Time value of Money - Present & Future value of Money

2 EXAMPLES

PART ISingle Sum.

Time Value of Money: Know this terminology and notation

FV       Future Value (1+i)Future Value Interest Factor [FVIF] 
PV       Present Value 1/(1+i)t  Present Value Interest Factor  [PVIF] 
   i        Rate per period  
   t        # of time periods  

Question: Why are (1+i) and (1+i)t  called interest factors?

Answer: 1. Start with simple arithmetic problem on interest:

How much will $10,000 placed in a bank account paying 5% per year be worth compounded annually?   

 Answer:   PrincipalInterest

     $10,000 + $10,000 x .05 = $10,500

 2. Factor out the $10,000.    

                        10,000 x (1.05) = $10,500

 3. This leaves (1.05) as the factor.

 1.  Find the value of $10,000 earning 5% interest per year after two years.

     Start with the amount after one year and multiply by the factor for each year.

                         [Amount after one year] x  (1.05)

=           [$10,000   x  (1.05)]   x  (1.05)

=          $10,000 x (1.05)2

 =         $11,025.

         So (1+i)t = (1+i)·(1+i)·(1+i)·(1+i)·(1+i)·(1+i)·(1+i)… ·(1+i) for “t” times

A.  Future Value

Find the value of $10,000 in 10 years. The investment earns 5% per year.

FV = $10,000·(1+i)·(1+i)·(1+i)·(1+i)·(1+i)·(1+i)·(1+i)·(1+i)·(1+i)·(1+i)           

FV = $10,000·(1.05)·(1.05)·(1.05)·(1.05)·(1.05)·(1.05)·(1.05)·(1.05)·(1.05)·(1.05)           

FV = $10,000 x (1.05)10

= $10,000 x 1.62889

= $16,289

Find the value of $10,000 in 10 years.  The investment earns 8% for four years and then earns 4% for the remaining six years.

FV = $10,000·(1+i)·(1+i)·(1+i)·(1+i)·(1+i)·(1+i)·(1+i)·(1+i)·(1+i)·(1+i)

FV = $10,000·(1.08)·(1.08)·(1.08)·(1.08)·(1.04)·(1.04)·(1.04)·(1.04)·(1.04)·(1.04)

FV = $10,000 x (1.08)4 x (1.04)6

                FV = $17,214.53

 B.Present Value:

Same idea, but begin at the end. Rearrange the Future value equation to look like this: 

  PV = FV÷ [(1+i)·(1+i)·(1+i)·(1+i)·(1+i)·(1+i)·(1+i)·(1+i)·(1+i)·(1+i)]                                PV = FV ÷ (1+i)t                                                                                                                       [2]

Example: How much do I need to invest at 8% per year, in order to have $10,000 in__.

a.  One year:              PV =10,000 ÷ (1.08) =     $9,259.26
b.  Two years:            PV = $10,000 ÷ (1.08) ÷ (1.08)

OR  $10,000 ÷ (1.08)= $8,573

            c.   Ten years              PV = $10,000 ÷ (1.08)10 =  $10,000 ÷ 2.1589 =   $4,632

 

C. Rate of Return

START WITH SAME RELATIONHSIP: FV = PV x (1+i)t

Solve for i.  (1+i)t =FV/PV.

1+i = (FV/PV)1/t

  i = (FV/PV)1/t-1.

Question:   An investor deposits $10,000. Ten years later it is worth $17,910.  What rate of return did the investor earn on the investment?

Solution:

          $17,910 = $10,000 x (1+i)10
    
    (1+i)10 = $17,910/10,000 =  1.7910

               (1+i) = (1.7910) 1/10  = 1.060

                     i = .060 = 6.0%