Mathematics Advanced • Year 12 • Module 7 • Lesson 10
Using Annuity Tables and Technology
Practise HSC-style writing on table lookup, interpolation, and the gap between technology output and full-marks working.
Reference PV table (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. Short-answer questions
1.1 Using the table, find the present value of an annuity paying $750 per year for 10 years at 6% per annum. 2 marks Band 3
1.2 The PV factor at r = 4%, n = 15 is 11.118, and at r = 5%, n = 15 is 10.380. Use linear interpolation to estimate the factor at r = 4.3%, n = 15. Hence find the PV of an annuity of $2,500 per year for 15 years at 4.3% p.a. 3 marks Band 3-4
1.3 The Excel formula =PV(0.05, 20, −5000) returns $62,311.05.
(a) Show, by direct formula evaluation, how this figure is obtained.
(b) Use the supplied table to confirm the same answer to within 3 decimal places of the factor. 4 marks Band 4
2. Extended response
2.1 A financial planner is comparing two methods of computing the PV of an ordinary annuity of $a per year at rate r per year for n years.
Method T (table): Look up the factor for r and n; if r is not in the table, interpolate linearly between the two nearest columns.
Method F (formula): Evaluate [1 − (1 + r)⁻ⁿ] / r directly.
(a) Use Method F to derive the exact PV factor at r = 5.5%, n = 20 (4 dp).
(b) Use Method T (interpolating between 5% and 6%) to estimate the same factor. Calculate the absolute and percentage error of the estimate.
(c) Explain mathematically (in 2-3 sentences) why Method T systematically over- or under-estimates the true factor, referring to the second derivative or concavity of the function f(r) = [1 − (1 + r)⁻ⁿ] / r.
(d) The HSC examiner asks the student to "find the PV using the formula." Discuss in 1-2 sentences why the marker would only award marks for Method F working, even when Method T produces a numerically close answer. 8 marks Band 5-6
Explicit marking criteria
Part (a) — 2 marks
• 1 mark — correct substitution r = 0.055, n = 20.
• 1 mark — exact factor rounded to 4 dp.
Part (b) — 2 marks
• 1 mark — correct interpolation (uses 5% factor 12.462 and 6% factor 11.470).
• 1 mark — correct absolute and percentage error.
Part (c) — 2 marks
• 1 mark — identifies concavity/curvature (states f is convex/concave).
• 1 mark — states the consequence for interpolation direction (over/under).
Part (d) — 2 marks
• 1 mark — argues that "by the formula" mandates explicit substitution.
• 1 mark — references the difference between procedural understanding and a numerical answer.
Your response:
Stuck on (c)? The PV-factor function in r is convex on r > 0 — the linear chord lies above the curve, so interpolation over-estimates the factor.How did this worksheet feel?
What I'll revisit before next class:
1.1 — $750/yr, 10 yrs, 6% (2 marks)
Sample response. Factor at r = 6%, n = 10 from table = 7.360. PV = 750 × 7.360 = $5,520.00.
Marking notes. 1 mark — correct factor lookup; 1 mark — correct PV. Common error: multiplying by n (= 10) instead of by the factor.
1.2 — Interpolation at r = 4.3% (3 marks)
Sample response. y = 11.118 + (4.3 − 4)/(5 − 4) × (10.380 − 11.118) = 11.118 + 0.3 × (−0.738) = 11.118 − 0.221 = 10.897. PV = 2,500 × 10.897 = $27,242.50.
Marking notes. 1 mark — correct interpolation setup with the right two anchors. 1 mark — correct factor. 1 mark — correct PV.
1.3 — Verifying Excel PV (4 marks)
(a) Sample response. r = 0.05; n = 20. PV = 5,000 × [1 − (1.05)⁻²⁰]/0.05 = 5,000 × (1 − 0.37689)/0.05 = 5,000 × 12.4622 = $62,311.05 ✓.
(b) Sample response. Table factor at r = 5%, n = 20 = 12.462. PV = 5,000 × 12.462 = $62,310.00 — agrees with Excel to within $1 (table is rounded to 3 dp).
Marking notes. (a) 1 mark — correct substitution; 1 mark — correct numerical result. (b) 1 mark — correct factor; 1 mark — agreement and explicit comment on rounding.
2.1 — Extended response (8 marks): sample Band-6 response with annotations
Sample Band-6 response.
Part (a). Using the formula with r = 0.055, n = 20:
factor = [1 − (1.055)⁻²⁰] / 0.055 = (1 − 0.34273) / 0.055 = 0.65727 / 0.055 = 11.9504 (4 dp). [1 mark — substitution; 1 mark — value.]
Part (b). Interpolating between the table columns at 5% (12.462) and 6% (11.470):
y = 12.462 + (5.5 − 5)/(6 − 5) × (11.470 − 12.462) = 12.462 + 0.5 × (−0.992) = 12.462 − 0.496 = 11.966. [1 mark — correct setup.]
Absolute error = 11.966 − 11.9504 = 0.0156. Percentage error = 0.0156 / 11.9504 = 0.13%. [1 mark — errors.]
Part (c). The PV-factor function f(r) = [1 − (1 + r)⁻ⁿ] / r is convex in r > 0 (its second derivative is positive). When we replace f by a straight line (a chord) between two table columns, the chord lies above the curve, so linear interpolation systematically over-estimates the true factor. The size of the over-estimate scales with the curvature and roughly with the square of the column gap. [1 mark — concavity; 1 mark — over-estimate direction.]
Part (d). An HSC question that says "find the PV using the formula" is testing the student's ability to perform the algebraic substitution and simplification, not just to obtain a numerical answer. A table look-up or Excel call produces the same number but bypasses the very mathematical reasoning the question is assessing, so the marker awards marks for the substitution and simplification steps regardless of how close the table answer is. [1 mark — "by the formula" requires explicit steps; 1 mark — links to the difference between procedural and computational understanding.]
Total: 8/8.
Band descriptors for marker.
Band 3: Computes (a) and (b) numerically but with arithmetic slips; (c) and (d) brief or absent. ≈ 2-3 marks.
Band 4: (a) and (b) correct; (c) attempts a convexity argument but does not state direction of error; (d) one weak sentence. ≈ 4-5 marks.
Band 5: (a)–(c) complete; (d) correct but does not explicitly contrast procedural with computational answers. ≈ 6-7 marks.
Band 6: All four parts correct; (c) explicitly invokes convexity or second-derivative argument; (d) discusses what "by the formula" requires and why technology cannot substitute. 8/8.