Future Value Formula
Future Value and Annuity Formulas
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
