Annuities
Also known as: annuity, regular payments, periodic payments
Grade 9-12
View on concept mapA series of equal payments made at regular intervals over a fixed period of time. Annuities model mortgages, car loans, retirement savings, pension payouts, and any financial plan involving regular payments.
Definition
A series of equal payments made at regular intervals over a fixed period of time. The future value and present value formulas calculate the total worth of these payment streams.
💡 Intuition
Imagine depositing \$100 every month into a savings account. Each deposit earns interest for a different amount of time—the first deposit earns interest for the full term, the last deposit barely earns any. An annuity formula adds up all these differently-growing deposits in one clean expression, instead of computing compound interest on each payment separately.
🎯 Core Idea
An annuity is a geometric series of compound interest calculations. Each payment grows at a different rate depending on when it was made. The formulas are closed-form sums of these geometric series.
Example
FV = 200 \cdot \frac{(1.005)^{240} - 1}{0.005} = 200 \cdot 462.04 = \$92{,}408
You deposited \48,000 total but earned \44,408 in interest.
Formula
FV = PMT \cdot \frac{(1 + i)^n - 1}{i}
Present value of ordinary annuity:
PV = PMT \cdot \frac{1 - (1 + i)^{-n}}{i}
where PMT = payment per period, i = interest rate per period, n = total number of payments.
Notation
PMT = payment amount per period, i = periodic interest rate (annual rate \div periods per year), n = total number of periods, FV = future value, PV = present value.
🌟 Why It Matters
Annuities model mortgages, car loans, retirement savings, pension payouts, and any financial plan involving regular payments. Knowing the formulas lets you calculate monthly payments, total interest paid, or how much to save for retirement.
💭 Hint When Stuck
Convert everything to the same time period first: divide the annual rate by 12 for monthly, and multiply years by 12 for total payments.
Formal View
Related Concepts
🚧 Common Stuck Point
The interest rate i in the formula is the PERIODIC rate, not the annual rate. For monthly payments at 6% annual, use i = 0.06/12 = 0.005, and n is the total number of months, not years.
⚠️ Common Mistakes
- Using the annual interest rate instead of the periodic rate: for monthly payments at 6% annual, i = 0.005 (not 0.06), and n = 240 months (not 20 years).
- Confusing ordinary annuity (payments at END of period) with annuity due (payments at BEGINNING of period). An annuity due has one extra compounding period, so multiply the ordinary annuity result by (1 + i).
- Forgetting that loan payments (like mortgages) use the present value formula—you're solving for PMT given PV, not the future value formula.
Go Deeper
Worked Examples
Step-by-step solved problems
Practice Problems
Test your understanding
Formula Explained
Notation, derivation, and common mistakes
FV = PMT \cdot \frac{(1 + i)^n - 1}{i}
Present value of ordinary annuity:
PV = PMT \cdot \frac{1 - (1 + i)^{-n}}{i}
where PMT = payment per period, i = interest rate per period, n = total number of payments.
Frequently Asked Questions
What is Annuities in Math?
A series of equal payments made at regular intervals over a fixed period of time. The future value and present value formulas calculate the total worth of these payment streams.
Why is Annuities important?
Annuities model mortgages, car loans, retirement savings, pension payouts, and any financial plan involving regular payments. Knowing the formulas lets you calculate monthly payments, total interest paid, or how much to save for retirement.
What do students usually get wrong about Annuities?
The interest rate i in the formula is the PERIODIC rate, not the annual rate. For monthly payments at 6% annual, use i = 0.06/12 = 0.005, and n is the total number of months, not years.
What should I learn before Annuities?
Before studying Annuities, you should understand: compound interest.
Prerequisites
Next Steps
Cross-Subject Connections
How Annuities Connects to Other Ideas
To understand annuities, you should first be comfortable with compound interest. Once you have a solid grasp of annuities, you can move on to present future value.