Shares and Dividends | ICSE Class 10 Maths Notes

This note covers shares and shareholders, face value and market value, premium, dividend and dividend rate, investment, dividend income, percentage return, and finding unknown quantities from the relationships between them.

What are shares, shareholders and share capital?

A share is a unit into which a company's share capital is divided. Share capital is the capital obtained through the issue of shares. A shareholder is a person who holds shares in the company.

Each share has a nominal value, also called its face value: the stated value assigned to that share. Multiplying the number of shares by their face value gives their total nominal value. This connects the idea of a share with a calculation.

How does a share represent ownership?

A dividend is a distribution of company profits to shareholders. Equity shares are ownership shares whose dividend is not fixed. The dividend on equity shares may vary from year to year depending upon the amount of profits available for distribution.

The rate of dividend is dividend expressed as a percentage of face value. A percentage expresses a quantity per hundred.

Holding a share and receiving a dividend are therefore different ideas. The share represents an ownership interest; the dividend is income received on that share. A stated dividend rate supplies information for a calculation, without establishing that the same rate will apply in another year.

How is total nominal value calculated?

The symbol ₹ means Indian rupees. In the following calculation, the number of shares is a count, while the value of each share is an amount in rupees. Multiplying these gives an amount in rupees, not a percentage.

In formulae, × and adjoining letters indicate multiplication, / denotes division, = means equality, + means addition, and − means subtraction. Brackets group operations.

Worked example 1. A company can issue 1,00,000 shares of ₹10 each. Find the total nominal value of these shares.

Answer: Total nominal value = number of shares × nominal value per share = 1,00,000 × ₹10 = ₹10,00,000.

The amount describes the shares at their stated nominal value. It does not supply a separate market price or a dividend rate. Those quantities must be identified independently before calculating the cost of a market purchase or the income from a dividend.

This distinction is useful throughout the topic: first identify what a quantity measures, then choose the operation. A count of shares, an amount paid, and income received cannot be used interchangeably merely because all concern the same holding.

How do face value and market value differ?

Market value is the price at which a share is bought or sold in the market at the stated time. Face value identifies its nominal amount. These values answer different questions: face value supplies the base for dividend calculations, while market value supplies the cost per share in a market purchase.

Which symbols will be used?

Let F denote face value per share and M denote market value per share, both in rupees. Let n denote the number of shares held. Let I denote investment, meaning the total amount spent to buy those shares.

Throughout the formulae, face value and purchase price are positive, and a holding contains a positive whole number of shares. Calculations use shares with the same face value, bought at the same stated price per share. Any comparison keeps its stated conditions in place.

Why must the two values stay separate?

For a holding of n shares, its total nominal value is n × F. Its purchase cost at market value is n × M. These products have the same unit, rupees, but they need not have the same value because they use different prices.

A question giving face value has not thereby given market value. Similarly, knowing how much was spent on shares does not reveal their total nominal value unless the number of shares and their face value can also be established.

Note: Use face value to calculate dividend. Use the actual purchase price per share to calculate investment. Label both before substituting values into a formula.

When reviewing a solution, read each amount together with its description. The word “value” alone is insufficient: “face value per share”, “market value per share” and “total nominal value” identify different quantities and prevent an otherwise plausible substitution from answering the wrong question.

What does a premium on a share mean?

A premium is the excess of a share's price over its face value. A share issued at a price above its nominal value is issued at a premium. The issue price is the price at which the company issues the share.

A share is at par when its price equals its face value. An issue price and a later market price describe different transactions. Identify which price the question supplies before calculating the excess over face value.

How are premium amount and premium percentage related?

Let P denote the premium amount per share in rupees, and let p denote the numerical premium percentage. The symbol % means “per hundred”, so p% means p/100. A premium percentage is calculated with face value as the base.

For a market price above face value, P = M − F. Conversely, M = F + P. To express this excess as a percentage of face value, use p = 100P/F. A percentage quotation must therefore be distinguished from an amount in rupees.

Worked example 2. A share of nominal value ₹100 is issued at ₹105. Find the premium amount per share and the premium percentage.

Answer: Premium amount = ₹105 − ₹100 = ₹5 per share. Premium percentage = (5/100) × 100 = 5%.

What does the premium change?

The premium increases the price above face value. It does not become part of the face value used to calculate dividend. The same distinction applies whether the excess is supplied directly as a rupee amount or indirectly as a percentage of face value.

If the premium is given as p% of face value, first obtain the premium amount as pF/100. Then add that amount to face value to obtain the price. Do not add the percentage number directly to a rupee amount.

Check the answer by subtracting face value from the resulting price. This should recover the premium amount. Dividing that difference by face value and multiplying by 100 should recover the quoted premium percentage. Both checks must refer to the same transaction and the same share.

Worked example 3. Jupiter Company Limited issues 35,000 equity shares of face value ₹10 each at a premium of ₹2 per share. The issue is fully subscribed and all money is received. Find the issue price, premium percentage and total amount received.

Answer: Issue price equals face value plus premium: $10 + 2 = 12$ rupees per share. The premium percentage is $\frac{2}{10}\times 100 = 20\%$.

Total amount received is $35{,}000\times 12 = 4{,}20{,}000$ rupees. Check: nominal capital is $35{,}000\times 10 = 3{,}50{,}000$ rupees and total premium is $35{,}000\times 2 = 70{,}000$ rupees. Their sum is ₹4,20,000.

How is the dividend on each share calculated?

The rate of dividend expresses dividend as a percentage of face value. Let r denote the numerical dividend percentage, so the rate is r%. Let d denote the dividend amount on each share in rupees for the stated period.

The calculation applies a percentage to a specified base. Since r% means r/100 and the base is F, multiply F by r/100. The resulting dividend per share is d = rF/100. This is a money amount, distinct from the percentage rate.

Result: Dividend is based on face value

1. Start with the face value F of one share, expressed in rupees.
2. Convert the dividend percentage r% into its fractional multiplier, r/100.
3. Multiply face value by that multiplier to obtain d = F × r/100.
4. State d as the dividend per share for the period to which the stated rate applies.

The price paid to acquire the share does not enter this calculation. If face value and dividend rate remain the same, dividend per share remains the same even when the purchase price differs. Purchase price matters when measuring return on the money invested.

What does a stated rate establish?

A stated dividend rate allows the dividend amount to be calculated for the specified period. It does not establish a permanent rate for equity shares. Their dividend is not fixed and may vary from year to year depending upon the profits available for distribution.

For an annual dividend calculation, the period is one year. Keep that period attached to the answer: dividend per share for the year and total annual dividend describe related quantities, but the first still needs multiplication by the number of shares.

Note: When r is the numerical percentage, divide by 100. If the rate has already been written as the fraction r/100, apply that fraction once; do not divide by 100 a second time.

A useful check is to read the calculation in words: dividend per share equals dividend fraction multiplied by face value per share. This shows both the correct base and the reason for the percentage conversion.

How are investment and the number of shares connected?

For shares bought at a common market price, investment equals the number of shares multiplied by the price paid for each. With the symbols already defined, this gives I = nM. The count n multiplies a rupee amount per share to produce the total purchase cost.

Result: Number of shares equals investment divided by price

When the full investment I is spent at price M per share, divide I = nM by M. Since M is positive, this gives n = I/M. The denominator is the price actually paid, rather than face value chosen simply because it appears in the question.

The calculation concerns whole shares, meaning complete units of shareholding. The data must give a whole-number holding when the entire investment is said to have been spent. A fractional quotient calls for checking the interpretation and arithmetic rather than reporting part of a share.

How should a purchase calculation be organised?

1. Identify the total amount invested and the price paid per share.
2. If the price is described through a premium, calculate the complete price first.
3. Divide the investment by that price to find the number of shares.
4. Multiply the resulting number by the price to check that it reproduces the investment.

Brokerage means a broker's charge for arranging a transaction. The purchase relationships used here contain no brokerage charge and use complete shares. Thus, I represents the purchase cost obtained directly by multiplying count by price.

Distinguish an amount actually invested from an amount merely available to spend. The equation I = nM describes money spent on the shares. It does not assert that every available sum can be spent exactly on complete shares at a given price.

Finding the number of shares is often the bridge between purchase information and income information. Once n has been established, it can multiply the dividend per share. Keep the market price in the purchase calculation and the face value in the dividend calculation so that these stages remain consistent.

