Chapter 7: Valuing Stocks


  • Pt = Stock Price at the End of Year t
  • rE = Equity Cost of Capital
  • n = Termination Date or Forecast Horizon
  • g = Expected Dividend Growth Rate
  • Divt = Dividends Paid in Year t
  • EPSt = Earnings per Share on Date t
  • PV = Present Value
  • EBIT = Earnings before Interest & Taxes
  • FCFt = Free Cash Flow on Date t
  • Vt = Enterprise Value on Date t
  • τc = Corporate Tax Rate
  • rwacc = Weighted Average Cost of Capital
  • gFCF = Expected Free Cash Flow Growth Rate
  • EBITDA = Earnings before Interest, Taxes, Depreciation, and Amortization

7-1: The Dividend-Discount Model

The Law of One Price implies that the price of a security should equal the present value of the expected cash flows an investor will receive from owning it. Using this, we derive the first method of valuing a stock: the dividend-discount model.

A One-Year Investor

There are 2 potential sources of cash flows from owning a stock; dividends and selling the share at some future date. The total amount received in dividends and from selling the stock will depend on the investor’s investment horizon. Let’s consider a 1-year investor.

They buy a stock for the current market price P0. Let Div1 be the total dividends paid at the end of the year. At the end of the year, the investor sells the share at the new market price P1.

Given these expectations, the investor will be willing to pay a price today up to the point that this transaction has a zero NPV. Since there’s some risk involved, we must discount these cash flows based on the equity cost of capital rE. This is the expected return of other investments available in the market with equivalent risk to the firm’s shares.


Dividend Yields, Capital Gains, and Total Returns

We can reinterpret the equation of P0 by multiplying by (1 + rE), divide by P­0, and subtract 1 from both sides. By doing this, we get the total return.

The dividend yield is the expected annual dividend of the stock divided by its current price. It’s the percentage return the investor expects to earn from the dividend paid by the stock.

The capital gain is what the investor will earn on the stock, P1P0. We divide it by the current stock price to express the capital gain as a percentage return, called the capital gain rate.

The sum of the dividend yield and capital gain rate is called the total return, which is what an investor will earn for a 1-year investment. The equation states that the expected total return of the stock should equal the expected return of other investments available in the market with equivalent risk.

If a stock offers a higher return than other securities with the same risk, investors would buy the stock and put upward pressure on the price. This increase in the price would lower the dividend yield and capital gain rate. The opposite is also true; lower return causes price to decrease, which increases total return until the equation is satisfied.

A Multiyear Investor

Suppose we hold the stock for 2 years.

Setting the stock price equal to the present value of the future cash flows in this case implies:


Note: There is no underlying difference between the 1-year equation and the 2-year equation. In the 1-year equation, there is no second term and the selling price of the stock is in the numerator of the first term as P1. However, 1 is determined by the expected future cash flows. In other words, we can get the second equation from the first one:

The Dividend-Discount Model Equation

We can continue this process for any number of years by replacing the final stock price with the value that the next holder of the stock would be willing to pay. Doing so leads to the general dividend-discount model for the stock price, where the horizon n is arbitrary:


This applies to a single n-year investor, who collects dividends for n years and then sells the stock, OR applies to a series of investors who hold the stock for shorter periods and then resell it.

For the special case in which the firm eventually pays dividends and is never acquired, it is possible to hold the shares forever. Consequently, we can let n go to infinity:


That is, the price of the stock is equal to the present value of the expected future dividends it will pay.

7-2: Applying the Dividend-Discount Model

Constant Dividend Growth

A common approximation to help estimate dividends in the future is to assume that, in the long run, dividends will grow at a constant rate g. The timeline for an investor who buys and holds a stock is:

Constant Dividend Growth Model


For another interpretation, we can rearrange is as follows:


We see that g equals the expected capital gain rate. In other words, with constant expected dividend growth, the expected growth rate of the share price matches the growth rate of dividends.

Dividends vs. Investment & Growth

With the equation above, we now know that a firm’s share price increases with the current dividend level Div1 and the expected growth rate g. To maximize share price, a firm wants to increase both. However, there’s a tradeoff; increasing growth may require investment but money spent on investment cannot be used to pay dividends.

