# Interest and Annuities

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INTRODUCTION

BANKING BUSINESS MAINLY CONSISTS OF ACCEPTING  DEPOSITS AND LENDING.BANK PAYS INTEREST TO THE DEPOSITORS ON LENDING  TO CUSTOMERS.  BANK CHARGES A CERTAIN INTEREST AT SPECIFIED RATE. •

INTEREST:   IT IS PAYABLE EITHER AT PERIODIC INTERVALS OR AT THE THE END OF THE LOAN PERIOD . CALCILATION OF INTEREST WILL BE BASED ON THE TERMS OF AGREEMENTN i.e WHETHER AT DEFINITE INTERVAL OR AT PERIOD END. SOPMETIMES CUSTOMER IS INTERESTED IN PAYINF A PART OF PRINCIPAL  ALONG WITH INTEREST. • •

ANNUITIES:   IT IS A SERIES OF FIXED PAYMENT REQUIRED TO BE PAID OVER THE COURSE OF FIXED PERIOD OF TIME.

SIMPLE  INTEREST

SIMPLE INTEREST IS THE INTEREST COMPUTED ON THE PRINCIPAL FOR THE ENTIRE PERIOD OF BORROWING. . INTEREST   AMOUNT IS ALWAYS SAME  FOR EVERY YEAR. ØI = P * r * t ØA = P + I ØSo, A = P ( 1+ rt) ØP = PRINCIPAL ØT= TIME ØR= PERCENTAGE OF PRINCIPAL CHARGEABLE AS INTEREST

QUESTION : SIMPLE INTEREST FOR A SUM OF Rs. 10,000 at 8.5% p.a. for 2 years: •Using , I = 10000* (8.5/100)*2 = Rs. 1700 •So, A = 10000 + 1700 = Rs. 11700

QUESTION : IF T IS 2 YEARS 6 MONTHS, THEN, T =21/2 YEARS = 5/2 YEARS •SIMPLE INTEREST WILL BE 10000*(8.5/100)*5/2 = Rs. 2125 •

QUESTION: MOHAN INVESTED 5000 IN MUTUAL FUND WITH INTEREST RATE @4.8% HOW MUCH INTEREST WOULD BE EARNED AFTER 2 YEARS?

•INTEREST CAN BE CALCULATED USING •I=P*R*T      •P=5000   R=4.8%   T=2 YEARS         I=5000*4.8%*2= 480 •MOHAN WOULD EARN 480 AFTER 2 YEARS

COMPOUND INTEREST

IF INTEREST IS CHARGED MORE THAN ONCE  DURING THE PERIOD OR INTEREST IS       REINVESTED , WE NEED TO COMPOUND THE INTEREST. BASICALLY IT IS INTEREST ON     INTEREST. ØA = P ( 1 + r )n    if the interest is compounded annually ØA  =  P (1 +r/t)nt if interest is compounded t number of times ØWhen  t becomes  infinity i.e continuously , A =  P* er n ØWhere  e = 2.71828N=  number of years

QUESTION :  HERE WE WILL CALCULATE QUARTERLY COMPOUNDING INTEREST FOR SUM OF RS. 10000 AT 8.5% FOR 2 YEARS. ( SAME AS ABOVE EXAMPLE) •NOW USING FORMULA, A = P *( 1+ r/t)nt  , time =4 (quarters in a year) •N = number of years i.e 2 years •= 10000*(1+.085/4)4*2  =  10000*(1.02125)8= 10000* 1.1832 =Rs 11832 •CI = 11832 – 10000 = Rs. 1832.

QUESTION: AVICHAL PUBLISHER BUY A MACHINE FOR 20,000 RATE OF DEP IS 10% FIND THE VALUE OF A MACHINE AFTER 3 YEARS? ALSO FIND THE AMOUNT OF DEPRECIATION?

SOL : USING =A=P*(I-r)  HERE  WE WILL SUBTRACT  AS THE VALUE OF MACHINE  WILL BE DEPRECIATED OVER 3 YEARS

P=20,000   RATE =10%  TIME=3 YEARS

=20000*(1-0.1)3    = 20000*.729

= Rs. 14,580

FIXED   INTEREST • •FIXED  RATE OF INTEREST MEANS RATE OF INTEREST IS FIXED. IT WILL NOT CHANGE  DURING THE ENTIRE PERIOD OF LOAN . •

FLOATING INTEREST RATE • •HERE RATE OF INTEREST WILL CHANGE DEPENDING UPON THE MARKET CONDITIONS.  RATE OF INTEREST CAN BE INCREASED OR DECREASED. •ALSO KNOWN AS VARIABLE RATES. •FIXED RATE IS NORMALLY  HIGHER THAN  FlOATING RATE .

ANNUITIES • • IT IS A SERIES OF FIXED PAYMENT REQUIRED TO BE PAID OVER THE COURSE OF FIXED PERIOD OF TIME. ØORDINARY  ANNUITIES –Payments  are required to be made at the end  of  each period . For  eg., Water Bill, Electricity Bill etc. ØANNUITY DUE  —Payments  are required to be made at the beginning   of  each period . For eg., Rent , fees etc.

CALCULATING THE PRESENT VALUE OF AN ORDINARY ANNUITY ØPV (ORDINARY ANNUITY)      =    C *(1+r)n – 1 / r( 1+r)n ØWhere,  C = Cash Flow per period ØR = rate of interest ØN = number of payments

QUESTION :  SUPPOSE YOU ARE RECEIVING Rs. 1000 EVERY YEAR FOR THE NEXT FIVE YEARS , AND YOU INVEST EACH PAYMENT AT 5% .

