**Common stock** is the types of stock or shares, which are for the general public and anybody can buy them. The values of Stock/share differs on the basis of the types of stock under consideration. There are two **Types of Stock**.

**Common Stock/Share**whose dividend income fluctuates**Preferred Stock/Share**whose dividend income is fixed on regular basis

**Preferred Stock/Share Pricing**

Preferred stock further has two types of valuation or pricing which are s follow

- Perpetual Investment Having Fixed Regular Dividends
- Finite Investment

**Perpetual Investment Having Fixed Regular Dividends:**

When an individual (or company) decides to purchase the stocks/shares of particular company & retain them forever then this represents perpetual investment that contains fix regular dividends cash flow stream. Following is the formula for stock/share pricing in this case

PV = P_{o}* = DIV_{1} / r_{PE}

Where PV = Present Market Value or estimated present price

DIV_{1} = Anticipated Future Dividend in the next Period (DIV_{1} = DIV_{2} = DIV_{3} . . .)

r_{PE} = The individual investor’s minimum required rate of return on preferred stock

**Example:**

The preferred stocks of Company XYZ are traded in the Karachi stock exchange and has market price of Rs 12. The company offers to pay fixed dividend of Rs 2 per share. The par value of every share is Rs 10. It is expected that after two years the price of preferred stock becomes Rs 12. The individual investing in that preferred stock requires expected rate of return of 15% which is more than risk free rate of return because the investor is investing in stocks which are risky.

Now for calculation of fair price of the preferred stock, following formula is used

PV = P_{o}* = DIV_{1} / r_{PE}

PV = 2 / 15%

PV = 13.33

The calculated intrinsic price of the preferred share is Rs 13.33 which is higher than its Market price of Rs 12. This difference shows that the worth of that preferred stock is more to the investor than its market price and that stock is undervalued. When the investor invests in that stock, he will gain additional value.

**Finite Investment of Preferred Stock:**

In case of finite investment in preferred stock, the investor purchases the stock & makes a decision to sell that stock after few days or years (n). Preferred stock/share pricing formula for such finite investment is similar to bond’s valuation formula, which is as follow:

PV = P_{o}* = DIV_{t} / (1+r_{PE})^{t} + P_{n} / (1+r_{PE})^{n}

Where P_{n} = final expected selling price, t = year, sum from t = 1 to n.

PV (Stock Price) = Dividend Value + Capital Gain/Loss

The dividend value is obtained from the dividend cash stream while difference between buying & selling price provide capital gain/loss.

**Example:**

By taking the similar above example

The preferred stocks of Company XYZ are traded in the Karachi stock exchange and has market price of Rs 12. The company offers to pay fixed dividend of Rs 2 per share. The par value of every share is Rs 10. It is expected that after two years the price of preferred stock becomes Rs 12. The individual investing in that preferred stock requires expected rate of return of 15% which is more than risk free rate of return because the investor is investing in stocks which are risky.

Following formula is used for calculation of fair price of preferred stock for finite investment

PV = P_{o}* = DIV_{t} / (1+r_{PE})^{t} + P_{n} / (1+r_{PE})^{n}

PV = 2 / (1+0.15) + 2 / (1+0.15)^{2} + 12 / (1+0.15)^{2}

PV = 1.74 + 1.51 + 9.07

PV = 12.32

In the above common example, the worth of perpetual investment in preferred stock is more than finite investment in preferred stock. The reason is that the present value of infinite stream of Rs 2 dividends is higher than the present value of the anticipated future selling price (Rs 13).

**Common Stock/Share Pricing:**

There are two cases of **Common Stock** / share pricing. One is related with finite investment for limited life. The other is for infinite investment.

**Finite Investment in Common Stock: **

This kind of investment is common. Cash flows need to be taken into account from variable dividends & Estimated Selling Price (P_{n}).

It is important to point out that P_{n} is based on DIV_{n} + 1. The price at any point in time must be based on the following year dividend. Its formula is same as for bond valuation equation.

**Perpetual Investment in Common Stock:**

Following is the formula of perpetual investment in common stock

PV = DIV_{1} / (1+r_{CE}) + DIV_{2} / (1+r_{CE})^{2} + . . . DIV_{n} / (1+r_{CE})^{n} + P_{n} / (1+r_{CE})^{n}

PV = Fair Price or Expected Price = Present Value of Stock

P_{n} = Anticipated future selling price

r_{CE} = Minimum Required Rate of Return of the investor investing in common stocks.

DIV_{1} = Anticipated future dividend at the end of the year 1

DIV_{2} = Anticipated future dividend at the end of the year 2 etc.

Also dividends are not certain & n = infinity

PV (Stock Price) = Dividend Value + Capital Gain/Loss

The Value of dividend is obtained from the dividend cash stream on the other hand difference between buying & selling price provide capital gain/loss.

This case is considered as an idealized one. The final cash flow in the form of P_{n} in the equation occurs at year n = Infinity. The present value of the last term P_{n} is almost equal to zero because when n = infinity then the discount factor (1+r_{E})^{n} in the denominator of the equation becomes very large so the last cash flow term occurring at year n should be ignored.

PV = DIV_{1} / (1+r_{E}) + DIV_{2} / (1+r_{E})^{2} + . . . DIV_{n} / (1+r_{E})^{n}

Or

PV = DIV_{t} / (1+r_{E})^{t}

Where t = year and sum from t = 1 to n

The above equation is impractical because of the reason that dividends must be forecasted for every year forever.

**Example:**

The common stock of Company XYZ is traded in the Karachi Stock Exchange. The market price of share is Rs 12. The forecasted dividend of the company for the first year will be Rs 2 and the second year is Rs 4 on the basis of balance sheet, income statement & cash flows statements of the company. The market price of the share after two years is anticipated to be Rs 12. The face value of share is RS 10. There is 10% of risk free rate of return but the expected minimum rate of return of the investor investing in the share of XYZ Company is Rs 20% because of the high riskiness.

In order to ascertain the fair price of the common stock of Company XYZ, following formula is used.

PV = DIV_{1} / (1+r_{CE}) + DIV_{2} / (1+r_{CE})^{2} + P_{n} / (1+r_{CE})^{2}

PV = 2 / 1.2 + 4 / (1.2)^{2} + 12 / (1.2)^{2}

PV = 1.67 + 2.78 + 8.33

PV = 12.78

**Infinite Investment Growth Models**

The above example is related to investment in common stocks for finite period. But in case of investment in common stocks for infinite period then the simple formula failed. There are two separate models that deal with the infinite investment in common stocks. These two models are as follow.

- Zero Growth Model
- Constant Growth Model

**Zero Growth Model:**

This model is used to calculate the fair price of the infinite investment in common stock. In this model it is assumed that the growth of perpetual dividends is zero. This means that there is constant or no growth in dividends which is similar concept as in infinite investment in preferred stocks. So DIV_{1} = DIV_{2} = DIV_{3}. For every time in future, there is cash flow stream of fixed regular dividends. This formula is similar to perpetuity formula of the preferred stock except one difference that the dividends of preferred stocks are declared by the Company but the dividends of the common stocks are estimated.

So the zero growth model formula for infinite investment in common stock becomes

PV = P_{o}* = DIV_{1} / (1+r_{CE}) + DIV_{1} / (1+r_{CE})^{2} + DIV_{1} / (1+r_{CE})^{3} + . . .

Or

PV = P_{o}* = DIV_{1} / r_{CE}

PV or P_{o}* is the expected present price which is theoretical. The price is based on DIV_{1} which is the expected future dividend for year 1 (same for all other coming years). The difference between the common stock & preferred Stock is that in case of common stock it is assumed that there is constant or zero growth but in case of preferred stock the company declared the fixed rate of dividend for the shareholders.

**Constant Growth Model:**

In constant growth model the next year dividend is forecasted and it is assumed that the dividends will grow constantly at inflationary growth rate “g”. In this model the rate of growth of inflation is considered as the growth rate of dividends in a constant manner. In other words the dividends received on the common stocks will continue to grow constantly at growth rate of inflation. For example if there is 10% inflation rate in the country than the dividends will grow at rate of 10%. Following is the formula of constant growth model.

PV = P_{o}* = DIV_{1} (1+g) / (1+r_{CE}) + DIV_{1} (1+g)^{2} / (1+r_{CE})^{2} + DIV_{1} (1+g)^{3} / (1+r_{CE})^{3} + . . .

PV = P_{o}* = DIV_{1} / (r_{CE} – g)

Where DIV_{1} = dividend for the first year.

**Example:**

Suppose an investor is making infinite investment in the common stock of Company XYZ. The expected rate of return (r_{CE}) of the investor is 20% because of the high risk associated. The par value of common share is Rs 10 and the current dividend declared by the Company is Rs 3.

In order to calculate the fair value at common stock by zero growth model following formula is used.

PV = P_{o}* = DIV_{1} / r_{CE}

PV = 3 / 0.20 = Rs 15

With constant growth model formula

PV = P_{o}* = DIV_{1} / (r_{CE} – g)

PV = 3 / (0.2 – 0.1) (g is assumed to be 10%)

PV = Rs 30

The present value calculated from constant growth model is higher than zero growth models because the perpetual growth is compounded 10% in constant growth model.

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