shares and dividends maths pdf
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Certain business organizations need to raise money from public. In India, such an organization needs to be registered under the Indian Companies Act. Such an organization is called a public limited company.

A company may need money to start business or to start a new project. The sum of money required is called capital. The required capital is divided into small equal parts, and each part is called share. The company prepares a detailed plan of the proposed project and frames rules and regulations regarding its functioning. They, then, draft a proposal, issue a prospectus, explaining the plan of the project and invite the public to invest money in their project. They, thus, pool up the required funds from the public, by assigning them shares of the company. The value of a share may be Re 1, Rs 10, Rs 100, Rs 1000, etc. The capital is raised by selling these shares. A person who purchases shares of the company becomes a shareholder of the company.

Value of shares

The original value of a share printed in the certificate of the share is called its face value or nominal value (in short, NV). The NV of a share is also known as register value, printed value and par value. The price at which the share is sold or purchased in the capital market through stock exchanges is called its market value (in short, MV).

A share is said to be:

At premium or Above par, if its market value is more than its face value.
At par, if its market value equals its face value.
At discount or Below par, if its market value is less than its face value.
The share of a company that is doing well or expected to do well is sold in the market at a price higher than its NV. In such a situation, we say the share is at premium or above par. For example, if a share of NV of Rs 10 is selling at Rs 16 then the share is at a premium of Rs 6. The share of a company that is neither doing well nor poorly is sold in the market at a price equal to its NV. For example, if a share of NV of Rs 100 is selling at Rs 100 then the share is at par. The share of a company that is doing poorly or may do poorly in the future is sold in the market at a price lower than its NV. In such a case, we say the share is at a discount or below par. For example, if a share of NV of Rs 100 is selling at Rs 80 then the share is at a discount of Rs 20.

Dividend, Rate of Dividend

The part of the annual profit of a company distributed among its shareholders is called dividend. The dividend is always reckoned on the face value of a share irrespective of its MV.

The rate of dividend is expressed as a percentage of the NV of a share per annum.

Meaning of the statement “r% Rs 100 at Rs M”

The statement r% Rs 100 shares at Rs M means the following:

The NV of a share is Rs 100.
The MV of a share is Rs M.
The dividend on a share is r% of NV, i.e., Rs r per annum.
An investment of Rs M gives an annual income of Rs r.
Rate of return per annum = Annual income from an investment of Rs 100 $latex =(\dfrac{Income}{Investment} \times 100) \%=(\dfrac{r}{M} \times 100)\% $
Look at the statement given below:

9% Rs 100 shares at Rs 120 means
Face value (NV) of 1 share = Rs 100.
Market value (MV) of 1 share = Rs 120.
The dividend on a share is 9% of its face value = 9% of Rs 100 = Rs 9
An investment of Rs 120 gives an annual income of Rs 9.
Rate of return per annum = Annual income from an investment of Rs 100 $latex =(\dfrac{Income}{Investment} \times 100) \%=(\dfrac{9}{120} \times 100) \%=7\dfrac{1}{2} \% $
Formula

1. Sum invested = No. of shares bought $latex \times $ MV of 1 share

2. No. of shares bought $latex =\dfrac{Sum \: invested}{MV \: of \: 1 \: share} $

Also, no. of shares bought $latex =\dfrac{Total \: dividend}{Dividend \: on \: 1 \: share}=\dfrac{Total \: income \: (profit)}{Income \: (profit)} $ from 1 share

3. Income (return or profit) = (No. of shares) $latex \times $ (rate of dividend) $latex \times $ (NV) =(No. of shares) $latex \times $ (Dividend on 1 share)

4. Return % = Income (profit) % $latex =\dfrac{Income}{Investment} \times 100 \% $

NOTE: The face value of a share remains the same. The market value of a share changes from time to time.

Examples

Example 1: Calculate the money required to buy: (i) 350, Rs 20 shares at a premium of Rs 7. (ii) 275, Rs 60 shares at a discount of Rs 10. (iii) 50, Rs 75 shares quoted at Rs 71.50.

Solution: (i) No. of shares = 350

NV = Rs 20

MV = Rs (20+7) = Rs 27

Therefore, money required to buy 350 shares = Rs (350 $latex \times $ 27)= Rs 9450

(ii) No. of shares = 275

NV= Rs 60

MV= Rs (60-10) = Rs 50

Therefore, money required to buy 275 shares = Rs (275 $latex \times $ 50) =Rs 13750

(iii) No. of shares = 50

NV= Rs 75

MV= Rs 71.50

Therefore, money required to buy 50 shares= Rs (50 $latex \times $ 71.50) = Rs 3575

Example 2: A man invests in shares for which we have the condition “7% Rs 100 shares at Rs 120”. What is the annual income of a person holding 150 such shares? Also, find his annual profit percentage.

