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Financial / Annuities 💵

You understand what these do — what trips you is the calculator. So this module is 20% formula, 80% punching it in correctly. Nail the four rules below and these stop being scary.

The one formula (regular monthly deposits)

"You deposit the same amount every month into an account that earns interest — what's it worth later?" That's an annuity / future value:

FV = PMT · [ (1 + i)ⁿ − 1 ] / i

PMT = the monthly payment · n = number of months · i = the annual rate % ÷ 1200 (that converts a yearly % into a monthly decimal).

⚠ The 4 calculator rules (this is the whole battle)

  1. Convert the rate: i = annual percent ÷ 1200. (1.6% → 1.6 ÷ 1200 = 0.0013333…) Keep ALL the decimals — don't round the rate, or the answer drifts.
  2. Add the 1 BEFORE the power. Type (1 + i) first, then raise it to ^n. This is the step you forget — and it makes everything after it wrong.
  3. Then in order: subtract 1 → divide by i → multiply by PMT. Don't reorder.
  4. Don't round until the very end. Round only the final dollar amount, to the penny.

Which one is it?

"deposit $X each month" … "future value after N months" → regular deposits.
FV = PMT·[(1+i)ⁿ − 1]/i  (i = rate%÷1200)
"invest $P once" … "compounded monthly … after N months/years" → a single lump sum.
A = P·(1 + i)ⁿ

Tell them apart by the verb: deposit every month (annuity) vs. invest once (compound lump sum).

Worked Example 1 — the $400/month account

You deposit $400 each month into an account earning 1.6% annual interest. What is the value after 96 months?

1PMT = 400, n = 96. Rate: i = 1.6 ÷ 1200 = 0.00133333… (keep the decimals)
2Power FIRST, 1 added before it: (1 + 0.00133333)96 = (1.00133333)96 ≈ 1.1364561
3Subtract 1: 1.1364561 − 1 = 0.1364561
4Divide by i: 0.1364561 ÷ 0.00133333 ≈ 102.3421
5Multiply by PMT: 102.3421 × 400 = $40,936.83
⚠ The error that bit you before: forgetting to add the 1 before the exponent (you raised the bare rate to the power and got basically zero). Type 1 + i in parentheses first, then ^96.

Worked Example 2 — the $200/month account

You deposit $200 each month at 1.4% annual interest for 72 months. Future value?

1i = 1.4 ÷ 1200 = 0.00116667. PMT = 200, n = 72.
2(1.00116667)72 ≈ 1.0875758
3− 1 = 0.0875758 → ÷ 0.00116667 ≈ 75.0649 → × 200 = $15,012.97

Now you Faded — fill the blanks

You deposit $1,000 each month at 1.2% annual interest for 24 months. Future value?

  1. i = 1.2 ÷ 1200 = __________
  2. (1 + i)²⁴ = __________ (add the 1 first!)
  3. − 1, ÷ i, × 1000 = __________
Check it ›
i = 0.001. (1.001)²⁴ ≈ 1.0242779. − 1 = 0.0242779 → ÷ 0.001 = 24.2779 → × 1000 = $24,278.03.

Practice set — punch them in carefully

Use the 4 rules. Write your rate conversion first, every time. Check the key to the penny.

  1. You deposit $250 each month at 1.8% annual interest for 48 months. Future value?
  2. You deposit $500 each month at 2.1% annual interest for 120 months. Future value?
  3. You deposit $150 each month at 3% annual interest for 36 months. Future value?
  4. You invest $5,000 once at 4% annual interest, compounded monthly, for 60 months. What is it worth? (careful — which formula?)
Answer key (worked) ›

1.

Annuity. i = 1.8/1200 = 0.0015. 250·[(1.0015)⁴⁸ − 1]/0.0015 = $12,432.90.

2.

Annuity. i = 2.1/1200 = 0.00175. 500·[(1.00175)¹²⁰ − 1]/0.00175 = $66,700.47.

3.

Annuity. i = 3/1200 = 0.0025. 150·[(1.0025)³⁶ − 1]/0.0025 = $5,643.08.

4.

Lump sum ("invest once") → A = P(1+i)ⁿ, NOT the annuity formula. i = 4/1200 = 0.0033333. A = 5000·(1.0033333)⁶⁰ = $6,104.98.
📝 Capture before you leave: put FV = PMT·[(1+i)ⁿ−1]/i on your sheet, and right under it the two reminders that actually save you: "i = rate% ÷ 1200" and "ADD the 1 before the power."
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