8/9/2019 Performance Analysis for Fixed Income Portfolios
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Performance Analysis for Fixed Income Portfolios
Paul Wild, 11.12.2002
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Agenda
Fixed Income Investment Process
Key Rate Duration Concept
Option Adjusted Spreads
Performance Decomposition
Discussion
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Starting Point
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Investment Process
Allocation of Funds
Duration Credit
Risk TypesRisk Types
Curve Sector / Rating
Risk Composition
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Modelling of Term Structure
Modelling of interest rate curve by several key rates along time
Linear Interpolation Zero Coupon Spot Rates of Government Bonds
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Option Adjusted Spreads (OAS)
“By how many basis points may the yield of a bond increase such that its return
advantage as compared to a government bond will vanish?”
Mathematically: Given the discount factors Df i from the zero coupon government
bond curve and the bond’s observed market price, we need to solve the followingequation for OAS
∑= ++
=
n
it
i
i
iOAS DF
CF
1 )1(PriceBondMarket
Option Adjusted Spread
0
0.5
1
1.5
2
2.5
3
3.5
4
ON 3m 6m 1y 2y 3y 4y 5y 6y 7y 8y 9y 10y 15y 20y 25y 30y
Time to Maturity
D i s c o u n t R a t e
Government Rate
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Option Adjusted Spreads (OAS)
OAS includes
credit spread for straight bonds
option price for callable/putable bonds
separation possible by considering probable redemption
suitable option pricing models required for more complex instruments (e.g.
convertibles)
Additional intermediate steps possible: Government --> Swap --> Credit
Analysis by further grouping: industry sectors, rating classes
Important: Development over time
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Interest Rate Risk
Key Rate Moves
4.8
4.9
5
5.1
5.25.3
5.4
5.5
5.6
1 2 3 4
Term (years)
S p o t R a t e
( % ) Key Rate 1
Key Rate 2Key Rate 3
Key Rate 4
Spot Rate
Calculation of portfolio (or single bond) sensitivity to change of each key rate leaving
all other things equal
Discounting of each bond’s future cash-flow (according to key rate curve) to get the
present value
Numerical approximation by calculating
i
i
i y P
P P
KRD ∆
−
−=
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Key Rate Duration Concept
Portfolio value change in response to key rate movements
As compared to the ‘Macauley Duration’ approach
the KRD concept is a ‘Multi-Factor’ model
KRD of Portfolio = Cap. Weighted sum of single bond’s KRDs
Shift of all key rates by same amount --> parallel shift of curve
Consequence: Sum of key rate durations equals Effective Duration
)(1
...)(1
)(1
2
2
21
1
1n
n
n y y
KRD y
y
KRD y
y
KRD
P
P ∆−
+++∆−
++∆−
+≈
∆
y D y P
P Mac ∆⋅⋅+
−≈∆
11
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Key Rate Duration Profile; Example
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Option Adjusted Spreads (OAS)
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Summary
Portfolio Positions + Curve
Decision Support
Cashflows and Discount Factors
Scenario Simulation(Curve and/or Spreads)
Solve for OAS
Key Rate Shifts and KRD
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Performance Analysis
Portfolio Return
Change of
Term Structure
Change of
Credit Spread Accretion Rolldown Effect
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Performance Analysis
Return Factor Description
Accretion Return Accretion return is calculated by holding a cashflow’s yieldconstant while moving the settlement date forward. Thereturn is due to the convergence of the price to par as the
time approaches its maturity.Rolldown Effect(Time Passages)
Results from the predictable change in the cashflow’s yieldas time elapses; reflects the change in the placementalong the yield curve
Changes in theterm structure
In the present model, the term structure results from a setof key rates and intermediate linear interpolation. Changes
of the term structure are detailed on the changes of eachkey rate and their effect is contributed to each cashflowaffected.
Change in OAS The OAS (on total bond level) is the spread to be added tothe government spot rate curve in order to obtain the
bond’s actual market price. This spread may change over time and thus gives a contribution to the bond’s return.
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Performance Analysis
t1 + Dtt1 timeTa Tb
Discounting Factor
KRa
KRb
DKRb
DKRa
DFbegin
DFend
DFtermstr DFTime
Dt
Decomposition of Discounting Factors
OAS uretermstruct Timebeginend
DF DF DF DF DF ∆+∆+∆+=
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Performance Analysis
OAS CF uretermstruct CF rolldownCF accretionCF
t t
begin
t
end
CF
returnreturnreturnreturn
DF CF
DF
CF
return
,,,,
1
)1(
)1(
1
1
+++=
−
+
+=
∆+
Algebraic manipulation results in additive decomposition of return
Calculation on level of each cashflow; capital weighted aggregation on bond position
and portfolio level
Correct overall return results on portfolio level
Re-calculation would be required at each time of transactions (purchases/sales)
since cashflow profile changes
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Performance Analysis
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Performance Analysis Cumulated
Return 2002
-1.00
-
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
T o t al
A c c r e t i on
R ol l d own
T er m T o t al
OA S C
h an g e
T er m ON
T er m 3 M
T er m 6 M
T er m 1 Y
T er m 2 Y
T er m 3 Y
T er m 4 Y
T er m 5 Y
T er m 6 Y
T er m 7 Y
T er m 8 Y
T er m 9 Y
T er m 1 0 Y
T er m 1 5 Y
T er m 2 0 Y
T er m 2 5 Y
T er m 3 0 Y
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