# Calculation of Interest and Annuities: JAIIB /DBF Paper 2 (Module A) Unit 1

## 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.

## ♦What is simple interest?

• Simple interest is a method of calculating interest charged on fixed deposit, savings account, and a loan. It is calculated on the principal amount. Simple interest is when an interest rate is charged on the principal amount on a daily/monthly/quarterly/annual basis and does not add any interest rate on the interest amount gathered on the principal amount.

#### Where Is the Concept of Simple Interest Used?

Simple interest may be used in the following financial situations:

Borrowing money: In case of a loan, you will need to pay interest on the amount you have borrowed.

Lending money: In case of a savings account, fixed deposit , or recurring deposit, you will receive the amount in the form of interest on your principal.

#### Simple Interest Formula

The formula for calculating simple interest is:

P x r x t ÷ 100

P = Principal

r = Rate of Interest

t = Term of the loan/deposit in years

This means that you are multiplying the principal amount with the rate of interest and the tenure of the loan or deposit. Make sure you enter the tenure in years and not months. If you are entering the tenure in months, then the formula will be:

P x r x t ÷ (100 x 12)

If you want to find the total amount – that is, the maturity value of a deposit or the total amount payable including principal and interest, then you can use this formula:

FV = P x (1 + (r x t))

Here, FV stands for Future Value. To get the interest payable or receivable, you can subtract the principal amount from the future value.

Let’s give you some examples to understand how much you will earn on your deposits, or how much you will have to pay on your loan if your bank uses simple interest.

Simple Interest Calculation in Deposits

Example 1: If you invest Rs.50,000 in a fixed deposit account for a period of 1 year at an interest rate of 8%, then the simple interest earned will be:

50,000 x 8 x 1 ÷ 100 = Rs.4,000

The interest you will receive at the end of the 1-year tenure will be Rs.4,000. Therefore, the maturity amount of the FD will be Rs.54,000.

Example 2: If you invest Rs.8 lakh in a fixed deposit account for a period of 5 years at an FD interest rate FD interest rate of 6.85%, then the simple interest earned will be:

8,00,000 x 6.85 x 5 ÷ 100 = Rs.2,74,000

The interest you will receive at the end of the 5-year tenure will be Rs.2.74 lakh. Therefore, the maturity amount of the FD will be Rs.10.74 lakh.

Simple Interest Calculation in Loans

Example 1: Say you borrowed Rs.5 lakh as personal loan from a lender on simple interest. The interest rate is 18% and the tenure is 3 years. The interest you will end up paying to the bank will be:

5,00,000 x 18 x 3 ÷ 100 = Rs.2,70,000

The interest you will be paying over the period of 3 years will be Rs.2.7 lakh. Therefore, the total repayment you will make to the bank will be Rs.7.7 lakh. On a monthly basis, this would come up to around Rs.21,389.

Example 2: Say you took a car loan on simple interest. The principal amount is Rs.12 lakh, the interest rate is 7%, and the tenure is 5 years. The interest you will end up paying will be:

12,00,000 x 7 x 5 ÷ 100 = Rs.4,20,000

The interest you will be paying over the period of 5 years will be Rs.4.2 lakh. Therefore, the total repayment you will make will be Rs.16.2 lakh. On a monthly basis, this would come up to around Rs.45,000.

## ♦Compound Interest

• Gone are the days of school mathematics, most of us easily forget, but a quick refresher may bring it all back. To understand compound interest in the easiest form, let’s take a look at what it means. Compound interest is a useful financial concept in which your interest earned is added to your principal. This amount then continues to earn more interest. So in this case, you also earn interest on the interest you’ve already earned. So your balance grows at an increasing rate. In a sense, you reinvest your interest, rather than receiving a pay-out.
• Year 1 – You earn interest on your Principal.
• Year 2 – You earn interest on your (Principal + Interest of Year 1).
• Year 3 – You earn interest on your (Principal + Interest of Year 1 + Interest of Year 2).

Compound interest is the basis of long-term growth of the stock market. It forms the basis of personal savings plans. Compound interest also affects inflation.

#### Types of Compound Interest

There are generally two types of compound interest used.

