Ex Post Relative Valuation Process Honing

Sector Optimization for Fixed-Income Portfolios Constrained By Value-at-Risk and Traditional

Risk Measures

Ron D’Vari, Juan C. Sosa, Kishore Yalamanchili

State Street Research and Management

8TH ANNUAL IAFE CONFERENCE

New York City

October 14-15, 1999

1999 IAFE, October 15, 1999

State Street Research & Management

2

Risk-constrained Optimization Facilitates Integration of Various Sector Views In Portfolio Construction

Ex Post•Relative Valuation•Process Honing

Ex Post•Monitoring•Attribution

Ex AnteExpectations

•Markets•Spreads•Risks

Portfolio Synthesis • Maximize Return• Minimize Risk

Research•Macro •Quantitative•Credit



3

Risk-Constrained Fixed-Income Sector Optimization Incorporating VaR and Traditional Risk Measures

Objectives Risk Models Risk-Constrained Optimization Results Conclusions



4

Objectives Validated, Comprehensive, and Flexible Risk Model

Tactical Sector Allocation and Optimization Model: Integration of tactical views of all research teams Incorporation of risk explicitly in the investment process

Comprehensive Tool to Synthesize Fixed-Income Portfolios: Maximize return under a set of probability weighted scenarios Constrain risk

Traditional measures such as relative duration, sector weights, duration contribution

Stress-scenarios incorporating outliers and extreme observations

Flexible historical Value-At-Risk (VaR) allowing for non-normal, time-variant distributions with fat tails

Generic Cell Structure for VaR and Sector OptimizationDuration/Sector Range 1 Range 2 Range 3 Range ... Range NTreasuryTreasury Futures

US AgenciesAgency Sector 1Agency Sector ...Agency Sector N

Structured ABSABSCMBSCBOCLO

Domestic Investment Grade CorporatesCorporate Sector 1Corporate Sector ...

Corporate Sector N Basic Mortgage Pass-throughs

Type/ Coupon Discount Cusp Premium30 Year Agency (N, S, and SS)15 Year Agency (N, S, and SS)Balloon Agency (N, S, and SS)Whole Loans (N, S, and SS)Mortgage Structured Products

High Yield CorporatesHigh Yield Sector 1High Yield Sector ...High Yield Sector N

NondollarCountry 1Country....Country N

Emerging MarketsCountry 1Country....Country N



6

Value-At-Risk Models

Objectives: Calculate distribution of returns and downside risk Comprehensive - include interest rate risk, curve and

spreads, for all major fixed-income sectors Flexible - allow specification of time window, decay factor,

and confidence level Accurate - account for non-normally distributed asset

classes such as MBS, high yield, and emerging market debt



7

Methodology• Domestic High Grade

- Weekly derived spread data from individual securities in the Government/Corporate/Mortgage universe

- Variance/Covariance• High Yield

- Weekly aggregate spread data for subsectors - GARCH with shocks

• Emerging Markets

- Weekly aggregate spread data for subsectors - GARCH with shocks

• Portfolio VaR and Relative VaR estimated via Structured Monte Carlo simulation with rolling correlation matrix



8

MethodologyWork-in-Progress

- Asset Backed Securities

- Nondollar

- Alternative specification for domestic assets (e.g. AR(1) to address mean reversion)



9

• Model choice has significant effect on the estimation of risk-return trade-offs, hence the optimal choice of portfolios

• Variance-Covariance VaR underestimates risk of non-normal assets (e.g. High Yield and EMBI)

Why Not Variance-Covariance?



10

Certain Fixed Income Sectors Exhibit Strongly Non-Normal Behavior

Weekly returns, from 07/05/1996 to 08/27/1999

Treasuries 7.69% 4.61% 0.59 -0.15 3.04Agencies 7.09% 3.71% 0.57 -0.18 3.26Corporates: Investment Grade 7.40% 4.63% 0.52 -0.29 3.79

Corporates: AAA/AA 7.37% 4.57% 0.52 -0.23 3.79Corporates: A 7.57% 5.26% 0.49 -0.34 4.07Corporates: BBB 7.59% 5.17% 0.51 -0.37 4.24

Mortgages: All 6.97% 4.59% 0.43 -0.63 6.84Mortgages: Discount Coupon (-100 bp) 7.16% 5.01% 0.43 -0.45 5.85Mortgages: Current Coupon 6.59% 4.09% 0.39 -1.18 9.93Mortgages: Premium Coupon (+100 bp) 6.32% 3.08% 0.43 -1.79 15.25Mortgages: 15-year 6.91% 3.80% 0.51 -1.19 11.89Mortgages: 30-year 6.98% 4.83% 0.41 -0.53 6.02Mortgages: GNMA 6.85% 4.61% 0.41 -0.48 5.44Mortgages: FNMA 7.00% 4.59% 0.44 -0.67 7.24

High Yield 8.33% 4.26% 0.79 -1.45 7.85EMBI 13.44% 13.20% 0.64 -0.13 8.52Full Portfolio 7.71% 4.31% 0.63 -0.20 4.08

Relative StatisticsAgencies relative to Treasuries -0.10% 1.12% -4.54 -0.47 7.20Corporates relative to Treasuries -0.29% 1.17% -4.51 -0.80 7.28Mortgages relative to Treasuries -0.45% 2.13% -2.55 -1.49 12.92High Yield relative to Treasuries 0.92% 6.30% -0.64 -0.88 6.95EMBI relative to Treasuries 6.16% 13.16% 0.09 -0.76 9.42

