Mathematics Advanced • Year 12 • Module 7 • Lesson 10
Using Annuity Tables and Technology
Build fluency in reading annuity factor tables, interpolating between table entries, and verifying against the closed-form formula.
Reference table (use throughout this worksheet) — PV factor [1 − (1+r)⁻ⁿ] / r:
| n \ r | 3% | 4% | 5% | 6% | 7% | 8% |
|---|---|---|---|---|---|---|
| 5 | 4.580 | 4.452 | 4.329 | 4.212 | 4.100 | 3.993 |
| 10 | 8.530 | 8.111 | 7.722 | 7.360 | 7.024 | 6.710 |
| 15 | 11.938 | 11.118 | 10.380 | 9.712 | 9.108 | 8.559 |
| 20 | 14.877 | 13.590 | 12.462 | 11.470 | 10.594 | 9.818 |
| 25 | 17.413 | 15.622 | 14.094 | 12.783 | 11.654 | 10.675 |
1. Quick recall
Answer each question in the space provided. 1 mark each
Q1.1 Write the PV factor formula: [1 − (1 + r)^____] / ____
Q1.2 In plain English, the PV factor 7.722 (r = 5%, n = 10) means: "If I receive $1 per period for ____ periods at ____% per period, the present value is $________ today."
Q1.3 Complete the linear interpolation formula:
y = y₁ + (x − x₁) / (____ − ____) × ( ____ − y₁ )
2. Worked example — interpolate at r = 4.8%, n = 15
Follow each line. Reasons appear in italics on the right.
Problem. Use linear interpolation on the table to estimate the PV factor for r = 4.8%, n = 15. Then find PV for a = $2,000.
Step 1 — Identify the two nearest table entries.
At r = 4%, factor = 11.118 ; at r = 5%, factor = 10.380.
Reason: 4.8% sits between 4% and 5%.
Step 2 — Apply linear interpolation.
y = 11.118 + (4.8 − 4) / (5 − 4) × (10.380 − 11.118)
= 11.118 + 0.8 × (−0.738) = 11.118 − 0.590 = 10.528.
Step 3 — Multiply by the payment.
PV = 2,000 × 10.528 = $21,056.
Step 4 — Verify against the formula.
Exact: 2,000 × [1 − (1.048)⁻¹⁵] / 0.048 = 2,000 × 10.5236 = $21,047.20.
Conclusion. Interpolated PV ≈ $21,056 (error: $8.80 or 0.04%).
3. Faded example — fill in the missing steps
Estimate the PV factor at r = 5.5%, n = 10, and compute PV for a = $400. 4 marks
Step 1 — Find the two nearest table values.
At r = ____%, factor = ____________ ; at r = ____%, factor = ____________.
Step 2 — Interpolate.
y = ____________ + (____ − ____) / (____ − ____) × (____________ − ____________)
= ____________ + ____________ × (____________) = ____________
Step 3 — Multiply by a.
PV = 400 × ____________ = $____________
Conclusion. Interpolated factor ≈ ____________ ; PV ≈ $____________.
4. Graduated practice
Show every line of working. Round factors to 3 dp and money to the nearest cent.
Foundation — direct table lookup (4 questions)
| Q | Setup | PV (= a × factor) |
|---|---|---|
| 4.1 1 | a = $1,000, r = 5%, n = 10 | |
| 4.2 1 | a = $1,500, r = 6%, n = 15 (lesson Activity 1) | |
| 4.3 1 | a = $800, r = 4%, n = 20 | |
| 4.4 1 | a = $2,500, r = 7%, n = 25 |
Standard — interpolation (6 questions)
Show your interpolation working before computing PV.
