CHAPTER 1 CALCULATIONS OF INTEREST & ANNUITIESACCOUNTING & FINANCE FOR BANKERS - JAIIB BUSINESS MATHEMATICS AND FINANCE 1 Log on https://www.myonlineprep.com/app/Registers/go for more free study materials, mock test and for JAIIB & CAIIB exam. | MODULE-A CHAPTER 1 CALCULATIONS OF INTEREST & ANNUITIES CONTENTS 1. Simple Interest 2. Compound Interest 3. Calculation of Equated Monthly Installments 4. Fixed and Floating Interest Rates 5. Interest Calculation using Products/Balances 6. Calculation of Annuities 7. Types of Annuities 8. Calculation of Annuities 2 9. Sinking Fund Log on https://www.myonlineprep.com/app/Registers/go for more free study materials, mock test and for JAIIB & CAIIB exam. | CHAPTER 1 CALCULATIONS OF INTEREST & ANNUITIES INTRODUCTION Interest is the cost of using somebody else’s money. When you borrow money, you pay interest. When you lend money, you earn interest. When borrowing: In order to borrow money, you’ll need to repay what you borrow. In addition, to compensate the lender for the risk of lending to you (and their inability to use the money anywhere else while you had it), you need to repay more than you borrowed. When lending: If you have extra money available, you can lend it out yourself or deposit it in a savings account and let the bank lend it out. In exchange, you’ll expect to earn interest – otherwise, you might be tempted to spend the money today because there’s little benefit to waiting (other than planning for your future). Annuity: is a lump sum of cash invested to produce a monthly stream of income for a fixed period or for life. The income can start now (immediate annuity) or in the future (deferred annuity). Funds are not protected or insured by the issuers. The size of the future monthly check CALCULATION OF INTREST & 3 isn’t always a given – it depends if the annuity is fixed or variable. ANNUITIES Log on https://www.myonlineprep.com/app/Registers/go for more free study materials, mock test and for JAIIB & CAIIB exam. | CHAPTER 1 CALCULATIONS OF INTEREST & ANNUITIES SIMPLE INTEREST Simple interest is paid by the borrower at the end of each year at a fixed rate (called rate of interest). In other words no interest is paid on the amount of interest. Another word, the simple annual interest rate is the interest amount per period, multiplied by the number of periods per year. The simple annual interest rate is also known as the nominal interest rate (not to be confused with nominal as opposed to real interest rates). The simple interest can be calculated as: Interest = Principal x Rate x Time i.e. I=PRT (where P is principal, R is rate of interest and T is time) Illustration A lends Rs.30000 to B at 10% interest rate. The annual interest would be Rs.30000 By Formula, I=PRT 300000 x (10/100) x 1 = 300000 x .10 x 1 = 30000 4 Hence total amount payable by the borrower to the lender = Principal + interest = (300000+30000) = 330000 Log on https://www.myonlineprep.com/app/Registers/go for more free study materials, mock test and for JAIIB & CAIIB exam. | CHAPTER 1 CALCULATIONS OF INTEREST & ANNUITIES AMOUNT OF INSTALMENTS Repayment of the loan can be made on a Yearly, Half-yearly, Quarterly, Monthly or even Weekly periodicity. Hence the total amount repayable can be divided by the units of time period in a year. Illustration 1. For example in the above case, the total loan repayable is if repayment is Yearly Lender = Principal + interest = (300000+30000) = 330000 2. If repayment is half-yearly (year/2), the amount of installment would be Lender = Principal + interest/2 = (300000+30000/2) = 315000 3. If repayment is quarterly (year/4), the amount of installment would be Lender = Principal + interest/4 = (300000+30000/4) = 307500 COMPOUND INTEREST Compound interest is the addition of interest to the principal sum of a loan or deposit, or in other words, interest on interest. It is the result of reinvesting interest, rather than paying it out, so that interest in the next period is then earned on the principal sum plus previously-accumulated interest. Compound interest 5 is standard in finance and economics. Log on https://www.myonlineprep.com/app/Registers/go for more free study materials, mock test and for JAIIB & CAIIB exam. | CHAPTER 1 CALCULATIONS OF INTEREST & ANNUITIES Compound interest may be contrasted with simple interest, where interest is not added to the principal, so there is no compounding. The simple annual interest rate is also known as the nominal interest rate (not to be confused with nominal as opposed to real interest rates) CALCULATION OF INTEREST The total accumulated value, including the principal sum P plus compounded interest I, is given by the formula: Fv=Pv(r/n)nt Where: P is the original principal sum P' is the new principal sum r is the nominal annual interest rate n is the compounding frequency t is the overall length of time the interest is applied (usually expressed in years). The total compound interest generated is: P′ = P + I I = P (1 + r/n) n t − P 6 Log on https://www.myonlineprep.com/app/Registers/go for more