JAIIB/Diploma in Banking & Finance Accounting & Finance for

Bankers

S.C.BansalMODULE-A

BUSINESS MATHEMATICS –

Presentation & MCQs

Why Mathematics in Banking

• To calculate interest on deposits and advances• To calculated yield on bonds in which banks

have to invest substantial amount.• To calculate depreciation• To decide on buying/selling rates of foreign

currencies• To calculate minimum capital required by the

bank• To appraise loan proposals

What level of maths is required

• In banking, very high level of maths is not needed

• We should know the following basic mathematical operations

• Addition,e.g. 24+33+9+56=122

• Subtraction,e.g.138-41-72=25

• Multiplication,e.g. 1.1*1.1=(1.1)2 =1.21

• Division,e.g.1/12=0.0833

Weightage of maths in JAIIB/DBF Exam

• Constitutes one of the four modules of the paper of Accounting & Finance( total 3 papers)

• Therefore, the weightage in the total exam is about 10%

• It is possible to get good score in questions related to this module, as only simple mathematics is involved and use of calculator is allowed

Can we cover entire syllabus in this class

• We can have conceptual clarity of the entire syllabus on business maths.

• You need to read the book thoroughly and solve problems

• You may get in touch with me whenever you need any clarification

• My mail id is bansalsc2006@yahoo.com

Simple interest

• Important symbols ; P=amount deposited initially, called Principal

• r=rate of interest. 12% per annum means that if you deposit Rs 100 for one year,you will get interest of Rs 12 at the end of the year.In our calculations,we will take r=12/100=0.12 p.a.

• T=number of years for which P is deposited• I=total interest receivable. I=P*r*T• A=amount receivable.A=P+I=P+(P*r*T)=P(1+rT)

Compound interest

• If you deposit Rs 100 @12%p.a.,it becomes Rs 112 at the end of one year.For next year,you should get interest on Rs112,which is 112*12/100=13.44.This is called compounding.In case of simple interest, you would have received interest of Rs 12 only for the 2nd year also.

• Compounding can be yearly,as shown above, or can be monthly,quarterly,half yearly etc.More frequent compounding means more interest for you.

• In yearly compounding, A=P(1+r) after 1year, P(1+r)2 after 2years,and so on.After T years, A=P(1+r)T

• If compounding is n times in a year, A=P(1+r/n)nT

• Rule of 72 is used to find the period in which our money doubles.

Discount factor

• We have seen that P becomes P(1+r)T in T years.Therefore,if somebody promises to give you Rs P(1+r)T after T years,you should know that it is worth only Rs P today.

• Amount receivable in future is to be multiplied by a number(always less than one) to arrive at the present worth of that amount.

• In above example,P(1+r)T is to be multiplied by 1/(1+r)T to arrive at present worth P. So ,The discount factor is 1/(1+r)T.

• E.g.,if rate of intt is 10%p.a., r=0.10. Therefore, discount factor is 1/1.10 for 1 year, 1/1.21for 2 years and so on.

Present value of money

• PV= Future amount* Discount Factor(DF)• DF = 1/(1+r)T

• E.g.,if rate of intt is 10%p.a., r=0.10. Therefore, discount factor is 1/1.10 for 1 year, 1/(1.10)2 =1/1.21 for 2 years and so on.

• In above example,PV of Rs 100 ,to be received after 2 years will be, 100*1/(1.10)2 =100/1.21=Rs 82.64.Similarly,PV of Rs 100,to be received after 5 years, will be100*1/(1.10)5

Future value of money

• Depending on the rate of interest, the amount you receive in future(A), will be more than the amount(P) available now.

• A=P(1+r)T ,when the compounding is yearly.• Therefore,FV=Present Amount*(1+r)T . We call (1+r)T

compounding factor.• E.g.,if rate of intt is 10%p.a., r=0.10. Therefore,

compounding factor is 1.10 for 1 year, (1.10)2 =1.21 for 2 years and so on.

• In above example,FV of Rs 100 , after 2 years will be, 100*(1.10)2 =100*1.21=Rs 121.Similarly,FV of Rs 100, after 5 years, will be100*(1.10)5

Annuities

• A series of fixed payments/receipts at a specified frequency, over a fixed period.

• E.g. Payment of Rs 1000 every year by LIC for next 20 years . Also, a Recurring deposit with bank for Rs 100 for 5 years.

• 2 types of Annuities. Ordinary Annuity; payment is at the end of the period. Annuity Due;payment is at the beginning of each period.

Present and Future value of Annuity

• For calculating PV of Annuity, PV of each payment is calculated and added.E.g. if Rs 100 is paid at the end of each year for 10 years, we calculate pv of each of these 10 payments of Rs 100 separately and add these 10 values.

• Similarly, for calculating FV of Annuity, FV of each payment is calculated and added.E.g. if Rs 100 is paid at the end of each year for 10 years, we calculate fv of each of these 10 payments of Rs 100 separately and add these 10 values.

Precaution while calculating PV and FV

• In the formulae, given in the books,we have to correctly arrive at r, i.e.the interest rate.E.g.the given intt rate is 12%p.a.If the payment is received yearly, r will be equal to 12/100=0.12.But if payment is received monthly, it will be 12/100*12=0.01.For quarterly payment, it will be 0.03 and for half yearly payment, it will be 0.06

Sinking fund

• Concept same as that of Annuity

• Suppose, you need a fixed amount(A) after,say, 5 years.You deposit an amount(C)every year with a bank.This becomes A after 5 years and can be used for repaying a debt or any other purpose.As the rate of intt and the FV is known, we can calculate C.