A Simple Model of Growth

If we define a firm’s dividend payout rate as the fraction of its earnings that the firm pays as dividends each year, we can write the firm’s dividend per share at date t as follows:

The firm can increase its dividends in 3 ways:

  1. Increase its earnings (net income).
  2. Increase its dividend payout ratio.
  3. Decrease shares outstanding.

To simplify things, let’s make the following assumptions:

  • Firms do not issue new or buy back shares. The number of shares outstanding is fixed.
  • If a firm does not invest, the firm does not grow, so earnings generated remains constant.

It can now do 2 things with earnings; pay dividends, or it retain/reinvest them. Therefore:

Change in Earnings=New Investment×Return on New Investment\text{Change in Earnings}=\text{New Investment}\times\text{Return on New Investment}

New investment equals earnings multiplied by the firm’s retention rate, the fraction of current earnings that the firm retains.

New Investment=Earnings×Retention Rate\text{New Investment}=\text{Earnings}\times\text{Retention Rate}

Using substitution, we can get an expression for the growth rate of earnings:

If the firm chooses to keep its dividend payout rate constant, then the growth in dividends will equal growth of earnings:

g=Retention Rate×Return on New Investmentg=\text{Retention Rate}\times\text{Return on New Investment}

This is sometimes known as the firm’s sustainable growth rate, the rate at which it can grow using only retained earnings.

Profitable Growth

A firm can increase its growth rate by retaining more earnings. But this means less will be paid out as dividends. So, if a firm wants to increase its share price, should it cut its dividends and invest more, or should it cut investment and pay more dividends?

The answer depends on profitability of the firm’s investments.

Cutting Dividends for Profitable Growth

A firm expects to have earnings per share of $6 in the coming year. The firm plans to pay out all of its earnings as dividends. With no expectation of growth, the firm’s current share price is $60. Suppose they cut the dividend payout rate to 75% for the foreseeable future and invest the retained earnings. The return on investment is 12%. Assume its equity cost of capital is unchanged, what effect would this new policy have on the firm’s stock price?

The firm’s dividend yield is $6/$60 = 10%. With no expected growth (g = 0%), we can estimate rE:


In other words, the expected return of other stocks in the market with equivalent risk must be 10%.

The reduction in the dividend payout rate to 75% will cause the dividend this coming year to fall to:


At the same time, since the firm now retains 25% of its earnings to invest, its growth rate will increase:

g=Retention Rate×Return on New Investment=25%×12%=3%g=\text{Retention Rate}\times\text{Return on New Investment}=25\%\times12\%=3\%

Assuming a constant growth rate, we can compute the share price to be:


The share price rises from $60 to $64.29 by cutting dividends to increase investment (implying the investment has positive NPV). The investment has a rate of return (12%) greater than the firm’s equity cost of capital (10%). But this is not always the case!

Unprofitable Growth

The same example as above, but now assume the return on investment is 8% instead of 12%. What will happen to the firm’s current share price in this case?

The dividend will still fall to $6 × 75% = $4.50. It’s growth rate under the new policy will now be:


The new share price is therefore:


Even though the firm grows under the new policy, the new investments has a negative NPV. The share price will fall if it cuts dividends to make new investments that return only 8% when its investors can earn 10% elsewhere with comparable risk. Thus, cutting the firm’s dividend to increase investment will raise the stock price if, and only if, the new investments have a positive NPV.

Changing Growth Rates

Successful young firms often have very high initial earnings growth rates. During this period of high growth, it’s not unusual for these firms to retain 100% of their earnings. As they mature, growth slows.

At that point, they begin to pay dividends. We cannot use the constant dividend growth model. They often pay no dividends when they’re young and their growth rates continues to change over time until they mature. However, we can use the dividend-discount model until the firm matures, and then use the constant growth model to calculate the future share price.

If the firm is expected to grow at a long-term rate g after year n + 1, then we know the following:


We can use this estimate of Pn as a terminal (continuation) value in the dividend-discount model.