free study materials, mock test and for JAIIB & CAIIB exam. | CHAPTER 1 CALCULATIONS OF INTEREST & ANNUITIES 1. Illustration Suppose a principal amount of Rs.1500 is deposited in a bank paying an annual interest rate of 4.3%, compounded quarterly. Then the balance after 6 years is found by using the formula above, with P = 1,500, r = 4.3% = 0.043, n = 4 and t = 6: So the new principal P′ after 6 years is approximately Rs.1, 938.84. Subtracting the original principal from this amount gives the amount of interest received: 2. Illustration Suppose a principal amount of Rs.1500 is deposited in a bank paying an annual interest rate of 4.3%, compounded half yearly Then the balance after 6 years is found by using the formula above, with P = 1,500, r = 4.3% = 0.043, n = 4 and t = 6: 7 Log on https://www.myonlineprep.com/app/Registers/go for more free study materials, mock test and for JAIIB & CAIIB exam. | CHAPTER 1 CALCULATIONS OF INTEREST & ANNUITIES So the new principal P′ after 6 years is approximately Rs.1, 921.24. The amount of interest received can be calculated by subtracting the principal from this amount. Rs. 1 , Rs. (1,921.24 − 1,500) = Rs. 421. (The interest is less compared with the previous case, as a result of the lower compounding frequency.) CONTINUOUS COMPOUNDING As n, the number of compounding periods per year, increases without limit, we have the case known as continuous compounding, in which case the effective annual rate approaches an upper limit of er − 1, where e is a mathematical constant that is the base of the natural logarithm. Continuous compounding can be thought of as making the compounding period infinitesimally small, achieved by taking the limit as n goes to infinity. See definitions of the exponential function for the mathematical proof of this limit. The amount after t periods of continuous compounding can be expressed in terms of the initial amount P0 as, P (t) = P0ert 8 Log on https://www.myonlineprep.com/app/Registers/go for more free study materials, mock test and for JAIIB & CAIIB exam. | CHAPTER 1 CALCULATIONS OF INTEREST & ANNUITIES RULE OF 72 In finance, the rule of 72, the rule of 70 and the rule of 69.3 are methods for estimating an investment's doubling time. The rule number (e.g., 72) is divided by the interest percentage per period to obtain the approximate number of periods (usually years) required for doubling. Although scientific calculators and spread sheet programs have functions to find the accurate doubling time, the rules are useful for mental calculations and when only a basic calculator is available. These rules apply to exponential growth and are therefore used for compound interest as opposed to simple interest calculations. They can also be used for decay to obtain a halving time. The choice of number is mostly a matter of preference: 69 is more accurate for continuous compounding, while 72 works well in common interest situations and is more easily divisible. There are a number of variations to the rules that improve accuracy. For periodic compounding, the exact doubling time for an interest rate of r per period is: Where T is the number of periods required. The formula above can be used for more than calculating the doubling time. If one wants to know the tripling time, for example, simply replace the constant 2 in the numerator with 3. As another example if one wants to know the number of periods it takes for the initial value to rise by 50%, replace the constant 2 with 1.5. 9 Log on https://www.myonlineprep.com/app/Registers/go for more free study materials, mock test and for JAIIB & CAIIB exam. | CHAPTER 1 CALCULATIONS OF INTEREST & ANNUITIES FLOATING INTEREST RATE A floating interest rate, also known as a variable or adjustable rate, refers to any type of debt instrument, such as a loan, bond, mortgage, or credit that does not have a fixed rate of interest over the life of the instrument. Floating interest rates typically change based on a reference rate (a benchmark of any financial factor, such as the Consumer Price Index). One of the most common reference rates to use as the basis for applying floating interest rates is the London Inter-bank Offered Rate, or LIBOR (the rates at which large banks lend to each other). The rate for such debt will usually be referred to as a spread or margin over the base rate: for example, a five-year loan may be priced at the six-month LIBOR + 2.50%. At the end of each six-month period, the rate for the following period will be based on the LIBOR at that point (the reset date), plus the spread. The basis will be agreed between the borrower and lender, but 1, 3, 6 or 12 month money market rates are commonly used for commercial loans. Typically, floating rate loans will cost less than fixed rate loans, depending in part on the yield curve. In return for paying a lower loan rate, the borrower takes the interest rate risk: the risk that rates will go up in future. In cases where the yield curve is inverted, the cost of borrowing at floating rates may actually be higher; in most cases, however, lenders require higher rates for longer-term fixed-rate loans, because they are bearing the interest rate risk (risking that the rate will go up, and they will get lower interest income than they would otherwise have had). 