# Multiscale Forecasting Approach of Property Insurance Income via Wavelet Method

Authors: Li-Wei Lin; Xiao-Hui Zhou
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

A multiscale prediction algorithm was presented in this paper based on the monthly data of property insurance income of Zhejiang Province from February 2013 to May 2018, and the correlation of the wavelet coefficients for property insurance income was also discussed at different time scales. By ADF tests, the stability of nonlinear approximation data and detail data were given. Furthermore, the prediction algorithm of nonlinear trend and detail data was studied according to the wavelet decomposition formula and the autoregressive moving average (ARMA) model. Finally, the short-term prediction results from June 2018 to January 2019 were given by the wavelet reconstruction formula. Compared to the prediction results of the traditional ARMA model, the relative error of the prediction results in this paper is smaller and better.


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## Body

## 1. Introduction

Our research mainly investigates the income of the finance and insurance industry in Zhejiang Province, China, using time series and wavelet forecasting methods to forecast. This study mainly combined related concepts such as mathematics, financial management, economics, and marketing. We broke the traditional concept of only pure economics, primarily to solve the accuracy of forecasting financial property insurance income. Property insurance plays an essential role in the insurance market, which is crucial for the management of property risk and maintaining economic and social stability and development in modern society. On the one hand, the level of property insurance income implicates the optimal allocation of financial resources; on the other hand, property insurance income provides a guarantee and support for the prosperity and stability of the capital market and the efficient operation of the financial market. Therefore, the analysis of the dynamic changes and correlation of property insurance income in different periods and the prediction of property insurance income play a guiding role in the rational allocation of financial resources and the stable operation of the financial market. However, the wavelet method is an effective method to analyse financial data in different periods.The wavelet method has become a new econometric method in economy and finance in recent years. Its unique feature is that it can deal with nonstationary data, localise in the time domain, and carry out multiscale analysis of economic and financial data based on the required time scale. Besides, a new forecasting method is also discussed by the wavelet method. For example, Long and Yan [1] have studied a composite forecasting algorithm for import and export volume based on a wavelet filter. Gallegati [2] investigated some examples of multiscale analysis of economic and financial data based on the wavelet method in the literature, for instance, Wavelet Applications in Economics and Finance. According to the wavelet method, Zhang [3] studied the detection of a financial crisis. A high-frequency prediction method for stock index and future is presented by wavelet analysis. Thus, the wavelet multiscale method has become a new topic for high-frequency data processing, and a lot of research has been done in this regard. However, its application for macroeconomic data needs further study, especially for a small sample. In this paper, an example of this problem is presented. Choose the monthly data of property insurance income in Zhejiang Province from February 2013 to May 2018, and 8-month short-term forecasting for property insurance income is discussed by wavelet multiscale method.The main idea is to decompose the economic or financial data onto different time scales by the wavelet multiscale method. Then the correlation between the nonlinear trend and the details on different time scales are discussed. Furthermore, combined with the autoregressive moving average (ARMA) model, the nonlinear trend and details are predicted on different time scales. Finally, the prediction results and figures are given for the discussion. Multiscale analysis results show that there is no significant linear correlation between the nonlinear trend and the details of different time scales. Moreover, the prediction results show that the wavelet multiscale method improves the prediction accuracy of the ARMA model. He et al. [4] referred to Comparative Study of Artificial Neural Networks and Wavelet Artificial Neural Networks for Groundwater Depth Data Forecasting with Various Curve Fractal Dimensions. Later, Guo et al. [5] referred to Air Pollution Forecasting Using Artificial and Wavelet Neural Networks with Meteorological Conditions. Guo and He [6] then referred to the Prediction of the Confirmed Cases and Deaths of Global COVID-19 Using Artificial Intelligence.


## 2. Theoretical Analysis and Literature Review

### 2.1. Property Insurance Income

Premium income is the consideration income that an insurance company collects from the policyholder for fulfilling the obligations stipulated in the insurance contract. It is the primary source of capital inflow for insurance companies and the main source of capital for insurers to fulfill their insurance liabilities. [7] Warrad et al. referred to Analysing the Dynamic Relationship between Foreign Direct Investment and Trade Balance. Fred E. Karlinsky and Fidei [8] referred to Is Florida Any Closer to Finding a Solution to Its Property Insurance Problems?Insurance property income is important for insurance companies in Zhejiang Province, China. Insurance income includes a series of property insurance income from real estate, accident insurance income, health insurance, car insurance, and so on.We know that our series of designs mainly predicts the accuracy of future property insurance income, which is why we do such prediction and research.


