Solutions — Annuities — Grade 12 General Maths

Fluency $300/month deposit, 0.4% monthly rate. Recurrence relation and balance after 2 months.

Recurrence relation: V_n+1 = V_n(1.004) + 300, V₀ = 0

V₁ = 0 × 1.004 + 300 = $300.00
V₂ = 300 × 1.004 + 300 = 301.20 + 300 = $601.20

Fluency $80 000 retirement fund, $1 200/month withdrawals, 0.35% monthly rate. Recurrence and balance after 1 withdrawal.

Recurrence relation: V_n+1 = V_n(1.0035) − 1 200, V₀ = 80 000

V₁ = 80 000 × 1.0035 − 1 200 = 80 280 − 1 200 = $79 080

Fluency $150/month deposits at 6% p.a. (0.5% monthly) for 24 months. Use the FV formula.

FV = D × ((1 + r)ⁿ − 1) / r
= 150 × (1.005²⁴ − 1) / 0.005
= 150 × (1.1272 − 1) / 0.005
= 150 × 0.1272 / 0.005
= 150 × 25.432
= $3 814.80

Fluency $200 000 fund, $1 500/month withdrawals, 0.3% monthly rate. Balance after 3 months.

V₀ = 200 000, W = 1 500, r = 0.003

V₁ = 200 000 × 1.003 − 1 500 = 200 600 − 1 500 = 199 100
V₂ = 199 100 × 1.003 − 1 500 = 199 697.30 − 1 500 = 198 197.30
V₃ = 198 197.30 × 1.003 − 1 500 = 198 791.89 − 1 500 = $197 291.89

Understanding $500/quarter deposits at 1% per quarter. Balance after 5 quarters by iteration.

V_n+1 = V_n(1.01) + 500, V₀ = 0

V₁ = 0 + 500 = 500.00
V₂ = 500 × 1.01 + 500 = 505 + 500 = 1 005.00
V₃ = 1 005 × 1.01 + 500 = 1 015.05 + 500 = 1 515.05
V₄ = 1 515.05 × 1.01 + 500 = 1 530.20 + 500 = 2 030.20
V₅ = 2 030.20 × 1.01 + 500 = 2 050.50 + 500 = $2 550.50

Understanding Monthly deposit to accumulate $10 000 in 24 months at 6% p.a. (0.5% monthly).

FV = D × ((1.005)²⁴ − 1) / 0.005
10 000 = D × 25.432
D = 10 000 / 25.432 = $393.21 per month

At $393.21 per month, total deposits = $393.21 × 24 = $9 437, with $563 coming from interest.

Understanding $150 000 fund, $2 000/month withdrawals, 0.5% monthly rate. How many months does it last?

Using the PV formula: PV = W × (1 − (1 + r)⁻ⁿ) / r
150 000 = 2 000 × (1 − 1.005⁻ⁿ) / 0.005
150 000 × 0.005 / 2 000 = 1 − 1.005⁻ⁿ
0.375 = 1 − 1.005⁻ⁿ
1.005⁻ⁿ = 0.625
−n × ln(1.005) = ln(0.625)
n = ln(0.625) / (−ln(1.005)) = 0.470 / 0.004988 ≈ 94.2

The fund lasts approximately 94 months (7 years 10 months), with a smaller final withdrawal in month 94.

Iteration confirms: by month 93 the balance is still positive; by month 94 the balance falls to near zero.

Understanding $320 000 fund, $1 800/month, 0.4% monthly. Will it last 20 years (240 months)?

Find the PV required to sustain $1 800/month for 240 months at 0.4%:
PV = 1 800 × (1 − 1.004⁻²⁴⁰) / 0.004
1.004²⁴⁰ ≈ 2.607, so 1.004⁻²⁴⁰ ≈ 0.3836
PV = 1 800 × (1 − 0.3836) / 0.004 = 1 800 × 154.10 = $277 380

Since the fund has $320 000 > $277 380, yes, the fund will last the full 20 years, with approximately $42 620 remaining after the final payment.

Problem Solving $400/month at 0.45% monthly toward a $15 000 car.

(a) Number of months to reach $15 000:
FV = D × (1.0045ⁿ − 1) / 0.0045
15 000 = 400 × (1.0045ⁿ − 1) / 0.0045
15 000 × 0.0045 / 400 = 1.0045ⁿ − 1
0.16875 = 1.0045ⁿ − 1
1.0045ⁿ = 1.16875
n = ln(1.16875) / ln(1.0045) = 0.15578 / 0.004490 ≈ 34.7

She reaches $15 000 after 35 months (just over 2 years 11 months).

Check: FV at n = 35 = 400 × (1.0045³⁵ − 1) / 0.0045 ≈ 400 × 37.82 ≈ $15 128 ✓

(b) Deposits vs interest:
Total deposits = 35 × $400 = $14 000
Interest earned = $15 128 − $14 000 = $1 128

Problem Solving $500 000 fund at 0.5% monthly. (a) How long at $3 000/month? (b) How much longer at $2 500/month?

(a) $3 000/month withdrawals:
500 000 = 3 000 × (1 − 1.005⁻ⁿ) / 0.005
500 000 × 0.005 / 3 000 = 1 − 1.005⁻ⁿ
0.8333 = 1 − 1.005⁻ⁿ
1.005⁻ⁿ = 0.1667
n = ln(0.1667) / (−ln(1.005)) = 1.7918 / 0.004988 ≈ 359 months (about 30 years)

(b) $2 500/month withdrawals:
Monthly interest on $500 000 = 500 000 × 0.005 = $2 500
The interest earned each month exactly equals the withdrawal amount.
The balance never decreases — the fund lasts indefinitely (in theory, forever).

By reducing the withdrawal by just $500/month, the fund goes from lasting 30 years to lasting forever. This illustrates why maintaining an income at or below the interest rate is the key to sustainable retirement funding.