Auxiliary function used by .ZI_projection

Description

Auxiliary function used by .ZI_projection

Usage

.F_func(rho, x, y, s, z, A, b, cl, p, B, xe, ka)

Arguments

Computes the empirical Conditional Value-at-Risk, Value-at-Risk and Mean Absolute Deviation for losses with given probabilities

Description

Computes the empirical Conditional Value-at-Risk, Value-at-Risk and Mean Absolute Deviation for losses with given probabilities

Usage

.RISK_post(returns, prob, alpha)

Arguments

Value

list of values

Computes
the solution of the linear program \min c^{T} x for Ax = b, x \ge 0

Description

which fulfills the condition |B(x - xhat)|^2 \to \min

Usage

.ZI_projection(clin, Amat, bmat, xhat, B, maxiter, tol, k)

Arguments

Value

list

Makes a diagonal matrix with values from x

Description

Function make_diag() takes a vector x and returns a matrix with diagonal consists the values of x.

Usage

.make_diag(x)

Arguments

Value

diagonal matrix y

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

Value

BDportfolio_optim returns a list with items:

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")) 
 

Portfolio optimization which finds an optimal portfolio with the smallest distance to a benchmark.

Description

PortfolioOptimProjection is a linear program for financial portfolio optimization. The function finds an optimal portfolio which has the smallest distance to a benchmark portfolio given by bvec. Solution is by the algorithm due to Zhao and Li modified to account for the fact that the benchmark portfolio bvec has the dimension of portfolio weights and the solved linear program has a much higher dimension since the solution vector to the LP problem consists of a set of primal variables: financial portfolio weights, auxiliary variables coming from the reduction of the mean-risk problem to a linear program and also a set of dual variables depending on the number of constrains in the primal problem (see Palczewski).

Usage

PortfolioOptimProjection (dat, portfolio_return,
risk=c("CVAR","DCVAR","LSAD","MAD"), alpha=0.95, bvec,
Aconstr=NULL, bconstr=NULL, LB=NULL, UB=NULL, maxiter=500, tol=1e-7)

Arguments

Value

PortfolioOptimProjection returns a list with items:

References

Palczewski, A., LP Algorithms for Portfolio Optimization: The PortfolioOptim Package, R Journal, 10(1) (2018), 308–327. DOI:10.32614/RJ-2018-028.

Zhao, Y-B., Li, D., Locating the least 2-norm solution of linear programs via a path-following method, SIAM Journal on Optimization, 12 (2002), 893–912. DOI:10.1137/S1052623401386368.

Examples

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;  
w_m <- rep(1/k,k) # benchmark portfolio, a vector of length k, 
port_ret = 0.05 # portfolio target return
alpha_optim = 0.95

# minimal constraints set: \sum theta_i = 1
# has to be in two inequalities: 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 <- PortfolioOptimProjection(dat, port_ret, risk="MAD",  
alpha=alpha_optim, w_m, Aconstr, bconstr, LB, UB, maxiter=200, tol=1e-7)

cat ( c("Projection optimal 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"))