Bonds

• A Bond is a form of debt raised by the issuer of the bond.

• Issuer of the bonds pays interest to the purchaser for using his money.

• Terms associated with bonds: Face value, Coupon rate, Maturity, Redemption value, Market value.

• Face value and redemption value may be different but these are fixed and known.

• Market value of the bond may be different form the face value and keeps changing.

Valuation of bonds

• The purchaser of the bonds gets regular interest payments as also the redemption amount on maturity.

• The interest on bond( also called coupon rate) is fixed at the time of its issue. But interest rate in the market keeps changing, and,therefore,market price of bond also changes.

• The market price or intrinsic value of a bond is different from the face value if the coupon rate is different from the market interest rate at that particular time.

• Market value is equal to PV of all the coupon receipts and redemption value discounted at the prevailing market rate.

Yield on bonds

• Current yield =coupon interest/current market price.

• E.g. if face value of a bond is Rs 50, coupon rate is 8% pa, and market price is Rs 40, then the current yield=4/40=0.1 or 10%

• Yield to Maturity(YTM) is that discount rate at which all future cash flows equal the present market value.

Theorems for bond valuation

• Effect of change in market interest rate

• Effect of maturity period

• Bond price is inversely proportional to YTM

• Interest rate elasticity= %age change in price/%age change in YTM

Capital budgeting

• Used to choose between various projects.• A capital project involves capital outflow( investment)

and capital inflows(net profit) over the life of the project.• PV of all cash inflows will be +ve and PV of all cash

outflows will be negative.PV will depend on the discount rate( cost of capital)

• Summation of all the PVs of cash inflows and outflows is called Net Present Value(NPV)

• IRR is that discount rate at which NPV of a project is zero.

• Other method used for capital budgeting is pay back period method.

Depreciation

• Concept of depreciation

• Straight line method;(cost-residual value)/ estiamted usful life

• Written Down Value method or declining balance mehtod : %age is fixed

Forex Arithmatics

• Earlier RBI used to fix buying and selling rates of Forex.Now LERMS( liberalised exchange rate management system) is used.

• Direct and indirect quotations.From 2-8-93 only direct quotations are being used.

• Cross rate/chain rule; e.g. if 1US$=Rs 48 and 1Euro=US$1.25, then 1Euro=Rs1.25*48

• Value date: Cash/ready,TOM, Spot, Forward• Premium and discount.• Factors affecting premium/discount

Capital adequacy

• Need for capital in banks.

• How much capital?

• Basel II norms

• RBI norms

Sample questions• 1.What is the Present Value of Rs. 115,000 to be received after 1

year at 10%?– 121,000– 100,500– 110,000– 104,545

• 2.At 10% p.a. 2 year discount factor is – 0.826– 1.00– 0.909– 0.814

• 3.If 1 year discount is 0.8333, what is the discount rate?– 10%– 20%– 30%– 15%

Sample questions

• 4.Present Value is defined as – Future cash flows discounted to the present at an appropriate

discount rate– Inverse of future cash flows– Present cash flows compounded into the future– Discounting of compounded future cash flows

• 5.Annuity is defined as – Equal cash flows at equal intervals forever– Equal cash flows at equal intervals for a specified period– Unequal cash flows at equal intervals for specified period– Unequal cash flows at equal intervals forever

Sample questions• 6.What is the N P V of the following at 15%• t = 0 t = 1 t = 2 -120,000 -100,000 300,000• 19,887• 80,000• 26,300• 40,000• 7.A bond holder of a company has one of the following

relationship with • It .Identify

– shareholder– depositor– creditor– employee

•

Sample questions

• 8.The yield to maturity is a rate of return which – gives the current yield– Is the discount rate at which the present value, of the coupons

• and the final payment at face value, equals the current price

– gives the return at maturity on the bond for the original holder– b) or c)

• 9.The relationship between the bond prices and interest rates is one of the

• Following– direct & linear– inverse & linear– direct and curvilinear– no relationship

Sample questions

• 13) What does the rate of return equal to if interest rates do not change during the pendency of the bond ?

– yield to maturity– coupon rate– compounded rate– current yield14.A Bond of face value Rs.5000 carries a coupon interest rate of 12%. It

is quoted in the market at Rs.4500. What is the current yield of the bond?

– 12%– 10%– 13.3%– 14.2%

Sample questions

• 15.Which of the following investment rules does not use the time value of the money concept?

• A.The payback period

• B.Internal rate of return

• C.Net present value

• D.All of the above use the time value concept

Sample questions• 16.A capital equipment costing Rs200,000 today hasRs 50,000

slavage value at the end of 5 years. If the straight line depreciation method is used, what is the book value of the equipment at the end of 2 years?

• Rs200,000• Rs170,000• Rs140,000• Rs50,00017.Cost of Car is Rs. 300,000, Depn. Rate is 10% on WDV. What is the

book value of car after 3 years.• 210,000• 220,00• 214,300• 218,700

Sample questions

• 18.If P=principal, r = rate of interest , n= number of instalments

• Then formula for equated monthly instalment (EMI) is

(p*r)(i+r)n

• (1+r)n – 1

Sample questions

• 19.If the rates in Mumbai are US $1=Rs.42.850 .In London market are US $ 1=Euros 0.7580 Therefore for one Euro we will get

• a) Rs.56.45

• b Rs.56.53

• c) Rs.56.38

• d) Rs.56.50