Financial mathematics: interest, depreciation, annuities

Q: In the context of compound interest, what do the terms Present Value and Future Value represent?

Present Value ($PV$) is the amount invested or held right now, while Future Value ($FV$) is the amount the money grows to after a certain time.

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

Welcome to Financial Mathematics!

Ever wondered how a small savings account grows into a large sum over time, or why a brand-new car loses value the moment you drive it off the lot? That is exactly what this chapter is about. In Topic 1: Number and Algebra, we explore the math behind money. Whether you are planning to be a business mogul or just want to manage your own bank account, these tools are some of the most practical skills you will ever learn in the IB DP.

Note: This chapter builds on Geometric Sequences and Series. If you remember how a sequence grows by multiplying by a common ratio, you are already halfway there!

1. Compound Interest (SL 1.4)

In the world of finance, Compound Interest is often called the "eighth wonder of the world." Unlike simple interest (where you only earn money on your initial deposit), compound interest allows you to earn interest on your interest.

How it works

Imagine you invest $ \$100 $ at a $ 5\% $ annual interest rate. After one year, you have $ \$105 $. In the second year, you don't just earn interest on the original $ \$100 $; you earn $ 5\% $ on the full $ \$105 $. This creates a geometric growth pattern.

The Vocabulary of Money

Present Value ($PV$): The amount of money you have or invest right now.
Future Value ($FV$): The amount of money you will have after a certain amount of time.
Interest Rate ($r$): The percentage at which the money grows (usually given per year).
Compounding Periods ($k$): How often the bank calculates the interest.
- Yearly: $k = 1$
- Half-yearly: $k = 2$
- Quarterly: $k = 4$
- Monthly: $k = 12$

Quick Tip: If a question doesn't specify a compounding period, assume it is compounded yearly.

Key Takeaway: Compound interest is an application of a geometric sequence where the common ratio is $ (1 + \frac{r}{100k}) $.

2. Annual Depreciation (SL 1.4)

While investments usually grow, physical assets like laptops, machinery, and cars usually lose value over time. This is called Depreciation. In this course, we focus on the reducing balance method, which is essentially "negative" compound interest.

The Logic of Depreciation

Instead of adding interest, we subtract a percentage of the value each year. This is a geometric decay. The value of the item will drop quickly at first and then more slowly as time goes on.

Example: A car worth $ \$20,000 $ depreciates at $ 15\% $ per year.
After 1 year: $ 20,000 \times 0.85 = \$17,000 $
After 2 years: $ 17,000 \times 0.85 = \$14,450 $

Don't worry if this seems tricky: Just remember that for depreciation, your "growth factor" or common ratio will always be less than 1.

3. Using Technology: The TVM Solver (SL 1.7)

The IB Applications and Interpretation course emphasizes using your Graphic Display Calculator (GDC). For complex financial problems, we use a tool called the TVM Solver (Time Value of Money).

The Variables You Need to Know

When you open the TVM Solver on your GDC (usually under the Finance menu), you will see these variables:

$N$: The total number of payment periods (usually years $\times$ payments per year).
$I\%$: The annual interest rate (Enter this as a percentage, e.g., enter $5$ for $5\%$, not $0.05$).
$PV$: Present Value.
$PMT$: Payment amount (used for annuities or loans).
$FV$: Future Value.
$P/Y$: Payments per year.
$C/Y$: Compounding periods per year.

The Golden Rule of Signs (Very Important!):
Think of money like a flow.
- If money is leaving your pocket (investing it or paying a bank), it is Negative ($-$).
- If money is entering your pocket (taking out a loan or receiving a payout), it is Positive ($+$).

Example: If you put $ \$500 $ into a savings account, you enter $ PV = -500 $. If the bank gives you back $ \$600 $ later, the $ FV $ will be $ +600 $.

4. Annuities and Amortization (SL 1.7)

These terms sound fancy, but they are just specific ways of moving money in regular intervals.

Annuities

An Annuity is a sequence of equal payments made at equal intervals.
Example: Saving $ \$200 $ every month into a retirement fund.
In the TVM Solver, you would set $ PMT = -200 $ and $ P/Y = 12 $.

Amortization

Amortization is the process of paying off a debt (like a mortgage or a car loan) over time with regular payments.
Example: You borrow $ \$10,000 $ to buy a car and pay it back over 5 years.
In the TVM Solver:
$ PV = 10000 $ (The money entered your pocket to buy the car)
$ FV = 0 $ (The goal is to owe nothing at the end)
$ PMT = $ (This is what you would solve for – it will be a negative number!)

Did you know? In the early years of a loan, most of your payment goes toward interest. As the balance drops, more of your payment goes toward the actual debt!

5. Common Mistakes to Avoid

Mixing up $N$ and years: If you pay monthly for 5 years, $ N $ is $ 5 \times 12 = 60 $, not $ 5 $.
Wrong Signs: Forgetting to make the $ PV $ negative when investing money. If you get an "Error" or a weird answer on your GDC, check your signs!
Percentage Form: Entering $ 0.05 $ instead of $ 5 $ for the interest rate in the TVM solver.
P/Y vs C/Y: Usually, these are the same (e.g., if you pay monthly, interest is usually compounded monthly). Read the question carefully to see if they differ.

Summary Checklist

- Can you identify if a problem is about growth (interest) or decay (depreciation)?
- Do you know which TVM variable to solve for?
- Are you comfortable with the "Money In / Money Out" sign convention?
- Can you use your GDC to find the future value of an annuity?

Remember: The GDC is your best friend in this chapter. Practice entering different scenarios until the TVM Solver feels like second nature!

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?

Compound interest allows you to earn interest on both your initial deposit and the interest previously earned, creating a geometric growth pattern.

In the context of compound interest, what do the terms Present Value and Future Value represent?

Present Value ($PV$) is the amount invested or held right now, while Future Value ($FV$) is the amount the money grows to after a certain time.

How does the compounding period ($k$) affect the frequency of interest calculations?

The compounding period ($k$) determines how many times per year interest is calculated, such as yearly ($k=1$), quarterly ($k=4$), or monthly ($k=12$).

What is annual depreciation using the reducing balance method?

Depreciation is the loss of value of an asset over time, modeled as geometric decay where a percentage of the value is subtracted each year.

In the GDC TVM Solver, what does the variable $N$ represent?

$N$ represents the total number of payment periods, calculated as the number of years multiplied by the payments per year.

How should you enter the interest rate ($I\%$) into the TVM Solver?

Enter the interest rate as a percentage value (e.g., enter $5$ for $5\%$), not as a decimal.

What is the "Golden Rule of Signs" when using the TVM Solver?

Money leaving your pocket (investments/payments) is negative ($-$), while money entering your pocket (loans/payouts) is positive ($+$).

What is an annuity?

An annuity is a sequence of equal payments made at equal intervals, such as saving a fixed amount every month.

What is amortization in financial mathematics?

Amortization is the process of paying off a debt (like a loan) through regular payments over a set period of time.

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