Yield curve scenario generation for portfolio optimisation

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Yield curve scenario generation for portfolio
optimisation Helgard Raubenheimer Machiel F.
Kruger MIF2005
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AGENDA

• Background
• Moment-matching scenario generation
• Generating yield curve scenarios
• Example
• Conclusion

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AGENDA

• Background
• Moment-matching scenario generation
• Generating yield curve scenarios
• Example
• Conclusion

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Background
Stochastic Programming
Liquid Asset Portfolio
Liquid Assets
Stochastic Programming and Liquid Asset Portfolio
Decision Making Under Uncertainty
Scenario Structure
Scenario Tree
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Background
Portfolio Management
Stochastic Programming
International fixed-income portfolio. Consiglio
and Zenios (2001)
Liquid Asset Portfolio
Asset/liability management Klaasens (1998)
Liquid Assets
Financial planning problems Mulvey and Vladimirou
(1992)
Stochastic Programming and Liquid Asset Portfolio
Fixed income securities Worzel et al. (1998)
Decision Making Under Uncertainty
Fixed income portfolio management Zenios et al.
(1998)
Scenario Structure
World wide asset and liability modeling Ziemba
and Mulvey (1998)
Scenario Tree
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Background
Mandatory by Law
Stochastic Programming
Banks Act, 94/1990
Liquid Asset Portfolio
Liquid Assets
Stochastic Programming and Liquid Asset Portfolio
Regulations Relating to Banks (SA, 2000)
Decision Making Under Uncertainty
Scenario Structure
Liquid Assets
Scenario Tree
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Background
Interest rate sensitive
Stochastic Programming
Low (credit) risk
Liquid Asset Portfolio
Low portfolio return
Liquid Assets
Stochastic Programming and Liquid Asset Portfolio
Decision Making Under Uncertainty
Scenario Structure
Scenario Tree
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Background
Manage Optimally
Stochastic Programming
Rebalance regularly
Liquid Asset Portfolio
Liquid Assets
Model rebalancing explicitly
Expert views
Stochastic Programming and Liquid Asset Portfolio
Risk factor movements
Decision Making Under Uncertainty
Legislation
Scenario Structure
Regulation
Scenario Tree
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Background
Representing uncertainty
Stochastic Programming
Suitable for computation
Liquid Asset Portfolio
Risk factor scenarios
Liquid Assets
Accurately approximate theoretical model
Stochastic Programming and Liquid Asset Portfolio
Arbitrage free
Decision Making Under Uncertainty
Scenario Structure
Scenario Tree
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Background
Stochastic Programming
Discrete approximation of the joint distribution
of random factors
Liquid Asset Portfolio
Scenario fan
Liquid Assets
Scenario tree
Stochastic Programming and Liquid Asset Portfolio
Decision Making Under Uncertainty
Scenario Structure
Scenario Tree
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Background
Stochastic Programming
Scenario fan Arbitrage (Thorlacius, 2000)
Liquid Asset Portfolio
Liquid Assets
Stochastic Programming and Liquid Asset Portfolio
Decision Making Under Uncertainty
Scenario Structure
Scenario Tree
t0 t1 t2 t3
tT
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Background
Stochastic Programming
Scenario fan Arbitrage (Thorlacius, 2000)
Liquid Asset Portfolio
Liquid Assets
Stochastic Programming and Liquid Asset Portfolio
Decision Making Under Uncertainty
Scenario Structure
Scenario Tree
t0 t1 t2 t3
tT
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Background
Stochastic Programming
Scenario Tree
Liquid Asset Portfolio
Liquid Assets
Stochastic Programming and Liquid Asset Portfolio
Decision Making Under Uncertainty
Scenario Structure
Scenario Tree
t0 t1 t2 t3
tT
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Background
Liquid assets
Stochastic Programming
Government Bonds
Liquid Asset Portfolio
Yield curve scenarios
Liquid Assets
Limited number to keep problem tractable
Stochastic Programming and Liquid Asset Portfolio
Decision Making Under Uncertainty
Scenario Structure
Scenario Tree
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Background
Yield Curve Scenario Tree
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AGENDA

• Background
• Moment-matching scenario generation
• Generating yield curve scenarios
• Example
• Conclusion

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Moment-matching scenario generation
Moment-matching
Generation method
Multiple-period
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Moment-matching scenario generation
Moment-matching
Matching principle moments
Generation method
Høyland and Wallace (2001)
Multiple-period
Non-linear programming
Multiple-period
Limited number of discrete outcomes
Satisfy specified statistical properties
Minimise some measure of distance
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Moment-matching scenario generation
Moment-matching
Generate a scenario tree
Generation method
Specified statistical properties
Multiple-period
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Moment-matching scenario generation
Moment-matching
V V1, , Vi
X11, ,X1n
P1
fi(X,P)
Generation method
X21, ,X2n
P1
Multiple-period
X
P
X
X1, ,Xn
Xs1, ,Xsn
Ps
t
t1
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Moment-matching scenario generation
Moment-matching

• Objective
• Construct X and P such that the specified
  statistical properties are matched as well as
  possible
• Minimising a measure of distance between the
  statistical properties of the constructed
  distribution and the specified statistical
  properties

Generation method
Multiple-period
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Moment-matching scenario generation
Moment-matching

• Specified statistical properties
• Central moments
• Co-moments
• In any period
• Marginal distribution
• Empirical data
• Worst case outcomes and expected outcomes
• Non convex problem
• Full Match valid specified values

Generation method
Multiple-period
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Moment-matching scenario generation
Moment-matching
Generation method
Multiple-period
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Moment-matching scenario generation
Moment-matching
Several smaller optimisations sequential
approach
Generation method
X1
X1
V
Multiple-period
X,P
X,P
Xs
V
X,P
X,P
V
t0 t1 t2 t3
tT
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AGENDA

