# Future Value Formula

Type

Formula

1 Future Value of a Present Sum

\[ FV=PV(1+i)^{n} \]
FV = PV * (1 + i)

^{n}2 Future Value of an Annuity

\[ FV=\dfrac{PMT}{i}((1+i)^n-1)(1+iT) \]

2.1 Future Value of an Ordinary Annuity

T = 0

T = 0

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

2.2 Future Value of an Annuity Due

T = 1

T = 1

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

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

T = 0 for an ordinary annuity

T = 1 for an annuity due

T = 0 for an ordinary annuity

T = 1 for an annuity due

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

4 Future Value of a Growing Annuity (g = i)

T = 0 for an ordinary annuity

T = 1 for an annuity due

T = 0 for an ordinary annuity

T = 1 for an annuity due

\[ FV=PMTn(1+i)^{n-1}(1+iT)\]

5 Future Value for Combined Future Value Sum and Cash Flow Annuity

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

5.1 Future Value for Combined Future Value Sum with an Ordinary Annuity

T = 0

T = 0

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

5.2 Future Value for Combined Future Value Sum with an Annuity Due

T = 1

T = 1

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

6 Future Value for Combined Future Value Sum with Growing Annuity (g < i)

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

7 Future Value for Combined Future Value Sum with Growing Annuity (g = i)

\[ FV=PV(1+i)^{n}+PMTn(1+i)^{n-1}(1+iT)\]

8 Future Value for Combined Future Value Sum and Annuity including Compounding, Time and Rate

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

9 Future Value for Combined Future Value Sum and Annuity with Continuous Compounding (m → ∞)

\[ FV=PVe^{rt}+\dfrac{PMT}{e^r-1}(e^{rt}-1)(1+(e^r-1)T)\]

9.1 Future Value for Combined Future Value Sum and Ordinary Annuity with Continuous Compounding (m → ∞)

\[ FV=PVe^{rt}+\dfrac{PMT}{e^r-1}(e^{rt}-1)\]

9.2 Future Value for Combined Future Value Sum and Annuity Due with Continuous Compounding (m → ∞)

\[ FV=PVe^{rt}+\dfrac{PMT}{e^r-1}(e^{rt}-1)e^r\]

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

\[ FV=\dfrac{PMT}{e^{r}-(1+g)}(e^{nr}-(1+g)^{n})(1+(e^{r}-1)T)\]

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

\[ FV=PMTne^{r(n-1)}(1+(e^{r}-1)T)\]

- FV = Future Value
- PV = Present Value
- PMT = Payment Amount
- i = interest rate per period (decimal form)
- n = number of periods when compounding is once per period
- n = mt when compounding frequency is different than period frequency
- m = compounding frequency per period
- t = number of periods

- g = interest rate growth per period (decimal form)
- T = Type
- T = 0 for Ordinary Annuity (end)
- T = 1 for Annuity Due (beginning)

- r = interest rate per period in decimal form
- e = Euler's number, a mathematical constant equal to approximately 2.71828