A heterogeneous agent approach vs. representative agent theory and the application of support vector machines to default risk analysis:
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| Beschreibung: | XX, 151 S. graph. Darst. |
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MARC
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| 100 | 1 | |a Moro, Rouslan Arthur |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a A heterogeneous agent approach vs. representative agent theory and the application of support vector machines to default risk analysis |c von Rouslan Arthur Moro |
| 264 | 1 | |c 2008 | |
| 300 | |a XX, 151 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 502 | |a Berlin, Humboldt-Univ., Diss., 2008 | ||
| 650 | 7 | |a Anlageverhalten |2 stw | |
| 650 | 7 | |a Capital Asset Pricing Model |2 stw | |
| 650 | 7 | |a Finanzmarkt |2 stw | |
| 650 | 7 | |a Kreditwürdigkeit |2 stw | |
| 650 | 7 | |a Risikopräferenz |2 stw | |
| 650 | 7 | |a Support Vector Machine |2 stw | |
| 650 | 7 | |a Theorie |2 stw | |
| 650 | 0 | 7 | |a Risikoanalyse |0 (DE-588)4137042-9 |2 gnd |9 rswk-swf |
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| 650 | 0 | 7 | |a Unternehmensbewertung |0 (DE-588)4078594-4 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Investor |0 (DE-588)4336776-8 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Insolvenz |0 (DE-588)4072843-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Risikoverhalten |0 (DE-588)4050133-4 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Support-Vektor-Maschine |0 (DE-588)4505517-8 |2 gnd |9 rswk-swf |
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| 689 | 2 | 0 | |a Unternehmensbewertung |0 (DE-588)4078594-4 |D s |
| 689 | 2 | 1 | |a Insolvenz |0 (DE-588)4072843-2 |D s |
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Datensatz im Suchindex
| _version_ | 1804138350344404992 |
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| adam_text | Contents
1 Introduction 1
2 The Pricing Kernel or the Stochastic Discount Factor 11
2.1 Expected Utility Asset Pricing.................. 11
2.2 Market Interactions, States and Preferences.......... 14
2.3 The Representative Agent.................... 28
2.4 Techniques for the Estimation of State Price Densities..... 38
2.4.1 Heston Model....................... 38
2.4.2 Estimation of the State Price Density with the Breeden-
Litzenberger Formula................... 40
2.4.3 The Estimation of the Subjective Probability Density . 47
2.4.4 The Estimation of the Pricing Kernel.......... 50
3 Heterogeneous Investor Hypothesis 59
3.1 The Pricing Kernel in the Black-Scholes World......... 59
3.2 Pricing Kernel Frontiers..................... 61
4 Empirical Pricing Kernels and Investor Preferences 67
4.1 Introduction............................ 69
4.2 Pricing kernels and utility functions............... 70
4.3 Estimation............................. 74
4.3.1 Estimation approaches for the pricing kernel...... 74
4.3.2 Estimation of the risk neutral density.......... 76
4.3.3 Estimation of the historical density........... 79
4.3.4 Empirical pricing kernels................. 83
4.4 Individual investors and their utility functions......... 89
4.4.1 Individual Utility Function................ 90
4.4.2 Market Aggregation Mechanism............. 92
4.4.3 The Estimation of the Distribution of Switching Points 94
4.5 Conclusion............................. 96
4.6 Acknowledgements........................ 98
xii Contents
5 Estimating Probabilities of Default With Support Vector Ma-
chines 99
5.1 Introduction............................ 101
5.2 Data................................ 104
5.3 Variable Selection......................... 105
5.4 Comparison of DA, Logistic Regression and SVM....... 109
5.5 Conversion of Scores into PDs.................. HI
5.6 Conclusion............................. 115
5.7 Acknowledgements........................ 115
5.8 Appendix............................. 116
6 Graphical Data Representation in Bankruptcy Analysis 121
6.1 Company Rating Methodology.................. 123
6.2 The SVM Approach .... ................... 127
6.3 Company Score Evaluation.................... 129
6.4 Variable Selection......................... 130
6.5 Conversion of Scores into PDs.................. 135
6.6 Colour Coding........................... 137
6.7 Conclusion............................. Ill
6.8 Acknowledgements........................ 141
List of Figures
2.1 The probabilities for unit average temperature ranges. The
graph represents the probability density, probability per centi-
grade................................ 28
2.2 The price per chance for different average temperature ranges. 26
2.3 The probabilities for the states described in terms of the total
wealth................................ 30
2.4 The price per chance for the states described in terms of the
total wealth............................. 31
2.5 Individual vs. market returns. Investor C with a lower risk
aversion (red line), investor D with a higher risk aversion (blue
line) and market returns (green line with the 45 degree slope). 32
2.6 The pay-offs of the call (left panel) and put options (right
panel). K is called the strike price................ 36
