Apply modelling to areas (financial forecasting, climate change)

9 Modelling & Simulations

Learning Objective

Apply modelling techniques to real‑world problems – for example financial forecasting, climate‑change prediction and simple operational simulations – and evaluate the reliability, limitations and ethical impact of the models you create.

1. What is a Model?

A model is a simplified representation of a real‑world system that helps us:

• understand how the system works,
• predict future behaviour, and
• make informed decisions.

2. Key Modelling Terminology (AO1)

3. Types of Models

• Physical models – tangible replicas (scale bridge, wind‑tunnel).
• Mathematical models – equations or algorithms (compound‑growth formula, differential equations).
• Statistical models – relationships derived from data (linear regression, time‑series).
• Simulation models – dynamic, often computer‑based, representations (climate‑system simulators, queue‑management).

4. Characteristics of Modelling Software (AO1)

• Variable handling: user‑defined inputs, instant recalculation.
• Built‑in solvers: linear/non‑linear equation solvers, optimisation engines.
• What‑if / Sensitivity analysis: scenario manager, goal‑seek, data tables.
• Visualisation: charts, heat‑maps, animated plots.
• Audit trail & documentation: cell‑reference tracking, comments, version control.
• Export options: CSV, PDF, image files for reporting.

These features are available in spreadsheet packages (Excel, Google Sheets) and specialised tools such as MATLAB, R, or dedicated climate‑modelling suites (e.g., GCMs).

5. Mathematical Modelling Process (AO2)

1. Define the problem & objectives – what are you trying to predict or decide?
2. Identify variables, parameters, constraints and assumptions.
3. Formulate equations or algorithms that relate the variables.
4. Implement the model in a spreadsheet or specialised software.
5. Validate the model using real data (compare predicted vs. observed).
6. Run simulations – what‑if analysis, goal‑seek, scenario testing.
7. Interpret & evaluate results – reliability, limitations and ethical issues.

6. What‑If Analysis & Goal‑Seek (AO2)

What‑if analysis explores how changing one or more inputs affects the output. Goal‑seek works backwards: you set a desired output and the software finds the input that achieves it.

7. Effectiveness of Spreadsheet Modelling (AO3)

8. Financial Forecasting (AO1‑AO3)

Key Concepts (AO1)

• Revenue growth rate (g)
• Cost behaviour – fixed vs. variable
• Discounted cash flow (DCF)
• Risk, uncertainty and sensitivity analysis

Core Formulas

Simple linear revenue model

Discounted cash flow (DCF)

Case Study – Forecasting Net Profit

Assumptions: revenue grows 10 % pa, costs increase 5 % pa. Extend the table to 2026‑2030 using the formulas above, then apply the DCF formula with a discount rate of 8 % to obtain the present value of future profits.

Scenario & Sensitivity Testing (AO2)

• Scenario 1 – Optimistic: revenue growth = 12 %, cost increase = 3 %.
• Scenario 2 – Pessimistic: revenue growth = 8 %, cost increase = 7 %.
• Use Excel’s Scenario Manager to store both sets of assumptions and compare the resulting NPV.

9. Climate‑Change Modelling (AO1‑AO3)

Components of a General Circulation Model (GCM)

• Atmospheric dynamics
• Ocean circulation
• Land‑surface processes
• Ice‑sheet dynamics
• Radiative forcing (greenhouse gases, aerosols)

Energy‑Balance Model (EBM)

• C – heat capacity of the Earth system (J K⁻¹)
• S – solar constant (W m⁻²)
• α – planetary albedo (dimensionless)
• O(T)=\sigma T^{4} – outgoing long‑wave radiation (Stefan‑Boltzmann law, σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴)

Representative Concentration Pathways (RCPs)

Running a Simple Climate Simulation in a Spreadsheet (AO2)

1. Columns: Year, S(1‑α), O(T), dT/dt, T.
2. Euler method: T₍ₙ₊₁₎ = Tₙ + (Δt/C)·[S(1‑α)‑O(Tₙ)].
3. Vary α (e.g., 0.30 → 0.25) to explore the impact of decreasing albedo (melting ice).
4. Plot T against year to visualise warming trends.

10. Low‑Tech Simulation Example: Queue Management (AO1‑AO3)

Purpose: estimate average waiting time and determine the number of tellers required.

• Variables: arrival rate λ (customers / hour), service rate μ (customers / hour per teller).
• Assumption: arrivals follow a Poisson distribution; service times are exponential (M/M/1 queue).
• Average waiting time in queue: \[ W_q = \frac{λ}{μ(μ‑λ)} \]

Spreadsheet implementation:

1. Input λ and μ.
2. Calculate utilisation ρ = λ/μ.
3. Compute Wq using the formula.
4. Use Goal‑Seek to find the minimum μ (i.e., number of tellers) that keeps Wq ≤ 5 minutes.

11. Critical Evaluation Checklist (AO3)

• Data quality: Are the historical data reliable and up‑to‑date?
• Model scope: Does the model include all relevant factors (seasonality, feedback loops, etc.)?
• Assumptions & simplifications: Are they justified? How might they bias results?
• Sensitivity analysis: Have you tested how changes in key parameters affect outputs?
• Ethical considerations: Could the model be misused (e.g., overly optimistic forecasts, under‑estimating climate risks)?
• Tool limitations: Spreadsheet size limits, rounding errors, lack of version control.
• Communication: Are results presented clearly with labelled tables, graphs and concise commentary?

12. Key Skills Required for the Examination (AO1‑AO3)

• Define and label variables, parameters, constraints and assumptions.
• Interpret model equations; manipulate formulas for compound growth, DCF, queue‑time, etc.
• Construct and use what‑if analysis, scenario manager and goal‑seek in a spreadsheet.
• Critically evaluate reliability, limitations and ethical implications of a model.
• Present findings using well‑labelled tables, line/bar charts and concise written commentary.

13. Practice Questions

1. Company X has current revenue £80 million and expects an annual growth rate g. Using Goal‑Seek, determine the value of g that will give a net profit of £60 million in 2026, assuming costs increase 5 % pa from a base of £50 million.
2. Calculate the present value of a £10 million cash flow expected in 3 years, discounted at 8 % using the DCF formula.
3. For the EBM with C = 1.0 × 10⁸ J K⁻¹, S = 1361 W m⁻², α = 0.30 and O(T)=σT⁴ (σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴), find the equilibrium temperature T (solve for T when dT/dt = 0).
4. A bank experiences an average arrival rate of 30 customers per hour. Using the M/M/1 formula, calculate the average waiting time if each teller can serve 40 customers per hour. Then use Goal‑Seek to find the minimum number of tellers required to keep the waiting time under 5 minutes.
5. Discuss two ethical issues that could arise from publishing an overly optimistic financial forecast.

14. Summary

Modelling equips you with a systematic approach to analyse complex systems. Mastering the modelling process, spreadsheet what‑if tools and the critical evaluation of assumptions enables you to:

• Produce reliable financial forecasts for budgeting, investment and risk assessment.
• Interpret climate‑change simulations that inform national and international policy.
• Design simple yet powerful simulations (e.g., queue management) that support operational decisions.

Balancing technical accuracy with ethical awareness is essential for success in the Cambridge AS/A‑Level IT 9626 examination.

Suggested Diagram

Flowchart of the modelling process: Problem definition → Data collection → Model building → Validation → What‑if analysis → Evaluation → Decision‑making.

Coverage Review (Topic 9 – Modelling & Simulations)