RiskPortfolios (version 2.1.2)

optimalPortfolio: Optimal portfolio

Description

Function wich computes the optimal portfolio's weights.

Usage

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

Arguments

Sigma

a \((N \times N)\) covariance matrix.

mu

a \((N \times 1)\) vector of expected returns. Default: mu = NULL.

semiDev

a vector \((N \times 1)\) of semideviations. Default: semiDev = NULL.

control

control parameters (see *Details*).

Value

A \((N \times 1)\) vector of optimal portfolio weights.

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', 'invvol', 'erc', 'maxdiv', 'riskeff' and 'maxdec' where:

  'mv' is used to compute the weights of the mean-variance portfolio. The weights are computed following this equation: $$w = \frac{1}{\gamma} \Sigma^{-1} \mu $$.

  'minvol' is used to compute the weights of the minimum variance portfolio.

  'invvol' is the inverse volatility 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: $$\% RC_i = \frac{ w_i {[\Sigma w]}_i}{w' \Sigma w} $$. Then we compute the optimal portfolio by solving the following optimization problem: $$w = argmin \left\{ \sum_{i=1}^N (\% RC_i - \frac{1}{N})^2 \right\} $$.

  'maxdiv' is used to compute the weights of the maximum diversification portfolio where: $$DR(w) = \frac{ w' \sigma}{\sqrt{w' \Sigma w} } \geq 1 $$ is used in the optimization problem.

  'riskeff' is used to compute the weights of the risk-efficient portfolio: $$w = {argmax}\left\{ \frac{w' J \xi}{ \sqrt{w' \Sigma w} }\right\} $$ where \(J\) is a \((N \times 10)\) matrix of zeros whose \((i,j)\)-th element is one if the semi-deviation of stock \(i\) belongs to decile \(j\),\(\xi = (\xi_1,\ldots,\xi_{10})'\).

  'maxdec' is used to compute the weights of the maximum-decorrelation portfolio: $$w = {argmax}\left\{ 1 - \sqrt{w' \Sigma w} \right\} $$ where \(R\) is the correlation matrix.

  Default: type = 'mv'.

  These portfolios are summarized in Ardia and Boudt (2015) and Ardia et al. (2017). Below we list the various references.

• constraint constraint used for the optimization, among 'none', 'lo', 'gross' and 'user', where: 'none' is used to compute the unconstraint portfolio, 'lo' is the long-only constraints (non-negative weighted), 'gross' is the gross exposure constraint, and 'user' is the set of user constraints (typically lower and upper boundaries. Default: constraint = 'none'. Note that the summability constraint is always imposed.

• LB lower boundary for the weights. Default: LB = NULL.

• UB lower boundary for the weights. Default: UB = NULL.

• w0 starting value for the optimizer. Default: w0 = NULL takes the equally-weighted portfolio as a starting value. When LB and UB are provided, it is set to mid-point of the bounds.

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

• gamma risk aversion parameter. Default: gamma = 0.89.

• ctr.slsqp list with control parameters for slsqp function.

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. (2015). Implied expected returns and the choice of a mean-variance efficient portfolio proxy. Journal of Portfolio Management 41(4), pp.66-81. 10.3905/jpm.2015.41.4.068

Ardia, D., Bolliger, G., Boudt, K., Gagnon-Fleury, J.-P. (2017). The Impact of covariance misspecification in risk-based portfolios. Annals of Operations Research 254(1-2), pp.1-16. 10.1007/s10479-017-2474-7

Choueifaty, Y., Coignard, Y. (2008). Toward maximum diversification. Journal of Portfolio Management 35(1), pp.40-51. 10.3905/JPM.2008.35.1.40

Choueifaty, Y., Froidure, T., Reynier, J. (2013). Properties of the most diversified portfolio. Journal of Investment Strategies 2(2), pp.49-70. 10.21314/JOIS.2013.033

Das, S., Markowitz, H., Scheid, J., Statman, M. (2010). Portfolio optimization with mental accounts. Journal of Financial and Quantitative Analysis 45(2), pp.311-334. 10.1017/S0022109010000141

DeMiguel, V., Garlappi, L., Uppal, R. (2009). Optimal versus naive diversification: How inefficient is the 1/n portfolio strategy. Review of Financial Studies 22(5), pp.1915-1953. 10.1093/rfs/hhm075

Fan, J., Zhang, J., Yu, K. (2012). Vast portfolio selection with gross-exposure constraints. Journal of the American Statistical Association 107(498), pp.592-606. 10.1080/01621459.2012.68282

Maillard, S., Roncalli, T., Teiletche, J. (2010). The properties of equally weighted risk contribution portfolios. Journal of Portfolio Management 36(4), pp.60-70. 10.3905/jpm.2010.36.4.060

