Hello, and welcome to today's mathematics lesson. Today, we are going to explore Chapter Three: Shares and Dividends. This is a fascinating topic that connects mathematics to the real world of business and investment. By the end of this lesson, you will understand how companies raise money through shares, how investors earn returns, and how to calculate everything from market values to dividend income.
Let us begin with the fundamental concept of what a share actually is. When a company needs a large amount of money to establish or expand its business, it rarely relies on just one person. Instead, the company divides its total estimated value into small, equal parts. Each of these parts is called a share. The value that is fixed for each share when it is first created is known as its nominal value, or face value.
The nominal value of a share is also referred to by several other names: register value, printed value, or face value. Here is something crucial to remember: the nominal value of a share never changes with time. It remains fixed at the amount printed on the share certificate.
Now, here is where things get interesting. Once shares are issued, they can be bought and sold in the market. The price at which a share trades in the market at any particular time is called its market value, or cash value. Unlike the nominal value, the market value of a share can and does change over time. It may increase or decrease depending on how well the company is performing, its profits, and overall market conditions.
The relationship between market value and nominal value tells us important information about a share's status. There are three possibilities.
First, if the market value equals the nominal value, we say the share is at par. This means M.V. = N.V.
Second, if the market value is greater than the nominal value, the share is said to be above par, or at a premium. This is written as M.V. > N.V. A premium indicates that investors are willing to pay more than the face value because they believe the company is doing well.
Third, if the market value is less than the nominal value, the share is below par, or at a discount. This is expressed as M.V. < N.V. A discount suggests that the company may be facing difficulties, or investors are less confident about its future.
Now, why do people buy shares? The answer lies in dividend. Dividend is the profit that a shareholder receives from the company out of its total profits. It is the return on their investment.
Two critical points about dividend: first, dividend is always expressed as a percentage of the nominal value of the share, not the market value. Second, the dividend rate does not depend on what you paid for the share in the market. Whether you bought a share at par, at a premium, or at a discount, the dividend calculation always uses the nominal value.
Let us now turn to the essential formulas you need to master this chapter.
The first formula calculates the total sum invested. Sum invested equals number of shares bought multiplied by market value of one share. In symbols: Sum invested = No. of shares × M.V. If shares are available at par, then market value equals nominal value, so the formula simplifies.
The second formula helps you find how many shares you can buy. Number of shares equals sum invested divided by market value of one share. Or: No. of shares = Sum invested / M.V.
The third formula calculates total dividend earned. Total dividend equals number of shares multiplied by rate of dividend multiplied by nominal value of a share. Written as: Total dividend = No. of shares × Rate of dividend × N.V.
The fourth formula gives the return percentage on your investment. Return percent equals income divided by investment, multiplied by 100 percent. That is: Return % = (Income / Investment) × 100%
Finally, remember this key equivalence: for a shareholder, income equals return equals profit, which equals the dividend paid by the company.
Let me walk you through some worked examples to see how these concepts work in practice.
First example: calculate the money required to buy 350 shares of nominal value 20 rupees each, at a premium of 7 rupees. Here, the nominal value is 20 rupees, and the premium is 7 rupees. So the market value equals 20 plus 7, which is 27 rupees. Money required for 350 shares equals 350 multiplied by 27, giving us 9,450 rupees.
Second example: Rakhee invested 12,500 rupees in shares of a company paying 6 percent dividend per annum. She bought 50 rupee shares for 62.50 rupees each. First, we find how many shares she bought: 12,500 divided by 62.50 equals 200 shares. The dividend on one share is 6 percent of the nominal value of 50 rupees, which equals 3 rupees. Therefore, her total income is 200 multiplied by 3, which equals 600 rupees.
Third example: Ramesh buys 100 rupee shares at a 20 rupee premium in a company paying 15 percent dividend. We need to find three things. First, the market value of 600 shares. Market value of one share equals 100 plus 20, which is 120 rupees. So market value of 600 shares equals 600 times 120, giving 72,000 rupees.
