Package 'RiskPortfolios'

Covariance matrix estimation

Description

Function which performs various estimations of covariance matrices.

Usage

covEstimation(rets, control = list())

Arguments

rets	a matrix $(T \times N)$ of returns.
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 covariance matrix, among 'naive', 'ewma', 'lw', 'factor','const', 'cor', 'oneparm', 'diag', 'large' and 'bs' where:

  'naive' is used to compute the naive (standard) covariance matrix.

  'ewma' is used to compute the exponential weighting moving average covariance matrix. The following formula is used to compute the ewma covariance matrix:

  $\Sigma := \frac{1-\lambda}{1-\lambda^T} \sum_{t=1}^{T} \lambda^{T-t} (r_t - \bar{r})(r_t - \bar{r})'$

  where $r_t$ is the $(N \times 1)$ vector of returns at time $t$ and $\bar{r}$ the sample mean. The exponential weights are normalized by $1-\lambda^T$ so that they sum to one over the $T$ available observations; the returns are centered. It is therefore the finite-sample, mean-corrected analogue of the RiskMetrics recursion $\Sigma_t := \lambda \Sigma_{t-1} + (1-\lambda) r_t r_t'$ started at zero, whose weights sum to $1-\lambda^T$ rather than to one. Note that the data must be sorted from the oldest to the latest. See RiskMetrics (1996)

  'factor' is used to compute the covariance matrix estimation using a K-factor approach. See Harman (1976).

  'lw' is a weighted average of the sample covariance matrix and a 'prior' or 'shrinkage target'. The prior is given by a one-factor model and the factor is equal to the cross-sectional average of all the random variables. See Ledoit and Wolf (2003).

  'const' is a weighted average of the sample covariance matrix and a 'prior' or 'shrinkage target'. The prior is given by constant correlation matrix. See Ledoit and Wolf (2002).

  'cor' is a weighted average of the sample covariance matrix and a 'prior' or 'shrinkage target'. The prior is given by the constant correlation covariance matrix given by Ledoit and Wolf (2003).

  'oneparm' is a weighted average of the sample covariance matrix and a 'prior' or 'shrinkage target'. The prior is given by the one-parameter matrix. All variances are the same and all covariances are zero. See Ledoit and Wolf (2004).

  'diag' is a weighted average of the sample covariance matrix and a 'prior' or 'shrinkage target'. The prior is given by a diagonal matrix. See Ledoit and Wolf (2002).

  'large' is an alias of 'lw', kept for backward compatibility. The shrinkage intensity of the market-prior estimator is derived so as to minimize the quadratic loss measured by the Frobenius norm, and is valid as the number of variables and/or the number of observations go to infinity; Monte-Carlo simulations show that it works well for values as low as 10. The shrinkage generally improves the conditioning of the estimate when variables outnumber observations, but it does not guarantee invertibility: the estimated intensity is truncated to $[0, 1]$ and can be zero, in which case the sample covariance matrix is returned unchanged and is singular whenever $T \le N$. The same caveat applies to 'cor', 'diag' and 'oneparm'. See Ledoit and Wolf (2003, 2004).

  'bs' is the Bayes-Stein estimator for the covariance matrix given by Jorion (1986).

  Default: type = 'naive'.

• lambda decay parameter, a single number in $(0, 1)$. It is excluded at one because the normalization by $1-\lambda^T$ would divide by zero. Default: lambda = 0.94.

• K number of factors to use when the K-factor approach is chosen to estimate the covariance matrix. Default: K = 1.

Value

A $(N \times N)$ covariance matrix.

Note

Part of the code is adapted from the Matlab code by Ledoit and Wolf (2014).

Author(s)

David Ardia, Kris Boudt and Jean-Philippe Gagnon Fleury.

References

Jorion, P. (1986). Bayes-Stein estimation for portfolio analysis. Journal of Financial and Quantitative Analysis 21(3), pp.279-292.

Harman, H.H. (1976) Modern Factor Analysis. 3rd Ed. Chicago: University of Chicago Press.

Ledoit, O., Wolf, M. (2002). Improved estimation of the covariance matrix of stock returns with an application to portfolio selection. Journal of Empirical Finance 10(5), pp.603-621.

