Practice Maths

Compound Interest and Investments

Key Terms

compound interest
interest calculated on both the original principal AND the accumulated interest already earned; produces exponential growth
compound interest formula
A = P(1 + r/n)nt — where P = principal, r = annual rate (as a decimal), n = compounding periods per year, t = time in years
compounding frequency
how often interest is added in a year — more frequent compounding (e.g. monthly > quarterly > annually) gives a slightly higher final amount for the same nominal rate
effective annual rate (EAR)
EAR = (1 + r/n)n − 1 — the equivalent simple annual rate that gives the same result; use to fairly compare investments with different compounding frequencies
financial calculator / TVM solver
a calculator tool (CAS, spreadsheet, or online) that solves for any unknown when given: N (periods), I% (annual rate), PV (present value), PMT (payment per period), FV (future value)
interest earned
I = A − P (the amount earned above the original principal)

Compound Interest Formula

Compound interest means interest is calculated on both the original principal and the interest already earned. This causes exponential growth.

A = P(1 + r/n)nt

• A = final amount ($)
• P = principal (initial investment, $)
• r = annual interest rate (as a decimal, e.g. 6% = 0.06)
• n = number of compounding periods per year (1=annual, 2=half-yearly, 4=quarterly, 12=monthly, 365=daily)
• t = time in years

Interest earned: I = A − P

Effective Annual Rate (EAR)

When interest compounds more than once per year, the effective annual rate is the equivalent simple annual rate that gives the same result:

EAR = (1 + r/n)n − 1

A higher compounding frequency increases the effective rate. Use EAR to compare investments with different compounding periods.

Hot Tip: In any TVM solver or financial calculator, money you pay out (e.g. an investment) is entered as a negative PV, and money you receive is positive. Getting this sign wrong gives a completely wrong answer with no error message. Always check the sign convention before solving.

Using a Financial Calculator or TVM Solver

Any financial calculator, CAS, or TVM solver uses the same five variables. Enter the four you know and solve for the one you don't:

  • N = total number of compounding periods (e.g. 5 years monthly → N = 60)
  • I% = annual interest rate as a percentage (e.g. enter 5.4 for 5.4%)
  • PV = present value — enter as a negative number if it represents money you are paying out (an investment)
  • PMT = payment per period (use 0 for a lump-sum investment with no regular deposits)
  • FV = future value — the amount you want to find (leave as 0 or blank)

Also set the compounding periods per year to match your problem (e.g. 12 for monthly). Getting the sign of PV wrong is the most common error — money you pay out is negative; money you receive is positive.

Worked Example 1

$8 000 is invested at 5.4% p.a. compounded monthly for 4 years. Find the final amount and interest earned.

Solution: P = 8000, r = 0.054, n = 12, t = 4.

A = 8000 × (1 + 0.054/12)12×4 = 8000 × (1.0045)48 ≈ 8000 × 1.2434 ≈ $9 947.13

Interest = 9947.13 − 8000 = $1 947.13

Worked Example 2

Compare two investments of $10 000 for 3 years: (A) 6% p.a. compounded annually vs (B) 5.85% p.a. compounded monthly. Which is better?

Option A: A = 10000 × (1.06)3 = $11 910.16. EAR = 6.00%

Option B: A = 10000 × (1 + 0.0585/12)36 ≈ $11 892.42. EAR = (1.004875)12 − 1 ≈ 6.002%

Option A returns slightly more despite the higher nominal rate due to less frequent compounding.

Full Lesson: Compound Interest and Investments

1. Why Compound Interest Matters

Understanding how money grows is one of the most practically useful skills in mathematics. Whether you are saving for a car, a home deposit, or retirement, or comparing bank account offers, compound interest is the engine that drives modern finance.

The key insight is this: with compound interest, you earn interest on your interest. Over short periods the difference from simple interest is small. Over decades, it is enormous. Albert Einstein allegedly called compound interest “the eighth wonder of the world.” Whether he said it or not, the mathematics backs it up.

2. Building the Compound Interest Formula

To understand where the formula comes from, let's build it step by step. Start with the simplest case: interest added once per year.

Suppose you invest $P at an annual interest rate of r, compounded once per year.

  • After Year 1: you earn interest of P × r on top of your original amount, so your new balance is P + Pr = P(1 + r)
  • After Year 2: you earn interest on the Year 1 balance (not just the original P): P(1 + r) × (1 + r) = P(1 + r)2
  • After Year 3: the same pattern continues: P(1 + r)3

The pattern is clear — after t years, the amount is A = P(1 + r)t.

Now, what if interest is added more than once per year?

If the annual rate is r but the bank adds interest n times per year (e.g. monthly means n = 12), then:

  • Each period uses a smaller rate: r/n (e.g. 6% annual compounded monthly → rate per month = 0.06/12 = 0.005)
  • In t years, the total number of compounding periods is nt (e.g. 3 years monthly → 36 periods)

Replacing r with r/n and t with nt gives the complete compound interest formula:

A = P(1 + r/n)nt

As n increases (more frequent compounding), A increases slightly but the gains get smaller each time. Monthly and daily compounding give very similar results — the difference between them is less than a dollar on most ordinary investments.

