Rule of 72

The Rule of 72 is a simple formula used to estimate the length of time required to double an investment. The rule of 72 is primarily used in off the cuff situations where an individual needs to make a quick calculation instead of working out the exact time it takes to double an investment. Also, one is more likely to remember the rule of 72 than the exact formula for doubling time or may not have access to a calculator that allows logarithms.

Example of Rule of 72

An individual is earning 6% on their money market account would like to estimate how long it would take to double their current balance. In order for this estimation to be remotely accurate, we must assume that there will be no withdrawals nor deposits into this account. We can estimate that it will take approximately 12 years to double the current balance after dividing 72 by 6.

Alternative Formula for Rule of 72

Alternatively, the rule of 72 can be applied to estimate the rate needed to double an investment within a specific time period. This alternative formula to the rule of 72 can be shown as

If we take a look at the prior example of Rule of 72, we can apply the same example to an individual wanting to estimate what their rate needs to be in order to double their money within a specific period of time. If an individual wants to estimate the rate needed to double their money within 12 years, this can be estimated as 6% from dividing 72 by 12 years.

Breakdowns of Rule of 72

The rule of 72 is generally used for quick estimates in situations where the rate is in the several percent range. As the rate gets too low or too high below and above approximately 8%, the estimate becomes less accurate.

Another issue with the rule of 72 is with large sums of money. If a company or individual has large sums involved, this doesn't necessarily affect the outcome of the formula, but the company or individual may choose to use the actual doubling time formula as each decision could affect their profitability on a larger scale.

Zero Coupon Bond Effective Yield

The zero coupon bond effective yield formula is used to calculate the periodic return for a zero coupon bond, or sometimes referred to as a discount bond.

A zero coupon bond is a bond that does not pay dividends (coupons) per period, but instead is sold at a discount from the face value. For example, an investor purchases one of these bonds at $500, which has a face value at maturity of $1,000. Although no coupons are paid periodically, the investor will receive the return upon sell assuming that the rates remain constant or upon maturity.

Zero Coupon Bond Effective Yield Formula vs. BEY Formula

The zero coupon bond effective yield formula shown up top takes into consideration the effect of compounding. For example, suppose that a discount bond has five years until maturity. If the number of years is used for n, then the annual yield is calculated. Considering that multiple years are involved, calculating a rate that takes time value of money and compounding into consideration is needed. An investment that pays 10% per year is not equivalent to a 10 year discount bond that pays a 100% return after ten years. The investment that pays 10% can be reinvested and by compounding the returns(or considering the time value of money), the total return after 10 years would be

which would equal 259%.

In contrast, the formula for the bond equivalent yield does not take compounding into consideration. For this reason, the formula for bond equivalent yield is primarily used to compare discount bonds of short maturity, specifically less than one year.

How is the Zero Coupon Bond Effective Yield Formula Derived?

The formula for calculating the effective yield on a discount bond, or zero coupon bond, can be found by rearranging the present value of a zero coupon bond formula:

This formula can be written as

This formula will then become

By subtracting 1 from the both sides, the result would be the formula shown at the top of the page.

Zero Coupon Bond Value

A zero coupon bond, sometimes referred to as a pure discount bond or simply discount bond, is a bond that does not pay coupon payments and instead pays one lump sum at maturity. The amount paid at maturity is called the face value. The term discount bond is used to reference how it is sold originally at a discount from its face value instead of standard pricing with periodic dividend payments as seen otherwise.

As shown in the formula, the value, and/or original price, of the zero coupon bond is discounted to present value. To find the zero coupon bond's value at its original price, the yield would be used in the formula. After the zero coupon bond is issued, the value may fluctuate as the current interest rates of the market may change.

Example of Zero Coupon Bond Formula

A 5 year zero coupon bond is issued with a face value of $100 and a rate of 6%. Looking at the formula, $100 would be F, 6% would be r, and t would be 5 years.

After solving the equation, the original price or value would be $74.73. After 5 years, the bond could then be redeemed for the $100 face value.

Example of Zero Coupon Bond Formula with Rate Changes

A 6 year bond was originally issued one year ago with a face value of $100 and a rate of 6%. As the prior example shows, the value at the 6% rate with 5 years remaining would be $74.73. In this example, we suppose that the interest rates have changed to 5% since it was originally issued.
The formula would be shown as

After solving the equation, the value would be $78.35.

