BDportfolio_optim {PortfolioOptim}

Portfolio Optimization by Benders decomposition

Description

BDportfolio_optim is a linear program for financial portfolio optimization. Portfolio risk is measured by one of the risk measures from the list c("CVAR", "DCVAR", "LSAD", "MAD"). Benders decomposition method is explored to enable optimization for very large returns samples (\sim 10^6).

The optimization problem is:
\min F({\theta^{T}} r)
over
\theta^{T} E(r) \ge portfolio\_return,
LB \le \theta \le UB,
Aconstr \theta \le bconstr,
where
F is a measure of risk;
r is a time series of returns of assets;
\theta is a vector of portfolio weights.

Usage

BDportfolio_optim(dat, portfolio_return,  
risk=c("CVAR", "DCVAR","LSAD","MAD"), alpha=0.95,  
Aconstr=NULL, bconstr=NULL, LB=NULL, UB=NULL, maxiter=500,tol=1e-8)

Arguments

dat

Time series of returns data; dat = cbind(rr, pk), where rr is an array (time series) of asset returns, for n returns and k assets it is an array with \dim(rr) = (n, k), pk is a vector of length n containing probabilities of returns.

portfolio_return

Target portfolio return.

risk

Risk measure chosen for optimization; one of "CVAR", "DCVAR", "LSAD", "MAD", where "CVAR" – denotes Conditional Value-at-Risk (CVaR), "DCVAR" – denotes deviation CVaR, "LSAD" – denotes Lower Semi Absolute Deviation, "MAD" – denotes Mean Absolute Deviation.

alpha

Value of alpha quantile used to compute portfolio VaR and CVaR; used also as quantile value for risk measures CVAR and DCVAR.

Aconstr

Matrix defining additional constraints, \dim (Aconstr) = (m,k), where k – number of assets, m – number of constraints.

bconstr

Vector defining additional constraints, length (bconstr) = m.

LB

Vector of length k, lower bounds of portfolio weights \theta; warning: condition LB = NULL is equivalent to LB = rep(0, k) (lower bound zero).

UB

Vector of length k, upper bounds for portfolio weights \theta.

maxiter

Maximal number of iterations.

tol

Accuracy of computations, stopping rule.

Value

BDportfolio_optim returns a list with items:

return_mean	vector of asset returns mean values.
mu	realized portfolio return.
theta	portfolio weights.
CVaR	portfolio CVaR.
VaR	portfolio VaR.
MAD	portfolio MAD.
risk	portfolio risk measured by the risk measure chosen for optimization.
new_portfolio_return	modified target portfolio return; when the original target portfolio return
	is to high for the problem, the optimization problem is solved for
	new_portfolio_return as the target return.

References

Benders, J.F., Partitioning procedures for solving mixed-variables programming problems. Number. Math., 4 (1962), 238–252, reprinted in Computational Management Science 2 (2005), 3–19. DOI: 10.1007/s10287-004-0020-y.

Konno, H., Piecewise linear risk function and portfolio optimization, Journal of the Operations Research Society of Japan, 33 (1990), 139–156.

Konno, H., Yamazaki, H., Mean-absolute deviation portfolio optimization model and its application to Tokyo stock market. Management Science, 37 (1991), 519–531.

Konno, H., Waki, H., Yuuki, A., Portfolio optimization under lower partial risk measures, Asia-Pacific Financial Markets, 9 (2002), 127–140. DOI: 10.1023/A:1022238119491.

Kunzi-Bay, A., Mayer, J., Computational aspects of minimizing conditional value at risk. Computational Management Science, 3 (2006), 3–27. DOI: 10.1007/s10287-005-0042-0.

Rockafellar, R.T., Uryasev, S., Optimization of conditional value-at-risk. Journal of Risk, 2 (2000), 21–41. DOI: 10.21314/JOR.2000.038.

Rockafellar, R. T., Uryasev, S., Zabarankin, M., Generalized deviations in risk analysis. Finance and Stochastics, 10 (2006), 51–74. DOI: 10.1007/s00780-005-0165-8.

Examples

library (Rsymphony)  
library(Rglpk) 
library(mvtnorm)
k = 3 
num =100
dat <-  cbind(rmvnorm (n=num, mean = rep(0,k), sigma=diag(k)), matrix(1/num,num,1)) 
# a data sample with num rows and (k+1) columns for k assets; 
port_ret = 0.05 # target portfolio return 
alpha_optim = 0.95 

# minimal constraints set: \eqn{\sum \theta_{i} = 1} 
# has to be in two inequalities: \eqn{1 - \epsilon <= \sum \theta_{i} <= 1 + \epsilon} 
a0 <- rep(1,k) 
Aconstr <- rbind(a0,-a0) 
bconstr <- c(1+1e-8, -1+1e-8) 

LB <- rep(0,k) 
UB <- rep(1,k) 

res <- BDportfolio_optim(dat, port_ret, "CVAR", alpha_optim, 
Aconstr, bconstr, LB, UB, maxiter=200, tol=1e-8) 

cat ( c("Benders decomposition portfolio:\n\n")) 
cat(c("weights \n")) 
print(res$theta) 

cat(c("\n mean = ", res$mu, " risk  = ", res$risk, 
"\n CVaR = ", res$CVaR, " VaR = ", res$VaR, "\n MAD = ", res$MAD, "\n\n"))