1. FFIEC Capital Markets Conference Portfolio Management and Theory Steve Mandel May 18-19, 2004

2. Portfolio Management Tools Security Level Portfolio Level Portfolio vs. Liabilities Nominal Yield/Risk Measures Scenario Analysis Effective Yield/Risk Measures Optimization Return Attribution

3. Nominal Yield/Risk Measures Nominal Yield Measures –Current Yield –Yield (to Maturity, to Worst) –Spread to Benchmark –Spread to Yield Curve Nominal Risk Measures –Years to Maturity (Average Life) –Nominal Duration (Macaulay, Modified) –Nominal Convexity

5. Nominal Yield Measures Current Yield FHLB 4.25 11/15/2010 Price (2/29/2004) = 103.369 Current Yield = 4.111%

6. Nominal Yield Measures Yield to Maturity – The discount rate at which the present value of the cash flows equals the full price of the bond.

7. Yield to Maturity

8. Nominal Yield Measures Spread to Benchmark - The difference between the yield of a security and the yield of a corresponding benchmark security stated in basis points (1 bp=.01%) The benchmark is typically an On-the-Run Treasury closest to the maturity of the security (or average life for an amortizing security)

9. Nominal Yield Measures Spread to Benchmark (Continued) Benchmark Security: US 5 2/15/2011 Yield of Bond:3.68 Yield of Benchmark Security: 3.47 Spread to Benchmark: 0.21 (21basis points)

10. Nominal Yield Measures Spread to Yield Curve - The difference between a security’s yield and the interpolated point on the yield curve corresponding to the security’s average life, stated in basis points (1 bp=.01%) –On-The-Run Treasury Curve –Off-The-Run Treasury Model Curve –Swap Curve

12. Nominal Yield Measures Spread to On-the-Run Treasury Yield Curve –Yield Curve: On-The-Run Tsy (2/27/2004) –Average Life of Security: 6.71 –Interpolated Point on Yield Curve: 3.302 –Yield of Security: 3.678 –Spread to Yield Curve:100x(3.68-3.30)=38bp

14. Yield Curves (2/27/2004) A -Treasury On-the-Run, B - Treasury Off-the-Run

15. Nominal Yield Measures Spread to Off-the-Run Treasury Yield Curve –Yield Curve: Off-The-Run Tsy (2/27/2004) –Average Life of Security: 6.71 –Interpolated Point on Yield Curve: 3.442 –Yield of Security: 3.678 –Spread to Yield Curve:100x(3.68-3.44)=24bp

16. Yield Curves (2/27/2004) A -Tsy On-the-Run, B - Tsy Off-the-Run, C - Swap

17. Nominal Yield Measures Spread to Swap Yield Curve –Yield Curve: Swap (2/27/2004) –Average Life of Security: 6.71 –Interpolated Point on Yield Curve: 3.820 –Yield of Security: 3.678 –Spread to Yield Curve:100x(3.68-3.82)= -14bp

18. Nominal Risk Measures Years to Maturity (Average Life): 6.71 Macaulay Duration - Percentage change in Price for a percentage change in Yield. (Average life of PV of Cash Flows)

19. Macaulay Duration

20. Nominal Risk Measures Years to Maturity Macaulay Duration Modified Duration - Percentage change in Price for a 100 basis point change in Yield. The tangent (first derivative) of the price/yield curve for a given yield. Modified Duration = 5.851/(1+3.678/200) = 5.746

21. Modified Duration

22. Nominal Risk Measures Years to Maturity Macaulay Duration Modified Duration Nominal Convexity - measures the degree to which the price/yield curve of a security differs from the tangent at the current yield.

23. Portfolio Management Tools Security Level Portfolio Level Portfolio vs. Liabilities Nominal Yield/Risk Measures Scenario Analysis Effective Yield/Risk Measures Optimization Return Attribution

24. Effective Yield/Risk Measures Effective Yield Measures –OAS –Yield Curve Margin* Effective Risk Measures –Effective Duration –Partial Durations –Effective Convexity –Spread Duration –Volatility Duration –Prepay Duration

25. Effective Yield Measures Option Adjusted Spread OAS –A security’s spread (in basis points) over the yield curve, after adjusting for the probability of any optional calls, puts, or prepayments and assuming a volatility (or set of volatilities) of future yields. –The spread over the yield curve’s forward rates (multiple rate paths are considered) that makes the present value of the cash flows equal to the full price.