### 2.2. Dynamic Relationship

The dynamic relationship method, also known as the proportional increase and decrease method, determines the planned index by using the general law of the development and change of a certain index in history, considering the changing factors during the planning period. This method is simple and easy to forecast for the market, but it still has the defect that the budget is out of touch with the target. Siemaszkiewicz [9] referred to the Dynamic Relationship between Precious Metals and Central European Stock Markets. Ann-Ni Soh and Meng Chang Jong [10] referred to Assessing the Dynamic Relationship of Stock Market, International Trade, and Economic Growth: Empirical Evidence from Wavelet Analysis. Blokhina [11] emphasised the importance of the stock market efficiency concept for investors and managers in the Vietnamese securities market.


### 2.3. Resource Allocation Theory

Resource allocation refers to the selection of relatively scarce resources to be compared for various uses. Hackman [12] proposed that resource allocation theory is mainly a process that facilitates decision-making by the entire organisational centre. The entire financial centre is dynamically affected by the entire environment; therefore, we further made a resource allocation action. Several studies have referred to the concept of organisational performance, and this resource allocation further affects the overall performance. Resource allocation is an important concept of team goals and interactions [13]. Cuijpers et al. [14] discussed the effects of dual identification and interteam conflict on multiteam system performance. According to some scholars, resource allocation affects the performance of the company, as long as the company’s performance is positively oriented towards its final effect and efficiency. Our research mainly used wavelets to predict future financial income, and we applied the theory of resource allocation to analyse and discuss financial income forecasting. Resource allocation has an important concept in economics; appropriate resource allocation to further achieve the overall income performance of property insurance is an important concept for the resource allocation theory. Research on this important issue has focused on identifying make decision choices on the best [15], risk information [16], and complex decision problems ([17] as shown in Table 1.Table 1
A summary of the antecedents to income of the finance and insurance industry in Zhejiang Province, China.
Illustrative researchContextTheoretical basisAntecedents to income of finance and insurance industry in Zhejiang provinceKey findings or propositionsDudzińska-Baryła [17]Between decision-makingProspect theoryComplex decision problemsAnalysis of complex decision problems is based on cumulative prospect theoryPutri et al. [16]Between different formats of risk informationProbabilistic mental model (PMM) theoryRisk informationThe findings suggest that risk information impact investment decisions: experimental test of PMM theory, a case of IndonesiaLiu et al. [15]Between organisation quality specificity immune evolution abilityFuzzy multiattribute decision-making methodMake decision choices on the best and optimal partner and the investee objectThe results show that the intuitively fuzzy multiattribute group decision-making of organisational quality specificity, immune evolution ability based on evidential reasoning


### 2.4. Wavelet Method

Assuming thatVj is a closed subspace sequence of L2R, and it satisfies:(1)
Vj⊂Vj+1, ∀j∈Z(2)
∩j∈ZVj=0, ∪j∈ZVj¯=L2R(3)
ft∈Vj⇔f2t∈Vj+1(4)
There exists a functionϕt such that the set ϕt−k,k∈Z forms an orthogonal basis of V0An orthogonal multiresolution analysis ofL2R is called to be generated by the subspace sequence Vj of L2R with the above properties, where Vj=closL2R<ϕj,k=2j/2ϕ2jt−k:k∈Z> and ϕj,k=2j/2ϕ2jt−k.Assuming that for everyj∈Z, Wj is an orthogonal complementary space of Vj in the space Vj+1, that is, Vj+1=Vj⊕Wj. Thus, ⊕j∈ZWj=L2R. If there exists a function ψt∈L2R such that the set ψt−k,k∈Z is an orthogonal basis of W0, where Wj=closL2R<ψj,k=2j/2ψ2jt−k:k∈Z>, then the function ϕt∈L2R is called to be a scaling function, and ψt∈L2R is the wavelet function corresponding to ϕt.For a stochastic processesXt, its approximate coefficients with scale parameter j and translation parameter k can be denoted by(1)cj,k=Xt,ϕj,kL2R=2j/2∫RXt⋅ϕ2jt−kdt.And its wavelet coefficients with scalej and translation k can be denoted by(2)dj,k=Xt,ψj,kL2R=2j/2∫RXt⋅ψ2jt−kdt.According to the multiresolution analysis, a given time seriesxt can be expressed as the sum of finite wavelet coefficients and approximate coefficients as follows:(3)xt≈∑kcJ,kϕJ,kt+∑kdJ,kψJ,kt+∑kdJ−1,kψJ−1,kt+⋯+∑kd1,kψ1,kt,where cJ,k=∫ϕJ,kxtdt is approximate (scale) coefficient and dj,k=∫ψj,kxtdt is the detail (wavelet) coefficient, j=1,2,⋯,J, J is the scale integer number such that 2J is less than the number of data points. However, cJ,k denotes the smooth coefficient that captures the trend of the raw data xt, and dJ,k,dJ−1,k,⋯,d1,k denotes the detail coefficients that capture the high-frequency volatility of the raw data xt from different scales.


