FUTURE VALUE
DEFINITION
Future value refers to the value of a sum of money at a particular point of time in the future. Due to the interest-earning capacity of money, a sum of money invested now will accumulate to a greater amount in the future.
CALCULATOR

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FORMULA AND DERIVATION
Consider a sum of money $$c$$ invested into a bank account that pays interest at the rate $$i$$. If $$c$$ is invested at the beginning of the year, an amount of $$c+ic$$ would be accumulated by the end of the year($$ic$$ is the amount of interest paid on $$c$$. So, the future value of $$c$$ after one year is
$FV=c+ic=c\left(1+i\right)$
In the second year, this amount of $$c\left(1+i\right)$$ will again accumulate at the rate $$i$$. Thus, the future value of this amount by the end of second year would be
$FV=c\left(1+i\right)\times \ \left(1+i\right)=c{\left(1+i\right)}^2$
Similarly, for a sum of money $$c$$ invested now, the future value at the end of $$n$$ years is
$FV=c{\left(1+i\right)}^n$
Future value of a series of cashflows:

Consider a series of cash flows ct occurring at times $$t=0,1,2,\dots \dots \dots n.$$ The future value of this series of cash flows will be the sum of future values of each cash flow.
$FV=\ c_o{\left(1+i\right)}^n+c_1{\left(1+i\right)}^{n-1}+c_2{\left(1+i\right)}^{n-2}\dots \dots \dots c_{n-1}{\left(1+i\right)}^1+c_n$ $FV=\ \sum^n_{t=1}{c_t}{\left(1+i\right)}^{n-t}$
EXAMPLES
Example 1

A cash flow of $$\textrm{₹}\ 100$$ occurs today. The future value of this cash flow at $$8\ \%$$ at the end of three years from now would be
$FV=c{\left(1+i\right)}^3$ $FV=100{\left(1+0.08\right)}^3=100{\left(1.08\right)}^3$ $FV=125.97$
Example 2

A cash flow of$$\mathrm{\textrm{₹}}\mathrm{\ 100}$$ will occur at the end of one year from today. The future value of this cash flow at $$8\ \%$$ at the end of three years from now would be
$FV=c{\left(1+i\right)}^{\left(n-1\right)}$ $FV=100{\left(1+0.08\right)}^{\left(3-1\right)}=100{\left(1.08\right)}^2$ $FV=116.64$
Since the cash flow is occurring at the end of the first year, It is accumulated only during the second and third year and hence the accumulation factor is$${\left(1+i\right)}^2$$ instead of $${\left(1+i\right)}^3$$

Example 3

The following payments are received at the beginning of every year for the next 6 years:-

$$\mathrm{\1000,\ \2000,\ \3000,\ \2500,\ \2000,\ \1500}$$

The future value of this series of cash flows at the rate of $$10\ \%$$ at the end of the sixth year would be
$FV=1000{\left(1.1\right)}^6+2000{\left(1.1\right)}^5+3000{\left(1.1\right)}^4+2500{\left(1.1\right)}^3+2000{\left(1.1\right)}^2+1500{\left(1.1\right)}^1$ $FV=16782.38$
Example 4

Consider the same series of cash flows as in example 3. But now, these cash flows occur at the end of every year. The future value at $$10\ \%$$ at the end of the sixth year would be
$FV=1000{\left(1.1\right)}^5+2000{\left(1.1\right)}^4+3000{\left(1.1\right)}^3+2500{\left(1.1\right)}^2+2000{\left(1.1\right)}^1+1500$ $FV=15256.71$