26. Option Adjusted Spread (OAS) Bonds Without Embedded Options – OAS is not dependent on volatility and will be close to nominal spread (small difference due to the shape of the yield curve). OAS depends on Price and Yield Curve Bonds With Embedded Options – OAS will depend on Price, Yield Curve and the volatility assumption. For callable bonds and mortgages the higher the volatility assumption the lower the OAS.

29. Effective Yield Measures Yield Curve Margin – OAS assuming a zero volatility. The spread over the yield curve’s forward rates that makes the present value of the cash flows equal to the full price. Option Cost = Yield Curve Margin – OAS

30. Effective Risk Measures Effective Duration –A measure of the sensitivity (percent change) of the Full Price of a security to a (100 bp) parallel shift of the Yield Curve. –Utilized to measure a security’s price sensitivity to a change in the general level of interest rates.

33. Effective Duration Calculation

34. Effective Risk Measures Effective Duration Partial Duration - A measure of the sensitivity (percent change) of the full price of a security to a move in a single “key rate” point of the Yield Curve. Utilized to measure a security’s sensitivity to a particular reshaping of the Yield Curve

37. Partial Duration Calculation Partial Duration (5Year) YC Point Partial Duration 1.337 2.174 3.300 5.548 10.738 20.347 30 -.053 Total 2.39

38. Effective Risk Measures Effective Duration Partial Duration Effective Convexity - measures the degree to which the price/parallel-shift curve of a security differs from the tangent at the current curve. A measure of the sensitivity of the Effective Duration of a security to a parallel shift of the Yield Curve so as to measure the sensitivity of price to “large” rate moves.

39. Effective Convexity Positive Convexity implies P/Y curve is above tangent. –Effective Duration goes up as rates come down. –P/Y curve gets steeper as rates come down. Negative convexity implies P/Y curve falls below tangent. –Effective Duration goes down as rates come down. –P/Y curve flattens as rates come down.

40. Effective Convexity Calculation

42. Effective Risk Measures Effective Duration Partial Duration Effective Convexity Volatility Duration - A measure of the sensitivity (percent change) of the full price of a security to changes in Volatility.

44. Term Structure of Volatilities

45. Volatility Durations

46. Effective Risk Measures Effective Duration Partial Duration Effective Convexity Volatility Duration Pre Pay Duration - The sensitivity of a (mortgage) security’s full price to changes in Prepayment Rates

47. Prepay Durations

48. Portfolio Management Tools Security Level Portfolio Level Portfolio vs. Liabilities Nominal Yield/Risk Measures Scenario Analysis Effective Yield/Risk Measures Optimization Return Attribution

49. Portfolio Risk Measures The Portfolio Risk Measures are analogous to the Security Measures in that they are measures of the sensitivity (percent change) of a Portfolio’s Market Value to various market changes. –They are calculated by taking a Market Weighted Average of the Individual Security Risk Measures.

50. Portfolio Risk Measures Effective Duration - A measure of the sensitivity (percent change) of the Market Value of a Portfolio to a parallel shift in the Yield Curve. –The Market Weighted Average of the individual securities Effective Durations

52. Portfolio Effective Duration Market Weighted Average of Individual Security Effective Durations = 3.93% or Percent MV Change (+/- 25 bp) on Portfolio

54. Portfolio Risk Measures Effective Duration Partial Durations - Measure the sensitivity of a Portfolio’s Market Value to reshapings of the Yield Curves –Market Weighted Average of Individual Security Partial Durations

56. Portfolio vs Benchmark/Liability Risk Measures Measures of the of the sensitivity of the ROR difference between the Portfolio and Benchmark (or Liabilities) to various market changes

59. Portfolio Management Tools Security Level Portfolio Level Portfolio vs. Liabilities Nominal Yield/Risk Measures Scenario Analysis Effective Yield/Risk Measures Optimization Return Attribution

60. Scenario Analysis Framework for evaluating the combined effect of Yield and Risk Measures for a range of assumptions. Nominal Return – Rate of Return on a security assuming it was purchased on a certain begin (settlement) date and sold on a certain horizon date. The return calculation takes into account the settlement full price, the horizon full price, intermediate cash flows from the security (coupon plus any principal payments) plus reinvestment of any intermediate cash flow payment to the horizon date.