Annualized Sharpe Ratio

Annualized Mean

Return

Annualized Standard Deviation

Weekly Skewness

Weekly Kurtosis



11

4-week 95% VaR for Government/Corporate/Mortgage Plus Alternative Sector, August 27, 1999

-6%

-5%

-4%

-3%

-2%

-1%

0%

0% 20% 40% 60% 80% 100%Fraction of Alternative Sector

High YieldEMBI

Variance-Covariance Methodology



12

Ratio of 4-week Simulated Expected Return to 4-week 95% VaR of Gov./Corp./Mtg. Plus Alternative Sector, August 27, 1999

Variance-Covariance Methodology

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0% 20% 40% 60% 80% 100%

Fraction of Alternative Sector

High Yield

EMBI



13

4-week 95% VaR for Government/Corporate/Mortgage Plus Alternative Sector, August 27, 1999

Variance-Covariance Methodology for G/C/M and GARCH-PJ for Alternative Sector

-4.5%

-4.0%

-3.5%

-3.0%

-2.5%

-2.0%

-1.5%

-1.0%

-0.5%

0.0%


High YieldEMBI



14

Ratio of 4-week Simulated Expected Return to 4-week 95% VaR of Gov/Corp/Mtg Plus Alternative Sector, August 27, 1999

Var-Covar Methodology for G/C/M and GARCH-PJ for Alternative Sector

0

0.1

0.2

0.3

0.4

0.5

0.6


High Yield

EMBI



15

• Return optimized over 6 month horizon

– Unchanged term structure and spreads returning to their mean

• Choice of Constraints

– Traditional: Duration

– Value at Risk (4 Week, 95 Percentile)

– Five Stress Scenarios: • Unchanged term structure and spreads (UNCH)

• Unchanged term structure and max spreads (MAXSPD)

• Unchanged term structure and min spreads (MINSPD)

• Max treasury yields and corresponding spreads (MAXTSY)

• Min treasury yields and corresponding spreads (MINTSY)

• Time Period: July 5, 1996 to Aug 27, 1999

Constrained Optimization Test Case



16

45

6473

56 59

91 95 102 96 9783

91 97112

96 100119 124

137122

156

183197

182 182

307

0

50

100

150

200

250

300

350

400

Ag

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1-3

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3-7

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(P

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15-Y

ear

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A C

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1-3

AA

A/A

A C

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3-7

AA

A/A

A C

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7-10

AA

A/A

A C

orp

ora

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10+

AA

A/A

A C

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A C

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A C

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A C

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A C

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A C

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BB

B C

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BB

B C

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3-7

BB

B C

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BB

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TO

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BB

Co

rp.

September 30, 1999 Max 75% Mean 25% Min

Monthly Spread History - Jan 1989 to Sept 1999Source: Salomon Yield Book and State Street Research



17

Optimization Test Case- Initial Data

Benchmark Weights (%) Benchmark Duration Contribution Initial Yield/Spreads

0-3 3-5 5-7 7-10 10-15 TOT 0-3 3-5 5-7 7-10 10-15 TOT 0-3 3-5 5-7 7-10 10-15 TOT

Treasury 14.55 7.34 4.83 2.84 7.49 37.06 0.25 0.28 0.29 0.27 0.87 1.97 5.69 5.89 6.05 6.36 6.32 --

Agencies 3.92 2.30 1.28 0.33 0.98 8.81 0.07 0.09 0.08 0.03 0.11 0.38 50 65 68 54 54 --

Total Govt 18.47 9.64 6.11 3.17 8.47 45.87 0.25 0.28 0.29 0.27 0.87 2.35 -- -- -- -- -- --

AAA/AA 1.50 1.52 1.35 0.35 0.52 5.23 0.03 0.06 0.08 0.03 0.06 0.26 73 81 91 97 103 --

A 1.97 2.20 2.88 0.79 1.39 9.25 0.04 0.09 0.18 0.07 0.16 0.54 91 103 121 119 139 --

BBB 1.16 1.52 1.96 0.67 0.85 6.16 0.02 0.06 0.12 0.06 0.09 0.36 131 154 167 163 173 --

Total Corp 4.63 5.25 6.19 1.81 2.76 20.63 0.07 0.15 0.27 0.10 0.22 1.17 -- -- -- -- -- --

DIS CC PREM TOT DIS CC PREM TOT DIS CC PREM --

GNMA 30 Year 3.75 1.15 1.70 6.60 0.23 0.05 0.05 0.34 105 116 120 --

Conventional 30 Year 15.57 2.32 1.98 19.87 0.83 0.10 0.06 0.99 107 113 124 --

GNMA 15 Year 0.23 0.11 0.09 0.43 0.01 0.00 0.00 0.02 99 100 118 --

Conventional 15 Year 4.91 1.02 0.67 6.60 0.20 0.04 0.01 0.25 102 102 117 --

Total Mortgage 24.46 4.60 4.44 33.50 1.27 0.00 0.00 1.59 -- -- -- -- -- --

HIGH YIELD 0.00 0.00 0.00 0.00 588 --

EMBI 0.00 0.00 0.00 0.00 767 --

Total 100 5.10



18

Optimization Test Case - Scenario Set

Scenario 1 Horizon Scenario 2 Horizon Scenario 3 HorizonDescription UNCH 6M Description MEANSPRD 6M Description MINSPRD 6MProbability 0.0% Probability 100% Probability 0.0%

