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Answer:

Present Value (PV) of the Growing Ordinary Annuity

$ 772,083.38

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Calculator Use

This comprehensive present value calculator uses multiple variables in the PV calculation:

  • The future value sum
  • Number of time periods
  • Interest rate
  • Compounding frequency
  • Cash flow payments
  • Growing annuities and perpetuities

You can enter 0 for any variable you'd like to omit when using this calculator. Also see our other present value calculators for additional present value calculations.

Period
Time period. Typcially a period will be a year but it can be any time interval as long as all inputs are in the same time unit.
Future Value (FV)
Future value of a lump sum.
Number of Periods (t)
Number of years or time periods.
Perpetuity
For a perpetual annuity t approaches infinity. For "Number of Periods (t)" enter p or perpetuity.
Interest Rate (R)
The annual nominal interest rate or stated rate per period, as a percentage.
Compounding (m)
The number of times compounding occurs per period. If a period is a year enter:
1 for annual compounding
4 for quarterly compounding
12 for monthly compounding
365 for daily compounding
Continuous Compounding
For frequency of compounding (m) approaches infinity. For "Compounding (m)" enter c or continuous.
Payment Amount (PMT)
The amount of the cash flow annuity payment each period.
Growth Rate (G)
If this is a growing annuity, enter the growth rate per period of payments in percentage form.
Payments per Period (Payment Frequency, q)
How often payments are made each period. If a period is a year enter:
1 for annual payments
4 for quarterly payments
12 for monthly payments
365 for daily payments
Payments at Period (Type)
Specify whether payments occur at the end of each payment period (ordinary annuity, in arrears) or if payments occur at the beginning of each payment period (annuity due, in advance)
Present Value (PV)
The present value of any future value lump sum plus future cash flows (payments)

Present Value Formula Derivation

The future value (FV) of a present value (PV) sum that accumulates interest at rate i over a single period of time is the present value plus the interest earned on that sum. The mathematical equation is

\( FV=PV+PVi \)

or

\( FV=PV(1+i) \)

For each period into the future the accumulated value increases by an additional factor (1 + i). Therefore, the future value accumulated over, say 3 periods, is given by

\( FV_{3}=PV_{3}(1+i)(1+i)(1+i)=PV_{3}(1+i)^{3} \)

or generally

\( FV_{n}=PV_{n}(1+i)^{n}\tag{1a} \)

Likewise we can solve for PV to get

\( PV_{n}=\dfrac{FV_{n}}{(1+i)^n}\tag{1b} \)

The equations we have are (1a) the future value of a present sum and (1b) the present value of a future sum at a periodic interest rate i where n is the number of periods in the future. Commonly this equation is applied with periods as years but it is less restrictive to think in the broader terms of periods. Dropping the subscripts from (1b) we have:

Present Value of a Future Sum

\( PV=\dfrac{FV}{(1+i)^n}\tag{1} \)

Present Value of an Annuity Formula Derivation

An annuity is a sum of money paid periodically, (at regular intervals). Let's assume we have a series of equal present values that we will call payments (PMT) for n periods at a constant interest rate i.  We can calculate FV of the series of payments 1 through n using formula (1) to add up the individual future values.

\( PV=\dfrac{PMT}{(1+i)^1}+\dfrac{PMT}{(1+i)^2}+\dfrac{PMT}{(1+i)^3}+...+\dfrac{PMT}{(1+i)^n}\tag{2a} \)

multiply both sides of this equation by (1 + i) to get

\( PV(1+i)=PMT+\dfrac{PMT}{(1+i)^1}+\dfrac{PMT}{(1+i)^2}+\dfrac{PMT}{(1+i)^3}+...+\dfrac{PMT}{(1+i)^{n-1}}\tag{2b} \)

subtracting the equation for PV (2a) from the equation for PV(1 + i) (2b) most terms cancel and we are left with

\( PV(1+i)-PV=PMT-\dfrac{PMT}{(1+i)^n} \)

pulling out like terms on both sides

\( PV((1+i)-1)=PMT\left[1-\dfrac{1}{(1+i)^n}\right] \)

cancelling 1's on the left

\( PVi=PMT\left[1-\dfrac{1}{(1+i)^n}\right] \)

and finally, after dividing through by i, the present value of an ordinary annuity, payments made at the end of each period, is

\( PV=\dfrac{PMT}{i}\left[1-\dfrac{1}{(1+i)^n}\right]\tag{2c} \)

