Practice Maths

Comparing Financial Options

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Key Terms

Simple interest
pays a fixed amount of interest each period, calculated only on the original principal: I = Prt.
Compound interest
earns interest on interest: A = P(1 + r/n)nt. The balance grows faster over time.
The Effective Annual Rate (EAR) converts any compounding frequency to an equivalent annual rate, enabling fair comparison.
For investments, a higher EAR is better. For loans, a lower EAR is better.
Always compare the total amount paid/received or total interest over the same period, not just the nominal rate.
📚 Key Formulas
Simple interest amount: A = P(1 + rt)
Compound interest amount: A = P(1 + r/n)nt
Effective Annual Rate: EAR = (1 + r/n)n − 1
Interest earned/paid: I = A − P
Feature Simple Interest Compound Interest
FormulaA = P(1 + rt)A = P(1 + r/n)nt
Interest basisPrincipal onlyPrincipal + accumulated interest
Growth shapeLinearExponential
Better for investorShort termsLong terms
Better for borrowerLonger termsShorter terms
Comparison toolTotal interestEAR

Growth comparison: $5 000 at 8% p.a. — simple vs compound over 10 years

Time (years) Amount ($) 0 1 2 3 4 5 6 7 8 9 10 $5000 $6000 $7000 $8000 $9000 Compound A = P(1+r/n)^nt Simple A = P(1+rt) Gap
Hot Tip When comparing accounts, always calculate EAR for compound interest accounts and compare to the simple interest rate directly. A nominal rate of 5.6% compounding monthly has an EAR of 5.75% — higher than a 5.7% simple interest account, even though 5.6% looks lower.

Worked Example 1 — Comparing two investment accounts

Question: Account A offers 6% p.a. simple interest. Account B offers 5.8% p.a. compounding monthly. Which gives a better return for a $10 000 investment over 3 years?

Account A (simple): A = 10000 × (1 + 0.06 × 3) = 10000 × 1.18 = $11 800.00

Account B (compound): A = 10000 × (1 + 0.058/12)36 = 10000 × (1.004833)36
(1.004833)36 ≈ 1.18982 → A ≈ $11 898.20

Conclusion: Account B earns $98.20 more despite having the lower nominal rate. Always compare final amounts or EAR, not just nominal rates.

Worked Example 2 — Choosing the best loan

Question: Sarah needs to borrow $8 000 for 2 years. Lender X charges 9% p.a. simple interest. Lender Y charges 8.5% p.a. compounding monthly. Which is the cheaper loan?

Lender X (simple): A = 8000(1 + 0.09 × 2) = 8000 × 1.18 = $9 440.00. Interest = $1 440.

Lender Y (compound): A = 8000 × (1 + 0.085/12)24 = 8000 × (1.007083)24
(1.007083)24 ≈ 1.18685 → A ≈ $9 494.80. Interest = $1 494.80.

Conclusion: Lender X is cheaper by $54.80, even though its nominal rate is higher. For loans, compound interest is costlier because interest accrues on the growing balance.

Introduction

In the real world, financial products come with different structures — some pay simple interest, others compound interest at various frequencies, and rates vary between providers. This lesson gives you the tools to systematically compare any two (or more) financial options and make the best choice, whether you are investing or borrowing.

Step-by-Step Approach to Comparison Problems

When asked to compare financial options, follow these four steps:

  1. Identify the type of interest for each option (simple or compound).
  2. Calculate the final amount (A) for each option using the same principal and same time period.
  3. Calculate the interest earned or paid (I = A − P) for each option.
  4. Compare and state a conclusion — for investments, higher final amount is better; for loans, lower total repayment is better.

Alternatively, if you only need to rank accounts (not find exact amounts), calculate the EAR for each compound account and compare directly to simple interest rates.

Example: Term deposit comparison

Bank A offers a 2-year term deposit at 4.5% p.a. simple interest. Bank B offers 4.3% p.a. compounding quarterly. For a $20 000 investment, which is better?

Bank A (simple): A = 20000(1 + 0.045 × 2) = 20000 × 1.09 = $21 800.00

Bank B (compound): A = 20000 × (1 + 0.043/4)8 = 20000 × (1.01075)8

(1.01075)8 = 1.08971 → A = 20000 × 1.08971 = $21 794.20

Conclusion: Bank A is slightly better by $5.80 for this time period and principal.

Using EAR for Quick Comparison

The Effective Annual Rate (EAR) is the most powerful comparison tool when the compounding periods differ. Once you have the EAR for each compound account, you can rank all options instantly without calculating final amounts.

Example: Ranking three accounts by EAR

Account 1: 6.0% p.a. simple interest
Account 2: 5.8% p.a. compounding monthly
Account 3: 5.9% p.a. compounding quarterly

Account 2 EAR = (1 + 0.058/12)12 − 1 = (1.004833)12 − 1 = 0.05957 = 5.957%

Account 3 EAR = (1 + 0.059/4)4 − 1 = (1.01475)4 − 1 = 0.06019 = 6.019%

Ranking (best first): Account 3 (6.019%) > Account 1 (6.0%) > Account 2 (5.957%)

Note: Account 3 beats Account 1 despite a lower nominal rate, because quarterly compounding boosts the effective return above 6%.

Real-World Context: Loans vs Investments

The same mathematics applies to both investments and loans, but the goal is reversed:

  • Investment: You want the highest final amount → choose the highest EAR.
  • Loan: You want the lowest total repayment → choose the lowest EAR.