Yield to Maturity

The yield to maturity formula is used to calculate the yield on a bond based on its current price on the market. The yield to maturity formula looks at the effective yield of a bond based on compounding as opposed to the simple yield which is found using the dividend yield formula.

Notice that the formula shown is used to calculate the approximate yield to maturity. To calculate the actual yield to maturity requires trial and error by putting rates into the present value of a bond formula until P, or Price, matches the actual price of the bond. Some financial calculators and computer programs can be used to calculate the yield to maturity.

Yield to Maturity and Present Value of a Bond

The yield to maturity is found in the present value of a bond formula:

For calculating yield to maturity, the price of the bond, or present value of the bond, is already known. Calculating YTM is working backwards from the present value of a bond formula and trying to determine what r is.

Example of Yield to Maturity Formula

The price of a bond is $920 with a face value of $1000 which is the face value of many bonds. Assume that the annual coupons are $100, which is a 10% coupon rate, and that there are 10 years remaining until maturity. This example using the approximate formula would be

After solving this equation, the estimated yield to maturity is 11.25%.

Example of YTM with PV of a Bond

Using the prior example, the estimated yield to maturity is 11.25%. However, after using this rate as r in the present value of a bond formula, the present value would be $927.15 which is fairly close to the price, or present value, of $920. Other examples may have a larger difference.

A higher yield to maturity will have a lower present value or purchase price of a bond. In this example, the estimated yield to maturity shows a present value of $927.15 which is higher than the actual $920 purchase price. Therefore, the yield to maturity will be a little higher than 11.25%.

Through trial and error, the yield to maturity would be 11.38%, which is found by adjusting each estimated rate until the present value equals the price of the bond.

Excel is helpful for the trial and error method by setting the spreadsheet so that all that is required to determine the present value is adjusting a fixed cell that contains the rate.

Total Stock Return

The formula for the total stock return is the appreciation in the price plus any dividends paid, divided by the original price of the stock. The income sources from a stock is dividends and its increase in value. The first portion of the numerator of the total stock return formula looks at how much the value has increased (P₁ - P₀). The denominator of the formula to calculate a stock's total return is the original price of the stock which is used due to being the original amount invested.

Total Stock Return Cash Amount

The formula shown at the top of the page is used to calculate the percentage return. The actual cash amount for the total stock return can be calculated using only the numerator of the percentage return formula.

For example, assume that an individual originally paid $1000 for a particular stock that has paid dividends of $20 and the ending price is $1020. The total return would be $40 which equals $1020 minus $1000, then plus $20.

Example of the Total Stock Return Formula

Using the prior example, the original price is $1000 and the ending price is $1020. The appreciation of the stock is then $20. The $20 in price appreciation can then be added to dividends of $20 which would equal a total return of $40. This can then be divided by the original price of $1000 which would equal a percentage return of 4%.

Alternative Total Stock Return Formula

The total stock return can also be calculated by adding the dividend yield to the capital gains yield. The capital gains yield may sometimes be shown as the percentage change in stock price. This alternative formula is derived from separating the stock appreciation and dividends in the formula shown at the top of the page which becomes the capital gains yield and the dividend yield.

Tax Equivalent Yield

The tax equivalent yield formula is used to compare the yield between a tax-free investment and an investment that is taxed. One of the most common examples of a tax-free investment is municipal bonds. Municipal bonds are generally issued by local governments to finance development in its local community.

Use of Tax Equivalent Yield Formula

Investors who pay higher taxes may want to consider investing in municipal bonds or equivalent investments. The higher tax bracket an individual is in, the more appealing a tax-free investment would become. The tax equivalent yield formula can be used to compare the yield on the tax-free investment vs. an investment that is taxed.

Example of Tax Equivalent Yield

It is important to note before proceeding to this tax equivalent yield formula example that it is assumed that both bonds have equal risk. That is to say, that with the exception of the differences in yield, all things are held constant. Differences in risk do not affect the formula itself, but should be considered in any investment decision and not just the yield.

