# Present Value of Annuity Calculator

## Calculator Use

Use this calculator to find the present value of annuities due, ordinary regular annuities, growing annuities and perpetuities.

- Period
- commonly a period will be a year but it can be any time interval you want as long as all inputs are consistent.
- Number of Periods (t)
- number of periods or years
- Perpetuity
- for a perpetual annuity t approaches infinity. Enter p, P, perpetuity or Perpetuity for t
- Interest Rate (R)
- is the annual nominal interest rate or "stated rate" per period in percent. r = R/100, the interest rate in decimal
- Compounding (m)
- is the number of times compounding occurs per period. If a period is a year then annually=1, quarterly=4, monthly=12, daily = 365, etc.
- Continuous Compounding
- is when the frequency of compounding (m) is increased up to infinity. Enter c, C, continuous or Continuous for m.
- Payment Amount (PMT)
- The amount of the annuity payment each period
- Growth Rate (G)
- If this is a growing annuity, enter the growth rate per period of payments in percentage here. g = G/100
- Payments per Period (Payment Frequency (q))
- How often will payments be made during each period? If a period is a year then annually=1, quarterly=4, monthly=12, daily = 365, etc.
- Payments at Period (Type)
- Choose if payments occur at the
or if payments occur at the*end of each payment period (ordinary annuity, in arrears, 0)**beginning of each payment period (annuity due, in advance, 1)* - Present Value (PV)
- the present value of any future value lump sum and future cash flows (payments)

## Present Value Annuity Formulas:

You can find derivations of present value formulas with our present value calculator.

### Present Value of an Annuity

where r = R/100, n = mt where n is the total number of compounding intervals, t is the time or number of periods, and m is the compounding frequency per period t, i = r/m where i is the rate per compounding interval n and r is the rate per time unit t. If compounding and payment frequencies do not coincide, r is converted to an equivalent rate to coincide with payments then n and i are recalculated in terms of payment frequency, q.

If type is ordinary, T = 0 and the equation reduces to the formula for
**present value of an ordinary annuity**

otherwise T = 1and the equation reduces to the formula for
**present value of an annuity due**

### Present Value of a Growing Annuity (g ≠ i)

where g = G/100

### Present Value of a Growing Annuity (g = i)

### Present Value of a Perpetuity (t → ∞ and n = mt)

When t approaches infinity, t → ∞, the number of payments approach infinity and we have a perpetual annuity with an upper limit for the present value. You can demonstrate this with the calculator by increasing t until you are convinced a limit of PV is essentially reached. Then enter P for t to see the calculation result of the actual perpetuity formulas.

### Present Value of a Growing Perpetuity (g < i) (t → ∞ and n = mt → ∞)

Likewise for a growing perpetuity, where we must have g<i, since (1 + i)^{n} grows faster than (1 + g)^{n}, that term goes to 0 and it reduces to

### Present Value of a Growing Perpetuity (g = i) (t → ∞ and n = mt → ∞)

Since n also goes to infinity (n → ∞) as t goes to infinity (t → ∞), we see that
**Present Value with Growing Annuity (g = i)** also goes to infinity

## Continuous Compounding (m ⇒ ∞)

Again, you can find these derivations with our present value formulas and our present value calculator.

### Present Value of an Annuity with Continuous Compounding

If type is ordinary annuity, T = 0 and we get the
**present value of an ordinary annuity with continuous compounding**

otherwise type is annuity due, T = 1 and we get the** present value of an annuity due with continuous compounding**

### Present Value of a Growing Annuity (g ≠ i) and Continuous Compounding (m → ∞)

### Present Value of a Growing Annuity (g = i) and Continuous Compounding (m → ∞)

### Present Value of a Perpetuity (t → ∞) and Continuous Compounding (m → ∞)

### Present Value of a Growing Perpetuity (g < i) (t → ∞) and Continuous Compounding (m → ∞)

### Present Value of a Growing Perpetuity (g = i) (t → ∞) and Continuous Compounding (m → ∞)

From our equation for
**Present Value of a Growing Perpetuity (g = i)** replacing i with e^{r}-1 we end up with the following formula but since n → ∞ for a perpetuity this will also always go to infinity.