Historical Context & Motivation
Money has been lent and borrowed for thousands of years, but the mathematics behind interest calculations has evolved dramatically over time. Ancient civilizations recognized that lending resources carried risk, and the lender deserved compensation for that risk — this compensation is what we call interest. The challenge was always the same: how do you fairly calculate what is owed when money grows over time? Early approaches used simple interest, where a fixed percentage was charged only on the original amount. But merchants and bankers soon realized that if interest itself could earn interest, wealth would grow much faster — and the concept of compound interest was born.
Today, financial mathematics underpins nearly every economic decision you'll encounter — from choosing a savings account to evaluating a car loan or planning for retirement. The IB SL 1.5 topic asks a deceptively simple question: how does money change in value over time, and how can we model that change precisely? In this lesson, you'll learn to answer that question using compound interest, depreciation, and annuities — with technology as your ally.
Core Principles & Definitions
Before diving into formulas, it's essential to understand the foundational ideas that connect compound interest, depreciation, and annuities. All three rely on the same mathematical engine — exponential growth or decay — applied in slightly different contexts. Think of them as three variations on the same theme: money multiplying (or shrinking) by a constant ratio each period.
Compound Interest
Depreciation
Annuities
Nominal vs. Effective Rate
Technology as a Tool
Visualizing Compound Growth vs. Depreciation
The diagram below compares three scenarios over 10 years: a $1,000 investment earning 8% compound interest annually, the same $1,000 earning 8% simple interest, and a $1,000 asset depreciating at 8% per year. Notice how the compound interest curve accelerates upward while the depreciation curve decelerates downward. Simple interest, by contrast, grows in a perfectly straight line.
In the graph, the cyan curve represents compound interest — notice how it bends upward more steeply as the years progress. This is because each year's interest is calculated on a larger and larger balance. The amber dashed line shows simple interest, which adds the same fixed dollar amount ($80) every year. The pink curve tracks depreciation: each year, 8% of the current value is removed, so the dollar amount lost decreases each year even though the percentage stays the same. This is why the pink curve flattens as it approaches zero but never actually reaches it.
Mathematical Framework
The three major formulas in SL 1.5 all build on the idea of multiplying a value by a constant ratio each compounding period. Let's define each formula carefully, then examine how they connect.
Compound Interest
The term (1 + r/k) is called the growth factor per period. When interest is compounded annually, k = 1 and the formula simplifies to FV = PV × (1 + r)n. If compounding occurs monthly, k = 12; quarterly, k = 4; daily, k = 365.
Depreciation
Annuities (using the GDC Finance Solver)
Detailed Breakdown — Compounding Frequency & Annuity Types
One of the trickiest aspects of financial mathematics is understanding how compounding frequency affects the final result. A bank might advertise an annual interest rate of 6%, but if that interest compounds monthly rather than annually, you actually earn slightly more. The table below shows how $10,000 grows in one year at a nominal rate of 6% under different compounding frequencies.
| Compounding Frequency | k (periods/year) | Rate per Period (r/k) | FV after 1 year | Effective Rate |
|---|---|---|---|---|
| Annually | 1 | 6.000% | $10,600.00 | 6.000% |
| Quarterly | 4 | 1.500% | $10,613.64 | 6.136% |
| Monthly | 12 | 0.500% | $10,616.78 | 6.168% |
| Daily | 365 | 0.01644% | $10,618.31 | 6.183% |
The cash flow diagram above is one of the most useful tools for understanding annuities. Each arrow represents a payment, and the dashed lines show how each payment grows as it earns interest until the end of the annuity. In an ordinary annuity (also called an annuity in arrears), payments happen at the end of each period. In an annuity due (annuity in advance), payments happen at the beginning — meaning every payment earns one extra period of interest. On the IB exam, you'll usually encounter ordinary annuities unless stated otherwise.
Worked Example — Saving for a Gap Year
Let's walk through a realistic problem that combines compound interest and annuities. Suppose you want to save $8,000 for a gap year trip. You open a savings account that pays 4.8% annual interest, compounded monthly. You plan to deposit the same amount at the end of every month for 3 years. How much should each monthly deposit be?
Comparing Financial Models — Strengths & Limitations
Each financial model has its place, but none is perfect. Understanding where each model works well — and where it breaks down — will help you choose the right approach on the IB exam and in real financial decisions.
| Model | Best Used For | Key Limitation |
|---|---|---|
| Compound Interest | Savings accounts, fixed-term deposits, investment growth projections | Assumes a constant interest rate — real rates fluctuate and may not compound as neatly as the formula suggests |
| Depreciation (reducing balance) | Vehicles, electronics, machinery — assets that lose value quickly at first | Value never reaches zero mathematically, but in reality, assets can become worthless. Doesn't account for sudden damage or obsolescence. |
| Annuities (savings) | Regular savings plans, retirement contributions, sinking funds | Assumes perfectly equal, uninterrupted payments — in real life, people miss payments or change amounts |
| Annuities (loans) | Mortgages, car loans, student loans with fixed repayments | Doesn't include fees, insurance, or variable rates. Real loan costs often exceed the model's prediction. |
Connection to Advanced Financial Theory
The compound interest and annuity formulas you learn in SL 1.5 are the building blocks of more advanced financial mathematics studied at university level and used daily in banking, insurance, and investment management. Understanding where these ideas lead can deepen your appreciation of the topic.
| SL 1.5 Concept | Advanced Extension | Real-World Application |
|---|---|---|
| Compound interest with fixed rate | Continuous compounding: FV = PV × e^(rt) | Modeling population growth, radioactive decay, and advanced pricing models in finance |
| Fixed-rate annuities | Variable-rate annuities, perpetuities (infinite annuities) | Valuing stocks that pay dividends forever, pension fund analysis |
| Reducing balance depreciation | Units-of-production and sum-of-years-digits methods | Corporate accounting and tax optimization for businesses |
| TVM solver on GDC | Net Present Value (NPV) and Internal Rate of Return (IRR) | Evaluating whether a business investment is worth pursuing |
One particularly elegant extension is continuous compounding, where interest is compounded not monthly or daily, but at every instant. This leads to the formula FV = PV × ert, where e ≈ 2.71828 is Euler's number. You won't be tested on this in SL, but recognizing that compound interest naturally leads to the exponential function shows how deeply connected financial mathematics is to the rest of the IB curriculum — especially logarithms and exponential models in Topic 2.
Practice Problems
Lesson Summary
Financial mathematics in SL 1.5 revolves around one central idea: the time value of money. The compound interest formula FV = PV × (1 + r/k)kn models exponential growth of an investment, where the key insight is that interest earns interest. Depreciation uses the same structure but with a decay factor (1 − d), modeling how assets like cars and electronics lose value over time. Annuities extend these ideas to sequences of equal, regular payments — whether saving toward a goal or repaying a loan.
The compounding frequency determines how often interest is calculated and added, affecting the effective interest rate. For IB exams, you are expected to use your GDC finance solver (TVM) for annuity problems, entering N, I%, PV, PMT, FV, P/Y, and C/Y correctly — remembering that cash outflows are negative and inflows are positive. Master these formulas and tools, and you'll have a powerful framework for understanding the real-world financial decisions that await you.