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# Simple and Compound Interest

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Though we had learnt about Simple and Compound Interests at school, because of the technological advantages and new gadgets over the years we have forgotten how to calculate it. This is my sincere effort to refresh the minds of interested persons about its concepts and how to calculate mannually.

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### Simple and Compound Interest

1. 1. SIMPLE AND COMPOUND INTEREST A Presentation By Himansu S M / 31-Jul-2013
2. 2. What is Interest?  When you borrow Money from someone  Or use somebody else’s Money  You have to pay a service charge to him.  This amount is paid back to the Lender along with the original amount borrowed.  This is sometimes known as the cost of Money which doesn’t belong to you, but you have used it. 31-Jul-2013 2(C) Himansu S M
3. 3. What is Interest?  This extra amount is called the “INTEREST”  The original amount borrowed is known as the “PRINCIPAL” or “CAPITAL” in different situations  The sum of both Principal and the interest is known as “AMOUNT” 31-Jul-2013 3(C) Himansu S M
4. 4. Types of Interest  There are basically TWO types of Interest  They are:  SIMPLE INTEREST  COMPOUND INTEREST 31-Jul-2013 4(C) Himansu S M
5. 5. Interest Calculation  To estimate or calculate the Interest we must have the following parameters as input:  A rate known as the Rate of Interest (RI or RoI) which is expressed in Percent per Year  A time period expressed in Years or Months or Days 31-Jul-2013 5(C) Himansu S M
6. 6. Interest Calculation  The Principal on which the Interest is to be calculated  And finally the Type of Interest (Methods of calculation are different)  For advance Business Applications the “Number of times of Interest Accrual in a year” is required  This is known as Compounding 31-Jul-2013 6(C) Himansu S M
7. 7. Simple Interest  Simple Interest is dependent on:  Rate of Interest  Time Period  Principal  And the Principal remains the same at the beginning of all the Periods  It means that the accrual of Interest is linear 31-Jul-2013 7(C) Himansu S M
8. 8. Compound Interest  Compound Interest is dependent on:  Rate of Interest  Time Period  Principal  And the Principal increases by the interest amount at the end of each Period  Interest for the next period is calculated on this increased Principal 31-Jul-2013 8(C) Himansu S M
9. 9. Compound Interest  It means that the Principal plus Interest of one period becomes the Principal for the next period  This goes on till the total time period for which the compound interest is calculated  This Period is called the period of compounding or the compounding interval 31-Jul-2013 9(C) Himansu S M
10. 10. Compound Interest  At the end of each such period the accrued interest is added to the Principal and this becomes the Principal for the next interval  In other words, the interest earns interest  Some call it reinvesting or cumulative 31-Jul-2013 © Himansu S M 10
11. 11. Compound Interest  It means that the accrual of Interest is NOT linear, but exponential  The compounding may be  Yearly, Half-Yearly,  Quarterly, Monthly,  Weekly, Daily,  Continuous (Infinitely Compounded) 31-Jul-2013 11(C) Himansu S M