## 2.1. Property Insurance Income

Premium income is the consideration income that an insurance company collects from the policyholder for fulfilling the obligations stipulated in the insurance contract. It is the primary source of capital inflow for insurance companies and the main source of capital for insurers to fulfill their insurance liabilities. [7] Warrad et al. referred to Analysing the Dynamic Relationship between Foreign Direct Investment and Trade Balance. Fred E. Karlinsky and Fidei [8] referred to Is Florida Any Closer to Finding a Solution to Its Property Insurance Problems?Insurance property income is important for insurance companies in Zhejiang Province, China. Insurance income includes a series of property insurance income from real estate, accident insurance income, health insurance, car insurance, and so on.We know that our series of designs mainly predicts the accuracy of future property insurance income, which is why we do such prediction and research.


## 2.2. Dynamic Relationship

The dynamic relationship method, also known as the proportional increase and decrease method, determines the planned index by using the general law of the development and change of a certain index in history, considering the changing factors during the planning period. This method is simple and easy to forecast for the market, but it still has the defect that the budget is out of touch with the target. Siemaszkiewicz [9] referred to the Dynamic Relationship between Precious Metals and Central European Stock Markets. Ann-Ni Soh and Meng Chang Jong [10] referred to Assessing the Dynamic Relationship of Stock Market, International Trade, and Economic Growth: Empirical Evidence from Wavelet Analysis. Blokhina [11] emphasised the importance of the stock market efficiency concept for investors and managers in the Vietnamese securities market.


## 2.3. Resource Allocation Theory

Resource allocation refers to the selection of relatively scarce resources to be compared for various uses. Hackman [12] proposed that resource allocation theory is mainly a process that facilitates decision-making by the entire organisational centre. The entire financial centre is dynamically affected by the entire environment; therefore, we further made a resource allocation action. Several studies have referred to the concept of organisational performance, and this resource allocation further affects the overall performance. Resource allocation is an important concept of team goals and interactions [13]. Cuijpers et al. [14] discussed the effects of dual identification and interteam conflict on multiteam system performance. According to some scholars, resource allocation affects the performance of the company, as long as the company’s performance is positively oriented towards its final effect and efficiency. Our research mainly used wavelets to predict future financial income, and we applied the theory of resource allocation to analyse and discuss financial income forecasting. Resource allocation has an important concept in economics; appropriate resource allocation to further achieve the overall income performance of property insurance is an important concept for the resource allocation theory. Research on this important issue has focused on identifying make decision choices on the best [15], risk information [16], and complex decision problems ([17] as shown in Table 1.Table 1
A summary of the antecedents to income of the finance and insurance industry in Zhejiang Province, China.
Illustrative researchContextTheoretical basisAntecedents to income of finance and insurance industry in Zhejiang provinceKey findings or propositionsDudzińska-Baryła [17]Between decision-makingProspect theoryComplex decision problemsAnalysis of complex decision problems is based on cumulative prospect theoryPutri et al. [16]Between different formats of risk informationProbabilistic mental model (PMM) theoryRisk informationThe findings suggest that risk information impact investment decisions: experimental test of PMM theory, a case of IndonesiaLiu et al. [15]Between organisation quality specificity immune evolution abilityFuzzy multiattribute decision-making methodMake decision choices on the best and optimal partner and the investee objectThe results show that the intuitively fuzzy multiattribute group decision-making of organisational quality specificity, immune evolution ability based on evidential reasoning


