Dynamic Asset Allocation for Hedging Downside Risk

Dynamic Asset Allocation for Hedging Downside Risk Gerd Infanger Stanford University Department of Management Science and Engineering and Infanger Investment Technology, LLC October 2009 Gerd Infanger, all rights reserved. 1

Outline Background and Concepts Stochastic Dynamic Programming for Multi-period Portfolio Optimization (Active Asset Allocation Management) Strategies for Hedging Downside Risk Evaluation via Back-testing October 2009 Gerd Infanger, all rights reserved. 2

Dynamic Asset Allocation The basic decision problem is how to allocate funds among various asset classes over time so as to achieve given investment objectives. An inherent trade-off exists between long-term growth and short-term value preservation. Optimal asset allocations change over time in response to: Changes in conditional return distributions (active management) Realized returns (attained wealth), depending on investor risk preferences The problem is best and most generally analyzed through multi-period portfolio optimization. October 2009 Gerd Infanger, all rights reserved. 3

Theoretical Background Samuelson (1969) and Merton (1969, 1990): The optimal investment strategy is independent of wealth and constant over time if: Asset return distribution is iid Utility function is CRRA (power) (If the utility function is logarithmic, non-iid asset returns result in a constant strategy as well) No transaction costs Mossin (1968), Hakanson (1971): Invest in each period as if all future investments were only in the risk-free asset if: Asset return distribution is iid Utility function is HARA (power, exponential and generalized logarithmic) No transaction costs Absence of any borrowing and short sales constraints More recently, analytical solutions have been obtained also for HARA utility functions with borrowing and short sale constraints (Cox and Huang (1999), Campbell and Viceira (2002)). October 2009 Gerd Infanger, all rights reserved. 4

Situations Requiring Numerical Solution Utility functions other than HARA (downside risk) Non i.i.d. return processes (active management) General side constraints Transaction costs October 2009 Gerd Infanger, all rights reserved. 5

Numerical Approaches to Dynamic Asset Allocation Stochastic programming Can efficiently solve the most general model. Successfully used for asset allocation and asset liability management. Stochastic dynamic programming (stochastic control) Discrete state space (e.g., Musumeci and Musumeci (1999), Brennan, Schwartz and Lagnado (1998)) When the state space is small, say, up to 3 or 4 state variables, value function approximation methods show promise. (e.g., De Farias and Van Roy (2003)) October 2009 Gerd Infanger, all rights reserved. 6

The WealthiOR TM Approach Applies stochastic dynamic programming to a rich problem representation (Infanger (2006)) Parameterized terminal utility functions representing various types of investor risk aversion (e.g., increasing and decreasing RRA) Downside risk control Normal, lognormal, and empirical return distributions (using bootstrapping from historical observations) (Possible extensions to a restricted class of autoregressive processes) Linear side constraints and bounds on holdings Arbitrary cash flows October 2009 Gerd Infanger, all rights reserved. 7

Increasing and Decreasing Relative Risk Aversion ART Decreasing RRA CRRA Increasing RRA CARA Represented as piecewise CARA approximation, see Infanger (2006) W October 2009 Gerd Infanger, all rights reserved. 8

Utility Functions for Hedging Downside Risk October 2009 Gerd Infanger, all rights reserved. 9

Stochastic Dynamic Programming Recursion October 2009 Gerd Infanger, all rights reserved. 10

In Practice, In-sample/Out-of-sample Approach October 2009 Gerd Infanger, all rights reserved. 11

Example of a Downside Risk Protected Strategy Start conservatively and take on riskier positions as wealth increases. Favorable returns Unfavorable returns Expected returns October 2009 Gerd Infanger, all rights reserved. 12

Example of a Downside Risk Protected Strategy (cont.) Scale back risk when losses push wealth towards target wealth. Strategy 2009 Strategy 2013 October 2009 Gerd Infanger, all rights reserved. 13

Robustness Analysis In order to test robustness, we first estimated a model of asset class returns based on historical data and computed the optimal dynamic strategy. We then simulated the performance of the obtained strategy using different means of asset class return by varying each estimated mean as a fraction of the estimated standard deviation. We compared the (% change of) certainty equivalent wealth of the dynamic strategy compared to the best fixed-mix strategy as a function of the mean disturbance in fractions of standard deviations. (For the effect of errors in the parameter estimation on (single-period) optimal portfolio choice, see Chopra and Ziemba (1993).) October 2009 Gerd Infanger, all rights reserved. 14

Robustness Analysis (cont.) The dynamic asset allocation model is robust with respect to wide ranges of changes in asset class means, with increasing improvement. Improvement in Certainty Equivalent Wealth, Dynamic vs Best Fixed Mix 20% 18% Horizon 10 Years, dynamic downside risk, quadratic and linear penalty 95% probability range 36 years 16 years 16% Improvement (% CEW) 14% 12% 10% 8% 6% 4% 2% 0% -0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 Deviation of means (standard deviations) October 2009 Gerd Infanger, all rights reserved. 15

