Comparing Financial Options — Solutions

Q1 — Comparing two simple interest accounts

Fluency

Formula: I = Prt

Bank A: P = $3000, r = 4.5% = 0.045, t = 2 years

I_A = 3000 × 0.045 × 2 = $270

Bank B: P = $3000, r = 4% = 0.04, t = 3 years

I_B = 3000 × 0.04 × 3 = $360

Bank B earns more interest: $360 > $270.

Bank B earns $90 more in total interest.

Note: Although Bank A has a higher rate, Bank B benefits from a longer term.

Q2 — Simple interest vs compound interest at 5% p.a.

Fluency

P = $8000, r = 5% p.a., t = 4 years

Simple interest:

I = Prt = 8000 × 0.05 × 4 = $1600

Total = $8000 + $1600 = $9600

Compound interest (annually):

A = P(1 + r)ⁿ = 8000 × (1.05)⁴

(1.05)⁴ = 1.21550625

A = 8000 × 1.21550625 = $9724.05

Difference: $9724.05 − $9600 = $124.05

Compound interest earns $124.05 more than simple interest.

Q3 — Ranking three savings accounts for $10 000 over 3 years

Understanding

P = $10 000, t = 3 years

(a) 5% p.a. simple interest:

I = 10000 × 0.05 × 3 = $1500

Balance = $11 500.00

(b) 4.8% p.a. compounding monthly:

r = 0.048/12 = 0.004 per month, n = 3 × 12 = 36

A = 10000 × (1.004)³⁶

(1.004)³⁶ = 1.15399 (approx)

A = 10000 × 1.15399 = $11 539.90

(c) 4.9% p.a. compounding quarterly:

r = 0.049/4 = 0.01225 per quarter, n = 3 × 4 = 12

A = 10000 × (1.01225)¹²

(1.01225)¹² = 1.15999 (approx)

A = 10000 × 1.15999 = $15 999.90

Wait — recalculating carefully:

(1.01225)¹²: ln(1.01225) = 0.012175, ×12 = 0.14610, e^0.14610 = 1.15724

A = 10000 × 1.15724 = $11 572.40

Ranking (highest to lowest final balance):

1st: Option (c) 4.9% quarterly — $11 572.40

2nd: Option (b) 4.8% monthly — $11 539.90

3rd: Option (a) 5% simple — $11 500.00

Despite Option (a) having the highest nominal rate, compound interest accumulates more effectively over 3 years.

Q4 — Comparing loan options: simple vs compound

Understanding

P = $15 000, t = 3 years

Loan A: 8% p.a. simple interest

I = 15000 × 0.08 × 3 = $3600

Total repayment = $15 000 + $3600 = $18 600

Loan B: 7.5% p.a. compounding monthly

r = 0.075/12 = 0.00625 per month, n = 36

A = 15000 × (1.00625)³⁶

(1.00625)³⁶ = 1.25058 (approx)

A = 15000 × 1.25058 = $18 758.70

Comparison:

Loan A total: $18 600

Loan B total: $18 758.70

Loan A (simple interest) has a lower total cost by $158.70.

Note: Although Loan B has a lower nominal rate (7.5% vs 8%), monthly compounding increases the effective cost above Loan A’s simple interest total over 3 years.

Q5 — Investing $20 000 for 5 years: simple vs compound

Understanding

P = $20 000, t = 5 years

Option A: 6% p.a. simple interest

I = 20000 × 0.06 × 5 = $6000

Final amount = $26 000

Interest earned = $6000

Option B: 5.7% p.a. compounding quarterly

r = 0.057/4 = 0.01425 per quarter, n = 5 × 4 = 20

A = 20000 × (1.01425)²⁰

(1.01425)²⁰: using (1 + 0.01425)²⁰ = 1.32897 (approx)

A = 20000 × 1.32897 = $26 579.40

Interest earned = $26 579.40 − $20 000 = $6579.40

Option B earns $579.40 more despite having a lower nominal rate.

This demonstrates the power of compound interest — quarterly compounding makes 5.7% effectively outperform 6% simple.

Q6 — Effective Annual Rate (EAR) for term deposits

Understanding

Formula: EAR = (1 + r/n)ⁿ − 1

Bank X: 3.8% p.a. compounding annually

EAR = (1 + 0.038/1)¹ − 1 = 0.038 = 3.800%

(Compounding annually, EAR equals the nominal rate.)

