Solutions: Compound Interest and Investments

Answer: $6 326.60

Using A = P(1 + r/n)^nt with P = 5000, r = 0.04, n = 1, t = 6:

A = 5000 × (1 + 0.04/1)^1×6 = 5000 × (1.04)⁶

(1.04)⁶ = 1.04 × 1.04 × 1.04 × 1.04 × 1.04 × 1.04 ≈ 1.26532

A = 5000 × 1.26532 = $6 326.60

Since n = 1 (annual compounding), the formula simplifies to A = P(1 + r)^t.
Answer: Interest = $2 360.16

P = 12 000, r = 0.036, n = 12, t = 5. Total periods: nt = 12 × 5 = 60.

Rate per period: r/n = 0.036 ÷ 12 = 0.003

A = 12 000 × (1.003)⁶⁰ ≈ 12 000 × 1.19668 ≈ $14 360.16

Interest earned = A − P = 14 360.16 − 12 000 = $2 360.16

Note: 3.6% p.a. compounded monthly means 0.3% per month. Over 60 months, the interest earned is significantly more than simple interest would give (12 000 × 0.036 × 5 = $2 160).
Answer: EAR ≈ 7.398%

Using EAR = (1 + r/n)ⁿ − 1 with r = 0.072, n = 4:

EAR = (1 + 0.072/4)⁴ − 1 = (1 + 0.018)⁴ − 1 = (1.018)⁴ − 1

(1.018)⁴ = 1.018 × 1.018 × 1.018 × 1.018 ≈ 1.07398

EAR = 1.07398 − 1 = 0.07398 = 7.398%

This means 7.2% p.a. compounded quarterly is equivalent to earning 7.398% simple interest annually — the extra 0.198% is the bonus from quarterly compounding.
Answer: t ≈ 9.0 years

We need A = 2P, so: 2P = P(1.08)^t

Dividing both sides by P: 2 = (1.08)^t

Taking logarithms of both sides: log(2) = t × log(1.08)

t = log(2) ÷ log(1.08) = 0.30103 ÷ 0.03342 ≈ 9.0 years

Check: (1.08)⁹ ≈ 1.9990 ≈ 2. ✓

Rule of 72 check: 72 ÷ 8 = 9 years — the Rule of 72 gives an excellent estimate here.
(a) Effective Annual Rates:

Bank A (5.5% p.a. compounded half-yearly, n = 2):

EAR = (1 + 0.055/2)² − 1 = (1.0275)² − 1 = 1.05576 − 1 ≈ 5.576%

Bank B (5.4% p.a. compounded monthly, n = 12):

EAR = (1 + 0.054/12)¹² − 1 = (1.0045)¹² − 1 ≈ 1.05541 − 1 ≈ 5.541%

(b) Final amounts after 3 years (P = $20 000):

Bank A: A = 20 000 × (1.0275)⁶ ≈ 20 000 × 1.17438 ≈ $23 487.68

Bank B: A = 20 000 × (1.0045)³⁶ ≈ 20 000 × 1.17313 ≈ $23 462.65

(c) Bank A gives a better return by $23 487.68 − $23 462.65 = $25.03 more over 3 years. Although Bank A has a lower nominal rate (5.5% vs — wait, Bank A is higher), its less frequent compounding means only a marginal advantage. Always compare EARs: Bank A EAR (5.576%) > Bank B EAR (5.541%).
Answer: P ≈ $27 481.50

We need to find the present value P given A = 50 000, r = 0.06, n = 12, t = 10.

Rearranging A = P(1 + r/n)^nt: P = A × (1 + r/n)^−nt

P = 50 000 × (1 + 0.06/12)⁻¹²⁰ = 50 000 × (1.005)⁻¹²⁰

(1.005)⁻¹²⁰ = 1 ÷ (1.005)¹²⁰ ≈ 1 ÷ 1.81940 ≈ 0.54963

P = 50 000 × 0.54963 ≈ $27 481.50

CAS approach: Finance Solver: N = 120, I% = 6, PV = ?, PMT = 0, FV = 50000, P/Y = C/Y = 12. Solve for PV ≈ −27 481.50 (negative indicates money paid out).
Answer: Priya has $2 200.23 more

Priya: P = 15 000, r = 0.048, n = 4, t = 8. Total periods: 4 × 8 = 32.

A = 15 000 × (1 + 0.048/4)³² = 15 000 × (1.012)³² ≈ 15 000 × 1.46077 ≈ $21 911.55

Zach: P = 12 000, r = 0.062, n = 12, t = 8. Total periods: 12 × 8 = 96.

Rate per period: 0.062 ÷ 12 ≈ 0.005167

A = 12 000 × (1.005167)⁹⁶ ≈ 12 000 × 1.64261 ≈ $19 711.32

Priya’s balance is larger: $21 911.55 − $19 711.32 = $2 200.23 more

Despite Zach’s higher interest rate, Priya’s larger principal ($15 000 vs $12 000) gives her the advantage.
Why the information is insufficient: The nominal rate alone does not determine the actual return — the compounding frequency is equally important. The same nominal rate compounded more frequently gives a higher effective return.

EAR calculations for the 5.0% option:

5.0% compounded annually: EAR = (1 + 0.05/1)¹ − 1 = 5.000%

5.0% compounded monthly: EAR = (1 + 0.05/12)¹² − 1 ≈ (1.004167)¹² − 1 ≈ 5.116%

EAR for 4.95% compounded monthly:

EAR = (1 + 0.0495/12)¹² − 1 ≈ (1.004125)¹² − 1 ≈ 5.065%

Conclusion: If the 5.0% is compounded annually (EAR = 5.000%), it is worse than the 4.95% monthly option (EAR = 5.065%). If it is compounded monthly (EAR = 5.116%), it is better. The compounding frequency must be disclosed to make a fair comparison.
(a) Effective Annual Rates:

Account X (5.5% p.a. compounded annually, n = 1): EAR = 5.500%

Account Y (5.3% p.a. compounded monthly, n = 12):

EAR = (1 + 0.053/12)¹² − 1 = (1.004417)¹² − 1 ≈ 5.434%

Account Z (5.2% p.a. compounded daily, n = 365):

EAR = (1 + 0.052/365)³⁶⁵ − 1 = (1.000142)³⁶⁵ − 1 ≈ 5.334%

(b) Present value needed for each (A = $100 000, t = 15 years):

Account X: P = 100 000 × (1.055)⁻¹⁵ ≈ 100 000 × 0.44793 ≈ $44 793

Account Y: P = 100 000 × (1.05434)⁻¹⁵ ≈ 100 000 × 0.45306 ≈ $45 306

Account Z: P = 100 000 × (1.05334)⁻¹⁵ ≈ 100 000 × 0.46010 ≈ $46 010

(c) Recommendation: Account X is the best choice for Lachlan. It has the highest EAR (5.500%) and therefore requires the smallest upfront investment ($44 793) — saving him $513 compared to Account Y and $1 217 compared to Account Z. Higher nominal rate with annual compounding can beat lower rates with more frequent compounding when the nominal rate difference is large enough.

Rate	Rule of 72	Exact (log)	Error
4%	18.0 yr	17.67 yr	+0.33 yr
6%	12.0 yr	11.90 yr	+0.10 yr
8%	9.0 yr	9.01 yr	−0.01 yr
12%	6.0 yr	6.12 yr	−0.12 yr