Financial Mathematics
Future and present value annuities, sinking funds
Future Value, Present Value and Annuities
An annuity is a series of equal payments at regular intervals.
• Future value annuity: F = x[(1+i)ⁿ − 1]/i (saving towards a goal)
• Present value annuity: P = x[1 − (1+i)⁻ⁿ]/i (loan repayment)
• Sinking fund: saving to replace an asset
Example
Future Value Annuity (Savings)
Save R500 per month for 5 years at 9% p.a. compounded monthly.
i = 0.09/12 = 0.0075, n = 60
F = 500[(1.0075)⁶⁰ − 1]/0.0075
= 500[1.5657 − 1]/0.0075
= 500 × 75.42
= R37 711.22
Example
Present Value Annuity (Loan)
A home loan of R800 000 at 11.5% p.a. compounded monthly over 20 years.
i = 0.115/12 ≈ 0.009583, n = 240
800 000 = x[1 − (1.009583)⁻²⁴⁰]/0.009583
800 000 = x × 93.06
x = R8 597.02 per month
Note
Remember
Always match the interest period and payment period. Convert annual rate to monthly: divide by 12. Convert years to months: multiply by 12. Use the timeline method to identify which formula applies. Sinking fund = future value annuity (saving to replace).
Key Vocabulary
AnnuityA series of equal payments at regular intervals
Future valueThe total value of an investment at the end of the period
Present valueThe current value of a future sum or payment stream
Sinking fundRegular savings to accumulate a specific future amount
Compounded monthlyInterest calculated and added every month
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Annuity
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