Simple and Compound Interest for Quantitative Techniques
Simple and Compound Interest for Quantitative Techniques
Principal: The Money lent or borrowed for a particular period is called the Principal, also known as the Sum of money. Denoted usually by P.
Interest: When a borrower pays extra money to the lender to compensate for the opportunity cost of the lender, is called the Interest. It is usually represented in the form of percentages. Usually denoted by R. It can be of two different kinds: Simple or Compound Interest.
Simple Interest: The interest is fixed over the period and doesn’t interest on itself.
Let the time be denoted by T.
Amount = S.I. + P
Example: Find the simple interest on Rs. 60000 at 6.5% per annum for 4.5 yrs.
Solution: P = 60000 ; R = 6.5% ; T= 4.5 yrs
S.I = P*R*T/100 = 60000*4.5*6.5/100 = 6*45*65 = Rs. 17550
If the Amount equals to double the Principal at simple interest in T yrs, then R = ( 100/T ) %
Example: If the Principal amount is Rs.100 which amounted to Rs. 200 in 3 yrs, then find the Rate of Interest.
Solution: The condition given is of double Principle hence we can use the above formula
R = (100/3) =33.33%
Alternate: S.I = 200 - 100
100=100(R*3/100)
1 = R*3/100
R = 100/3 = 33.33%
If the Amount equals to triple the Principal amount in T yrs at R%, then R = (200/T) %.
If the Amount equals to n times the Principal amount in T yrs at R%, then R =((n-1)*100/T) %
Compound Interest: Concept of interest on interest is applicable in case of compound interest. Hence the lender earns interest on completion of one time period and further interest on the earlier earned interest.
When money is compounded Annually:
The compound interest formula can be varied for the different time by substituting R by R/n and T by T*n where n is how frequent compounding is being done.
For Example:
When money is compounded Semi-Annually:
When money is compounded Quarterly:
When money is compounded Monthly:
Compound Interest = Compound Amount - Principal
Example: Find the compound interest on Rs. 10000 at 5% per annum for 2 yrs, compounded annually.
Solution: C.I = P[ (1+R/100)2 -1] = 10000 [(1+5/100)2-1]
= 10000 [(21/20)2 -1] = 500
Example: If I invest Rs. 2000 for 3 yrs at 25% p.a. and earn compound interest. Which compounding will yield me a better amount yearly or half-yearly.
Solution: Principal = Rs. 2000, Time =3 yrs and Rate of interest= 25%
C.Amount (yearly) = P (1+R/100)^T = 2000 (1+ 25/100)3
= 2000 (5/4)3 = 2000 * 1.95 = 3906.25
C.Amount (half-yearly) = P (1+ R/2*100)^T*2 = 2000 (1+ 25/200)3*2
= 2000 (9/8)6 = 4054.57
Compounding done semi annualy will amount more in the given investment.
When Rates are different for different years lets say R1, R2 and R3 for three consecutive years then,
When the Principal becomes x in T years, then the rate of the compound is
When the Principal amount A1 in T years while it amounts to A2 in (T+1) years at CI, Then
If a person takes a loan from the bank at R% and agrees to pay it back in equal instalments for T years, then the value of each instalment is:
Where X is the instalment amount.
Rule of 72: If the amount equals to double the Principal at R% in T yrs then we can use the formula R = 72/T or T = 72 / R to find either R and T.
Suppose C.I of 7th year = P(1+R/100)7 - P(1+R/100)6
= P(1+R/100)6 *R/100
Hence using the above example we can say
Compound Interest on nth year:
Difference between Compound Interest and Simple Interest if calculated for the same number of years and Same rate of interest
T = 1yr;
C.I = P[(1+R/100)-1] = PR/100
S.I = PR/100
Difference =C.I – S.I = PR/100 – PR/100 = 0
T= 2 yrs;
C.I = P(1+R/100)2 – P
S.I = 2PR / 100
C.I – S.I = [P(1+R/100)2 – P] - 2PR / 100
= PR2/1002 (on solving previous equation)
T = 3 yrs;
C.I = P(1+R/100)3 – P
S.I = 3PR / 100
C.I – S.I = [P(1+R/100)3 – P] - 3PR / 100
= PR2/1002 [R/100 +3]
T = n yrs;
C.I – S.I = (C.I – S.I)2nd year * (R/100 + 3)
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