Calculation of Interest and Annuities: Jaiib /DBF Paper 2 (Module A) Unit 1:
Dear bankers,
As we all know that is Calculation of Interest and Annuities for JAIIB Exam. JAIIB exam conducted twice in a year. So, here we are providing the Calculation of Interest and Annuities (Unit-1), Business Mathematics and Finance (Module A), Accounting Finance for Bankers-2.
♦Introduction
People will earn money for their livelihood i.e. to spend on rent, food, clothing, education etc. along with this they need money to meet some extra expenditure like marriage in family, purchasing of vehicle, house or set up their own business and so on. Some people will manage with their own money, but most people have to borrow money for such contingencies.
Meaning of Interest:
Interest can be defined as the price paid by a borrower for the use of a lender’s money.
It is compensation paid to the depositor.
In another words, it is excess of money paid or received on deposits or borrowings.
Interest is the price paid by a borrower for the use of a lender’s money. If you borrow (or lend) some money from (or to) a person for a particular period you would pay (or receive) more money than your initial borrowing (or lending).
♦Reasons for Charging Interest:
There are a variety of reasons for charging the interest, they are-
Time value of money:
- Time value of money means that the value of a unity of money is different in different time periods. The sum of money received in future is less valuable than it is today.
- In other words the worth of rupees received after some time will be less than a rupee received today.
- Since a rupee received today has more value, rational investors would prefer current receipts to future receipts. If they postpone their receipts, they will certainly charge some money i.e. interest.
Opportunity Cost:
- The lender has a choice between using his money in different investments. If he chooses one, he forgoes the return from all others.
- In other words, lending incurs an opportunity cost due to the possible alternative uses of the lent money.
Inflation:
- Most economies generally exhibit inflation. Inflation is a fall in the purchasing power of money.
- Due to inflation, a given amount of money buys fewer goods in the future than it will now. The borrower needs to compensate the lender for this.
Liquidity Preference:
- People prefer to have their resources available in a form that can immediately be converted into cash rather than a for that takes time or involves expenditure to realize
Risk Factor:
- There is always a risk that the borrower will go bankrupt or otherwise default on the loan.
- Risk is one determinable factor in fixing rate of interest.
- A lender generally charges more interest rate (risk premium) for taking more risk.
♦Types of Interest
Interest can be of two types:
- Simple Interest
- Compound Interest
Simple Interest:
- SI is interest earned on only the original amount, called Principal, lent over a period of time at a certain rate.
- Formula for SI-=PRT /100,
- For ex: Rs.1000 deposited for one year at the rate of 8% p.a. interest will be Rs.8.
Compound Interest
- CI is interest earned on any previous interests earned as well as on the Principal lent. It is Interest on interest.
- Formula : CI= P(1+i)^n- P A= P(1+i)^n
- For ex: Rs. 1000 deposited for in year at the rate of 8% CI p.a. interest will be- Rs.80
Compound Interest conversion:
Conversion period | Description | No. of conversion |
1Day | Compounded daily | 365 |
1 Month | Compounded Monthly | 12 |
3 Month | Compounded Quarterly | 4 |
6 Month | Compounded Semiually | 6 |
12 Month | Compounded yearly | 1 |
Compound Vs Simple Interest:
Basis | Simple Interest | Compound Interest |
Calculation | Easy to understand simple to calculate | Difficult to calculate |
Constancy | Principal money remains same for all the years | Principal varies |
Suitability | Suitable for short term deposits | Suitable for long term deposits |
Formula | SI= PRT/ 100 | CI= P(1+i)n – p |
Accrual | Interest will not accrue to the principal | Interest accrues every time to principal |
Types of Interest for Bank Deposits:
Type of Deposit | Type of Interest |
Saving Account | Simple |
Current Account | No Interest |
Fixed Deposit | Simple |
Reinvestment | Compound |
Recurring Deposit | Compound |
Effective Rate of Return:
- If interest is compounded more than once a year the effective interest rate for a year exceeds the per annum interest rate, then ERR>NR
- When compounding is done annually- ERR=NR ERR= (1+i)n -1
- For Ex: The effective rate of return for 10% CI p. a, when compounded semi -annually, quarterly and monthly Will be-
- Half yearly= (1+.05)2 -1 = 10.25%
- Quarterly= (1+.025)4 -1 = 10.38%
- Monthly= (1+.0083)12 – 1= 10.47%
♦Daily Product Method
Introduction:
Finance is the life blood of trade, commerce and industry. Now-a-days, banking sector acts as the backbone of modern business. Development of any country mainly depends upon the banking system. With a bank people can open saving account, current account, fixed deposit account and recurring deposit account.