Ledoit, O., Wolf, M. (2003). Honey, I Shrunk the Sample Covariance Matrix. Journal of Portfolio Management 30(4), pp.110-119.

Ledoit, O., Wolf, M. (2004). A well-conditioned estimator for large-dimensional covariance matrices. Journal of Multivariate Analysis 88(2), pp.365-411.

RiskMetrics (1996) RiskMetrics Technical Document. J. P. Morgan/Reuters.

Examples

# Load returns of assets or portfolios
data("Industry_10")
rets = Industry_10

# Naive covariance estimation
covEstimation(rets)

# Ewma estimation of the covariance with default lambda = 0.94
covEstimation(rets, control = list(type = 'ewma'))

# Ewma estimation of the covariance with default lambda = 0.90
covEstimation(rets, control = list(type = 'ewma', lambda = 0.9))

# Factor estimation of the covariance with dafault K = 1
covEstimation(rets, control = list(type = 'factor'))

# Factor estimation of the covariance with K = 3
covEstimation(rets, control = list(type = 'factor', K = 3))

# Ledot-Wolf's estimation of the covariance
covEstimation(rets, control = list(type = 'lw'))

# Shrinkage of the covariance matrix using constant correlation matrix
covEstimation(rets, control = list(type = 'const'))

# Shrinkage of the covariance matrix towards constant correlation matrix by
# Ledoit-Wolf.
covEstimation(rets, control = list(type = 'cor'))

# Shrinkage of the covariance matrix towards one-parameter matrix
covEstimation(rets, control = list(type = 'oneparm'))

# Shrinkage of the covariance matrix towards diagonal matrix
covEstimation(rets, control = list(type = 'diag'))

# Shrinkage of the covariance matrix for large data set (alias of 'lw')
covEstimation(rets, control = list(type = 'large'))

# Bayes-Stein estimation of the covariance
covEstimation(rets, control = list(type = 'bs'))

---

Industry Portfolios

Description

A matrix containing daily returns of 10 industry portfolios for the year 2014.

Usage

data("Industry_10")

Format

A matrix (of size 252 x 10) containing daily returns of 10 industry portfolios.

Note

Data are available from Kenneth French's website.

Source

http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.

---

Estimation of mean returns

Description

Function which is used to compute the estimation of the mean returns.

Usage

meanEstimation(rets, control = list())

Arguments

rets	a $(T \times N)$ matrix of past returns.
control	control parameters (see *Details*).

Details

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

• type method used to estimate the mean returns, among 'naive', 'ewma', 'bs' and 'mart' where:

  'naive' is used to compute the arithmetic mean of the returns.

  'ewma' is used to compute the exponential weighted moving average mean of the returns. The data must be sorted from the oldest to the latest. See RiskMetrics (1996).

  'bs' is used to compute the Bayes-Stein estimation. See Jorion (1986).

  'mart' is used to compute the Martellini (2008) implied returns.

  Default: type = 'naive'.

• lambda decay parameter, a single number in $(0, 1]$; $\lambda = 1$ is the no-decay limit and gives equal weights. Default: lambda = 0.94.

Value

A $(N \times 1)$ vector of expected returns.

Author(s)

David Ardia, Kris Boudt and Jean-Philippe Gagnon Fleury.

References

Jorion, P. (1986). Bayes-Stein estimation for portfolio analysis. Journal of Financial and Quantitative Analysis 21(3), pp.279-292.

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

RiskMetrics (1996) RiskMetrics Technical Document. J. P. Morgan/Reuters.

Examples

# Load returns of assets or portfolios
data("Industry_10")
rets = Industry_10

# Naive estimation of the mean
meanEstimation(rets)

# Naive estimation of the mean
meanEstimation(rets, control = list(type = 'naive'))

# Ewma estimation of the mean with default lambda = 0.94
meanEstimation(rets, control = list(type = 'ewma'))

# Ewma estimation of the mean with lambda = 0.9
meanEstimation(rets, control = list(type = 'ewma', lambda = 0.9))

# Martellini's estimation of the mean
meanEstimation(rets, control = list(type = 'mart'))

# Bayes-Stein's estimation of the mean
meanEstimation(rets, control = list(type = 'bs'))

---

Optimal portfolio

Description

Function which 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*).