An example of the tax equivalent yield formula would be an investor who must decide between a bond that pays 6% that is taxed and a bond that pays 4% but earnings are tax free. His marginal tax rate is 33%. In order to compare the yield of these two investments, the equation for this example using the tax equivalent yield formula would be

After solving the formula, the equivalent yield for 4% would be 6.06%. This rate is higher than the 6% rate from the bond that is taxed and will give a higher after-tax return.

Present Value of Stock - Zero Growth

The formula for the present value of a stock with zero growth is dividends per period divided by the required return per period. The present value of stock formulas are not to be considered an exact or guaranteed approach to valuing a stock but is a more theoretical approach.

The present value of a stock formula used above is specific to stocks that have zero growth, or no growth. It is important to remember that the period used for both dividends and the required return must match. For example, if one is using annual dividends, then the annual return must be used.

Use of the PV of a Stock with Zero Growth

As stated above, the present value of a stock with no growth formula is more conceptual than forcefully implemented in every circumstance. The general idea is that a stock is essentially like any other form of investment and should be valued based on future cashflows. In practice, investors and analysts may take various factors besides only dividends into consideration. Examples of other considerations may be expected future earnings based on news, appreciation of a stock due to retained earnings, and overall economy (inflation, production, and capital). Also, there are various methods or models used in trying to value a stock.

How is the PV of a Stock with Zero Growth Derived?

The present value of a stock is broadly considered the sum of the discounted future cash flows. Dividends are considered the future cash flows as the appreciation of a stock is not realized unless sold. Since the stock is held with no maturity date, one could consider a stock to be a perpetuity, in that its dividends are to be received infinitely.

The formula for the present value of a stock with no growth shown at the top of the page theorizes that the stock is a perpetuity where dividends will be received on an ongoing basis for an unending period of time. Dividends would be denoted as cash flows in the perpetuity formula.

Present Value of Stock - Constant Growth

The formula for the present value of a stock with constant growth is the estimated dividends to be paid divided by the difference between the required rate of return and the growth rate.

The present value of a stock with constant growth is one of the formulas used in the dividend discount model, specifically relating to stocks that the theory assumes will grow perpetually. The dividend discount model is one method used for valuing stocks based on the present value of future cash flows, or earnings.

How is the Present Value of Stock with Constant Growth Derived?

As previously stated, the present value of a stock with constant growth is based on the dividend discount model, which sums the discount of each cash flow to its present value. The formula shown at the top of the page for stocks with constant growth uses the present value of a growing perpetuity formula, based on the underlying theoretical assumption that a stock will continue indefinitely, or in perpetuity. This assumption is not without scrutiny, however the present value of a growing perpetuity can be used as a comparable measure along with other stock valuation methods for companies that are stable and tend to have a calculable outcome of steady growth.

Growth Rate in the Present Value of Stock Formula

The growth rate used for calculating the present value of a stock with constant growth can be estimated as

Multiplying the retention ratio by the return on equity can then be reduced to retained earnings divided average stockholder's equity.

It is important to note that in practice, growth can not be infinitely negative nor can it exceed the required rate of return. A fair amount of stock valuation requires non-mathematical inference to determine the appropriate method used.

Required Rate of Return in the Present Value of Stock Formula

The required rate of return variable in the formula for valuing a stock with constant growth can be determined by a few different methods.

One method for finding the required rate of return is to use the capital asset pricing model.

The capital asset pricing model method looks at the risk of a stock relative to the risk of the market to determine the required rate of return based on the return on the market.

Another method that can be used is to determine the required rate of return based on the present value of dividends. This method also uses the present value of a growing perpetuity formula and rearranges the formula to calculate the required rate of return. After rearranging the formula, it is shown as

which is the dividend yield + growth rate.

The arbitrage pricing theory can also be used which is similar to the capital asset pricing model but uses various risk factors and the betas for each risk factor to determine the total risk premium for the stock.

Risk Premium

The formula for risk premium, sometimes referred to as default risk premium, is the return on an investment minus the return that would be earned on a risk free investment. The risk premium is the amount that an investor would like to earn for the risk involved with a particular investment.

The US treasury bill (T-bill) is generally used as the risk free rate for calculations in the US, however in finance theory the risk free rate is any investment that involves no risk.