Meaning of Saving Account: (SA)
A deposit account held at a bank or other financial institution that provides principal security and a modest interest rate.
Savings account funds are considered one of the most liquid investments outside of demand accounts and cash.
RBI Guidelines on Saving Account
Savings bank account interest calculation by banks in India as per the new RBI guidelines is based on daily products, i.e. the balances outstanding as at the end of the day. The old method which banks used to calculate interest on savings interest was based on minimum balance kept in the a/c from 10th of any month and last working day of that month. But as per the revised RBI guidelines; the old method was changed (with effect from 01st April 2010). RBI deregulated the interest calculated on savings account which permitted banks to set their own rate of interest on the savings bank account.
New Vs Old method of Interest calculation on SB
Let’s understand the savings interest calculation (New Vs. Old) with an example. We’ll assume a/c statements as given in below are in INR.
Date | Particular | Debit | Credit | Balance |
1-1-2013 | By Op Balance | 300000 | 300000 | |
12-1-2013 | To down payment | 50000 | 250000 | |
20-1-2013 | By Cheque(000456) | 80000 | 330000 | |
28-1-2013 | To ABC Co, | 90000 | 240000 |
Calculating Interest as per the OLD method:
Following formula was applicable till 31st March 2010:
Savings Interest (Old Method) = Minimum balance between 10th and last working day of that month*Rate of Interest* 1/ No. of Months in a Year
So old rate of interest you’d received = 240000*4*1/12*100= 800 INR
Calculating Interest as per the NEW Method:
Applicable From 01 Apr 2010 as per the RBI guidelines:
Date | Particular | Debit | Credit | Balance | No. of Days (B) | Product (A*B) |
1-1-2013 | By Op Balance | 300000 | 300000 | 11 | 3300000 | |
12-1-2013 | To down payment | 50000 | 250000 | 8 | 2000000 | |
20-1-2013 | By Cheque(000456) | 80000 | 330000 | 8 | 2640000 | |
28-1-2013 | To ABC Co, | 90000 | 240000 | 4 | 960000 | |
31 | 8900000 |
Savings Interest (New Method):
(Total Products*Rate of Interest/ 365)
So, the interest you’ll receive = 89, 00,000*0.04/365 = 975.34 INR
The new guideline has bought happiness to the account holder since he’ll see more balance in the account at the end of the month.
♦Equated Monthly Installment
Meaning of EMI (Equated Monthly Installment)
An Equated Monthly Installment (EMI) is “A fixed payment amount made by a borrower to a lender at a specified date. Equated monthly installments are used to pay off both interest and principal each month, so that over a specified number of years, the loan is paid off in full.” common types of loans, such as real estate mortgages, the borrower makes fixed periodic payments can be paid with the help of EMI. EMIs differ from variable payment plans, in which the borrower is able to pay higher payment amounts at his or her discretion. In EMI plans, borrowers are usually only allowed one fixed payment amount each month.
Factors to be known about EMI:
- The repayment of a loan is done by paying an EMI to the bank. The EMI depends on three factors: loan amount, interest rate and the duration of the loan.
- The EMI is decided when the loan is sanctioned and remains constant throughout the period of the loan, provided there is no change in any of the factors on the basis of which it is calculated.
- The EMI has an interest and a principal portion. Through the principal, the borrower repays the loan each month. Through the interest, he pays the bank the interest due on the outstanding loan amount.