2.7 The call prices as a function of strike prices. DAX, 24 March
2000, maturity is 0.075 year or 19 days.............. 43
2.8 The call prices as a function of strike prices smoothed with
B-Splines. DAX, 24 March 2000, maturity is 0.075 year or 19
days................................. 45
2.9 The state price density estimated as the second derivative of
the call prices approximated with B-Splines. DAX, 24 March
2000, maturity is 0.075 year or 19 days.............. 46
2.10 The state price density q estimated as the second derivative of
the call prices approximated with B-Splines (red). The density
of 19 day DAX returns p estimated on one year of data prior
to 24 March 2000 (blue). The pricing kernel which is the ratio
I (green). DAX, 24 March 2000, maturity is 0.075 year or 19
days. The ODAX market, r = 3.64%............... 47
xiv ___________________List of Figures
2.11 The state price density q estimated as the second derivative of
the call prices approximated with B-Splines (red). The density
of 59 day DAX returns p estimated on one year of data prior
to 24 March 2000 (blue). The pricing kernel which is the ratio
2 (green). DAX, 24 March 2000, maturity is 0.233 year or 59
days. The ODAX market, r = 3.757%.............. 48
2.12 The state price density estimated with the Heston model. DAX,
24 March 2000, the maturity is 0.5 year or 177 days. Private
trading data of the Sal. Oppenheim Bank............ 49
2.13 The density of 59 day DAX returns p estimated on one year
of data prior to 24 March 2000 with B-splines (blue) and as a
weighted combination of two log-normal density functions (red). 52
2.14 DAX index price development over one year prior to 24 March
2000................................. 54
2.15 The 59 day DAX returns over one year prior to 24 March 2000. 55
2.16 Estimated 59 day risk neutral (red) and subjective (blue) den-
sities and the pricing kernel (green) on 24 March 2000.....57
3.1 Estimated 59 day risk neutral (red) and subjective (blue) den-
sities under the assumption of their log-normality and the re-
sulting pricing kernel (green) on 24 March 2000......... 60
3.2 Estimated frontiers of the pricing kernel for 59 day maturity
claims on 24 March 2000. q is assumed to be log-normal while
p estimated with a kernel estimator................ 64
3.3 The switching behaviour of investors between bearish and
bullish attitudes. Investor A with the switching point 2.4 is
more opimistic than investor B since he switches earlier from
bearish to bullish attitudes.................... 66
4.1 up: Utility function in the Black Seholes model for T = 0.5
years ahead and drift ^ = 0.1, volatility a = 0.2 and interest
rate r = 0.03. down: Market utility function on 06/30/2000
for T = 0.5 years ahead...................... 73
4.2 Risk neutral density on 24/03/2000 half a year ahead...... 80
4.3 Implied volatility surface on 24/03/00.............. 80
4.4 Historical density on 24/03/2000 half a year ahead....... 82
4.5 DAX, 1998 - 2004............ ............. 84
4.6 Empirical pricing kernel on 24/03/2000 (bullish market). ... 85
4.7 Empirical pricing kernel on 24/03/2000 (bullish), 30/07/2002
(bearish) and 30/06/2004 (unsettled or sidewards market). . . 86
4.8 Market utility functions on 24/03/2000 (bullish), 30/07/2002
(bearish) and 30/06/2004 (unsettled or sidewards market). . . 87
List of Figures
4.9 Relative risk aversions on 24/03/2000 (bullish), 30/07/2002
(bearish) and 30/06/2004 (unsettled or sidewards market). . . 88
4.10 Common utility functions (solid) and their pricing kernels (dot-
ted) (upper: quadratic, middle: power, lower panel: Kahne-
man and Tversky utility function)................ 89
4.11 Market utility function (solid) with bearish (dashed) and bullish
(dotted) part of an individual utility function 4.5 estimated in
the unsettled market of 30/06/2004................ 91
4.12 Left panel: the market utility function (red) and the fitted
utility function (blue). Right panel: the distribution of the
reference points. 24 March 2000, a bullish market........ 96
4.13 Left panel: the market utility function (red) and the fitted
utility function (blue). Right panel: the distribution of the
reference points. 30 July 2002, a bearish market......... 97
4.14 Left panel: the market utility function (red) and the fitted
utility function (blue). Right panel: the distribution of the
reference points. 30 June 2004, an unsettled market....... 97
5.1 A classification example. The boundary between the classes of
solvent (black triangles) and insolvent companies (white rect-
angles) was estimated using DA and logit regression (two in-
distinguishable lines) and an SVM (a non-linear curve).....102
5.2 One year PDs evaluated for several financial ratios on the
Deutsche Bundesbank data. The ratios are the net income
change K21; net interest ratio K24: interest coverage ratio K29
and the logarithm of total assets K33...............103
5.3 Median AR for DA (rectangles), Logit (circles) and SVM (tri-
angles) for models with different numbers of predictors. At
each step a model with the highest median AR is selected. . . 107
5.4 Left panel: the AR for different radial basis coefficients r. Ca-
pacity is fixed at c = 10. Right panel: the AR for different
capacities c. The radial basis coefficient r is fixed at r = 5.
