**Module- A**

**Simple Interest**

‘Simple’ interest or **‘flat rate’ **interest is the amount of interest paid each year in a fixed percentage of the amount borrowed or lent at the start.

Formula for calculating simple interest :

### Interest = Principal x Rate x Time (PRT), where:

**‘Interest’ **is the total amount of interest paid

**‘Principal’ **is the amount lent or borrowed

**‘Rate’ **is the percentage of the principal charged as interest each year.

**‘Time’ **is the time in years of the loan.

**Example :**

Principal: ‘P’ = Rs. 50,000, Interest rate: ‘R’ = 10% = 0.10, Repayment time: T = 3 years. Find the amount of interest paid.

Interest = PRT

= 50,000×0.10×3

= Rs. 15,000/-

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**Compound Interest**

Compound interest is paid on the original principal and accumulated part of interest. Formula for calculating compound interest :

### P = A(1 +r/n)^nt, where

P = the principal

A = the amount deposited

r = the rate (expressed as fraction, e.g. 6 per cent = 0.06) n = number of times per year that interest is compounded t = number of years invested

Frequently compounding of Interest. If the interest is compounded : Annually = P (1 + r)

Quarterly = P (1 + r/4)^4 Monthly = P (1 + r/12)^12

**Example :**

The compound interest on Rs. 30,000 at 7% per annum is Rs. 4347. The period (in years) is: Amount = Rs. (30000 + 4347) = Rs. 34347.

Let the time be *n *years. Then

30000(1+7/100)^n = 34347 (107/100)^n = 34347/30000 (107/100)^n = 11449/10000 (107/100)^n = (107/100)^2

*n *= 2 years.

**The Rule of 72: **Allows you to determine the number of years before your money doubles whether in debt or investment. Divide the number 72 by the percentage rate.

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**EQUATED MONTHLY INSTALMENTS (EMIs)**

Equated Monthly Installment (EMI) refers to the monthly payment a borrower makes on his loan. Though it is a combination of interest payment and principal repayment, the total monthly amount is calculated in such a way that it remains constant all through the repayment tenure. In Equated Monthly Installments (EMIs), the principal and the interest thereon is repaid through equal monthly installment over the fixed tenure of the loan. The benefit of an EMI for borrowers is that they know precisely how much money they will need to pay toward their loan each month, making the personal budgeting process easier.

**Formula :**

E = P×r×(1 + r)n/((1 + r)n – 1)

E is EMI

where P is Principle Loan Amount

r is rate of interest calculated in monthly basis it should be = Rate of Annual interest/12/100 if its 10% annual ,then its 10/12/100=0.00833

n is tenure in number of months

**Example :**

For 100000 at 10% annual interest for a period of 12 months, it comes to : 100000*0.00833*(1 + 0.00833)12/((1 + 0.00833)12 – 1) = 8792

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**Present Value**

Present value describes how much a future sum of money is worth today. Three most influential components of present value are : time, expected rate of return, and the size of the future cash flow. The concept of *present value *is one of the most fundamental and pervasive in the world of finance. It is the basis for stock pricing, bond pricing, financial modeling, banking, insurance, pension fund valuation. It accounts for the fact that money we receive today can be invested today to earn a return. In other words, present value accounts for the time value of money.

The formula for present value is:

### PV = CF/(1+r)n

Where:

CF = cash flow in future period

r = the periodic rate of return or interest (also called the discount rate or the required rate of return) n = number of periods

### Example :

Assume that you would like to put money in an account today to make sure your child has enough money in 10 years to buy a car. If you would like to give your child 10,00,000 in 10 years, and you know you can get 5% interest per year from a savings account during that time, how much should you put in the account now?

### PV = 10,00,000/ (1 + .05)10 = 6,13,913/-

Thus, 6,13,913 will be worth 10,00,000 in 10 years if you can earn 5% each year. In other words, the present value of 10,00,000 in this scenario is 6,13,913.