12. 12. Comparison [ @ 10% pa ] Simple Interest Year Principal Interest 1 100 10 2 100 10 3 100 10 4 100 10 5 100 10 6 100 10 7 100 10 8 100 10 TOTAL 100 80 Compound Interest Year Principal Interest 1 100 10 2 110 11 3 121 12.1 4 133.1 13.3 5 146.4 14.6 6 161.1 16.1 7 177.2 17.7 8 194.9 19.5 TOTAL 100 114.5 31-Jul-2013 12(C) Himansu S M
13. 13. Interests Graph 0 5 10 15 20 25 30 35 40 45 Year1 Year2 Year3 Year4 Year5 Year6 Year7 Year8 Year9 Year10 Year11 Year12 Year13 Year14 Year15 Year16 Simple Compound 31-Jul-2013 13(C) Himansu S M
14. 14. Amount Graph 0 50 100 150 200 250 300 350 400 450 500 Year0 Year1 Year2 Year3 Year4 Year5 Year6 Year7 Year8 Year9 Year10 Year11 Year12 Year13 Year14 Year15 Year16 Simple Compound 31-Jul-2013 14(C) Himansu S M
15. 15. Formula for Interest Calculation  Let’s assume:  Principal = P  Amount = A  Total Interest = I  Interest Rate = i expressed in % pa  Time Period = t expressed in Years  Frequency of Compounding = n expressed in no. Of times in a Year 31-Jul-2013 15(C) Himansu S M
16. 16. Formula for Simple Interest  A = P + I  Example:  If P = 100,  I = 50  Then A = 100 + 50 = 150 31-Jul-2013 16(C) Himansu S M
17. 17. Formula for Simple Interest  I = P * t * i / 100  Example:  If P = 150  i = 12 % pa,  And t = 3 Yrs  Then I = 150 * 3 * 12/100 = 54 31-Jul-2013 17(C) Himansu S M
18. 18. Formula for Simple Interest  A = P * (1 + t * i / 100)  Example:  If P = 150  i = 12 % pa,  And t = 3 Yrs  Then A = 150 * (1 + 3 * 12 / 100)  150 * 1.36 = 204 31-Jul-2013 18(C) Himansu S M
19. 19. Formula for Simple Interest  Out of the Five Basic Variables:  Principal  Amount  Interest  Time Period and  Rate of Interest,  If we know any Three, then rest can be calculated by manipulating the formula 31-Jul-2013 19(C) Himansu S M
20. 20. Formula for Compound Interest  Pls note that the “Simple Interest” CAN be directly calculated, but the “Compound Interest” CAN’T be directly calculated.  First the Amount is calculated and then the difference of Amount & Principal is the “Interest”  A = P + I  I = A – P 31-Jul-2013 20(C) Himansu S M
21. 21. Formula for Compound Interest  A = P * ( 1 + i / 100 / n ) ^ ( t * n )  [ the symbol ^ denotes “to the power of” or “raised to” ]  i = Rate of interest  t = Time period  n = Compounding frequency  P = Principal  A = Amount 31-Jul-2013 21(C) Himansu S M
22. 22. Formula for Compound Interest  Example: let’s take the same example as our previous slide – the graph of comparison  P = 100  t = 8 yrs  i = 10 % pa.  n = 1 time every Year  See next slide- 31-Jul-2013 22(C) Himansu S M
23. 23. Formula for Compound Interest  A = 100 * ( 1 + 10 / 100 / 1 ) ^ ( 8 * 1 )  = 100 * ( 1.10 ) ^ 8  = 100 * 2.144  = 214.4  So I = A – P = 214.4 – 100 = 114.4  Which matches our result. 31-Jul-2013 23(C) Himansu S M
24. 24. Formula for Compound Interest  How to find Rate of Interest:  If A, P, t are given  For simplicity let’s assume n=1  Then the formula is:  i = [ { ( A / P ) ^ (1 / t ) } – 1 ] * 100 31-Jul-2013 24(C) Himansu S M
