# Time Value of Money

Time value of money is the concept that a dollar received today (referred to in finance as time 0 or t=0) is worth more than a dollar that will be received in future say, after one year (referred to as t=1).

If interest rate is 5%, the dollar received at t=0 can earn interest of $0.05 ($1 multiplied by the interest rate) per year. After one year, dollar received at t=0 is worth $1.05 ($1 plus accumulated interest of $0.05) which is $0.05 more than the dollar received at t=1.

Time value of money is one of the core concepts in finance. Net present value, internal rate of return, and valuation of a share of common stock or bond, etc. are all applications of time value of money.

The value of any amount today i.e. at t=0 is called the present value, the value of any sum at some time in future is called the future value and these two values are connected by the interest rate and time. In the above example, $1 received today is the present value and $1.05 that it is worth after 1 year given a 5% interest rate is the future value. The relationship can be represented algebraically as follows:

$$ Future\ Value \\ =\ Present\ Value\ +\ Interest \\ =\ Present\ Value\ +\ Present\ Value\ \times\ Interest\ Rate \\ =\ Present\ Value\ \times\ (1\ +\ Interest\ Rate) $$

The above expression is for one year. Let’s assume we need to compare a dollar received at t=0 and one received after two years (i.e. t=2). Value of $1 received at t=0 after first year is $1.05 as illustrated above, after the second year, the value is $1.1025 (t=1 value of $1.05 plus interest earned in second year of $1.05 × 5%). Value of a dollar n years in future given an interest rate i can be worked out using the following equation:

$$ Future\ Value=Present\ Value\ \times{(1+i)}^n $$

The above comparison can also be made by finding the present value of $1 received at t=1 today i.e. at t=0. We just need to make an algebraic adjustment to the above equation to get:

$$ Present\ Value=\frac{Future\ Value}{{(1+i)}^n} $$

Crunching the numbers shows that $1 received after 1 year (i.e. at t=1) is worth $0.9524 today:

$$ Present\ Value=\frac{$1}{(1+5\%)^1}=$0.9524 $$