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:

  Every supplied bound enters the optimization itself. Under the 'gross' constraint the mean-variance, minimum-variance and maximum-decorrelation problems are recast as quadratic programs in $(w^+, w^-)$ with $w = w^+ - w^-$, which turns $\|w\|_1 \le c$ into a linear constraint and makes them exactly solvable.

  'mv' is used to compute the weights of the mean-variance portfolio, that is the solution of

  $\max_w \left\{ w' \mu - \frac{\gamma}{2} w' \Sigma w \right\} \quad s.t. \quad w'1 = 1$

  and, in the unconstrained case, is given in two-fund form by

  $w = \frac{\Sigma^{-1} 1}{1' \Sigma^{-1} 1} + \frac{1}{\gamma} \left( \Sigma^{-1} \mu - \frac{1' \Sigma^{-1} \mu}{1' \Sigma^{-1} 1} \Sigma^{-1} 1 \right)$

  that is, the minimum-variance portfolio plus $1/\gamma$ times a self-financing speculative portfolio. Note that the summability constraint is imposed here as it is for the other portfolios, so the risk aversion $\gamma$ governs the tilt away from the minimum-variance portfolio rather than the leverage.

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

  'invvol' is the inverse volatility portfolio. It is computed in closed form and is always long-only with weights summing to one, so the 'lo' and 'gross' constraints hold automatically. Bounds do not enter its definition: if LB/UB are supplied and bind, the weights are projected onto $\{LB \le w \le UB, w'1 = 1\}$ and a warning is issued.

  '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:

  $w = argmin \left\{ \sum_{i=1}^N (\% RC_i - \frac{1}{N})^2 \right\}$

  . Without bounds, or under the long-only constraint, the solution is obtained exactly by solving $y_i [\Sigma y]_i = 1/N$ for $y > 0$ and rescaling to $w = y / (y'1)$, which attains a zero objective. See Spinu (2013) and Chaves et al. (2012). This is necessary because the objective above is not convex and admits local minima on the boundary at which whole groups of assets receive a zero weight; a general-purpose optimizer started at the equally-weighted portfolio converges to them whenever the correlation matrix has sizable negative entries. Under 'user' or 'gross' constraints the problem is solved numerically from several starting values.

  '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 $j$,$\xi = (\xi_1,\ldots,\xi_{10})'$.

  Note that this portfolio carries additional bounds that stabilize the optimization: whenever LB (resp. UB) is not supplied it is set to $1/(2N)$ (resp. $2/N$), under every value of constraint. The risk-efficient portfolio is therefore never the summability-only problem, it is always long-only, and gross.c can never bind on it since $\|w\|_1 = 1$. Supply LB and UB explicitly to override these defaults.

  'maxdec' is used to compute the weights of the maximum-decorrelation portfolio:

  $w = {argmax}\left\{ 1 - \sqrt{w' R w} \right\}$

  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 unconstrained 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 upper 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. For the non-convex problems ('erc' under 'user'/'gross' constraints, 'maxdiv' and 'riskeff') w0 is one of several starting values that are tried, the best solution being returned.

• gross.c gross exposure constraint, that is the bound imposed on $\|w\|_1$. Since the weights sum to one, it cannot be smaller than one. Default: gross.c = 1.6.

• gamma risk aversion parameter, a single positive number. Default: gamma = 0.89.

• ctr.slsqp list with control parameters for the slsqp function. Default: list(xtol_rel = 1e-18, ftol_rel = 1e-12, check_derivatives = FALSE, maxeval = 2000). xtol_rel is below double precision and so cannot be met on its own; ftol_rel supplies an attainable stopping criterion, and the optimizer warns if it stops on maxeval instead of converging. It is used only by the non-convex problems – 'erc' under 'user' or 'gross' constraints, 'maxdiv' and 'riskeff'. The mean-variance, minimum-variance and maximum-decorrelation portfolios are quadratic programs under every constraint, including 'gross', and are solved exactly by solve.QP.

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 alternative to cap-weighted 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

Chaves, D., Hsu, J., Li, F., Shakernia, O. (2012). Efficient algorithms for computing risk parity portfolio weights. Journal of Investing 21(3), pp.150-163.

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.