Bank Y: 3.75% p.a. compounding monthly

EAR = (1 + 0.0375/12)¹² − 1

= (1 + 0.003125)¹² − 1

= (1.003125)¹² − 1

= 1.03819 − 1 = 0.03819 = 3.819%

Comparison:

Bank X EAR: 3.800%

Bank Y EAR: 3.819%

Bank Y is the better option — despite having a lower nominal rate (3.75% vs 3.8%), monthly compounding gives Bank Y a higher EAR of 3.819%.

Q7 — Ranking three accounts for $5000 over 3 years

Understanding

P = $5000, t = 3 years

(a) Simple interest at 6% for 3 years:

I = 5000 × 0.06 × 3 = $900

Balance = $5900.00

(b) Compound 5.5% p.a. monthly for 3 years:

r = 0.055/12 = 0.004583… per month, n = 36

A = 5000 × (1.004583)³⁶

(1.004583)³⁶ = 1.17894 (approx)

A = 5000 × 1.17894 = $5894.70

(c) Compound 5.8% p.a. annually for 3 years:

A = 5000 × (1.058)³

(1.058)³ = 1.058 × 1.058 × 1.058 = 1.18397 (approx)

A = 5000 × 1.18397 = $5919.85

Ranking (highest to lowest):

1st: Option (c) 5.8% annually — $5919.85

2nd: Option (a) 6% simple — $5900.00

3rd: Option (b) 5.5% monthly — $5894.70

Over only 3 years, the higher rate of 5.8% compound annually outperforms 5.5% monthly compound, and the simple rate of 6% sits between them.

Q8 — Pawnbroker vs bank: comparing interest on a $2000 loan for 6 months

Understanding

P = $2000, t = 6 months

Pawnbroker: 3% per month simple interest

Monthly rate = 3%, t = 6 months

I = P × r × t = 2000 × 0.03 × 6 = $360

Total repayment = $2000 + $360 = $2360

Bank: 18% p.a. compounding monthly

r = 0.18/12 = 0.015 per month, n = 6

A = 2000 × (1.015)⁶

(1.015)⁶ = 1.09344 (approx)

A = 2000 × 1.09344 = $2186.88

Interest = $2186.88 − $2000 = $186.88

Comparison:

Pawnbroker interest: $360

Bank interest: $186.88

The bank charges $173.12 less interest. The pawnbroker’s 3% per month equals 36% p.a., which is double the bank’s 18% p.a. rate.

Q9 — Find the principal needed to reach $30 000 in 5 years

Problem Solving

Target: A = $30 000, t = 5 years

Option A: 4.5% p.a. compounding quarterly

A = P(1 + r/n)^nt

30000 = P × (1 + 0.045/4)^4×5

30000 = P × (1.01125)²⁰

(1.01125)²⁰ = 1.25084 (approx)

P = 30000 ÷ 1.25084 = $23 983.54

Option B: 4.8% p.a. simple interest

A = P(1 + rt)

30000 = P × (1 + 0.048 × 5)

30000 = P × (1 + 0.24)

30000 = P × 1.24

P = 30000 ÷ 1.24 = $24 193.55

Option A requires less investment: $23 983.54 compared to $24 193.55.

Option A requires $210.01 less upfront, even though it has a lower nominal rate, because compounding quarterly produces greater growth over 5 years.

Q10 — Retirement planning: comparing $50 000 over 20 years

Problem Solving

P = $50 000, t = 20 years

(a) 5% p.a. simple interest:

I = 50000 × 0.05 × 20 = $50 000

Final amount = $100 000

(b) 4.5% p.a. compounding monthly:

r = 0.045/12 = 0.00375, n = 20 × 12 = 240

A = 50000 × (1.00375)²⁴⁰

(1.00375)²⁴⁰: ln(1.00375) = 0.003743, ×240 = 0.89832, e^0.89832 = 2.45513

A = 50000 × 2.45513 = $122 756.50

(c) 5% p.a. compounding annually:

A = 50000 × (1.05)²⁰

(1.05)²⁰ = 2.65330 (standard value)

A = 50000 × 2.65330 = $132 664.89

Ranking (highest to lowest):

1st: Option (c) 5% annually — $132 664.89

2nd: Option (b) 4.5% monthly — $122 756.50

3rd: Option (a) 5% simple — $100 000.00

Recommendation: Option (c) is best. Over 20 years, compound interest dramatically outperforms simple interest — option (c) produces $32 664.89 more than simple interest. The power of compounding is amplified by the long time horizon. Option (b) earns more than simple interest but less than annual compounding at the same rate because 4.5% monthly compounding does not overcome the 0.5% rate disadvantage over 20 years.