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  1. 1. Risk Management for Mutual Fund Portfolios An Analysis of Linear Rebalancing Strategies Mehmet ÖNAL David WIMBERLY
  2. 2. Introduction <ul><li>We seek to apply formal risk management methodology to the optimization of mutual fund portfolios </li></ul><ul><li>Our solution methodology is based on the CVaR approach outlined in Krokhmal et. al (2002) </li></ul>
  3. 3. Introduction <ul><li>The problem is to find a strategy to allocate available fund to some number of accounts </li></ul><ul><li>These accounts have daily returns </li></ul>
  4. 4. Introduction <ul><li>The solution approach outlined in Krokhmal et al (2002) is to maximize expected daily rate of return subject to constraints on risk measure </li></ul>
  5. 5. Introduction <ul><li>Krokhmal et al (2002) use several risk measures </li></ul><ul><ul><li>Mean Absolute Deviation (MAD) </li></ul></ul><ul><ul><li>Conditional Drawdown at Risk (CDaR) </li></ul></ul><ul><ul><li>Maximum Loss </li></ul></ul><ul><ul><li>Conditional Value at Risk (CVaR) </li></ul></ul><ul><li>In our work we choose CVaR to be our risk measure </li></ul>
  6. 6. Problem Formulation <ul><li>Maximize </li></ul><ul><li>Subject to </li></ul>
  7. 7. Problem Formulation <ul><li>where </li></ul>
  8. 8. Problem Formulation <ul><li>We can reduce this program to a linear program with scenarios of equal probability of occurrence </li></ul><ul><li>Each scenario is a vector of daily returns of accounts i=1,2,…,n </li></ul>
  9. 9. Problem Formulation <ul><li>Maximize </li></ul><ul><li>Subject to </li></ul>
  10. 10. Problem Formulation <ul><li>where </li></ul>
  11. 11. Problem Formulation <ul><li>As the days pass we obtain more information on the performances of the accounts </li></ul><ul><li>We suggest resolving this optimization as the daily data become available, i.e., as the scenarios to consider increase </li></ul>
  12. 12. Solution Approach <ul><li>Starting with a sufficient number of scenarios (in this work it is one year), we suggest re-optimizing (rebalancing the portfolio) in every 20 business days with the updated scenarios </li></ul>
  13. 13. Solution Approach time Scenarios start on this day optimization after the first year with 260 scenarios reoptimization after the second month with 300 scenarios …………… reoptimization after one moth with 280 scenarios
  14. 14. Solution Approach <ul><li>Begin with a sufficiently large number of scenarios </li></ul><ul><li>While you are controlling the funds </li></ul><ul><ul><li>Run optimization on the in sample set </li></ul></ul><ul><ul><li>Observe the performance of the portfolio for the following 20 days </li></ul></ul><ul><ul><li>Update the in-sample set by adding 20 business days’ data </li></ul></ul>
  15. 15. Solution Approach <ul><li>We have approximately 5 years’ data to test the performance of this algorithm </li></ul><ul><li>The data was obtained from Theta Research, Inc., a mutual fund research firm which monitors mutual fund managers and their portfolio results </li></ul>
  16. 16. Solution Approach <ul><li>We first optimize with the scenarios obtained in the first year (in-sample set: data of 260 days) </li></ul><ul><li>We then regularly rebalance the portfolio every 20 business days, increasing the size of the in-sample set in each iteration </li></ul>
  17. 18. Results <ul><li>We did all our calculations in MATLAB </li></ul><ul><li>Optimal accounts found in the last in-sample optimization and their historical performances in the last 5 years are presented in the next slides </li></ul><ul><li>The results were obtained with CVaR α <-0.995, α =0.90 (recall that we worked on the loss function) </li></ul>
  18. 21. Results <ul><li>Historical performance of our portfolio is shown below </li></ul>
  19. 22. Results <ul><li>We were able to increase the performance of our methodology if we make the CVaR constraint tighter and constrain that no more than 15 % of the funds can be allocated to any account </li></ul>
  20. 23. <ul><li>Historical performance of our portfolio with the additional constraints </li></ul>
  21. 24. Conclusion <ul><li>Despite some issues with the data set, we were able to construct an efficient frontier and an optimal portfolio with the in-sample data </li></ul><ul><li>We were able to run out-of-sample calculations and reached an overall result of a 5% loss on the first run </li></ul><ul><li>On the second run, we were able to improve this to a breakeven position </li></ul>
  22. 25. <ul><li>We observed that we were dealing with funds which were all fund of funds </li></ul><ul><li>It appeared the managers were all benchmarking the S&P 500 </li></ul><ul><li>But our results were better than the S&P 500 by a considerable margin </li></ul>