• Background
• Moment-matching scenario generation
• Generating yield curve scenarios
• Example
• Conclusion

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Generating yield curve scenarios
Yield Curve
Generation method
Single-period
Multiple-period
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Generating yield curve scenarios
Yield Curve
Term structure of zero rates
Generation method
Suitable for pricing government bonds
Single-period
Multiple-period
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Generating yield curve scenarios
Yield Curve
Yield Curve
Generation method
Generation method
Single-period
Single-period
Multiple-period
Multiple-period
S St
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Generating yield curve scenarios
Yield Curve
Yield Curve
Generation method
Generation method
Single-period
Single-period
Multiple-period
Multiple-period
St (St,m1,,St,md)
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Generating yield curve scenarios
Yield Curve

• Xt(Xt,m1,,Xt,md) log changes,
• Xt,mj ln(St,mj/St-1,mj)
• Generate scenarios for log changes
• St St-1expXt

Yield Curve
Generation method
Generation method
Single-period
Single-period
Multiple-period
Multiple-period
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Generating yield curve scenarios
Yield Curve
Sequential approach
Yield Curve
Generation method
Generation method
First four moments
Single-period
Single-period
Correlations
Multiple-period
Multiple-period
Conditional statistical properties
VAR(k) CCC GARCH(p,q) Bollerslev (1990)
Yield curve forecasting - Audrino and Trojani
(2003)
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Generating yield curve scenarios
Yield Curve

• s number of scenarios
• Xt matrix of outcomes
• Xt,i row vector of outcomes of ith random
  variable
• Pt row vector of scenario probabilities
• MOMt matrix of specified moment
• MOMt,ji ,, jth central moment of variable i
• R constant correlations
• MOMt,ji(X,P) and Rij(X,P) mathematical function
• Assume that skewness and kurtosis are time
  invariant

Yield Curve
Generation method
Generation method
Single-period
Single-period
Multiple-period
Multiple-period
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Generating yield curve scenarios
Yield Curve
Yield Curve
St,i St-1,iexpXt,i
Generation method
Generation method
Single-period
Single-period
Multiple-period
Multiple-period
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Generating yield curve scenarios
Yield Curve
For each outcome
Yield Curve
Generation method
Generation method
Calculate condition properties
Single-period
Single-period
MOM1,i EXtFt-1
Multiple-period
Multiple-period
MOM2,i VarXtFt-1
Condition correlation, skewness and kurtosis,
time invariant
Non-linear - Eyeball test
Test for arbitrage freeness
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AGENDA

• Background
• Moment-matching scenario generation
• Generating yield curve scenarios
• Example
• Conclusion

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Example
Yield Curve
Term structure data (NS, JIBAR, FRA, Swap,
2-10Y)
Yield Curve
NS used by Central Banks (BIS, 1999)
3M, 12M, 60M, 120M
Nelson and Siegel
Monthly log changes
Diagonal VAR(1)-CCC-GARCH(1,1)
Mean reversion
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Scn11 P 0.28488
Scn12 P 0.43565
Scn1 P 0.33423
Scn13 P 0.27947
Scn21 P 0.35074
Scn22 P 0.34421
Scn2 P 0.33838
Scn23 P 0.30505
Scn31 P 0.33272
Scn32 P 0.29949
Scn3 P 0.32739
Scn33 P 0.36779
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Scn11 P 0.28488
Scn12 P 0.43565
Scn1 P 0.33423
Scn13 P 0.09431
Scn21 P 0.35074
Scn0
Scn22 P 0.34421
Scn2 P 0.33838
Scn23 P 0.30505
Scn31 P 0.33272
Scn32 P 0.29949
Scn3 P 0.32739
Scn33 P 0.36779
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Scn1 P 0.278694
Y3M ? Y12M ? Y60M ? Y120M ?
Scn2 P 0.187438
Y3M ? Y12M ? Y60M ? Y120M ?
Scn0
Scn3 P 0.126786
Y3M ? Y12M ? Y60M ? Y120M ?
Scn4 P 0.221952
Y3M ? Y12M ? Y60M ? Y120M ?
Scn5 P 0.040747
Y3M ? Y12M ? Y60M ? Y120M ?
Scn6 P 0.144383
Y3M ? Y12M ? Y60M ? Y120M ?
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Scn51 P 0.189778
Y3M ? Y12M ? Y60M ? Y120M ?
Scn52 P 0.272094
Y3M ? Y12M ? Y60M ? Y120M ?
Scn53 P 0.158889
Y3M ? Y12M ? Y60M ? Y120M ?
Scn5 P 0.040747
Scn54 P 0.141778
Y3M ? Y12M ? Y60M ? Y120M ?
Scn55 P 0.116407
Y3M ? Y12M ? Y60M ? Y120M ?
Scn56 P 0.121053
Y3M ? Y12M ? Y60M ? Y120M ?
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Scn521 P 0.252466
Y3M ? Y12M ? Y60M ? Y120M ?
Scn522 P 0.053514
Y3M ? Y12M ? Y60M ? Y120M ?
Scn523 P 0.215672
Y3M ? Y12M ? Y60M ? Y120M ?
Scn52 P 0.272094
Scn524 P 0.108585
Y3M ? Y12M ? Y60M ? Y120M ?
Scn525 P 0.227123
Y3M ? Y12M ? Y60M ? Y120M ?
Scn526 P 0.142641
Y3M ? Y12M ? Y60M ? Y120M ?
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AGENDA

• Background
• Moment-matching scenario generation
• Generating yield curve scenarios
• Example
• Conclusion

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Conclusion

• Believable yield curve scenarios
• Adequate for pricing bonds
• Stochastic Programming Problem formulation
• Ongoing process

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