The interest rate of endowment insurance is a kind of insurance closely related to people's lives. Usually, insurance companies will provide a variety of old-age insurance plans for the insured to choose from, and the amount of insurance premiums and pensions is listed in detail in the plans. For example, an insurance company's insurance pointed out that under the agreement that 200 yuan pays a pension every month to 60 years old, if a man is insured at the age of 25, the monthly pension is 2282 yuan; If you are insured at the age of 35, the monthly pension is 1056 yuan; If you are insured at the age of 45, the monthly pension will be 420 yuan. Try to determine the interest rate of the insurance premium paid in these three cases. Remarks: The model for calculating the old-age insurance according to the average life expectancy of men is designed to be 75 years old. Through analyzing the given insurance scheme and combining with the actual situation of endowment insurance, this paper puts forward a calculation method which is beneficial to the insured. The following is a brief analysis of the scheme given in the title: Scheme 1: I will start insurance at the age of I: from the age of 25 to 59, I will start to receive a pension at the age of 60 until my death, and I will pay a certain amount to my family at one time. Option 2: Insure from the age of 35 to the age of 59, receive a pension at the age of 60 until death, and pay a certain amount to family members in one lump sum. Option 3: Insure from the age of 45 to the age of 59, receive a pension at the age of 60 until death, and pay a certain amount to family members in one lump sum. In view of this scheme, we need to find an index to determine the favorable scheme, so we introduce the guaranteed interest rate (defined as the ratio of the difference between the total income (including interest) and the total insured amount (including interest) to the total insured amount); In this way, the future funds will be converted into present value, which reflects the interests of the insured and the insurance company and which scheme is more beneficial to the insured. It needs to be explained here: a. It means that the insured is profitable; B, representing the equivalent exchange between the insured and the insurance company; C. it means that the insurance company is profitable. In addition, the greater the value, the better for the insured. We calculated the value of scheme 1 as 0.00485, scheme 2 as 0.0046 1 and scheme 3 as 0.004 13. According to our definition of rights, Option 1 is more beneficial to the insured. One question reiterates that endowment insurance is a kind of insurance closely related to people's lives. Usually, insurance companies will provide a variety of old-age insurance plans for the insured to choose from, and the amount of insurance premiums and pensions is listed in detail in the plans. For example, an insurance company's insurance pointed out that under the agreement that 200 yuan pays a pension every month to 60 years old, if a man is insured at the age of 25, the monthly pension is 2282 yuan; If you are insured at the age of 35, the monthly pension is 1056 yuan; If you are insured at the age of 45, the monthly pension will be 420 yuan. Try to determine the interest rate of the insurance premium paid in these three cases. Remarks: Based on the average life expectancy of men is 75 years, the following reasonable assumptions are made according to the topic and the actual situation, which makes the problem simple and easy to solve. 1. Assume that the time to pay insurance premium and receive pension is the beginning and end of each month respectively. 2. Assume that life expectancy is the last month to receive a pension. 3. The monthly interest rate of the bank does not change with time. 4. It is more beneficial for the insured to understand that in different schemes, the difference between the total pension (including interest) and the total insurance (including interest) at the time of death accounts for a larger proportion of the total insurance. Description of three symbols: insurable interest; : interest on insurance income; : Insurance income (total amount received+interest); : collect the total amount; : insurance premium (total insured amount+interest); : total insured amount; A: After the death of the insured, the insurance company will pay the family members in one lump sum. Analysis of Four Problems This problem is a multi-factor fuzzy description problem in the actual social background. To solve this problem, we need to go through the following processes: 1. According to our hypothetical conditions, the benefit to the insured depends on the ratio of the difference between the total pension (including interest) and the total insured amount (including interest) to the total insured amount. The definition is as follows: insurance interest rate = ……………………………………………………… (1) The insurance premium rate of the above formula is the solution to the problem we are seeking, which is the optimal solution, that is, the higher the insurance interest rate, the better it will be for the insured. According to the hypothesis and its definition, there are the following situations: (1), indicating that the insured is profitable; (2), said the insured and the insurance company equivalent exchange; (3) It means that the insurance company is profitable. 