Business Mathematics: Stocks and Bonds - Stocks and Dividends

One way to invest money is to buy stock or shares in a corporation.

A shareholder is a part owner of the corporation and receives part of the corporation’s profits in the form of dividends.

The price of the share, called par value, is set by the company when the stock is first sold to the public.

When the stock is resold on the stock market, its price is determined by what the buyer is willing to pay and the seller is willing to accept.

This value, called the market value, is published in the financial section of most major newspapers.

Typical stock quotations are shown in the table below:

Table: Stock Quotations

Example
Referring to Table above, find (a) the total dividend paid for the past year to an investor owning 300 shares of stock A, and (b) the price of 100 shares of stock B purchased at the high price for that day.

The two kinds of stock sold are common and preferred.

The dividend on preferred stock is fixed when the stock is issued.

Stock D in the table above is a preferred stock (see the abbreviation “pf”) paying a dividend of 5.50% of the par value (usually $100). Dividends to preferred stockholders are paid before dividends are paid to common stockholders. Most preferred stock is cumulative. When a company declares dividends in any given year, the holders of cumulative preferred stock will receive all dividends not declared in previous years plus the current year’s dividend. On noncumulative preferred stock, declared dividends are paid only for the current year.

Example
The ABC Corporation has issued 2,000 shares of $100 par value common stock and 500 shares of 8%, $100 par value preferred stock. If the company declared $12,000 in dividends for the year, what will be the dividend per share of common stock?


Example
Paul White owns 120 shares of a cumulative %,$100 par value preferred stock. He has not received dividends for the previous 2 years. How much dividend will he receive this year if the company declares dividends?

Shares are usually traded in groups of 100 shares, called round lots. An odd lot refers to fewer than 100 shares. An odd-lot differential fee of execution price. A commission, based on the dollar value of the stocks traded, is paid to a stockbroker, who acts as an agent performing transactions between buyer and seller. Commission rates depend on the brokerage firm.

Example
An investor bought 60 shares of stock selling at What was the broker’s commission if the commission rate was 1.4%?


Example
Mrs. Smith sold 130 shares of stock selling at Find the broker's commission if the commission rate was 1.6%.

Several criteria may be used to evaluate investments in stocks. The annual yield, often expressed as a percent, is the ratio of the annual dividend per share to the price:

 

capital gain is the net proceeds less the total cost:
Capital gain = net proceeds – total cost

The total gain is the sum of total dividends plus the capital gain:
Total gain = total dividends + capital gain

Investors usually base their decisions of whether to buy or sell stock on the estimated total gain for a specified period of time. This estimate ignores commission, SEC charges, and transfer taxes.

Example
Find the rate of annual yield on a common stock if the semiannual dividend is $1.35 and the price per share is $18.

Example
An investor bought a common stock at $22 per share. Her quarterly dividend was 45¢ share. She sold the stock after 2 years at $37.50 per share. What is (a) the total gain per share and (b) the percent of total gain relative to cost?

Solved Problems:

7.1 Find the cost of 85 shares of stock B in Table 7.1 at the closing price of the day.
 

Solution


7.2 Find the gross proceeds from the sale of 215 shares of stock A in Table 7.1 at the low price of the day.
 

Solution


7.3 Find the broker’s commission on the purchase of 20 shares of stock C in Table 7.1 at the high price of the day, if the commission rate is 2.2%.
 

Solution


7.4 Find the broker’s commission in Prob. 7.3 assuming that the transaction was a sale.
 

Solution


7.5 The Sherlock Corporation declares a dividend of $42,540 to its 5,000 shares of 5.5%, $100 par value preferred stock and its 8,000 shares of common stock. What is the dividend per share of (a) preferred stock and (b) common stock?
 

Solution
(a)   Dividend per share of preferred stock: 5.5% × $100 = $5.50
(b)   Since dividends are paid first to preferred stock, the amount available to common stock is calculated as follows:
Dividend to common stock = total dividend – dividend to preferred stock


7.6 The Ruben Corporation declares dividends as follows from 1998 to 2001:
 

Its outstanding stock consists of 3,000 shares of 7%, $100 par value cumulative preferred stock and 5,000 shares of common stock. No dividends are in arrears as of the end of 1997. What amount of dividends is paid to each class of stock each year and what are the total dividends for each class for the 4 years?
 

Solution
The normal preferred dividend is
7% × $100 × 3,000 shares = $21,000

Dividends

* Since the normal preferred dividend is $21,000, the amount in arrears at the end of 1998 is $21,000 - $11,000 (the declared dividend). Note that this means that none of the 1998 dividend is available to common stock.
** The normal preferred dividend is added to the amount in arrears to determine the dividend available to cumulative preferred stock.

† The amount of the 1999 dividend available to common stock is the total 1999 dividend less the amount applied to preferred stock:
$37,000 – $31,000 = $6,000
‡ $21,000 – $8,000 = $13,000 in arrears
§ $45,000 – $34,000 = $11,000

7.7 Solve the above problem, assuming that the preferred stock is noncumulative.
 

Solution

Dividends

* The normal preferred dividend is still $21,000, so none of the 1998 dividend is available to common stock.
** Since the stock is noncumulative, only the normal preferred dividend amount is available to preferred stock.
† Total dividend – dividend to preferred stock = dividend to common stock.

7.8 Bundy Products, Inc. declares a dividend of $45,000 to its 2,000 shares of 8%, $100 par value preferred stock and its 6,000 shares of common stock. After each common share has received $2.50, preferred shareholders participate with the common shareholders in the distribution of the remaining dividend, in the ratio of shares in each class. Find the total dividend and the dividend per share for each class of stock.
 

Solution

7.9 Solve the above problem assuming that the preferred stock is not participating in the distribution of the remaining dividends.
 

Solution


7.10   Determine the rate of annual yield for stock A in Table 7.1, if bought at the low price of the day.
 

Solution


7.11   Three years ago Jeff purchased 100 shares of stock D at the table at top. Assuming that dividends have been constant for the 3-year period, what is (a) the total gain and (b) the annual rate of gain based on cost?
 

Solution


7.12   Two years ago, an investor bought 200 shares of stock E at Table 7.1. Calculate (a) the investor’s total loss and (b) the annual rate of loss based on cost.
 

Solution