- The EMIs are structured in such a way that the interest portion forms a major part of the payment that is made in the initial years. In the later years, the principal component becomes high.
- The EMI can change in the case of an alteration in interest rates or if there is a prepayment. It is also possible to keep the EMI constant and increase or decrease the tenure of the loan to reflect the changes in interest rates or loan prepayment.
Formula for EMI Calculation:
Following is formula used to determine the EMI.
EMI = [P x R x (1+R)^N]/[(1+R)^ (N-1)]
Where,
E = Installment Amount
P = Principal Loan Amount
R = Rate of Interest; for EMI, Rate of Interest has to be divided by 12
N = Number of installments
Let us understand calculation of EMI with following example-
For Ex: Mr. Goyal has taken personal loan of Rs. 100000 for 12 months at CI of 10% p.a. Calculate the EMI that has to be paid by him = Rs. 8885.19
♦Fixed and Floating Interest Rates
There are two different modes of Interest. They are-
- Fixed Rates
- Floating Rates also called as variable rates.
What is Fixed Interest Rate?
- People who opt for Fixed Interest Rate mean that they have to repay the home loan is fixed and equal instalments as per the loan tenure. The advantage of fixed interest rate is that it would not change even if there are fluctuations or changes in the Indian financial market conditions or trends. Fixed Interest rate becomes the first preference when the financial market is down. Consumers take the opportunity by blocking or fixing the interest rate as per their preference. In simple terms, if you think that financial market will not drop down below a certain point or foresee a rise in the interest rates, then choosing fixed interest rate shall be the best option to avail.
What is Floating Interest Rate?
- Interest rate which is volatile and keeps on changing as per market scenario is termed as Floating Interest Rate. This type of interest rate depends on the base rate offered by several lenders, so whenever the base rate changes, the interest rate gets automatically revised. As compared to fixed interest rate, floating rates are comparatively cheaper. Fixed interest rates are 1%-2.5% higher than the floating interest rate. The increase and decrease in the floating interest rate is temporary, as it varies as per the market trends. As home loan is a long-term association with the lender, sometimes it becomes difficult to plan for the financials.
Comparison between Fixed and Floating Interest Rate
Fixed Interest Rate | Floating Interest Rate |
Higher Interest Rate | Lower Interest Rate |
Not affected by financial market conditions | Affected by changes in the financial market |
Fixed EMIs | EMIs change as per interest rate or MCLR |
Budget planning possible | Difficult to budget or manage financials |
Sense of security | Generates savings |
Suitable for short/medium term (3-10 years) | Suitable for long term (20-30 years) |
Lesser risk | Higher risk |
♦Front-End and Back-End Interest Rates
What Is the Front-End Ratio?
- The front-end ratio, also known as the mortgage-to-income ratio, is a ratio that indicates what portion of an individual’s income is allocated to mortgage payments. The front-end ratio is calculated by dividing an individual’s anticipated monthly mortgage payment by his/her monthly gross income. The mortgage payment generally consists of principal, interest, taxes, and mortgage insurance (PITI). Lenders use the front-end ratio in conjunction with the back-end ratio to determine how much to lend.
What Is the Back-End Ratio?
The back-end ratio, also known as the debt-to-income ratio, is a ratio that indicates what portion of a person’s monthly income goes toward paying debts. Total monthly debt includes expenses, such as mortgage payments (principal, interest, taxes, and insurance), credit card payments, child support, and other loan payments.
♦Annuities
What are Annuities?
At some point in your life, you may have had to make a series of fixed payments over a period of time –such as rent or car payments-or have received a series of payments over a period of time, such as bond coupons. These are called annuities.
- Ordinary Annuity: Payment are required at the end of each period. For an illustration, straight bonds usually make coupon payments at the end of every six months until the bond’s maturity date.
- Annuity Due: Payments are required at the beginning of each period. Rent is an illustration of annuity due. You are usually required to pay rent when you first move in at the beginning of the month, and then on the first of each month thereafter.
Since the present and future value calculations for ordinary annuities and annuities due are slightly different, we will first discuss the present and future value calculation for ordinary annuities.
