Financial Mathematics — SL 1.5 Financial mathematics (compound interest, depreciation, annuities) (technology-supported) - IB Mathematics Applications and Interpretation

SECTION 1

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.

~2000 BCE • Babylonian Lending

Clay tablets from ancient Mesopotamia show that Babylonian merchants charged interest on grain and silver loans, using simple interest calculations at rates of about 20–33% per year.

1494 • Pacioli's Summa

Italian mathematician Luca Pacioli published tables for compound interest in his landmark work on accounting and mathematics, laying the groundwork for modern financial calculations.

1613 • Richard Witt's Tables

English mathematician Richard Witt published the first comprehensive compound interest tables, enabling bankers to calculate future values without laborious manual arithmetic.

1683 • Bernoulli & the Number e

Jacob Bernoulli studied what happens when you compound interest infinitely often, discovering the mathematical constant e ≈ 2.71828 — a number that appears throughout mathematics and science.

1900s–Today • Technology-Supported Finance

Spreadsheets, graphing calculators, and financial software now handle complex annuity and depreciation calculations instantly, making financial mathematics accessible to everyone.

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.

SECTION 2

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

Interest is calculated on both the original principal and all previously accumulated interest. Your money grows exponentially because interest earns interest.

Depreciation

Assets like cars or computers lose value over time. Depreciation models this decline, often as a fixed percentage lost each year — the mirror image of compound growth.

Annuities

A series of equal, regular payments made over time — such as monthly loan repayments or retirement fund contributions. Each payment earns (or costs) compound interest.

Nominal vs. Effective Rate

A nominal rate is the stated annual rate. The effective rate accounts for how often interest is compounded (monthly, quarterly, etc.) and reflects what you truly earn or owe.

Technology as a Tool

The IB expects you to use your GDC (graphing display calculator) or financial solver for complex problems, especially annuities. Understanding the formulas helps you interpret results correctly.

✦

KEY TAKEAWAY

Think of compound interest like a snowball rolling downhill. At the top, it's small, but as it rolls, it picks up more snow — and the bigger it gets, the more snow it picks up each rotation. That's exactly how compound interest works: the larger your balance grows, the more interest it generates in the next period. Depreciation is the reverse — imagine the snowball slowly melting as it rolls through a warm valley.

SECTION 3

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.

After 10 years at 8%, compound interest yields $2,159 — that's $359 more than simple interest ($1,800). Meanwhile, depreciation reduces the asset to just $434. The widening gap between the compound and simple interest curves illustrates the power of earning interest on interest.

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.

SECTION 4

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

COMPOUND INTEREST (FUTURE VALUE)

FV = PV × (1 + r/k)^(kn)

FV = future value, PV = present value (initial amount), r = nominal annual interest rate (as a decimal), k = number of compounding periods per year, n = number of years.

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

DEPRECIATION (REDUCING BALANCE)

V = V₀ × (1 − d)^n

V = value after n years, V₀ = initial value, d = annual depreciation rate (as a decimal), n = number of years. Note (1 − d) < 1, so the value decreases each year.

Annuities (using the GDC Finance Solver)

FUTURE VALUE OF AN ORDINARY ANNUITY

FV = PMT × [((1 + i)^N − 1) / i]

PMT = regular payment per period, i = interest rate per period (r/k), N = total number of payments (k × n). In IB exams, you'll typically use the TVM (Time Value of Money) solver on your GDC rather than memorizing this formula.

PRESENT VALUE OF AN ORDINARY ANNUITY

PV = PMT × [(1 − (1 + i)^(−N)) / i]

This formula calculates the lump sum today that is equivalent to receiving (or paying) a series of equal payments in the future. This is essential for loan calculations — the PV is the loan amount, and PMT is the monthly repayment.

💡

GDC Finance Solver Variables

On your GDC (e.g., TI-84 or Casio), the TVM solver uses these variables: N (total payments), I% (annual interest rate as a percent), PV (present value), PMT (payment per period), FV (future value), P/Y and C/Y (payments and compoundings per year). Cash flowing out of your pocket is entered as a negative number; cash flowing in is positive.

