# Present Value vs Future Value

Present value is the equivalent value today of some amount to be received or paid in future and future value is the accumulated value in future of an amount received or paid today. The equivalency arises because a cash flow that occur at time 0 can accumulate interest.

If interest rate is 5%, the dollar received at t=0 can earn interest of \$0.05 (\$1 × 5%) 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.

This interplay of money today and some future date is called the time value of money. It 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.

## Future Value

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.

If FV stands for future value, PV stands for present value, I stands for total interest expense and i stands for the interest rate, the relationship can be represented algebraically as follows:

FV
= PV + I
= PV + PV × i
= PV × (1 + i)

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). Before delving into the actual calculations, we need to determine one thing i.e. whether the interest earned in the first year will earn interest too. If yes, it is called the compound interest and the 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%). However, if the interest doesn't accumulate further interest, it is called simple interest and the future value is \$1.1 (i.e. \$1.05 + interest for one more year of \$0.05).

In case of compound interest, value of a dollar n years in future given an interest rate i can be worked out using the following equation:

FV = PV × (1 + i)n

The factor (1+i)n is called the future value factor.

In case of simple interest, value of a dollar after n periods is given by the following equation:

FV = PV × (1 + i × n)

## Present Value

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 = 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 = \$1 = \$0.9524 (1 + 5%)1