A share of stock represents a part of ownership of a corporation. Each stock has it's own ticker symbol, which is a unique abbreviation assigned to publicly traded companies used when its trades are reported on the ticker. The ticker is a real time electronic display of trading activity. These ticker symbols usually consist of three or fewer characters on the NYSE, while those on the NASDAQ usually have four or more characters. Common stock is a share of ownership in the corporation, which gives its owner rights to any common dividends as well as rights to vote on the election of directors, mergers, or other major events. Dividends are usually periodic payments that allow the stock carrier to share in profits of the corporation.

The Dividend-Discount Model

As the valuation principle states, we must determine expected cash flows that an investor will receive from owning stock in order to value it. To begin, we will look at investors with a one-year investment horizon.

For a one-year investor, there are two potential sources of cash flows from owning stock:
1. The firm might pay out cash to its shareholders in the form of dividends. 2. The investor might generate cash by selling the shares at a future date.
Assuming all dividends are paid at year end, the timeline for such an investment would have a negative initial stock price at period 0 and at period 1, you would have the price you sell the stock for plus your dividend payment. Because such cash flows are risky, the risk free interest rate cannot be used to discount them. Instead we use the cost of capital. The cash flow must be discounted based on the equity cost of capital, which is the expected return of other investments available in the market with equivalent risk to the firm's shares. With that, the equation to solve for the stock price looks something like this: Po = (Div1 + P1) / (1+r) Po is the original price of stock, with Div1 being the dividend payment and P1 being the selling price and r equaling the equity cost of capital rate.

Reworking the equation can give us other beneficial numbers, such as the Total Return (r) = ((Div1 + P1)/Po) - 1. In this equation, the dividend yield (or the expected annual dividend of the stock divided by it's current price) is represented with the (Div1/Po). The capital gain (the amount by which the selling price of an asset exceeds its initial purchase price) is found by taking (P1 - Po). We divide the capital gain by the current stock price Po to get the capital gain rate, which expresses the capital gain as a percentage return. Therefore, the sum of dividend yield and the captial gain rate is referred to as total return of the stock. The expected total return of the stock should equal the expected return of other investments available in the market with equivalent risks.

For a multi-year investment, setting the stock price equal to the present value of the future cash fows implies the following equation: Po = [Div1/(1+r)] + [(Div2 + P2)/((1+r)^2)]

With that said, we see that the formula for the stock price of a two year investment is the same as that for a sequence of two one-year investments. This gives us basis for our dividend-discount model, which is a model that values shares of a firm according to the present value of the future dividends the firm will pay. Therefore, in the above equation, the denominator in the second part of the equation can be raised to the N-th power to represent an arbitrary horizon. This tells us that the price of stock is equal to the present value of all of the expected future dividends it will pay.

Estimating Dividends (Dividend-Discount Model)

The value of a stock is expressed in terms of the expected future dividends the firm will pay. However, estimating these dividends is difficult. A common practice is to assume that dividends will grow in the long run at a constant rate. The constant dividend growth model points this out to us. It is a model for waluing a stock by viewing its dividends as a constant growth perpetuity. This model looks like this: Po = Div1 / (r-g) where g is the growth rate. With constant expected dividend growth, the expected growth rate of the share price matches the growth rate of the dividends. If we define a firms dividend payout rate as a fraction of its earnings that the firm pays as dividends each year, we can refer to the firm's dividend pare share by taking the Earnings and dividing it by the shares outstanding, then taking that quantity and multiplying it by the dividend payout rate. A firm's dividends can increase by increasing the earnings, increasing the dividend payout rate, or by decreasing the number of shares outstanding. A firm can either pay its earnings out as dividends or retain them for reinvestment. New investment equals the firm's earnings multiplied by its retention rate (the fraction of a firm's current earnings that the firm retains). However, cutting the firm's dividend to increase investment will raise the stock price only if the new investments have a positive Net Present Value. What if firms often pay no dividends when they are young or if their growth rate continues to change over time until they mature? We must use the dividend-discount model in an adjusted way to account for future share price of the stock once a firm matures and its expected growth rate stabilizes: Pn will represent the final cash flow in the model, which can be attained by: Pn = (DivN+1) / (r-g) Here, the dividend is for an arbitrary year N plus 1.

