Insurance is a risk transfer from the insured to the insurer. In general insurance companies are grouped into two types that life insurance and general insurance. For measure risk in general insurance the method used is using a measure of risk. In the study of risk management, there is one method forming risk measure known a distortion function. The purpose of this study is prove theorems of properties a measure of coherent and consistent risk of distortion. In this study explain the formation of a measure of risk distortion using a distortion function, indicates that if the distortion function is a concave function and shows the consistency of risk distortion measures preserve second order stochastic dominance and show coherence and consistency several of distortion risk measures. The results of this study concave distortion function is a necessary condition and sufficient condition for coherence and a strictly concave distortion function is a necessary condition and sufficient condition for strict ordering consistent with preserve second order stochastic dominance.

To assess the success of development in agriculture can be done by looking at the level of welfare of farmers, which can be measured from the Farmer Exchange Rate (NTP). NTP is the ratio between the price index value received by farmers and the price index value paid by farmers. NTP is suspected to have a relationship with various factors, where to determine the influence of these factors can be done through modeling both as response variables and predictor variables. NTP values in several sectors can be observed from an object of research in a certain period of time. Therefore, panel data regression can be used in modeling the relationship between NTP and the factors that influence it. The purpose of this research is to analyze the factors that are thought to influence the Food Crop Farmer Exchange Rate (NTTP) by using panel data regression. The factors referred to are land area, harvested area, production, productivity, GRDP of the agricultural sector, inflation, and the consumer price index.. The data used comes from the five largest rice-producing provinces in Indonesia according to data from the Ministry of Agriculture in 2020. This research data is sourced from the website of the Badan Pusat Statistik (BPS) and the Indonesian Ministry of Agriculture for the 2008-2017 time period. The independent variabels in the study were land area, harvested area, production, productivity, agricultural sector GDP, inflation, and the consumer price index, while the dependent variabel was NTTP. The results of the regression analysis, it can be concluded that the Common Effect Model is the best model of the NTTP panel regression in 5 provinces of Indonesia with an R-Squared value of 53.25% and an error value of 7.55% accuracy of the estimation results using MAPE. This shows that the factors that are thought to affect NTTP such as Productivity, Inflation, and CPI have a significant influence, while the variabels of Land Area, Harvest Area, Production, GRDP of the Agricultural Sector are not significant in the regression model and the rest is influenced by other factors. outside of this research. The value of the MAPE accuracy error rate shows a percentage below 10% which means the forecast value is very accurate.

Model epidemik merupakan salah satu bentuk model matematika di bidang epidemologi. Penyakit diare adalah salah satu penyakit menular yang dapat dicegah melalui treatment. Tujuan penelitian ini adalah untuk menjelaskan terbentuknya model epidemik penyebaran penyakit diare,  menganalisis kestabilan model, dan membuat simulasi numerik. Penelitian ini menggunakan metode linierisasi untuk melinierkan model nonlinier. Metode matriks next generation  untuk menentukan Basic reproduction number  dan metode runge kutta orde empat untuk melakukan simulasi model. Hasil dari penelitian ini, diperoleh model epidemik penyakit diare berbentuk Model SIRT (Susceptible, Infected, Treatment, Recovered) dengan fungsi insidensi Holling Tipe 2. Selanjutnya, diperoleh dua titik ekulibrium dan diperlihatkan bahwa  berperan penting dalam proses penyebaran penyakit. Jika   maka titik ekuilibrium bebas penyakit stabil asimtotik sehingga populasi akan terbebas dari wabah penyakit. Sebaliknya jika  maka titik ekuilibrium endemik stabil asimtotik sehingga penyakit akan selalu ada dalam populasi. Berdasarkan nilai indeks sensitivitas menunjukkan bahwa parameter laju kontak efektif dan laju kelahiran  adalah parameter yang paling sensitif (berbanding lurus) terhadap perubahan nilai . Selanjutnya, simulasi model diberikan untuk memperlihatkan ilustrasi terhadap analisa kestabilan model

A life insurance contract contains the amount of funds that must be paid by insured as a responsibility for a received compensation. There funds are called as premium. Payment of the premium which paid with one payment at the beginning of the contract time called as net single premium. One factor that influenced the calculation of life insurance premiums is a life probability. In general, a life probability constructed by the assumption that death only involves two conditions, life and death. Yet, there are another condition for the insured that also affect a person’s death condition which is sick. The objecktive of this research is to determine net single premium of term life insurance formula using transition probability as a life probability. The first will constructed transition from three condition which are health, sick, and death as stochastic process. Transition probability will be determined by solving Chapman Kolmogorov system differential equation. Then the probability transition that determined will be used for calculate net single premium from term life insurance. Net single premium will be determined by using expectation value of present value of benefit random variables. From this research get formula of net single premium of term life insurance contains discount function, transition probability, and force of mortality of someone.

In the case of nonlife insurance, insurance companies are very potential to get losses if claims submitted by customers (policyholders) exceeds the reserves of budgeted claims. It is the risk that have to managed properly by insurance companies . One possible disadvantage is the aggregate loss model. The aggregate loss model is a random variable that states the total of all losses incurred in an insurance policy block. This kind of loss can be modeled using a collective risk approach where the number of claims is a discrete random variable and the size of claim is a continuous random variable. The purpose of this study is to determine risk measure of standard deviation premium principle, value at risk (VaR), and conditional tail expectation (CTE) of the aggregate loss model. Standard deviation premium principle risk measure of aggregate loss model is determined analytically by substituted it expected value and varians. Meanwhile, VaR risk measure is determined using numerical method by Monte Carlo method, then the quantile value and it confidence interval for the actual value will estimate. In the CTE calculation, based on the loss data obtained in the Monte Carlo method, the CTE value is estimated by calculating the average loss that exceeds the VaR value. If the data size is large enough, the CTE value estimation will converge to the actual value.Keywords: Aggregate Loss Model, Standard Deviation Premium Principle, Value at Risk (VaR), Conditional Tail Expectation (CTE).