Yield/Spread Changes Yield/Spread Changes Yield/Spread Changes

0-3 3-5 5-7 7-10 10-15 TOT 0-3 3-5 5-7 7-10 10-15 TOT 0-3 3-5 5-7 7-10 10-15 TOT

Treasury Yield 0.00 0.00 0.00 0.00 0.00 -- 0.00 0.00 0.00 0.00 0.00 -- 0.00 0.00 0.00 0.00 0.00 --

Agencies (Spread) 0.00 0.00 0.00 0.00 0.00 -- -29 -33 -24 -17 -21 -- -51 -58 -42 -38 -34 --

Total Govt -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- --

AAA/AA 0.00 0.00 0.00 0.00 0.00 -- -32 -34 -36 -29 -32 -- -51 -50 -56 -54 -52 --

A 0.00 0.00 0.00 0.00 0.00 -- -37 -39 -45 -32 -45 -- -59 -59 -68 -59 -69 --

BBB 0.00 0.00 0.00 0.00 0.00 -- -55 -64 -62 -36 -48 -- -85 -96 -100 -79 -80 --

Total Corp -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- --

DIS CC PREM -- DIS CC PREM -- DIS CC PREM --

GNMA 30 Year 0.00 0.00 0.00 -- -53 -59 -57 -- -81 -91 -93 --

Conventional 30 Year 0.00 0.00 0.00 -- -49 -55 -65 -- -78 -89 -104 --

GNMA 15 Year 0.00 0.00 0.00 -- -47 -49 -60 -- -83 -85 -103 --

Conventional 15 Year 0.00 0.00 0.00 -- -47 -50 -61 -- -83 -82 -95 --

Total Mortgage -- -- -- -- -- -- -- -- -- -- -- --

HIGH YIELD 0.00 -- -153 -- -286 --

EMBI 0.00 -- -282 -- -530 --



19

Optimization Test Case - Scenario Set

Scenario 4 Horizon Scenario 5 Horizon Scenario 6 HorizonDescription MAXSPRD 6M Description MAXTSY 6M Description MINTSY 6MProbability 0.0% Probability 0.0% Probability 0.0%

Yield/Spread Changes Yield/Spread Changes Yield/Spread Changes

0-3 3-5 5-7 7-10 10-15 TOT 0-3 3-5 5-7 7-10 10-15 TOT 0-3 3-5 5-7 7-10 10-15 TOT

Treasury Yield 0.00 0.00 0.00 0.00 0.00 -- 0.79 0.92 1.01 0.95 1.00 -- -1.36 -1.57 -1.69 -1.46 -1.25 --

Agencies (Spread) 8 5 6 12 13 -- -7 -19 -3 -4 -3 -- -26 -26 -15 -2 -30 --

Total Govt -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- --

AAA/AA 14 15 8 17 10 -- -47 -46 -46 -37 -37 -- 4 -4 -3 7 1 --

A 3 10 6 21 7 -- -55 -56 -60 -45 -63 -- -5 -12 -8 9 -5 --

BBB 6 5 10 41 26 -- -76 -84 -84 -50 -66 -- -23 -37 -12 22 1 --

Total Corp -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- --

DIS CC PREM -- DIS CC PREM -- DIS CC PREM --

GNMA 30 Year 10 7 17 -- -58 -70 -70 -- -1 -17 -17 --

Conventional 30 Year 23 18 13 -- -53 -60 -70 -- 9 -9 -34 --

GNMA 15 Year 45 37 31 -- -62 -66 -79 -- -12 -13 -31 --

Conventional 15 Year 34 27 17 -- -59 -64 -72 -- -2 -2 -16 --

Total Mortgage -- -- -- -- -- -- -- -- -- -- -- --

HIGH YIELD 190 -- -204 -- 116 --

EMBI 256 -- -246 -- 60 --



20

Under No Constraints Emerging Market is Asset of Choice

Unconstrained Optimization(Garch-PJ Approach for EMBI and HYLD)

Optimized Portfolio Weights Optimized Relative Weights Optimized Duration Contributions Relative Duration Contribution

0-3 3-5 5-7 7-10 10-15 TOT 0-3 3-5 5-7 7-10 10-15 TOT 0-3 3-5 5-7 7-10 10-15 TOT 0-3 3-5 5-7 7-10 10-15 TOT

Treasury 0.00 0.00 0.00 0.00 0.00 0.00 -14.55 -7.34 -4.83 -2.84 -7.49 -37.06 0.00 0.00 0.00 0.00 0.00 0.00 -0.25 -0.28 -0.29 -0.27 -0.87 -1.97

Agencies 0.00 0.00 0.00 0.00 0.00 0.00 -3.92 -2.30 -1.28 -0.33 -0.98 -8.81 0.00 0.00 0.00 0.00 0.00 0.00 -0.07 -0.09 -0.08 -0.03 -0.11 -0.38

Total Govt 0.00 0.00 0.00 0.00 0.00 0.00 -18.47 -9.64 -6.11 -3.17 -8.47 -45.87 0.00 0.00 0.00 0.00 0.00 0.00 -0.25 -0.28 -0.29 -0.27 -0.87 -2.35