Total Payout and Free Cash Flow Valuation Models

Because the dividend-discount model is limited, we must factor in other valuation models. Sometimes firms wont pay out dividends but will purchase stock share back, know as share repurchase. The total payout model is then a method that values shares of a firm by discounting the firm's total payouts to the equity holders and then divides by the current number of shares outstanding. This equation looks as follows: Po = PV(of Future Total Dividends and Repurchases) / Shares Outstanding The discounted free cash flow model goes a step further than the total payout model by determining the total value of the firm to all investors both equity holders and debt holders. It is a method for estimating a firm's enterprise value by discounting its future free cash flows: Enterprise Value = Market Value of Equity + Debt - Cash with this, we must measure the cash generated by the firm before any payments of debt or equity holders are considered. We do this with: Free Cash Flow = EBIT x (1 - tax rate) + Depreciation - Cap. Expenditures - Inc. in Net Working Capital A key difference between the discounted free cash flow model and the earlier models we have seen is the discount rate. Previously, we used the equity cost of capital because we were discounting the cash flows to equity holders. Here, we are discounting free cash flows that will be paid to both debt and equity holders. Therefore, we use the firm's weighted average cost of capital, the cost of capital that reflects the risk of the overall business, which is combined risk of the firm's equity and debt. Incorporating the WACC into the equation, we get a setup that looks something like this: Vn = FCFn+1/(WACC - g) = [(1 + g) / (WACC - g)] x FCF

Valuation Based on Similar Firms

The method of comparables is an estimate of the value of a firm based on the value of other, comparable firms or other investments taht are expected to generate very similar cash flows in the future. It is yet another method we use to value a firm. Because identical companies do not exist, it makes the method of comparables a little difficult. However, we can adjust for differences in scale between firms by expressing their value in terms of a valution multiple, which is a ratio of the value to some measure of the firm's scale. The most common valuation multiple is the price-earnings ratio. We can compute a firm's price-earnings ration by using either trailing earnings (earnings over the prior 12 months) or forward earnings (earnings expected over the coming 12 months), with the resulting ratio being called the trailing P/E or the forward P/E. To calculate the forward P/E: Po / EPS = (Div1/EPS1) / (r - g) = Dividend Payout Rate / (r - g).
It is also common to use valuation multiples based on the firm's enterprise value. Because the enterprise value represents the entire value of the firm before the firm pays its debt, to form an appropriate multiple, we must divide it by a measure of earnings or cash flows before interest payments are made. Common multiples are EBIT or EBITDA. This valuation multiple is higher for firms with growth rates and low capital requirements.
However, multiples have limits. They are only as close of a measure as the similarity between firms. Using multiples is often viewed as taking a shortcut to the discounted cash flow methods of valuation.
It is a good idea to use multiple techniques, because no single technique will yield a final answer regarding a stock's true value.

Information, Competition, and Stock Prices

When discussing stocks, the way we use a valuation model will depend on the quality of our information, and the model will tell us the most about the variable for which our prior information is the least reliable. Market price often provides very accurate information regarding true value of its shares. Also, the idea that competition among investors works to eliminate all positive NPV trading opportunities is referred to as the efficient markets hypothesis. This implies that securities will be fairly priced, based on their future cash flows, given all information that is available to investors and no other information is utilized. Some information is easily utilized and in public records, while some is not publicly available.
Positive NPV trading opportunities are hard to come by and may be disappointing. However, if stocks are fairly priced according to our valuation models, then investors who buy stocks can expect to receive future cash flows taht fairly compensate them for their risk of investing. Some key steps to follow include:

## Chapter 9 – Valuing Stock

## Introduction

A share of stock represents a part of ownership of a corporation. Each stock has it's ownticker symbol, which is a unique abbreviation assigned to publicly traded companies used when its trades are reported on the ticker. The ticker is a real time electronic display of trading activity. These ticker symbols usually consist of three or fewer characters on the NYSE, while those on the NASDAQ usually have four or more characters.Common stockis a share of ownership in the corporation, which gives its owner rights to any common dividends as well as rights to vote on the election of directors, mergers, or other major events. Dividends are usually periodic payments that allow the stock carrier to share in profits of the corporation.## The Dividend-Discount Model

As the valuation principle states, we must determine expected cash flows that an investor will receive from owning stock in order to value it. To begin, we will look at investors with a one-year investment horizon.For a one-year investor, there are two potential sources of cash flows from owning stock:

1.

The firm might pay out cash to its shareholders in the form of dividends.2. The investor might generate cash by selling the shares at a future date.Assuming all dividends are paid at year end, the timeline for such an investment would have a negative initial stock price at period 0 and at period 1, you would have the price you sell the stock for plus your dividend payment. Because such cash flows are risky, the risk free interest rate cannot be used to discount them. Instead we use the cost of capital. The cash flow must be discounted based on the

equity cost of capital, which is the expected return of other investments available in the market with equivalent risk to the firm's shares. With that, the equation to solve for the stock price looks something like this:Po = (Div1 + P1) / (1+r)

Po is the original price of stock, with Div1 being the dividend payment and P1 being the selling price and r equaling the equity cost of capital rate.

Reworking the equation can give us other beneficial numbers, such as the Total Return (r) = ((Div1 + P1)/Po) - 1. In this equation, the

dividend yield(or the expected annual dividend of the stock divided by it's current price) is represented with the (Div1/Po). Thecapital gain(the amount by which the selling price of an asset exceeds its initial purchase price) is found by taking (P1 - Po). We divide the capital gain by the current stock price Po to get thecapital gain rate, which expresses the capital gain as a percentage return. Therefore, the sum of dividend yield and the captial gain rate is referred to astotal returnof the stock. The expected total return of the stock should equal the expected return of other investments available in the market with equivalent risks.For a multi-year investment, setting the stock price equal to the present value of the future cash fows implies the following equation:

Po = [Div1/(1+r)] + [(Div2 + P2)/((1+r)^2)]

With that said, we see that the formula for the stock price of a two year investment is the same as that for a sequence of two one-year investments.

This gives us basis for our

dividend-discount model, which is a model that values shares of a firm according to the present value of the future dividends the firm will pay. Therefore, in the above equation, the denominator in the second part of the equation can be raised to the N-th power to represent an arbitrary horizon. This tells us that the price of stock is equal to the present value of all of the expected future dividends it will pay.## Estimating Dividends (Dividend-Discount Model)

The value of a stock is expressed in terms of the expected future dividends the firm will pay. However, estimating these dividends is difficult. A common practice is to assume that dividends will grow in the long run at a constant rate.The

constant dividend growth modelpoints this out to us. It is a model for waluing a stock by viewing its dividends as a constant growth perpetuity. This model looks like this:Po = Div1 / (r-g)

where g is the growth rate.

With constant expected dividend growth, the expected growth rate of the share price matches the growth rate of the dividends. If we define a firms

dividend payout rateas a fraction of its earnings that the firm pays as dividends each year, we can refer to the firm's dividend pare share by taking the Earnings and dividing it by the shares outstanding, then taking that quantity and multiplying it by the dividend payout rate. A firm's dividends can increase by increasing the earnings, increasing the dividend payout rate, or by decreasing the number of shares outstanding.A firm can either pay its earnings out as dividends or retain them for reinvestment. New investment equals the firm's earnings multiplied by its

retention rate(the fraction of a firm's current earnings that the firm retains). However, cutting the firm's dividend to increase investment will raise the stock price only if the new investments have a positive Net Present Value.What if firms often pay no dividends when they are young or if their growth rate continues to change over time until they mature? We must use the dividend-discount model in an adjusted way to account for future share price of the stock once a firm matures and its expected growth rate stabilizes: Pn will represent the final cash flow in the model, which can be attained by:

Pn = (DivN+1) / (r-g)

Here, the dividend is for an arbitrary year N plus 1.

## Total Payout and Free Cash Flow Valuation Models

Because the dividend-discount model is limited, we must factor in other valuation models. Sometimes firms wont pay out dividends but will purchase stock share back, know asshare repurchase.Thetotal payout modelis then a method that values shares of a firm by discounting the firm's total payouts to the equity holders and then divides by the current number of shares outstanding. This equation looks as follows:Po = PV(of Future Total Dividends and Repurchases) / Shares Outstanding

The

discounted free cash flow modelgoes a step further than the total payout model by determining the total value of the firm to all investors both equity holders and debt holders. It is a method for estimating a firm's enterprise value by discounting its future free cash flows:Enterprise Value = Market Value of Equity + Debt - Cash

with this, we must measure the cash generated by the firm before any payments of debt or equity holders are considered. We do this with:

Free Cash Flow = EBIT x (1 - tax rate) + Depreciation - Cap. Expenditures - Inc. in Net Working Capital

A key difference between the discounted free cash flow model and the earlier models we have seen is the discount rate. Previously, we used the equity cost of capital because we were discounting the cash flows to equity holders. Here, we are discounting free cash flows that will be paid to both debt and equity holders. Therefore, we use the firm's

weighted average cost of capital, the cost of capital that reflects the risk of the overall business, which is combined risk of the firm's equity and debt. Incorporating the WACC into the equation, we get a setup that looks something like this:Vn = FCFn+1/(WACC - g) = [(1 + g) / (WACC - g)] x FCF

## Valuation Based on Similar Firms

Themethod of comparablesis an estimate of the value of a firm based on the value of other, comparable firms or other investments taht are expected to generate very similar cash flows in the future. It is yet another method we use to value a firm. Because identical companies do not exist, it makes the method of comparables a little difficult. However, we can adjust for differences in scale between firms by expressing their value in terms of avalution multiple, which is a ratio of the value to some measure of the firm's scale. The most common valuation multiple is the price-earnings ratio. We can compute a firm's price-earnings ration by using eithertrailing earnings(earnings over the prior 12 months) orforward earnings(earnings expected over the coming 12 months), with the resulting ratio being called thetrailing P/Eor theforward P/E. To calculate the forward P/E: Po / EPS = (Div1/EPS1) / (r - g) = Dividend Payout Rate / (r - g).It is also common to use valuation multiples based on the firm's enterprise value. Because the enterprise value represents the entire value of the firm before the firm pays its debt, to form an appropriate multiple, we must divide it by a measure of earnings or cash flows before interest payments are made. Common multiples are EBIT or EBITDA. This valuation multiple is higher for firms with growth rates and low capital requirements.

However, multiples have limits. They are only as close of a measure as the similarity between firms. Using multiples is often viewed as taking a shortcut to the discounted cash flow methods of valuation.

It is a good idea to use multiple techniques, because no single technique will yield a final answer regarding a stock's true value.

## Information, Competition, and Stock Prices

When discussing stocks, the way we use a valuation model will depend on the quality of our information, and the model will tell us the most about the variable for which our prior information is the least reliable. Market price often provides very accurate information regarding true value of its shares. Also, the idea that competition among investors works to eliminate all positive NPV trading opportunities is referred to as theefficient markets hypothesis. This implies that securities will be fairly priced, based on their future cash flows, given all information that is available to investors and no other information is utilized. Some information is easily utilized and in public records, while some is not publicly available.Positive NPV trading opportunities are hard to come by and may be disappointing. However, if stocks are fairly priced according to our valuation models, then investors who buy stocks can expect to receive future cash flows taht fairly compensate them for their risk of investing. Some key steps to follow include: