PRESENT VALUE
DEFINITION
Present value refers to the value of a future cash flow as of now, or the current worth of a future cash flow. Since money has interest-earning capacity, a sum of money in present time will accumulate to a greater amount in the future. So, present value is simply the amount of money that is to be invested now, for it to accumulate to a higher value, at a particular point of time in the future. In other words, present value is the value of a future cash flow, if it had occurred now.
CALCULATOR

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FORMULA AND DERIVATION
Consider a sum of money, $$a$$, invested into a bank account that pays interest at the rate $$i$$. If $$a$$ is invested at the beginning of the year then it will accumulate to $$a\left(1+i\right)$$ by the end of the year. Consider this accumulated amount as $$c$$. So,
$a\left(1+i\right)=c$
The above equation can be explained as follows:-
The future value of $$a$$ is $$c$$
or
The present value of $$c$$ is $$a$$.

The second explanation is of greater interest to us. It tells us that if an amount of $$c$$ is required at the end of the year, we need to invest an amount, $$a$$, at the beginning of the year. This means that an amount of $$c$$, due one year from now, is currently worth $$a$$, and thus $$a$$ is the present value of $$c$$.
$a\left(1+i\right)=c$ $\Rightarrow a=\frac{c}{\left(1+i\right)}$
So, the formula for the present value of a cash flow $$c$$, due one year from now, is
$PV=\frac{c}{1+i}$ $PV=cv$
Where, $$v=\frac{1}{1+i}={\left(1+i\right)}^{-1}$$

Similarly, the formula for the present value of a cash flow $$c$$, due $$n$$ years from now is
$PV=\frac{c}{{\left(1+i\right)}^n}=c{\left(1+i\right)}^{-n}={cv}^n$
Present value of a series of cashflows

Consider a series of cash flows$$c_t$$ occurring at times $$t=0,1,2,3,\dots .n$$. The present value of this series of cash flows will be the sum of present values of each cash flow.
$PV=c_0+c_1{\left(1+i\right)}^{-1}+c_2{\left(1+i\right)}^{-2}\dots \dots \dots c_n{\left(1+i\right)}^{-n}$ $=c_o+c_1v^1+c_2v^2\dots \dots \dots c_nv^n$ $=\sum^n_{t=0}{c_tv^t}$
$$c_0$$ is not multiplied by the discount factor, $$v$$, because the present value of a cash flow occurring now, is simply the value of that cash flow.

EXAMPLES
Example 1

A cash flow of $$\textrm{₹}100$$ is due at the end of the fifth year from now. A rate of $$10\%$$ p.a. is considered to discount the cash flows to their present values.

Here,
$c=100$ $n=5$ $i=10\%p.a$

$PV={cv}^n$ $PV=100v^n=100{(1+0.1)}^{(-5)}$ $PV=100{(1.1)}^{(-5)}$ $PV=62.09$
This means that if $$\textrm{₹}62.09$$ is invested today, it will be worth $$\textrm{₹}100$$ in 5 years’ time, at $$10\%$$ p.a.

Example 2

A cash flow of $$\textrm{₹}100$$ is due at the beginning of the fifth year from now. The discount rate is $$10\%$$ p.a.
$PV={cv}^{\left(n-1\right)}$ $PV=100v^4=100{(1+0.1)}^{(-4)}$ $PV=100{(1.1)}^{(-4)}$ $PV=68.3$
Here, the cash flow is discounted only for 4 years because the cash flow is occurring at the beginning of the fifth year, which is the same as a cash flow of equal amount occurring at the end of the fourth year.

Example 3

$$\textrm{₹}1000$$, $$\textrm{₹}2000$$, $$\textrm{₹}3000$$, $$\textrm{₹}2500$$, $$\textrm{₹}2000$$, $$\textrm{₹}1500$$ are received for the next 6 year at the end of every year. Considering a discount rate of $$10\%$$ p.a. the present value of this series of cash flows would be
$PV=1000v+2000v^2+3000v^3+2500v^4+2000v^5+1500v^6$ $PV=8612.02$
Example 4

Consider the same cash flows as in example 3. But now, the cash flows arise at the beginning of every year. The discount rate is $$10\%$$ p.a.
$PV=1000+2000v+3000v^2+2500v^4+1500v^5$ $PV=9473.22$
The cash flow of $$\textrm{₹}1000$$ is not discounted because it is occurring at present time (at the beginning of the year, i.e. now). So, its present value is just $$\textrm{₹}1000$$.