Risk Premium of the Market

The risk premium of the market is the average return on the market minus the risk free rate. The term "the market" in respect to stocks can be connoted as an entire index of stocks such as the S&P500 or the Dow. The market risk premium can be shown as:

The risk of the market is referred to as systematic risk. In contrast, unsystematic risk is the amount of risk associated with one particular investment and is not related to the market. As an investor diversifies their investment portfolio, the amount of risk approaches that of the market. Systematic and unsystematic risk and their relation to returns is where the many clichés about diversifying your investment portfolio is derived.

Risk Premium on a Stock Using CAPM

The risk premium of a particular investment using the capital asset pricing model is beta times the difference between the return on the market and the return on a risk free investment.

As noted earlier, the return on the market minus the return on a risk free investment is called the market risk premium. From here, the capital asset pricing model can be rewritten as

Price to Sales Ratio

The formula for price to sales ratio, sometimes referenced as the P/S Ratio, is the perceived value of a stock by the market compared to the revenues of the company. The price to sales ratio is calculated by dividing the stock price by sales per share. Sales per share uses the weighted average of shares for the time period evaluated, which is generally one year.

Revenues and sales are synonymous terms and can be found on a company's income statement. The price of the stock is the price listed on the stock exchange, or secondary market.

Use of P/S Ratio Formula

The price to sales formula can be used in lieu of the price to earnings ratio in situations where the company has a net loss. Also, some may give more relevance to the price to sales than the P/E ratio due to earnings can be manipulated based on accounting practices. However, it is important when evaluating an investment to look at all aspects of a company. Evaluating the changes of a company with multiple formulas at once may shed light on issues that may not be found by looking at each formula individually.

Issues with the P/S Ratio Formula

As with the P/E and P/BV formula, the price to sales ratio uses the share price in the numerator of the formula. According to efficient market theory, the market functions based on all information such that the price of a stock accurately denotes the inherent value of the stock at any given moment. However, opponents of the efficient market theory would suggest that the price of a stock is based on the perceived value of the stock.

Whether a stock price is based on perception or a rational calculated value plays on deeper philosophical questions of whether any individual or the market can possess the veridical value of a stock, assuming there is such.

Those that oppose efficient market theory would suggest that when using the price to sales formula an investor would need to keep in mind that the price of the stock is based on its perceived value by the market. This can become an issue because a stock that has a low price to sales may be considered by an individual as a good investment due to being undervalued whereas another individual may consider the investment to be poor compared to other companies in the same industry with a higher price to sales ratio.

Price Earnings Ratio

The formula for the price to earnings ratio, also referred to as the P/E Ratio, is the price per share divided by earnings per share. The price to earnings ratio is used as a quick calculation for how a company's stock is perceived by the market to be worth relative to the company's earnings. A higher price to earnings ratio implies that the market values the stock as a better investment than if there was a lower price to earnings ratio, ceteris paribus. The increased perceived worth is due to news, speculation, or analysis from investors that the stock has a higher growth potential for the future.

The price to earnings ratio varies across different industries and also different countries. When comparing the price to earnings ratio among companies, it is important to compare within the same industry and country. Some industries are generally considered to have high growth expectations for the future as opposed to other industries that have a steady and established growth rate.

Earnings per Share in the P/E Ratio

Earnings per share in the price to earnings ratio is a company's net income divided by the weighted average of outstanding shares. It is important to consider that a company's net income can vary depending on the company's accounting methods. When comparing two different companies that use different inventory valuation and depreciation methods, the price to earnings ratio for both companies can not be exactly compared, as all things are not held constant.

Issues with the P/E Ratio

There are a few issues to consider when using the price to earnings ratio. As previously stated, different companies may use different accounting methods which can affect net income. This, in turn, will affect the price to earnings ratio when trying to compare companies.

Another issue with the price to earnings ratio is companies with a net loss. The denominator of the formula, earnings per share, relies on a company having a net income as opposed to a loss.

Also, the price to earnings ratio is self-referencing, in that it is calculated based on the price of the stock. A specific company may have a high price to earnings ratio, but based on stock valuation methods, some may consider the stock to be overvalued. In this situation, a stock with a higher price to earnings ratio may actually be a poor investment. The price to earnings ratio is simply a perception of the market by investors, and relies on efficient market theory as the sine qua non of its absolute accuracy.