Dividend-Discount Model with Constant Long-Term Growth


Limitations of the Dividend-Discount Model

  • There is uncertainty when it comes to forecasting a firm’s future dividends. This is important because even small changes in the expected dividend growth rate can lead to large changes in the estimated stock price.
  • Difficult to know which estimate of the dividend growth is more reasonable to use. Forecasting dividends requires forecasting the firm’s earnings, dividend payout rate, and future share count.
  • But future earnings will depend on interest expense (which depends on how much the firm borrows), and its share count & dividend payout rate will depend on whether the firm uses some earnings to repurchase shares. Borrowing & repurchase decisions are at management’s discretion, they can be difficult to forecast reliably.

7-3: Total Payout & Free Cash Flow Valuation Models

There are two alternative approaches to valuing the firm’s shares that avoid some of the difficulties of the dividend-discount model.

Share Repurchases and the Total Payout Model

Share repurchases are becoming popular. It’s where the firm uses excess cash to buy back its own stock, instead of paying dividends to investors. There are two consequences for the dividend-discount model. First, as cash is used to repurchase sales, the less there is to pay dividends. Second, it reduces the share count which increases earnings & dividends per share.

A more reliable method of determining share price when a firm repurchases shares is the total payout model, which values all of the firm’s equity, rather than a single share.

P0=PV(Future Total Dividends and Repurchases)Shares Outstanding0P_0=\frac{PV(\text{Future Total Dividends }\text{and }\text{Repurchases})}{\text{Shares Outstanding}_0}

We can apply the same simplifications that we obtained by assuming constant growth to the total payout method. The only change is that we discount total dividends and share repurchases and use the growth rate of total earnings (rather than earnings per share) when forecasting the growth of the firm’s total payouts. This method can be more reliable & easier to apply when the firm uses share repurchases.

The Discounted Free Cash Flow Model

The discounted free cash flow model goes further and begins by determining the total value of the firm to all investors (both equity and debt holders). We begin by estimating the firm’s enterprise value:

Enterprise Value=Market Value of Equity+DebtCash\text{Enterprise Value}=\text{Market Value of Equity}+\text{Debt}-\text{Cash}

The enterprise value is the value of the firm’s underlying business, unencumbered by debt & separate from any cash/marketable securities. We can interpret the enterprise value as the net cost of acquiring the firm’s equity, taking its cash, paying off all debt, and thus owning the unlevered business. The advantage of the discounted free cash flow model is that it allows us to value a firm without explicitly forecasting its dividends, share repurchases, or its use of debt.

Valuing the Enterprise

To calculate a firm’s enterprise, we must compute the present value of the free cash flow (FCF) that the firm has available to pay all investors, both debt & equity holders.

When we are looking at the entire firm, it is natural to define the firm’s net investment as its capital expenditures in excess of depreciation:

Net Investment=Capital ExpendituresDepreciation\text{Net Investment}=\text{Capital Expenditures}-\text{Depreciation}

We can loosely interpret net investment as investment intended to support the firm’s growth, above and beyond the level needed to maintain the firm’s existing capital. We can rewrite the formula to:

A firm’s enterprise value V0 is computed by calculating the present value of the firm’s free cash flow:

Discounted Free Cash Flow Model

V0=PV(Future Free Cash Flow of Firm)V_0=PV(\text{Future Free Cash Flow of Firm})

Given the enterprise value, we can estimate the share price by solving for the value of equity and then dividing by the total number of outstanding shares:

P0=V0+Cash0Debt0Shares Outstanding0P_0=\frac{V_0+\text{Cash}_0-\text{Debt}_0}{\text{Shares Outstanding}_0}

The difference between the discounted free cash flow model and the dividend-discount model is that in the dividend-discount model, the firm’s cash and debt are included indirectly through the effect of interest income and expenses on earnings. In the discounted free cash flow model, we ignore interest income and expenses because free cash flow is based on EBIT, but then adjust for cash & debt directly.

Implementing the Model

To find the present value of the future free cash flows of a firm, we must discount it using the weighted average cost of capital (WACC) rwacc which is the cost of capital the firm must pay to all of its investors (debt & equity holders). This replaced rE. If the firm has no debt, then rwacc­­ = rE. But if it does, r­wacc is an average of the firm’s debt and equity cost of capital. Since debt is generally less risky than equity, r­wacc is generally less than rE. We forecast the firm’s free cash flow up to some horizon, together with a terminal (continuation) value of the enterprise:


Often, the terminal value is estimated by assuming a constant long-run growth rate gFCF for free cash flows beyond year n, so that:

Vn=FCFn+1rwaccgFCF=(1+gFCFrwaccgFCF)×FCFnV_n=\frac{FCF_{n+1}}{r_{wacc}-g_{FCF}}=\left(\frac{1+g_{FCF}}{r_{wacc}-g_{FCF}}\right)\times FCF_n

The gFCF is typically based on the expected long-run growth rate of the firm’s revenues.

Connection to Capital Budgeting

Because the firm’s free cash flow is equal to the sum of the free cash flows from the firm’s current and future investments, we can interpret the firm’s enterprise value as the total NPV that the firm will earn from continuing its existing projects and initiating new ones. Hence, the NPV of any individual project represents its contribution to the firm’s enterprise value.


Many forecasts and estimates are necessary to estimate the free cash flows of the firm. On one hand, this gives us flexibility to incorporate many specific details about the future prospects of the firm. On the other, some uncertainty surrounds each assumption.

It is therefore important to conduct a sensitivity analysis to translate this uncertainty into a range of potential values for the stock. A sensitivity analysis shows how our final calculated value changes with respect to changes in one of the input variables in our model.

Summary of Different Valuation Methods

7-4: Valuation Based on Comparable Firms

Another application of the Law of One Price is the method of comparables. In the method of comparables (or “comps”), rather than value the firm’s cash flows directly, we estimate the value of the firm based on the value of other, comparable firms or investments that we expect will generate very similar cash flows in the future.

Valuation Multiples

Of course, identical companies do not exist. They are likely to be of different sizes and scales. We can adjust for differences in scale between firms by expressing their value in terms of a valuation multiple, which is a ratio of the value to some measure of the firm’s scale. Analogy: You can find the average price per square foot, and then use that to value different sized buildings.

The Price-Earnings Ratio

The most common valuation multiple is the price-earnings (P/E) ratio.

P/E Ratio=Share PriceEarnings per Share\text{P/E Ratio}=\frac{\text{Share Price}}{\text{Earnings per Share}}

To interpret the P/E multiple, consider the stock price formula for constant dividend growth:


If we divide both sides by EPS, we have the following formula:

Forward P/E=P0EPS1=Div1/EPS1rEg=Dividend Payout RatiorEg\text{Forward P/E}=\frac{P_0}{EPS_1}=\frac{Div_1/EPS_1}{r_E-g}=\frac{\text{Dividend Payout Ratio}}{r_E-g}

This is a firm’s forward P/EI, which is the P/E multiple computed based on its forward earnings (expected earnings over the next 12 months). We can also compute the firm’s trailing P/E (earnings over the prior 12 months). For valuation purposes, forward P/E is preferred.

Trailing P/E=P0EPS0=(1+g0)P0EPS1=(1+g0)(Forward P/E)\text{Trailing P/E}=\frac{P_0}{EPS_0}=\frac{(1+g_0)P_0}{EPS_1}=(1+g_0)(\text{Forward P/E}) 

This implies if 2 stocks have same payouts, EPS growth rates, and risk (same r­E), then they’ll have the same P/E. It also shows that firms/industries with high growth rates, which generate cash well in excess of their investment needs so that they can maintain high payout rates, should have high P/E multiples.

Example: Valuation Using the Price-Earnings Ratio

Suppose a firm has EPS of $1.38. If the average P/E of a comparable stocks is 21.3, estimate the value for the firm by using the P/E valuation method. What are the assumptions underlying this estimate?

We estimate share price by multiplying its EPS by the P/E of comparable firms. Thus, P0 = $1.38 × 21.3 = $29.39. This assumes that the firm will have similar future risk, payout rates, and growth rates.