SOLUTION :  NOW USING THE ABOVE FORMULA  ,

= 1000* (1+.05)5 -1 / .05(1+.05)5

= 1000 * 0 .27628  / .05(1.27628)

= 1000 *0.27628 /.063814

=1000* 4.3295

=4329.45

CALCULATION OF FUTURE VALUE OF AN ORDINARY ANNUITY ØTHIS WILL BE CALCULATED USING THE FOLLOWING FORMULA Ø  FV (ORDINARY  ANNUITY)     =  C* ( 1+ i)n – 1 / i ØWhere, C = CASH FLOW PER PERIOD Ø I =  INTEREST RATE ØN = NUMBER OF PAYMENTS

QUESTION : USING THE ABOVE QUESTION  WE WILL CALCULATE THE FUTURE VALUE OF AN ORDINARY ANNUITY.

SOLUTION :   = 1000* (1 +.05)5 – 1 / 0.05

= 1000 * 5.53

=5525.63

CALCULATION OF  THE PRESENT VALUE OF AN ANNUITY DUE Ø P V ( ANNUITY DUE) = C *[ ( 1 + r )n – 1 / r ( 1 + r )n ] * (1 + r )

QUESTION : SUPPOSE YOU MAKE YOUR FIRST RENT  PAYMENT OF Rs. 1000(INTEREST BEING SAME 5 %)  AT THE BEGINNING OF THE MONTH . NOW YOU HAVE TO EVALUATE THE PRESENT VALUE OF YOUR 5 MONTH LEASE ON THAT SAME DAY.

USING THE ABOVE FORMULA

= 1000  *  [ (1 + .05)5 – 1 / .05(1 +.05)5 ] *1.05

= 1000* [.27628/.063814] *1.05

= 1000 *4.3295 *1.05 = 4545.92

CALCULATION OF FUTURE VALUE OF AN ANNUITY DUE ØFV ( ANNUITY DUE ) = C *[(1+r)n  – 1/ i ] * (1 + i)

QUESTION : SUPPOSE YOU MAKE PAYMENT FOR Rs. 1000 AT THE BEGINNING OF THE PERIOD RATHER THAN THE END ( INTEREST RATE IS STILL 5 %)

SOLUTION : USING THE ABOVE FORMULA WE WILL CALCULATE

= 1000* [ (1 +.05)5 – 1 / .05] * (1 + .05)

= 1000*5.53*1.05

=  5806.5

RULE 72

IT HELPS US TO FIND OUT THE NUMBER OF YEARS BEFORE THE MONEY GETS DOUBLED.

IT IS CALCULATED BY DIVIDING 72 BY r

EXAMPLE :  IF WE WANT TO KNOW A CASH FLOW OF Rs. 500 AT 6% p.a. INTEREST WILL BE DOUBLED IN HOW MANY  YEARS . THEN WE CAN SIMPLY CALCULATE IT BY DIVIDING 72 BY 6  i.e. in 12 years .

REPAYMENT OF DEBT

A DEBT IS TO BE REPAID AS PER THE TERMS OF THE CONTRACT WITH LENDER.  IN BANKING INDUSTRY  IN INDIA, FOLLOWING METHODS OF REPAYMENT ARE COMMON.

EQUATED MONTHLY/ QUARTERLY  INSTALLMENT COVERING BOTH PRINCIPAL AND INTEREST (EMIs) ØTHE FORMULA FOR CALCULATION OF EMI  IS Ø EMI = (P * r )* ( 1 +r)n / ( 1 + r)n – 1 ØWhere,   P = PRINCIPAL ( AMOUNT OF LOAN ) Ø          R = Rate of interest Ø         N = number of installments in the tenure

QUESTION : FOR A LOAN OF Rs. 100000 AT AN INTEREST RATE OF 12 % p.a. TO BE REPAID IN 12 MONTHS.

P = 100000

R = 12%/ 12 = 1% i.e .01

N= 12

EMI = (100000*.01) *(1+.01)12 / (1 +.01)12 – 1

= 1000* 1.126825 /.126825

= 8885

THUS THE EMI  =  Rs. 8885

BULLET / BALLOON REPAYMENT

HERE , IF THE ENTIRE LOAN AMOUNT IS REPAID AT THE END OF THE PERIOD WITH ACCUMULATED INTEREST ,THE AMOUNT CAN BE CALCULATED USING COMPOUND INTEREST FORMULA . BUT IF INTEREST IS PAID  PERIODICALLY AS AND WHEN APPLIED AND PRINCIPAL AMOUNT OF THE LOAN IS PAID AT THE END OF THE CONTRACT PERIOD. USUALLY A SINKING FUND IS CREATED TO REPAY THE LOAN UNDER THIS METHOD SO THAT THE FUNDS ARE READILY AVAILABLE FOR REPAYMENT  AND THE CASH  FLOWS ARE NOT BURDENED AT THE TIME OF REPAYMENT. ØF = A [(1+ i)n– 1 / i] ØWhere F is the future value of an annuity A Ø    R = rate of interest Ø    N = number of years Ø

QUESTION : HOW MUCH MONEY WILL A STUDENT OWE AT GRADUATION IF SHE BORROWS Rs. 3000 PER YEAR AT 5% INTEREST DURING EACH OF HER FOUR YEARS OF SCHOOL?

SO, USING  F = A [(1+i)n– 1 / i]

F = 3000[(1+.05)4– 1/.05

Rs. 12930