Intelligent forecasts and evaluation of financial risk assets based on Grey Markov chain model

Abstract

In order to analyze the change trend in risk financial assets of the retail business, this paper which is combined with the advantages of grey prediction system and Markov chain, establishes the “grey differential-Markov chain” intelligent prediction and evaluation model according to business characteristics. Meanwhile, the modeling steps are introduced, and the solution methods, model validation results, model prediction and evaluation methods are introduced in detail through specific cases. Through specific analysis, this paper aims to show that the model has high prediction accuracy and wide application value in the field of financial retail business.

Keywords

Grey model Markov chain state transition probability risk asset ratio

1 Introduction

In the financial retail business, especially in the field of housing mortgage, auto consumption loans and credit cards, the focus of financial institutions is the level of credit risk, that is, whether the customers repay the loan on time [1]. No repayment schedule will form overdue accounts, and overdue accounts equal to financial institutions risk assets. If the risk assets are not properly controlled, it will seriously affect the value of financial institutions, which is also the core of the new Basel capital accord.

In order to understand the credit risk, in the practice of risk asset control, an important task is to predict and assess the future situation of risky assets according to the historical repayment condition of the customer group. The methods familiar to all are Credit Metrics, Credit Portfolio View, KMV, Credit Risk+ and Kamakura, etc., but these methods are mostly designed for enterprise customers, the methods specifically designed for retail financial services, which are also known as the prediction methods specifically targets the individual for customer groups are extremely rare. Therefore, this paper proposes a new idea, namely through the forecast of risk asset ratio (the proportion of risk assets = month overdue accounts/accounts receivable), that we can realize the implementation of effective management and control of credit risk caused by financial risk assets in retail business.

Due to the fact that the risk assets ratio is influenced by some certain comprehensively accidental factors such as personal credit, social factor, accidental factor and etc., in the continuously changing state, therefore resulting in a complex, and fluctuating random sequences. In Liu and Wang’s paper, the grey system method was introduced into the prediction of non-performing loan ratio [2], and this paper took China postal savings bank as the empirical research object and the model is well tested by the precision test.

Liu chose the amount of non-performing loans in the different quarters in 2004–2005, and used GM (1, 1) model to anticipate the trend of non-performing loans [3]. Wang and Wu divided the data into groups, established differential equations by using the sequences of different seasons, and obtained the final prediction formula, which proved that the method was better than the conventional seasonal prediction method [4]. Cheung [5] and Dueker [6] had studied the foreign exchange rate by using the Markov chain model; Lin and so on used the method to study the Dow Jones Industrial Average component stock return moving average model [7]; Zhang and Wang used the Markov chain state transition matrix to analyze and predict overdue loans of commercial banks, and provide reference for the effective overdue loan management of banks [8]; Liu divided the loans into five categories by their different levels, and then uses the absorption Markov chain to predict the future state of the loan risk [9], therefore proving that the method is an effective quantitative analysis method.

As the traditional GM (1, 1) model is only applicable to the original series which is decreased exponentially, the predicted geometry of which is a smooth curve, resulting in its poor prediction accuracy of random numbers, while Markov chain is applicable to the dynamic process of random fluctuation. Therefore, it can just make up for the defect of GM (1, 1) model. However, in the Markov chain, it is required that prediction objects should enjoy the characteristics of Markov chain and have a smooth process, while the proportion of risk assets is changing with time or show a change trend of non-stationary process, therefore, if we adapt the GM (1, 1) model to combine the data and then to find its change trend, we shall make up for the shortage of Markov chain. Therefore, if we can predict the proportion of risk assets assessment by the combination of GM (1, 1) and Markov chain based on the advantages of the construction of the “Grey Markov chain model”, we should be able to combine the advantages and avoid the disadvantages to improve the prediction accuracy.

