# Future Value Calculator

This is a comprehensive future value calculator that takes into account any present value lump sum investment, periodic cash flow payments, compounding, growing annuities and perpetuities. You can **enter 0 for the variables you want to ignore** or if you prefer specific future value calculations see our other future value calculators.

- Period
- commonly a period will be a year but it can be any time interval you want as long as all inputs are consistent.
- Present Value (PV)
- is the present value or principal amount
- 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 periodic 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)** - Future Value (FV)
- is a future value lump sum

## Future 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 used in the future value calculator is

or

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

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

### Future Value of a Present Sum

\( FV=PV(1+i)^{n}\tag{1} \)## Future Value 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) and are paid once each period for n periods at a constant interest rate i. The future value calculator will calculate FV of the series of payments 1 through n using formula (1) to add up the individual future values.

\( FV=PMT+PMT(1+i)^1+PMT(1+i)^2+...+PMT(1+i)^{n-1}\tag{2a} \)In formula (2a), payments are made at the end of the periods. The first term on the right side of the equation, PMT, is thelast payment of the seriesmade at the end of the last period which is at the same time as the future value. Therefore, there is no interest applied to this payment. The last term on the right side of the equation, PMT(1+i)^{n-1}, is thefirst payment of the seriesmade at the end of the first period which is only n-1 periods away from the time of our future value.

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

\( FV(1+i)=PMT(1+i)^1+PMT(1+i)^2+PMT(1+i)^3+...+PMT(1+i)^{n}\tag{2b} \)subtracting equation (2a) from (2b) most terms cancel and we are left with

\( FV(1+i)-FV=PMT(1+i)^n-PMT \)pulling out like terms on both sides

\( FV((1+i)-1)=PMT((1+i)^n-1) \)cancelling 1's on the left then dividing through by i, the future value of an ordinary annuity, payments made at the end of each period, is

\( FV=\dfrac{PMT}{i}((1+i)^n-1)\tag{2c} \)For an annuity due, payments made at the beginning of each period instead of the end, therefore payments are now 1 period further from the FV. We need to increase the formula by 1 period of interest growth. This could be written as

\( FV_{n}=PV_{n}(1+i)^{(n+1)} \)but factoring out the (1 + i)

\( FV_{n}=PV_{n}(1+i)^{n}(1+i) \)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 FV for an annuity due. This can be written more generally as

### Future Value of an Annuity

\( FV=\dfrac{PMT}{i}((1+i)^n-1)(1+iT)\tag{2} \)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 and we set T = 1.

### Future Value of an Ordinary Annuity

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

### Future Value of an Annuity Due

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

## Future Value Growing Annuity Formula Derivation

You can also calculate a growing annuity with this future value calculator. In a growing annuity, each resulting future value, after the first, increases by a factor (1 + g) where g is the constant rate of growth. Modifying equation (2a) to include growth we get

\( FV=PMT(1+g)^{n-1}+PMT(1+i)^1(1+g)^{n-2}+PMT(1+i)^2(1+g)^{n-3}+...+PMT(1+i)^{n-1}(1+g)^{n-n}\tag{3a} \)In formula (3a), payments are made at the end of the periods. The first term on the right side of the equation, PMT(1+g)^{n-1}, was thelast payment of the seriesmade at the end of the last period which is at the same time as the future value. When we multiply through by (1 + g) this period has the growth increase applied (n - 1) times. The last term on the right side of the equation, PMT(1+i)^{n-1}(1+g)^{n-n}, is thefirst payment of the seriesmade at the end of the first period and growth is not applied to the first PMT or (n-n) times.

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

\( FV\dfrac{(1+i)}{(1+g)}=PMT(1+i)^1(1+g)^{n-2}+PMT(1+i)^2(1+g)^{n-3}+PMT(1+i)^3(1+g)^{n-4}+...+PMT(1+i)^{n}(1+g)^{n-n-1}\tag{3b} \)subtracting equation (3a) from (3b) most terms cancel and we are left with

\( FV\dfrac{(1+i)}{(1+g)}-FV=PMT(1+i)^{n}(1+g)^{n-n-1}-PMT(1+g)^{n-1} \)with some algebraic manipulation, multiplying both sides by (1 + g) we have

\( FV(1+i)-FV(1+g)=PMT(1+i)^{n}-PMT(1+g)^{n} \)pulling like terms out on both sides

\( FV(1+i-1-g)=PMT((1+i)^{n}-(1+g)^{n}) \)cancelling

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

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

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

\( FV=\dfrac{PMT}{(i-g)}((1+i)^{n}-(1+g)^{n})(1+iT)\tag{3} \)### Future Value of a Growing Annuity (g = i)

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

\( FV=PMT(1+i)^{n-1}+PMT(1+i)^1(1+i)^{n-2}+PMT(1+i)^2(1+i)^{n-3}+...+PMT(1+i)^{n-1}(1+i)^{n-n} \)combining terms we have

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

### Future Value of a Perpetuity or Growing Perpetuity (t → ∞)

For g < i, for a perpetuity, perpetual annuity, or growing perpetuity, the number of periods t goes to infinity therefore n goes to infinity and, logically, the future value in equations (2), (3) and (4) go to infinity so no equations are provided. The future value of any perpetuity goes to infinity.

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

We can combine equations (1) and (2) to have a future value formula 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.