Martellini, L. (2008). Towards the design of better equity benchmarks. Journal of Portfolio Management 34(4), Summer,pp.34-41. 10.3905/jpm.2008.709978

Examples

# NOT RUN {
# Load returns of assets or portfolios
data("Industry_10")
rets = Industry_10

# Mean estimation
mu = meanEstimation(rets)

# Covariance estimation
Sigma = covEstimation(rets)

# Semi-deviation estimation
semiDev = semidevEstimation(rets)

# Mean-variance portfolio without constraint and gamma = 0.89
optimalPortfolio(mu = mu, Sigma = Sigma)

# Mean-variance portfolio without constraint and gamma = 1
optimalPortfolio(mu = mu, Sigma = Sigma, 
  control = list(gamma = 1))

# Mean-variance portfolio without constraint and gamma = 0.89
optimalPortfolio(mu = mu, Sigma = Sigma, 
  control = list(type = 'mv'))

# Mean-variance portfolio without constraint and gamma = 0.89
optimalPortfolio(mu = mu, Sigma = Sigma, 
  control = list(type = 'mv', constraint = 'none'))

# Mean-variance portfolio with the long-only constraint and gamma = 0.89
optimalPortfolio(mu = mu, Sigma = Sigma, 
  control = list(type = 'mv', constraint = 'lo'))

# Mean-variance portfolio with LB and UB constraints
optimalPortfolio(mu = mu, Sigma = Sigma, 
  control = list(type = 'mv', constraint = 'user', LB = rep(0.02, 10), UB = rep(0.8, 10)))

# 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'))

# 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))

# Minimum volatility portfolio without constraint
optimalPortfolio(Sigma = Sigma, 
  control = list(type = 'minvol'))

# Minimum volatility portfolio without constraint
optimalPortfolio(Sigma = Sigma, 
  control = list(type = 'minvol', constraint = 'none'))

# Minimim volatility portfolio with the long-only constraint
optimalPortfolio(Sigma = Sigma, 
  control = list(type = 'minvol', constraint = 'lo'))
  
# Minimim volatility portfolio with LB and UB constraints
optimalPortfolio(Sigma = Sigma, 
  control = list(type = 'minvol', constraint = 'user', LB = rep(0.02, 10), UB = rep(0.8, 10)))

# Minimum volatility portfolio with the gross constraint 
# and the gross constraint parameter = 1.6
optimalPortfolio(Sigma = Sigma, 
  control = list(type = 'minvol', constraint = 'gross'))

# 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))
    
# Inverse volatility portfolio
optimalPortfolio(Sigma = Sigma, 
  control = list(type = 'invvol'))

# Equal-risk-contribution portfolio with the long-only constraint
optimalPortfolio(Sigma = Sigma, 
  control = list(type = 'erc', constraint = 'lo'))
  
# Equal-risk-contribution portfolio with LB and UB constraints
optimalPortfolio(Sigma = Sigma, 
  control = list(type = 'erc', constraint = 'user', LB = rep(0.02, 10), UB = rep(0.8, 10)))

# Maximum diversification portfolio without constraint
optimalPortfolio(Sigma = Sigma, 
  control = list(type = 'maxdiv'))

# Maximum diversification portoflio with the long-only constraint
optimalPortfolio(Sigma = Sigma, 
  control = list(type = 'maxdiv', constraint = 'lo'))
  
# Maximum diversification portoflio with LB and UB constraints
optimalPortfolio(Sigma = Sigma, 
  control = list(type = 'maxdiv', constraint = 'user', LB = rep(0.02, 10), UB = rep(0.8, 10)))

# Risk-efficient portfolio without constraint
optimalPortfolio(Sigma = Sigma, semiDev = semiDev, 
  control = list(type = 'riskeff'))

# Risk-efficient portfolio with the long-only constraint
optimalPortfolio(Sigma = Sigma, semiDev = semiDev, 
  control = list(type = 'riskeff', constraint = 'lo'))
  
# Risk-efficient portfolio with LB and UB constraints
optimalPortfolio(Sigma = Sigma, semiDev = semiDev, 
  control = list(type = 'riskeff', constraint = 'user', LB = rep(0.02, 10), UB = rep(0.8, 10)))
  
# Maximum decorrelation portfolio without constraint
optimalPortfolio(Sigma = Sigma, 
  control = list(type = 'maxdec'))

# Maximum decorrelation portoflio with the long-only constraint
optimalPortfolio(Sigma = Sigma, 
  control = list(type = 'maxdec', constraint = 'lo'))
  
# Maximum decorrelation portoflio with LB and UB constraints
optimalPortfolio(Sigma = Sigma, 
  control = list(type = 'maxdec', constraint = 'user', LB = rep(0.02, 10), UB = rep(0.8, 10)))
# }