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Annuity

An annuity is a series of equal payments made at regular time intervals, where interest compounds each period. Common examples include monthly mortgage payments, retirement fund contributions, and car loan installments.

An annuity is a finite sequence of equal cash flows occurring at equally spaced intervals over a specified period, subject to a fixed rate of compound interest per period. The two primary calculations associated with annuities are the future value (the total accumulated amount after all payments and interest) and the present value (the equivalent lump sum today that would match the annuity's worth). An ordinary annuity assumes payments occur at the end of each period, while an annuity due assumes payments at the beginning.

Key Formula

FV = P \cdot \frac{(1 + r)^n - 1}{r}$$ $$PV = P \cdot \frac{1 - (1 + r)^{-n}}{r}
Where:
  • FVFV = future value of the annuity
  • PVPV = present value of the annuity
  • PP = payment amount per period
  • rr = interest rate per period (as a decimal)
  • nn = total number of payment periods

Worked Example

Problem: You deposit $200 at the end of each month into a savings account that earns 6% annual interest, compounded monthly. What is the future value of the annuity after 5 years?
Step 1: Identify the variables. The payment is $200 per month, the annual rate is 6%, and the total duration is 5 years.
P=200,r=0.0612=0.005,n=5×12=60P = 200, \quad r = \frac{0.06}{12} = 0.005, \quad n = 5 \times 12 = 60
Step 2: Substitute into the future value formula.
FV=200⋅(1+0.005)60−10.005FV = 200 \cdot \frac{(1 + 0.005)^{60} - 1}{0.005}
Step 3: Calculate (1.005)60(1.005)^{60}. Using a calculator, this is approximately 1.34885.
(1.005)60≈1.34885(1.005)^{60} \approx 1.34885
Step 4: Compute the fraction and multiply by the payment amount.
FV=200⋅1.34885−10.005=200⋅0.348850.005=200⋅69.77=13,954FV = 200 \cdot \frac{1.34885 - 1}{0.005} = 200 \cdot \frac{0.34885}{0.005} = 200 \cdot 69.77 = 13{,}954
Answer: The future value of the annuity is approximately 13,954.Notethatyoucontributed13,954. Note that you contributed12,000 in total (200×60),soabout200 × 60), so about1,954 came from compound interest.

Visualization

Why It Matters

Annuity calculations are at the core of personal finance. Whenever you take out a car loan, pay a mortgage, or contribute to a retirement account, the payments follow an annuity structure. Understanding the formulas helps you compare financial products, figure out how much you need to save each month, or determine the true cost of a loan.

Common Mistakes

Mistake: Using the annual interest rate directly instead of converting to the per-period rate
Correction: If payments are monthly and the annual rate is 6%, you must divide by 12 to get r=0.005r = 0.005 per month. The same applies to nn—convert years to the number of payment periods.
Mistake: Confusing future value and present value formulas
Correction: Use the future value formula when you want to know what your payments will grow to. Use the present value formula when you need today's equivalent of a series of future payments, such as finding the fair price of a loan.

Related Terms