10 Log on https://www.myonlineprep.com/app/Registers/go for more free study materials, mock test and for JAIIB & CAIIB exam. | CHAPTER 1 CALCULATIONS OF INTEREST & ANNUITIES FIXED INTEREST RATE A fixed interest rate loan is a loan where the interest rate doesn't fluctuate during the fixed rate period of the loan. This allows the borrower to accurately predict their future payments. A fixed interest rate is based on the lender's assumptions about the average discount rate over the fixed rate period. For example, when the discount rate is historically low, fixed rates are normally higher than variable rates because interest rates are more likely to rise during the fixed rate period. Conversely, when interest rates are historically high, lenders normally offer a discount to borrowers to fix their interest rate over time, as rates are more likely to fall during the fixed rate period. FRONT-END AND BACK-END INTEREST RATE In case of front-end interest where deduction of interest is done from principal amount and after deducting the interest from the principal amount, the net amount is disbursed to borrower. For example when a customer approaches for a Gold Loan to a bank, and if that bank disburses the amount on gold loan after deducting interest for agreed period, say (monthly, quarterly, yearly etc.) months, and net amount after deduction of interest on principal is disbursed to customer such interest is called front-end interest However in banking practice other than gold loan, normal practice for interest calculation is back-end interest, in which full loan amount is disbursed to borrower and interest is calculated on the basis of 11 agreed monthly, quarterly, yearly basis. Log on https://www.myonlineprep.com/app/Registers/go for more free study materials, mock test and for JAIIB & CAIIB exam. | CHAPTER 1 CALCULATIONS OF INTEREST & ANNUITIES CALCULATION OF BANK INTEREST ON DAILY BASIS From April 1, 2010, interest on all savings bank account deposits is being calculated on a daily basis, thereby earning account holders higher interest income. This is due to the fact that the Reserve Bank of India has instructed banks to change the mechanism of interest income calculation. The calculation is done on the 'daily balance method' Earlier, banks would pay interest on the lowest available balance in the account between the tenth and the last day of the month. Any deposit in the account between the tenth and the end of the month, would not earn the account holder any interest as it is not part of the interest rate calculation. Any withdrawal between the same periods would result in lower interest income as the lowest balance would be taken into account for the calculation. Illustration Example: Manisha had an account balance of Rs 85,000 on April 10. He received a payment of Rs 300,000 on April 17 from the sale of some mutual fund units. On April 29, he made a down payment of Rs 320,000 to a builder for a property. This resulted in her account balance reducing to Rs 65,000. For the interest income calculation for the month of April, the bank would take Rs 65,000 as the base and pay her interest on that amount. So interest due to Manisha would be on Rs 65,000 for 30 days @ 3.5% p.a. which would be Rs 187. In spite of having a high account balance for most period of the month, Manisha lost interest income for 12 the month. Log on https://www.myonlineprep.com/app/Registers/go for more free study materials, mock test and for JAIIB & CAIIB exam. | CHAPTER 1 CALCULATIONS OF INTEREST & ANNUITIES Under this method of interest rate calculation, the best thing Manisha could do is ensure that all transactions are done between the first and ninth of any month so that he would get benefit of interest. This required proper planning. New method of interest rate calculation Interest will be paid @3.5% p.a. on the daily balance in the account at the end of the day. Here, the account holder will get interest on the actual day end balance. Under this method, Manisha’s interest income calculation would be: For the first 14 days of April, interest to be paid would be calculated on Rs 85,000; For the next 14 days of April, interest to be paid would be calculated on Rs 385,000 and; For the balance 2 days, interest to be paid would be calculated on Rs 65,000. So the total interest due to Manisha would be Rs 643. Under this method, Manisha's interest income is higher by Rs 456! Besides, she did not have to plan her withdrawals and deposits as he would receive interest on the actual account balance. As a savings bank account holder, you should be pleased with the latest change. Who would not like to see higher balance on account of higher interest income? PS: Interest rate calculation Formula 13 Daily interest = Amount (Daily balance) * Interest (3.5/100) / days in the year Log on