## 2.4. Wavelet Method

Assuming thatVj is a closed subspace sequence of L2R, and it satisfies:(1)
Vj⊂Vj+1, ∀j∈Z(2)
∩j∈ZVj=0, ∪j∈ZVj¯=L2R(3)
ft∈Vj⇔f2t∈Vj+1(4)
There exists a functionϕt such that the set ϕt−k,k∈Z forms an orthogonal basis of V0An orthogonal multiresolution analysis ofL2R is called to be generated by the subspace sequence Vj of L2R with the above properties, where Vj=closL2R<ϕj,k=2j/2ϕ2jt−k:k∈Z> and ϕj,k=2j/2ϕ2jt−k.Assuming that for everyj∈Z, Wj is an orthogonal complementary space of Vj in the space Vj+1, that is, Vj+1=Vj⊕Wj. Thus, ⊕j∈ZWj=L2R. If there exists a function ψt∈L2R such that the set ψt−k,k∈Z is an orthogonal basis of W0, where Wj=closL2R<ψj,k=2j/2ψ2jt−k:k∈Z>, then the function ϕt∈L2R is called to be a scaling function, and ψt∈L2R is the wavelet function corresponding to ϕt.For a stochastic processesXt, its approximate coefficients with scale parameter j and translation parameter k can be denoted by(1)cj,k=Xt,ϕj,kL2R=2j/2∫RXt⋅ϕ2jt−kdt.And its wavelet coefficients with scalej and translation k can be denoted by(2)dj,k=Xt,ψj,kL2R=2j/2∫RXt⋅ψ2jt−kdt.According to the multiresolution analysis, a given time seriesxt can be expressed as the sum of finite wavelet coefficients and approximate coefficients as follows:(3)xt≈∑kcJ,kϕJ,kt+∑kdJ,kψJ,kt+∑kdJ−1,kψJ−1,kt+⋯+∑kd1,kψ1,kt,where cJ,k=∫ϕJ,kxtdt is approximate (scale) coefficient and dj,k=∫ψj,kxtdt is the detail (wavelet) coefficient, j=1,2,⋯,J, J is the scale integer number such that 2J is less than the number of data points. However, cJ,k denotes the smooth coefficient that captures the trend of the raw data xt, and dJ,k,dJ−1,k,⋯,d1,k denotes the detail coefficients that capture the high-frequency volatility of the raw data xt from different scales.


## 3. Resources of Data and Its Decomposition

To avoid the impact of COVID-19, the monthly property insurance income data of Zhejiang Province in China were chosen, and 64 data points were noted from February 2013 to May 2018. The data are derived from the Zhejiang Provincial Bureau of Statistics (https://tjj.zj.gov.cn/). The property insurance income of Zhejiang Province from February 2013 to May 2018 is shown in Figure 1 (denoted by PII: 100 million yuan). The monthly property insurance income data clearly shows an increasing trend with cyclical fluctuations in Figure 1. The maximum property insurance income was 87.67 yuan in February 2018, while the minimum value was 28.93 yuan in March 2013. The mean μ of data was 54.2063. By computing, the kurtosis was 0.3033, and the skewness was 0.5243, where kurtosis and skewness were computed by EXt−μ3 and EXt−μ4, respectively.Figure 1
Property insurance income of Zhejiang Province from February 2013 to May 2018 (unit: 100 million yuan).Based on discrete wavelet transform, the monthly property insurance income data is decomposed onto a scale-by-scale basis. According to the length of the data, a db3 wavelet filter was chosen with a filter length of 6. By the decomposition and reconstruction algorithm, the data were decomposed to scale 2 for our prediction, as presented in Section4. Note that the lower time scale corresponds to a higher frequency band. In detail, approximate coefficient A2 shows the nonlinear trend for the period from 4 to 8 months in Figure 2. The data a2 also has an apparent periodicity. The wavelet coefficient D1 of scale 1 implicates the high-frequency volatility with the period from 2 to 4 months. The coefficient D2 of scale 2 implicates the high-frequency volatility with the period from 4 to 8 months. The oscillation of d1 is relatively violent, and the oscillation of d2 is relatively slow. This shows that the oscillation of the detail data decreases with the increasing scale. That means the fluctuation amplitude and frequency decrease as time increases. This phenomenon was more pronounced if the raw data were decomposed on more time scales.Figure 2
Multiscale analysis of property insurance income of Zhejiang Province.On the other hand, the correlation of the nonlinear trend data a2 and the detail data d1 and d2 were estimated in Table2. The data a2, d1, and d2 showed no significant linear correlation. These results were consistent with the orthogonal relationship between the spatial sequences based on the wavelet analysis. Thus, the nonlinear trend data a2 and the detail data d1 and d2 may be predicted.Table 2
Correlation between the nonlinear trend a2 and the detailed data d1 and d2.
a2d1d2a21.0000−0.01520.0145d1−0.01521.00000.0181d20.01450.01811.0000Note. The correlation coefficients between trend data a2 and detail data d1 and d2 are given in the table. Each scale is corresponding to a specific time period. For example, the first time scale is from 2 to 4 months, and the second is from 4 to 8 months.