SECTION 5

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%

Effect of compounding frequency on $10,000 at 6% nominal rate over 1 year

An ordinary annuity has payments made at the end of each period. Each $200 payment earns interest for a different length of time. The first payment earns interest for 4 periods, the last payment earns none. The future value is the sum of all grown payments: $1,020.20.

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.

SECTION 6

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?

Finding the Monthly Payment for a Savings Goal

1
Step 1 — Identify Given Values

We know FV = $8,000, the annual interest rate I% = 4.8%, compounding is monthly so k = 12, and the time is n = 3 years. Since payments are at the end of each month, this is an ordinary annuity. PV = 0 (starting from nothing).

FV = 8000, I% = 4.8, N = 36, PV = 0, P/Y = C/Y = 12
2
Step 2 — Calculate Total Number of Payments

N = k × n = 12 × 3 = 36 total monthly payments. The interest rate per period is i = 0.048 / 12 = 0.004.

N = 36, i = 0.004 per month
3
Step 3 — Apply the Annuity Formula (or Use GDC)

Using FV = PMT × [((1 + i)^N − 1) / i], we substitute: 8000 = PMT × [((1.004)³⁶ − 1) / 0.004]. First calculate (1.004)³⁶ = 1.15468… so the bracket becomes (1.15468 − 1) / 0.004 = 0.15468 / 0.004 = 38.670.

8000 = PMT × 38.670
4
Step 4 — Solve for PMT

Dividing both sides by 38.670: PMT = 8000 / 38.670 ≈ 206.88. On your GDC finance solver, enter N = 36, I% = 4.8, PV = 0, FV = 8000, P/Y = 12, C/Y = 12, then solve for PMT. The calculator returns PMT = −206.88 (negative because it's money leaving your account).

PMT ≈ $206.88 per month
5
Step 5 — Verify and Interpret

Over 36 months, you deposit 36 × $206.88 = $7,447.68 in total. The remaining $8,000 − $7,447.68 = $552.32 is interest earned on your growing balance. You need to deposit approximately $207 per month to reach your $8,000 goal in 3 years.

Total deposited: $7,447.68 | Interest earned: $552.32

SECTION 7

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.

Strengths and limitations of the four main financial models in SL 1.5

✦

KEY TAKEAWAY

Think of financial models like weather forecasts. A 5-day forecast is quite reliable, but a 30-day forecast is much less accurate. Similarly, financial formulas give you a solid estimate, especially over shorter periods with stable conditions. Over longer time horizons or in volatile markets, the real outcome may diverge from the model's prediction. The formulas aren't wrong — they're simplifications of a more complex reality. The IB expects you to understand this distinction and comment on it when asked.

SECTION 8

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

How SL 1.5 concepts connect to university-level financial mathematics

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 × e^rt, 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.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

Explain why an investment earning 5% compounded monthly will grow to a larger amount after 10 years than the same investment earning 5% compounded annually. In your explanation, use the concept of 'interest earning interest.'

Show answer

When interest is compounded monthly, the interest earned each month is added to the balance, and the next month's interest is calculated on this larger amount. With annual compounding, the interest sits 'idle' for a full year before it starts earning additional interest. Over 12 months, the monthly-compounded account has 12 mini-bursts of growth rather than one. The effective annual rate for 5% compounded monthly is (1 + 0.05/12)¹² − 1 ≈ 5.116%, which is higher than the 5.000% effective rate from annual compounding. Over 10 years, this small difference accumulates significantly.

Question 2: BASIC CALCULATION

You invest $5,000 in a savings account that pays 3.6% annual interest, compounded quarterly. What is the value of the investment after 6 years? Round to the nearest cent.