AAA/AA 0.00 0.00 0.00 0.00 0.00 0.00 -1.50 -1.52 -1.35 -0.35 -0.52 -5.23 0.00 0.00 0.00 0.00 0.00 0.00 -0.03 -0.06 -0.08 -0.03 -0.06 -0.26

A 0.00 0.00 0.00 0.00 0.00 0.00 -1.97 -2.20 -2.88 -0.79 -1.39 -9.25 0.00 0.00 0.00 0.00 0.00 0.00 -0.04 -0.09 -0.18 -0.07 -0.16 -0.54

BBB 0.00 0.00 0.00 0.00 0.00 0.00 -1.16 -1.52 -1.96 -0.67 -0.85 -6.16 0.00 0.00 0.00 0.00 0.00 0.00 -0.02 -0.06 -0.12 -0.06 -0.09 -0.36

Total Corp 0.00 0.00 0.00 0.00 0.00 0.00 -4.63 -5.25 -6.19 -1.81 -2.76 -20.63 0.00 0.00 0.00 0.00 0.00 0.00 -0.07 -0.15 -0.27 -0.10 -0.22 -1.17

DIS CC PREM TOT DIS CC PREM TOT DIS CC PREM TOT DIS CC PREM TOT

GNMA 30 Year 0.00 0.00 0.00 0.00 -3.75 -1.15 -1.70 -6.60 0.00 0.00 0.00 0.00 -0.23 -0.05 -0.05 -0.34

Conventional 30 Year 0.00 0.00 0.00 0.00 -15.57 -2.32 -1.98 -19.87 0.00 0.00 0.00 0.00 -0.83 -0.10 -0.06 -0.99

GNMA 15 Year 0.00 0.00 0.00 0.00 -0.23 -0.11 -0.09 -0.43 0.00 0.00 0.00 0.00 -0.01 0.00 0.00 -0.02


Total Mortgage 0.00 0.00 0.00 0.00 -24.46 -4.60 -4.44 -33.50 0.00 0.00 0.00 0.00 -1.27 0.00 0.00 -1.59

HIGH YIELD 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

EMBI 100.0 100.0 100 100 4.46 4.46 4.46 4.46

Total 100 0.0 4.46 -0.64

Dur Return 4WK 95% VaR 4WK 95% Rel. VaR

Yrs Sc1 Sc2 Sc3 Sc4 Sc5 Sc6 Comp (%) (%)Benchmark (%) 5.1 3.88 5.20 6.01 3.38 0.37 11.36 5.20 -1.86% --

Portfolio (%) 4.46 6.81 22.97 43.11 -1.69 13.12 11.50 22.97 -4.24% -4.42%Difference (yrs or bp) -0.6 293 1777 3710 -507 1275 14 1777 -238 --Constraint (yrs or bp) -- -- -- -- -- -- -- -- -- --



21

Duration Constraint Alone Is Inadequate

Duration-Constrained Optimization(Garch-PJ Approach for EMBI and HYLD)


0-3 3-5 5-7 7-10 10-15 TOT 0-3 3-5 5-7 7-10 10-15 TOT 0-3 3-5 5-7 7-10 10-15 TOT 0-3 3-5 5-7 7-10 10-15 TOT

Treasury 0.00 0.00 0.00 0.00 0.00 0.00 -14.55 -7.34 -4.83 -2.84 -7.49 -37.06 0.00 0.00 0.00 0.00 0.00 0.00 -0.25 -0.28 -0.29 -0.27 -0.87 -1.97

Agencies 0.00 0.00 0.00 0.00 0.00 0.00 -3.92 -2.30 -1.28 -0.33 -0.98 -8.81 0.00 0.00 0.00 0.00 0.00 0.00 -0.07 -0.09 -0.08 -0.03 -0.11 -0.38

Total Govt 0.00 0.00 0.00 0.00 0.00 0.00 -18.47 -9.64 -6.11 -3.17 -8.47 -45.87 0.00 0.00 0.00 0.00 0.00 0.00 -0.25 -0.28 -0.29 -0.27 -0.87 -2.35

AAA/AA 0.00 0.00 0.00 0.00 0.28 0.28 -1.50 -1.52 -1.35 -0.35 -0.23 -4.95 0.00 0.00 0.00 0.00 0.03 0.03 -0.03 -0.06 -0.08 -0.03 -0.03 -0.23

A 0.00 0.00 0.00 0.00 0.48 0.48 -1.97 -2.20 -2.88 -0.79 -0.91 -8.76 0.00 0.00 0.00 0.00 0.06 0.06 -0.04 -0.09 -0.18 -0.07 -0.10 -0.49

BBB 0.00 0.00 0.00 0.00 0.54 0.54 -1.16 -1.52 -1.96 -0.67 -0.31 -5.62 0.00 0.00 0.00 0.00 0.06 0.06 -0.02 -0.06 -0.12 -0.06 -0.03 -0.30

Total Corp 0.00 0.00 0.00 0.00 1.31 1.31 -4.63 -5.25 -6.19 -1.81 -1.45 -19.33 0.00 0.00 0.00 0.00 0.15 0.15 -0.07 -0.15 -0.27 -0.10 -0.07 -1.02


GNMA 30 Year 0.00 0.00 0.00 0.00 -3.75 -1.15 -1.70 -6.60 0.00 0.00 0.00 0.00 -0.23 -0.05 -0.05 -0.34


GNMA 15 Year 0.00 0.00 0.00 0.00 -0.23 -0.11 -0.09 -0.43 0.00 0.00 0.00 0.00 -0.01 0.00 0.00 -0.02