Price to Book Value

The Price to Book Ratio formula, sometimes referred to as the market to book ratio, is used to compare a company's net assets available to common shareholders relative to the sale price of its stock. The formula for price to book value is the stock price per share divided by the book value per share.

The stock price per share can be found as the amount listed as such through the secondary stock market.

The book value per share is considered to be the total equity for common stockholders which can be found on a company's balance sheet.

Use of Price to Book Value Formula

The price to book value formula can be used by investors to show how the market perceives the value of a particular stock to be. A ratio over one implies that the market is willing to pay more than the equity per share. A ratio under one implies that the market is willing to pay less.

A price to book value of less than one can imply that the company is not running up to par. This, along with other factors, could also lead to a hostile takeover.

Issues with the Price to Book Value Formula

One may argue that a ratio under one implies that the company is perceived as being a worse investment than if it were above one. On the other hand, another may argue that the stock is underpriced and a favorable investment. Due to these discrepancies of opinion, using other stock valuation methods along with or apart from the price to book value formula may be beneficial.

Another issue with the price to book value formula is that there are many underlying factors that can affect the formula such as issuing new stock, paying dividends, and stock repurchases. It is possible for a company to manipulate this ratio by various means which is why it is important to not use any one particular financial formula in isolation.

Preferred Stock

The formula shown is for a simple straight preferred stock that does not have additional features, such as those found in convertible, retractable, and callable preferred stocks.

A preferred stock is a type of stock that provides dividends prior to any dividend paid to common stocks. Apart from having preference for dividend payouts, preferred stocks generally will have preference of asset allocation upon insolvency of the company, compared to common stocks. Because of these preferences, preferred stock is generally considered to be more secure than common stock and similar to a debt financial instrument, i.e., a bond. Despite the similarities, bonds do have preference for the same reasons and are generally considered more secure, ceteris paribus.

The formula for the present value of a preferred stock uses the perpetuity formula. A perpetuity is a type of annuity that pays periodic payments infinitely. As previously stated, preferred stocks in most circumstances receive their dividends prior to any dividends paid to common stocks and the dividends tend to be fixed. With this, its value can be calculated using the perpetuity formula.

Example of Preferred Stock Value Formula

An individual is considering investing in straight preferred stock that pays $20 per year in dividends. It has been determined that based on risk, the discount rate would be 5%. The price the individual would want to pay for this security would be $20 divided by .05(5%) which is calculated to be $400.

Alternative Formula

The formula could be reworked to find the rate or return by dividing the fixed dividend payout by the price.

For example, if the price is $40 per share and the annual dividend is $4, the rate would be .10 or 10%.

Net Asset Value

The net asset value formula is used to calculate a mutual fund's value per share. A mutual fund is a pool of investments that are divided into shares to be purchased by investors. Each share contains a weighted portion of each investment in the collective pool. The premise of grouping in this manner is to minimize risk by diversifying.

It is important to note that net asset value does not look at future dividends and growth as do other stock and bond valuation methods. The formula for net asset value only looks at the fund's per share value based on its net assets.

The net asset value is determined by the mutual fund company and priced according to this formula. Stock and bond valuation methods are not used due to mutual funds being sold directly from the company and not through an exchange or on the secondary market. Stocks, on the other hand, are sold through bid and ask pricing on the secondary market which requires an investor to determine a share's value to them based on expected future earnings, in which they bid accordingly.

Use of the Net Asset Value Formula

As already stated, the net asset value formula is used by a mutual fund company to determine the price of a share of a specific mutual fund. An individual investor may not find the net asset value formula particularly useful besides for the sake of knowledge. However, it is important to know how net asset value is calculated, in that expected future earnings are not considered.

Example of Net Asset Value Formula

A simple, perhaps unrealistic, example of calculating net asset value would be a mutual fund with assets of $1 million, liabilities of $100,000, and 100,000 outstanding shares. Putting this information into the variables of the net asset value formula would show

which would return $9 per share.

Holding Period Return

The formula for the holding period return is used for calculating the return on an investment over multiple periods.