2. The relationship between the main elements The insured must also consider the interest generated by the invested funds, and the interest generated at this time () is actually a part of the total insurance amount (). We might as well assume that: insurance income () = received amount+interest, that is: 1. If a person is insured at the age of 25, according to formula (3), the insurance premium represents the corresponding interest. According to formula (2), the insurance income is: ................. (5) At this time: a =10000; In which: the amount received at the age of J; Represents the corresponding interests. Substitute Formula (4) and Formula (5) into Formula (1). According to Formula (3), the insurance premium is based on Formula (2), and the insurance income is: ... (8) At this time, A =10000; In which: the amount received at the age of J; Represents the corresponding interests. Substituting Formula (4) and Formula (5) into Formula (1), we can get: ... (9) Here we can get: =0.0046 1 Scheme 3: Under the agreement that a man will be insured from the age of 45 to the age of 60, then the monthly pension will be 420 yuan until his death, and the insurance company will pay his family in one lump sum after his death. According to formula (3), the insurance premium represents the corresponding interest. According to formula (2), the insurance income is: ... (8) At this time: a =10000; In which: the amount received at the age of J; Represents the corresponding interests. Substituting Formula (4) and Formula (5) into Formula (1), we can get: ....................... (9) Here we can get: =0.004 13. Comparing the above three values, we can see that the first insurance method is more favorable. Test of six models This model deduces the general model of the problem from the actual problem and explains the results with the defined model. According to our above model and analysis, we can know that it can be positive, negative or zero, and its meaning is as mentioned above; For the first question, the values of Scheme I, Scheme II and Scheme III are 0.00485 and 0.0046 1 and 0.004 13 respectively. Obviously, according to the meaning of value defined by us, at this time, Scheme I is more beneficial to the insured than Scheme II and Scheme III; Scheme 1 is a special form of our general model (e=0), and we use the data given by the problem to optimize it when testing the model. For the first scheme, we have made the following tests: In the process of analysis, we should clearly know that if the life expectancy of the insured does not reach the life expectancy of the insurance company, then the value we seek should be less than the value that reaches the life expectancy. When it is realized in C language, the value of a person at the age of 74 is less than 0.00485, which means that they are all beneficial to the insured (the definition of value is known); When we take someone's life span as 74 years old, the value is positive at this time, and the corresponding value is negative when the life span is 73 years old; At this time, there is a negative value. According to our analysis, it means that it is beneficial to the insurance company at this time. Scheme 1 conforms to the actual situation, so our model is established for Scheme 1. From the above test, we can see that the interest rate of Scheme 1 is higher than that of Scheme 2 and Scheme 3. This is in line with our actual situation and the final conclusion of our model. So the model is available under certain conditions. Model evaluation This model not only calculates the optimal insurance in the scheme given by the title, but also considers the influence of the amount of insurance funds on the insurance profit, and introduces quantification. But this model is based on our assumptions, such as: the bank's interest rate can not be unchanged for many years, regardless of the annual death probability of people. In this way, the influence of these aspects on the model can be considered in the improvement of the model. This model is of great significance to the actual insurance problem, which can not only be used as a reference tool for insurance companies, but also provide certain information for policyholders. In this paper, the sensitivity analysis of model change caused by life change is also carried out. However, there are some shortcomings: the model has no graphics, tables and other parts, which can not make the problem more clear and intuitive. References: [1] Jiang Qiyuan, Xie Jinxing, Alfred. Mathematical model (3rd edition) [M]. Beijing: Higher Education Press, February 2004 [2] Tang, He Mingfeng. Introduction to Mathematical Models (2nd Edition). Beijing: Higher Education Press, May 2002 [3] Appendix: C Source Program: # include