Spinu, F. (2013). An algorithm for computing risk parity weights. SSRN working paper. doi:10.2139/ssrn.2297383

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

---

RiskPortfolios: Computation of risk-based portfolios in R

Description

RiskPortfolios (Ardia et al., 2017) is an R package for constructing risk-based portfolios dedicated to portfolio managers and quantitative analysts. It provides a set of functionalities to build mean-variance, minimum variance, inverse-volatility weighted (Leote et al., 2012), equal-risk-contribution (Maillard et al. 2010), maximum diversification (Choueifaty and Coignard, 2008), and risk-efficient (Amenc et al., 2011) portfolios. Optimization is achieved with the R functions solve.QP and slsqp. Long or gross constraints can be added to the optimizer. As risk-based portfolios are mainly based on covariances, the package also provides a large set of covariance matrix estimators. A simulation study relying on the package is described in Ardia et al. (2017).

Functions

• optimalPortfolio: Optimization of portfolios.

• meanEstimation: Computation of expected returns.

• covEstimation: Computation of covariance matrix.

• semidevEstimation: Computation of semi-deviation.

Update

The latest version of the package is available at https://github.com/ArdiaD/RiskPortfolios.

Note

By using RiskPortfolios you agree to the following rules: (1) You must cite Ardia et al. (2017) in working papers and published papers that use RiskPortfolios (use citation("RiskPortfolios")), (2) you must place the URL https://CRAN.R-project.org/package=RiskPortfolios in a footnote to help others find RiskPortfolios, and (3) you assume all risk for the use of RiskPortfolios.

Author(s)

David Ardia, Kris Boudt and Jean-Philippe Gagnon-Fleury.

References

Amenc, N., Goltz, F., Martellini, L., Retowsky, P. (2011). Efficient indexation: An alternative to cap-weighted 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

Ardia, D., Boudt, K., Gagnon-Fleury, J.-P. (2017). RiskPortfolios: Computation of risk-based portfolios in R. Journal of Open Source Software 2(10), 171. doi:10.21105/joss.00171

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.

See Also

Useful links:

• https://github.com/ArdiaD/RiskPortfolios

• Report bugs at https://github.com/ArdiaD/RiskPortfolios/issues

---

Estimation of the semideviation

Description

Function which computes the semideviation.

Usage

semidevEstimation(rets, control = list())

Arguments

rets	a $(T \times N)$ matrix of past returns.
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 semideviation vector, among 'naive' and 'ewma' where:

  'naive' is used to compute the simple semideviation.

  'ewma' is used to compute the exponential weighted moving average semideviation. The data must be sorted from the oldest to the latest. See RiskMetrics (1996).

  The semideviation for one stock is computed as follows. First we select the returns which are smaller than the average of the past returns; we get a new vector of dimension $K \times 1, K \le T$. Then, the weight $w_i$ for each observation at its corresponding time $t$ is computed as $w = \lambda^{t}$. We obtain a $K \times 1$ vector. The vector of weights is then normalized. Finally, the semideviation is obtained as the weighted standard deviation.

  Note that the weights are normalized over the $K$ selected observations only, so that 'naive' returns $\sqrt{K^{-1} \sum (r_t - \bar{r})^2}$ over the returns below the mean, and not the $1/T$ version of the semideviation. The two differ by a factor $\sqrt{T/K}$, which is close to $\sqrt{2}$ for symmetric returns.

  Default: type = 'naive'.

• lambda decay parameter, a single number in $(0, 1]$; $\lambda = 1$ is the no-decay limit and gives equal weights. Default: lambda = 0.94.

Value

A $(N \times 1)$ vector of semideviations.

Author(s)

David Ardia, Kris Boudt and Jean-Philippe Gagnon Fleury.

References

RiskMetrics (1996) RiskMetrics Technical Document. J. P. Morgan/Reuters.

Examples

# Load returns of assets or portfolios
data("Industry_10")
rets = Industry_10

# Naive semideviation estimation
semidevEstimation(rets)

# Naive estimation of the semideviation
semidevEstimation(rets, control = list(type = 'naive'))

# Ewma estimation of the semideviation. Default lambda = 0.94
semidevEstimation(rets, control = list(type = 'ewma'))

# Ewma estimation of the semideviation. lambda = 0.9
semidevEstimation(rets, control = list(type = 'ewma', lambda = 0.9))

---