2 Establishment of Grey Markov model

2.1 Grey number series forecast

Suppose time series x⁽⁰⁾ has n observed values: x⁽⁰⁾ ={ x⁽⁰⁾ (1), x⁽⁰⁾ (2), …, x⁽⁰⁾ (n) }, and it can form a new series by accumulating all the obser-ved values: x⁽¹⁾ ={ x⁽¹⁾ (1), x⁽¹⁾ (2), …, x⁽¹⁾ (n) } [10, 11]. $x^{(1)} (t) = \sum_{k = 1}^{t} x^{(0)} (k)$ (1) in the formula: t = 1,2, …, n. Therefore we can get a new series x⁽¹⁾ (t), when combined with the original series x⁽⁰⁾, its random possibility is largely reduced. Its change trend can be described approximately by the following differential equation: $\frac{{dx}^{(1)}}{dt} + {ax}^{(1)} = u$ (2)

Structure accumulation matrix B and constant term vector Yn, namely: $B = [\begin{matrix} - 0.5 (x^{(1)} (1) + x^{(1)} (2)) & 1 \\ - 0.5 (x^{(1)} (2) + x^{(1)} (3)) & 1 \\ \dots & \dots \\ - 0.5 (x^{(1)} (n - 1) + x^{(1)} (n)) & 1 \end{matrix}]$ (3) $Y_{n} = [\begin{matrix} x^{(0)} (2) & x^{(0)} (3) & \dots & x^{(0)} (n) \end{matrix}]$ (4)

Using the least squares method of solving linear differential equation $\hat{a}$ , then we get: $\hat{a} = (\begin{matrix} a \\ u \end{matrix}) = (B^{T} B)^{- 1} B^{T} Y_{n}$ (5)

$\hat{a}$ is called the specific parameters; u is called endogenous control grey number.

If we put the grey parameter $\hat{a}$ into the differential equation, then we get: $x^{(1)} (t + 1) = (x^{(0)} (1) - \frac{u}{a}) e^{- at} + \frac{u}{a}$ (6)

Seek derivative of the x⁽¹⁾ (t + 1), then we get: ${\overset{⌢}{x}}^{(0)} (k) = x^{(1)} (t + 1) x - x^{(1)} (t)$ (7)

2.2 Markov probability transfer matrix

State division: dividing state is to divide ${\hat{x}}^{(0)} (k)$ into n states according to the forecast curve’s divarication degree from the original number series. For a non-smooth random number series x⁽⁰⁾ (t + 1) which is adaptable to the Markov chain [12], any state of it, namely θ_i can be expressed as following: θ_i = [θ _1i _,θ _2i ], θ _1i = ${\hat{x}}^{(0)} (k)$ +A_i,θ _2i = ${\hat{x}}^{(0)} (k)$ +B_i, (i = 1, 2, …, n).

As ${\hat{x}}^{(0)} (k)$ changes with the value t, so do θ _1i and θ _2i , therefore θ_i is dynamic. The state partition number n and grey element θ _1i , θ _2i can be determined according to the prediction accuracy and the number of raw data.

Transfer matrix of initial probability for computing state: ${Pi}_{j} = {Mi}_{j} (m) / m_{i} (i, j = 1, 2, \dots, n)$ (8)

In the formula (8), M_ij (m) is the number of raw data samples transferred from state θ_i to state θ_j, m_i is the number of raw data samples in θ_i state. The state transition probability matrix is: $A (m) = [\begin{matrix} p_{11} & p_{12} & \dots & p_{1 n} \\ p_{21} & p_{22} & \dots & p_{2 n} \\ \dots & \dots & \dots & \dots \\ p_{n 1} & p_{n 2} & \dots & p_{nn} \end{matrix}]$ (9)

A (m) reflects the law of transfer between states of the system. The state transition probability P_ij (m) reflects the probability that the state θ_i shifts from the M step to the state θ_j. This is the basis of Markov probability matrix prediction. By examining A (m), the future state of the system can be predicted [13].

Absolute error: $ɛ (k) = x^{(0)} (k) - {\hat{x}}^{(0)} (k)$ , relative error: e (k) = ɛ (k) ÷ x⁽⁰⁾ (k), relative accuracy q = 1 - e (k).