The returns on an investment may be shown on an annual, quarterly, or monthly basis. An individual may be tempted to incorrectly add the percentages of return to find the return over the multiple periods. By incorrectly adding the periodic returns, the effect of compounding is not taken into consideration. For example, a return of 10% the first year followed by a return of 10% the next year does not equate to an aggregate of 20%. By using the holding period return formula, the amount gained would be 21%. The extra 1% can be attributed to the effect of compounding through earning 10% in the second year on the 10% that was earned in the first year. This is sometimes called earning interest on interest.

Example of Holding Period Return Formula

An example of the holding period return formula would be an investment in an asset that has an annual appreciation of 10%, 5%, and -2% over three years. As stated in the prior section, simply adding the annual appreciation of each year together would be inaccurate as the 5% earned in year two would be on the original value plus the 10% earned in the first year. After putting the annual percentages into the holding period return formula, the correct calculation would be:

After solving this equation, the holding period return would be 13.19% for all three years.

Alternative Holding Period Return Formula

If the returns per period are the same, the holding period return formula can be reduced to

where r is the periodic rate and n is the number of periods. This alternative formula is very similar to the annual percentage yield formula, in that both formulas calculate the yield, which takes into consideration the effect of compounding.

If the periodic rates are unknown, the holding period return could be calculated with the following formula

Earnings include dividends. The appreciation of an asset, also referred to as capital gains, would be the increase in value of the asset which would be calculated by subtracting the initial value of the investment from the ending value.

Estimated Earnings

The formula for estimated earnings is forecasted sales minus forecasted expenses. The formula above is a simple way of restating how to calculate net income, i.e. earnings, based on its estimated components. However, the practice of calculating estimated earnings is far more complex.

It is important to note that the expenses in the estimated earnings formula should include interest and taxes.

Estimated Earnings with Profit Margin Formula

Another simple way to state the estimated earnings formula is

This formula of projected sales times the projected net profit margin can be used because net profit margin is net income divided by sales. By multiplying net sales and net profit margin, sales will be canceled out which leaves net income(earnings). Again, this is a general concept but calculating the actual estimates are more complex.

Calculating Estimated Earnings

Internally, a company may calculate estimated earnings for a particular project or company changes with pro forma income statements. Any changes of a company's operations should be taken into consideration when traversing from what earnings is known today to what earnings is estimated to come, and pro forma statements can provide this information.

Estimated earnings can also be calculated by trends. By looking at prior trends, regression analysis can be used to determine an estimate of earnings.

Scenario analysis can also be used as a method to calculate estimated earnings. Scenario analysis looks at different situations that may occur, ranging from the worst case scenario to the best case scenario to avoid analyzing in broad strokes. The state of the entire economy should be factored into estimate calculations. Scenario analysis can include many factors including giving weight to the overall economy.

There are various ways to calculate the variables of the estimated earnings formula. The different methods can also be used in combination to narrow any outlier estimates that vary dramatically and don't paint a true picture of what earnings will be.

Equity Multiplier

The formula for equity multiplier is total assets divided by stockholder's equity. Equity multiplier is a financial leverage ratio that evaluates a company's use of debt to purchase assets.

Use of Equity Multiplier Formula

The equity multiplier formula is used in the return on equity DuPont formula for the financial leverage portion of DuPont analysis. Broadly speaking, financial leverage is used in financial analysis to evaluate a company's use of debt.

To understand how the equity multiplier formula is related to debt, it should be noted that in finance, a company's assets equal debt plus equity. Debt is not specifically referenced in the equity multiplier formula, but it is an underlying factor in that total assets in the numerator of the formula for the equity multiplier includes debt. This can be shown by restating total assets in the equity multiplier formula as debt plus equity.

Alternative Equity Multiplier Formula

An alternative formula for the equity multiplier is the reciprocal of the equity ratio. As previously stated, a company's assets are equal to debt plus equity. Therefore, the equity ratio calculates the equity portion of a company's assets. This ratio in the denominator of the formula can also be found by subtracting one minus the debt ratio.

Earnings Per Share

The formula for earnings per share, or EPS, is a company's net income expressed on a per share basis. Net income for a particular company can be found on its income statement. It is important to note that the earnings per share formula only references common stock and any preferred stock dividends is subtracted from the net income, if applicable.