4.5 Estimate the PV factor for r = 4.5%, n = 20. 2 marks
4.6 Estimate the PV factor for r = 6.5%, n = 10, and PV for a = $500. 2 marks
4.7 Estimate the PV factor for r = 5.5%, n = 15, and PV for a = $1,200. 2 marks
4.8 Estimate the PV factor for r = 7.5%, n = 25, and PV for a = $3,000. 2 marks
4.9 Use the FV factor formula [(1 + r)ⁿ − 1] / r. Find the FV factor at r = 7.5%, n = 10 by linear interpolation between r = 7% (factor 13.816) and r = 8% (factor 14.487). Compute FV for a = $500. 2 marks
4.10 For r = 5%, n = 10, compute the exact PV factor from the formula and check it matches the table entry 7.722 to 3 decimal places. 2 marks
Extension — error analysis (2 questions)
4.11 Interpolate the PV factor at r = 4.5%, n = 20 (you did this in 4.5) and compare with the exact value from the formula. State the absolute and percentage errors. 3 marks
4.12 Explain in 1-2 sentences why interpolation error grows when the gap between table columns is large (e.g. 5% vs 10%) but stays small when the gap is 1%. 2 marks
5. Self-check the easy 3
Tick once you've checked your method works.
How did this worksheet feel?
What I'll revisit before next class:
Q1.1 — PV factor formula
[1 − (1 + r)^−n] / r.
Q1.2 — Meaning of factor 7.722
"If I receive $1 per period for 10 periods at 5% per period, the present value is $7.722 today."
Q1.3 — Interpolation formula
y = y₁ + (x − x₁) / (x₂ − x₁) × (y₂ − y₁).
Q3 — Faded example: r = 5.5%, n = 10, a = $400
At r = 5%, factor = 7.722; at r = 6%, factor = 7.360. y = 7.722 + (5.5 − 5)/(6 − 5) × (7.360 − 7.722) = 7.722 + 0.5 × (−0.362) = 7.722 − 0.181 = 7.541. PV = 400 × 7.541 = $3,016.40. Exact: 400 × 7.538 = $3,015.20 — interpolation error ≈ $1.20 (0.04%).
Q4.1
PV = 1,000 × 7.722 = $7,722.
Q4.2
PV = 1,500 × 9.712 = $14,568.
Q4.3
PV = 800 × 13.590 = $10,872.
Q4.4
PV = 2,500 × 11.654 = $29,135.
Q4.5 — r = 4.5%, n = 20
y = 13.590 + (4.5 − 4)/(5 − 4) × (12.462 − 13.590) = 13.590 + 0.5 × (−1.128) = 13.590 − 0.564 = 13.026.
Q4.6 — r = 6.5%, n = 10
y = 7.360 + 0.5 × (7.024 − 7.360) = 7.360 − 0.168 = 7.192. PV = 500 × 7.192 = $3,596.00.
Q4.7 — r = 5.5%, n = 15
y = 10.380 + 0.5 × (9.712 − 10.380) = 10.380 − 0.334 = 10.046. PV = 1,200 × 10.046 = $12,055.20.
Q4.8 — r = 7.5%, n = 25
y = 11.654 + 0.5 × (10.675 − 11.654) = 11.654 − 0.490 = 11.164. PV = 3,000 × 11.164 = $33,492.
Q4.9 — FV interpolation at r = 7.5%, n = 10
y = 13.816 + 0.5 × (14.487 − 13.816) = 13.816 + 0.336 = 14.152. FV = 500 × 14.152 = $7,076.
Q4.10 — Verify table entry 7.722
Exact: [1 − (1.05)⁻¹⁰] / 0.05 = (1 − 0.6139) / 0.05 = 0.3861 / 0.05 = 7.7217 ≈ 7.722 ✓.
Q4.11 — Error analysis
Interpolated (Q4.5): 13.026. Exact: [1 − (1.045)⁻²⁰]/0.045 = (1 − 0.4146)/0.045 = 13.008. Absolute error ≈ 0.018; percentage error ≈ 0.018/13.008 = 0.14%. Very small for a 1% column gap.
Q4.12 — Why error grows with gap
The PV factor is a curved (concave-up) function of r, but linear interpolation assumes a straight line between table entries. For tight column gaps the chord and the curve are nearly the same; for wide gaps the chord under-estimates (or over-estimates) the curve more strongly, and the error grows roughly with the square of the gap size.