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 wantas 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 end of each payment period (ordinary annuity, in arrears, 0) or if payments occur at the 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
\( PV=\dfrac{PMT}{i}\left[1-\dfrac{1}{(1+i)^n}\right](1+iT) \)
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
\( PV=\dfrac{PMT}{i}\left[1-\dfrac{1}{(1+i)^n}\right] \)
otherwise T = 1and the equation reduces to the formula for present value of an annuity due
\( PV=\dfrac{PMT}{i}\left[1-\dfrac{1}{(1+i)^n}\right](1+i) \)
Present Value of a Growing Annuity (g ≠ i)
where g = G/100
\( PV=\dfrac{PMT}{(i-g)}\left[1-\left(\dfrac{1+g}{1+i}\right)^n\right](1+iT) \)
Present Value of a Growing Annuity (g = i)
\( PV=\dfrac{PMTn}{(1+i)}(1+iT) \)
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.
\( PV=\dfrac{PMT}{i}(1+iT) \)
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
\( PV=\dfrac{PMT}{(i-g)}(1+iT) \)
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
\( PV=\dfrac{PMTn}{(1+i)}(1+iT)\rightarrow\infty \)
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
\( PV=\dfrac{PMT}{(e^r-1)}\left[1-\dfrac{1}{e^{rt}}\right](1+(e^r-1)T) \)
If type is ordinary annuity, T = 0 and we get the present value of an ordinary annuity with continuous compounding
\( PV=\dfrac{PMT}{(e^r-1)}\left[1-\dfrac{1}{e^{rt}}\right] \)
otherwise type is annuity due, T = 1 and we get the present value of an annuity due with continuous compounding
\( PV=\dfrac{PMT}{(e^r-1)}\left[1-\dfrac{1}{e^{rt}}\right]e^r \)
Present Value of a Growing Annuity (g ≠ i) and Continuous Compounding (m → ∞)
\( PV=\dfrac{PMT}{e^{r}-(1+g)}\left[1-\dfrac{(1+g)^{n}}{e^{nr}}\right](1+(e^{r}-1)T) \)
Present Value of a Growing Annuity (g = i) and Continuous Compounding (m → ∞)
\( PV=\dfrac{PMTn}{e^{r}}(1+(e^r-1)T) \)
Present Value of a Perpetuity (t → ∞) and Continuous Compounding (m → ∞)
\( PV=\dfrac{PMT}{(e^r-1)}(1+(e^r-1)T) \)
Present Value of a Growing Perpetuity (g < i) (t → ∞) and Continuous Compounding (m → ∞)
\( PV=\dfrac{PMT}{e^{r}-(1+g)}(1+(e^{r}-1)T) \)
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 er-1 we end up with the following formula but since n → ∞for a perpetuity this will also always go to infinity.
\( PV=\dfrac{PMTn}{e^{r}}(1+(e^r-1)T)\rightarrow \infty \)