Total Mortgage 0.08 0.00 0.00 0.08 -24.38 -4.60 -4.44 -33.42 0.00 0.00 0.00 0.00 -1.26 -0.08 0.00 -1.58

HIGH YIELD 19.01 19.01 19.01 19.01 0.90 0.90 0.90 0.90

EMBI 79.7 79.7 80 80 3.55 3.55 3.55 3.55

Total 100 0.0 4.60 -0.50



Portfolio (%) 4.6 6.58 20.92 38.26 -1.88 12.31 10.88 20.92 -3.34% -3.62%Difference (yrs or bp) -0.5 270 1572 3225 -526 1194 -48 1572 -148 --Constraint (yrs or bp) 0.5 -- -- -- -- -- -- -- -- --



22

VaR Constraint Leads to Reasonable Allocation

VaR Constrained Optimization(Garch-PJ Approach for EMBI and HYLD)


0-3 3-5 5-7 7-10 10-15 TOT 0-3 3-5 5-7 7-10 10-15 TOT 0-3 3-5 5-7 7-10 10-15 TOT 0-3 3-5 5-7 7-10 10-15 TOT

Treasury 1.22 1.02 0.42 4.38 3.43 10.48 -13.33 -6.32 -4.41 1.54 -4.06 -26.6 0.02 0.04 0.03 0.41 0.40 0.90 -0.23 -0.24 -0.27 0.15 -0.47 -1.07

Agencies 1.77 0.18 0.07 0.76 0.37 3.15 -2.15 -2.12 -1.21 0.42 -0.61 -5.7 0.03 0.01 0.00 0.07 0.04 0.15 -0.04 -0.08 -0.07 0.04 -0.07 -0.22

Total Govt 2.99 1.20 0.49 5.14 3.80 13.63 -15.48 -8.44 -5.62 1.97 -4.67 -32.2 0.05 0.05 0.03 0.48 0.44 1.05 -0.20 -0.23 -0.26 0.22 -0.43 -1.29

AAA/AA 2.81 0.47 5.37 0.08 0.30 9.03 1.31 -1.05 4.03 -0.27 -0.22 3.8 0.06 0.02 0.34 0.01 0.03 0.46 0.03 -0.04 0.25 -0.02 -0.03 0.19

A 5.06 3.90 8.22 0.77 0.08 18.04 3.09 1.70 5.34 -0.02 -1.31 8.8 0.10 0.16 0.52 0.07 0.01 0.86 0.06 0.07 0.34 0.00 -0.15 0.32

BBB 3.65 0.99 7.92 1.06 0.10 13.72 2.50 -0.54 5.96 0.39 -0.76 7.6 0.07 0.04 0.50 0.10 0.01 0.72 0.05 -0.02 0.37 0.04 -0.08 0.36

Total Corp 11.52 5.36 21.51 1.91 0.48 40.78 6.89 0.11 15.33 0.10 -2.29 20.1 0.24 0.21 1.35 0.18 0.05 2.03 0.16 0.07 1.09 0.07 -0.16 0.86


GNMA 30 Year 10.60 6.46 1.41 18.48 6.85 5.31 -0.29 11.9 0.65 0.30 0.05 0.99 0.42 0.24 -0.01 0.65


GNMA 15 Year 0.81 0.16 0.85 1.83 0.58 0.05 0.76 1.4 0.04 0.01 0.02 0.06 0.03 0.00 0.02 0.05

Conventional 15 Year 0.32 0.21 3.64 4.17 -4.59 -0.81 2.97 -2.4 0.01 0.01 0.07 0.09 -0.19 -0.03 0.06 -0.15

Total Mortgage 19.07 7.28 7.52 33.87 -5.39 2.68 3.08 0.4 1.09 0.33 0.19 1.61 -0.18 -18.74 -7.09 0.02

HIGH YIELD 9.34 9.34 9.34 9.3 0.44 0.44 0.44 0.44

EMBI 2.4 2.4 2 2.4 0.11 0.11 0.11 0.11

Total 100 0.0 5.24 0.14



Portfolio (%) 5.24 4.52 7.32 9.41 3.05 2.46 11.95 7.32 -1.83% -0.20%Difference (yrs or bp) 0.14 64 212 340 -33 209 59 212 2 --Constraint (yrs or bp) -- -- -- -- -- -- -- -- -- -20



23

Addition of Duration Constraint Modifies Solution Modestly

Duration and VaR Constrained Optimization(Garch-PJ Approach for EMBI and HYLD)


0-3 3-5 5-7 7-10 10-15 TOT 0-3 3-5 5-7 7-10 10-15 TOT 0-3 3-5 5-7 7-10 10-15 TOT 0-3 3-5 5-7 7-10 10-15 TOT

Treasury 3.41 3.51 1.06 3.34 2.54 13.9 -11.14 -3.83 -3.77 0.50 -4.95 -23.2 0.06 0.13 0.06 0.32 0.30 0.87 -0.19 -0.15 -0.23 0.05 -0.58 -1.10

Agencies 1.16 2.45 0.00 0.01 0.45 4.1 -2.76 0.15 -1.28 -0.33 -0.53 -4.7 0.02 0.09 0.00 0.00 0.05 0.17 -0.05 0.01 -0.08 -0.03 -0.06 -0.21