The training and validation data sets are bootstrapped 100
times without overlapping from the data for 1992-1998. Each
training and validation set contains 400 solvent, and 400 insol-
vent companies...........................108
5.5 The improvement in AR of (i) SVM over DA, (ii) SVM over
Logit and (iii) Logit over DA for the models with the highest
median AR as they were selected by the BSP. The training
data: 1995; testing data: 1998..................110
xvi List of Figures
5.6 The power of a model: beta errors as a function of alpha errors.
An SVM has a higher power than DA or Logit since it has
smaller beta errors for the same alpha errors. Predictors were
selected by the BSP. The training data: 1995; testing data:
1998.................................HI
5.7 Moriotonisation of PDs with the pool adjacent violator algo-
rithm. The thin line denotes PDs estimated with the fc-NN
method with uniform weights and k = 3 before monotonisa-
tion and the bold line after monotonisation. Here y = 1 for
insolvencies, y = 0 for solvent companies.............113
5.8 Smoothing and monotonisation of binary data (y = 1, default
or y = 0, non-default ) represented as circles with a fc-NN
method and a pool adjacent violator (PAV) algorithm. The
estimated PD equals, up to the scale, the first derivative of
the cumulative PD.........................114
5.9 One year probabilities of default estimated with an SVM for
1995.................................114
5.10 The separating hyperplane x w + b = 0 and the margin in
a linearly non-separable case. The observations marked with
bold crosses and zeros are support vectors. The hyperplanes
bounding the margin zone equidistant from the separating hy-
perplane are represented as xTw + 6=1 and xTw + b = -1. • 117
5.11 Mapping from a two-dimensional data space into a three-dimensional
space of features K2 - K3....................119
6.1 A classification example. The boundary between the classes
of solvent (black triangles) and insolvent (white squares) com-
panies was estimated using DA, the logit regression (two in-
distinguishable linear boundaries) and an SVM (a non-linear
boundary) for a subsample of the Bundesbank data. The back-
ground corresponds to the PDs computed with an SVM. ... 124
6.2 One year cumulative PDs evaluated for several financial ratios
from the Deutsche Bundesbank data. The ratios are net in-
come change, K21, net interest ratio, K24, interest coverage
ratio, K29 and logarithm of total assets, K33. The fc-nearest-
neighbours procedure was used with the size of the window
being around 8% of all observations. The total number of
observations is 553500.......................124
6.3 One year probabilities of default for different rating grades. . . 126
List of Figures xvii
6.4 The separating hyperplane x w + b = 0 and the margin in
a non-separable case. The observations marked with bold
crosses and zeros are support vectors. The hyperplanes bound-
ing the margin zone equidistant from the separating hyper-
plane are represented as x w + b = 1 and x w + b = — 1. . . . 127
6.5 Mapping from a two-dimensional data space into a three-dimensional
space of features K2 h- K3 using a quadratic kernel func-
tion K(xj,Xj) = (xjxj)2. The three features correspond to
the three components of a quadratic form: £ = x , £2 =
v2.x i:r2 and £3 = x , thus, the transformation is ^(. t l, £2) =
(x , y/2x X2, xffi. The data separable in the data space with
a quadratic function will be separable in the feature space
with a linear function. A non-linear SVM in the data space is
equivalent to a linear SVM in the feature space. The number
of features will grow fast with the dimensionality d and the
degree of the polynomial kernel p, which equals 2 in our exam-
ple, making the closed-form representation of $ such as here
practically impossible.......................128
6.6 The power curves for a perfect, random and some real classi-
fication models. The AR is the ratio of two areas A/B. It lies
between 0 for a random model with no predictive power and
1 for a perfect model........................130
6.7 The relationship between an accuracy measure (AR) and the
coefficient r in the SVM formulation. Higher r s correspond
to less complex models. The median ARs were estimated on
100 bootstrapped subsamples of 500 solvent and 500 insolvent
companies both in the training and validation sets. A bivariate
SVM with the variables K5 arid K29 was used. We will be
using r = 4 in all SVMs discussed in this chapter........131
6.8 Accuracy ratios for univariate SVM models. Box-plots are
estimated based on 100 random subsamples. The AR for the
model containing only random variable K10 is zero.......133
6.9 Accuracy ratios for bivariate SVM models. Each model in-
cludes variable K5 and one of the remaining. Box-plots are
estimated based on 100 random subsamples...........133
6.10 Accuracy ratios for SVM models with eight variables. Each
model includes variables K5, K29, K7, K33, K18, K21. K24
and one of the remaining. Box-plots are estimated based on
100 random subsamples......................134
xviii List of Figures
6.11 Median improvement in AR. SVM vs. DA (the upper line)
and SVM vs. logit regression (the lower line). Box-plots are
estimated based on 100 random subsamples for the case of DA.
