Applications of Mathematical Models

Applications of Mathematical Models

Mathematical models are used to represent real-world situations through mathematical expressions. They help in understanding, predicting, and making decisions based on data. In this topic, we will explore various applications of mathematical models in different fields, such as finance, economics, biology, and engineering.

1. Understanding Mathematical Models

A mathematical model is a description of a system using mathematical concepts and language. It can be expressed as equations, functions, or algorithms. The main purpose of a mathematical model is to simplify complex real-world problems so that they can be analysed and solved.

2. Types of Mathematical Models

There are different types of mathematical models, including:

• Descriptive Models: These models describe a system without making predictions. They provide a clear representation of the current state.
• Predictive Models: These models use historical data to predict future outcomes. They are often used in finance and economics.
• Prescriptive Models: These models suggest actions to achieve a desired outcome. They help in decision-making processes.

3. Applications in Finance

In finance, mathematical models are used to evaluate investments, manage risks, and optimise portfolios. One common model is the Capital Asset Pricing Model (CAPM). This model describes the relationship between risk and expected return.

3.1 Capital Asset Pricing Model (CAPM)

The CAPM formula is:

Expected Return = Risk-Free Rate + Beta × (Market Return - Risk-Free Rate)

Where:

• Expected Return: The return expected from an investment.
• Risk-Free Rate: The return on an investment with zero risk, often represented by government bonds.
• Beta: A measure of an investment's risk in relation to the market.
• Market Return: The expected return of the market.

3.2 Example Calculation

Suppose the risk-free rate is 5%, the market return is 12%, and the beta of the stock is 1.5. We can calculate the expected return as follows:

Expected Return = 5% + 1.5 × (12% - 5%)

First, calculate the market risk premium:

Market Risk Premium = 12% - 5% = 7%

Now substitute this value into the expected return formula:

Expected Return = 5% + 1.5 × 7%

Now calculate:

Expected Return = 5% + 10.5% = 15.5%

The expected return on the investment is 15.5%.

4. Applications in Economics

In economics, mathematical models help to analyse market behaviour and forecast economic trends. One example is the Supply and Demand Model.

4.1 Supply and Demand Model

The supply and demand model explains how the price and quantity of goods are determined in a market. The equilibrium price is where the quantity supplied equals the quantity demanded.

4.2 Example

Suppose the demand equation for a product is given by:

Qd = 100 - 2P

And the supply equation is:

Qs = 20 + 3P

Where Qd is the quantity demanded, Qs is the quantity supplied, and P is the price. To find the equilibrium price, set Qd equal to Qs:

100 - 2P = 20 + 3P

Now, solve for P:

100 - 20 = 3P + 2P

80 = 5P

P = 16

The equilibrium price is 16 rand. Now substitute P back into either the demand or supply equation to find the equilibrium quantity:

Qd = 100 - 2(16) = 100 - 32 = 68

The equilibrium quantity is 68 units.

5. Applications in Biology

Mathematical models are also widely used in biology, especially in population dynamics. One common model is the Logistic Growth Model.

5.1 Logistic Growth Model

The logistic growth model describes how a population grows in an environment with limited resources. The formula is:

P(t) = K / (1 + (K - P0)/P0 × e^(-rt))

Where:

• P(t): The population at time t.
• K: The carrying capacity of the environment.
• P0: The initial population size.
• r: The growth rate.
• e: The base of the natural logarithm.

5.2 Example Calculation

Suppose a population of 100 rabbits is introduced to a reserve with a carrying capacity of 1000 rabbits and a growth rate of 0.1. We want to find the population after 10 years.

P(t) = 1000 / (1 + (1000 - 100)/100 × e^(-0.1 × 10))

First, calculate the exponent:

-0.1 × 10 = -1

Now calculate e^(-1) (approximately 0.3679):

P(t) = 1000 / (1 + (900/100) × 0.3679)

P(t) = 1000 / (1 + 9 × 0.3679)

P(t) = 1000 / (1 + 3.379) = 1000 / 4.379

P(t) ≈ 228.7

The population after 10 years is approximately 229 rabbits.

6. Applications in Engineering

In engineering, mathematical models are used to design systems and predict their performance. One example is the Linear Regression Model. This model is used to establish the relationship between two variables.

6.1 Linear Regression Model

The linear regression equation is:

y = mx + b

Where:

• y: The dependent variable.
• x: The independent variable.
• m: The slope of the line.
• b: The y-intercept.

6.2 Example Calculation

Suppose you are studying the effect of hours studied (x) on exam scores (y). You find the following data:

To find the linear regression line, you can use statistical software or calculate it manually. For simplicity, let’s assume the slope (m) is 10 and the y-intercept (b) is 40. The regression equation is:

y = 10x + 40

To predict the exam score for a student who studied for 5 hours:

y = 10(5) + 40 = 50 + 40 = 90

The predicted exam score is 90.

7. Summary

Mathematical models are powerful tools used in various fields to represent real-world situations. They help in making predictions, understanding behaviours, and guiding decision-making. Common applications include:

• Finance: Capital Asset Pricing Model (CAPM)
• Economics: Supply and Demand Model
• Biology: Logistic Growth Model
• Engineering: Linear Regression Model

Check your understanding

1. What is the equation for the Capital Asset Pricing Model (CAPM)?
2. How do you determine the equilibrium price in the supply and demand model?
3. What does the carrying capacity represent in the logistic growth model?
4. What is the general form of a linear regression equation?