**Future Value**

\( FV=PV(1+i)^{n}+\dfrac{PMT}{i}((1+i)^n-1)(1+iT)\tag{5} \) As in formula (2.1) if T = 0, payments at the end of each period, we have the formula for **future value with an ordinary annuity**

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

### Future Value when i = 0

In the case where i = 0, g must also be 0, and we look back at equations (1) and (2a) to see that the combined future value formula can reduce to

### Future Value with Growing Annuity (g < i)

rewritten from formula (3)

\( FV=PV(1+i)^{n}+\dfrac{PMT}{(i-g)}((1+i)^{n}-(1+g)^{n})(1+iT)\tag{6} \)### Future Value with Growing Annuity (g = i)

rewritten from formula (4)

\( FV=PV(1+i)^{n}+PMTn(1+i)^{n-1}(1+iT)\tag{7} \)**Note on Compounding m, Time t, and Rate r**

Formula (5) can be expanded to account for compounding.

\( FV=PV(1+\frac{r}{m})^{mt}+\dfrac{PMT}{\frac{r}{m}}((1+\frac{r}{m})^{mt}-1)(1+(\frac{r}{m})T)\tag{8} \)where n = mt and i = 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" is a broad term.

Related to the calculator inputs, r = R/100 and g = G/100. If compounding and payment frequencies do not coincide in these calculations, r and g are 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 future value of a present sum and the second part is the future value of an annuity.

**Future Value with Perpetuity or Growing Perpetuity (t → ∞ and n = mt → ∞)**

For a perpetuity, perpetual annuity, the number of periods t goes to infinity therefore n goes to infinity and, logically, the future value in equation (5) goes to infinity so no equations are provided. The future value of any perpetuity goes to infinity.

## Continuous Compounding (m → ∞)

Calculating future value with continuous compounding, again looking at formula (8) 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 = r/m and n = mt.

\( FV=PV(1+\frac{r}{m})^{mt}+\dfrac{PMT}{\frac{r}{m}}((1+\frac{r}{m})^{mt}-1)(1+(\frac{r}{m})T)\tag{8} \)The effective rate is i_{eff} = ( 1 + ( r / m ) )^{m} - 1 for a rate r compounded m times per period. It can be proven mathematically that as m → ∞, the effective rate of r with continuous compounding reaches the upper limit equal to e^{r} - 1. [i_{eff} = e^{r} - 1 as m → ∞] Removing the m and changing r to the effective rate of r, e^{r} - 1:

formula (5) or (8) becomes

\( FV=PV(1+e^r-1)^{t}+\dfrac{PMT}{e^r-1}((1+e^r-1)^{t}-1)(1+(e^r-1)T) \)cancelling out 1's where possible we get the final formula for future value with continuous compounding

### Future Value with Continuous Compounding (m → ∞)

\( FV=PVe^{rt}+\dfrac{PMT}{e^r-1}(e^{rt}-1)(1+(e^r-1)T)\tag{9} \)for an **ordinary annuity**

for an **annuity due**

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

We can modify equation (3a) for continuous compounding, replacing i's with e^{r} - 1 and we get:

which reduces to

\( FV=PMT(1+g)^{n-1}+PMTe^{r}(1+g)^{n-2}+PMTe^{2r}(1+g)^{n-3}+PMTe^{3r}(1+g)^{n-4}+...+PMT(e^{(n-1)r})(1+g)^{n-n}\tag{10a} \)Multiplying (10a) by e^{r}/(1+g)

subtracting (10a) from (10b) most terms cancel out leaving

\( \dfrac{FVe^{r}}{1+g}-FV=PMT(e^{nr})(1+g)^{n-n-1}-PMT(1+g)^{n-1} \)multiplying through by (1+g)

\( FVe^{r}-FV(1+g)=PMTe^{nr}-PMT(1+g)^{n} \)factoring out like terms on both sides then solving for FV by dividing both sides by (e^{r} - (1 + g)) we have

Adding on the term to account for whether we have a growing annuity due or growing ordinary annuity we multiply by the factor (1 + (e^{r}-1)T)

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

Starting with equation (4) replacing i's with e^{r} - 1 and simplifying we get:

**Example Future Value Calculations:**

An example you can use in the future value calculator. You have $15,000 savings and will start to save $100 per month in an account that yields 1.5% per year compounded monthly. You will make your deposits at the end of each month. You want to know the value of your investment in 10 years or, the future value of your savings account.

- 1 Period = 1 Year
- Present Value Investment PV = 15,000
- Number of Periods t = 10 (years)
- Rate per period R = 1.5% (r = 0.015)
- Compounding 12 times per period (monthly) m = 12
- Growth Rate per Period G = 0
- Payment Amount PMT = 100.00
- Payments per Period q = 12 (monthly)

Using equation (7) we have

\( FV=15,000(1+0.015/12)^{12*10}+\dfrac{100}{0.015/12}((1+0.015/12)^{12*10}-1)(1+(0.015/12)*0) \)\( FV=15,000(1.00125)^{120}+\dfrac{100}{0.00125}((1.00125)^{120}-1) \)\( FV=17,425.88+92,938.03-80,000= $30,361.91 \)FV = 17,425.88 + 92,938.03 - 80,000 = $30,361.91

At the end of 10 years your savings account will be worth $30,363.91

Suppose you find a bank that offers you daily compounding (365 times per year).

Category: Annuity

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