Each model includes variables K5, K29, K7. K33, K18, K21,
K24 and one of the remaining...................134
6.12 PD and cumulative PD estimated with the SVM for a sub-
sample of 200 observation from the Bundesbank data. The
variables were included into the model that achieved the high-
est AR: K5, K29, K7, K33, K18, K21, K24 and K9. The higher
is the score, the higher is the rank of a company.........136
6.13 The luminance and saturation dimensions of the IILS colour
space. We will keep luminance and saturation constant and
encode the information about PDs with hue...........137
6.14 The hue dimension of the IILS colour space...........138
6.15 Probability of default estimated for a random subsample of 500
failing and 500 surviving companies plotted for the variables
K33 and K29. An SVM of high complexity with the radial
basis kernel 0.5E1/2 was used........*...........139
6.16 Probability of default estimated for a random subsample of 500
failing and 500 surviving companies plotted for the variables
K33 and K29. An SVM of average complexity with the radial
basis kernel 4E1/2 was used....................139
6.17 Probability of default estimated for a random subsample of 500
failing and 500 surviving companies plotted for the variables
K33 and K29. An SVM of low complexity with the radial basis
kernel 100S1/2 was used......................140
6.18 Probability of default plotted for the variables K21 and K29.
The boundaries of five risk classes are shown, which correspond
to the rating classes: BBB and above (investment grade), BB,
B+, B, B- and lower........................140
List of Tables
1.1 The estimates of the probability of death from various causes. 6
2.1 Total wealth in different states of the world and their proba-
bilities................................ 17
2.2 Endowments of investors with initial wealth and contingent
claims before trading starts.................... 17
2.3 Investors utility functions and their characteristics....... 18
2.4 The distibution of the wealth after trading.............18
2.5 Expected utilities of investors before and after trading..... 19
2.6 Certainty equivalent of the investors expected utility before
and after trading and its gain................... 20
2.7 Certainty equivalent of the expected utility for highly risk
averse investors with 7 equal 3.6 and 7.2 before and after trad-
ing and its gain........................... 20
2.8 The distibution of the wealth for two identical investors with
7 equal 1.2 after trading. Note the perfect diversification, each
investor holds the market portfolio. The share of the investor
Ai is 52.98%, of the investor A2 47.02%............. 21
2.9 Equilibrium prices of the contingent claims on the states, prob-
abilities of the states and prices per chance for the example
with the investors A and B.................... 21
2.10 The probabilities for unit average temperature ranges...... 23
2.11 The wealth of the investors in the present and in the future
contingent on the average temperature before the trade..... 24
2.12 The wealth of the investors in the present and in the future
contingent on the average temperature after the trade.....25
2.13 The state prices and prices per chance for contingent claims
on average temperature...................... 25
2.14 The gains from trade........................ 25
2.15 Investors relative risk aversion coefficients and expected re-
turns vs. market expected returns................ 30
2.16 Parameters of the log-normal densities.............. 52
xx List of Tables
3.1 Parameters of the estimated log-normal densities, 24 March
2000................................. 59
3.2 Coefficients of the power functions representing pricing kernel
frontiers, 24 March 2000...................... 65
4.1 Models and the time periods used for their estimation.....81
4.2 Market regimes in 2000, 2002 and 2004 described by the return
So/So-a for periods A = 1.0y,2.0y................ 84
5.1 Summary Statistics. qa is an a quantile. IQR is the interquar-
tile range..............................105
5.2 The distribution of the data over the years for solvent and
insolvent companies for the period 1992 1998 for the observa-
tions without missing variables..................105
5.3 Variables included in the DA, Logit and SVM models that
produced the highest ARs. 1 denotes a variable that was se-
lected. The values in parenthesis are the median Alt achieved
for the model reported.......................107
5.4 Forecasting accuracy improvement for each pair of models and
the median AR for an SVM (the highest AR among the three
models). 100 bootstrapped training and 100 bootstrapped
testing samples are used. All figures are reported as percentage
of the ideal AR (100%)......................109
5.5 Forecasting accuracy improvement for each pair of models and
the AR estimated for an SVM (the highest AR among the three
models). All data for the given years are used. All figures are
reported as percentage of the ideal AR( 100%).........HO
5.6 One year PDs of the rating classes represented in Figure 5.9,
the number and percentage of observations in each class for
1995. The total number of observations is 28549. The classes
are denoted using the Moody s notation.............115
6.1 Summary statistics for the Bundesbank data. qa is an a quan-
tile. IQR is the interquartile range................132
|
| adam_txt |
Contents
1 Introduction 1
2 The Pricing Kernel or the Stochastic Discount Factor 11
2.1 Expected Utility Asset Pricing. 11
2.2 Market Interactions, States and Preferences. 14
2.3 The Representative Agent. 28
2.4 Techniques for the Estimation of State Price Densities. 38
2.4.1 Heston Model. 38
2.4.2 Estimation of the State Price Density with the Breeden-
Litzenberger Formula. 40
2.4.3 The Estimation of the Subjective Probability Density . 47
2.4.4 The Estimation of the Pricing Kernel. 50
3 Heterogeneous Investor Hypothesis 59
3.1 The Pricing Kernel in the Black-Scholes World. 59
3.2 Pricing Kernel Frontiers. 61
4 Empirical Pricing Kernels and Investor Preferences 67
4.1 Introduction. 69
4.2 Pricing kernels and utility functions. 70
4.3 Estimation. 74
4.3.1 Estimation approaches for the pricing kernel. 74
4.3.2 Estimation of the risk neutral density. 76
4.3.3 Estimation of the historical density. 79
4.3.4 Empirical pricing kernels. 83
4.4 Individual investors and their utility functions. 89
4.4.1 Individual Utility Function. 90
4.4.2 Market Aggregation Mechanism. 92
4.4.3 The Estimation of the Distribution of Switching Points 94
4.5 Conclusion. 96
4.6 Acknowledgements. 98
xii Contents
5 Estimating Probabilities of Default With Support Vector Ma-
chines 99
5.1 Introduction. 101
5.2 Data. 104
5.3 Variable Selection. 105
5.4 Comparison of DA, Logistic Regression and SVM. 109
5.5 Conversion of Scores into PDs. HI
5.6 Conclusion. 115
5.7 Acknowledgements. 115
5.8 Appendix. 116
6 Graphical Data Representation in Bankruptcy Analysis 121
6.1 Company Rating Methodology. 123
6.2 The SVM Approach .'. 127
6.3 Company Score Evaluation. 129
6.4 Variable Selection. 130
6.5 Conversion of Scores into PDs. 135
6.6 Colour Coding. 137
6.7 Conclusion. Ill
6.8 Acknowledgements. 141
List of Figures
2.1 The probabilities for unit average temperature ranges. The
graph represents the probability density, probability per centi-
grade. 28
2.2 The price per chance for different average temperature ranges. 26
2.3 The probabilities for the states described in terms of the total
wealth. 30
2.4 The price per chance for the states described in terms of the
total wealth. 31
2.5 Individual vs. market returns. Investor C with a lower risk
aversion (red line), investor D with a higher risk aversion (blue
line) and market returns (green line with the 45 degree slope). 32
2.6 The pay-offs of the call (left panel) and put options (right
panel). K is called the strike price. 36
2.7 The call prices as a function of strike prices. DAX, 24 March
2000, maturity is 0.075 year or 19 days. 43
2.8 The call prices as a function of strike prices smoothed with
B-Splines. DAX, 24 March 2000, maturity is 0.075 year or 19
days. 45
2.9 The state price density estimated as the second derivative of
the call prices approximated with B-Splines. DAX, 24 March
2000, maturity is 0.075 year or 19 days. 46
2.10 The state price density q estimated as the second derivative of
the call prices approximated with B-Splines (red). The density
of 19 day DAX returns p estimated on one year of data prior
to 24 March 2000 (blue). The pricing kernel which is the ratio
I (green). DAX, 24 March 2000, maturity is 0.075 year or 19
days. The ODAX market, r = 3.64%. 47
xiv _List of Figures
2.11 The state price density q estimated as the second derivative of
the call prices approximated with B-Splines (red). The density
of 59 day DAX returns p estimated on one year of data prior
to 24 March 2000 (blue). The pricing kernel which is the ratio
2 (green). DAX, 24 March 2000, maturity is 0.233 year or 59
days. The ODAX market, r = 3.757%. 48
2.12 The state price density estimated with the Heston model. DAX,
24 March 2000, the maturity is 0.5 year or 177 days. Private
trading data of the Sal. Oppenheim Bank. 49
2.13 The density of 59 day DAX returns p estimated on one year
of data prior to 24 March 2000 with B-splines (blue) and as a
weighted combination of two log-normal density functions (red). 52
2.14 DAX index price development over one year prior to 24 March
2000. 54
2.15 The 59 day DAX returns over one year prior to 24 March 2000. 55
2.16 Estimated 59 day risk neutral (red) and subjective (blue) den-
sities and the pricing kernel (green) on 24 March 2000.57
3.1 Estimated 59 day risk neutral (red) and subjective (blue) den-
sities under the assumption of their log-normality and the re-
sulting pricing kernel (green) on 24 March 2000. 60
3.2 Estimated frontiers of the pricing kernel for 59 day maturity
claims on 24 March 2000. q is assumed to be log-normal while
p estimated with a kernel estimator. 64
3.3 The switching behaviour of investors between "bearish" and
"bullish" attitudes. Investor A with the switching point 2.4 is
more opimistic than investor B since he switches earlier from
bearish to bullish attitudes. 66
4.1 up: Utility function in the Black Seholes model for T = 0.5
years ahead and drift ^ = 0.1, volatility a = 0.2 and interest
rate r = 0.03. down: Market utility function on 06/30/2000
for T = 0.5 years ahead. 73
4.2 Risk neutral density on 24/03/2000 half a year ahead. 80
4.3 Implied volatility surface on 24/03/00. 80
4.4 Historical density on 24/03/2000 half a year ahead. 82
4.5 DAX, 1998 - 2004.". 84
4.6 Empirical pricing kernel on 24/03/2000 (bullish market). . 85
4.7 Empirical pricing kernel on 24/03/2000 (bullish), 30/07/2002
(bearish) and 30/06/2004 (unsettled or sidewards market). . . 86
4.8 Market utility functions on 24/03/2000 (bullish), 30/07/2002
(bearish) and 30/06/2004 (unsettled or sidewards market). . . 87
List of Figures
4.9 Relative risk aversions on 24/03/2000 (bullish), 30/07/2002
(bearish) and 30/06/2004 (unsettled or sidewards market). . . 88
4.10 Common utility functions (solid) and their pricing kernels (dot-
ted) (upper: quadratic, middle: power, lower panel: Kahne-
man and Tversky utility function). 89
4.11 Market utility function (solid) with bearish (dashed) and bullish
(dotted) part of an individual utility function 4.5 estimated in
the unsettled market of 30/06/2004. 91
4.12 Left panel: the market utility function (red) and the fitted
utility function (blue). Right panel: the distribution of the
reference points. 24 March 2000, a bullish market. 96
4.13 Left panel: the market utility function (red) and the fitted
utility function (blue). Right panel: the distribution of the
reference points. 30 July 2002, a bearish market. 97
4.14 Left panel: the market utility function (red) and the fitted
utility function (blue). Right panel: the distribution of the
reference points. 30 June 2004, an unsettled market. 97
5.1 A classification example. The boundary between the classes of
solvent (black triangles) and insolvent companies (white rect-
angles) was estimated using DA and logit regression (two in-
distinguishable lines) and an SVM (a non-linear curve).102
5.2 One year PDs evaluated for several financial ratios on the
Deutsche Bundesbank data. The ratios are the net income
change K21; net interest ratio K24: interest coverage ratio K29
and the logarithm of total assets K33.103
5.3 Median AR for DA (rectangles), Logit (circles) and SVM (tri-
angles) for models with different numbers of predictors. At
each step a model with the highest median AR is selected. . . 107
5.4 Left panel: the AR for different radial basis coefficients r. Ca-
pacity is fixed at c = 10. Right panel: the AR for different
capacities c. The radial basis coefficient r is fixed at r = 5.
