Background

Problem 1. Maximizing sum of expected return and weighted CVaR Deviation

Simplified Problem Statement

Mathematical Problem Statement

Problem dimension and solving time

Solution in MATLAB Environment

Problem 2. Maximizing sum of expected return and weighted Variance

Simplified Problem Statement

Mathematical Problem Statement

Problem dimension and solving time

Solution in MATLAB Environment

Problem 3. Maximizing sum of expected return and weighted Mean Absolute Deviation

Simplified Problem Statement

Mathematical Problem Statement

Problem dimension and solving time

Solution in MATLAB Environment

 

 

Background

This case study investigates two investment strategies for a portfolio of hedge funds.  The first strategy rebalances the portfolio by solving an optimization problem.  The second  strategy, called "20-best", every time period selects 20 hedge funds with the highest return (over available data) and give them equal weight. Rebalancing in both strategies is done on a monthly basis. Out-of-sample numerical experiments illustrate  investment strategies performance in a simulated historical environment.  We solve the portfolio optimization problem using three risk measures: CVaR, Variance,  and MAD (code can handle any risk measure included in PSG). Also, we take into account the market-neutrality (i.e., beta-zero) constraint. Performance of  strategies is compared with SP500 benchmark. The setup of this case study is described in  Krokhmal et al [1].

 

The dataset contains monthly rates of returns for 302 hedge funds (66 monthly returns from December 1995 to May 2001, for funds with market capitalization of at least $5,000,000 as of December 1995).  Also, monthly data for the benchmark portfolio (SP500) are included in the dataset.

 

The case study consecutively run 54 passes corresponding to 54 out-of-sample months. The first pass uses  the first 12 months of historical returns as in-sample scenarios  to generate scenarios (in-sample) for the optimization problem described in Formal Problem Statement. Each scenario is a vector of monthly returns for all funds considered in the optimization problem (all scenarios are assigned equal probabilities). Optimization gives new portfolio positions. To evaluate the out-of-sample performance of this reoptimized portfolio we multiply the new positions by the 13-th month return. I.e., we do paper-trading in the out-of sample 13-th month.

To implement the "20-best" strategy, 302 hedge funds rates of returns are calculated for the first 12 months. Then we include in the "20-best" portfolio the  20 hedge funds with the largest returns over 12 months . All 20 funds have equal monetary values (equally weighted portfolio with weights=1/20). Finally, we calculate new value of the benchmark SP500 for the13-th month.  

 

During the second pass we repeat the described procedure including in the in-sample period the 13-th month and using the next, 14-th, month as out-of-sample. This procedure is continued until we hit the last month (May 2001) in the database.

 

The performance of the resulted portfolios is presented graphically and in form of tables. The tables present portfolios dynamics, and portfolio performance statistics including annual return, maximum drawdown and Sharpe ratio. The graph compares the portfolios out-of-sample performance.

 

Three risk measures are compared in this case study:

 

The best total out-of-sample rate of return (205.01%) belongs to  portfolio4 with  Mean Absolute Deviation risk measure .

The best out of sample Sharpe ratio (3.83) belongs to portfolio6 with Variance risk measure

 

References

[1] Krokhmal, P., Uryasev, S., and G. Zrazhevsky. Risk Management for Hedge Fund Portfolios: A Comparative Analysis of Linear Portfolio Rebalancing Strategies. Journal of Alternative Investments, V.5, #1, 2002, 10-29.

 

Problem 1

Maximizing sum of expected return and weighted CVaR Deviation.

 

Simplified Problem Statement

 

Maximize Linear + CVaR_dev

 subject to:

Linearmulti = Const1 (budget constraint)

-Const2 ≤ Linearmulti ≤ Const2 (market-neutrality constraint)

Box constraints (lower bounds on positions)

 

where

 

CVaR_dev = CVaR Deviation

Linearmulti = Set of Linear Functions

Box constraints = constraints on individual decision variables

 

Mathematical Problem Statement

 

See (CS.1) - (CS.4):

 

Problem dimension and solving time

 

Example solution of one optimization problem with confidence level 0.2 in CVaR Deviation:

 

 

Solution in MATLAB Environment

 

Solved with riskrprog PSG subroutine (General (Text) Format of PSG in MATLAB):

 

Input Files to run CS:

 

Problem 2

Maximizing sum of expected return and weighted Variance.

 

Simplified Problem Statement

 

Maximize Linear + Variance

 subject to:

Linearmulti = Const1 (budget constraint)

-Const2 ≤ Linearmulti ≤ Const2 (market-neutrality constraint)

Box constraints (lower bounds on positions)

 

where

 

Linearmulti = Set of Linear Functions

Box constraints = constraints on individual decision variables

 

Mathematical Problem Statement

 

See (CS.5) - (CS.8):

 

Problem dimension and solving time

 

Example solution of one optimization problem:

 

 

Solution in MATLAB Environment

 

Solved with riskprog PSG subroutine (General (Text) Format of PSG in MATLAB):

 

Input Files to run CS:

 

Problem 3

Maximizing sum of expected return and weighted Mean Absolute Deviation.

 

Simplified Problem Statement

 

Maximize Linear + Meanbs_dev

 subject to:

Linearmulti = Const1 (budget constraint)

-Const2 ≤ Linearmulti ≤ Const2 (market-neutrality constraint)

Box constraints (lower bounds on positions)

 

where

 

Meanbs_pen = Mean Absolute Deviation

Linearmulti = Set of Linear Functions

Box constraints = constraints on individual decision variables

 

Mathematical Problem Statement

 

See (CS.9) - (CS.12):

 

Problem dimension and solving time

 

Example solution of one optimization problem:

 

 

Solution in MATLAB Environment

 

Solved with riskprog PSG subroutine (General (Text) Format of PSG in MATLAB):

 

Input Files to run CS: