Shares and Dividends — chapter notes

A company divides its capital into shares. A shareholder owns part of the company and may receive part of its profit as a dividend.

• Face value (nominal value, FV): printed on the share; it never changes.
• Market value (MV): the price at which the share is bought or sold today; it changes with demand.
• Brokerage and fractional shares are not in the ICSE 2027 syllabus: every problem here buys a whole number of shares and pays no brokerage.

• MV $=$ FV: the share is at par.
• MV $>$ FV: at a premium; premium $=$ MV $-$ FV.
• MV $<$ FV: at a discount; discount $=$ FV $-$ MV.
• A premium or discount per cent is a per cent of the face value: a ₹50 share at a 16% premium costs $50 + 8 = 58$.

The dividend is declared as a rate $r\%$ of the face value, whatever price the shareholder paid.

Example: 450 shares of ₹10 at 18%: $450 \times 1.80 = 810$. Doubling the number of shares doubles the income; the market value does not affect it.

Shares are bought at the market value:

• Premium: ₹15,600 in ₹100 shares at ₹130 buys 120 shares; at 9% the income is ₹1,080.
• Discount: ₹22,500 in ₹50 shares at ₹45 buys 500 shares; at 7.5% the income is ₹1,875.
• Income given: for ₹2,400 from ₹20 shares paying 15%, divide by the dividend on one share (₹3): 800 shares.

The return (yield) is the income as a per cent of the money invested:

• Bought at par: return $=$ rate of dividend.
• Bought at a premium: return $<$ rate of dividend.
• Bought at a discount: return $>$ rate of dividend.

Example: ₹100 shares at ₹125 paying 10% return $\dfrac{10}{125} \times 100 = 8\%$.

• The better investment is the one with the higher return, not the higher rate of dividend: ₹100 shares at ₹160 paying 13% (8.125%) beat ₹10 shares at ₹16 paying 12% (7.5%).
• With the same sum invested, the higher return gives the higher income.
• Price for a return: $\text{MV} = \dfrac{\text{FV} \times r}{\text{return}}$; e.g. 9% on ₹100 for 7.2% needs ₹125.
• Rate for a return: $r = \dfrac{\text{return} \times \text{MV}}{\text{FV}}$; e.g. ₹10 shares at ₹15 returning 8% pay 12%.

1. Old income $=$ shares $\times$ dividend on one share.
2. Sale proceeds $=$ shares sold $\times$ selling price.
3. New shares $=$ proceeds $\div$ new market value.
4. New income from the new face value and rate.
5. Change in income $=$ new income $-$ old income.

The gain or loss on the sale (selling price $-$ buying price, per share) is a different question from the change in income. A higher rate of dividend does not guarantee a higher income: the new price matters.