https://www.myonlineprep.com/app/Registers/go for more free study materials, mock test and for JAIIB & CAIIB exam. | CHAPTER 1 CALCULATIONS OF INTEREST & ANNUITIES ANNUITIES An annuity is a series of payments made at equal intervals. Examples of annuities are regular deposits to a savings account, monthly home mortgage payments, monthly insurance payments and pension payments. Annuities can be classified by the frequency of payment dates. The payments (deposits) may be made weekly, monthly, quarterly, yearly, or at any other regular interval of time. TYPES OF ANNUITIES 1. ANNUITY-IMMEDIATE OR ORDINARY ANNUITIES If the payments are made at the end of the time periods, so that interest is accumulated before the payment, the annuity is called an annuity-immediate or ordinary annuity. Mortgage payments are annuity-immediate, interest is earned before being paid. 2. ANNUITIES DUE An annuity-due is an annuity whose payments are made at the beginning of each period. Deposits in savings, rent or lease payments, and insurance premiums are examples of annuities due. 3. CONTINGENT ANNUITIES Annuities that provide payments that will be paid over a period known in advance are annuities 14 certain or guaranteed annuities. Annuities paid only under certain circumstances are contingent annuities. A common example is a life annuity, which is paid over the remaining lifetime of the Log on https://www.myonlineprep.com/app/Registers/go for more free study materials, mock test and for JAIIB & CAIIB exam. | CHAPTER 1 CALCULATIONS OF INTEREST & ANNUITIES annuitant. Certain and life annuities are guaranteed to be paid for a number of years and then become contingent on the annuitant being alive. 4. FIXED ANNUITIES These are annuities with fixed payments. If provided by an insurance company, the company guarantees a fixed return on the initial investment. Fixed annuities are not regulated by the Securities and Exchange Commission. 5. VARIABLE ANNUITIES Registered products that are regulated by The Insurance Regulatory and Development Authority (IRDA) They allow direct investment into various funds that are specially created for Variable annuities. Typically, the insurance company guarantees a certain death benefit or lifetime withdrawal benefits. 6. DEFERRED ANNUITIES An annuity which begins payments only after a period is a deferred annuity. An annuity which begins payments without a deferral period is an immediate annuity. 7. ANNUITY CERTAIN OR GUARANTEED ANNUITY If the number of payments is known in advance, the annuity is an annuity certain or guaranteed 15 annuity. Log on https://www.myonlineprep.com/app/Registers/go for more free study materials, mock test and for JAIIB & CAIIB exam. | CHAPTER 1 CALCULATIONS OF INTEREST & ANNUITIES CALCULATION OF ANNUITIES 1. FUTURE VALUE OF AN ORDINARY ANNUITY The future value of an annuity formula is used to calculate what the value at a future date would be for a series of periodic payments. The future value of an annuity formula assumes that 1. The rate does not change 2. The first payment is one period away 3. The periodic payment does not change If the rate or periodic payment does change, then the sum of the future value of each individual cash flow would need to be calculated to determine the future value of the annuity. If the first cash flow, or payment, is made immediately, The future value of annuity due formula would be used. 16 Log on https://www.myonlineprep.com/app/Registers/go for more free study materials, mock test and for JAIIB & CAIIB exam. | CHAPTER 1 CALCULATIONS OF INTEREST & ANNUITIES Illustration Ayush decides to save by depositing Rs.1000 into an account per year for 5 years. The first deposit would occur at the end of the first year. If a deposit was made immediately, then the future value of annuity due formula would be used. The effective annual rate on the account is 2%. If he would like to determine the balance after 5 years, he would apply the future value of an annuity formula to get the following equation. Here, P = 1000 r = 2% n=5 As per formula, = The balance after the 5th year would be Rs.5204.04. 17 Log on https://www.myonlineprep.com/app/Registers/go for more free study materials, mock test and for JAIIB & CAIIB exam. | CHAPTER 1 CALCULATIONS OF INTEREST & ANNUITIES 2. PRESENT VALUE OF AN ORDINARY ANNUITY The present value of annuity formula determines the value of a series of future periodic payments at a given time. The present value of annuity formula relies on the concept of time value of money, in that one rupee present day is worth more than that same rupee at a future date. Illustration Mr. Reddy wants to determine today's value of a future payment series with cash flow schedule as follow, receiving Rs. 1000 into an account per year for 5 years with the effective annual rate on the account is Since Mr. Reddy wants to determine today’s value of future payment series, we will use formula 18 that calculates the present value of an ordinary annuity. This is the formula we would use as part Log on https://www.myonlineprep.com/app/Registers/go