2. Multi-Level Risk-Controlled Sector Optimization for Opportunistic Global Fixed-Income Portfolios Ron D'Vari, Juan C. Sosa, KishoreYalamanchilli State Street Research & Management CIFEr, New York March 27th, 2000

3. State Street Research, Multilevel Risk-Controlled Optimization2 Risk-constrained Optimization Facilitates Integration of Various Sector Views In Portfolio Construction Feedback Relative Valuation Process Honing Results Monitoring Attribution Expectations Markets Currencies Spreads Risks Portfolio Synthesis Maximize Return Constrain Risk to Tolerance Impose Compliance Research Macro Quantitative Credit Nondollar Emerging

4. State Street Research, Multilevel Risk-Controlled Optimization3 Agenda Problem Statement Why Multi-level Optimization? Proposed Multilevel Risk-Constrained Optimization Algorithm –Sub-level: Nondollar Sectors vs. Domestic Index –Top Level: Domestic Sectors + Customized Nondollar Portfolio Brief Review of Our Past Research Sector Structure Risk Model for G-13 Nondollar Government Bond Markets Sample Optimization Results Conclusion

5. State Street Research, Multilevel Risk-Controlled Optimization4 Risk-Constrained Optimization Problem Definition

6. State Street Research, Multilevel Risk-Controlled Optimization5 Problem Definition (Cont’d) Maximize Relative or Absolute Return Under a Single View or Probably Weighted View –Requires explicit views on all sectors –Coordinated effort by all research teams –Could blend short (tactical) and long term views (strategic) Subject To Constraints –Relative or Absolute Conditional VaR at CL= X < CVaR Limit –Under performance under defined scenarios < Scenario Return Limit –Traditional Relative or Absolute Risk Measures Duration Curve Risk Measures Duration Contributions from Various Sectors

7. State Street Research, Multilevel Risk-Controlled Optimization6 Why Multi-Level Optimization Avoid ill-conditioned problem of joint risk characterization –Simultaneous optimization of a large number of subsectors Requires a large length of time histories for joint risk characterization Can lead to numerical instabilities and noise Dependency of returns tend to be stable when considering –Cells within each sector, and –Different sectors as aggregate Dependency of returns tend to be noisy when considering –Cells that fall in diverse sectors e.g. 5-year AAA CMBS spread vs. 10-year JGB yield)

8. State Street Research, Multilevel Risk-Controlled Optimization7 Proposed Multilevel Optimization Algorithm Sub-level: –Optimize Relative Return of Nondollar Sectors to Domestic Index (Decision Variables: N W i, C w i ) –Subject to: Relative CVaR Limit Allotted to Nondollar –Example (3-Month, 97.5% Confidence Level): Total Relative CVaR =100bp Allotted Nondollar CVaR at Total Portfolio Level = 30bp Allotted Nondollar Relative CVaR at Sector Level = 300bp (All numbers measured w.r.t. domestic index) Hedge limits (i.e., 0.95 <[ C W i / N W i ]< 1.05) Scenario constraints Other constraints (related to guideline or research view) –Country weight or duration contribution

9. State Street Research, Multilevel Risk-Controlled Optimization8 Proposed Multilevel Optimization Algorithm (Cont’d) Top-level: –Optimize Relative Return vs. Overall Index with Decision Variables as Domestic Sector Weights ( D W i ) Weight of Customized Nondollar Portfolio ( N w customized index ) –Specific opportunistic countries can be segregated and optimized at the top level (e.g. Greece) –Subject to: Relative CVaR Limit vs.. Overall benchmark Scenario constraints Traditional constraints

10. State Street Research, Multilevel Risk-Controlled Optimization9 Brief Review of Our Past Research Risk Models Used Rolling GARCH-PJ Sample Domestic Results

11. State Street Research, Multilevel Risk-Controlled Optimization10 Methodology Requirements for Risk Models Accuracy –Nonstationary –Non-normal Robustness Feasible automation and maintenance

12. State Street Research, Multilevel Risk-Controlled Optimization11 Risk Models Used Rolling Variance-Covariance –G-13 Government Bond Yields –Investment Grade Corporate and ABS OASs GARCH and Garch-t –Mortgage passthroughs –G-13 currencies Garch-PJ –Used for high yield and emerging markets Univariate GARCH with Persistent Jumps Rolling white noise correlation matrix Exogenized jump frequencies

13. State Street Research, Multilevel Risk-Controlled Optimization12 Rolling GARCH-PJ (univariate) A variation of GARCH(1,1) that features Bernoulli-style jumps  s t = a 0 + e t, where e t = sqrt(h t )u t + j t, with u t ~ N(0,1) i.i.d. h t = g 0 + g 1 e 2 t-1 + g 2 h t-1 j t ~ N(  j,  j 2 ) with probability p 0 with probability 1-p Jump occurrences in this model will induce a volatility spike in subsequent days Bernoulli, rather than Poisson jumps, simplify and speed up the parameter estimation procedure VaR estimates are also produced via simulation Jump frequencies are also allowed to depend on exogenous or past data

14. State Street Research, Multilevel Risk-Controlled Optimization13 0255075100125150175200225250 -3 -2 0 1 2 3 Brazil: Daily Series of 1-day spread changes, Jan/01/98-Jan/22/99 and 90%&99% Var-Covar VaR estimates

15. State Street Research, Multilevel Risk-Controlled Optimization14 0255075100125150175200225250 -3 -2 0 1 2 3 4 Brazil: Daily Series of 1-day spread changes, Jan/01/98-Jan/22/99 and 90%&99% GARCH(1,1) VaR estimates

16. State Street Research, Multilevel Risk-Controlled Optimization15 0255075100125150175200225250 -3 -2 0 1 2 3 4 Brazil: Daily Series of 1-day spread changes, Jan/01/98-Jan/22/99 and 90%&99% GARCH-PJ(1,1) VaR estimates

17. State Street Research, Multilevel Risk-Controlled Optimization16 0255075100125150175200225250 -3 -2 0 1 2 3 4 Brazil: Daily Series of 1-day spread changes, Jan/01/98-Jan/22/99 and 90%&99% GARCH-PJ(1,1) w/ Exogenized Jumps VaR estimates

18. State Street Research, Multilevel Risk-Controlled Optimization17 Model Choice The skewness and kurtosis of the standardized innovations support GARCH-PJ Brazil 1992-1999:Skewness Kurtosis Rolling Var-Covar 5.94 99.67 GARCH 2.96 47.20 GARCH-PJ * 0.16 3.50 GARCH-PJ Exo* 0.12 3.42 *jump days excluded

19. State Street Research, Multilevel Risk-Controlled Optimization18 Multivariate ARCH Issues Multivariate ARCH models suffer from estimation problems, deriving from the inclusion of correlation parameters Our ad-hoc approach: a 3-month sample correlation matrix estimated from (non- jump) standardized innovations

20. State Street Research, Multilevel Risk-Controlled Optimization19 Sample Domestic Results (Cont’d)

21. State Street Research, Multilevel Risk-Controlled Optimization20 Sample Domestic Results (Cont’d)

22. State Street Research, Multilevel Risk-Controlled Optimization21 Sample Domestic Results