Lesson Ordinary Annuity saving plans and geometric progressions

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Ordinary Annuity saving plans and geometric progressions


Geometric progressions are widely used in  financial mathematics.
Particularly, they are working tools in calculating and analyzing  saving plans.
You can learn from this lesson  how to apply geometric progressions to the  'Ordinary Annuity'  type saving plans.
An  annuity  is a sequence of equal periodic deposits.  The periodic deposits may be made annually, quarterly, monthly, or daily.
If the deposit is made at the end of each payment period and the interest is credited at the same time, the annuity is called  'ordinary annuity'.
Another type of an annuity is  'annuity due'  saving plan when the deposit is made at the beginning of each payment period and the interest is compounded at the end
of the payment period.  This type of saving plan is considered in the lesson  Annuity Due saving plans and geometric progressions  under the current topic in this site.

Problem 1 ('Ordinary Annuity' saving plan)

Mr. Robertson has an ordinary annuity saving plan at a credit union. At the end of each month he deposits $100 at a credit union account paying 6% interest compounded
monthly. What is Mr. Robertson's balance at the end of  15  years?

Solution

During 15 years, Mr. Robertson have to make 180 payments (180 = 12*15).
This ordinary annuity saving plan works in such a way that the first deposit earns interest at the monthly rate of  6%2F12% = 0.5%  for  179  months.  For the last month
of the period of 180 months the interest is not credited.
The second deposit earns interest for  178  months, the third deposit earns interest for  177  months, and so on.
The 179-th deposit earns interest for  1  month, and the last, 180-th deposit is not credited, so it contributes to the final balance without interest.

Using the formula for compound interest, you can write the total balance of all  180  deposits as

Total = 100 + 100%2A1.005%5E1 + 100%2A1.005%5E2 + . . . + 100%2A1.005%5E179.

This is the sum of the  180  terms of the geometric progression with the first term  100  and the common ratio  1.005.
For the formula for the sum of a geometric progression see the lesson  Geometric progressions  in this site.

So, Mr. Robertson's balance at the end of the 15-th year is

Total = 100*%28%281.005%5E180-1%29%2F%281.005-1%29%29 = 29081.87.

Answer. Mr. Robertson's balance at the end of the 15-th year is  $29081.87.


Below is one more problem on the ordinary annuity saving plan.

Problem 2 ('Ordinary Annuity' saving plan)

It was just described in the previous  Problem 1  how an  ordinary annuity  saving plan works.
During the time period of  t  years,  an individual deposits a fixed amount of  P  dollars to the account at the end of each month.  The account
is credited at the end of each month except the last month at the monthly interest rate  i = r%2F12%  where  r  is per annum interest rate.
Derive a general formula for the balance at the end of the ordinary annuity saving plan and find the balance if
    a)  P = $100,  t = 10 years,  r = 4%;
    b)  P = $500,  t = 30 years,  r = 4%.

Solution

The saving plan spans the time period of  n=12*t  months.
The first deposit will earn interest at the monthly rate of  i=r%2F12%  for  n-1=12*t-1  months (each month except the last one) .  This means that the amount of money
generated by the first deposit will get the value of  P%2A%281%2Bi%29  at the end of the first month;  the value of  P%2A%281%2Bi%29%5E2  at the end of the second month; the value of  P%2A%281%2Bi%29%5E3  at the end of the third month,  and so on.  The amount of money generated by the first deposit will get the value of  P%2A%281%2Bi%29%5E%28n-1%29at the end of the saving plan.

Similarly, the second deposit will earn interest at the monthly rate of  i=r%2F12%  for  n-2=12*t-2  months.  This means that the amount of money at the account
generated by the second deposit will reach the value of  P%2A%281%2Bi%29%5E%28n-2%29  at the end of the saving plan.

The third deposit will earn interest at the monthly rate of  i=r%2F12%  for  n-3  months.  It will generate the amount of money  P%2A%281%2Bi%29%5E%28n-3%29  at the end of the saving plan.

The last,  n-th payment,  is deposited at the end of the saving plan and does not generate interest.  So, the last payment  P  contributes the same amount  P  to the
balance.

Thus, at the end of the saving plan the total amount of money in the account will be

P + P%2A%281%2Bi%29 + P%2A%281%2Bi%29%5E2 + . . . + P%2A%281%2Bi%29%5E%28n-2%29 + P%2A%281%2Bi%29%5E%28n-1%29.

This is the sum of the geometric progression with the first term  P , the common difference  1%2Bi%29  and the number of terms equal to  n=12*t.

Applying the formula for the sum of a geometric progression  from the lesson  Geometric progressions  you get the general formula for the balance
of the annuity due saving plan

Total = P * %28%281%2Bi%29%5En-1%29%2Fi =P * %28%281%2Bi%29%5E%2812%2At%29-1%29%2Fi = P * +%28+%281%2B%28r%2F12%29%29%5E%2812%2At%29-1%29%2F%28r%2F12%29++.

Now, let us apply this formula to cases  a)  and  b).

a)  There are  n = 12*t = 12*10 = 120 monthly payment periods in this case.  The monthly rate is  i = r%2F12 = 0.04%2F12 = 0.003333.
     Total = 100*%281.003333%5E120-1%29%2F0.003333 = 100*0.490832682%2F0.003333 = 14,724.98.

      Answer. The balance is equal to $14,724.98.


b)  There are  n = 12*t = 12*30 = 360 monthly payment periods in this case.  The monthly rate is  i = r%2F12 = 0.04%2F12 = 0.003333.
     Total = 500*%281.003333%5E360-1%29%2F0.003333 = 500*2.313498%2F0.003333 = 347,024.70.

      Answer. The balance is equal to $347,024.70.


You can compare these results with the similar calculations for the  annuity due  saving plans of the lesson  Annuity Due saving plans and geometric progressions 
in this site to see the difference.


My other lessons in this site associated with annuity saving plans and retirement plans are

    - Geometric progressions
    - The proofs of the formulas for geometric progressions

    - Annuity Due saving plans and geometric progressions
    - Solved problems on Ordinary Annuity saving plans
    - Finding present value of an annuity, or an equivalent amount in today's dollars
    - Withdrawing a certain amount of money periodically from a compounded saving account
    - Miscellaneous problems on retirement plans

OVERVIEW of my lessons on geometric progressions with short annotations is in the lesson  OVERVIEW of lessons on geometric progressions.

Use this file/link  ALGEBRA-II - YOUR ONLINE TEXTBOOK  to navigate over all topics and lessons of the online textbook  ALGEBRA-II.

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