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

FFIEC Capital Markets Conference

Portfolio Management

and Theory

Steve Mandel

May 18-19, 2004

Portfolio Management Tools

Security LevelPortfolio Level

Portfolio vs. Liabilities

Nominal Yield/RiskMeasures

ScenarioAnalysis

EffectiveYield/RiskMeasures

Optimization

ReturnAttribution

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

Nominal Yield Measures

• Current Yield

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

111.4100369.103

25.4

Pr x

iceFlat

CouponldCurrentYie


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

Yield to MaturityCashflow

Date Years Nominal PV5/15/2004 0.211 2.23125 2.214

11/15/2004 0.711 2.125 2.0715/15/2005 1.211 2.125 2.033

11/15/2005 1.711 2.125 1.9975/15/2006 2.211 2.125 1.960

11/15/2006 2.711 2.125 1.9255/15/2007 3.211 2.125 1.890

11/15/2007 3.711 2.125 1.8565/15/2008 4.211 2.125 1.823

11/15/2008 4.711 2.125 1.7905/15/2009 5.211 2.125 1.757

11/15/2009 5.711 2.125 1.7265/15/2010 6.211 2.125 1.695

11/15/2010 6.711 2.125 1.66411/15/2010 6.711 100 78.302

Yield 3.678 Full Price 104.703


• 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)


• 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)


• 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


• 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

302.3947.2985.3510

571.6947.2

x

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


• 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

442.3388.3452.350.675.6

50.671.6388.3

x

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


• 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

820.3772.3829.350.675.6

50.671.6772.3

x

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)

n

ii

n

iii

PV

PVTrationMacaulayDu 1

iT

ii yield

CashflowPV

2)200

1(

Macaulay DurationDate Years Nominal PV(3.678) Years*PV

5/15/2004 0.211 2.231 2.214 0.46711/15/2004 0.711 2.125 2.071 1.4725/15/2005 1.211 2.125 2.033 2.462

11/15/2005 1.711 2.125 1.997 3.416

5/15/2006 2.211 2.125 1.960 4.33511/15/2006 2.711 2.125 1.925 5.2195/15/2007 3.211 2.125 1.890 6.070

11/15/2007 3.711 2.125 1.856 6.8885/15/2008 4.211 2.125 1.823 7.675

11/15/2008 4.711 2.125 1.790 8.4325/15/2009 5.211 2.125 1.757 9.158

11/15/2009 5.711 2.125 1.726 9.8565/15/2010 6.211 2.125 1.695 10.525

11/15/2010 6.711 2.125 1.664 11.16711/15/2010 6.711 100.000 78.302 525.496

Yield 3.678 Full Price 104.703 612.639Macaulay Duration 5.851

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

2001Yield

rationMacaulayDurationModifiedDu

Modified Duration

P/Y Curve & Modified Duration Tangent

90

95

100

105

110

115

120

125

1 2 3 4 5 6

Yield

Price

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.





ScenarioAnalysis


Optimization

ReturnAttribution

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

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.

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.

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

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.

Effective Duration Calculation

%350.2200095.105

373.104608.105

200Pr

PrPr 2525

nochange

bpbp

iceFull

iceFulliceFull


• 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

Partial Duration Calculation

• Partial Duration (5Year)

548.200095.105

943.104231.105

200)(

,5

25,525,5

NoChangeYr

bpYrbpYr

FP

FPFPYC

PointPartial

Duration

1 .337

2 .174

3 .300

5 .548

10 .738

20 .347

30 -.053

Total 2.39

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.

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.

Effective Convexity Calculation

18.3100095.10525.

)095.1052(373.104608.105

100Pryield)in (change

]Pr[2PrPr

2

2

nochange

nochangeupdown

iceFull

iceFulliceFulliceFull



• Partial Duration

• Effective Convexity

• Volatility Duration - A measure of the sensitivity (percent change) of the full price of a security to changes in Volatility.