Enterprise Value Multiples

Since enterprise value represents the total value of the firm’s underlying business rather than just the value of equity, using enterprise value is advantageous when comparing firms with different leverage. Common multiples to consider are enterprise value to EBIT, EBITDA, and free cash flow. Enterprise value to EBITDA is the most practiced. If expected free cash flow growth is constant, then:


As with the P/E multiple, this valuation multiple is higher for firms with high growth rates and low capital requirements (so that free cash flow is high in proportion to EDITDA).

Example: Valuation Using an Enterprise Value Multiple

Suppose Firm A has EPS of $2.3, EBITDA of $30.7M, 5.4M shares outstanding, and debt of $125M (net of cash). You believe Firm B is comparable, but it has no debt. If Firm B has a P/E of 13.3 and an enterprise value to EBITDA multiple of 7.4, estimate the value of Firm A’s shares using both multiples.

Using Firm B’s P/E, we estimate a share price of P0 = $2.3 × 13.3 = $30.59. Using the enterprise value to EBITDA multiple, we estimate Firm A’s enterprise value V0 = $30.7M × 7.4 = $227.2M. We subtract debt, and divide by outstanding shares to get share price P0 = ($227.2M - $125M)/5.4M Shares. Because of the large difference in leverage between the firms, we would expect the second estimate (based on enterprise value) to be more reliable.

Other Multiples

Other valuation multiples are possible. Enterprise value as a multiple of sales can be useful if it’s reasonable to assume the firms will maintain similar margins in the future. For firms with substantial tangible assets, the ratio of price to book value of equity per share is sometimes used. Some multiples are industry specific. In the cable TV industry, you can consider enterprise value per subscriber.

Limitations of Multiples

Of course, firms are not identical. The usefulness of a valuation multiple will depend on the nature of the differences between firms and the sensitivity of the multiples to these differences. Differences can come differences in accounting rules between different countries, or even public vs. private firms. Differences can come from accounting numbers simply being misstated. Thus, a key shortcoming of the comparables approach is that it does not consider the important differences among firms.

Another limitation of comparables is that they provide only information regarding the value of the firm relative to the other firms in the comparison set. Using multiples will not help us determine if an entire industry is overvalued, for example.

Comparison with Discounted Cash Flow Methods

Using a valuation multiple based on comparables is best viewed as a “shortcut” to the discounted cash flow (DCF) methods of valuation. In addition to its simplicity, the multiples approach has the advantage of being based on actual prices of real firms, rather than what may be unrealistic forecasts of future cash flows.

DCF methods have the advantage that they allow us to incorporate specific information about the firm’s cost of capital or future growth. In addition, DCF methods make explicit the future performance the firm must achieve in order to justify its current value.

Stock Valuation Techniques: The Final Word

In the end, no single technique provides a final answer regarding a stock’s true value. All approaches require assumptions or forecasts that are too uncertain to provide a definitive assessment of the firm’s value. Most real-world practitioners use a combination of these approaches and gain confidence if the results are consistent across a variety of methods. Also, when information changes, our predictions change. This is why stock prices are changing constantly in the stock market.

7-5: Information, Competition, and Stock Prices

The models described link the firm’s expected future cash flows, its cost of capital (determined by risk), and the value of its shares.

But what if the actual market price doesn’t match our estimate? Are we wrong, or is the market?

Information in Stock Prices

If your valuation model suggests a stock is worth $30 per share when it is trading for $20 per share in the market, the discrepancy is equivalent to knowing that thousands of investors—many of them professionals who have access to the best information—disagree with your assessment. This knowledge should make you reconsider your original analysis.

The triangle above tells us that, given accurate information about any 2 variables, we can make inferences about the 3rd variable. Thus, the way we use a valuation model depends on the quality of information: The model tells us the most about the variable for which we have the least information.

For a publicly traded firm, its market price should already provide very accurate information, aggregated from a multitude of investors, regarding the true value of its shares. Therefore, in most situations, a valuation model is best applied to tell us something about the firm’s future cash flows or cost of capital, based on its current stock price.

Competition and Efficient Markets

Investors compete, and if someone has information that buying a stock has a positive NPV, then their attempts to purchase it will drive up the stock’s price. The opposite is also true, when investors sell. This is known as the efficient markets hypothesis. It implies securities will be fairly priced, based on their future cash flows, given all information that is currently available to investors. The accuracy of this hypothesis depends on how many investors have access to the information impacting a firm.

