Annuities Payment

1. A Perspective about Compounding &
Discounting of Annuities
Special reference to Loan Schedules
Presenter : Mr. Puneet Sharma
IIMT College of Management

2. Introduction
• An Annuity is a series of Payment made over a
due course of time.
• About 80% of Finance and its relevant studies
are based on the present/future value of the
annuities
• The repayments on loans are calculated by
deriving the present value of the future
payments i.e discounting the annuities to
present time by a specific rate of interest
• Discounting and accumulating is based on the
power of compounding principle of Finance

3. Objective
• The research is focused to gain an insight on
the interest rate and the total interest charged
by banks as has been communicated by them.
• I have taken example of Home loan for
deriving to the requisite results.

4. Study
• Let us see how calculation of Equated Monthly
Installments are done for a Loan
• Eg: If a Loan of 1,00,000 is taken for 5 years at
10% Rate of Interest and the EMI is payable on a
Monthly basis.
• The calculated EMI comes down to Rs. 2103.56.
The Question arises how, the answer to this is as
follows
• Let the EMI = E so,
1,00,000 = 12*E*(Present Value Annuity
factor payable monthly)

5. • In other words we can say that the present value
of the 60 payments to be made in future
computed at a ROI of 10% should be 1,00,000.
• The PV Annuity Factor comes down to 3.961542
• So EMI Comes down to 1,00,000/(12*3.961542) =
2103.56
Now lets see what banks charge

6. • SBI – EMI for the same loan = 2,124.7
• ICICI Bank – EMI for the same loan = 2,125
• Axis Bank – 2,126
• Federal Bank – 2,125
(Source: Banks official website)

7. • As rightfully seen the charged amount by all
the banks are the same and pretty over the
actual amount.
Rs. 2,125 – Rs. 2,103.56 = Rs. 21.44
The total amount paid extra over the period of 5
years for a loan of Rs.1,00,000 come down to Rs.
1284.4

8. • Total Interest paid = Rs.27,500
• Total Interest should have been paid =
Rs.26,216
• Extra interest paid = Rs.1,284
The customer ends up paying this much more
amount, however the amount is not of much
significant for small loans and short tenure but Home
loans are generally of longer tenure. Lets see an
example of a 20 year loan.

9. • Eg: A Loan of Rs. 50,00,000 at the ROI of 10%
P.A for 20 years
– What Banks charge is an EMI of = Rs. 48,251
– What actually should be charged = Rs. 46,831.76
• (based on the PV of annuity factor calculation)
- Extra amount paid by the customer per month
= Rs. 1419.24
- So he ends up paying Rs.17,030.88 extra
annually, and Rs. 3,40,617.6 extra amount in 20
years

10. Findings
• The question arises as to why this difference
occurs. The answer is the difference in the
rate of interest charged either monthly,
quarterly, half yearly or yearly.
• To state it simply an effective rate of interest
of 10% P.A is not equal to an effective rate of
interest of 5% for half year. It would be an
effective rate of 4.88% half yearly.

11. • We simply do not divide the rate of interest by
the time period.
• In the above example if suppose Mr. X has
borrowed Rs. 100 for 1 year at the rate of
10%P.A and he wishes to pay it in 6 Months he
will not pay Rs.105 i.e 100 Rs. Principal and Rs.
5 Interest, he would pay Rs. 104.88 i.e an
interest of Rs. 4.88, based on the principle of
compounding.

12. • Let’s see it with an example 10% P.A
= 100*(1+i)^n = 110
(where i=10% and n=1year)
So for half year 100*(1+i)^n = 104.88
100*(1.1)^(1/2)=104.88

13. • So from the above we can derive that if Mr. X
wants to repay the loan in 1 month he would
pay
• 100*(1+i)^(1/12) = Rs.100.7974
As against what we perceive Mr. X should pay
100*(1+i/12) = Rs.100.8333

14. Conclusion
• We hence can derive to a formula for converting
between rate of interest payable Pthly as
• Half yearly = (1+i)=(1+i2/2)^2
• Quarterly = (1+i) = (1+i4/4)^4
• Monthly = (1+i) = (1+i12/12)^12
(Where i2, i4, & i12 are expressions for Half
Yearly, Quarterly and Monthly respectively

15. • Eg @ 10%
– Half yearly
• (1.1) = (1+i2/2)^2
–So i2/2={(1.1)^(1/2)}-1
»(where i2/2 is the effective rate of
interest paid half yearly)
»So it comes down to 0.0488

16. • Eg @ 10%
– Quarterly
• (1.1) = (1+i4/4)^4
–So i4/4={(1.1)^(1/4)}-1
»(where i4/4 is the effective rate of
interest paid quarterly)
»So it comes down to 0.0241

17. • Eg @ 10%
– Monthly
• (1.1) = (1+i12/12)^12
–So i12/12={(1.1)^(1/12)}-1
»(where i12/12 is the effective rate of
interest paid monthly)
»So it comes down to 0.007974