Total Govt 4.58 5.96 1.06 3.35 2.99 17.9 -13.90 -3.68 -5.05 0.17 -5.48 -27.9 0.08 0.23 0.06 0.32 0.35 1.04 -0.17 -0.05 -0.23 0.05 -0.53 -1.31

AAA/AA 1.37 1.72 3.24 0.01 0.07 6.4 -0.13 0.20 1.89 -0.34 -0.44 1.2 0.03 0.07 0.20 0.00 0.01 0.31 0.00 0.01 0.12 -0.03 -0.05 0.04

A 2.16 5.24 6.24 2.44 0.14 16.2 0.18 3.04 3.35 1.65 -1.25 7.0 0.04 0.21 0.39 0.22 0.02 0.88 0.00 0.12 0.21 0.15 -0.14 0.34

BBB 2.05 3.34 5.12 1.65 0.04 12.2 0.89 1.81 3.16 0.98 -0.82 6.0 0.04 0.13 0.32 0.15 0.00 0.65 0.02 0.07 0.20 0.09 -0.09 0.29

Total Corp 5.57 10.30 14.59 4.10 0.25 34.8 0.94 5.05 8.41 2.29 -2.51 14.2 0.11 0.41 0.92 0.37 0.03 1.84 0.04 0.26 0.65 0.27 -0.19 0.67


GNMA 30 Year 6.57 7.63 0.20 14.4 2.82 6.48 -1.50 7.8 0.40 0.35 0.01 0.76 0.17 0.30 -0.05 0.42

Conventional 30 Year 17.06 3.16 0.07 20.3 1.49 0.84 -1.91 0.4 0.91 0.13 0.00 1.04 0.08 0.04 -0.06 0.06

GNMA 15 Year 0.02 0.02 0.01 0.0 -0.21 -0.09 -0.08 -0.4 0.00 0.00 0.00 0.00 -0.01 0.00 0.00 -0.01

Conventional 15 Year 0.34 0.02 0.46 0.8 -4.57 -1.00 -0.21 -5.8 0.01 0.00 0.01 0.02 -0.19 -0.03 0.00 -0.22

Total Mortgage 23.99 10.83 0.74 35.6 -0.47 6.23 -3.70 2.1 1.33 0.48 0.02 1.83 0.06 -23.51 -10.81 0.24

HIGH YIELD 10.89 10.9 10.89 10.9 0.51 0.51 0.51 0.51

EMBI 0.8 0.8 1 0.8 0.04 0.04 0.04 0.04

Total 100.0 0.0 5.26 0.16



Portfolio (%) 5.26 4.63 7.27 9.15 3.07 2.44 11.85 7.27 -1.83% -0.20%Difference (yrs or bp) 0.16 75 207 314 -31 207 49 207 3 --Constraint (yrs or bp) 0.5 -- -- -- -- -- -- -- -- -20



24

Duration and Scenario Constraints AloneCould Lead to Extreme Solutions

Duration and Stress-Scenario Constrained Optimization(Garch-PJ Approach for EMBI and HYLD)


0-3 3-5 5-7 7-10 10-15 TOT 0-3 3-5 5-7 7-10 10-15 TOT 0-3 3-5 5-7 7-10 10-15 TOT 0-3 3-5 5-7 7-10 10-15 TOT

Treasury 0.00 0.00 0.00 0.00 0.00 0.0 -14.55 -7.34 -4.83 -2.84 -7.49 -37.1 0.00 0.00 0.00 0.00 0.00 0.00 -0.25 -0.28 -0.29 -0.27 -0.87 -1.97

Agencies 0.00 0.00 0.00 0.00 0.00 0.0 -3.92 -2.30 -1.28 -0.33 -0.98 -8.8 0.00 0.00 0.00 0.00 0.00 0.00 -0.07 -0.09 -0.08 -0.03 -0.11 -0.38

Total Govt 0.00 0.00 0.00 0.00 0.00 0.0 -18.47 -9.64 -6.11 -3.17 -8.47 -45.9 0.00 0.00 0.00 0.00 0.00 0.00 -0.25 -0.28 -0.29 -0.27 -0.87 -2.35

AAA/AA 0.00 0.00 0.00 0.00 0.00 0.0 -1.50 -1.52 -1.35 -0.35 -0.52 -5.2 0.00 0.00 0.00 0.00 0.00 0.00 -0.03 -0.06 -0.08 -0.03 -0.06 -0.26

A 0.00 0.00 0.00 0.00 0.00 0.0 -1.97 -2.20 -2.88 -0.79 -1.39 -9.2 0.00 0.00 0.00 0.00 0.00 0.00 -0.04 -0.09 -0.18 -0.07 -0.16 -0.54

BBB 0.00 0.00 0.00 0.00 0.00 0.0 -1.16 -1.52 -1.96 -0.67 -0.85 -6.2 0.00 0.00 0.00 0.00 0.00 0.00 -0.02 -0.06 -0.12 -0.06 -0.09 -0.36

Total Corp 0.00 0.00 0.00 0.00 0.00 0.0 -4.63 -5.25 -6.19 -1.81 -2.76 -20.6 0.00 0.00 0.00 0.00 0.00 0.00 -0.07 -0.15 -0.27 -0.10 -0.22 -1.17


GNMA 30 Year 33.03 29.28 0.00 62.3 29.28 28.13 -1.70 55.7 2.03 1.34 0.00 3.37 1.80 1.28 -0.05 3.03