Watch out: compound interest on loans accumulates faster than you might expect. Always calculate the total amount owed, not just compare rates.

💡 Common Mistake: Students sometimes compare nominal rates directly (e.g., 5.6% vs 5.8%) without accounting for compounding frequency. A 5.6% monthly compounding account can easily outperform a 5.8% annual compounding account. Always convert to EAR or calculate final amounts before comparing.

Summary

To compare financial options: (1) identify simple vs compound; (2) calculate A or EAR for each; (3) compare total interest earned/paid; (4) state a clear conclusion with justification. For investments, choose the highest return. For loans, choose the lowest total repayment.

Mastery Practice

See Answers ➔
  1. Fluency

    Compare these two simple interest accounts for a $5 000 investment over 4 years. Which earns more interest, and by how much?

    1. (a) Account P: 5.5% p.a. simple interest. Calculate the final amount.
    2. (b) Account Q: 6.0% p.a. simple interest. Calculate the final amount.
    3. (c) How much more does the better account earn?
  2. Fluency

    $4 000 is invested for 3 years at 5% p.a. Compare simple and compound interest (annual compounding).

    1. (a) Calculate the final amount under simple interest.
    2. (b) Calculate the final amount under compound interest (annual compounding).
    3. (c) By how much does compound interest exceed simple interest?
  3. Understanding

    Priya has $12 000 to invest for 5 years. She is comparing three accounts.

    Account X: 4.8% p.a. simple interest
    Account Y: 4.5% p.a. compounding quarterly
    Account Z: 4.6% p.a. compounding monthly
    1. (a) Calculate the EAR for Account Y and Account Z.
    2. (b) Calculate the final amount in each account after 5 years.
    3. (c) Which account should Priya choose? Justify with figures.
  4. Understanding

    Jordan needs to borrow $6 000 for 2 years. Two lenders have made offers.

    Lender A: 10% p.a. simple interest
    Lender B: 9.5% p.a. compounding monthly
    1. (a) Calculate the total amount owed to each lender after 2 years.
    2. (b) Calculate the total interest charged by each lender.
    3. (c) Which lender should Jordan choose? Explain why the lower nominal rate is not necessarily the better deal.
  5. Understanding

    Mei is comparing two car loans for $18 000 over 4 years.

    Loan A: 7.2% p.a. compounding monthly
    Loan B: 7.5% p.a. simple interest
    1. (a) Find the total amount owed under Loan A after 4 years.
    2. (b) Find the total amount owed under Loan B after 4 years.
    3. (c) Which loan has the lower total cost? By how much?
    4. (d) If Mei could pay off the loan in 1 year instead of 4, would the answer change? Explain.
  6. Understanding

    Alex wants to have $30 000 saved in 6 years for a house deposit. He is choosing between two investment accounts.

    Option 1: 5.4% p.a. simple interest
    Option 2: 5.0% p.a. compounding monthly
    1. (a) If Alex invests $20 000 today, how much will he have in each account after 6 years?
    2. (b) Which option gets him closer to his $30 000 goal?
    3. (c) How much does he still need to save, on top of his investment, with the better option?
  7. Understanding

    $5 000 is available to invest for 4 years. Compare the following three options and rank them from best to worst.

    Option A: 6.0% p.a. simple interest
    Option B: 5.8% p.a. compounding semi-annually
    Option C: 5.75% p.a. compounding monthly
    1. (a) Calculate the final amount for each option.
    2. (b) Calculate the interest earned for each option.
    3. (c) Rank the three options from best to worst return. Was the highest nominal rate the best option?
  8. Understanding

    Blake needs $2 000 urgently and is considering two sources.

    Pawnbroker: charges 8% per month simple interest (can repay after 6 months)
    Bank personal loan: 24% p.a. compounding monthly (same 6-month term)
    1. (a) Calculate the total cost of the pawnbroker loan after 6 months.
    2. (b) Calculate the total cost of the bank loan after 6 months.
    3. (c) What is the annual equivalent rate of the pawnbroker’s 8% per month? Comment on how this compares to the bank rate.
  9. Problem Solving

    The Nguyen family is considering three options to fund a $40 000 home renovation, to be repaid over 5 years.

    Mortgage redraw: 4.8% p.a. compounding monthly (secured against home)
    Personal loan: 12.5% p.a. compounding monthly
    Credit card: 19.99% p.a. compounding monthly
    1. (a) Calculate the total amount owed under each option at the end of 5 years.
    2. (b) Calculate the total interest charged by each option.
    3. (c) How much more does the most expensive option cost compared to the cheapest?
    4. (d) Explain why the difference in interest grows so dramatically between these three options despite using the same principal and time period.
  10. Problem Solving

    Retirement planning: Elena (age 45) has $80 000 to invest for 20 years until she retires. She is comparing two superannuation-style options.

    Conservative fund: 4.5% p.a. compounding annually (lower risk)
    Growth fund: 7.0% p.a. compounding annually (higher risk)
    1. (a) Calculate the value of her investment in each fund after 20 years.
    2. (b) How much more does the growth fund return compared to the conservative fund?
    3. (c) Suppose Elena instead chooses a fixed-rate term deposit at 5.2% p.a. simple interest for 20 years. Calculate its final value and compare to both funds.
    4. (d) Justify which option Elena should choose, explaining the role of compounding, time, and risk in your answer.