Show answer

Using FV = PV ×

(1 + r/k)^{kn}

: PV = 5000, r = 0.036, k = 4, n = 6. Rate per period = 0.036/4 = 0.009. Total periods = 4 × 6 = 24. FV = 5000 × (1.009)²⁴ = 5000 × 1.23996… ≈ $6,199.79. The investment grows by approximately $1,199.79 in interest over 6 years. You can verify this on your GDC by entering N = 24, I% = 3.6, PV = −5000, PMT = 0, P/Y = 4, C/Y = 4, then solving for FV.

Question 3: INTERMEDIATE

A laptop is purchased for $1,800 and depreciates at 22% per year (reducing balance). After how many complete years will the laptop first be worth less than $400? Show your working.

Show answer

Using V = V₀ × (1 − d)ⁿ: 400 = 1800 × (0.78)ⁿ. Dividing: (0.78)ⁿ = 400/1800 = 0.2222… Taking logarithms of both sides: n × log(0.78) = log(0.2222), so n = log(0.2222)/log(0.78) = (−0.6532)/(−0.1079) ≈ 6.054. Since we need complete years and n = 6 gives (0.78)⁶ = 0.2252 → $405.36 (still above $400), we need n = 7 complete years. Check: (0.78)⁷ ≈ 0.1757 → $316.18 < $400. The laptop first falls below $400 after 7 complete years.

Question 4: APPLIED

Priya takes out a car loan of $24,000 at 6.5% annual interest, compounded monthly, to be repaid in equal monthly installments over 5 years. (a) Calculate the monthly payment. (b) Find the total amount Priya pays over the life of the loan. (c) How much of that total is interest?

Show answer

(a) Using the GDC TVM solver: N = 60, I% = 6.5, PV = 24000, FV = 0, P/Y = 12, C/Y = 12. Solving for PMT gives PMT ≈ −469.71. Alternatively, using PV = PMT × [(1 −

(1 + i)^{−N}

)/i] with i = 0.065/12 ≈ 0.005417 and N = 60: 24000 = PMT × [(1 −

(1.005417)^{−60}

)/0.005417] = PMT × 51.095, so PMT = 24000/51.095 ≈ $469.71. (b) Total paid = 60 × $469.71 = $28,182.60. (c) Total interest = $28,182.60 − $24,000 = $4,182.60. Priya pays about 17.4% of the principal in interest over 5 years.

Question 5: CRITICAL THINKING

Marcus deposits $300 at the end of each month into a savings account earning 5.4% annual interest, compounded monthly. At the same time, he buys a motorcycle for $9,000 that depreciates at 15% per year (reducing balance). After how many complete years will the value of his savings first exceed the value of his motorcycle? Justify your answer.

Show answer

This requires comparing two models year by year. Savings: FV = 300 × [(

(1.0045)^N

− 1)/0.0045] where N = 12n (months). Motorcycle: V = 9000 × (0.85)ⁿ. At n = 1: Savings after 12 months ≈ 300 × 12.2975 ≈ $3,689.25; Motorcycle = 9000 × 0.85 = $7,650. At n = 2: Savings ≈ 300 × 25.2064 ≈ $7,561.92; Motorcycle = $6,502.50. At n = 3: Savings ≈ 300 × 38.7543 ≈ $11,626.30; Motorcycle = $5,527.13. So the savings ($7,561.92) do not yet exceed the motorcycle ($6,502.50) at… wait — at n = 2, savings = $7,561.92 > $6,502.50. Actually, let's recalculate more carefully. At 24 months: FV = 300 × [((1.0045)²⁴ − 1)/0.0045] = 300 × [(1.11367 − 1)/0.0045] = 300 × 25.26 ≈ $7,577. Motorcycle at year 2: 9000 × 0.85² = $6,502.50. Since $7,577 > $6,502.50, the savings first exceed the motorcycle after 2 complete years. At n = 1, savings ($3,689) < motorcycle ($7,650), confirming n = 2 is the answer.

SUMMARY

Lesson Summary

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.