GNMA 15 Year 0.00 0.00 0.00 0.0 -0.23 -0.11 -0.09 -0.4 0.00 0.00 0.00 0.00 -0.01 0.00 0.00 -0.02


Total Mortgage 33.03 29.28 0.00 62.3 8.57 24.68 -4.44 28.8 2.03 1.34 0.00 3.37 0.76 -31.70 -29.28 1.78

HIGH YIELD 0.00 0.0 0.00 0.0 0.00 0.00 0.00 0.00

EMBI 37.7 37.7 38 37.7 1.68 1.68 1.68 1.68

Total 100.0 0.0 5.05 -0.05



Portfolio (%) 5.05 6.37 14.00 22.33 2.88 6.89 12.04 14.00 -2.46% -1.92%Difference (yrs or bp) -0.1 249 880 1632 -50 652 68 880 -60 --Constraint (yrs or bp) 0.5 50 -- 50 50 50 50 -- -- --



25

VaR and Stress-Scenario Constraints Combined Lead to Reasonable Overall Allocation

VaR and Stress-Scenario Constrained Optimization(Garch-PJ Approach for EMBI and HYLD)


0-3 3-5 5-7 7-10 10-15 TOT 0-3 3-5 5-7 7-10 10-15 TOT 0-3 3-5 5-7 7-10 10-15 TOT 0-3 3-5 5-7 7-10 10-15 TOT

Treasury 2.72 2.64 0.07 1.36 1.63 8.4 -11.83 -4.71 -4.76 -1.48 -5.86 -28.6 0.05 0.10 0.00 0.13 0.19 0.47 -0.21 -0.18 -0.29 -0.14 -0.68 -1.50

Agencies 2.00 0.83 0.00 1.13 0.26 4.2 -1.92 -1.47 -1.27 0.80 -0.72 -4.6 0.04 0.03 0.00 0.10 0.03 0.20 -0.03 -0.06 -0.08 0.07 -0.08 -0.18

Total Govt 4.73 3.47 0.08 2.49 1.88 12.6 -13.75 -6.17 -6.03 -0.68 -6.59 -33.2 0.08 0.13 0.00 0.23 0.22 0.67 -0.17 -0.15 -0.29 -0.04 -0.65 -1.67

AAA/AA 2.67 2.04 0.00 0.53 0.11 5.4 1.17 0.52 -1.34 0.18 -0.41 0.1 0.05 0.08 0.00 0.05 0.01 0.19 0.02 0.02 -0.08 0.02 -0.05 -0.07

A 3.37 2.75 0.02 1.53 2.21 9.9 1.40 0.55 -2.86 0.74 0.82 0.6 0.07 0.11 0.00 0.14 0.25 0.57 0.03 0.02 -0.18 0.07 0.09 0.03

BBB 4.39 10.11 20.06 0.12 0.23 34.9 3.23 8.59 18.10 -0.55 -0.63 28.7 0.09 0.40 1.26 0.01 0.03 1.79 0.07 0.34 1.14 -0.05 -0.07 1.42

Total Corp 10.43 14.90 20.08 2.18 2.54 50.1 5.80 9.66 13.90 0.37 -0.22 29.5 0.21 0.59 1.26 0.20 0.29 2.55 0.14 0.44 0.99 0.09 0.07 1.38


GNMA 30 Year 16.15 8.72 0.19 25.1 12.40 7.57 -1.51 18.5 0.99 0.40 0.01 1.40 0.76 0.35 -0.05 1.06


GNMA 15 Year 0.28 1.00 0.08 1.4 0.05 0.89 -0.01 0.9 0.01 0.04 0.00 0.05 0.00 0.03 0.00 0.04


Total Mortgage 18.52 9.74 4.12 32.4 -5.94 5.14 -0.32 -1.1 1.12 0.44 0.09 1.64 -0.15 -18.08 -9.65 0.05

HIGH YIELD 4.83 4.8 4.83 4.8 0.23 0.23 0.23 0.23

EMBI 0.0 0.0 0 0.0 0.00 0.00 0.00 0.00

Total 100 0.0 5.09 -0.01



Portfolio (%) 5.09 4.39 6.98 8.60 3.52 2.50 12.12 6.98 -1.79% -0.20%Difference (yrs or bp) -0 51 178 259 14 213 76 178 6 --Constraint (yrs or bp) -- 50 -- 50 50 50 50 -- -- -20



26

VaR and Stress-Scenario Constraints Combined Lead to Reasonable Overall Allocation

Duration, VaR and Stress-Scenario Constrained Optimization(Garch-PJ Approach for EMBI and HYLD)


0-3 3-5 5-7 7-10 10-15 TOT 0-3 3-5 5-7 7-10 10-15 TOT 0-3 3-5 5-7 7-10 10-15 TOT 0-3 3-5 5-7 7-10 10-15 TOT

Treasury 2.24 2.55 2.01 3.14 3.27 13.2 -12.31 -4.79 -2.82 0.30 -4.22 -23.8 0.04 0.10 0.12 0.30 0.38 0.94 -0.21 -0.18 -0.17 0.03 -0.49 -1.03

Agencies 2.39 0.01 0.01 2.11 3.00 7.5 -1.53 -2.29 -1.27 1.78 2.03 -1.3 0.04 0.00 0.00 0.20 0.34 0.58 -0.03 -0.09 -0.08 0.16 0.23 0.20

