Mathematics Advanced • Year 12 • Module 7 • Lesson 10
Using Annuity Tables and Technology
Apply annuity tables, interpolation, and spreadsheet formulas to realistic financial decisions.
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 |
Problem 1 — Lump-sum appraisal of a rent stream
A real-estate analyst values a 20-year commercial lease that pays $40,000 per year (ordinary annuity). The investor's required return is 6% p.a.
Set up: What are we solving for?
(i) Use the table to find the PV. 2 marks
(ii) Verify the table value of 11.470 by direct formula evaluation (4 dp). 2 marks
(iii) If the analyst's required return rises to 7%, re-evaluate PV and state the dollar change. 1 mark
Problem 2 — Mortgage repayment from the table
A homebuyer needs the PV factor at r = 4.6% (annual) and n = 25 to value a mortgage stream.
Set up: What are we solving for?
(i) Use linear interpolation between r = 4% and r = 5% (n = 25) to estimate the factor. 2 marks
(ii) Verify against the formula and state the absolute error. 2 marks
(iii) Use your interpolated factor to find the PV of $18,000/year for 25 years at 4.6%. 2 marks
Problem 3 — Sustainable withdrawal from a $250,000 nest egg
Tahir retires with $250,000. He plans 15 years of equal end-of-year withdrawals from an account earning 5% p.a.
Set up: What are we solving for?
(i) Use the table factor (n = 15, r = 5%) to compute the sustainable annual withdrawal a (to the nearest dollar). 2 marks
(ii) If the return drops to 3%, recompute a using the table. State the percentage cut to Tahir's annual income. 2 marks
(iii) Write the Excel formula that would compute the same withdrawal at 5% (use =PMT(rate, nper, −PV) or rearrange). State what the negative sign means. 2 marks
Stuck on (iii)? Negative PV in Excel finance functions denotes a cash outflow at time 0.Problem 4 — Technology comparison
A small business needs to check several annuity scenarios. For r = 8% and n = 25:
Set up: What are we solving for?
(i) Use the table to find PV and FV for a = $1,000. 2 marks
(ii) Write down the Excel formulas: =PV(0.08, 25, −1000) and =FV(0.08, 25, −1000). What numerical answers would Excel give, and why might they differ slightly from your table answers? 2 marks
(iii) Explain in 1-2 sentences why the HSC marker still requires formula working even when Excel can produce the same answer. 2 marks
Problem 5 — Why a tight column gap matters
Two students estimate the PV factor at r = 4.5%, n = 20. Student A interpolates between 4% and 5% (table values 13.590 and 12.462). Student B interpolates between 3% and 6% (table values 14.877 and 11.470).
Set up: What are we solving for?
(i) Find Student A's interpolated factor. 1 mark
(ii) Find Student B's interpolated factor. 1 mark
(iii) Compute the exact factor from the formula. State each student's absolute error. 3 marks
(iv) In one sentence, advise future students how to choose the column gap when interpolating from a table. 1 mark
How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Lump-sum appraisal of a lease
Set up. We are converting a rent stream into a single price an investor would pay today.
(i) Factor (n = 20, r = 6%) = 11.470. PV = 40,000 × 11.470 = $458,800.
(ii) Formula: [1 − (1.06)⁻²⁰]/0.06 = (1 − 0.31180)/0.06 = 0.68820/0.06 = 11.4699 ≈ 11.470 ✓.
(iii) At r = 7%: factor = 10.594. PV = 40,000 × 10.594 = $423,760. Dollar change = −$35,040 (PV falls by about 7.6%).
Problem 2 — Mortgage interpolation
Set up. We are estimating a factor not in the table, then verifying with the formula and applying it to a payment stream.
(i) y = 15.622 + (4.6 − 4)/(5 − 4) × (14.094 − 15.622) = 15.622 + 0.6 × (−1.528) = 15.622 − 0.917 = 14.705.
(ii) Formula: [1 − (1.046)⁻²⁵]/0.046 = (1 − 0.32669)/0.046 = 0.67331/0.046 = 14.6373. Absolute error ≈ 0.068 (≈ 0.46%).
(iii) PV = 18,000 × 14.705 ≈ $264,690. (Using the exact factor: 18,000 × 14.637 = $263,472.)
Problem 3 — Sustainable withdrawal
Set up. We are solving 250,000 = a × factor for a, then re-running at a different rate.
(i) Factor (n = 15, r = 5%) = 10.380. a = 250,000 / 10.380 ≈ $24,084/year.
(ii) Factor (n = 15, r = 3%) = 11.938. a = 250,000 / 11.938 ≈ $20,941/year — a cut of about 13% in annual income.
(iii) Excel: =PMT(0.05, 15, −250000) returns ≈ $24,083.89. The negative sign in front of 250,000 marks the lump sum as a cash outflow at time 0 (paid into the account); Excel then returns the positive cash inflow per period (the withdrawal).
Problem 4 — Technology comparison
Set up. We are cross-checking table-derived values against Excel finance functions.
(i) PV factor (n = 25, r = 8%) = 10.675. PV = 1,000 × 10.675 = $10,675. FV factor = [(1.08)²⁵ − 1]/0.08 = (6.8485 − 1)/0.08 = 73.106. FV = 1,000 × 73.106 = $73,106.
(ii) Excel =PV(0.08, 25, −1000) returns about $10,674.78; =FV(0.08, 25, −1000) returns about $73,105.94. Tiny differences come from the table being rounded to 3 dp.
(iii) The HSC tests the underlying mathematical reasoning, not the ability to call a function. Showing the formula substitution and algebraic steps demonstrates that the student understands what the calculator is doing, which is what the marker is paying for.
Problem 5 — Why a tight column gap matters
Set up. We are comparing interpolation accuracy when the two anchor columns are close vs far.
(i) Student A: y = 13.590 + 0.5 × (12.462 − 13.590) = 13.590 − 0.564 = 13.026.
(ii) Student B: y = 14.877 + (4.5 − 3)/(6 − 3) × (11.470 − 14.877) = 14.877 + 0.5 × (−3.407) = 14.877 − 1.704 = 13.173.
(iii) Exact: [1 − (1.045)⁻²⁰]/0.045 ≈ 13.008. Student A error ≈ 0.018 (0.14%). Student B error ≈ 0.165 (1.27%) — almost 10× worse.
(iv) Always interpolate between the two adjacent table columns (the smallest gap that brackets your target rate) to minimise the curvature error.