Finance

Simple Returns

A simple return, written as R, tells us how much an investment gained or lost during a certain period compared with the amount invested at the beginning. In very simple terms, it answers the question: “How much did I make or lose compared with what I originally paid?” A simple return can come from two places. The first is the change in the price of the asset, called a capital gain or loss. The second is any cash received while owning the asset, such as a dividend from a stock or an interest payment from a bond.

For a stock, the simple return formula is:

R = ((pE - pB) + D) / pB

In this formula, R means simple return, pB means the price per share at the beginning of the period, pE means the price per share at the end of the period, and D means the dividend per share received during the period.

The easiest way to understand the formula is to read it as: take the change in price, add the dividend received, and divide everything by the price you originally paid.

A share is one unit of ownership in a company. The price per share tells us how much one share costs. For example, if a stock has a price per share of $40, buying one share costs $40. Buying 5 shares costs $200, while buying 20 shares costs $800. The number of shares changes the total amount invested, but the price per share is still $40.

A dividend is cash paid by a company to its shareholders. Dividends are usually stated as a dividend per share. For example, if a company pays a dividend of $1.50 per share, a person who owns 1 share receives $1.50. Someone who owns 10 shares receives $15, and someone who owns 100 shares receives $150.

The total dividend can therefore be calculated as:

Total Dividend = Dividend Per Share × Number of Shares

For example, suppose a company pays a dividend of $1.50 per share and you own 30 shares. Your total dividend is:

$1.50 × 30 = $45

Therefore, you receive $45 in total dividends.

The dividend yield is different from the dividend itself. A dividend is an amount of money, while the dividend yield is a percentage. Dividend yield tells us how large the dividend is compared with the beginning price of the share.

For the holding period used in the simple-return calculation:

Dividend Yield = D / pB

Suppose a share costs $40 at the beginning of the period and pays a dividend of $2 per share. The dividend yield is:

$2 / $40 = 0.05 = 5%

Therefore, the dividend yield is 5%. This means that the dividend received is equal to 5% of the amount originally invested.

Buying more shares changes the total dollar amount of the dividend, but it does not normally change the dividend yield percentage if every share was bought at the same price and receives the same dividend per share. For example, if one share costs $40 and pays a $2 dividend, the dividend yield is 5%. If you buy 10 shares, you invest $400 and receive $20 in total dividends:

$20 / $400 = 0.05 = 5%

If you buy 100 shares, you invest $4,000 and receive $200 in total dividends:

$200 / $4,000 = 0.05 = 5%

The dollar amounts become larger because you own more shares, but the dividend yield remains 5%.

A capital gain happens when the price of an asset increases. A capital loss happens when its price decreases. The capital gain or loss, expressed as a percentage of the beginning price, is:

Capital Gain/Loss = (pE - pB) / pB

For example, suppose you buy a share for $40 and later sell it for $50. The price increased by:

$50 - $40 = $10

The capital gain is:

$10 / $40 = 0.25 = 25%

Therefore, you earned a 25% capital gain from the increase in the share price.

If instead you bought the share for $40 and later sold it for $35, the price decreased by $5. That would be a capital loss rather than a capital gain.

A simple return combines the capital gain or loss with the dividend yield. Therefore:

Simple Return = Capital Gain or Loss + Dividend Yield

Example: One Share

Suppose you buy one share for a beginning price of:

pB = $50

One year later, the share price has increased to:

pE = $60

During the year, the company also pays:

D = $2

in dividends per share.

The simple return is:

R = (($60 - $50) + $2) / $50

First, calculate the price change:

$60 - $50 = $10

Then add the dividend:

$10 + $2 = $12

You made a total gain of $12 from an original investment of $50:

R = $12 / $50

R = 0.24 = 24%

Therefore:

R = 24%

The 24% simple return has two components. The capital gain is:

($60 - $50) / $50 = $10 / $50 = 0.20 = 20%

The dividend yield is:

$2 / $50 = 0.04 = 4%

Adding them together gives:

20% + 4% = 24%

Therefore:

