For a zero coupon bond there is no interest, so the value is the present value of the face value received at maturity.

$V = \frac{M}{(1+k)^n} = \frac{1{,}000}{(1.12)^{10}}$

Since $(1.12)^{10} = 3.1058$,

$V = \frac{1{,}000}{3.1058} = \text{Rs } 321.97$

Value of the bond = Rs 321.97

Using Gordon's constant growth model:

$D_1 = D_0(1+g) = 25(1.05) = \text{Rs } 26.25$

$P_0 = \frac{D_1}{k_e - g} = \frac{26.25}{0.15 - 0.05} = \frac{26.25}{0.10} = \text{Rs } 262.50$

Value of the stock at present = Rs 262.50 per share.

By discounting the expected future cash flows (dividends) to their present value.

Perpetual Bond

A perpetual bond (also called a consol or irredeemable bond) is a bond that has no maturity date. It never repays the principal (face value); the issuer pays a fixed coupon (interest) at regular intervals forever. Since the principal is never returned, the only cash flows to the investor are the periodic interest payments, continuing indefinitely.

Because the bond has no maturity, its value depends only on the coupon and the required rate of return.

Valuation of a perpetual bond:

$V = \frac{I}{k_d}$

where:

• $V$ = value of the bond
• $I$ = annual coupon (interest) payment
• $k_d$ = required rate of return (discount rate)

Example: A perpetual bond pays Rs 120 interest annually and the required return is 12%. $V = \frac{120}{0.12} = \text{Rs } 1{,}000$

Valuation of a Zero Coupon Bond

A zero coupon bond pays no periodic interest. It is issued at a discount and the investor receives only the face value at maturity. The return comes entirely from the difference between the purchase price and the maturity value.

Since there are no coupon payments, the value is simply the present value of the face value discounted at the required rate for the number of years to maturity.

$V = \frac{M}{(1 + k_d)^n}$

where:

• $V$ = value (price) of the bond
• $M$ = face (maturity) value
• $k_d$ = required rate of return
• $n$ = number of years to maturity

Example: A zero coupon bond of face value Rs 1,000 matures in 5 years; required return is 10%. $V = \frac{1000}{(1.10)^5} = \frac{1000}{1.6105} = \text{Rs } 620.92$

Valuation of a Coupon Bond with Finite Maturity

A coupon bond with finite maturity pays a fixed coupon at regular intervals and repays the face value at a fixed maturity date. Its value is the present value of all future cash flows — the stream of coupon payments plus the maturity value.

$V = \sum_{t=1}^{n} \frac{I}{(1 + k_d)^t} + \frac{M}{(1 + k_d)^n}$

This can also be written as:

$V = I \times \left[ \frac{1 - (1 + k_d)^{-n}}{k_d} \right] + \frac{M}{(1 + k_d)^n}$

where:

• $I$ = periodic coupon (interest) payment
• $M$ = face value repaid at maturity
• $k_d$ = required rate of return
• $n$ = number of years to maturity

The first term is the present value of the annuity of coupons; the second term is the present value of the lump sum face value.

Example: A bond of face value Rs 1,000 pays 10% coupon annually and matures in 3 years; required return is 12%.

$V = \frac{100}{(1.12)^1} + \frac{100}{(1.12)^2} + \frac{100}{(1.12)^3} + \frac{1000}{(1.12)^3}$

$V = 89.29 + 79.72 + 71.18 + 711.78 = \text{Rs } 951.97$

Summary of the Three Cases

Key point: In all three cases, the value of a bond is the present value of its expected future cash flows, discounted at the required rate of return. The difference lies only in the pattern of cash flows — perpetual, single lump sum, or annuity plus lump sum.

a. Value of common stock

Given: $D_0 = Rs. 10,\quad g = 20\% = 0.20,\quad k_e = 12\% = 0.12$

The constant dividend growth model is: $P_0 = \frac{D_0(1+g)}{k_e-g}$

It requires (k_e > g). Here (12% < 20%), so (k_e-g = -8%). Therefore the dividend growth model is not applicable. The value of the common stock cannot be determined using this model. If applied mechanically: $P_0 = \frac{10(1.20)}{0.12-0.20} = \frac{12}{-0.08} = -Rs. 150$ which is not economically meaningful.

b. Value of bond

Assuming annual coupon payments: $C = 10\% \times 1000 = Rs. 100,\quad r = 15\% = 0.15,\quad n = 10,\quad F = Rs. 1000$

$P_0 = C \times \frac{1-(1+r)^{-n}}{r} + \frac{F}{(1+r)^n}$

$P_0 = 100 \times \frac{1-(1.15)^{-10}}{0.15} + \frac{1000}{(1.15)^{10}}$

$P_0 = 100 \times 5.0188 + 1000 \times 0.2472$

$P_0 = 501.88 + 247.18 = Rs. 749.06$

Thus, the value of the bond is Rs. 749.06.

Differences between preferred stock and common stock:

1. Rate of dividend: Preferred stock carries a fixed rate of dividend, whereas common stock carries a fluctuating (variable) rate of dividend depending on the profit earned.
2. Priority of dividend and repayment: Preferred stockholders get dividend before common stockholders and are repaid their capital before common stockholders at the time of liquidation.

Any two characteristics of a bond:

1. Fixed interest (coupon) payment — the issuer pays a stated rate of interest at fixed intervals until maturity.
2. Fixed maturity date and face value — the principal (face value) is repaid to the bondholder on a predetermined maturity date.

Meaning of Ordinary Share

An ordinary share (also called equity share or common stock) is a unit of ownership capital that represents the residual ownership of a company. The holder of an ordinary share is a part-owner of the company, bears the highest risk, and receives dividend only after all other claims (debentures, loans and preference shares) have been paid. In Nepal, ordinary shares are issued under the Companies Act, 2063 and are the most common form of share capital.

Features of Ordinary Shares

1. Residual claim on income and assets — Ordinary shareholders are paid dividend only after preference shareholders have been paid their fixed dividend. In case of winding up, they receive the remaining assets only after all creditors and preference shareholders are settled. Hence they are the last claimants.

2. Ownership and control — Ordinary shares represent real ownership of the company. Holders enjoy voting rights in the general meeting (one share, one vote) and thus control the management by electing the board of directors. They also have the right to vote on major decisions.

3. Variable (fluctuating) dividend — The dividend on ordinary shares is not fixed. It depends on the profit earned and the dividend policy of the company. In a profitable year the dividend may be high; in a loss year no dividend may be paid at all.

4. No fixed rate of return but high earning potential — Ordinary shares carry no guaranteed return, but they offer the possibility of high returns through increased dividend and capital appreciation. The market value of ordinary shares rises when the company performs well.

5. Permanent capital / no maturity — Ordinary share capital is permanent capital. It is not repayable during the life of the company and is returned only at the time of winding up. Shareholders can, however, sell their shares in the secondary market to convert them into cash.

6. Limited liability — The liability of ordinary shareholders is limited to the face value of the shares held by them. Their personal property is not at risk for the debts of the company.

7. Pre-emptive right — Existing ordinary shareholders usually have the right to be offered new shares first, in proportion to their existing holding, so that their ownership proportion is not diluted.

a) Value of perpetual bond

Given:

• Par value (M) = Rs. 1,000
• Coupon rate = 12%
• Annual interest (I) = 12% of Rs. 1,000 = Rs. 120
• Required rate of return (k_d) = 15%

Formula for a perpetual (irredeemable) bond:

$V_d = \frac{I}{k_d}$

Substituting:

$V_d = \frac{120}{0.15} = \text{Rs. } 800$

Conclusion: Since the required return (15%) is greater than the coupon rate (12%), the bond sells at a discount. Its value is Rs. 800.

b) Value of stock at present

Given:

• Last year's dividend (D₀) = Rs. 20
• Required rate of return (k_e) = 14%
• Growth rate (g) = 8%

Step 1: Expected dividend for next year

$D_1 = D_0(1+g) = 20(1.08) = \text{Rs. } 21.60$

Step 2: Value of stock using the Gordon growth model

$P_0 = \frac{D_1}{k_e - g}$

$P_0 = \frac{21.60}{0.14 - 0.08} = \frac{21.60}{0.06} = \text{Rs. } 360$

Conclusion: The present value of the stock is Rs. 360 per share.

The value of a common share under the zero-growth model is determined by capitalising the constant annual dividend at the required rate of return:

$P_0 = \frac{D}{K_e}$

where $P_0$ = value of the share, $D$ = constant annual dividend, and $K_e$ = required rate of return (capitalisation rate). Since dividends never grow, the share is valued as a perpetual annuity.

A zero-coupon bond pays no periodic interest; its value is simply the present value of the face value receivable at maturity.

$V_0 = \frac{FV}{(1+k_d)^n} = FV \times PVIF_{k_d,\,n}$

where $FV$ = face value, $k_d$ = required rate of return (discount rate), $n$ = years to maturity.

Valuation of Common Stock

Common stock (ordinary share) represents ownership in a company. Unlike bonds or preference shares, its dividend is not fixed and its holding period is uncertain, so its value is estimated rather than determined exactly. The following procedures are adopted:

1. Book Value Method

Book value per share is found from the balance sheet:

$\text{Book Value per Share} = \frac{\text{Total Net Assets} - \text{Preference Claims}}{\text{Number of Common Shares Outstanding}}$

This method uses historical figures and ignores the earning capacity and market position of the firm, so it is only a rough guide.

2. Dividend Capitalisation Method (Dividend Discount Model)

The value of a share is the present value of all expected future dividends.

a. Zero growth (perpetual constant dividend):

$P_0 = \frac{D}{k_e}$

where $D$ = annual dividend per share, $k_e$ = required rate of return.

b. Constant growth (Gordon's model):

$P_0 = \frac{D_1}{k_e - g} = \frac{D_0(1+g)}{k_e - g}$

where $g$ = constant growth rate of dividends, $D_1$ = next year's dividend. This model requires $k_e > g$.

c. Variable (multiple) growth: Dividends growing at different rates in different periods are discounted year by year, and the terminal value at the end of the abnormal growth period is added.

3. Earnings Capitalisation Method

Value is based on the earning power of the firm:

$P_0 = \frac{EPS}{k_e} \quad \text{or} \quad P_0 = \text{EPS} \times \text{P/E ratio}$

This method is useful when the company retains most of its earnings and pays little dividend.

4. Market Value / P/E Multiple Method

The share is valued by applying the industry's price–earnings multiple to the company's EPS. This reflects investor confidence and market sentiment.

Example

Zero growth: A company pays a dividend of Rs 12 per share every year. Investors require a 15% return. Find the value of the stock.

$P_0 = \frac{D}{k_e} = \frac{12}{0.15} = \text{Rs 80 per share}$

Constant growth: The same company's dividend just paid is $D_0$ = Rs 10, expected to grow at 6% forever, and $k_e$ = 15%.

$D_1 = 10(1.06) = \text{Rs 10.60}$ $P_0 = \frac{10.60}{0.15 - 0.06} = \frac{10.60}{0.09} = \text{Rs 117.78 per share}$

Earnings capitalisation: If EPS = Rs 20 and the capitalisation rate is 12.5%,

$P_0 = \frac{20}{0.125} = \text{Rs 160 per share}$

Conclusion: The dividend discount model is the most widely used procedure because it directly links the share's value to the cash flows (dividends) an investor actually receives. In practice, analysts use it together with the earnings and market value methods to cross-check the estimate.

Authorized capital is only the upper limit; book value depends on issued capital and reserves.

(a) Total book value of shares

$\text{Total book value} = \text{Issued capital} + \text{Retained earnings} = 8{,}00{,}000 + 2{,}00{,}000 = \text{Rs } 10{,}00{,}000$

(b) Book value per share

$\text{Number of shares} = \frac{8{,}00{,}000}{100} = 8{,}000$

$\text{Book value per share} = \frac{10{,}00{,}000}{8{,}000} = \text{Rs } 125$

(c) If the remaining shares are issued at Rs 120 each

Remaining authorized capital = $20{,}00{,}000 - 8{,}00{,}000 = 12{,}00{,}000$, so new shares = $12{,}00{,}000 / 100 = 12{,}000$.

Amount raised = $12{,}000 \times 120 = \text{Rs } 14{,}40{,}000$.

$\text{Total equity} = 10{,}00{,}000 + 14{,}40{,}000 = \text{Rs } 24{,}40{,}000$

$\text{Total shares} = 8{,}000 + 12{,}000 = 20{,}000$

$\text{Book value per share} = \frac{24{,}40{,}000}{20{,}000} = \text{Rs } 122$