The training and validation data sets are bootstrapped 100
times without overlapping from the data for 1992-1998. Each
training and validation set contains 400 solvent, and 400 insol-
vent companies.108
5.5 The improvement in AR of (i) SVM over DA, (ii) SVM over
Logit and (iii) Logit over DA for the models with the highest
median AR as they were selected by the BSP. The training
data: 1995; testing data: 1998.110
xvi List of Figures
5.6 The power of a model: beta errors as a function of alpha errors.
An SVM has a higher power than DA or Logit since it has
smaller beta errors for the same alpha errors. Predictors were
selected by the BSP. The training data: 1995; testing data:
1998.HI
5.7 Moriotonisation of PDs with the pool adjacent violator algo-
rithm. The thin line denotes PDs estimated with the fc-NN
method with uniform weights and k = 3 before monotonisa-
tion and the bold line after monotonisation. Here y = 1 for
insolvencies, y = 0 for solvent companies.113
5.8 Smoothing and monotonisation of binary data (y = 1, 'default'
or y = 0, 'non-default') represented as circles with a fc-NN
method and a pool adjacent violator (PAV) algorithm. The
estimated PD equals, up to the scale, the first derivative of
the cumulative PD.114
5.9 One year probabilities of default estimated with an SVM for
1995.114
5.10 The separating hyperplane x'w + b = 0 and the margin in
a linearly non-separable case. The observations marked with
bold crosses and zeros are support vectors. The hyperplanes
bounding the margin zone equidistant from the separating hy-
perplane are represented as xTw + 6=1 and xTw + b = -1. • 117
5.11 Mapping from a two-dimensional data space into a three-dimensional
space of features K2 - K3.119
6.1 A classification example. The boundary between the classes
of solvent (black triangles) and insolvent (white squares) com-
panies was estimated using DA, the logit regression (two in-
distinguishable linear boundaries) and an SVM (a non-linear
boundary) for a subsample of the Bundesbank data. The back-
ground corresponds to the PDs computed with an SVM. . 124
6.2 One year cumulative PDs evaluated for several financial ratios
from the Deutsche Bundesbank data. The ratios are net in-
come change, K21, net interest ratio, K24, interest coverage
ratio, K29 and logarithm of total assets, K33. The fc-nearest-
neighbours procedure was used with the size of the window
being around 8% of all observations. The total number of
observations is 553500.124
6.3 One year probabilities of default for different rating grades. . . 126
List of Figures xvii
6.4 The separating hyperplane x' w + b = 0 and the margin in
a non-separable case. The observations marked with bold
crosses and zeros are support vectors. The hyperplanes bound-
ing the margin zone equidistant from the separating hyper-
plane are represented as x' w + b = 1 and x' w + b = — 1. . . . 127
6.5 Mapping from a two-dimensional data space into a three-dimensional
space of features K2 h- K3 using a quadratic kernel func-
tion K(xj,Xj) = (xjxj)2. The three features correspond to
the three components of a quadratic form: £\ = x\, £2 =
v2.x'i:r2 and £3 = x\, thus, the transformation is ^(.'t'l, £2) =
(x\, y/2x\X2, xffi. The data separable in the data space with
a quadratic function will be separable in the feature space
with a linear function. A non-linear SVM in the data space is
equivalent to a linear SVM in the feature space. The number
of features will grow fast with the dimensionality d and the
degree of the polynomial kernel p, which equals 2 in our exam-
ple, making the closed-form representation of $ such as here
practically impossible.128
6.6 The power curves for a perfect, random and some real classi-
fication models. The AR is the ratio of two areas A/B. It lies
between 0 for a random model with no predictive power and
1 for a perfect model.130
6.7 The relationship between an accuracy measure (AR) and the
coefficient r in the SVM formulation. Higher r's correspond
to less complex models. The median ARs were estimated on
100 bootstrapped subsamples of 500 solvent and 500 insolvent
companies both in the training and validation sets. A bivariate
SVM with the variables K5 arid K29 was used. We will be
using r = 4 in all SVMs discussed in this chapter.131
6.8 Accuracy ratios for univariate SVM models. Box-plots are
estimated based on 100 random subsamples. The AR for the
model containing only random variable K10 is zero.133
6.9 Accuracy ratios for bivariate SVM models. Each model in-
cludes variable K5 and one of the remaining. Box-plots are
estimated based on 100 random subsamples.133
6.10 Accuracy ratios for SVM models with eight variables. Each
model includes variables K5, K29, K7, K33, K18, K21. K24
and one of the remaining. Box-plots are estimated based on
100 random subsamples.134
xviii List of Figures
6.11 Median improvement in AR. SVM vs. DA (the upper line)
and SVM vs. logit regression (the lower line). Box-plots are
estimated based on 100 random subsamples for the case of DA.
