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Annuity

An annuity is a series of regular small payments or savings made with a view to getting a high future value amount. In other words, small savings made to achieve one big lump sum payment at the end. You may treat this like saving for one’s retirement, or for one’s future living.

The process of calculating this lumpsum amount involves using compound interest and geometric sequence formula.

 

Let’s consider the following example.

Samuel deposits $200 per month in an annuity scheme that pays 12% p.a. The scheme runs for 10 years, and Samuel has made the last payment a month ago, and he’s now ready to withdraw the final amount. How much is this amount?

Since Samuel deposits $200 every month, and gets a compound interest of 12% p.a., we need to convert the annual interest rate to a monthly interest rate.

r = 12% p.a.  =  0.12 p.a.  =   0.12/12  =  0.01 p.m.

P = 200, and n = 10 years or 120 months.

Samuel’s first payment of $200 will attract a compound interest of 1% per month. This amount will increase to 200 (1 + 0.01)1 at the end of the first month.

Since his first deposit will be invested for the entire 10 years (or 120 months), his first month’s amount A1 will be 200 (1 + 0.01)120.

Similarly his second month’s payment of $200 will attract compound interest of 1% p.m. for the next 119 months, and this amount A2 will be 200 (1 + 0.01)119.

For the third month, the amount A3 will be 200 (1 + 0.01)118, and so on.

The total value at the end of 10 years (i.e. 120 months) is the sum of all these individual amounts A1, A2, A3, A4, and so on all the way till A120.

So final amount A = A1 + A2 + A3 + A4 +. . . . . . . . . . . +A120

A  = 200 (1.01)120 + 200 (1.01)119 + 200 (1.01)118 + 200 (1.01)117 + . . . . . . . . . + 200 (1.01)2 + 200 (1.01)1

= 200 [(1.01)120 + (1.01)119 + (1.01)118 + (1.01)117 + . . . . . . . . . + (1.01)2 + (1.01)1]

 

The numbers in the square brackets form a geometric series given by the formula –

Sn  =   {a (r^n - 1)}/{r - 1} ; |r| >1, where

a is the first term (= 1.01 here), and the common ratio r = 1.01.

So first we need to get the value of this geometric series, and then multiply it with 200 to get the final amount.

Substituting these values of a and r in the geometric series formula, we get

S120  =   {1.01 (1.01^120~ - ~1)}/{1.01~-~1}  =  232.3391

So now the amount A = 200 x 232.3391  =  $46467.82 (to the nearest cent).

Hence Samuel will get a lump sum amount of $46467.82 after 10 years of depositing $200 every month.

 

A quick way to calculate annuity:

The entire calculations shown above can be done in one step using the formula –

A  =    {M(1+r) ((1+r)^n - 1)}/{r} where

  • A is the final lump sum amount at the end of time period n,
  • M is monthly investment made,
  • r is the rate of interest per time period, and
  • n is the total time period (in years, or months) the investment is made.

Using this formula, the final amount for Samuel at the end of 10 years will be    {200(1+0.01) ((1+0.01)^120~-~1)}/{0.01}  

=  $46467.82

 

There is a variation to the above formula to calculate the lump sum amount accumulated at the start of the final time period.

In this case, if Samuel wanted to know his lump sum (or annuity) at the start of his last month, we’d use this formula.

A  =    {M ((1~+~r)^n~-~1)}/{r}  

=    {200 ((1~+~0.01)^120~-~1)}/{0.01}  

     =  $46007.74   (to the nearest cent)