Calculating the Future Value of an Ordinary Annuity
Period
commonly a period will be a year but it can be any time interval you want as long as all inputs are consistent.
Number of Periods (t)
number of periods or years Perpetuity for a perpetual annuity t approaches infinity. Enter p, P, perpetuity or Perpetuity for t
Interest Rate (R)
is the annual nominal interest rate or “stated rate” per period in percent. r = R/100, the interest rate in decimal
Compounding (m)
is the number of times compounding occurs per period. If a period is a year then annually=1, quarterly=4, monthly=12, daily = 365, etc.
Continuous Compounding
is when the frequency of compounding (m) is increased up to infinity. Enter c, C, continuous or Continuous for m.
Payment Amount (PMT)
The amount of the annuity payment each period
Growth Rate (G)
If this is a growing annuity, enter the growth rate per period of payments in percentage here. g = G/100
Payments per Period (Payment Frequency (q))
How often will payments be made during each period? If a period is a year then annually=1, quarterly=4, monthly=12, daily = 365, etc.
Future Value of an Annuity
FV=PMTi[(1+i)n−1](1+iT)FV=PMTi[(1+i)n−1](1+iT)
where r = R/100, n = mt where n is the total number of compounding intervals, t is the time or number of periods, and m is the compounding frequency per period t, i = r/m where i is the rate per compounding interval n and r is the rate per time unit t. If compounding and payment frequencies do not coincide, r is converted to an equivalent rate to coincide with payments then n and i are recalculated in terms of payment frequency, q.
If type is ordinary, T = 0 and the equation reduces to the formula for future value of an ordinary annuity
FV=PMTi[(1+i)n−1]FV=PMTi[(1+i)n−1]
otherwise T = 1 and the equation reduces to the formula for future value of an annuity due
FV=PMTi[(1+i)n−1](1+i)
Calculating The Present Value of an Ordinary Annuity
An ordinary annuity is a series of equal payments, with all payments being made at the end of each successive period. An example of an ordinary annuity is a series of rent or lease payments. The present value calculation for an ordinary annuity is used to determine the total cost of an annuity if it were to be paid right now.
The formula for calculating the present value of an ordinary annuity is:
P = PMT [(1 – (1 / (1 + r)n)) / r]
Where:
P = The present value of the annuity stream to be paid in the future
PMT = The amount of each annuity payment
r = The interest rate
n = The number of periods over which payments are to be made
For example, ABC International has committed to a legal settlement that requires it to pay 50,000 per year at the end of each of the next ten years. What would it cost ABC if it were to instead settle the claim immediately with a single payment, assuming an interest rate of 5%? The calculation is:
P = 50,000 [(1 – (1/(1+.05)10))/.05]
P = 386,087
As another example, ABC International is contemplating the acquisition of a machinery asset. The supplier offers a financing deal under which ABC can pay 500 per month for 36 months, or the company can pay 15,000 in cash right now. The current market interest rate is 9%. Which is the better offer? The calculation of the present value of the annuity is:
P = 500 [(1 – (1/(1+.0075)36))/.0075]
P = 15,723.40
In the calculation, we convert the annual 9% rate to a monthly rate of 3/4%, which is calculated as the 9% annual rate divided by 12 months. Since the up-front cash payment is less than the present value of the 36 monthly lease payments, ABC should pay cash for the machinery.
While this formula can be quite useful, it can yield misleading results if actual interest rates vary during the analysis period.
Calculating The Future value of an Annuity Due
Future value is the value of a sum of cash to be paid on a specific date in the future. An annuity due is a series of payments made at the beginning of each period in the series. Therefore, the formula for the future value of an annuity due refers to the value on a specific future date of a series of periodic payments, where each payment is made at the beginning of a period. Such a stream of payments is a common characteristic of payments made to the beneficiary of a pension plan. These calculations are used by financial institutions to determine the cash flows associated with their products.