## 4. Multiscale Forecasting Method and Results

According to the results of multiscale decomposition in Section3, the orthogonal relationships among the nonlinear trend data a2 and the detail data d1 and d2 guaranteed that these data can be predicted. In this section, based on the wavelet method and the ARMA model, a multiscale forecasting method is presented to predict the 8-month property insurance income. The main idea was to construct the ARMA model for the nonlinear trend data a2 and the detail data d1 and d2. Based on the multiresolution analysis and (3), the raw data xt was rewritten as follows:(4)xt≈A2t+D2t+D1t,where A2t=∑kc2,kϕJ,kt, D1t=∑kd1,kψ1,kt, and D2t=∑kd2,kψ2,kt. Moreover, (4) is equivalent to J=2 in (3). Considering the computing speed and error, the historical data of property insurance income in Zhejiang Province from February 2013 to May 2018 were decomposed to scale 2 for our prediction.Multiscaling Forecasting Algorithm: In step one, the nonlinear trend dataA2t and the detail data D1t and D2t were obtained by taking the wavelet transform on the monthly property insurance income data of Zhejiang Province from February 2013 to May 2018.In step two, the ADF test was taken for the nonlinear trend dataA2t and the detail data D1t and D2t.In step three, according to the results of the ADF test, an ARIMA model was constructed for the nonlinear trend dataA2t, and ARMA models for the detail data D1t and D2t were established.In step four, based on the above models, the predicted valuesA2^t, D1^t, and D2^t for the nonlinear trend data A2t and the detail data D1t and D2t, were computed, respectively. Then the predicted value x^t for the property insurance income xt was obtained by (4), that is, x^t=A2^t+D2^t+D1^t.By taking the wavelet transform on the monthly property insurance income data of Zhejiang Province from February 2013 to May 2018, the nonlinear trend dataA2t and the detail data D1t and D2t are shown in Figure 3. The nonlinear trend data A2t is shown at the top of Figure 3, and the detail data D1t is shown at the bottom of Figure 3. In the middle of Figure 3, the detail data D2t is given. The detail data D1t and D2t implicates the high-frequency volatility with the periods from 2 to 4 months and from 4 to 8 months, respectively. By observing the figures of the nonlinear trend data A2t and the detail data D1t and D2t, the nonlinear trend data A2t is nonstationary, while the detail data D1t and D2t are stationary. According to the ADF test of the nonlinear trend data A2t and the detail data D1t and D2t, the results of their stability are shown in Table 3.Figure 3
Real and forecast values of the nonlinear trends A2 from February 2013 to January 2019.Table 3
Stability test forA2t, D1t, and D2t.
VariableTest value of ADFThe cutoff value for each significant levelResult1% level5% level10% levelA21.942958−2.608490−1.946996−1.612934NonstableD1−6.765018−2.609324−1.947119−1.612867StableD2−2.740476−2.609324−1.947119−1.612867StableΔA2−4.444665−2.608490−1.946996−1.612934StableAccording to the results of the ADF test for the dataA2t, D1t, and D2t, the wavelet detail data D1t and D2t of the original property insurance income are stable at each significant level. But the nonlinear trend A2t is nonstable, and its difference ΔA2t of order 1 is stable. The data ΔA2t and detail data D1t and D2t can be ordered to establish the corresponding ARMA model by their autocorrelation function and partial autocorrelation function, respectively. According to the AIC or SC information guidelines, the orders p,q that minimised their AIC or SC are chosen for identifying the ARMA model. Thus, the corresponding predicted data can be obtained by their identified ARMA model.Generally, for a time seriesxt and white noise processes εt, the ARMA (p, q) model can be simplified in the following form:(5)ΦLxt=ΘLεt,where ΦL=1−ϕ1L−ϕ2L2−⋯−ϕpLp, ΘL=1−θ1L−θ2L2−⋯−θqLq, and L is the lag operator such that Lkxt=xt−k and Lkεt=εt−k. By (5), to establish an ARMA (p, q) model is equivalent to determine the operators ΦL and ΘL.The difference for the nonlinear trendsA2t with order 1 can be denoted by ΔA2t=A2t+1−A2t, and both its autocorrelation and partial autocorrelation functions show drag-tail characteristics. According to the AIC guidelines, choose p=12,q=4, and AIC=−2.6388, and the operator ΦL and ΘL can be identified as follows:(i)
ΦΔA2tL=1−1.2L+0.4649L2−0.5478L3+1.369L4−1.306L5+0.5022L6−0.5984L7+1.372L8−1.269L9+0.4897L10−0.5822L11+0.3033L12(ii)