Per Share

The denominator of the earnings per share is the weighted average of outstanding shares of common stock. When the amount of common shares changes mid-year, the "per share" portion requires additional calculation. The per share portion is weighted based on the length of time each number of shares is in effect.

An example of the weighted average would be a company who has 100,000 outstanding common shares for 9 months and due to issuing new common stocks, has 120,000 outstanding shares for the remaining 3 months. The weight for 100,000 share would be 9/12(.75) and the weight for 120,000 shares would be 3/12(.25).

To calculate the weighted average from the example:
(.75)100,000 + (.25)120,000 = 75,000 + 30,000 = 105,000

In Relation to Dividends

Earnings and dividends are not one in the same. Companies, many times, retain some of their earnings for future growth. The amount of earnings that are paid out in dividends to common stockholders can be found from using the dividend payout ratio formula. A company that pays 20% of their net income to common stockholders will have a 20% payout ratio.

A company that pays all their earnings out to common stockholders will have a 100% payout ratio. Stocks that pay out all of their earnings to common stockholders are considered zero growth stocks, as none of the earnings the company receives is retained for future growth. In these circumstances, earnings will equal dividends. Likewise, earnings per share will be equal to dividends per share.

Use of Earnings Per Share Formula

The earnings per share formula is used in other formulas such as the P/E ratio formula and, on occasion, stock valuation.

Dividends Per Share

The formula for dividends per share, or DPS, is the annual dividends paid divided by the number of shares outstanding.

Per Share

The denominator of the dividends per share formula generally uses the annual weighted average of outstanding shares. The weighted average is also used with the earnings per share formula. However, there are key differences between these two formulas. The numerator for earnings per share is net income, or earnings. The numerator for the dividends per share formula is dividends. Earnings is effectively a continuous process throughout the year whereas dividends are paid at a given moment.

How often a company pays a dividend may warrant consideration for how to calculate the per share portion of the formula when using financial analysis for investments.

An unlikely figurative example would be a company who paid dividends in January with 2,000 outstanding shares and issued 20,000 additional shares in December. The result of the dividends per share formula would vary greatly depending on which method is used for determining the number of shares outstanding. Considering that the dividend yield formula uses dividends per share, it would vary greatly as well.

However, another hypothetical company pays dividends monthly and has issued common shares periodically throughout the year. One may consider using the weighted average in this example.

As with most financial formulas, perspective is important.

Use of DPS Formula

Dividends per share and the formula provided may be used by individuals who are evaluating various stocks to invest in and prefer companies who pay dividends. This formula alone does not necessarily provide an overall outlook on a company as some companies retain their earnings for growth instead of paying dividends. A company with a low dividend payout ratio, i.e. a company who pays a smaller percentage of their net income to stockholders, will reinvest their net income which may lead to an increase in the value of the company due to expansion.

Dividends per share is also used in other financial formulas, including dividend yield and dividend payout ratio.

Dividend Yield (Stock)

The formula for the dividend yield is used to calculate the percentage return on a stock based solely on dividends. The total return on a stock is the combination of dividends and appreciation of a stock.

The dividends paid for a company can be found on the statement of retained earnings, which can then be used to calculate dividends per share.

Use of Dividend Yield Formula

The dividend yield formula can be used by investors who are looking for increasing or declining trends of the dividend yield. On a broader level, a company that is paying less in dividends relative to its price may be having problems or it could be retaining more of a percentage of its net income for growth. When evaluating a stock, it is important to consider the overall company and how much net income it is retaining as reinvesting its net income could lead to growth and an appreciation of the stock price.

The formula for dividend yield may be of greater interest to investors who rely on dividends from their investments. However, a lower dividend yield does not imply lower dividends as the price could have substantially increased. As stated before, a trend of a declining dividend yield should only warrant investigation and not an immediate dismissal of the investment.

Example of Dividend Yield Formula

An example of the dividend yield formula would be a stock that has paid total annual dividends per share of $1.12. The original stock price for the year was $28. If an individual investor wants to calculate their return on the stock based on dividends earned, he or she would divide $1.12 by $28. Using the formula for this example, the dividend yield would be 4%.