2.3 Determine the change range of the predicted value

By investigating the transition probability matrix, the future transfer state θ_j of the system can be determined according to the maximum probability of transfer, and the grey element θ _1i , θ _2i are determined, therefore, the variation range of the prediction value is determined as [θ _1i , θ _2i ]

The most probable predictive value is:

${\hat{\hat{x}}}^{(0)} (k) = {\hat{x}}^{(0)} (t) + \sum^{} \frac{1}{2} θ_{j} P_{j i}$ (10)

3 The application of the model

Table 1 shows the data of risk asset ratio of the auto loan business of a financial institution from June 2012 to September 2014, a total of 28 months’ data. T stands for the risk assets ratio in June 2016, and we can get a result by accumulating the data in Table 1, and the result can be seen in Table 2.

Table 1
Risk asset ratio of the auto loan business of a financial institution

t 1 2 3 4 5 6 7

x 36.1 40.2 36.5 36.4 31 33.4 31.9

t 8 9 10 11 12 13 14

x 31.1 24.5 28.6 30.6 30.4 26.8 24.3

t 22 23 24 25 26 27 28

x 51.3 45.8 45.8 43.4 38.4 35.9 37.8

t 15 16 17 18 19 20 21

x 29.5 29.6 35.3 39 40.1 42.7 51.6

t	1	2	3	4	5	6	7
x	36.1	40.2	36.5	36.4	31	33.4	31.9
t	8	9	10	11	12	13	14
x	31.1	24.5	28.6	30.6	30.4	26.8	24.3
t	22	23	24	25	26	27	28
x	51.3	45.8	45.8	43.4	38.4	35.9	37.8
t	15	16	17	18	19	20	21
x	29.5	29.6	35.3	39	40.1	42.7	51.6

Table 2

The proportion of risk assets of automobile loan business in a financial institution

t	1	2	3	4	5	6	7
x	36.1	76.3	112.8	130.5	180.2	213.6	245.5
t	8	9	10	11	12	13	14
x	276.6	301.5	330	360	391	417.5	441.8
t	15	16	17	18	19	20	21
x	471.3	500.8	536.1	580.3	615.2	658	710
t	22	23	24	25	26	27	28
x	760.8	806.6	853	896	934.3	970.2	1008

According to formula (3) and formula (4), the accumulated matrix B and constant term vector are constructed: $B = [\begin{matrix} - 56.2 & 1 \\ - 94.5 & 1 \\ \dots & \dots \\ - 989.1 & 1 \end{matrix}]$ (11) $Y_{n} = [\begin{matrix} 40.2 & 36.5 & \dots & 37.8 \end{matrix}]$ (12)

According to formula (5), Solving grey parameter by least square method $\hat{a}$ , we shall get: a= −0.014; u= 29.005.

According to formula (6), we get GM (1, 1) model, x⁽¹⁾ (t + 1)= 2107.886e^0.014t −2071.786.

According to formula 6, conduct the reductive derivation for x⁽¹⁾ (t + 1), the predicted and predicted errors of each period are shown in Table 3.

Table 3

Prediction value and prediction error in GM (1, 1) model

No.	Anticipation value	Actual value	Absolute error	Relative error
1	29.72	36.1	6.38	17.68%
2	30.14	40.2	10.1	25.03%
3	30.56	36.5	5.94	16.27%
4	30.99	36.4	5.41	14.86%
5	31.43	31	0.43	1.39%
6	31.87	33.4	1.53	4.57%
7	32.32	31.85	0.47	1.48%
8	32.78	31.1	1.68	5.39%
9	33.24	24.5	8.74	35.67%
10	33.71	28.6	5.11	17.86%
11	34.18	30.6	3.58	11.71%
12	34.67	30.4	4.27	14.03%
13	35.15	26.8	8.35	31.17%
14	35.65	24.3	11.4	46.71%
15	36.15	29.5	6.65	22.55%
16	36.66	29.6	7.06	23.86%
17	37.18	35.3	1.88	5.32%
18	37.7	39	1.3	3.32%
19	38.23	40.1	1.87	4.65%
20	38.77	42.7	3.93	9.19%
21	39.32	51.6	12.3	23.80%
22	39.88	51.3	11.4	22.27%
23	40.44	45.8	5.36	11.71%
24	41.01	45.8	4.79	10.46%
25	41.59	43.4	1.81	4.18%
26	42.17	38.4	3.77	9.82%
27	42.77	35.9	6.87	19.13%
28	43.37	37.8	5.57	14.73%