25. 25. Formula for Compound Interest  Example: Let’s take the Last example  A = 214.4  P = 100  t = 8  n = 1  i = [{( 214.4 / 100 ) ^ ( 1 / 8 )} – 1 ] * 100  = ( 2.144 ^ 0.125 – 1 ) * 100  = ( 1.10 -1 ) * 100 = 0.10 * 100 = 10 % pa 31-Jul-2013 25(C) Himansu S M
26. 26. Some Norms  Simple Interest is rarely used in today’s world  Business, Banks, Statistics, Finance, Dem -ography, Population, Accounting, every- where the Compounding Interest / Growth / Increase are used. 31-Jul-2013 26(C) Himansu S M
27. 27. Some Norms  If the compounding interval is not mentioned then it is assumed to be “Yearly”  The compounding interval is NEVER more than a Year, it means the value “n” is never less than 1  So mentioning only the rate of interest without the compounding interval is incomlete information 31-Jul-2013 27(C) Himansu S M
28. 28. Compounding Interval  The more the Compounding Frequency,  Or the less the Compounding Interval,  The more is the Effective Annual Interest.  The Formula for Calculation is:  A = P * ( 1 + i / 100 / n ) ^ ( t * n )  And the Effective Annual Interest is:  I = A - P 31-Jul-2013 28(C) Himansu S M
29. 29. Compounding Interval  Example: Let’s Say:  P = 100  i = 12 % pa  t = 1 Year  n = 1 (Yearly), 2 (Half-Yearly), 4 (Quarterly), 12 (Monthly), 52 (Weekly), 365 (Daily), etc.  Let’s Calculate: 31-Jul-2013 29(C) Himansu S M
30. 30. Compounding Interval  For n = 1, Yearly Compounding  A = 100 * ( 1 + 12 / 100 / 1 ) ^ ( 1 * 1 )  = 100 * ( 1.12 ^ 1 ) = 112  Effective i = ( 112 – 100 ) / 100  = 12 / 100 = 12 % pa. 31-Jul-2013 30(C) Himansu S M
31. 31. Compounding Interval  For n = 2, Half-Yearly Compounding  A = 100 * ( 1 + 12 / 100 / 2 ) ^ ( 1 * 2 )  = 100 * ( 1.06 ^ 2 ) = 112.36  Effective i = ( 112.36 – 100 ) / 100  = 12.36 / 100 = 12.36 % pa. 31-Jul-2013 31(C) Himansu S M
32. 32. Compounding Interval  For n = 4, Quarterly Compounding  A = 100 * ( 1 + 12 / 100 / 4 ) ^ ( 1 * 4 )  = 100 * ( 1.03 ^ 4 ) = 112.55  Effective i = ( 112.55 – 100 ) / 100  = 12.55 / 100 = 12.55 % pa. 31-Jul-2013 32(C) Himansu S M
33. 33. Compounding Interval  For n = 12, Monthly Compounding  A = 100 * ( 1 + 12 / 100 / 12 ) ^ ( 1 * 12 )  = 100 * ( 1.01 ^ 12 ) = 112.68  Effective i = ( 112.68 – 100 ) / 100  = 12.68 / 100 = 12.68 % pa. 31-Jul-2013 33(C) Himansu S M
34. 34. Compounding Interval  For n = 52, Weekly Compounding  A = 100 * ( 1 + 12 / 100 / 52 ) ^ ( 1 * 52 )  = 100 * ( 1.0023 ^ 52 ) = 112.734  Effective i = ( 112.68 – 100 ) / 100  = 12.734 / 100 = 12.734 % pa. 31-Jul-2013 34(C) Himansu S M
35. 35. Compounding Interval  For n = 365, Daily Compounding  A = 100 * ( 1 + 12 / 100 / 365 ) ^ ( 1 * 365 )  = 100 * ( 1.00033 ^ 365 ) = 112.747  Effective i = ( 112.68 – 100 ) / 100  = 12.747 / 100 = 12.747 % pa. 31-Jul-2013 35(C) Himansu S M
36. 36. Continuous Compounding  What if we keep increasing value of “n” further to say very high no. or infinity,  That means the interval getting smaller and smaller to say zero.  This is known as Infinitely compounding, for which the formula is:  A = P * e ^ ( i * t ) 31-Jul-2013 36(C) Himansu S M
37. 37. Continuous Compounding  Where e = base of natural logarithm = 2.71828  But there are a few tricks:  The “ I ” is expressed in decimal –  Example: 12 % is 0.12  The “ t ” is expressed in multiples of the period of interest rate –  Example: if RoI is per annum then 3 yrs 6 months shall be 3.5 yrs. 31-Jul-2013 37(C) Himansu S M
