ffiec capital markets conference portfolio management and theory steve mandel may 18-19, 2004
TRANSCRIPT
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
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.
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
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)
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)
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
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
302.3947.2985.3510
571.6947.2
x
Yield Curves (2/27/2004)A -Treasury On-the-Run, B - Treasury Off-the-Run
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
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
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
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.
Portfolio Management Tools
Security LevelPortfolio Level
Portfolio vs. Liabilities
Nominal Yield/RiskMeasures
ScenarioAnalysis
EffectiveYield/RiskMeasures
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 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
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
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.
Term Structure of Volatilities
Volatility Durations
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
Prepay Durations
Portfolio Management Tools
Security LevelPortfolio Level
Portfolio vs. Liabilities
Nominal Yield/RiskMeasures
ScenarioAnalysis
EffectiveYield/RiskMeasures
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.
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
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
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
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
Portfolio Management Tools
Security LevelPortfolio Level
Portfolio vs. Liabilities
Nominal Yield/RiskMeasures
ScenarioAnalysis
EffectiveYield/RiskMeasures
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 Constant CPR & Nominal Spread
Rolling Yield - Model Prepay Projections & Constant OAS
Portfolio Management Tools
Security LevelPortfolio Level
Portfolio vs. Liabilities
Nominal Yield/RiskMeasures
ScenarioAnalysis
EffectiveYield/RiskMeasures
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 Management Tools
Security LevelPortfolio Level
Portfolio vs Liabilities
Nominal Yield/RiskMeasures
ScenarioAnalysis
EffectiveYield/RiskMeasures
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)
Portfolio Management Tools
Security LevelPortfolio Level
Portfolio vs Liabilities
Nominal Yield/RiskMeasures
ScenarioAnalysis
EffectiveYield/RiskMeasures
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)
Portfolio Management Tools
Security LevelPortfolio Level
Portfolio vs Liabilities
Nominal Yield/RiskMeasures
ScenarioAnalysis
EffectiveYield/RiskMeasures
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