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A Note on Optimal Value of Loans
Kaňková, Vlasta
People try to gain (in the last decades) own residence (a flat or a little house). Since young people do not posses necessary financial resources, bank sector offers them a mortgage. Of course, the aim of any bank is to profit from such a transaction. Therefore, according to their possibilities, the banks employ excellent experts to analyze the financial situation of potenitial clients. Consequently, the banks know what could be a maximal size of the loan (in dependence on the debtor's position, salary and age) and what is reasonable size of installments. The aim of this contribution is to analyze the situation from the second size. In particular, the aim is to investgate the possibilities of the debtors not only on the dependence on their present - day situation, but also on their future private and subjective decisions and on possible “unpleasant” events. Moreover, consequently according to these indexes, the aim of this contribution is to suggest a method for a recognition of a “safe” loan and simultaneously to offer tactics to state a suitable environment for future time.The stochastic programming theory will be employed to it.
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Optimal Value of Loans via Stochastic Programming
Kaňková, Vlasta
A question of mortgage leads to serious and complicated problems of financial mathematics. On one side is a bank with an aim to have a “good” profit, on the other side is the client trying to invest money safely, with possible “small” risk.Let us suppose that a young married couple is in a position of client. Young people know that an expected and also unexpected unpleasant financial situation can happen. Many unpleasant financial situation can be caused by a random factor. Consequently stochastic methods are suitable to secure against them. The aim of the suggested model is not only to state a maximal reasonable value of loans, but also to endure unpleasant financial period. To this end we employ stochastic optimization theory. A few suitable models will be introduced. The choice of the model depends on environment of the young people. Models will be with “deterministic” constraints, probability constraints, but also with stochastic dominance constraints. The suggested models will be analyzed both from the numerical point of view and from possible method solution based on data. Except static one-objective problem we suggest also multi–objective models.
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Transient and Average Markov Reward Chains with Applications to Finance
Sladký, Karel
The article is devoted to Markov reward chains, in particular, attention is primarily focused on the reward variance arising by summation of generated rewards. Explicit formulae for calculating the variances for transient and average models are reported along with sketches of algorithmic procedures for finding policies guaranteeing minimal variance in the class of policies with a given transient or average reward. Application of the obtained results to financial models is indicated.
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Scenario Generation via L-1 Norm
Kaňková, Vlasta
Optimization problems depending on a probability measure correspond to many economic and financial situations. It can be very complicated to solve these problems, especially when the underlying probability measure belongs to continuous type. Consequently, the underlying continuous probability measure is often replaced by discrete one with finite number of atoms (scenario). The aim of the contribution is to deal with the above mentioned approximation in a special form of stochastic optimization problems with an operator of the mathematical expectation in the objective function. The stability results determined by the help of the Wasserstein metric (based on the L_1 norm) are employed to generate approximate distributions
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Second Order Optimality in Transient and Discounted Markov Decision Chains
Sladký, Karel
The article is devoted to second order optimality in Markov decision processes. Attention is primarily focused on the reward variance for discounted models and undiscounted transient models (i.e. where the spectral radius of the transition probability matrix is less than unity). Considering the second order optimality criteria means that in the class of policies maximizing (or minimizing) total expected discounted reward (or undiscounted reward for the transient model) we choose the policy minimizing the total variance. Explicit formulae for calculating the variances for transient and discounted models are reported along with sketches of algoritmic procedures for finding second order optimal policies.
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