Annuity Calculator

The difference comes down to timing — specifically, when each payment occurs within the period:

• Ordinary annuity (annuity in arrears) — Payments occur at the end of each period. This is the most common type. Mortgage payments, bond coupons, and most loan payments are ordinary annuities.
• Annuity due (annuity in advance) — Payments occur at the beginning of each period. Rent payments and insurance premiums are common examples — you pay at the start of the month or year.

Why does this matter financially? Because an annuity due always has a higher present value and future value than an otherwise identical ordinary annuity. The reason is simple: each payment in an annuity due is received (or made) one period earlier, which means each payment has one extra period to earn interest (for future value) or one fewer period of discounting (for present value).

The mathematical relationship between the two is elegant:

• FV (annuity due) = FV (ordinary) × (1 + r)
• PV (annuity due) = PV (ordinary) × (1 + r)

This means the annuity due values are always exactly (1 + r) times larger. At a 6% rate, an annuity due is worth 6% more than the corresponding ordinary annuity. This difference compounds significantly over long time horizons, which is why getting the timing right matters in financial modeling.

The present value of an ordinary annuity formula calculates what a stream of future payments is worth right now:

PV = PMT × [(1 − (1 + r)⁻ⁿ) / r]

Where:

• PMT = Payment amount per period
• r = Interest rate per period (annual rate divided by compounding frequency)
• n = Total number of periods (years times compounding frequency)

Example: Suppose you will receive $1,000 per year for 10 years, and the discount rate is 8%. The present value is:

PV = $1,000 × [(1 − (1.08)⁻¹⁰) / 0.08] = $1,000 × 6.7101 = $6,710.08

This means that receiving $1,000 per year for 10 years is equivalent to receiving approximately $6,710 as a lump sum today. You'd be indifferent between the two if your required return is 8%.

For an annuity due, multiply the result by (1 + r) because each payment arrives one period earlier. In the example above, the annuity due PV would be $6,710.08 × 1.08 = $7,246.89.

The present value formula is the foundation of bond pricing (coupon payments are an annuity), loan calculations (monthly payments are an annuity), and retirement planning (how much do you need today to fund annual withdrawals).

The future value of an ordinary annuity tells you how much a series of regular payments will grow to after compounding:

FV = PMT × [((1 + r)ⁿ − 1) / r]

This formula answers the classic retirement question: “If I save $500 per month for 30 years at 7% annual return, how much will I have?”

Example with monthly compounding: $500/month, 7% annual rate (0.5833% monthly), 360 months:

FV = $500 × [((1.005833)³⁶⁰ − 1) / 0.005833] = $566,764

You contributed $180,000 in total ($500 × 360), so $386,764 — nearly 68% of the final balance — came purely from compound interest. This is the power of compounding.

Key insight: The future value formula shows why starting early matters so much. The first payments have the most time to compound. A payment made in year 1 compounds for 29 years, while a payment in year 30 barely compounds at all. This is why financial advisors stress that time in the market beats timing the market.

For an annuity due, the future value is FV (ordinary) × (1 + r), because each payment has one extra period to grow. In the example above, an annuity due would produce about $570,070 — an extra $3,306 just from paying at the start of each month instead of the end.

Compounding frequency determines how often interest is calculated and added to the balance within each year. The more frequently interest compounds, the more total interest you earn (or owe), because each compounding event creates interest on previously earned interest.

Common compounding frequencies and their effect:

• Annual (1x/year) — Interest is calculated once per year. Simplest to calculate, lowest effective yield.
• Quarterly (4x/year) — Interest compounds every 3 months. Common for corporate bonds and some savings accounts.
• Monthly (12x/year) — Interest compounds every month. Standard for mortgages, auto loans, and most consumer lending.
• Daily (365x/year) — Interest compounds every day. Used by some savings accounts and credit cards.

Practical example: A 6% annual rate with $1,000 annual payments over 20 years:

• Annual compounding: FV = $36,786
• Monthly compounding ($83.33/month): FV = $38,929

The monthly version produces about 5.8% more wealth from the same total contributions, purely because interest compounds more frequently and payments are made more often (giving each payment slightly more time to grow on average).

When modeling annuities, always match the payment frequency to the compounding frequency. If you make monthly payments, use a monthly rate (annual rate / 12) and monthly periods (years × 12). Mixing annual rates with monthly payments without adjusting is a common and costly mistake.

Annuity math is the backbone of DCF valuation and many other corporate finance applications. Here's how the connection works:

1. Terminal value as a perpetuity (infinite annuity):

In a DCF model, the terminal value is often calculated using the Gordon Growth Model: TV = FCF × (1 + g) / (r − g). This is actually the formula for a growing perpetuity — an annuity that never stops and grows at rate g. It's the limiting case of the annuity present value formula as n approaches infinity.

2. Loan and lease modeling:

Companies carry debt with fixed payments — these are annuities. When you calculate the present value of a company's debt obligations for enterprise value, you're using the annuity PV formula.

3. Capital budgeting:

When a company evaluates a project that generates equal annual cash flows, the NPV calculation simplifies to an annuity PV minus the initial investment. Finance teams use annuity factors daily to assess project viability.

4. Bond valuation:

A bond's price is the PV of its coupon payments (an annuity) plus the PV of its face value (a lump sum). The annuity component typically represents 30-70% of a bond's total value, depending on the coupon rate and time to maturity.

Mastering annuity calculations gives you the mathematical intuition to understand and build more complex financial models. Every DCF, LBO, and bond pricing model is built on these formulas.

The total interest earned on an annuity is the difference between the future value and the total amount of payments you contributed. It represents the cumulative return generated by compounding over time.

Total Interest = Future Value − Total Payments

Why this number is more important than most people realize:

• Early years — Interest earned is relatively small because the balance is still low. Most of each period's growth comes from your new payment, not from compounding.
• Later years — The interest component dominates. In a 30-year annuity at 7%, the interest earned in the final year alone can exceed several years of contributions combined.
• The crossover point — There's a moment in every annuity where cumulative interest earned surpasses cumulative contributions. For a 7% annual return, this crossover typically happens around year 15-18.

For lenders (the flip side): If you're the one paying an annuity (like a mortgage), the total interest represents your cost of borrowing. A 30-year mortgage at 7% on a $300,000 loan generates roughly $419,000 in total interest — meaning you pay back more than double the original loan amount. This is why accelerating payments (or refinancing to a lower rate) can save enormous amounts of money.

The year-by-year growth table in this calculator makes the compounding curve visible. Watch how the balance growth accelerates in later years — that's compound interest working exponentially, not linearly.