Portfolio optimisation with VaR or CVaR constraints using linear programming

Question

I would like to optimize a portfolio allocation (maximizing the exposure or the expected return), but with VaR or CVaR contraints. (some parts of my portfolio cannot exceed a certain VaR)

How can I achieve that? Is there a way to turn the problem into a linear programming problem? or to approximate the results?

Any links or ideas are welcome.

Answer 1

In my experience, a VaR or CVaR portfolio optimization problem is usually best specified as minimizing the VaR or CVaR and then using a constraint for the expected return. As noted by Alexey, it is much better to use CVaR than VaR. The main benefit of a CVaR optimization is that it can be implemented as a linear programming problem. Another option I have tried is the technique in this paper:

http://www.math.uwaterloo.ca/~tfcolema/articles/bank_article.pdf

Another option is the two-step heuristic where one first finds the mean-variance efficient frontier and then you could calculate whatever are the relevant portfolio statistics on only the mean-variance efficient portfolios. In this way you could exclude portfolios that have too high a VaR or CVaR (or mixed CVaR deviation) for your consideration.

However, as you say you are particularly concerned about the VaR or CVaR of certain parts of your portfolio. As noted above, VaR constraints for different different groups of assets would require non-linear constraints. However, CVaR constraints for different assets could be calculated using linear constraints (though it would also be possible to implement a relatively slower methodology using non-linear constraints). For guidance on how to implement this as a linear constraint, it might help to follow

http://www.soa.org/library/proceedings/arch/2008/arch-2008-iss1-cox-lin.aspx

with the only difference that you would want to calculate the CVaR over the relevant groups of securities.

Answer 2

The VaR constraint is convex and quadratic and can be handled with any solver supports quadratic constraints, like Guribi, cplex (from IBM) or xpress (from FICO).

The CVaR can be formulated as a linear program if you are able to perform monte-carlo simulations on the returns. Briefly, the LP model is

\begin{eqnarray*} c &\ge& \alpha + {1 \over (1-\beta)|J|} \sum_{j\in J} z_j \\ z_j &\ge& \sum_{i \in I} r_{ij} x_i - \alpha \hspace{0.2in} \forall j \in J \end{eqnarray*} Where c is the cvar at $\beta$ confidence, $I$ is the set of investments, $x_i$ is the level of investment in $i$, $J$ is the set of monte-carlo scenarios, $r_{ij}$ is the unexpected loss of investment $i$ in simulation $j$. $\alpha$ is the loss of the $100 \cdot \beta$ percentile scenario, and ${1 \over (1-\beta)|J|} \sum_j z_j$ is the average unexpected loss (in excess of alpha) of the worst $(1-\beta)|J|$ scenarios.

The resulting LP instances are very dense and large, so it requires delayed column and constraint generation for non-trivial problems.

Answer 3

You can find a good example on CVaR optimization in the book "Portfolio Optimization with R/Rmetrics" By Diethelm Wuertz, Yohan Chalabi, William Chen, Andrew Ellis.

#load library fPortfolio
library(fPortfolio)

#use indicies LPP2005, see http://www.pictet.com/en/home/lpp_indices.html
lppData  <-  100*LPP2005.RET[,1:6]

#create portfolio specification
frontierSpec  <- portfolioSpec();

#optimization criteria - CVaR
setType(frontierSpec)  <- "CVAR"

#set optimization algorithm
setSolver(frontierSpec)  <- "solveRglpk"

#set confidence level CVaR
setAlpha(frontierSpec)  <- 0.05

#number of portfolios in efficient frontier
setNFrontierPoints(frontierSpec)  <- 25

#optimize, without shortselling
frontier <- portfolioFrontier(data = lppData, spec = frontierSpec, constraints="LongOnly");

#build efficient frontier graph
tailoredFrontierPlot(object=frontier,mText="Mean-CVaR Frontier (Long only)",risk="CVaR");
weightedReturnsPlot(frontier)

I don't recommend you to use VaR optimization for two reasons:

1. VaR is not a subbaditive risk measure, therefore your portfolio could be highly undiversified.
2. It's more challenging computationally than CVaR optimization

Answer 4

I think what you're looking for is a type of solver called a second-order cone program (SOCP) solver. This is just like a quadratic program (QP) solver, except the constraints can be quadratic as well as the objective function. There is an open-source implementation in python via the CVXOPT module.

Answer 5

You could define your optimization problem as a typical linear risk/return optimization problem(1), then use some predicted return as the return component and VaR or cVaR as the risk component. This will not be linear, however you can use an evolutionary algorithm, or some other exotic search algorithm, to maximize for return given a limit on VaR or cVar. The tricky part is to make a good utility/fitness function, i.e., a smooth one.

In essence what you will be doing is to generate a bunch of random feasible portfolios, making sure to throwing out any that have too large of a VaR. The remaining feasible portfolios are ranked based on the utility/fitness function. Keep some of the best once (survival of the fittest) and based on these make small random alterations to the portfolios (mutations). Then re-rank the portfolios. Redo this for several generations. Eventually you should be optimizing based on VaR. There are many tools and frameworks that can do this for you. I used Encog last time I did something like it.

(1) Utility(portfolio) = PredictedReturn(portoflio) - VaR(portoflio)