Each model includes variables K5, K29, K7. K33, K18, K21,
K24 and one of the remaining.134
6.12 PD and cumulative PD estimated with the SVM for a sub-
sample of 200 observation from the Bundesbank data. The
variables were included into the model that achieved the high-
est AR: K5, K29, K7, K33, K18, K21, K24 and K9. The higher
is the score, the higher is the rank of a company.136
6.13 The luminance and saturation dimensions of the IILS colour
space. We will keep luminance and saturation constant and
encode the information about PDs with hue.137
6.14 The hue dimension of the IILS colour space.138
6.15 Probability of default estimated for a random subsample of 500
failing and 500 surviving companies plotted for the variables
K33 and K29. An SVM of high complexity with the radial
basis kernel 0.5E1/2 was used.*.139
6.16 Probability of default estimated for a random subsample of 500
failing and 500 surviving companies plotted for the variables
K33 and K29. An SVM of average complexity with the radial
basis kernel 4E1/2 was used.139
6.17 Probability of default estimated for a random subsample of 500
failing and 500 surviving companies plotted for the variables
K33 and K29. An SVM of low complexity with the radial basis
kernel 100S1/2 was used.140
6.18 Probability of default plotted for the variables K21 and K29.
The boundaries of five risk classes are shown, which correspond
to the rating classes: BBB and above (investment grade), BB,
B+, B, B- and lower.140
List of Tables
1.1 The estimates of the probability of death from various causes. 6
2.1 Total wealth in different states of the world and their proba-
bilities. 17
2.2 Endowments of investors with initial wealth and contingent
claims before trading starts. 17
2.3 Investors' utility functions and their characteristics. 18
2.4 The distibution of the wealth after trading.18
2.5 Expected utilities of investors before and after trading. 19
2.6 Certainty equivalent of the investors' expected utility before
and after trading and its gain. 20
2.7 Certainty equivalent of the expected utility for highly risk
averse investors with 7 equal 3.6 and 7.2 before and after trad-
ing and its gain. 20
2.8 The distibution of the wealth for two identical investors with
7 equal 1.2 after trading. Note the perfect diversification, each
investor holds the market portfolio. The share of the investor
Ai is 52.98%, of the investor A2 47.02%. 21
2.9 Equilibrium prices of the contingent claims on the states, prob-
abilities of the states and prices per chance for the example
with the investors A and B. 21
2.10 The probabilities for unit average temperature ranges. 23
2.11 The wealth of the investors in the present and in the future
contingent on the average temperature before the trade. 24
2.12 The wealth of the investors in the present and in the future
contingent on the average temperature after the trade.25
2.13 The state prices and prices per chance for contingent claims
on average temperature. 25
2.14 The gains from trade. 25
2.15 Investors' relative risk aversion coefficients and expected re-
turns vs. market expected returns. 30
2.16 Parameters of the log-normal densities. 52
xx List of Tables
3.1 Parameters of the estimated log-normal densities, 24 March
2000. 59
3.2 Coefficients of the power functions representing pricing kernel
frontiers, 24 March 2000. 65
4.1 Models and the time periods used for their estimation.81
4.2 Market regimes in 2000, 2002 and 2004 described by the return
So/So-a for periods A = 1.0y,2.0y. 84
5.1 Summary Statistics. qa is an a quantile. IQR is the interquar-
tile range.105
5.2 The distribution of the data over the years for solvent and
insolvent companies for the period 1992 1998 for the observa-
tions without missing variables.105
5.3 Variables included in the DA, Logit and SVM models that
produced the highest ARs. "1" denotes a variable that was se-
lected. The values in parenthesis are the median Alt achieved
for the model reported.107
5.4 Forecasting accuracy improvement for each pair of models and
the median AR for an SVM (the highest AR among the three
models). 100 bootstrapped training and 100 bootstrapped
testing samples are used. All figures are reported as percentage
of the ideal AR (100%).109
5.5 Forecasting accuracy improvement for each pair of models and
the AR estimated for an SVM (the highest AR among the three
models). All data for the given years are used. All figures are
reported as percentage of the ideal AR( 100%).HO
5.6 One year PDs of the rating classes represented in Figure 5.9,
the number and percentage of observations in each class for
1995. The total number of observations is 28549. The classes
are denoted using the Moody's notation.115
6.1 Summary statistics for the Bundesbank data. qa is an a quan-
tile. IQR is the interquartile range.132 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Moro, Rouslan Arthur |