As a final check, attach units to the division. Rupees divided by rupees per share gives a number of shares. It does not directly give income or a percentage return.

Worked example 4. Sunrise Company Ltd. has registered capital of ₹40,00,000 divided into shares of ₹10 each. Find the number of shares represented by this capital.

Answer: Divide total nominal capital by face value per share: $n = \frac{40{,}00{,}000}{10} = 4{,}00{,}000$ shares. Check: $4{,}00{,}000\times 10 = 40{,}00{,}000$ rupees. This gives 4,00,000 shares; it is a nominal-capital calculation, not a market purchase.

Worked example 5. Rahul Limited buys a building from Handa Limited for ₹5,40,000, paying by issuing shares of face value ₹100 each at par. How many shares must it issue?

Answer: At par, the issue price equals the face value, ₹100 per share. Number of shares equals amount payable divided by issue price: $n = \frac{5{,}40{,}000}{100} = 5{,}400$ shares.

Check: $5{,}400\times 100 = 5{,}40{,}000$ rupees, matching the building price. The payment therefore requires 5,400 shares.

Worked example 6. Rahul Limited instead issues its ₹100 shares at a premium of 20% to pay Handa Limited for the ₹5,40,000 building. Find the issue price and number of shares required.

Answer: Premium per share is $\frac{20}{100}\times 100 = 20$ rupees. Issue price is $100 + 20 = 120$ rupees per share. Divide the amount payable by this price: $n = \frac{5{,}40{,}000}{120} = 4{,}500$ shares.

Check: $4{,}500\times 120 = 5{,}40{,}000$ rupees. Thus, 4,500 shares settle the payment. The count is smaller than at par because each share is issued for a larger amount.

Worked example 7. Jindal and Company buys a machine from High Life Machine Limited for ₹3,80,000. It pays ₹20,000 in cash and the balance with shares of face value ₹100 each issued at par. Find the number of shares issued.

Answer: First deduct the cash payment: $3{,}80{,}000 - 20{,}000 = 3{,}60{,}000$ rupees remain payable in shares. At par, issue price is ₹100 per share, so $n = \frac{3{,}60{,}000}{100} = 3{,}600$ shares.

Check the entire payment: $3{,}600\times 100 + 20{,}000 = 3{,}80{,}000$ rupees. The required issue is 3,600 shares.

Worked example 8. Jindal and Company pays ₹20,000 in cash for its ₹3,80,000 machine and settles the balance by issuing ₹100 shares at a premium of 20%. Find the number of shares required.

Answer: Amount payable in shares is $3{,}80{,}000 - 20{,}000 = 3{,}60{,}000$ rupees. Premium per share is $\frac{20}{100}\times 100 = 20$ rupees, so issue price is $100 + 20 = 120$ rupees per share.

Number of shares is $n = \frac{3{,}60{,}000}{120} = 3{,}000$. Check: $3{,}000\times 120 + 20{,}000 = 3{,}80{,}000$ rupees. Therefore, 3,000 shares, together with the cash payment, meet the machine's full price.

How is total dividend income calculated?

Dividend income is the total dividend received on the holding for the stated period. Let D denote this total in rupees. If each of the n shares receives dividend d, multiplying the per-share amount by the count gives D = nd.

Result: Total income equals shares multiplied by dividend per share

1. The dividend rate is r%, equivalent to the fraction r/100.
2. For face value F, dividend per share is d = rF/100.
3. A holding of n such shares receives n times the dividend on one share.
4. Substituting the expression for d gives D = nrF/100.

Thus, D = nrF/100. This form shows all the ingredients: number of shares, dividend percentage and face value. The market value does not replace face value in the product, even if market value was used earlier to determine n.

What if the question gives investment instead of share count?

Use the purchase relationship n = I/M first. Substituting this into the income formula gives D = IrF/(100M). The formula combines the purchase and dividend stages without changing the role of either price.

Working in separate stages can make a solution easier to check. Find the number of shares, find dividend per share, and multiply them. Alternatively, use the combined expression, provided the meanings of investment, market price and face value are clear.

Use dividend amounts for the same period throughout. If the rate and per-share dividend are annual, D is annual income. A calculation using one share's dividend cannot be labelled as the income of the entire holding unless the holding consists of that single share.

Check the total by dividing it by n. This should recover the per-share dividend d. Applying the dividend fraction to F should give the same d independently, providing a check on both the share count and the percentage calculation.

How does percentage return differ from dividend rate?

Percentage return measures dividend income relative to the amount invested. Let y denote its numerical percentage, so the return is y%. For income D and investment I, the relationship is y = 100D/I.

The dividend rate r% uses face value as its base. The return y% uses the amount actually invested. Both are percentages, but they describe different comparisons. Stating the base of each percentage is more reliable than assuming that equal-looking percentage language means equal values.

Result: Return depends on the ratio of face value to purchase price

1. Total dividend income for the holding is D = nrF/100.
2. The investment at market price M per share is I = nM.
3. Substitute into the return formula: y = 100 × (nrF/100)/(nM).
4. Cancel the common positive share count and the percentage conversion to obtain y = rF/M.

Therefore, y = rF/M. For a fixed face value and dividend rate, the percentage return depends on purchase price. Increasing the number of shares at the same price increases both investment and income in the same proportion, leaving this percentage unchanged.

What happens at par or at a premium?

At par, M = F. Substituting into y = rF/M gives y = r, so the percentage return equals the dividend percentage. At a premium, M is greater than F. For a positive dividend rate, F/M is less than one, so y is less than r.

The lower return percentage at a premium does not mean that the dividend per share has changed. It means that the same dividend amount is being compared with a higher purchase cost. This conclusion assumes the same face value and dividend rate.

The formula measures return from dividend income for the stated period. It contains neither a selling price nor a calculation of profit from selling the shares. An answer based on this formula should therefore be described as a dividend return.

To check a result, calculate dividend per share and divide it by purchase price per share, then multiply by 100. This should give the same percentage as using total dividend income and total investment for the holding.

How can the formulae be rearranged and checked?

An inverse calculation finds a missing starting quantity from a known result. The income and return relationships can be rearranged without introducing another rule. First identify the unknown, choose an equation containing it, and ensure that every other quantity in that equation is supplied or can be found.

Which rearrangements are useful?

For the share-count relationship, start with D = nrF/100. Multiply by 100 and divide by rF. This requires a positive dividend rate, since division by zero is undefined. The resulting n must also be compatible with a whole-share holding.

For market price, start with y = rF/M. Multiplying by M and dividing by positive y gives M = rF/y. Here r and y are both numerical percentages. Their percentage conversion factors have already cancelled in the underlying return formula.

How do you verify the recovered value?

1. Read the question again and label each given quantity, including whether an amount is per share or for the whole holding.
2. State the original relationship before rearranging it, keeping percentage rates distinct from fractional multipliers.
3. Calculate the missing quantity and attach its correct description: shares, rupees, rupees per share or percentage.
4. Substitute the result into the original relationship and check that it reproduces the supplied income, investment or return.

Consistency checks should use the conditions of the question. For a positive dividend on shares bought at a premium, return should be below the dividend rate. For shares bought at par, those percentages should agree. Neither check replaces the calculation; each tests its interpretation.

A missing datum should remain missing until a stated relationship determines it. Do not choose an unstated dividend rate, treat face value as market value without justification, or insert a sale price into a dividend-income calculation.

These checks connect meaning with algebra. A rearrangement can be mechanically correct yet applied to the wrong quantity. Writing the unknown in words beside its symbol helps establish that the formula answers the question actually asked.

Glossary

• Share — A unit into which a company's share capital is divided, representing an ownership interest.
• Shareholder — A person who holds shares in a company and has an ownership interest through them.
• Share capital — The capital a company obtains through the issue of its shares.
• Face value — The nominal amount assigned to each share, used as the base for dividend calculations.
• Nominal value — Another name for the face value assigned to a share.
• Market value — The price at which a share is bought or sold in the market at the stated time.
• Issue price — The price at which the company issues a share to a subscriber.
• At par — A description of a share priced at exactly its face value.
• Premium — The excess of a share's price over its face value, expressed as an amount or percentage.
• Dividend — A distribution of company profits to shareholders, calculated here using the stated rate and face value.
• Rate of dividend — Dividend expressed as a percentage of the face value of the share.
• Investment — The total amount spent to acquire the holding of shares at the stated purchase price.
• Dividend income — The total dividend received on all shares in a holding for the stated period.
• Percentage return — Dividend income expressed as a percentage of the amount invested in the shares.
• Equity shares — Ownership shares whose dividend is not fixed and may vary with profits available for distribution.

Common errors and misconceptions

• Misconception: Dividend is calculated on market value because that is the price paid. Correct: Apply the dividend rate to face value. Market price is used to calculate purchase cost and dividend return.
• Misconception: Face value and market value are interchangeable names for one amount. Correct: Face value is the nominal amount assigned to the share; market value is its price in the market at the stated time.
• Misconception: A premium is added to face value before calculating dividend. Correct: Premium increases the price above face value. Dividend remains based on face value at the given rate.
• Misconception: Dividend percentage and percentage return must agree. Correct: They use different bases. They agree at par; for a positive dividend rate, a purchase at a premium gives a lower dividend return percentage.
• Misconception: Divide investment by face value to find shares bought in the market. Correct: Divide by the price actually paid per share, then check that the holding consists of complete shares.
• Misconception: Multiplying face value directly by the numerical percentage gives dividend per share. Correct: Divide that percentage by 100 before multiplying, or use an already converted fractional rate once.
• Misconception: Dividend per share is the income on the whole holding. Correct: Multiply the per-share dividend by the number of shares to obtain total dividend income for the same period.
• Misconception: A stated equity dividend rate establishes next year's income. Correct: Equity dividend is not fixed and may vary from year to year depending upon profits available for distribution.

Exam-style questions with model answers

Q1. Distinguish between face value and market value, stating the calculation in which each is used. [2 marks]

1. Face value is the nominal amount assigned to a share. The stated dividend percentage is applied to this amount to calculate dividend per share.
2. Market value is the share's market price at the stated time. It determines purchase cost and is used when calculating dividend return on that purchase.

Q2. A share of nominal value ₹100 is issued at ₹105. Calculate the premium per share and the premium percentage. [2 marks]

1. The premium amount is the excess of issue price over face value: ₹105 − ₹100 = ₹5 per share.
2. The percentage uses face value as its base: (₹5/₹100) × 100 = 5%. Thus, the share is issued at a premium of 5%.

Q3. A company can issue 1,00,000 shares of nominal value ₹10 each. State what the nominal value means, calculate the total nominal value, and explain whether these data determine dividend income. No dividend rate is supplied. [3 marks]

1. The nominal value is the face value assigned to each share. Here it is ₹10 per share, rather than a stated market purchase price.
2. Total nominal value equals number of shares multiplied by nominal value per share: 1,00,000 × ₹10 = ₹10,00,000.
3. Dividend income cannot be calculated from these data. The number of shares and face value are supplied, but the dividend rate needed to calculate income is missing.

Q4. A holding contains n shares, each of face value F rupees, with an annual dividend rate of r%. Here n is a positive whole number, F is positive, and r is the numerical dividend percentage. Derive the dividend per share and total annual dividend income. [4 marks]

1. The rate r% means the fraction r/100. This fraction must be applied to face value because dividend is calculated on the nominal value of a share.
2. Let d be the annual dividend per share in rupees. Then d = F × r/100 = rF/100.
3. Let D be total annual dividend income in rupees. The n shares together receive n times the per-share dividend, so D = nd.
4. Substituting d gives D = nrF/100 rupees. The answer describes the entire holding's annual dividend, while d describes the annual dividend on one share.

Q5. An investor buys n complete shares at market price M rupees each, without brokerage. Each has face value F rupees and pays an annual dividend of r%. Here n, M and F are positive, and r is the numerical dividend percentage. Derive the annual percentage return and explain why changing n alone does not change it. [5 marks]

1. Let I denote investment in rupees. Multiplying the number of shares by purchase price per share gives I = nM, with no additional transaction charge.
2. Let D denote annual dividend income in rupees. Dividend on one share is rF/100 rupees, so the holding receives D = nrF/100.
3. Let y be the numerical annual return percentage. Income as a percentage of investment gives y = 100D/I.
4. Substituting the expressions gives y = 100 × (nrF/100)/(nM). Cancelling the common factors yields y = rF/M, so the annual return is y%.
5. The positive share count cancels. Changing n alone multiplies income and investment in the same proportion, leaving their ratio and the return percentage unchanged.

