Managing Risk Using Multi-Stage Stochastic...
Transcript of Managing Risk Using Multi-Stage Stochastic...
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Managing Risk Using Multi-Stage Stochastic Optimization
MS&E348 Professor Gerd Infanger
Winter 2011/2012
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Winter 2011/2012 MS&E348/Infanger 2
Conduits Freddie Mac Fannie Mae
Wall Street Debt
Investors
Debt
different maturity callable, non callable
Mortgages Mortgage Pools Securities
Interest rate risk, prepayment risk, (credit risk insured)
Funding Mortgage Pools Through Issuing Debt
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Winter 2011/2012 MS&E348/Infanger 3
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Winter 2011/2012 MS&E348/Infanger 4
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Winter 2011/2012 MS&E348/Infanger 5
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Winter 2011/2012 MS&E348/Infanger 6
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Winter 2011/2012 MS&E348/Infanger 7
Time
Value
Funding Decision Rule
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Winter 2011/2012 MS&E348/Infanger 8
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Winter 2011/2012 MS&E348/Infanger 9
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Winter 2011/2012 MS&E348/Infanger 10
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Winter 2011/2012 MS&E348/Infanger 11
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Winter 2011/2012 MS&E348/Infanger 12
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Winter 2011/2012 MS&E348/Infanger 13
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Winter 2011/2012 MS&E348/Infanger 14
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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Winter 2011/2012 MS&E348/Infanger 15
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Winter 2011/2012 MS&E348/Infanger 16
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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Winter 2011/2012 MS&E348/Infanger 17
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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Winter 2011/2012 MS&E348/Infanger 18
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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Winter 2011/2012 MS&E348/Infanger 19
Interest rate
Price
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Winter 2011/2012 MS&E348/Infanger 20
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Winter 2011/2012 MS&E348/Infanger 21
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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Winter 2011/2012 MS&E348/Infanger 22
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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Winter 2011/2012 MS&E348/Infanger 23
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 $M NPV
Freq
uenc
y
Dur. And conv. Constr. Max exp. ret. Min downside risk
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Winter 2011/2012 MS&E348/Infanger 24
Optimal Solution -- Duration Gap
-6.00
-4.00
-2.00
0.00
2.00
4.00
6.00
8.00
10.00
0.00 2.00 4.00 6.00 8.00 10.00 12.00
Interest rate (%)
Dur
atio
n G
ap (%
)
95% max exp. ret. Dur. and conv constr. Min. downside risk
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Winter 2011/2012 MS&E348/Infanger 25
Model Normal, Multi-Stage, Min Downside Risk
OBJECTIVE 1518.888
X1.L rate amount product j1y10 j1m06n j1y05n j1y05nc1 j1y05nc3
w10001 0.066 0.266 0.235 0.218 0.281
X2.L rate amount product j2y10 j2m03n j2y01n j2y03nc1 j2y05n j2y10n j2y10nc1 j2y10nc3 j2y30n
w20001 0.036 0.017 0.148 0.24 0.079w20002 0.077 0.102 0.165w20003 0.063 0.117 0.149w20004 0.099 0.266w20005 0.055 0.031 0.032 0.237 0.185w20006 0.089 0.266w20007 0.075 0.014 0.252w20008 0.047 0.118 0.3 0.019 0.047w20009 0.057 0.012 0.201 0.053w20010 0.085 0.266
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Winter 2011/2012 MS&E348/Infanger 26
Model Normal, Multi-Stage, Duration Constrained OBJECTIVE -7871.49
X1.L rate amount product j1y10 j1m06n j1y05n j1y05nc1 j1y07n j1y10n
w10001 6.60% 0.122, 0.083, 0.534, 0.239 0.022
X2.L rate amount product j2y10 j2m03n j2m06n j2y01n j2y03nc1 j2y05n j2y05nc1 j2y07nc1 j2y10n j2y10nc1 j2y10nc3
w20001 3.60% 0.466 0.117 0.073w20002 7.70% 0.011 0.112w20003 6.30% 0.064 0.058w20004 9.90% 0.152w20005 5.50% 0.506 0.043 0.044 0.058 0.004w20006 8.90% 0.064 0.059w20007 7.50% 0.122w20008 4.70% 0.567 0.089w20009 5.70% 0.072 0.051w20010 8.50% 0.316 0.34
CX12.L rate call amount product j1y10 j1y05nc1
w20004 9.90% 0.029w20010 8.50% 0.534
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Winter 2011/2012 MS&E348/Infanger 27
Out of Sample Efficient Frontier Normal
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0 1 2 3 4 5
Risk ($M NPV)
Expe
cted
Ret
urn
($M
NPV
)
Single-stage Multi-stage Multi-stade dur. constr.
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Winter 2011/2012 MS&E348/Infanger 28
Out of Sample Risk-Return Profile Normal
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0 1 2 3 4 5 6 7 8 9 10
Risk STD ($M NPV)
Expe
cted
Ret
urn
($M
NPV
)
Single-stage Multi-stage Multi-stage dur.const.
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Winter 2011/2012 MS&E348/Infanger 29
Out of Sample Efficient Frontier Normal, Larger Sample
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0 1 2 3 4 5
Risk ($M NPV)
Expe
cted
Ret
urn
($M
NPV
)
Single-stage Multi-stage Multi-stage dur. const.
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Winter 2011/2012 MS&E348/Infanger 30
Out of Sample Risk-Return Profile Normal, Larger Sample
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0 1 2 3 4 5 6 7 8 9 10
Risk STD ($M NPV)
Expe
cted
Ret
urn
($M
NPV
)
Single-stage Multi-stage Multi-stage dur. Const.
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Winter 2011/2012 MS&E348/Infanger 31
In Sample vs. Out of Sample Efficient Frontier Normal, Larger Sample
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0 1 2 3 4 5
Risk ($M NPV)
Expe
cted
Ret
urn
($M
NPV
)
Multi-stage Multi-stage in sample
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Winter 2011/2012 MS&E348/Infanger 32
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Winter 2011/2012 MS&E348/Infanger 33
Using parallel DECIS problem L5 solved in less than 10 min!
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Winter 2011/2012 MS&E348/Infanger 34