter = 1.2. optimalPortfolio(Sigma = Sigma, control = list(type = "maxdiv", constraint = "gross", gross.c = 1.2)) #Computes the risk-efficient portfolio without constraint. optimalPortfolio(Sigma = Sigma, semiDev = semiDev, control = list(type = "riskeff")) #Computes the risk-efficient portfolio without constraint. optimalPortfolio(Sigma = Sigma, semiDev = semiDev, control = list(type = "riskeff", constraint = "none")) #Computes the risk-efficient portfolio with the long only constraint. optimalPortfolio(Sigma = Sigma, semiDev = semiDev, control = list(type = "riskeff", constraint = "lo")) #Computes the risk-efficient portfolio with the gross constraint and the gross parameter = 1.6. optimalPortfolio(Sigma = Sigma, semiDev = semiDev, control = list(type = "riskeff", constraint = "gross")) #Computes the risk-efficient portfolio with the gross constraint and the gross parameter = 1.2. optimalPortfolio(Sigma = Sigma, semiDev = semiDev, control = list(type = "riskeff", constraint = "gross", gross.c = 1.2))

ExpectedReturns documentation built on May 31, 2017, 4:03 a.m.