Time Value of Money (PV and FV)

                                                                Time Value of Money

Some terminology:

•          PV = Present value ($) (unit is very important)
•          FV = Future value ($)
•          n = number of periods (#)
•          r = interest rate (%) > 0 (assumption) (read the book called theory of interest rate written in 1930 by Irvin Fischer)
•          Remember, no uncertainty for now!

Simple future value (FV):

Importance of time lines!

                           

                        

                                                                       n = 10

The main insight:

·                                   A dollar today is worth more than a dollar tomorrow.

For Example:

If a bank pays you 10% interest per year and you are given a choice between two plans:

1. A.        Receive $100 today.
2. B.      Receive $100 one year from now

Which would you prefer, why? (Suppose number of period is 1)

Solution:   Prefer A:

A:  

     0                1

     $100

FV = 100(1+0.1) ^1

      = $110

B:

0                           1

                      $100

PV = 100/ (1+0.1) ^1

      = $90.91

Future value (Concept):

Future Value (FV) = Initial payment ($100) + Accumulated interest ($10)

Future value (Formula):

FV = P + r*P (P = Initial Payment)

     = P*(1+r) ^n (where 1 + r is a future value factor i.e. if you multiply it with any number you will get the future value)

Relating to the above question: What is the future value of $100 two years from now?

Solution:

FV = P*(1+r) ^n

      = 100 * (1+0.1)^2

      = $121

It is very important to note that from where is that extra $1 came. What I mean is for 1 year, FV is $110 and for 2 year it is $121. So what is going on. lets see:

so it is clear that in the first year the interest is calculated for $100 and in this case it is 10% of $100 and it is $10 and in the next year again you get the interest of $10 dollar that is 10% of $10 which is $1.

Future Value (Power of Compounding)

what are the future values of investing $100 at 10% versus 5% for 100 years?

Solution:

    FV = 100(1+0.1)^100           FV = 100(1+0.05)^100

           = $1,378,061                        = $13,150

See the difference. It is huge. If you weren't compounding then for $100 you would have earned $10 interest and for 100 years it would be $1000 + $100 = $1100. So, compounding is very much complicated.

     

Ram bought the Island from Native Americans for $24 in 1626. Suppose that Native Americans would have earned 6% on their investment all these years. How much would have they today?

Solution:

    PV = $24       r = 6%

      n = 2013-1626 = 387 years.

Then, 

               FV = 24*(1+0.06)^387

                     = $149,135,522,178

 Present Value (Concept):

We know, 

            FV = PV*(1+r) ^n

so,  

            PV = FV/(1+r)^n

Suppose you will inherit $121,000 two years from now and the interest rate is 10%. what is the value today to you?

Solution:

Now,

             PV = FV/(1+r)^n

                  = 121,000(1+0.1)^2

                  =$100,000

what is happening is shown below!