Second, his annual income. Using our formula: 600 shares multiplied by 15 percent multiplied by 100 rupees nominal value equals 9,000 rupees.
Third, his percentage income. This is income divided by investment times 100 percent: 9,000 over 72,000 times 100 percent equals 12.5 percent. Notice how the return percentage is lower than the dividend rate because he paid a premium for the shares.
Here is a powerful relationship that connects dividend rate, nominal value, return rate, and market value. Rate of dividend multiplied by nominal value equals profit percent multiplied by market value. Or: Dividend% × N.V. = Return% × M.V. This formula is incredibly useful when you need to find any one of these quantities given the other three.
Let us see this in action. A man buys an 80 rupee share in a company paying 20 percent dividend. He buys at such a price that his profit is 16 percent on his investment. At what price did he buy the share?
Using our relationship: 20 percent multiplied by 80 equals 16 percent multiplied by market value, which is 16. So 16 equals 0.16 times market value. Therefore, market value equals 100 rupees. Even though the nominal value is only 80 rupees, he paid 100 rupees because he wanted a 16 percent return rather than the full 20 percent.
Another important application is comparing different investments. Which is better: 12 percent 100 rupee shares at 120, or 8 percent 100 rupee shares at 90?
For the first investment: profit percent on 120 equals 12 percent on 100. This gives profit percent equals 10 percent.
For the second investment: profit percent on 90 equals 8 percent on 100. This gives profit percent equals approximately 8.9 percent.
Therefore, the first investment is better because it gives a higher return percentage. Notice that even though the first investment has a higher dividend rate and costs more, it still yields better returns. What matters is your return relative to what you actually invested.
Let us consider a more complex scenario involving selling shares and reinvesting. Mr. Ram Gopal invested 8,000 rupees in 7 percent 100 rupee shares at 80. After a year, he sold these shares at 75 each and invested the proceeds, including his dividend, in 18 percent 25 rupee shares at 41.
First, his dividend for the first year. Number of shares equals 8,000 divided by 80, which is 100 shares. Dividend on one share is 7 percent of 100, which is 7 rupees. Total dividend equals 700 rupees.
Second, his annual income in the second year. Proceeds from selling shares: 100 times 75 equals 7,500 rupees. Plus dividend of 700 rupees gives total 8,200 rupees to reinvest. Number of new shares: 8,200 divided by 41 equals 200 shares. Dividend on one new share: 18 percent of 25 equals 4.50 rupees. Annual income: 200 times 4.50 equals 900 rupees.
Third, the percentage increase in his return. Increase is 900 minus 700 equals 200 rupees. Percentage increase on original investment: 200 over 8,000 times one hundred percent equals two point five percent.
Now let us recap the key takeaways from this chapter.
First, the nominal value or face value of a share is fixed and never changes, while the market value fluctuates based on company performance and market conditions.
Second, shares can be at par, at a premium, or at a discount, depending on whether market value equals, exceeds, or is less than nominal value.
Third, dividend is always calculated as a percentage of nominal value, never market value, and represents the shareholder's income from the company.
Fourth, the fundamental formulas: sum invested equals number of shares times market value; total dividend equals number of shares times dividend rate times nominal value; and return percentage equals income over investment times 100 percent.
Fifth, the powerful relationship: dividend percent times nominal value equals return percent times market value, which connects all four key quantities.
Sixth, when comparing investments, always calculate the actual return percentage on your investment, not just the stated dividend rate.
Shares and dividends form a practical bridge between pure mathematics and financial literacy. The calculations you have learned today are not just exam topics, they are real skills used by investors every day. Master these formulas, understand the relationships between nominal and market values, and you will be well equipped to analyze any share investment problem.
Thank you for your attention, and I wish you success in your studies. Keep practicing, and remember: every calculation brings you closer to understanding the mathematics behind wealth creation.