Financial applications: compound interest and depreciation

Q: What is the fundamental difference between simple interest and compound interest?

Unlike simple interest, compound interest calculates "interest on interest," where earned interest is added to the principal for subsequent periods.

Q: In the context of compound interest, what do the variables $PV$, $FV$, and $r$ represent?

$PV$ is the Present Value (principal), $FV$ is the Future Value , and $r$ is the nominal annual interest rate .

Q: What does the variable $k$ represent in compound interest, and what is its value for quarterly compounding?

$k$ represents the number of compounding periods per year ; for quarterly compounding, $k = 4$.

Q: What is the definition of inflation in a financial context?

Inflation is the rate at which the general level of prices for goods and services rises, effectively reducing the purchasing power of money.

Q: In a GDC Finance Solver, what do $N$ and $I\%$ represent?

$N$ is the total number of compounding periods ($n \times k$), and $I\%$ is the annual interest rate entered as a percentage.

Mathematics - Analysis and Approaches · IB Diploma Programme (DP) - SL & HL · 5 min read · Updated: 7 Sept 2026

Introduction to Financial Mathematics

Welcome! In this chapter, we explore how money grows and how assets lose value over time. Whether you are saving for a future goal, taking out a loan, or buying a car, the concepts of compound interest and depreciation are essential. These are not just exam topics; they are life skills that help you understand the "time value of money."

Context: This chapter is a practical application of Geometric Sequences (Topic 1.3). In a geometric sequence, we multiply by a common ratio $r$ at each step. In finance, that ratio is determined by the interest or depreciation rate.

1. Compound Interest

Unlike simple interest (where you only earn interest on your original deposit), compound interest is "interest on interest." At the end of each period, the interest you earned is added to your balance, and in the next period, you earn interest on that new, larger amount.

The Compound Interest Variables

To solve these problems, you need to identify these five variables:

$PV$ (Present Value): The amount of money you start with (the principal).
$FV$ (Future Value): The amount of money you will have after a certain amount of time.
$r\%$: The nominal annual interest rate.
$n$: The number of years.
$k$: The number of compounding periods per year.

Compounding Periods ($k$)

Interest isn't always calculated once a year. The syllabus requires you to understand different frequencies:

Yearly: $k = 1$
Half-yearly: $k = 2$
Quarterly: $k = 4$
Monthly: $k = 12$

The Golden Rule: The more often you compound ($k$ increases), the more interest you earn over time!

The Formula

While you won't be asked to derive this, you should be comfortable using it:
$FV = PV \times (1 + \frac{r}{100k})^{kn}$

Example: If you invest $\$1000$ at $5\%$ interest compounded quarterly for $3$ years:
$PV = 1000$, $r = 5$, $k = 4$, $n = 3$.
$FV = 1000 \times (1 + \frac{5}{100 \times 4})^{4 \times 3} = 1000 \times (1.0125)^{12}$

Quick Takeaway: Compound interest grows exponentially. Even a small interest rate can lead to large sums if left for a long time.

2. Annual Depreciation

Depreciation is the opposite of interest. It describes how an asset (like a laptop or a car) loses value over time. In IB Mathematics AA, we focus on the "reducing balance method," which is essentially "negative" compound interest.

The Depreciation Formula

The formula is almost identical to compound interest, but we subtract the rate because the value is going down:
$FV = PV \times (1 - \frac{r}{100})^n$

Example: A car is bought for $\$20,000$ and depreciates at a rate of $15\%$ per year. What is its value after $5$ years?
$PV = 20000$, $r = 15$, $n = 5$.
$FV = 20000 \times (1 - 0.15)^5 = 20000 \times (0.85)^5$

Common Mistake: Students often try to calculate $15\%$ of the original price and subtract it every year. Don't do this! Depreciation is calculated on the current value each year, not the original price.

3. Real Value and Inflation

Inflation is the rate at which the general level of prices for goods and services rises. If you have $\$100$ today, it might buy $20$ pizzas. If inflation is high, that same $\$100$ might only buy $15$ pizzas in five years.

The Real Value of an investment tells you what your future money is actually "worth" in today's prices.

How to Calculate Real Value

To find the real value of an investment, you "adjust" the Future Value ($FV$) by the inflation rate ($r_{inf}$):
$\text{Real Value} = \frac{FV}{(1 + \frac{r_{inf}}{100})^n}$

Analogy: Think of inflation as a "tax" on your purchasing power. Even if your bank account balance goes up, if prices go up faster, you are technically getting poorer!

4. Using Technology (The GDC)

On Paper 2 and Paper 3, you are expected to use your Graphic Display Calculator (GDC). Most calculators have a "Finance Solver" or "TVM (Time Value of Money) Solver."

GDC Variable Key:

$N$: Total number of compounding periods (usually $n \times k$).
$I\%$: Annual interest rate (enter as a percentage, e.g., $5$ for $5\%$).
$PV$: Present Value (usually entered as a negative if you are "giving" the money to a bank).
$PMT$: Payment (set to $0$ for simple compound interest questions).
$FV$: Future Value.
$P/Y$ and $C/Y$: Payments per year and Compounding periods per year (usually the same value, $k$).

Pro-tip: If the GDC gives you a negative number for $FV$, don't panic! It just means money is flowing back to you. In financial math, the "sign" ($+$ or $-$) indicates the direction of the cash flow.

5. Summary and Tips for Success

Key Takeaways:

Identify your variables: Before starting any calculation, list $PV, FV, r, n,$ and $k$.
Check the compounding period: Is it monthly ($k=12$) or quarterly ($k=4$)? This is the most common place to lose marks.
Depreciation = Decay: Always use a minus sign in the formula or ensure your multiplier is less than $1$.
Inflation: Remember that "Real Value" is always lower than the "Future Value" if inflation is positive.

Don't worry if this seems tricky at first! The formulas are very consistent. Once you practice identifying whether a question is asking for a future value, a present value, or a rate, the math becomes a simple matter of substitution or calculator entry.

Quick Review: If an investment compounds half-yearly at $4\%$ for $10$ years, what value of $n$ and $k$ would you use?
Answer: $n = 10$, $k = 2$. Total periods ($N$) = $20$.

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the fundamental difference between simple interest and compound interest?

Unlike simple interest, compound interest calculates "interest on interest," where earned interest is added to the principal for subsequent periods.

In the context of compound interest, what do the variables $PV$, $FV$, and $r$ represent?

$PV$ is the Present Value (principal), $FV$ is the Future Value, and $r$ is the nominal annual interest rate.

What does the variable $k$ represent in compound interest, and what is its value for quarterly compounding?

$k$ represents the number of compounding periods per year; for quarterly compounding, $k = 4$.

What is the mathematical formula for calculating Future Value ($FV$) with compound interest?

$FV = PV \times (1 + \frac{r}{100k})^{kn}$

What is the formula for calculating the value of an asset after annual depreciation?

$FV = PV \times (1 - \frac{r}{100})^n$

What is the definition of inflation in a financial context?

Inflation is the rate at which the general level of prices for goods and services rises, effectively reducing the purchasing power of money.

How do you calculate the Real Value of a future sum of money given an inflation rate?

$\text{Real Value} = \frac{FV}{(1 + \frac{r_{inf}}{100})^n}$

In a GDC Finance Solver, what do $N$ and $I\%$ represent?

$N$ is the total number of compounding periods ($n \times k$), and $I\%$ is the annual interest rate entered as a percentage.

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