ΘΔA2tL=1+0.09943L+0.02434L2−0.08236L3−0.9763L4An ARMA(12,4) forΔA2t can be established as follows:(6)ΦΔA2tLΔA2t=ΘΔA2tLεtΔA2t.The data fit of this model is presented in Figure3, and the predicted values A2^t show the characteristics of the nonlinear trend A2t, goodness of fit R2=0.9380, and mean square error MSE=0.02937, where the real value of A2t is depicted by the blue real line and the predicted value of the model is indicated by the star line ∗. According to the initial value A21 and the prediction model (6), forecasting data from 65 to 72 show the predicted trend A2t from June 2018 to January 2019 in Figure 3, as illustrated by the star line ∗.For the detail dataD1t, both its autocorrelation and partial autocorrelation functions show drag-tail characteristics. According to the AIC guidelines, p=11,q=9, and AIC=−0.2691 were chosen, and the operators ΦD1tL and ΘD1tL can be identified as follows:(i)
ΦD1tL=1+L+0.913L2+0.9332L3+0.9439L4+1.063L5+1.109L6+1.111L7+1.103L8+1.153L9+1.136L10+1.086L11(ii)
ΘD1tL=1−1.092L−0.2529L2−0.2271L3+0.8583L4−0.4449L5−0.05491L6−0.1796L7+1.065L8−0.666L9An ARMA(11,9) forD1t can be established as follows:(7)ΦD1tLD1t=ΘD1tLεtD1t.The data fit of this model is presented in Figure4, and the predicted values D1^t show the oscillating information of the detail data D1t, goodness of fit R2=0.9716, and mean square error MSE=0.2636, where the real value of D1t is depicted by the blue real line and the predicted value of the model is indicated by the star line ∗. According to the initial value D11 and the prediction model (7), forecasting data from 65 to 72 show the detail data D1t from June 2018 to January 2019 in Figure 4, as illustrated by the star line ∗.Figure 4
Real and forecast values of the detail data D1 from February 2013 to January 2019.For the detail dataD2t, both its autocorrelation and partial autocorrelation functions show drag-tail characteristics. According to the AIC guidelines, choose p=11,q=11, AIC=−2.5240, and the operators ΦD2tL and ΘD2tL can be identified as follows:(i)
ΦD2tL=1−0.8035L+1.093L2−0.8917L3+1.062L4−0.8771L5+L6−0.8069L7+0.9944L8−0.7839L9+1.019L10−0.8015L11(ii)
ΘD2tL=1+1.485L−0.2338L2−0.7508L3+0.07579L4+0.1357L5−0.4524L6−0.7002L7+0.8016L8+1.403L9+0.1628L10−0.221L11An ARMA(11,11) forD2t can be established as follows:(8)ΦD2tLD1t=ΘD2tLεtD2t.The data fit of this model is presented in Figure5, and the predicted values D2^t show the oscillating information of the detail data D2t, goodness of fit R2=0.9685, and mean square error MSE=0.02857, where the real value of D2t is depicted by the blue real line and the predicted value of the model is indicated by the star line ∗. According to the initial value D21 and the prediction model (8), forecasting data from 65 to 72 show the detail data D2t from June 2018 to January 2019 in Figure 5, seen by a star line ∗.Figure 5
Real and forecast values of the detail data D2 from February 2013 to January 2019.Combined with the predicted nonlinear trendsA2^t and the predicted detailed data D1^t and D2^t can be presented from the following formula:(9)x^t=A2^t+D2^t+D1^t.The forecasting value of property insurance income from June 2018 to January 2019 was calculated, as shown in Figure6, in which the blue solid line shows the true value of property insurance income and the star line ∗ shows the forecast value of property insurance income. In Figure 6, the predicted and the real values show a good prediction effect, and the growth trend and the fluctuation error between the predicted value and the real value are all small. The total predicted values from June 2018 to January 2019 listed in Table 3 illustrate the accuracy of the predicted values. In Table 4, the true values and the forecast values predicted by the traditional ARMR model and the wavelet ARMA model of property insurance income from June 2018 to January 2019 are all shown, respectively. For comparison with the traditional ARMA model, the relative errors are calculated in the last column in Table 4, where the relative error derived from the wavelet ARMA model is in parentheses. By comparison, it is not difficult to find that the relative error predicted by the wavelet ARMA model is smaller than that of conventional ARMA models, except for in September 2018. The maximum relative error of the wavelet ARMA model was 7.75%, and the minimum was 0.16%, while the conventional ARMR model was 17.70%. Therefore, the prediction values of the wavelet methods outperformed the prediction results of conventional ARMA models in general.Figure 6
Real and forecast values of property insurance income in Zhejiang Province from February 2013 to May 2018.Table 4
Real value and forecasting value of the property insurance income from June 2018 to January 2019.
TimeReal valueForecasting value of ARMA modelForecasting value by wavelet ARMARelative errorJune 201864.668.852264.24646.58% (0.55%)July 201868.156.046462.824517.70% (7.75%)August 201866.271.307163.19567.71% (4.54%)September 201867.567.012263.15000.72% (6.44%)October 201872.482.514973.310513.97% (1.26%)November 201862.861.142061.83592.64% (1.54%)December 20187270.893874.24761.54% (3.12%)January 201984.580.695984.36904.50% (0.16%)