The formula for calculating the future value of an annuity due (where a series of equal payments are made at the beginning of each of multiple consecutive periods) is:
P = (PMT [((1 + r)n – 1) / r])(1 + r)
Where:
P = The future value of the annuity stream to be paid in the future
PMT = The amount of each annuity payment
r = The interest rate
n = The number of periods over which payments are to be made
This value is the amount that a stream of future payments will grow to, assuming that a certain amount of compounded interest earnings gradually accrue over the measurement period. The calculation is identical to the one used for the future value of an ordinary annuity, except that we add an extra period to account for payments being made at the beginning of each period, rather than the end.
For example, the treasurer of ABC Imports expects to invest 50,000 of the firm’s funds in a long-term investment vehicle at the beginning of each year for the next five years. He expects that the company will earn 6% interest that will compound annually. The value that these payments should have at the end of the five-year period is calculated as:
P = (50,000 [((1 + .06)5 – 1) / .06])(1 + .06)
P = 298,765.90
As another example, what if the interest on the investment compounded monthly instead of annually, and the amount invested were 4,000 at the end of each month? The calculation is:
P = (4,000 [((1 + .005)60 – 1) / .06])(1 + .005)
P = 280,475.50
The .005 interest rate used in the last example is 1/12th of the full 6% annual interest rate.
Calculating the Present value of An annuity Due
The present value of an annuity due is used to derive the current value of a series of cash payments that are expected to be made on predetermined future dates and in predetermined amounts. The calculation is usually made to decide if you should take a lump sum payment now, or to instead receive a series of cash payments in the future (as may be offered if you win a lottery).
The present value calculation is made with a discount rate, which roughly equates to the current rate of return on an investment. The higher the discount rate, the lower the present value of an annuity will be. Conversely, a low discount rate equates to a higher present value for an annuity.
The formula for calculating the present value of an annuity due (where payments occur at the beginning of a period) is:
P = (PMT [(1 – (1 / (1 + r)n)) / r]) x (1+r)
Where:
P = The present value of the annuity stream to be paid in the future
PMT = The amount of each annuity payment
r = The interest rate
n = The number of periods over which payments are made
This is the same formula as for the present value of an ordinary annuity (where payments occur at the end of a period), except that the far right side of the formula adds an extra payment; this accounts for the fact that each payment essentially occurs one period sooner than under the ordinary annuity model.
For example, ABC International is paying a third party $100,000 at the beginning of each year for the next eight years in exchange for the rights to a key patent. What would it cost ABC if it were to pay the entire amount immediately, assuming an interest rate of 5%? The calculation is:
P = ($100,000 [(1 – (1 / (1 + .05)8)) / .05]) x (1+.05)
P = $678,637
The factor used for the present value of an annuity due can be derived from a standard table of present value factors that lays out the applicable factors in a matrix by time period and interest rate. For a greater level of precision, you can use the preceding formula within an electronic spreadsheet.
♦Repayment of a Debt
A debt is required to be repaid as per the terms of the contract with lender. In banking industry in India, the following three methods of repayment are common.
- Equal monthly/quarterly installment of principal plus the interest applied during the period.
- Equated monthly/ quarterly installment covering both the principal and the interest.
- Bullet/balloon repayment under which the entire loan amount is repaid at the end of the period with accumulated interest. Alternatively, the interest is paid periodically, as and when applied, and the principal amount of the loan is paid at the end of the contract period.
Equal monthly/ quarterly installment of principal plus the interest applied during the period.
Your friend has borrowed Rs 1000 from you and wants to repay you on a payment basis rather than the whole amount all at once the end of the year. The important point here is that he will owe you less in principal each month. The applicable rate of interest 8% p.a. means 0.667% per month.
The principal payment each month will be 83.33 (1000 divided by 12)
First month: Interest = 1000*0.667%*1=6.67 plus 83.33 for a total payment of Rs 90. The principal owed at the end of the month is 916.67.
Second month: Interest= Rs 916.67* 0.667%*1= Rs 6.11 plus Rs 83.33 for a total payment of Rs 89.44
Third Month: Interest=Rs 833.34*0.667%*1= Rs 5.56 plus Rs 83.33 for a total payment of Rs 88.89.
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