for more free study materials, mock test and for JAIIB & CAIIB exam. | CHAPTER 1 CALCULATIONS OF INTEREST & ANNUITIES of a bond pricing calculation. The PV of an ordinary annuity calculates the present value of the coupon payments that we will receive in the future. P = 1000 (Cash flow per period) r = 5% n=5 Assumption The formula shown has assumptions, in that it must be an ordinary annuity. These assumptions are that 1) The periodic payment does not change 2) The rate does not change 3) The first payment is one period away 19 Log on https://www.myonlineprep.com/app/Registers/go for more free study materials, mock test and for JAIIB & CAIIB exam. | CHAPTER 1 CALCULATIONS OF INTEREST & ANNUITIES 3. FUTURE VALUE OF ANNUITY DUE The future value of annuity due formula is used to calculate the ending value of a series of payments or cash flows where the first payment is received immediately. The first cash flow received immediately is what distinguishes an annuity due from an ordinary annuity. An annuity due is sometimes referred to as an immediate annuity. The future value of annuity due formula calculates the value at a future date. The use of the future value of annuity due formula in real situations is different than that of the present value for an annuity due. For example, suppose that an individual or company wants to buy an annuity from someone and the first payment is received today. To calculate the price to pay for this particular situation would require use of the present value of annuity due formula. However, if an individual is wanting to calculate what their balance would be after saving for 5 years in an interest bearing account and they choose to put the first cash flow into the account today, the future value of annuity due would be used. 20 Log on https://www.myonlineprep.com/app/Registers/go for more free study materials, mock test and for JAIIB & CAIIB exam. | CHAPTER 1 CALCULATIONS OF INTEREST & ANNUITIES Illustration Ayush would like to calculate their future balance after 5 years with today being the first deposit. The amount deposited per year is Rs.1,000 and the account has an effective rate of 3% per year. It is important to note that the last cash flow is received one year prior to the end of the 5th year. For this example, we would use the future value of annuity due formula to come to the following equation: After solving, the balance after 5 years would be Rs.5468.41. 4. PRESENT VALUE OF ANNUITY DUE For the present value of an annuity due formula, we need to discount the formula one period forward as the payments are held for a lesser amount of time. When calculating the present value, we assume that the first payment was made today. We could use this formula for calculating the present value of your future rent payments as 21 specified in a lease you sign with your landlord. Let's say for Example 4 that you make your first Log on https://www.myonlineprep.com/app/Registers/go for more free study materials, mock test and for JAIIB & CAIIB exam. | CHAPTER 1 CALCULATIONS OF INTEREST & ANNUITIES rent payment at the beginning of the month and are evaluating the present value of your five- month lease on that same day. Your present value calculation would work as follows: Present value of an annuity due formula PV If cash flow schedule are same as above example, SINKING FUND A fund created, by gradual periodic deposits, with the objective of getting a targeted amount to pay off future debts, is called a sinking fund. The sinking funds can be created for a no. of purposes such as repayment of debt in lump sum, redemption of bonds, replacement of worn out equipment, buying of a new equipment etc. 22 This can be done by knowing the future value of an annuity, by using the following formula Log on https://www.myonlineprep.com/app/Registers/go for more free study materials, mock test and for JAIIB & CAIIB exam. | CHAPTER 1 CALCULATIONS OF INTEREST & ANNUITIES Illustration In 10 years, a Rs. 40,000 machine will have a salvage value of Rs. 4,000. A new machine at that time is expected to sell for Rs. 52,000. In order to provide funds for the difference between the replacement cost and the salvage value, a sinking fund is set up into which equal payments are placed at the end of each year. If the fund earns 7 per cent compounded annually, how much should each payment be? r = 0.07 , n = 10 F = (52,000-4,000) = 48,000 48,000 = A[(1+07)10 -1/0.07] 48,000 = Ax13.82 A = 48,000/13.82 = 3,474.12 23 Log on https://www.myonlineprep.com/app/Registers/go for more free study materials, mock test and for JAIIB & CAIIB exam. | CHAPTER 1 CALCULATIONS OF INTEREST & ANNUITIES OUR SPECIALISATION JAIIB/ CAIIB CLASSROOM PREPARATION BEST FACULTY WITH 10+ YEAR EXP. JAIIB & CAIIB PREPRATION BEST STUDY MATERIALS (HIGHLY ACCLAIMED) TOPIC WISE MOCK TEST FULL LENTH MOCK TEST RESULT ASSESSMENT WITH GRAPHICAL REPRESENTATION APP BASED LEARNING AND PREPARATION 24 Log on https://www.myonlineprep.com/app/Registers/go for more free study materials, mock test and for JAIIB & CAIIB exam. |