Solution: Given,

Rate of dividend=7%

Nominal value (NV) = Rs 100

Market value (MV) = Rs 120

No. of shares= 150

Therefore, Income =No. of shares $latex \times $ rate of dividend $latex \times $ NV= $latex Rs \: (150 \times \dfrac{7}{100} \times 100)=Rs \: 1050 $

First we need to find the sum invested to find the profit percentage.

Investment=No. of shares $latex \times $ MV= $latex Rs \: (150 \times 120)=Rs \: 18000 $

Therefore, Required profit percentage $latex =\dfrac{Income}{Investment} \times 100 \% =\dfrac{Rs \: 1050}{Rs \: 18000} \times 100=5\dfrac{5}{6} \% $

Example 3: Which is a better investment: 16% at 80 or 20% at 120?

Solution: 16% at 80 means MV of 1 share is Rs 80, NV of 1 share is Rs100 and dividend paid is 16%.

Similarly, 20% at 120 means MV of 1 share is Rs 120, NV of 1 share is Rs 100 and dividend paid is 20%.

Case 1:

Income on Rs 80=16% of Rs 100=Rs 16

Therefore, income on $latex Re \: 1=Rs \: \dfrac{16}{80}= Rs \: 0.20 $

Case 2:

Income on Rs 120=20% of Rs 100= Rs 20

Therefore, income on $latex Re \: 1=Rs \: \dfrac{20}{120}= Rs \: 0.16 $

Therefore, the first investment is better.

Example 4: A company declares semiannual dividend of 6%. A man has 500 shares of NV Rs 25 each. Find his annual income.

Solution: Total NV of shares= Rs(25 $latex \times $ 500)= Rs 12500

Semiannual dividend = 6% of Rs 12500= $latex \dfrac{6}{100} \times 12500=Rs \: 750 $

Therefore, his annual income= Rs (750 $latex \times $ 2)=Rs 1500

Example 5: Divide Rs 29184 into two parts such that if one part is invested in 12%, Rs 100 shares at 4% discount and the other in 15%, Rs 100 shares at 8% premium, the annual incomes are equal.

Solution: Let the investment in 12%, Rs 100 shares at 4% discount be $latex Rs \: x $.

Then, investment in 15%, Rs 100 shares at 8% premium be $latex Rs \: (29184-x) $.

MV of Rs 100 shares at 4% discount= Rs (100-4% of 100) = Rs 96

Annual income on 1 share of Rs 96=Rs (12% of 100) = Rs 12

Annual income on $latex Rs \: x=Rs \: {12x}{96}=Rs \: {x}{8} $

MV of Rs 100 shares at 8% premium= Rs (100+8% of 100) = Rs 108

Annual income on 1 share of Rs 108=Rs (15% of 100) = Rs 15

Annual income on $latex Rs \: (29184-x) = Rs \: \dfrac{15(29184-x)}{108} =Rs \: \dfrac{5(29184-x)}{36} $

$latex \therefore $ $latex \dfrac{x}{8}=\dfrac{5(29184-x)}{36} $

$latex \Rightarrow \dfrac{x}{8}+ \dfrac{5x}{36}= \dfrac{12160}{3} $

So,$latex x=15360 $

So, the first part is Rs 15360.

Second part= Rs (29184-15360) = Rs 13824

Example 6: Mukul invests Rs 9000 in a company paying a dividend of 6% per annum when a share of NV Rs 100 stands at Rs 150. What is his annual income? If he sells 505 of his shares when the price rises to Rs 200, what is his gain in this transaction?

Solution: No. of shares bought by Mukul=$latex \dfrac{Investment}{MV}=\dfrac{9000}{150}=60 $

His annual income on 1 share=6% of NV=6% of Rs 100= Rs 6

His total annual income=60 $latex \times $ Rs 6= Rs 360

Since, 50% of shares= 50% of 60 =30

Money received on selling these shares =30 $latex \times $ Rs 200=Rs 6000

Also, cost of these shares=30 $latex \times $ Rs 150=Rs 4500

Therefore, Mukul’s gain= Rs (6000-4500) = Rs 1500

Example 7: A man wants to buy 62 shares available at Rs 132(NV being Rs 100).

How much will he have to invest?
If the dividend is 7.5%, what will be his annual income?
If he wants to increase his annual income by Rs 150, how many extra shares should he buy?
Solution: 1. He will have to invest= 62 $latex \times $ Rs 132= Rs 8184

2. Dividend on 1 share= 7.5% of Rs 100=Rs 7.50

Therefore, his annual income = 62 $latex \times $ Rs 7.50= Rs 465

3. The man wants to increase his income by Rs 150 and income on 1 share= Rs 7.50

Therefore, no. of extra shares he must buy= $latex\dfrac{150}{7.50} = 20 $
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