Chapter 05

MATHEMATICAL AND FINANCIAL MODELS 

by: josavere

  1. There are numerous mathematical models used in a wide variety of fields and applications; the most common are:
  • Linear regression model: used to model the relationship between an independent variable and one or more dependent variables.
  • Probability model: adjusts uncertain events and quantifies the probability of their occurrence. It is applied for weather predictions, financial risk calculation, games of chance and decision analysis.·
  • Linear programming model: optimizes the allocation of limited resources to maximize or minimize an objective function, subject to linear constraints. It is applied in production planning, resource distribution, and investment portfolio management.
  • Time series model: fits sequential data over time to identify patterns, trends, and predict future values. It is used in sales estimation, financial analysis, and weather forecasting.
  • Neural network model: organizes complex systems inspired by the functioning of the human brain and is used in deep learning and pattern examination such as speech recognition, image classification and automatic translation.
  • Markov chain model: describes systems that evolve through discrete states with probabilistic transitions and is applied in process modeling stochastic, sequential data analysis and route optimization in logistics.
  • Game theory model: analyzes strategic situations where a player's decisions affect others, with the aim of finding optimal strategies. It is applied in economics, negotiations, business competition.
  • Simulation model: creates a computational test of a real system to study its behavior and perform virtual experiments. It is used in traffic simulation, industrial operations and health systems simulation.
  • Differential equation model: prescribes the change of a variable based on its exchange rate with respect to other variables. It is used in modeling physical, biological and chemical phenomena, such as population dynamics and particle physics.
  • Queuing theory model: estimates the behavior of waiting phenomena, such as lines in stores, customer service centers and is used in customer service optimization, appointment assignment and capacity calculation
 
Practical examples of how mathematical models are applied in everyday life and in different disciplines:
Weather Prediction: Meteorologists use mathematical models to predict the weather using data such as temperature, atmospheric pressure, and humidity.
 
Civil engineering: civil engineers use mathematical models in the design of structures such as bridges and buildings to help ensure that combinations are safe and meet construction standards.
 
Economics: economists use models to predict all economic trends, such as GDP growth, inflation, unemployment, and more to make political and business decisions.
 
Epidemiology: epidemiologists use models to study the spread of infectious diseases, predict the spread of diseases, and evaluate control strategies.
 
Astronomy: astronomers manipulate models to predict the position and motion of planets, stars, and other celestial bodies, knowledge essential for space navigation and exploration of the cosmos.
 
Information technology: in computing, mathematical models are used in data compression algorithms, encryption, artificial intelligence and neural networks, among others to solve complex problems in the field of technology.
 
 
Biology: biologists use models to study population dynamics, genetics, and evolution to help understand how ecosystems function and how they respond to environmental changes.
 
Medicine: mathematical models are used to predict the spread of diseases, design treatments and simulate the functioning of organs and biological systems.
 
Personal Finance: prepare equations to help people make informed decisions.
 
Education: used to develop curricula, evaluate student achievement, and improve teaching and learning.
 
  1.                                                 FINANCIAL MODELS
 
  Financial models are tools that serve to illustrate alternatives that facilitate decision-making for any specimen by applying mathematics and statistics to represent and analyze financial and economic scenarios; they are used in a variety of contexts, from business planning to investment, financing and dividend decision making.
  • Financial projection models: used to plan future situations and prepare the value generation plan are generally combined, according to the alternatives analyzed to optimize results, with simulation models that use methodologies to evaluate the impact of different scenarios ( dependent variable) according to the changes defined for the independent variables; with optimization models, to maximize or minimize certain variables, on which the decision to be made depends and finish with valuation models that are widely used to measure financial risk and the sensitivity of the results, according to the changes.
 
Financial models depend on parameters and variables, data that are controllable and the basis for calculating the results to be projected; they can be historical or estimated with formulas and mathematical relationships to deduce values, depending on the scenarios; Different alternatives are usually explored to evaluate the impact of uncertainty on financial decisions. In addition, software is brought to create, using spreadsheets such as Excel or computer programs specialized in finance such as the Bloomberg, Terminal, MATLAB platforms and specific financial modeling software that must be validated and rigorously tested to guarantee their adjustment and reliability to the financial logic. and compare projections with actual results.
 