Public, Easily Interpretable Information

Information that is available to all investors news reports, financial statements, corporate press releases, or in other public data sources. If the impact of this information on the firm’s future cash flows can be readily ascertained, then all investors can determine the effect of this information on the firm’s value. Thus, we expect fierce competition & the stock price to react nearly instantly to such news. The efficient markets hypothesis holds very well with respect to this type of information.

Private or Difficult-to-Interpret Information

Some information is not publicly available, such as the information an analyst takes time gathering from employees, competitors, suppliers, or customers that is relevant to the firm’s future cash flows.

Even publicly available information may be difficult to interpret.

When private information is relegated to the hands of a relatively small number of investors, they can profit by trading on their information. In this case, the efficient markets hypothesis will not hold in the strongest form. However, as these informed traders begin to trade, they will tend to move prices, so over time prices will begin to reflect their information as well.

If the profit opportunity is large, others will gain the expertise by devoting the needed resources. Thus, in the long run, we should expect that the degree of inefficiency in the market will be limited by the costs of obtaining information.

Forms of Market Efficiency

  • Strong-Form Market Efficiency: All information, even private information (such as those known only by managers), is immediately incorporated into stock prices. Unrealistic.
  • Semistrong-Form Market Efficiency: All publicly available information is incorporated very quickly into stock prices.
  • Weak-Form Market Efficiency: Only the history of past prices is already reflected in the stock price.

Lessons for Investors & Corporate Managers

Consequences for Investors

As in other markets, investors should be able to identify positive-NPV trading opportunities in securities markets only if some barrier or restriction to free competition exists. An investor’s competitive advantage may take several forms; expertise, access to special information, lower trading costs, etc. Either way, the source of the positive-NPV trading opportunity must be something hard to replicate or your gains will be competed away.

Since positive-NPV trading opportunities are hard to come by, then investors who buy stocks can be reassured to know that they will be fairly compensated for the risk of their investment with future cash flows. This means even uninformed investors can invest with confidence (in competitive markets with reliable information).

Implications for Corporate Managers

If stocks are fairly valued according to the models described, then firm value is determined by the cash flows that it can pay to its investors. This result has several key implications for corporate managers:

  • Focus on NPV and free cash flow: To boost stock price, make investments that increase the present value of the firm’s free cash flow.
  • Avoid accounting illusions: With efficient markets, the accounting consequences of a decision do not directly affect the value of the firm and should not drive decision making. Similarly, investors must beware accounting numbers inconsistent with stock prices & need to be suspicious of manipulations.
  • Use financial transactions to support investment: With efficient markets, the firm can sell its shares at a fair price to new investors. Thus, the firm should not be constrained from raising capital to fund positive NPV investment opportunities.

The Efficient Markets Hypothesis vs. No Arbitrage

An important distinction can be drawn between the efficient markets hypothesis and the notion of a normal market, which is based on the idea of arbitrage. An arbitrage opportunity is a situation in which two securities (or portfolios) with identical cash flows have different prices. Because anyone can earn a sure profit in this situation by buying the low-priced security and selling the high-priced one, we expect that investors will immediately exploit and eliminate these opportunities. Thus, in a normal market, arbitrage opportunities will not be found.

When the NPV of investing is zero, the price of every security equals the present value of its expected cash flows when discounted at a cost of capital that reflects its risk. So, the efficient markets hypothesis implies that securities with equivalent risk should have the same expected return. If they don’t have the same expected return, this does not imply an arbitrage opportunity because each security’s cash flows are still unique. However, as investors try buy the securities with abnormally high expected returns and sell securities with abnormally low expected returns, we would see prices adjust so the equivalent risk securities had equivalent expected returns.

The efficient markets hypothesis is therefore incomplete without a definition of “equivalent risk.” Furthermore, because investors must forecast the riskiness of securities, and may do so differently, there is no reason to expect the efficient markets hypothesis to hold perfectly; it is best viewed as an idealized approximation for highly competitive markets.

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