Present value and discounting

Present value is defined as the current value of future cash flows discounted at the appropriate discount rate. 

Discounting is the process of finding out the present value of some future amount. Finding out the present value is discounting and the reverse is called compounding.

Present value for a single period case:

Value after one year = Present value * (100+i)%

Present value for a multiple period case:

PV = FV / (1+i)^n

where, 

PV = present value

FV = future value 

i = rate of interest

n = number of years or interval of time periods 

Also, by the use of the PVIF Table,

PV = FV * FVIF (i , n )

where, FVIF (i, n) = Future value interest factor at i% of interest and n years

Discounting periods ad their implication on present value:

The present value of a sum of money has an inverse relation with the time period and interest rate. As the time period to receive a future sum of money increases, the present value of the sum of money will decline. 

The relationship also holds with an interest rate, that is, the larger the interest rate, the lower will be the present value of a future sum. 

Determining the discount rate:

i = (FV / PV)^1/n -1, 

where 

PV = present value

FV = future value 

i = rate of interest

n = number of years or interval of time periods 

Finding the number of periods:

We just have to use logarithms from the same above future value and present value formula, after dividing FV by PV and solving it.

Example:

    FV = PV (1+i)^n

or, (1+i)^n = FV / PV

Taking log on both sides, we get, 

    n log (1+i) = log( FV / PV)

or, n = log (FV / PV) /  log (1 +i) 

Tabular solution:

Example:

Q. If PV = Rs. 2000, FV = 4000 i.e double of PV, with an annual interest rate of 8 %, what will be the n ( number of periods)?

Solution:

    PV = FV * PVIF (i, n)

or, 2000 = 4000 * PVIF (8%, n)

or, 0.5 = PVIF (8%, n)

Now, 

Looking at the PVIF table at an 8 percent interest rate, the factor 0.5 is close to the 9-year factor (0.5002). So, the respective time period that Rs. 2000 doubles at 8 percent annual interest rate that is 9 years. 

Hence n = 9 years.