Term Structure of Volatilities

Volatility Durations



• Partial Duration

• Effective Convexity

• Volatility Duration

• Pre Pay Duration - The sensitivity of a (mortgage) security’s full price to changes in Prepayment Rates

Prepay Durations





ScenarioAnalysis


Optimization

ReturnAttribution

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.


• 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

Portfolio Effective Duration

• Market Weighted Average of Individual Security Effective Durations = 3.93%

or • Percent MV Change (+/- 25 bp) on Portfolio

%93.3200938,101

922,100926,102

200)( 2525

NoChange

bpbp

MV

MVMV



• 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

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





ScenarioAnalysis


Optimization

ReturnAttribution

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.

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

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.

Rolling Yield CalculationConstant Nominal Spread

Rolling Yield CalculationConstant Nominal Spread

• BeginningPrice of Security = 103.366Yield of Security = 3.678Interpolated Yield Curve = 3.442Nominal Spread to Curve = .236

• HorizonInterpolated Yield Curve = 3.165Nominal Spread to Curve = .236Yield of Security = 3.401Price of Security = 104.362

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

100Pr

PrRePrPr

iceBegFull

iceBegFullinvinCoupiceHorFullROR

%02.5100700.104

700.104022.0356.4578.105

ROR

Rolling Yield – Constant Nominal Spread

Rolling Yield CalculationConstant OAS

• For a bond without embedded options assuming constant nominal spread produces results similar to those produced by the more accurate constant OAS method.

Rolling Yield – Constant OAS

Rolling Yield CalculationConstant 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.

Rolling Yield Constant CPR & Nominal Spread

Rolling Yield - Model Prepay Projections & Constant OAS





ScenarioAnalysis


Optimization

ReturnAttribution

Scenario AnalysisParallel Shifts – 3 Months Horizon

Pct ROR for Assets vs LiabilitiesParallel Shifts – 3 Months Horizon

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.

Principal Components Scenarios

Principal ComponentsCombination Scenarios

Scenario Analysis Principal Comp Comb Scenarios- 3 Month Horizon

Pct ROR for Assets vs LiabilitiesPrinc Component Scenarios – 3 Months Horizon



Portfolio vs Liabilities


ScenarioAnalysis


Optimization

ReturnAttribution

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

Portfolio OptimizationDuration Target Example

• Universe – All Securities in Citigroup Treasury Index

• Objective – Maximize Average Yield

• Constraints– Average Duration = 5– Total Market Value = $50mm

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.

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.

Portfolio Optimization Cash Matching Example 1

• Universe – All non callable securities in Citigroup Treasury Index

• Objective – Minimize Cost• Constraints - Cash Match Liability Schedule

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).

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.

Portfolio OptimizationImmunization 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

Portfolio OptimizationContingent 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.

Portfolio Optimization Assets vs Liabilities

• Optimize Assets

• Optimize Liabilities

• Optimize Assets and Liabilities Simultaneously (Dual Optimization)





ScenarioAnalysis


Optimization

ReturnAttribution

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.

The Yield BookReturn Attribution Model

• 3 Steps To Portfolio Management Process– Select Duration and Yield Curve Exposure– Select Sector Weights– Select Specific Issues

The Yield BookReturn 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

Return Attribution Example

• Bank Assets Vs. Bank Liabilities– Month of March 2004– Assume constant OAS on Commercial Loans

and Bank Liabilities

Total Return without AttributionBank Assets vs Bank Liabilities

Individual Security Return AttributionNon-Callable Bond

Individual Security Return AttributionMortgage Backed Security

Individual Security Return AttributionMortgage Backed Security – further detail

Return AttributionPortfolio Vs. Benchmark (Liabilities)





ScenarioAnalysis


Optimization

ReturnAttribution

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.

Scenario Returns - Assets vs Liabilities Principal Component Scenarios

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

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

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

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

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

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