38. 38. Continuous Compounding  Calculation Example:  Let’s take same example of P = 100 & i = 12 % pa. If t = 1 yr what is I?  A = P * e ^ ( i * t )  = 100 * 2.71828 ^ (0.12 * 1 )  = 100 * 2.71828 ^ 0.12  = 100 * 1.127497 = 112.7497  So I = 112.7497 – 100 = 12.7497 31-Jul-2013 38(C) Himansu S M
39. 39. Continuous Compounding  We see here as the compounding frequency increases or interval decreases the effective annual rate increases  So, in the limiting case it is the highest yielding of all the other frequencies of compounding.  Theoretically it is the highest compound interest. 31-Jul-2013 39(C) Himansu S M
40. 40. Continuous Compounding Interval “ n ” Value RoI (Say) Effective Annual RoI (%) Yearly 1 12 % pa 12.0000 Half-Yearly 2 12 % pa 12.3600 Quarterly 4 12 % pa 12.5509 Monthly 12 12 % pa 12.6825 Weekly 52 12 % pa 12.7341 Daily 365 12 % pa 12.7475 Continuous Infinity 12 % pa 12.7497 31-Jul-2013 © Himansu S M 40
41. 41. Exercise  You might have seen Bank Ads depicting 14 – 17 % effective annualised yield after say 5-7-10 yrs. Let’s calculate that as an exercise and see how they do it.  Let’s say ABCD Bank gives 10% pa to Senior citizens for Fixed Deposits (all bank interests are quarterly compounded). Now let’s calculate the effective annualised yield for 5-7-10 yrs: 31-Jul-2013 © Himansu S M 41
42. 42. Exercise  t = 5 yrs  n = 4 Quarterly Compounded  i = 10 % pa.  So A = ( 1 + 10% / 100 / 4 ) ^ ( 5 * 4 )  = ( 1 + 0.025 ) ^ 20 = 1.025 ^ 20  = 1.6386 So I = 0.6386 in 5 yrs.  Effective i = 0.6386 / 5  = 12.772 % pa 31-Jul-2013 © Himansu S M 42
43. 43. Exercise  t = 7 yrs  n = 4 Quarterly Compounded  i = 10 % pa.  So A = ( 1 + 10% / 100 / 4 ) ^ ( 7 * 4 )  = ( 1 + 0.025 ) ^ 20 = 1.025 ^ 28  = 1.9965 So I = 0.9965 in 7 yrs.  Effective i = 0.9965 / 7  = 14.236 % pa 31-Jul-2013 © Himansu S M 43
44. 44. Exercise  t = 10 yrs  n = 4 Quarterly Compounded  i = 10 % pa.  So A = ( 1 + 10% / 100 / 4 ) ^ ( 10 * 4 )  = ( 1 + 0.025 ) ^ 20 = 1.025 ^ 40  = 2.685 So I = 1.685 in 10 yrs.  Effective i = 1.685 / 10  = 16.85 % pa 31-Jul-2013 © Himansu S M 44
45. 45. Conclusion  With the help of Compound Interest one can trick others.  Compound interest can give very high returns after longer periods.  Example: Say a young man gets a lumpsum from his father at the age of 25. If he puts it for 12 % pa till he retires at 60, let’s compare the Simple & Compound Interest 31-Jul-2013 © Himansu S M 45
46. 46. Conclusion  Simple Interest:  12 % pa * 35 yrs = 420 % or 4.2 times  The Amount will be 1 + 4.2 = 5.2 times  Compound Interest:  ( 1 + 12 / 100 ) ^ 35 = 52.8 times  Hence the return is 10 times in 35 yrs.  Such is the POWER OF COMPOUNDING! 31-Jul-2013 © Himansu S M 46
47. 47. Disclaimer!  This is my sincere effort to guide the interested persons how to calculate the compound interest.  Anyone can use a scientific calculator or any calculator where “x^y” function is built- in.  One can use a PC with excel worksheet to do complex calculations. 31-Jul-2013 © Himansu S M 47
48. 48. THANK YOU Himansu S M / 31-Jul-2013 31-Jul-2013 (C) Himansu S M 48