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Arithmetic Annuity   
First Payment Progression Payment N Interest Rate Increasing Decreasing
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Given an interest rate of 8% and a first payment amount of 1000 arithmetically increasing by 100 for 4 periods, calculate the Present Value (PV) and Accumulated Value (AV) of an Increasing Arithmetic Annuity Immediate:

Ian|i  =  Arithmetic Payment x (än|i - nvn)
  i

Calculate d
d  =  i
  1 + i

d  =  0.08
  1 + 0.08

d  =  0.08
  1.08

d = 0.0740740740741

Calculate Present Value of Annuity Factor (PVA) given i = 0.08, n = 4, and v = 0.925925925926
ä4|0.08  =  (1 - 0.9259259259264)
  0.0740740740741

ä4|0.08  =  (1 - 0.735029852796)
  0.0740740740741

ä4|0.08  =  0.264970147204
  0.0740740740741

ä4|0.08 = 3.5771

Now Calculate the Present Value of an Increasing Arithmetic Annuity:
Ia4|0.08  =  Arithmetic Payment x (ä4|0.08 - nvn)
  i

Ia4|0.08  =  100 x (3.5771 - 4(0.925925925926)4)
  0.08

Ia4|0.08  =  100 x (3.5771 - 4(0.735029852796))
  0.08

Ia4|0.08  =  100 x (3.5771 - 2.94011941119)
  0.08

Ia4|0.08  =  100 x 0.636980588814
  0.08

Ia4|0.08  =  63.6980588814
  0.08

Ia4|0.08 = 796.225736018

Calculate the Accumulated Value of an Increasing Arithmetic Annuity:
Isn|i  =  Arithmetic Payment x (sn|i - n)
  i

sn|i  =  (1 + i)n - 1
  d

sn|i  =  (1 + 0.08)4 - 1
  0.0740740740741

sn|i  =  1.084 - 1
  0.0740740740741

sn|i  =  1.36048896 - 1
  0.0740740740741

sn|i  =  0.36048896
  0.0740740740741

s4|0.08 = 4.86660096

Calculate AV given i = 0.08, n = 4
Isn|i  =  1000 x (sn|i - n)
  0.08

Isn|i  =  1000 x (4.86660096 - 4)
  0.08

Isn|i  =  1000 x (0.86660096)
  0.08

Isn|i  =  866.60096
  0.08

Isn|i = 10832.512







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Tags:  accumulated value, annuity, future value, interest, present value


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