Dividend Payout Ratio

The dividend payout ratio is the amount of dividends paid to stockholders relative to the amount of total net income of a company. The amount that is not paid out in dividends to stockholders is held by the company for growth. The amount that is kept by the company is called retained earnings.

Net income shown in the formula can be found on the company's income statement.

This formula is used by some when considering whether to invest in a profitable company that pays out dividends versus a profitable company that has high growth potential. In other words, this formula takes into consideration steady income versus reinvestment for possible future earnings, assuming the company has a net income.

Alternative Formula

I.

The retention ratio and the dividend payout ratio together equal 1 or 100% of net income. The premise is that whatever amount not paid in dividends is kept by the company to reinvest for expansion.

A simple example would be a company who pays out 100% of their net income in dividends. In this situation, net income would be equal to dividends. Using the formula for this example, the dividend payout ratio would be 1 or 100%. The retention ratio would be 0 or 0% as they do not retain and reinvest any of their earnings for growth. Using the alternative formula 1 - 0 would be 1.

Alternatively, a company who pays no dividends would have a 0 dividend payout ratio and a 1 retention ratio, which means that the company reinvests all of their net income for growth.

II.

The dividend payout ratio formula can also be restated on a "per share" basis. If the dividend per share and earnings per share is known, the dividend payout ratio can be calculated using the same concept of dividends paid divided by earnings, or net income.

Diluted Earnings per Share

Diluted earnings per share, or Diluted EPS, is a firm's net income divided by the sum of it's average shares and other convertible instruments.

A company's net income can be found on its income statement.

A company's average shares refers to the weighted average of common shares throughout the year. The weights of each factor would be the length of time each quantity of common shares is outstanding.

As a simple example, suppose that a company has 100 outstanding common shares for 9 months and 120 outstanding common shares for 3 months. The weights of 9 months and 3 months would be .75 and .25 respectively. This represents 3/4's of a year and 1/4 of a year. Thus, the formula for the average shares portion of the diluted earnings per share for this example would be .75(100) + .25(120), which would equal a weighted average of 105 common shares for the entire year.

Other Convertible Instruments and Diluted EPS

The term "convertible instruments" refers to any financial instrument that could possibly be converted into a common shares.

For reference, a few examples of convertible instruments that may be considered in the diluted earnings per share formula are stock options and convertible preferred stocks, but there are many others and anything than has the availability to be converted to a common share could be included.

Use of Diluted EPS

The diluted earnings per share is used by investors in replace of earnings per share to account for financial instruments that can be converted to shares. This conservative approach to calculating earnings per share may be used in lieu of the simple EPS formula as any convertible instrument could be converted at any time which could cause the real events to widely deviate from the simple EPS future estimates. This is especially important to consider when other financial formulas use earnings per share in its calculations.

Current Yield

Current yield is a bond's annual return based on its annual coupon payments and current price (as opposed to its original price or face). The formula for current yield is a bond's annual coupons divided by its current price.

Use of the Current Yield Formula

The current yield formula is used to determine the yield on a bond based on its current price. The current yield formula can be used along with the bond yield formula, yield to maturity, yield to call, and other bond yield formulas to compare the returns of various bonds.

The current yield formula may also be used with risk ratings and calculations to compare various bonds. As a general rule in financial theory, one would expect a higher premium, or return, for a riskier investment. If two bonds are held constant in respect to their risk, a higher return would be preferable.

Current Yield vs. Other Bond Formulas

There are various formulas that are used to compare the yields on bonds. Current yield, as its name implies, is the current or 'here and now' annual yield based solely on coupons. This is the difference between the bond yield and current yield. The bond yield looks at the original price of the bond or face value.

The current yield differs from the yield to maturity in that the yield to maturity looks at all future inflows, including a higher or lower face value than its current price, to determine the yield based on a present value equal to the current price of the bond. The formula for current yield only looks at the current price and one year coupons.

Example of the Current Yield Formula

An example of the current yield formula would be a bond that was issued at $1,000 that has an aggregate annual coupon of $100. The bond yield on this particular bond would be 10%. Suppose that the same bond is currently selling for $900 based on today's market rates. Recall that if the price of a bond goes down, the market rates or bond rate has gone up. For this example, the current yield formula would be shown as

which would return a current yield of 11.11%.