61. Scenario Analysis Security: FHLB 4.25 11/15/2010 Settlement Date: 2/29/2004 Horizon Date: 2/28/2005 Yield Curve Assumption: No Change Pricing Assumption: Constant Spread to Yield Curve

62. Rolling Yield Scenario Return calculation assuming that at the horizon the security will have the same spread (nominal or OAS) to the yield curve as the beginning spread. For a positive yield curve assuming Rolling Yield decreases the horizon yield and increases the expected return.

63. Rolling Yield Calculation Constant Nominal Spread

64. Beginning Price of Security = 103.366 Yield of Security = 3.678 Interpolated Yield Curve = 3.442 Nominal Spread to Curve =.236 Horizon Interpolated Yield Curve = 3.165 Nominal Spread to Curve =.236 Yield of Security = 3.401 Price of Security = 104.362

65. Return Calculation Beg Full Price = 103.366 + 1.334 = 104.700 Hor Full Price = 104.362 + 1.216 = 105.578 Coupon = 2.231 + 2.125 = 4.356 Reinv =.022

66. Rolling Yield – Constant Nominal Spread

67. Rolling Yield Calculation Constant OAS For a bond without embedded options assuming constant nominal spread produces results similar to those produced by the more accurate constant OAS method.

69. Rolling Yield – Constant OAS

70. Rolling Yield Calculation Constant OAS For a bond without embedded options assuming constant nominal spread produces results similar to those produced by the more accurate constant OAS method. For a bond with embedded options such as callable bonds and mortgage backed securities using constant OAS produces significantly different and more accurate results especially for large yield curve shifts.

71. Rolling Yield Constant CPR & Nominal Spread

72. Rolling Yield Constant CPR & Nominal Spread

73. Rolling Yield - Model Prepay Projections & Constant OAS

74. Portfolio Management Tools Security Level Portfolio Level Portfolio vs. Liabilities Nominal Yield/Risk Measures Scenario Analysis Effective Yield/Risk Measures Optimization Return Attribution

75. Scenario Analysis Parallel Shifts – 3 Months Horizon

77. Pct ROR for Assets vs Liabilities Parallel Shifts – 3 Months Horizon

78. Principal Component Scenarios Statistically likely re-shapings of the Yield Curve derived through analysis of 15 years of monthly movements in the Off- The-Run Treasury Yield Curve. These scenarios model 95% of observed movements in the Yield Curve. That is 95% of the monthly movements can be represented as a linear combination of the Principal Component Scenarios.

79. Principal Components Scenarios

80. Principal Components Combination Scenarios

81. Scenario Analysis Principal Comp Comb Scenarios- 3 Month Horizon

82. Pct ROR for Assets vs Liabilities Princ Component Scenarios – 3 Months Horizon

83. Portfolio Management Tools Security Level Portfolio Level Portfolio vs Liabilities Nominal Yield/Risk Measures Scenario Analysis Effective Yield/Risk Measures Optimization Return Attribution

84. Portfolio Optimization A methodology utilizing mathematical procedures such as Linear Programming to optimize portfolios given: –Universe of available securities –A Portfolio Objective –Series of Portfolio Constraints

85. Portfolio Optimization Duration Target Example Universe – All Securities in Citigroup Treasury Index Objective – Maximize Average Yield Constraints –Average Duration = 5 –Total Market Value = $50mm

88. Portfolio Optimization Tips Remember optimizer is just a very powerful (but dumb) tool which can quickly evaluate all possible combinations to identify the optimal solution. The solution is only as good as the formulation of objective and constraints. Since the objective was to max YTM the optimizer incorrectly selected a callable bond trading to call.

91. Portfolio Optimization Tips The number of securities in the optimal portfolio are equal to the number of binding constraints. To increase the number of securities in a portfolio add more constraints such as per issue limits. As you add more (binding) constraints the value of the objective function gets marginally worse.

93. Portfolio Optimization Cash Matching Example 1 Universe – All non callable securities in Citigroup Treasury Index Objective – Minimize Cost Constraints - Cash Match Liability Schedule

97. Optimization Tips Minimum Cost for cash matching liabilities can be reduced by marginally increasing risk –Increasing reinvestment rate assumption –Lowering quality of portfolio (agencies, corporates etc., control risk with issue/sector limits) –Allowing callable bonds and mortgages (control risk by imposing a series of scenario dependent cash flow constraints).

