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Managing RiskUsing Multi-Stage Stochastic Optimization
MS&E348 Winter 2005/2006Prof. Gerd Infanger
Winter 2005/2006 MS&E348/Infanger 2
ConduitsFreddie MacFannie Mae
Wall StreetDebt
Investors
Debt
different maturitycallable, non callable
MortgagesMortgage Pools Securities
Interest rate risk, prepayment risk,(credit risk insured)
Funding Mortgage PoolsThrough Issuing Debt
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Time
Value
Funding Decision Rule
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Initial Yield Curves
0
1
2
3
4
5
6
7
8
9
10
0 60 120 180 240 300 360
Maturity (Months)
Yiel
d (%
)
Normal Flat Steep
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Efficient Frontier Normal
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0
12.0
13.0
14.0
15.0
0.0 1.0 2.0 3.0 4.0 5.0
Risk ($M NPV)
Exp.
Ret
urn
($M
NPV
)
Single-stage Multi-stage
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Model Normal, Single-Stage
OBJECTIVE 3095.051
X1.L rate amount product j1y10 j1m06n j1y07n
w10001 6.60% 0.853 0.147
X2.L rate amount product j2y10 j2m03n
w20001 3.60% 0.853w20002 7.70% 0.853w20003 6.30% 0.853w20004 9.90% 0.853w20005 5.50% 0.853w20006 8.90% 0.853w20007 7.50% 0.853w20008 4.70% 0.853w20009 5.70% 0.853w20010 8.50% 0.853
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Model Normal, Multi-Stage
OBJECTIVE 1854.769
X1.L rate amount product j1y10 j1m06n j1y05nc1
w10001 6.60% 0.785 0.215
X2.L rate amount product j2y10 j2m03n j2y03n j2y03nc1 j2y05nc1 j2y10n j2y10nc1 j2y10nc3 j2y30n
w20001 3.60% 1w20002 7.70% 0.597 0.188w20003 6.30% 0.125 0.42 0.24w20004 9.90% 0.785w20005 5.50% 0.508 0.423 0.069w20006 8.90% 0.785w20007 7.50% 0.785w20008 4.70% 0.251 0.749w20009 5.70% 0.301 0.035 0.261 0.188w20010 8.50% 0.785
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Interest rate
Price
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Risk-Return Profile, Multi, Normal
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5
Risk ($M NPV)
Exp.
Ret
. ($M
NPV
)
Downside Delta-Gamma
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Risk-Return Profile, Multi, Normal
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6 7 8 9 10
Risk STD ($M NPV)
Exp.
Ret
. ($M
NPV
)
Downside Delta-Gamma
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Histogram
0
100
200
300
400
500
600
-20 -17 -14 -11 -8 -5 -2 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49
Returns $M NPV
Freq
uenc
y
Dur. And conv. Constr. Max exp. ret. Min downside risk
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Optimal Solution -- Duration Gap
-6.00
-4.00
-2.00
0.00
2.00
4.00
6.00
8.00
10.00
0.00 2.00 4.00 6.00 8.00 10.00 12.00
Interest rate (%)
Dur
atio
n G
ap (%
)
95% max exp. ret. Dur. and conv constr. Min. downside risk
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Model Normal, Multi-Stage, Min Downside Risk
OBJECTIVE 1518.888
X1.L rate amount product j1y10 j1m06n j1y05n j1y05nc1 j1y05nc3
w10001 0.066 0.266 0.235 0.218 0.281
X2.L rate amount product j2y10 j2m03n j2y01n j2y03nc1 j2y05n j2y10n j2y10nc1 j2y10nc3 j2y30n
w20001 0.036 0.017 0.148 0.24 0.079w20002 0.077 0.102 0.165w20003 0.063 0.117 0.149w20004 0.099 0.266w20005 0.055 0.031 0.032 0.237 0.185w20006 0.089 0.266w20007 0.075 0.014 0.252w20008 0.047 0.118 0.3 0.019 0.047w20009 0.057 0.012 0.201 0.053w20010 0.085 0.266
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Model Normal, Multi-Stage, Duration ConstrainedOBJECTIVE -7871.49
X1.L rate amount product j1y10 j1m06n j1y05n j1y05nc1 j1y07n j1y10n
w10001 6.60% 0.122, 0.083, 0.534, 0.239 0.022
X2.L rate amount product j2y10 j2m03n j2m06n j2y01n j2y03nc1 j2y05n j2y05nc1 j2y07nc1 j2y10n j2y10nc1 j2y10nc3
w20001 3.60% 0.466 0.117 0.073w20002 7.70% 0.011 0.112w20003 6.30% 0.064 0.058w20004 9.90% 0.152w20005 5.50% 0.506 0.043 0.044 0.058 0.004w20006 8.90% 0.064 0.059w20007 7.50% 0.122w20008 4.70% 0.567 0.089w20009 5.70% 0.072 0.051w20010 8.50% 0.316 0.34
CX12.L rate call amount product j1y10 j1y05nc1
w20004 9.90% 0.029w20010 8.50% 0.534
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Out of Sample Efficient Frontier Normal
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0 1 2 3 4 5
Risk ($M NPV)
Expe
cted
Ret
urn
($M
NPV
)
Single-stage Multi-stage Multi-stade dur. constr.
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Out of Sample Risk-Return Profile Normal
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0 1 2 3 4 5 6 7 8 9 10
Risk STD ($M NPV)
Expe
cted
Ret
urn
($M
NPV
)
Single-stage Multi-stage Multi-stage dur.const.
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Out of Sample Efficient Frontier Normal, Larger Sample
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0 1 2 3 4 5
Risk ($M NPV)
Expe
cted
Ret
urn
($M
NPV
)
Single-stage Multi-stage Multi-stage dur. const.
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Out of Sample Risk-Return Profile Normal, Larger Sample
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0 1 2 3 4 5 6 7 8 9 10
Risk STD ($M NPV)
Expe
cted
Ret
urn
($M
NPV
)
Single-stage Multi-stage Multi-stage dur. Const.
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In Sample vs. Out of Sample Efficient Frontier Normal, Larger Sample
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0 1 2 3 4 5
Risk ($M NPV)
Expe
cted
Ret
urn
($M
NPV
)
Multi-stage Multi-stage in sample
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Using parallel DECIS problem L5 solved in less than 10 min!
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