Table 4

Division criteria of 5 state regions

State	θ ₁	θ ₂	θ ₃	θ ₄	θ ₅
θ _1j	θ₁₁= ${\hat{x}}^{(0)} (k)$	θ₁₂= ${\hat{x}}^{(0)} (k)$ +5	θ₁₃= ${\hat{x}}^{(0)} (k)$	θ₁₄= ${\hat{x}}^{(0)} (k)$ −4	θ₁₅= ${\hat{x}}^{(0)} (k)$ −8.5
θ _2j	θ₂₁= ${\hat{x}}^{(0)} (k)$ +5	θ₂₂= ${\hat{x}}^{(0)} (k)$ +12.5	θ₂₃= ${\hat{x}}^{(0)} (k)$ −4	θ₂₄= ${\hat{x}}^{(0)} (k)$ −8.5	θ₂₅= ${\hat{x}}^{(0)} (k)$ −12

It can be seen from Table 3 that the maximum absolute error is 12.3 percentage points, the average absolute error is 5.2 percentage points, the maximum relative error is 46.71%, and the average relative error is 15.2%. The model can fit the change trend of the original data, but the prediction error has a larger swing range.

According to the predicted curve ${\hat{x}}^{(0)} (k)$ as the center point, we divide into 5 strip regions parallel to the curve ${\hat{x}}^{(0)} (k)$ (see Table 4), each region forming a state, which indicates that the predicted value deviates from the original data. The number of the state can be determined according to the original number of samples in each state.

The number of states above can also be divided into more states, but the original sample point and the mean feature of each state should be considered. If the number of samples is small and the data is different, it is not appropriate to be classified as a state.

It can be seen that the original data is not obvious exponential change, showing an increasing trend. The original sample points falling into the θ₁ state are ordinal numbers 6, 18, 19, 20, 24, 25, respectively, a total of 6 sample points. By formula 7, the next state of θ₁ can be shifted to state θ₁, θ₂, θ ₃ , θ ₄ , θ ₅ probability.

The possibility are 3/6, 1/6, 2/6, 0 and 0. Other states are in turn analogous to each other. Because the last data of the original series is uncertain, the last data is deleted. We can get the state next traction probability matrix A (m): $A (m) = [\begin{matrix} 3 / 6 & 1 / 6 & 2 / 6 & 0 & 0 \\ 1 / 7 & 5 / 7 & 1 / 7 & 0 & 0 \\ 2 / 6 & 0 & 1 / 6 & 2 / 6 & 1 / 6 \\ 0 & 0 & 2 / 6 & 3 / 6 & 1 / 6 \\ 0 & 0 & 0 & 1 & 0 \end{matrix}]$ (13)

According to one step probability transfer matrix of each state, the prediction value and prediction error of each period are calculated in Table 5.