3. Compounding Frequencies Compared

Consider $1 000 at 12% p.a. for 1 year under different compounding frequencies:

Frequency n A after 1 year
Annual1$1 120.00
Half-yearly2$1 123.60
Quarterly4$1 125.51
Monthly12$1 126.83
Daily365$1 127.47

4. Effective Annual Rate (EAR)

When comparing investments advertised with different compounding frequencies, the nominal rate (the stated annual rate) can be misleading. The EAR gives the true annual return:

EAR = (1 + r/n)n − 1

Example: 6% p.a. compounded monthly has EAR = (1 + 0.06/12)12 − 1 = (1.005)12 − 1 ≈ 0.06168 = 6.168% p.a.

This means earning 6% nominal compounded monthly is equivalent to earning 6.168% simple interest annually. Always compare EARs when choosing between investments.

5. Using a Financial Calculator or TVM Solver

Most exam boards allow a financial calculator or CAS with a TVM (Time Value of Money) solver. The process is the same regardless of brand:

  1. Identify the unknown. You will know four of the five variables (N, I%, PV, PMT, FV) and solve for the fifth.
  2. Calculate N. Multiply years by compounding periods per year. For 5 years compounded monthly: N = 5 × 12 = 60.
  3. Enter I%. Use the annual rate as a percentage (e.g. type 5.4 for 5.4% p.a.). Your calculator handles the per-period conversion when you set the compounding frequency.
  4. Enter PV as a negative number if it represents money you are paying out (an investment). This is the most common source of sign errors.
  5. Set PMT = 0 for a lump-sum investment with no regular deposits.
  6. Set the compounding periods per year to match your problem (1 = annual, 4 = quarterly, 12 = monthly, 365 = daily).
  7. Solve for FV to find the future value, or solve for whichever variable the question asks for.

These steps apply to any TVM solver — whether on a CAS calculator, a financial calculator, or an online tool. The key is entering the correct sign for PV and matching the compounding periods to your rate.

6. Common Errors

  • Forgetting to convert the annual rate: if the rate is 6% p.a. and compounding is monthly, the rate per period is 0.06 ÷ 12 = 0.005, not 0.06.
  • Confusing n (periods per year) with t (years). The exponent is nt, the total number of periods.
  • Using percentage instead of decimal: r = 6% means r = 0.06 in the formula.
  • Not rounding correctly: financial answers should be given to the nearest cent (2 decimal places).

Mastery Practice

  1. Annual Compounding Fluency

    Calculate the final amount when $5 000 is invested at 4% p.a. compounded annually for 6 years.

  2. Monthly Compounding — Interest Earned Fluency

    $12 000 is invested at 3.6% p.a. compounded monthly for 5 years. Find the total interest earned.

  3. Effective Annual Rate Fluency

    Calculate the effective annual rate (EAR) for a nominal rate of 7.2% p.a. compounded quarterly. Give your answer as a percentage correct to 3 decimal places.

  4. Doubling Time Fluency

    An investment doubles in value. Using A = P(1 + r)t with annual compounding and r = 0.08, find the time t (in years) for the investment to double. Give your answer correct to 1 decimal place.

  5. Comparing Two Banks Understanding

    Bank A offers 5.5% p.a. compounded half-yearly. Bank B offers 5.4% p.a. compounded monthly. You want to invest $20 000 for 3 years.

    1. Calculate the EAR for each bank.
    2. Find the final amount at each bank after 3 years.
    3. Which bank gives a better return, and by how much?

  6. Present Value Understanding

    How much would you need to invest now at 6% p.a. compounded monthly to have $50 000 in 10 years? Use the formula P = A × (1 + r/n)−nt or a financial calculator.

  7. Investment Race Understanding

    Priya invests $15 000 at 4.8% p.a. compounded quarterly. Zach invests $12 000 at 6.2% p.a. compounded monthly. After 8 years, who has the larger balance, and by how much?

  8. Misleading Advertising Understanding

    An advertisement states: “Earn 5.0% p.a. interest!” without mentioning the compounding frequency. Explain why this information is insufficient to compare this investment with one offering 4.95% p.a. compounded monthly. Calculate the EAR for both options at the two extremes of compounding (annual and monthly for the 5% option) to support your answer.

  9. House Deposit Strategy Problem Solving

    Lachlan wants to have $100 000 saved in 15 years. He compares three investment accounts:

    • Account X: 5.5% p.a. compounded annually
    • Account Y: 5.3% p.a. compounded monthly
    • Account Z: 5.2% p.a. compounded daily
    1. Find the EAR for each account.
    2. Calculate how much Lachlan needs to invest now (as a lump sum) in each account to reach his $100 000 goal.
    3. Recommend the best account for Lachlan and justify your answer numerically.

  10. The Rule of 72 Problem Solving

    The “Rule of 72” is a mental shortcut: the approximate time (in years) to double an investment ≈ 72 ÷ (annual interest rate as a percentage).

    1. Use the Rule of 72 to estimate the doubling time for rates of 4%, 6%, 8%, and 12% p.a. (compounded annually).
    2. For each rate, calculate the exact doubling time using logarithms.
    3. Evaluate the accuracy of the Rule of 72 across these rates and identify where it is most and least accurate.
    4. At what annual rate compounded annually would $5 000 grow to $20 000 in 20 years? Use logarithms or a financial calculator.
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