FFIEC Capital Markets Conference
Portfolio Management
and Theory
Steve Mandel
May 18-19, 2004
Portfolio Management Tools
Nominal
Yield/Risk
Measures
Return
Attribution
Security Level
Portfolio Level
Portfolio vs. Liabilities
Optimization
Effective
Yield/Risk
Measures
Scenario
Analysis
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 11/15/2010
Price (2/29/2004) = 103.369
Current Yield = 4.111%
Coupon
4.25
CurrentYie ld 

x100  4.111
Flat Pr ice 103.369
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 Maturity
Date
5/15/2004
11/15/2004
5/15/2005
11/15/2005
5/15/2006
11/15/2006
5/15/2007
11/15/2007
5/15/2008
11/15/2008
5/15/2009
11/15/2009
5/15/2010
11/15/2010
11/15/2010
Yield
Years
0.211
0.711
1.211
1.711
2.211
2.711
3.211
3.711
4.211
4.711
5.211
5.711
6.211
6.711
6.711
3.678
Cashflow
Nominal
PV
2.23125
2.214
2.125
2.071
2.125
2.033
2.125
1.997
2.125
1.960
2.125
1.925
2.125
1.890
2.125
1.856
2.125
1.823
2.125
1.790
2.125
1.757
2.125
1.726
2.125
1.695
2.125
1.664
100
78.302
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
 6.71  5 
2.947  
 x3.985  2.947   3.302
 10  5 
– Interpolated Point on Yield Curve: 3.302
– Yield of Security: 3.678
– Spread to Yield Curve:100x(3.68-3.30)=38bp
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
 6.71  6.50 
3.388  
 x3.452  3.388  3.442
 6.75  6.50 
– Interpolated Point on Yield Curve: 3.442
– Yield of Security: 3.678
– Spread to Yield Curve:100x(3.68-3.44)=24bp
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
 6.71  6.50 
3.772  
 x3.829  3.772  3.820
 6.75  6.50 
– Interpolated Point on Yield Curve: 3.820
– Yield of Security: 3.678
– Spread to Yield Curve:100x(3.68-3.82)= -14bp
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
MacaulayDu ration 
 T PV
i 1
n
i
i
 PV
i
i
Cashflowi
PVi 
yield 2Ti
(1 
)
200
Macaulay Duration
Date
5/15/2004
11/15/2004
5/15/2005
11/15/2005
Years
0.211
0.711
1.211
1.711
5/15/2006
11/15/2006
5/15/2007
11/15/2007
5/15/2008
11/15/2008
5/15/2009
11/15/2009
5/15/2010
11/15/2010
11/15/2010
Yield
2.211
2.711
3.211
3.711
4.211
4.711
5.211
5.711
6.211
6.711
6.711
3.678
Nominal PV(3.678) Years*PV
2.231
2.214
0.467
2.125
2.071
1.472
2.125
2.033
2.462
2.125
1.997
3.416
2.125
1.960
2.125
1.925
2.125
1.890
2.125
1.856
2.125
1.823
2.125
1.790
2.125
1.757
2.125
1.726
2.125
1.695
2.125
1.664
100.000
78.302
Full Price 104.703
Macaulay Duration
4.335
5.219
6.070
6.888
7.675
8.432
9.158
9.856
10.525
11.167
525.496
612.639
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.
ModifiedDu ration 
MacaulayDu ration
Yield
1
200
Modified Duration = 5.851/(1+3.678/200) = 5.746
Modified Duration
P/Y Curve & Modified Duration Tangent
125
120
115
110
Price
105
100
95
90
1
2
3
4
Yield
5
6
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
Nominal
Yield/Risk
Measures
Return
Attribution
Security Level
Portfolio Level
Portfolio vs. Liabilities
Optimization
Effective
Yield/Risk
Measures
Scenario
Analysis
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
 Full Pr ice 25bp  Full Pr ice 25bp 