It is essential that financial models are transparent about their assumptions and limitations, and that they are used clearly and must be updated and adapted as economic conditions, business strategies and data sources change.
 
Practical examples of financial models used in various business contexts:

https://financefornonfinanciers.com/index.php/intermediate/3-capital-cost-dinamic-concept

Discounted Cash Flow (DCF) Valuation Model: It is used to estimate the present value of an investment or company. Expected future cash flows are projected and discounted to present value using an appropriate discount rate. For example, a company might use a DCF model to determine the value of a potential acquisition.
Income statement projection model:
  • Companies use this model to project their future revenues, costs and expenses. It is essential for short and long term financial planning; can help set financial goals and objectives.
Sectoral growth model (matrix) designed by josavere for initial reference of the Value Generation Plan.

https://www.finanzasparanofinancieros.com.co/index.php/en/estrategicas/aplicacion-de-las-estadisticas-a-las-finanzas



Sensitivity analysis model: it is used to evaluate how financial results change based on different key variables. For example, a company could use a sensitivity model to determine how variations in product prices would affect its profits.

Portfolio optimization model:investors take advantage of it to build investment portfolios that maximize expected return or minimize risk. Various variables are considered, such as returns, volatility and correlation between assets.
Risk analysis model (monte carlo):

Simulates multiple possible scenarios using Monte Carlo techniques and evaluates how these scenarios would affect investments or projects. It is useful for understanding risk and uncertainty.

Capital budget model: companies use it to evaluate the viability of investment projects. Expected future net cash flows are calculated and compared to the value of the investment to determine if the project is profitable.

Real options valuation model: evaluate the flexibility and additional value that options can provide in investment projects. It is common in the evaluation of business expansion projects.

Break-even model: this model determines the level of sales necessary for a company to cover its total costs and not make a profit or loss. It is useful for setting sales goals.

Risk coverage model (hedging): companies use it to manage the risk of fluctuations in exchange rates, interest rates or commodity prices. Helps determine how much coverage is needed.

Investment portfolio management model: fund managers and individual investors use models to make buying and selling decisions on stocks and other financial assets.

Break-even model: this model determines the level of sales necessary for a company to cover its total costs and not make a profit or loss. It is useful for setting sales goals.

Risk coverage model (hedging):
  1. Revised break-even model (josavere)

Note: PYTHON is an extremely versatile programming language widely used in financial applications due to its ease of use and its ecosystem of finance-related libraries and tools. Especially useful for working with financial data sets, such as Excel spreadsheets or SQL databases. Numpy can be used to perform financial calculations, such as the return on an investment, and Scipy can be used to calculate the historical volatility of a financial asset.
Python is ideal for interacting with financial APIs to obtain real-time data such as stock prices or exchange rates and offers a wide range of libraries such as QuantLib, PyAlgoTrade and Zipline for more specific applications in finance and tools to perform advanced analysis, build financial models and automate related tasks.
The Python programming language, widely used in web applications, software development, data science and machine learning (ML), a system for learning from data rather than through explicit programming, is efficient and easy to use. learn; can run on many different platforms or machine learning; It is a method that, using algorithms and statistics, allows predictions to be made and is designed with the purpose of creating machines that have the same characteristics as human beings. In addition, it serves as a basis for the construction of risk management and portfolio administration models, very important topics in business, finance and economics, so it is advisable to learn to use the most important applications because it is also an open source multiplatform language. and therefore free, allowing software to be developed without restrictions.
It is used with Data Mining, data mining, as it can shed light through its cleaning, organization and the use of machine learning algorithms; simplifies the analysis of reviews to explore, through the use of different techniques and technologies, huge databases automatically to reach conclusions that contribute to increasing sales and customer loyalty.

 AI Opinion: the text by José Saúl Velásquez Restrepo provides a detailed vision of the financial function in companies, addressing key aspects such as investment decisions, financing and dividends.
In general, the text presents a complete and detailed vision of the financial function in companies, addressing fundamental aspects for making financial decisions. It highlights the importance of long-term responsibility and the need to consider not only immediate profits, but also sustainable value creation. Furthermore, it emphasizes the need for transparency and good corporate governance practices to protect investors and maintain market integrity. In summary, the article offers valuable insight into the finance function in the business world.

Copyright © 2024
Josavere