## 5. Conclusion

In the paper, based on the multiscale decomposition of the wavelet method, nonlinear growth trend A2 and detailed data D1 and D2 were obtained by dealing with Zhejiang Province property insurance income monthly data from 2013 02 to 2018 05. These data did not show any significant linear correlation with each other. This result fits with the orthogonal relationship of wavelet spaces, and it also provides empirical evidence for multiscale forecasting methods. For the nonlinear growth trend A2 and detailed data D1 and D2, the corresponding ARMA models were established for forecasting the values from June 2018 to January 2019, respectively. The forecasting value of property insurance income from June 2018 to January 2019 was obtained using (9), and the relative error predicted by the wavelet ARMA model was smaller than that depicted by conventional ARMA models. In general, the prediction values of the wavelet methods outperformed the prediction results of conventional ARMA models. This work is an example of wavelet methods applied in the macroeconomic data and provides a reference for predictive algorithms based on small amounts of data especially. In our future study, we will further explore whether the wavelet method still has the advantage of predicting other small amounts of sample data. [18–26].


## 6. Managerial and Practical Implications

Our research focuses on forecasting the income of the property and casualty insurance industry in the entire Zhejiang Province of China. Since the application of wavelets to forecast can accurately see the overall resource allocation benefits, we suggest that subsequent researchers can apply the wavelet formula to construct, which is rarely applied by scholars for economic and financial forecasting. Some financial companies can also apply and forecast data in this way, which will have an overall benefit to the company. In this study, we disintegrated the traditional hypotheses, which have been applied by a few people in financial forecasting.



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