Total Govt 4.62 2.56 2.02 5.25 6.27 20.7 -13.85 -7.09 -4.09 2.08 -2.20 -25.1 0.08 0.10 0.12 0.49 0.72 1.52 -0.17 -0.18 -0.17 0.22 -0.15 -0.83

AAA/AA 2.70 0.82 0.02 0.02 0.34 3.9 1.20 -0.70 -1.33 -0.33 -0.18 -1.3 0.06 0.03 0.00 0.00 0.04 0.13 0.02 -0.03 -0.08 -0.03 -0.02 -0.14

A 5.04 4.06 0.03 0.11 0.02 9.2 3.06 1.85 -2.86 -0.69 -1.37 0.0 0.10 0.16 0.00 0.01 0.00 0.28 0.06 0.07 -0.18 -0.06 -0.16 -0.26

BBB 3.39 5.87 15.77 0.08 0.03 25.1 2.23 4.35 13.81 -0.58 -0.83 19.0 0.07 0.23 0.99 0.01 0.00 1.30 0.05 0.17 0.87 -0.05 -0.09 0.94

Total Corp 11.13 10.75 15.82 0.21 0.39 38.3 6.50 5.50 9.63 -1.60 -2.38 17.7 0.23 0.43 0.99 0.02 0.04 1.71 0.16 0.28 0.73 -0.08 -0.17 0.54


GNMA 30 Year 18.05 10.03 0.27 28.4 14.30 8.88 -1.43 21.8 1.11 0.46 0.01 1.58 0.88 0.41 -0.05 1.24


GNMA 15 Year 0.38 0.23 0.33 0.9 0.15 0.12 0.24 0.5 0.02 0.01 0.01 0.03 0.01 0.00 0.01 0.02


Total Mortgage 21.67 10.28 1.69 33.6 -2.79 5.68 -2.75 0.1 1.30 0.47 0.04 1.81 0.03 -21.20 -10.23 0.22

HIGH YIELD 4.91 4.9 4.91 4.9 0.23 0.23 0.23 0.23

EMBI 2.5 2.5 2 2.5 0.11 0.11 0.11 0.11

Total 100.0 0.0 5.37 0.27



Portfolio (%) 5.37 4.50 7.18 9.11 3.44 2.13 12.44 7.18 -1.88% -0.20%Difference (yrs or bp) 0.28 62 198 310 6 176 108 198 -3 --Constraint (yrs or bp) 0.5 50 -- 50 50 50 50 -- -- -20



27

VaR-Constrained Efficient Frontier

0

0.5

1

1.5

2

2.5

3

3.5

4

10 20 30 40 50

Relative VaR Tolerance (bp)

Exce

ss R

etur

n (%

)



28

VaR-Constrained Efficient Frontier

Relative VaR Tolerance10bp 20bp 30bp 40bp 50bp Benchmark

WeightDur.

Cont. WeightDur.

Cont. WeightDur.

Cont. WeightDur.

Cont. WeightDur.

Cont. Weight

Dur. Cont.

Treasury 17.9 1.4 10.5 0.9 9.5 0.6 6.1 0.6 9.4 0.5 37.1 2.0

Agencies 4.6 0.2 3.2 0.2 4.4 0.2 1.9 0.1 5.7 0.4 8.8 0.4

Total Govt 22.5 1.6 13.6 1.1 13.9 0.8 8.0 0.6 15.1 0.9 45.9 2.4

AAA/AA 6.2 0.3 9.0 0.5 5.8 0.3 3.5 0.2 7.1 0.6 5.2 0.3

A 10.1 0.3 18.0 0.9 11.7 0.6 10.0 0.6 10.0 0.7 9.2 0.5

BBB 23.9 1.2 13.7 0.7 21.1 1.3 26.1 1.6 15.7 1.1 6.2 0.4

Total Corp 40.2 1.8 40.8 2.0 38.6 2.3 39.6 2.3 32.8 2.4 20.6 1.2

GNMA 30 Year 1.1 0.0 1.8 0.1 0.9 0.0 0.2 0.0 0.0 0.0 0.4 0.3

Conventional 30 Year 3.4 0.1 4.2 0.1 2.4 0.1 1.5 0.1 1.0 0.0 6.6 1.0

GNMA 15 Year 11.4 0.6 18.5 1.0 14.5 0.8 15.6 0.9 10.0 0.6 6.6 0.0

Conventional 15 Year 17.3 0.9 9.4 0.5 12.9 0.7 15.2 0.8 18.9 1.0 19.9 0.3

Total Mortgage 33.2 1.6 33.9 1.6 30.7 1.5 32.5 1.7 30.0 1.6 33.5 1.6

HIGH YIELD 3.4 0.2 9.3 0.4 12.4 0.6 13.5 0.6 13.4 0.6 0.0 0.0

EMBI 0.7 0.0 2.4 0.1 4.4 0.2 6.4 0.3 8.7 0.4 0.0 0.0

Total 100.0 5.2 100.0 5.2 100.0 5.4 100.0 5.6 100.0 5.9 100.0 5.1



29

Conclusions

VaR Model Choice Is Significant in Assessing Risk-Reward in Portfolios

• Risk-Constrained Optimization Facilitates Integration of Various Sector Views In Portfolio Construction

Experience Is Required to Select Suitable Model Parameters and Choice of Constraints

Ex Post Relative Valuation Process Honing

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