| author_facet | Moro, Rouslan Arthur |
| author_role | aut |
| author_sort | Moro, Rouslan Arthur |
| author_variant | r a m ra ram |
| building | Verbundindex |
| bvnumber | BV035178350 |
| classification_rvk | QC 020 |
| ctrlnum | (OCoLC)552450045 (DE-599)BVBBV035178350 |
| discipline | Wirtschaftswissenschaften |
| discipline_str_mv | Wirtschaftswissenschaften |
| format | Thesis Book |
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| spelling | Moro, Rouslan Arthur Verfasser aut A heterogeneous agent approach vs. representative agent theory and the application of support vector machines to default risk analysis von Rouslan Arthur Moro 2008 XX, 151 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Berlin, Humboldt-Univ., Diss., 2008 Anlageverhalten stw Capital Asset Pricing Model stw Finanzmarkt stw Kreditwürdigkeit stw Risikopräferenz stw Support Vector Machine stw Theorie stw Risikoanalyse (DE-588)4137042-9 gnd rswk-swf Wertpapiermarkt (DE-588)4189708-0 gnd rswk-swf Unternehmensbewertung (DE-588)4078594-4 gnd rswk-swf Investor (DE-588)4336776-8 gnd rswk-swf Insolvenz (DE-588)4072843-2 gnd rswk-swf Risikoverhalten (DE-588)4050133-4 gnd rswk-swf Support-Vektor-Maschine (DE-588)4505517-8 gnd rswk-swf (DE-588)4113937-9 Hochschulschrift gnd-content Risikoanalyse (DE-588)4137042-9 s Support-Vektor-Maschine (DE-588)4505517-8 s DE-604 Investor (DE-588)4336776-8 s Risikoverhalten (DE-588)4050133-4 s Wertpapiermarkt (DE-588)4189708-0 s DE-188 Unternehmensbewertung (DE-588)4078594-4 s Insolvenz (DE-588)4072843-2 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016985157&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Moro, Rouslan Arthur A heterogeneous agent approach vs. representative agent theory and the application of support vector machines to default risk analysis Anlageverhalten stw Capital Asset Pricing Model stw Finanzmarkt stw Kreditwürdigkeit stw Risikopräferenz stw Support Vector Machine stw Theorie stw Risikoanalyse (DE-588)4137042-9 gnd Wertpapiermarkt (DE-588)4189708-0 gnd Unternehmensbewertung (DE-588)4078594-4 gnd Investor (DE-588)4336776-8 gnd Insolvenz (DE-588)4072843-2 gnd Risikoverhalten (DE-588)4050133-4 gnd Support-Vektor-Maschine (DE-588)4505517-8 gnd |
| subject_GND | (DE-588)4137042-9 (DE-588)4189708-0 (DE-588)4078594-4 (DE-588)4336776-8 (DE-588)4072843-2 (DE-588)4050133-4 (DE-588)4505517-8 (DE-588)4113937-9 |
| title | A heterogeneous agent approach vs. representative agent theory and the application of support vector machines to default risk analysis |
| title_auth | A heterogeneous agent approach vs. representative agent theory and the application of support vector machines to default risk analysis |
| title_exact_search | A heterogeneous agent approach vs. representative agent theory and the application of support vector machines to default risk analysis |
| title_exact_search_txtP | A heterogeneous agent approach vs. representative agent theory and the application of support vector machines to default risk analysis |
| title_full | A heterogeneous agent approach vs. representative agent theory and the application of support vector machines to default risk analysis von Rouslan Arthur Moro |
| title_fullStr | A heterogeneous agent approach vs. representative agent theory and the application of support vector machines to default risk analysis von Rouslan Arthur Moro |
| title_full_unstemmed | A heterogeneous agent approach vs. representative agent theory and the application of support vector machines to default risk analysis von Rouslan Arthur Moro |
| title_short | A heterogeneous agent approach vs. representative agent theory and the application of support vector machines to default risk analysis |
| title_sort | a heterogeneous agent approach vs representative agent theory and the application of support vector machines to default risk analysis |
| topic | Anlageverhalten stw Capital Asset Pricing Model stw Finanzmarkt stw Kreditwürdigkeit stw Risikopräferenz stw Support Vector Machine stw Theorie stw Risikoanalyse (DE-588)4137042-9 gnd Wertpapiermarkt (DE-588)4189708-0 gnd Unternehmensbewertung (DE-588)4078594-4 gnd Investor (DE-588)4336776-8 gnd Insolvenz (DE-588)4072843-2 gnd Risikoverhalten (DE-588)4050133-4 gnd Support-Vektor-Maschine (DE-588)4505517-8 gnd |
| topic_facet | Anlageverhalten Capital Asset Pricing Model Finanzmarkt Kreditwürdigkeit Risikopräferenz Support Vector Machine Theorie Risikoanalyse Wertpapiermarkt Unternehmensbewertung Investor Insolvenz Risikoverhalten Support-Vektor-Maschine Hochschulschrift |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016985157&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT mororouslanarthur aheterogeneousagentapproachvsrepresentativeagenttheoryandtheapplicationofsupportvectormachinestodefaultriskanalysis |