• Periodic Compounding – Under this method, the interest rate is applied at intervals and generated. This interest is added to the principal. Periods here would mean annually, bi-annually, monthly, or weekly.
• Continuous Compounding – This method uses a natural log-based formula and calculates interest at the smallest possible interval. This interest is added back to the principal. This can be equalled to the constant rate of growth for all natural growth. This figure was born out of physics. It uses Euler’s number which is a famous irrational number which is known to more than 1 trillion digits of accuracy. Euler’s number is denominated by the letter “E”.

#### Periodic Compound Interest Formula Overview

There are two formulas you can use to calculate compound interest, depending on what result you wish to find out. You can find out the following:

• The total value of the deposit.
• The total compound interest earned.

Value of the Deposit

Formulas can be a deterrent to many. If you aren’t savvy with math, your eyes turn away from these codes or just skip them altogether. But once it’s explained, it’s pretty simple to understand. To calculate the total value of your deposit, the formula is as follows:

P (1+ i/n)nt

P = Principal invested.

i = Nominal Rate of Interest.

n = Compounding Frequency or number of compounding periods in a year.

t = Time, meaning the length of time the interest is applicable, generally in years.

Simply put, you calculate the interest rate divided by the number of times in a year the compound interest is generated. For instance, if your bank compounds interest quarterly, there are 4 quarters in a year, so n = 4. This result must be multiplied to the power of the deposit period. For example, if your deposit is for 10 years, t = 10. This whole result should be multiplied by the principal you invested. The result generated will equal the total accumulated value of your deposit. You can find out how much your deposit is worth currently after accumulating interest.

Total Compound Interest Earned

To find out how much interest was earned, you can use the following formula for Compound Interest.

P[(1+ i/n))nt-1]

Compound Interest Equation and Calculation

To understand the compound interest equation further, we can break it down in simpler terms. If you decide to invest in a fixed deposit with compound interest, this is how you will earn interest every year.

 Period Deposit Balance Investment P Year 1 P + iP Year 2 (P+ iP) + i(P+iP)

To collapse this formula, we can pull out factors of (1+i). Simply substitute iP with (1+i) to get the following:

 Period Deposit Balance Investment P Year 1 P(1+i) Year 2 P(1+i)2 Year 3 P(1+i)3

Formula for Annual Compound Interest

To calculate the compound interest for a number of years together, we need to multiply P(1+i) to the power of the number of years of the deposit. So we end up with this formula:

P (1+ i/n)n

This formula can be used to calculate compound interest that is compounded annually. This means you receive interest only once a year. It is added to your principal, and you continue to earn interest on the new amount.

This formula can be used to calculate compound interest that is compounded annually. This means you receive interest only once a year. It is added to your principal, and you continue to earn interest on the new amount.

Half-Yearly, Quarterly, Monthly Compound Interest Formula

If you are earning interest multiple times in a year, you need to factor in this number into the equation. So the formula generated is:

P (1+ i/n)nt

This formula can also be used for instances where the interest is compounded once every two years. In this case, n = 0.5, as each year is calculated as half.

Examples of Compound Interest

For example, Rs. 10,000 is invested in a fixed deposit for 10 years. The interest is compounded every quarter which means 4 times in a year. The interest paid by the bank is 5%. To find out your nominal rate of interest, you need to divide 5 by 100 which equals 0.05. Now, we look at the formula and substitute the letters with the relevant numbers.

Calculating the Total Value of the Deposit

P (1+ i/n)nt

Step 1: 10,000 (1+0.05/4)4×10

Step 2: 10,000(1+0.0125)40

Step 3: 10,000 (1.0125)40

Step 4: 10,000 (1.64361946349)

Step 5: 16436.1946349

We can round of this total to Rs. 16,436.19. So the compound interest earned after 10 years is Rs. 6,436.19.

Calculating the Interest Earned

We can also arrive at this figure using the formula for compound interest earned. We can substitute the numbers for letters as seen below:

P[(1+ i/n)nt -1]

Step 1: 10,000 [(1+0.05/4)4×10 -1]

Step 2: 10,000 [(1+0.0125)40-1]

Step 3: 10,000 [(1.0125)40-1]

Step 4: 10,000 [(1.64361946349) -1]

Step 5: 10,000 (0.664361946349

Step 5: 6436.1946349

We can now add this interest earned to the principal amount to find out the value of the deposit. The maturity value will be Rs. 16,436.19.

## ♦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.