optimalPortfolio {RiskPortfolios} | R Documentation |
Optimal portfolio
Description
Function wich computes the optimal portfolio's weights.
Usage
optimalPortfolio(Sigma, mu = NULL, semiDev = NULL, control = list())
Arguments
Sigma |
a |
mu |
a |
semiDev |
a vector |
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'
,'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 portfoliow
, 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 stocki
belongs to decilej
,\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. WhenLB
andUB
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.
Value
A (N \times 1)
vector of optimal portfolio weights.
Author(s)
David Ardia, Kris Boudt 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. (2015). Implied expected returns and the choice of a mean-variance efficient portfolio proxy. Journal of Portfolio Management 41(4), pp.66-81. doi: 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. doi: 10.1007/s10479-017-2474-7
Choueifaty, Y., Coignard, Y. (2008). Toward maximum diversification. Journal of Portfolio Management 35(1), pp.40-51.
Choueifaty, Y., Froidure, T., Reynier, J. (2013). Properties of the most diversified portfolio. Journal of Investment Strategies 2(2), pp.49-70.
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.
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.
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.
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(4), Summer,pp.34-41.
Examples
# 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)))