Example: International Asset Allocation, Hedging Downside Risk Asset classes: Equities: US, EU, Japan, Asia Long-term bonds, short-term bonds, cash Benchmark: MSCI World 40%, Long-term bonds 60% Model calibration via implied means Horizon of 5 years with monthly rebalancing Active monthly return predictions, based on multiple factors Based on proprietary technique used in Infanger s active management of about $150M of asset allocation funds In actual trading active management adds about 100 basis points per year over benchmark at comparable risk Strategy evaluation via back-testing October 2009 Gerd Infanger, all rights reserved. 16

Performance Measures Results from 11 overlapping back-tests over 5 years from 1993-2007 Wealth Target = Max (0%, Money Market minus 1%) Months 660 Exp. Return Underperf Underperf Turnover (% per annum) (# of months) (% of months) (% per month) Target 2.49% Benchmark 7.83% 177 26.82% w/o predictions DR 5.04% 9 1.36% 3.56% w/o predictions QL 4.55% 3 0.45% 4.21% Active 10 DR 8.23% 16 2.42% 7.81% Active 10 QL 8.37% 7 1.06% 7.90% Active 20 DR 8.62% 14 2.12% 13.25% Active 20 QL 8.47% 3 0.45% 12.78% We will show results for 2008 later October 2009 Gerd Infanger, all rights reserved. 17

Case Active 10 DR, Number of Underperformances Number of Underperformances 8 7 Number of Underperformances 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Month of Strategy Portfolio Benchmark October 2009 Gerd Infanger, all rights reserved. 18

Case Active 10 DR, Frequency and Length of Underperformances Downside risk (frequency of years below target) 3 2 Frequency 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Length of underperformance (months) Portfolio Benchmark October 2009 Gerd Infanger, all rights reserved. 19

Case Active 10 DR, Cumulative Performance 2000-2004 Cumulative Performance 1.4 1.2 1 Value of Portfolio 0.8 0.6 0.4 0.2 0 0 12 24 36 48 60 Years starting 12_31_99 Portfolio Benchmark Target October 2009 Gerd Infanger, all rights reserved. 20

Case Active 10 DR, Asset Allocation 2000-2004 Asset Allocation A_05_00 Asset weights (%) 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 Time StxUS StxEU StxJA StxAS BdEU BdSh Cash October 2009 Gerd Infanger, all rights reserved. 21

Case Active 10 DR, Results for 2008 Cumulative Performance Cumulative Performance 1.4 1.4 1.2 1.2 1 1 Value of Portfolio 0.8 0.6 Value of Portfolio 0.8 0.6 0.4 0.4 0.2 0 2008 0.2 0 2008 0 12 24 36 48 60 0 12 24 36 48 60 Years starting 12_31_03 Years starting 12_31_04 Portfolio Benchmark Lower Bound Portfolio Benchmark Lower Bound Cumulative Performance Cumulative Performance 1.2 1.2 1 1 0.8 0.8 Value of Portfolio 0.6 Value of Portfolio 0.6 0.4 0.4 0.2 0 2008 0.2 0 2008 0 12 24 36 48 60 0 12 24 36 48 60 Years starting 12_31_05 Years starting 12_31_06 Portfolio Benchmark Lower Bound Portfolio Benchmark Lower Bound October 2009 Gerd Infanger, all rights reserved. 22

Case Active 10 DR, Results for 2008 (cont.) Asset Allocation A_05_03 Asset Allocation A_05_04 Asset weights (%) Asset weights (%) 2008 2008 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 Time StxUS StxEU StxJA StxAS BdEU BdSh Cash 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 Time StxUS StxEU StxJA StxAS BdEU BdSh Cash Asset Allocation A_05_05 Asset Allocation A_05_06 Asset weights (%) Asset weights (%) 2008 2008 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 Time StxUS StxEU StxJA StxAS BdEU BdSh Cash 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 Time StxUS StxEU StxJA StxAS BdEU BdSh Cash October 2009 Gerd Infanger, all rights reserved. 23

Case Active 10 DR, Results for 2008 (cont.) 2008 # of Underperformances 60 Mean Month Portfolio Benchmark Sample Portfolio Benchmark 1 0 3 5 1.009 0.974 2 0 3 5 1.013 0.965 3 0 3 5 1.014 0.947 4 0 3 5 1.014 0.968 5 0 3 5 1.012 0.970 6 0 3 5 1.014 0.932 7 0 3 5 1.020 0.933 8 0 3 5 1.028 0.958 9 0 4 5 1.027 0.924 10 0 5 5 1.027 0.884 11 0 5 5 1.042 0.877 12 0 5 5 1.053 0.866 Sum 0 43 Average 5.34% -13.37% October 2009 Gerd Infanger, all rights reserved. 24

Summary A numerical approach to dynamic asset allocation based on stochastic dynamic programming handles a very rich problem representation. The approach may be used successfully for controlling downside risk in the short-run, while obtaining benchmark level expected returns in the long-run. As demonstrated through back-testing, downside risk was reduced considerably, while the cost of the downside risk protection was more than offset by employing active management based on predictions. The approach had performed well during the crisis of 2008 with no underperformances versus the target observed. This was due to the downside risk protection and the active predictions as well. October 2009 Gerd Infanger, all rights reserved. 25