Q6. A share has face value F rupees, purchase price M rupees and a positive annual dividend rate r%. All quantities are positive. Compare its annual dividend return percentage with r% when M equals F and when M is greater than F. [3 marks]

1. Let y denote the numerical annual dividend return percentage. Dividend per share is rF/100 rupees, giving y = 100 × (rF/100)/M = rF/M.
2. When M = F, the share is bought at par. Substitution gives y = r, so the return percentage equals the dividend percentage.
3. When M is greater than F, the share is bought at a premium. Since F/M is less than one and r is positive, y is less than r.

Q7. A complete-share holding earns annual dividend income D rupees. Each share has face value F rupees and an annual dividend rate r%. The shares were bought at M rupees each without brokerage. All quantities are positive and the data give a whole-number holding. Derive expressions for the number of shares, investment and annual return percentage. [5 marks]

1. Let d denote annual dividend per share in rupees. Applying the dividend fraction to face value gives d = rF/100.
2. Let n denote the number of shares. Since total income equals n times per-share dividend, n = D/d = 100D/(rF).
3. Let I denote investment in rupees. Multiply count by purchase price: I = nM = 100DM/(rF).
4. Let y denote the numerical annual return percentage. Then y = 100D/I. Substituting the investment expression and cancelling gives y = rF/M.
5. Check the holding by substituting n into nrF/100; it reproduces D. The question's whole-number condition permits complete shares, and the positive rate permits division by rF.

Q8. A share has face value F rupees and a positive annual dividend rate r%. Its annual dividend return at purchase is y%, where y is positive and smaller than r. Here r and y are numerical percentages, and there is no brokerage. Find the purchase price and express its premium amount. [4 marks]

1. Let M denote purchase price per share in rupees. The dividend-return relationship is y = rF/M because dividend per share is rF/100 rupees.
2. Multiply the equation by M and divide by positive y. The purchase price is therefore M = rF/y rupees per share.
3. Since y is smaller than r, the ratio r/y is greater than one. Therefore M is greater than F, establishing that the purchase is at a premium.
4. Let P denote premium amount per share. Subtract face value from price: P = rF/y − F rupees per share.

Key takeaways

• A share is a unit of share capital; its holder is a shareholder with an ownership interest in the company.
• Face value supplies the base for dividend calculations, while the price actually paid supplies the cost per share.
• A premium is the excess over face value; calculate its percentage by comparing that excess with face value.
• Dividend per share equals face value multiplied by the dividend percentage divided by 100, using the stated period.
• Total dividend income equals dividend per share multiplied by the number of shares held for the calculation.
• Percentage return compares dividend income with investment; it can differ from the dividend percentage because the bases differ.
• At par, return and dividend percentages agree; at a premium, a positive dividend gives a lower return percentage.
• Check inverse calculations in the original formula, preserve units, and ensure that a share count represents complete shares.

Test yourself

Which value is the base for calculating dividend, and which price determines purchase cost?

Dividend uses face value as its base. Purchase cost uses the price actually paid per share, multiplied by the number of shares bought.

A ₹100 share is issued at ₹105. What is its premium amount?

The premium is ₹5 per share, obtained by subtracting the ₹100 face value from the ₹105 issue price.

Why is total nominal value insufficient to calculate dividend income when no dividend rate is supplied?

Total nominal value provides the percentage base. The dividend rate is still needed to determine what fraction of that amount becomes dividend income.

What changes in the dividend calculation when more shares with the same face value and dividend rate are held?

The dividend per share remains the same. Total dividend income increases in proportion to the number of those shares held.

For a positive dividend rate, how does buying at a premium affect dividend return compared with the dividend percentage?

The dividend return percentage is lower because dividend is calculated on face value but compared with a purchase price greater than face value.

Why does increasing the share count alone leave dividend return percentage unchanged?

At the same purchase price and dividend per share, income and investment increase in the same proportion. Their ratio therefore remains unchanged.

How can you check a number of shares calculated from investment and purchase price?

Multiply the calculated whole-number holding by the price per share. The product should equal the amount stated to have been invested.

Does a stated equity dividend rate establish the rate for the following year?

No. Equity dividend is not fixed and may vary from year to year depending upon the profits available for distribution.