Full Pr icenochange

  200




105.608  104.373
 200  2.350%
105.095
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)
( FP5Yr , 25bp  FP5Yr , 25bp )
FP5Yr , NoChange
 200 
105.231  104.943
 200  .548
105.095
YC
Point
Partial
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
 Full Pr icedown  Full Pr iceup  2[ Full Pr icenochange] 


2
(change in yield)  Full Pr icenochange

 100




105.608  104.373  (2 105.095)
100  3.18
2
.25 105.095
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
Nominal
Yield/Risk
Measures
Return
Attribution
Security Level
Portfolio Level
Portfolio vs. Liabilities
Optimization
Effective
Yield/Risk
Measures
Scenario
Analysis
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
( MV 25bp  MV 25bp )
MVNoChange
 200 
102,926  100,922
 200  3.93%
101,938
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
Nominal
Yield/Risk
Measures
Return
Attribution
Security Level
Portfolio Level
Portfolio vs. Liabilities
Optimization
Effective
Yield/Risk
Measures
Scenario
Analysis
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 Calculation
Constant Nominal Spread
Rolling Yield Calculation
Constant Nominal Spread
• 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
Return Calculation
HorFull Pr ice  Coup  Pr in  Re inv  BegFull Pr ice
ROR 
100
BegFull Pr ice
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
ROR 
105.578  4.356  0  .022  104.700
100  5.02%
104.700
Rolling Yield – Constant Nominal Spread
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.
Rolling Yield – Constant OAS
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.
Rolling Yield
Constant CPR & Nominal Spread
Rolling Yield
Constant CPR & Nominal Spread
Rolling Yield - Model Prepay
Projections & Constant OAS
Portfolio Management Tools
Nominal
Yield/Risk
Measures
Return
Attribution
Security Level
Portfolio Level
Portfolio vs. Liabilities
Optimization
Effective
Yield/Risk
Measures
Scenario
Analysis
Scenario Analysis
Parallel Shifts – 3 Months Horizon
Pct ROR for Assets vs Liabilities
Parallel 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 OffThe-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 Components
Combination Scenarios
Scenario Analysis
Principal Comp Comb Scenarios- 3 Month Horizon
Pct ROR for Assets vs Liabilities
Princ Component Scenarios – 3 Months Horizon
Portfolio Management Tools
Nominal
Yield/Risk
Measures
Return
Attribution
Security Level
Portfolio Level
Portfolio vs Liabilities
Optimization
Effective
Yield/Risk
Measures
Scenario
Analysis
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 Optimization
Duration 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 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
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.
Portfolio Optimization
Assets vs Liabilities
• Optimize Assets
• Optimize Liabilities
• Optimize Assets and Liabilities
Simultaneously (Dual Optimization)
Portfolio Management Tools
Nominal
Yield/Risk
Measures
Return
Attribution
Security Level
Portfolio Level
Portfolio vs Liabilities
Optimization
Effective
Yield/Risk
Measures
Scenario
Analysis
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 Book
Return Attribution Model
• 3 Steps To Portfolio Management Process
– Select Duration and Yield Curve Exposure
– Select Sector Weights
– Select Specific Issues
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
Return Attribution Example
• Bank Assets Vs. Bank Liabilities
– Month of March 2004
– Assume constant OAS on Commercial Loans
and Bank Liabilities
Total Return without Attribution
Bank Assets vs Bank Liabilities
Individual Security Return Attribution
Non-Callable Bond
Individual Security Return Attribution
Mortgage Backed Security
Individual Security Return Attribution
Mortgage Backed Security – further detail
Return Attribution
Portfolio Vs. Benchmark (Liabilities)
Portfolio Management Tools
Nominal
Yield/Risk
Measures
Return
Attribution
Security Level
Portfolio Level
Portfolio vs Liabilities
Optimization
Effective
Yield/Risk
Measures
Scenario
Analysis
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