For an annuity due, payments made at the beginning of each period instead of the end, therefore payments are now 1 period closer to the PV.  We need to discount each future value payment in the formula by 1 period. This could be written on (1b) as

\( PV_{n}=\dfrac{FV_{n}}{(1+i)^{n-1}} \)

but factoring out the (1 + i)

\( PV_{n}=\dfrac{FV_{n}}{(1+i)^{n}}(1+i) \)

So, multiplying each payment in equation (2a), or the right side of equation (2c), by the factor (1 + i) will give us the equation of PV for an annuity due.  This can be written more generally as

Present Value of an Annuity

\( PV=\dfrac{PMT}{i}\left[1-\dfrac{1}{(1+i)^n}\right](1+iT)\tag{2} \)

where T represents the type. (similar to Excel formulas) If payments are at the end of the period it is an ordinary annuity and we set T = 0.  If payments are at the beginning of the period it is an annuity due an we set T = 1.

Present Value of an Ordinary Annuity

if T = 0, payments are at the end of each period and we have the formula for present value of an ordinary annuity

\( PV=\dfrac{PMT}{i}\left[1-\dfrac{1}{(1+i)^n}\right]\tag{2.1} \)

Present Value of an Annuity Due

if T = 1, payments are at the beginning of each period and we have the formula for present value of an annuity due

\( PV=\dfrac{PMT}{i}\left[1-\dfrac{1}{(1+i)^n}\right](1+i)\tag{2.2} \)

Present Value Growing Annuity Formula Derivation

In a growing annuity, each payment, after the first, is increased by a factor g such that payment 2 is PMT(1 + g), payment 3 is PMT(1 + g)(1 + g), payment 4 is PMT(1 + g)(1 + g)(1 + g), etc.  Modifying equation (2a) to include growth we get

\( PV=\dfrac{PMT}{(1+i)^1}+\dfrac{PMT(1+g)^1}{(1+i)^2}+\dfrac{PMT(1+g)^2}{(1+i)^3}+\dfrac{PMT(1+g)^3}{(1+i)^4}+...+\dfrac{PMT(1+g)^{n-1}}{(1+i)^n}\tag{3a} \)

Multiply PV by (1+i)/(1+g) to get

\( PV\dfrac{(1+i)}{(1+g)}=\dfrac{PMT}{(1+g)^1}+\dfrac{PMT}{(1+i)^1}+\dfrac{PMT(1+g)^1}{(1+i)^2}+\dfrac{PMT(1+g)^2}{(1+i)^3}+...+\dfrac{PMT(1+g)^{n-2}}{(1+i)^{n-1}}\tag{3b} \)

subtracting equation (3a) from (3b) most terms cancel and we are left with

\( PV\dfrac{(1+i)}{(1+g)}-PV=\dfrac{PMT}{(1+g)}-\dfrac{PMT(1+g)^{n-1}}{(1+i)^{n}} \)

with some algebraic manipulation, multiplying both sides by (1 + g) we have

\( PV(1+i)-PV(1+g)=PMT-\dfrac{PMT(1+g)^{n}}{(1+i)^{n}} \)

pulling like terms out on both sides

\( PV(1+i-1-g)=PMT\left[1-\left(\dfrac{1+g}{1+i}\right)^n\right] \)

cancelling the 1's on the left then dividing through by (i-g) we finally get

Present Value of a Growing Annuity (g ≠ i)

\( PV=\dfrac{PMT}{(i-g)}\left[1-\left(\dfrac{1+g}{1+i}\right)^n\right] \)

Similar to equation (2), to account for whether we have a growing annuity due or growing ordinary annuity we multiply by the factor (1 + iT)

\( PV=\dfrac{PMT}{(i-g)}\left[1-\left(\dfrac{1+g}{1+i}\right)^n\right](1+iT)\tag{3} \)

Present Value of a Growing Annuity (g = i)

If g = i you'll notice that (1 + g) terms cancel in equation (3a) and we get

\( PV=\dfrac{PMT}{(1+i)}+\dfrac{PMT}{(1+i)}+\dfrac{PMT}{(1+i)}+...+\dfrac{PMT}{(1+i)} \)

since we now have n instances of PMT/(1+i) we can reduce the equation. Also accounting for an annuity due or ordinary annuity, multiply by (1 + iT), and we get

\( PV=\dfrac{PMTn}{(1+i)}(1+iT)\tag{4} \)