Table 5

The prediction value of Grey Markov Chain Model and its error

Number	Anticipation value	Actual value	Absolute error	Relative error
1	36.05	36.1	0.05	0.00
2	36.46	40.2	3.74	0.09
3	36.89	36.5	0.39	0.01
4	37.32	36.4	0.92	0.03
5	28.14	31	2.86	0.09
6	33.91	33.4	0.51	0.02
7	29.03	31.85	2.82	0.09
8	29.49	31.1	1.61	0.05
9	22.99	24.5	1.51	0.06
10	28.21	28.6	0.39	0.01
11	30.89	30.6	0.29	0.01
12	29.17	30.4	1.23	0.04
13	29.65	26.8	2.85	0.11
14	25.40	24.3	1.10	0.05
15	30.65	29.5	1.15	0.04
16	30.65	29.5	1.15	0.04
17	31.16	29.6	1.56	0.05
18	33.89	35.3	1.41	0.04
19	39.74	39	0.74	0.02
20	40.27	40.1	0.17	0.00
21	40.81	42.7	1.89	0.04
22	45.65	51.6	5.95	0.12
23	46.20	51.3	5.10	0.10
24	46.76	45.8	0.96	0.02
25	43.05	45.8	2.75	0.06
26	43.63	43.4	0.23	0.01
27	38.88	38.4	0.48	0.01
28	37.27	35.9	1.37	0.04

From the table above, the maximum absolute error value is 5.95%, the average absolute error value is 1.6%, the maximum relative error is12%, and the average value is 4%.

From the analysis above, the prediction results with the simple Grey Markov model GM (1, 1) compared the prediction results, the prediction error of the former swing was significantly reduced (see Table 6), the prediction accuracy is increased by 14.2 percentage points.

Table 6

Comparison of model prediction errors

Model	Average absolute error	average relative error %	maximum relative error %	average relative precision %
GM(1,1)	5.2	15.2	46.7	84.2
Grey Markov	1.6	4	12	98.4

According to GM (1, 1) model, the financial assets of the financial institutions in October 2014 and November were 43.9%and 44.6%, respectively.

The calculation of October 2014 (twenty-ninth) the most likely prediction value: When the state probability matrix A(m) is investigated, for the state of ${\hat{x}}^{(0)} (28)$ lies in θ ₄ and for P₄₄ is the maximum number. Therefore, October 2009 is most likely to be in θ ₄ , and its range of variation is [39.9%, 35.4%]. According to formula (10), the most likely forecast for the financial institution in October 2009 is ${\hat{\hat{x}}}^{(0)} (29)$ = 38.4%.

Calculate the most probable forecast for November 2014. To predicts the most probable prediction in November 2014. We must first find the two step transition probability of the state probability matrix A (m): $A (m) = [\begin{matrix} 0.38 & 0.2 & 0.24 & 0.11 & 0.06 \\ 0.22 & 0.53 & 0.17 & 0.04 & 0.02 \\ 0.22 & 0.06 & 0.25 & 0.39 & 0.08 \\ 0.11 & 0 & 0.22 & 0.53 & 0.14 \\ 0 & 0 & 0.33 & 0.5 & 0.17 \end{matrix}]$ (14)

According to the above transition probability, the financial institution may continue to be in θ ₄ in November 2014, the range of variation is [40.6%, 36.1%], the most probable forecast for November 2014 is ${\hat{\hat{x}}}^{(0)} (30)$ = 39.7%.

From the above results, if the environment and the system is not changed, the proportion of risk assets of the agency in the next two months will slow in the adverse trend rise, prompting the control work of the organization to conduct asset quality risk. In order to reduce the proportion of risk assets fundamentally, the key is to adjust the parameter a from a minus number to a positive number, and then to make the prediction curve ${\hat{x}}^{(0)} (k)$ change from upward to downward by increasing and decreasing, and this adjustment is not spontaneous and endogenous, but it must be realized through the adjustment of system and mechanism of external environment.

4 Conclusions

The results show that, compared with the single GM prediction model or Markov chain model, the prediction accuracy is improved greatly in the process of predicting and evaluating the financial risk assets. At the same time, this study also found that the grey horse prediction model has the following defects:

The model needs history data information. But too much historical data information will result in less prediction errors in the long-term, and the short-term prediction error will be larger; the appropriate number of historical data will result in larger prediction error in the long term, but the short-term prediction error will be smaller.

The prediction accuracy is closely related to the range and number of state changes.

The longer the future time is, the greater the prediction error will be. In order to improve the accuracy, you must continue to increase the latest information, and then carry out the next forecast on the basis of the new statistical samples.

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