Simple Return = 24%

Example: Different Number of Shares

Now suppose you buy 25 shares instead of one share. The beginning price is still $50 per share, so the total amount invested is:

$50 × 25 = $1,250

At the end of the period, each share is worth $60. Therefore, the total value of the 25 shares is:

$60 × 25 = $1,500

The increase in the value of the shares is:

$1,500 - $1,250 = $250

The dividend is $2 per share, and you own 25 shares, so the total dividend received is:

$2 × 25 = $50

Your total gain is therefore:

$250 + $50 = $300

Compared with your original investment of $1,250:

R = $300 / $1,250

R = 0.24 = 24%

The number of shares increased your total dollar profit from $12 to $300, but the percentage return stayed at 24%.

This shows an important difference between amounts and percentages. The price per share tells us the price of one unit. The dividend per share tells us how much dividend is paid for one unit. The number of shares tells us how many units we own. The total dividend tells us how much dividend money we receive from all our shares together. The dividend yield expresses the dividend as a percentage of the beginning price. The simple return expresses the total gain or loss as a percentage of the beginning investment.

The Coca-Cola example in the original notes follows exactly the same idea. A share was bought at the end of 2008 for $45.27 and sold at the end of 2009 for $57.00. During 2009, a dividend of $1.64 per share was received. The simple return was therefore:

R = (($57.00 - $45.27) + $1.64) / $45.27

First, calculate the increase in the share price:

$57.00 - $45.27 = $11.73

Then add the dividend:

$11.73 + $1.64 = $13.37

Now compare the total gain of $13.37 with the original price of $45.27:

R = $13.37 / $45.27

R = 0.2953 ≈ 29.5%

Therefore:

Simple Return ≈ 29.5%

This return can again be separated into two components. The increase in Coca-Cola’s share price produced a capital gain of approximately 25.9%.

Capital Gain:

($57.00 - $45.27) / $45.27

$11.73 / $45.27 = 0.259 ≈ 25.9%

The dividend produced a dividend yield of approximately 3.6%.

Dividend Yield:

$1.64 / $45.27 = 0.0362 ≈ 3.6%

Adding the two together gives:

25.9% + 3.6% = 29.5%

Therefore, the total simple return was approximately 29.5%.

The idea of a simple return is not limited to stocks. It can be used for other assets as well. For example, if you buy a bond, you may receive an interest payment instead of a dividend. You can still calculate the simple return by taking the change in the bond’s price, adding the interest payment received, and comparing the total with the bond’s beginning price.

In general, the simple return of an asset can be understood as:

Simple Return = (Capital Gain or Loss + Cash Flow Received) / Beginning Price

For a stock, the cash flow may be a dividend. For a bond, the cash flow may be an interest payment. The basic idea stays the same: compare everything you gained or lost during the period with the amount you had invested at the beginning.

Finally, simple returns may also be called arithmetic returns or holding-period returns. In this context, these terms refer to the same basic concept.

Notes

• Share = one unit of ownership in a company.
• Price per share = the price of one share.
• Number of shares = how many shares you own.
• Dividend = cash paid by a company to shareholders.
• Dividend per share (D) = dividend paid for each individual share.
• Total Dividend = Dividend Per Share × Number of Shares.
• Dividend Yield = dividend received compared with the beginning price.
• Dividend Yield formula = D / pB.
• Capital Gain = increase in an asset’s price.
• Capital Loss = decrease in an asset’s price.
• Capital Gain/Loss formula = (pE - pB) / pB.
• pB = price at the beginning of the period.
• pE = price at the end of the period.
• D = dividend per share received during the period.
• R = simple return.
• Simple Return formula = ((pE - pB) + D) / pB.
• Simple Return = Capital Gain/Loss + Dividend Yield.
• More shares = larger total dollar investment.
• More shares = larger total dollar dividend.
• More shares do not automatically mean a higher percentage return.
• For stocks, cash flow is commonly a dividend.
• For bonds, cash flow is commonly an interest payment.
• Simple return, arithmetic return, and holding-period return are names for the same concept in this discussion.
• The easiest way to remember simple return is: price change + cash received, divided by beginning price.