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In risk models, the ruin probabilities and ‎Lundberg‎ bound are calculated despite knowing the statistical distribution of random variables. In the present paper, for collective risk model and discrete time risk model of insurance company for independent and identically distributed claims with light-tailed distribution, the infinite time ruin probabilities are computed using Lundberg bound, moreover the general forms of density functions of random variables of claim sizes are derived. 

‎‎‎For some special cases in the  discrete time risk model, the density functions of claim sizes have the shifted geometric distribution, and for the collective risk model, they always have an exponential distribution.‎ ‎

‎‎Presenting the numerical examples of infinite time ruin probabilities and the simulated values of these probabilities and the Lundberg bound are the final results of this article.

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Risk means a situation in which there is a possibility of deviation from a predicted result. Insurance is one of the methods of risk exposure that leads to the transfer of all or part of the risk from the insured to the insurer. Insurance policies are usually classified into two types: personal and non-life (non-life) insurance. According to this classification, any insurance policy that deals with the health of the insured person or persons is a personal insurance policy, otherwise it will be a nonlife insurance policy. Many insurances in the insurance industry are classified as non-life insurances. Fire, automobile, engineering, shipping, oil and energy insurances are examples of these insurances. Explanation and calculation of three issues in risk models are very important: the ruin probability, the time of ruin and the amount of ruin. In this article, the main and well-known results that have been obtained so far in the field of non-life insurance; Emphasizing the possibility of ruin, it is given along with various examples. 

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Insurance companies are modeled with mathematical and statistical models in terms of their random structure. In this paper, the individual risk model of insurance company with different interest rates in a period of time is considered and assumed that the interest rates have the probability transition matrix with finite and countable state. The finite and infinite time ruin probabilities are computed using the conditional probability on the first claim of density function. Moreover, the upper bounds for the infinite time ruin probability are obtained using the mathematical induction. In the numerical examples, the ruin probabilities for heavy tailed distributions are compared with the obtained probabilities in Bazyari (2022) for the classical individual risk model and also, the infinite time ruin probabilities for light tailed distributions are compared with Lundberg's inequality. The results show that the existence of interest rate with probability transition matrix and having finite state leads to decrease the ruin probabilities.