R/optimal.portfolio.expected.shortfall.R

In rhochreiter/scenportopt: Scenario-based Portfolio Optimization

##### Scenario-based Portfolio Optimization (scenportopt)
##### (c)2013-2014 Ronald Hochreiter <ron@hochreiter.net>
##### http://www.finance-r.com/

### Portfolio Optimization minimizing Conditional Value at Risk (CVaR)
# Implementation based on [Rockafellar and Uryasev 2001]

# maximize { b - sum(s) { ( p[s] * z[s] ) / alpha } }
# for(s) { loss[s] == sum(a) { (data[s,a] * x[a]) } }
# for(s) { b - loss[s] - z[s] <= 0 } // z >= b - loss
# for(s) { z[s] >= 0 }

optimal.portfolio.expected.shortfall <- function(model) {  
  ### Variables: b, x[asset], loss[scenario], z[scenario]
  n_var <- 1 + model$assets + 2*model$scenarios
  ix_b <- 1
  ix_x <- 2
  ix_loss <- ix_x + model$assets
  ix_z <- ix_loss + model$scenarios
  
  ### Objective function
  
  # maximize { b - sum(s) { ( p[s] * z[s] ) / alpha } }
  Objective <- list()
  Objective$linear <- rep(0, n_var)
  Objective$linear[ix_b] <- 1
  for (s in 0:(model$scenarios-1)) { Objective$linear[ix_z+s] <- -model$scenario.probabilities[s+1]/model$alpha }
  
  ### Constraints
  Constraints <- list(n=n_var, A=NULL, b=NULL, Aeq=NULL, beq=NULL)
  
  # sum(a) { x[a] } == sum.portfolio
  Constraints <- linear.constraint.eq(Constraints, c((ix_x):(ix_x+model$assets-1)), model$sum.portfolio)

  # sum(a) { x[a] * mean[a] } => min.mean
  if(!is.null(model$min.mean)) { Constraints <- linear.constraint.iq(Constraints, c((ix_x):(ix_x+model$assets-1)), -model$min.mean, -1*model$asset.means) }

#   # sum(a) { x[a] * mean[a] } <= max.mean
#   if(!is.null(model$max.mean)) { 
#     Constraints <- linear.constraint.iq(Constraints, c((ix_x):(ix_x+model$assets-1)), model$max.mean, model$asset.means) 
#   } else {
     # sum(a) { x[a] * mean[a] } == fix.mean
     if(!is.null(model$fix.mean)) { Constraints <- linear.constraint.eq(Constraints, c((ix_x):(ix_x+model$assets-1)), model$fix.mean, model$asset.means) }
#   }
  
  ### CVaR constraints
  
  # for(s) { loss[s] == sum(a) { (data[s,a] * x[a]) } }
  for (s in 0:(model$scenarios-1)) { Constraints <- linear.constraint.eq(Constraints, c((ix_x:(ix_x+model$assets-1)), ix_loss+s), 0, c(as.vector(model$data[(s+1),]), -1)) }
  
  # for(s) { b - loss[s] - z[s] <= 0 } // z >= b - loss
  for (s in 0:(model$scenarios-1)) { Constraints <- linear.constraint.iq(Constraints, c(ix_b, ix_loss+s, ix_z+s), 0, c(1,-1,-1)) }
  
  ### Bounds
  Bounds <- list()
  
  # all variables unbounded
  M <- 1e9
  Bounds$lower <- rep(-M, n_var)
  Bounds$upper <- rep(M, n_var)
  
  # portfolio constrained to model parameters
  Bounds$lower[(ix_x):(ix_x+model$assets-1)] <- model$asset.bound.lower
  Bounds$upper[(ix_x):(ix_x+model$assets-1)] <- model$asset.bound.upper
  
  # for(s) { z[s] >= 0 }
  Bounds$lower[(ix_z):(ix_z+model$scenarios-1)] <- 0
  
  ### Solve optimization problem using modopt.linprog  
  solution <- linprog(-Objective$linear, Constraints$A, Constraints$b, Constraints$Aeq, Constraints$beq, Bounds$lower, Bounds$upper)

  ### Add optimal portfolio to model  
  portfolio <- list()
  portfolio$x <- solution$x[ix_x:(ix_x+model$assets-1)]
  portfolio$x <- round(portfolio$x, model$precision)  
  model$portfolio <- portfolio
  return(model) 
}