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**Future Value**

The value of an asset or cash at a specified date in the future that is equivalent in value to a specified sum today. It refers to a method of calculating how much the present value (PV) of an asset or cash will be worth at a specific time in the future. There are two ways to calculate FV:

- For an asset with simple annual interest: = Original Investment x (1+(interest rate*number of years))
- For an asset with interest compounded annually: = Original Investment x ((1+interest rate)^number of years)

- 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}

- A=P(1+r)

=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}

**Example:**

- 10,000 invested for 5 years with simple annual interest of 10% would have a future value of

FV = 10000(1+(0.10*5))

= 10000(1+0.50)

= 10000*1.5

= 15000

- 10,000 invested for 5 years at 10%, compounded annually has a future value of : FV = 10000(1+0.10)^5)

= 10000(1.10)^5

= 10000*1.61051

= 16105.10

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**Annuities**

Annuities are essentially a series of fixed payments required from you or paid to you at a specified frequency over the course of a fixed time period. The most common payment frequencies are yearly, semi-annually (twice a year), quarterly and monthly. There are two basic types of annuities: ordinary annuities and annuities due.

**Ordinary Annuity: **Payments are required at the end of each period. For example, straight bonds usually pay 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 example 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.

- 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.

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**Present Value and Future Value of an 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.

The present value an annuity is the sum of the periodic payments each discounted at the given rate of interest to reflect the time value of money.

PV of an Ordinary Annuity = R (1 − (1 + i)^-n)/i PV of an Annuity Due = R (1 − (1 + i)^-n)/i × (1 + i) Where,

i is the interest rate per compounding period;

n are the number of compounding periods; and R is the fixed periodic payment.

In the formulae, given, 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

**Example :**

- Calculate the present value on Jan 1, 2015 of an annuity of 5,000 paid at the end of each month of the calendar year 2015. The annual interest rate is 12%.

**Solution**

We have,

Periodic Payment R = 5,000 Number of Periods n = 12 Interest Rate i = 12%/12 = 1% Present Value

PV = 5000 × (1-(1+1%)^(-12))/1%

= 5000 × (1-1.01^-12)/1%

= 5000 × (1-0.88745)/1%

= 5000 × 0.11255/1%

= 5000 × 11.255

= 56,275.40

- A certain amount was invested on Jan 1, 2015 such that it generated a periodic payment of 10,000 at the beginning of each month of the calendar year 2015. The interest rate on the investment was 13.2%. Calculate the original investment and the interest earned.

**Solution**

Periodic Payment R = 10,000 Number of Periods n = 12

Interest Rate i = 13.2%/12 = 1.1%

Original Investment = PV of annuity due on Jan 1, 2015

= 10,000 × (1-(1+1.1%)^(-12))/1.1% × (1+1.1%)

= 10,000 × (1-1.011^-12)/0.011 × 1.011

= 10,000 × (1-0.876973)/0.011 × 1.011

= 10,000 × 0.123027/0.011 × 1.011

= 10,000 × 11.184289 × 1.011

= 1,13,073.20

Interest Earned = 10,000 × 12 − 1,13,073.20

= 1,20,000 – 1,13,073.20

= 6926.80

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**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.

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### Understanding Formula for EMI, Annuities

- Let us take case of a home loan of Rs 1lac at 12%p.a. ,repayable in 180 installments (here p=1,00,000and r=12/100*12=.01)
- In the 1
^{st}month, bank will charge interest equal to p*r=Rs 1000 and so, the outstanding amount will become Rs 1,01,000. - What happens if the EMI is fixed at p*r, which is Rs 1000?This EMI will meet only the interest applied and so the principal will remain unchanged at Rs 1,00,000.This process will continue and the loan will remain outstanding for ever. Therefore, EMI has to be slightly more than p*r so that some amount can go towards reducing the principal amount
- If EMI has to be more than p*r, we should multiply p*r by a fig which is more than 1.
- This fig is (1+r)
^{n}/ (1+r)^{n}-1.You will observe that denominator in less than numerator by 1 only. E.g., if numerator is 4.3210, the denominator will be 3.3210 .So, this fig is always more than 1. - As you know, (1+r)
^{n}is an important fig in business maths, and if the above concept is clear, you will never have difficulty in remembering EMI formula - Once you are comfortable with EMI formula, you can derive yourself the formula for PV and FV of Annuities.
- Home loan is like an ordinary annuity in which payment takes place at the end of each month for an amount equal to EMI, and p is like the present value of annuity. Therefore, in a question, if periodic payment ,n and r are given, you can calculate PV. FV is calculated by multiplying PV by (1+r)
^{n}. - In case of annuity due, the payments are at the beginning of the period and not at the end as is the case with ordinary annuity. Therefore, both PV and FV will be more than what is arrived in case of ordinary annuity. The multiplying factor is (1+r)

- In the 1

**Bond Value Debt**