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

For a perpetuity, perpetual annuity, time and the number of periods goes to infinity therefore n goes to infinity.  As n increases the 1/(1 + i)n term in formula (2) goes to 0 leaving

\( PV=\dfrac{PMT}{i}(1+iT)\tag{5} \)

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 in formula (3) and it reduces to

\( PV=\dfrac{PMT}{(i-g)}(1+iT)\tag{6} \)

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

\( PV=\dfrac{PMTn}{(1+i)}(1+iT)\rightarrow\infty\tag{7} \)

Present Value Formula for Combined Future Value Sum and Cash Flow (Annuity):

We can combine equations (1) and (2) to have a present value equation that includes both a future value lump sum and an annuity. This equation is comparable to the underlying time value of money equations in Excel.

Present Value

\( PV=\dfrac{FV}{(1+i)^n}+\dfrac{PMT}{i}\left[1-\dfrac{1}{(1+i)^n}\right](1+iT)\tag{8} \)

As in formula (2.1) if T = 0, payments at the end of each period, we have the formula for present value with an ordinary annuity

\( PV=\dfrac{FV}{(1+i)^n}+\dfrac{PMT}{i}\left[1-\dfrac{1}{(1+i)^n}\right]\tag{8.1} \)

As in formula (2.2) if T = 1, payments at the beginning of each period, we have the formula for present value with an annuity due

\( PV=\dfrac{FV}{(1+i)^n}+\dfrac{PMT}{i}\left[1-\dfrac{1}{(1+i)^n}\right](1+i)\tag{8.2} \)

Present Value when i = 0

In the case where i = 0 and we look back at equations (1) and (2a) to see that the combined present value formula can reduce to

\( PV=FV+PMTn(1+iT) \)

Present Value with Growing Annuity (g ≠ i)

rewritten from formula (3)

\( PV=\dfrac{FV}{(1+i)^n}+\dfrac{PMT}{(i-g)}\left[1-\left(\dfrac{1+g}{1+i}\right)^n\right](1+iT)\tag{9} \)

Present Value with Growing Annuity (g = i)

rewritten from formula (4)

\( PV=\dfrac{FV}{(1+i)^n}+\dfrac{PMTn}{(1+i)}(1+iT)\tag{10} \)

Note on Compounding m, Time t, and Rate r

Formula (8) can be expanded to account for compounding (m).

\( PV=\dfrac{FV}{(1+\frac{r}{m})^{mt}}+\dfrac{PMT}{\frac{r}{m}}\left[1-\dfrac{1}{(1+\frac{r}{m})^{mt}}\right](1+(\frac{r}{m})T)\tag{11} \)

where n = mt and \(i = \frac{r}{m}\).  t is the number of periods, m is the compounding intervals per period and r is rate per period t. (this is easily understood when applied with t in years, r the nominal rate per year and m the compounding intervals per year) When written in terms of i and n, i is the rate per compounding interval and n is the total compounding intervals although this can still be stated as "i is the rate per period and n is the number of periods" where period = compounding interval. "Period" can be a broad term.

Related to the calculator inputs, r = R/100 and g = G/100. If compounding (m) and payment frequencies (q) do not coincide in these calculations, r is converted to an equivalent rate to coincide with payments then n and i are recalculated in terms of payment frequency, q. The first part of the equation is the present value of the future sum and the second part is the present value of an annuity.

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

For a perpetuity, perpetual annuity, the number of periods t goes to infinity therefore n goes to infinity.  The FV term in equation (11) goes to 0 and the 1/(1 + i)n in the second term also goes to 0 leaving just formula (5)

\( PV=\dfrac{PMT}{i}(1+iT)\tag{5} \)

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, this term in formula (9) reduces to formula (6)

\( PV=\dfrac{PMT}{(i-g)}(1+iT)\tag{6} \)

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) (10) goes to infinity and we are back at equation (7)

\( PV=\dfrac{PMTn}{(1+i)}(1+iT)\rightarrow\infty\tag{7} \)

Continuous Compounding (m → ∞)

We look back to formula (11) for present value where m is the compounding per period t, t is the number of periods and r is the compounded rate with \(i = \frac{r}{m}\) and n = mt.

\( PV=\dfrac{FV}{(1+\frac{r}{m})^{mt}}+\dfrac{PMT}{\frac{r}{m}}\left[1-\dfrac{1}{(1+\frac{r}{m})^{mt}}\right](1+(\frac{r}{m})T)\tag{11} \)

The effective rate is ieff = ( 1 + ( r / m ) )m - 1 for a rate r compounded m times per period.  It can be proven mathematically that as m → ∞, ieff (the effective rate of r with continuous compounding) reaches the upper limit equal to er - 1.