98. Portfolio Optimization Cash Matching Example 2 Universe – Treasury, Agency, and Mortgage securities from the Citigroup BIG Index Objective – Minimize Cost Constraints - Cash Match Liabilities for each of 7 Principal Component Scenarios.

101. Portfolio Optimization Immunization Example Universe – All non callable securities in Citigroup Treasury Index Objective – Maximize IRR of Portfolio Constraints – –Market Value = PV of Liabilities at estimated IRR of Portfolio –Duration of Portfolio = Duration of Liabilities at estimated IRR rate Iterate until IRR of Portfolio equals estimated IRR

109. Portfolio Optimization Contingent Immunization Example Add a buffer of additional Market Value to the portfolio beyond the minimum required for an immunized portfolio. Use that buffer for additional flexibility to tilt the duration of the portfolio away from the duration of the liabilities so as to maximize return per market view of manager. Impose constraints to insure that in the event of adverse market moves there is sufficient remaining Market Value to immunize.

113. Portfolio Optimization Assets vs Liabilities Optimize Assets Optimize Liabilities Optimize Assets and Liabilities Simultaneously (Dual Optimization)

114. Portfolio Management Tools Security Level Portfolio Level Portfolio vs Liabilities Nominal Yield/Risk Measures Scenario Analysis Effective Yield/Risk Measures Optimization Return Attribution

115. Return Attribution Return Attribution dissects the return of fixed income securities, trades, portfolios, indices and other benchmarks such as liability portfolios. The goal is to “explain” returns by decomposing the total return of each security into components corresponding to the effect of various market changes such as yield curve movement, volatility changes, sector spread changes etc.

116. The Yield Book Return Attribution Model 3 Steps To Portfolio Management Process –Select Duration and Yield Curve Exposure –Select Sector Weights –Select Specific Issues

117. The Yield Book Return Attribution Model Create a Matched Treasury Portfolio for each Security Run a series of Scenario Analysis type RORs for the MTP and for the security. Each scenario analysis run introduces one new market factor. Capture the return due to each factor by dividing it’s ROR by the prior ROR

119. Return Attribution Example Bank Assets Vs. Bank Liabilities –Month of March 2004 –Assume constant OAS on Commercial Loans and Bank Liabilities

120. Total Return without Attribution Bank Assets vs Bank Liabilities

121. Individual Security Return Attribution Non-Callable Bond

122. Individual Security Return Attribution Mortgage Backed Security

123. Individual Security Return Attribution Mortgage Backed Security – further detail

124. Return Attribution Portfolio Vs. Benchmark (Liabilities)

125. Portfolio Management Tools Security Level Portfolio Level Portfolio vs Liabilities Nominal Yield/Risk Measures Scenario Analysis Effective Yield/Risk Measures Optimization Return Attribution

126. Combining Scenario Analysis, Optimization and Return Attribution Use Scenario Analysis and Portfolio Optimization to Rebalance Assets to better match the return profile of the liabilities. Use Return Attribution to analyze results.

127. Scenario Returns - Assets vs Liabilities Principal Component Scenarios

128. Portfolio Optimization Constraints –Trade Only Treasury, Agency, Mortgage Securities –ROR of Assets must be greater than ROR of Liabilities across all Principal Component Scenarios –Max 10mm per issue Objective - Minimize Transactions

133. Return Attribution Major Components Yield Curve Effects Sector Weighting Effects Issue Selection Effect Sample Bank Investment Portfolio vs Treasury, Agency and Mortgage components of Citigroup BIGINDEX

135. Sector Weighting Effect Measures the effect of sector under-weighting /over-weighting decisions For each sector calculated by; [(Weight of Sector in Portfolio) - (Weight of Sector in Index)] multiplied by [(Spread Advantage of Sector in Index) - (Spread Advantage of Entire Index)] Example: Treasury Sector [.1976 -.3307]x[-.007 - (-.040)] = [-.1331]x[.033] = -.004

136. Issue Selection Effect Measures the effect of security selection within each sector For each sector calculated by; [Weight of Sector in Portfolio] multiplied by [(Spread Advantage of Sector in Portfolio) - (Spread Advantage of Sector in Index)] Example: Agency Sector [.3940] x [-.224 – (-.026)] =.3940 x (-.198) = -.078