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Adoption of deep learning Markov model combined with copula function in portfolio risk measurement

Liangxiong Li
Liangxiong Li
 and 
Bishr Muhamed Muwafak
Bishr Muhamed Muwafak
   | Dec 13, 2021
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Published Online: Dec 13, 2021
Page range: 901 - 916
Received: Jun 17, 2021
Accepted: Sep 24, 2021
DOI: https://doi.org/10.2478/amns.2021.2.00112
Keywords
© 2021 Liangxiong Li et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Model optimisation based on different functions.
Model optimisation based on different functions.

Fig. 2

Solving process of portfolio VaR. VaR, value at risk.
Solving process of portfolio VaR. VaR, value at risk.

Fig. 3

Basic structure of RNN model. RNN, recurrent neural network.
Basic structure of RNN model. RNN, recurrent neural network.

Fig. 4

Descriptive statistical results of the exponential logarithmic yield of each industry. (A) banking industry; (B) insurance industry; (C) securities industry; (D) trust industry).
Descriptive statistical results of the exponential logarithmic yield of each industry. (A) banking industry; (B) insurance industry; (C) securities industry; (D) trust industry).

Fig. 5

Distribution characteristics of different industry sequences.
Distribution characteristics of different industry sequences.

Fig. 6

PIT statistical results. (A) banking industry; (B) insurance industry; (C) securities industry; (D) trust industry. PIT, probability integral transform.
PIT statistical results. (A) banking industry; (B) insurance industry; (C) securities industry; (D) trust industry. PIT, probability integral transform.

Fig. 7

Iteration results for different function parameters.
Iteration results for different function parameters.

Fig. 8

Iteration results of parameters of different function. (A) state 1; (B) state 2.
Iteration results of parameters of different function. (A) state 1; (B) state 2.

Fig. 9

Transition and state conditional probability of mixed Copula model based on HMM. (A) the dynamic transition graph of the state; (B) the probability graph of low dependence; and (C) the probability graph of high dependence.
Transition and state conditional probability of mixed Copula model based on HMM. (A) the dynamic transition graph of the state; (B) the probability graph of low dependence; and (C) the probability graph of high dependence.

Fig. 10

Influence of different parameter values on average prediction error of the model. (A) The influence curve of training duration and input data coverage and (B) the influence curve between the number of neurons in the hidden layer and the width of input data.
Influence of different parameter values on average prediction error of the model. (A) The influence curve of training duration and input data coverage and (B) the influence curve between the number of neurons in the hidden layer and the width of input data.

Fig. 11

Stock price prediction results based on deep learning Markov model.
Stock price prediction results based on deep learning Markov model.

Estimation of edge distribution parameters of sequence exponential logarithmic rate of yield based on GAS model.

Distribution α β w
A Laplace distribution 0.065 0.989 3.677 × 103
B Laplace distribution 0.043 0.995 4.223 × 103
C Laplace distribution 0.038 0.996 4.019 × 103
D Partial t distribution 0.039 0.997 3.567 × 103

Tail dependence of various industries in different states.

State A B C D
A 1 — 0.443 0.431 0.112
2 — 0.223 0.210 0.153
B 1 0.293 — 0.311 0.241
2 0.286 — 0.132 0.112
C 1 0.257 0.210 — 0.413
2 0.162 0.132 — 0.215
D 1 0.206 0.232 0.241 —
2 0.103 0.134 0.213 —

Estimation of edge distribution parameters of sequential exponential logarithmic yield sequence based on GARCH model.

Distribution μ ω α β θ AIC
A Normal 0.002 7.226 × e−6 0.052 0.960 5.311 −5.442
T 0.002 8.039 × e−6 0.043 0.970 −5.559
B Normal 0.002 1.322 × e−6 0.062 0.950 5.825 −5.297
T 0.002 1.264 × e−6 0.037 0.952 −5.384
C Normal 0.001 2.339 × e−6 0.038 0.970 5.797 −5.338
T 0.001 2.407 × e−6 0.051 0.963 −5.458
D Normal 0.002 3.389 × e−6 0.036 0.953 5.265 −5.165
T 0.002 3.667 × e−6 0.055 0.954 −5.364

Parameter estimates for single and mixed Copula functions.

Function θ w Log-likelihood function AIC BIC
Gumbel Copula 1.711 — 1720.59 −3432.65 −3433.29
Clayton Copula 1.121 — 1593.28 −3087.24 −3177.52
Frank Copula 4.552 — 1633.74 −3006.88 −3014.87
Mixed Copula 1.538 0.439 1957.33 −3926.01 −3800.65
HMM-mixed Copula — — 2059.63 −4088.39 −4010.98

Descriptive statistical results of the yield sequence.

Mean value Maximum Minimum Standard deviation Skewness Kurtosis J_B statistic P value
A 0.026 1.578 −1.826 1.552 0.031 9.668 3,511.921 0.000
B 0.041 2.065 −2.052 1.938 0.082 6.268 897.682 0.000
C 0.009 2.559 −2.597 2.356 0.043 6.842 1,123.972 0.000
D 0.007 2.113 −2.273 2.054 −0.741 6.557 1,250.747 0.000
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