Present Value with Continuous Compounding (m → ∞)

Removing the m and changing r to the effective rate of r, er - 1, in formula (11), formulas (8) & (11) for Present Value become

\( PV=\dfrac{FV}{(1+e^{r}-1)^{t}}+\dfrac{PMT}{e^{r}-1}\left[1-\dfrac{1}{(1+e^{r}-1)^{t}}\right](1+(e^{r}-1)T) \)

cancelling out 1's where possible we get the final formula for present value with continuous compounding

\( PV=\dfrac{FV}{e^{rt}}+\dfrac{PMT}{(e^r-1)}\left[1-\dfrac{1}{e^{rt}}\right](1+(e^r-1)T)\tag{12} \)

for an ordinary annuity

\( PV=\dfrac{FV}{e^{rt}}+\dfrac{PMT}{(e^r-1)}\left[1-\dfrac{1}{e^{rt}}\right]\tag{12.1} \)

for an annuity due

\( PV=\dfrac{FV}{e^{rt}}+\dfrac{PMT}{(e^r-1)}\left[1-\dfrac{1}{e^{rt}}\right]e^r\tag{12.2} \)

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

We can modify equation (3a) for continuous compounding, replacing i's with er - 1 and we get:

\( PV=\dfrac{PMT}{(1+e^{r}-1)^1}+\dfrac{PMT(1+g)^1}{(1+e^{r}-1)^2}+\dfrac{PMT(1+g)^2}{(1+e^{r}-1)^3}+\dfrac{PMT(1+g)^3}{(1+e^{r}-1)^4}+...+\dfrac{PMT(1+g)^{n-1}}{(1+e^{r}-1)^n} \)

which reduces to

\( PV=\dfrac{PMT}{e^{1r}}+\dfrac{PMT(1+g)^1}{e^{2r}}+\dfrac{PMT(1+g)^2}{e^{3r}}+\dfrac{PMT(1+g)^3}{e^{4r}}+...+\dfrac{PMT(1+g)^{n-1}}{e^{nr}}\tag{13a} \)

Multiplying (13a) by er/(1+g)

\( \dfrac{PVe^{1r}}{(1+g)}=\dfrac{PMT}{(1+g)}+\dfrac{PMT}{e^{1r}}+\dfrac{PMT(1+g)^1}{e^{2r}}+\dfrac{PMT(1+g)^2}{e^{3r}}+...+\dfrac{PMT(1+g)^{n-2}}{e^{(n-1)r}}\tag{13b} \)

subtracting (13a) from (13b) most terms cancel out leaving

\( \dfrac{PVe^{1r}}{(1+g)}-PV=\dfrac{PMT}{(1+g)}-\dfrac{PMT(1+g)^{n-1}}{e^{nr}} \)

multiplying through by (1+g)

\( PVe^{r}-PV(1+g)=PMT-\dfrac{PMT(1+g)^{n}}{e^{nr}} \)

solving this equation for PV and adding on the term to account for whether we have a growing annuity due or growing ordinary annuity we multiply by the factor (1 + (er-1)T)

\( PV=\dfrac{PMT}{e^{r}-(1+g)}\left[1-\dfrac{(1+g)^{n}}{e^{nr}}\right](1+(e^{r}-1)T)\tag{13} \)

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

Starting with equation (4) replacing i's with er - 1 and simplifying we get:

\( PV=\dfrac{PMTn}{e^{r}}(1+(e^r-1)T)\tag{14} \)

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

As t → ∞, ert → ∞ and formula (12) becomes

\( PV=\dfrac{PMT}{(e^r-1)}(1+(e^r-1)T)\tag{15} \)

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

As t → ∞, n → ∞ and enr in formula (13) grows fastest causing this term to go to 0 and we are left with:

\( PV=\dfrac{PMT}{e^{r}-(1+g)}(1+(e^{r}-1)T)\tag{16} \)

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) (7) 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.

\( PV=\dfrac{PMTn}{e^{r}}(1+(e^r-1)T)\rightarrow\infty\tag{17} \)


 

Cite this content, page or calculator as:

Furey, Edward "Present Value Calculator"; from http://www.calculatorsoup.com - Online Calculator Resource.

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