# Chapter 35 Security valuation.

WHAT IS IT?

Security valuation is the use of analytic methods to determine the intrinsic value (as opposed to the observed market price) of a security. The intrinsic value is then compared to the market price to determine whether the security appears attractively priced.

There are several classes of valuation methods for assessing a security's intrinsic value, including:

* Discounted Cash Flow Methods

* Market Based Methods

* Asset Based Methods

The focus of this chapter is valuing equity securities (stock). Equities are most commonly valued using the first two categories: discounted cash flow methods and market based methods. Less often used are asset based methods that entail valuing a company based upon its underlying net assets (assets less liabilities, or book value). Asset based methods are more likely to be used when the company is expected to be liquidated.

DISCOUNTED CASH FLOW METHODS

As with any investment, the intrinsic value of a stock can be determined by calculating the present value of the future cash flows expected to be received:

[V.sub.0] = [C.sub.1]/[(1 + r).sup.1] + [C.sub.2]/[(1 + r).sup.2] + [C.sub.3]/[(1 + r).sup.3] + ... + [C.sub.n]/[(1 + r).sup.n]

Here, [V.sub.0] is the current value, C is the cash flow for each period (1, 2, 3 ...), r is the required rate of return, and n is the number of periods. Dividing by [(1 + r).sup.n] is the mathematical formula for taking the present value of each cash flow. For example, if an investor expects to receive \$1 one year from today and has a desired rate of return of 10% (r = 0.10), the present value would be \$0.9091 (\$1.0/1.1). If the investor expects to receive \$1 in two years, the present value would be \$0.8265 (\$1/1.1 (2)). Rather than using the formula to determine the present value, a present value table (see Appendix A) can be used to determine the present value of each cash flow or a financial calculator can be used.

Assume an investor is considering investing in an entity that promises a return of \$1 per year for three years. At the end of the three years the entity is expected to liquidate and pay the investor an additional \$10. If the investor requires a 10% rate of return for this investment, how much is it worth today? The present value would be:

[V.sub.0] = \$1/[(1.1).sup.1] + \$1/[(1.1).sup.2] + \$1/[(1.1).sup.3]

= \$0.9091 + \$0.8265 + \$8.2644 = \$10

So a fair price for this investment would be \$10.00. If the market price were \$9, the investment would be undervalued given the investor's expectations and required return. If the investment is purchased for \$9, the expected return would be greater than the required return of 10%. (1) If the market price was above \$10, the investment would be considered overvalued and the expected return would be less than 10%.

In practice, valuing the cash flows from a stock is not as easy as in this stylized example. Future cash flows must first be estimated. There are several cash flows that can be used in a discounted cash flow approach. Two of the most common are dividends and free cash flow to equity. Dividends are cash flows that are actually distributed by the firm to shareholders. Free cash flow to equity is the cash flow the company generates from its operations after deducting any needed investments in the new assets. Free cash flow to equity may be distributed to shareholders as dividends or kept by the firm for future use. The next section presents discounted dividend valuation and the subsequent section presents discounted free cash flow valuation. Finally, this chapter addresses the use of earnings in place of cash flow in valuation models (termed capitalization of earnings).

It is also important to understand the role of the discount rate (desired or required rate of return) in the present value equations used throughout this chapter. It is the rate at which an investor is indifferent between receiving a smaller amount of money today or a larger amount in the future. In the investment world, this number is highly dependent on the general level of interest rates.

Discounted Dividend Models

As the actual cash flows an investor will receive from stock holdings, dividends are an appropriate cash flow to value. Modifying the general cash flow model to consider the present value of dividends (D):

[V.sub.0] = [D.sub.1]/[(1 + r).sup.1] + [D.sub.2]/[(1 + r).sup.2] + [D.sub.3]/[(1 + r).sup.3] + ... + [D.sub.n]/[(1 + r).sup.n]

A share of stock's current value can be determined by forecasting future dividends per share and discounting them back to the current value at the required rate of return, r. Absent evidence to the contrary, a company is generally considered to be a going concern with dividend streams that continue into perpetuity.

That a company will continue paying dividends forever may seem an aggressive assumption. However, the further out in time a dividend occurs, the less it is worth in today's dollars. Consider a stock that is expected to pay a dividend of \$1 at the end of each of the next 30 years and assume that an investor requires a 10% rate of return. The present value of each dividend is:
```Year       Dividend       Present
Value

1            1.00          0.9091
2            1.00          0.8264
3            1.00          0.7513
4            1.00          0.6830
5            1.00          0.6209
6            1.00          0.5645
7            1.00          0.5132
8            1.00          0.4665
9            1.00          0.4241
10           1.00          0.3855
11           1.00          0.3505
12           1.00          0.3186
13           1.00          0.2897
14           1.00          0.2633
15           1.00          0.2394
16           1.00          0.2176
17           1.00          0.1978
18           1.00          0.1799
19           1.00          0.1635
20           1.00          0.1486
21           1.00          0.1351
22           1.00          0.1228
23           1.00          0.1117
24           1.00          0.1015
25           1.00          0.0923
26           1.00          0.0839
27           1.00          0.0763
28           1.00          0.0693
29           1.00          0.0630
30           1.00          0.0573
Total       30.00          9.4269
```

In present value terms, each subsequent dividend in the table above is worth incrementally less than the previous dividend. By year 30, the \$1 dividend is worth less than six cents in today's dollars.

Only 30 years of dividends were presented in this example; forecasting dividends in perpetuity would be inconvenient, to say the least. Fortunately, the process can be simplified for value estimation. If it is assumed that the dividends are constant forever (as they were for 30 years in the table above), the present value of future dividends can be tersely stated as:

[V.sub.0] = D/r

This is the formula for a no-growth dividend discount model. It is based on the formula for a perpetual annuity (or perpetuity) where the dividend does not increase over time. So, if a company is expected to pay a constant \$1 dividend forever and an investor requires a 10% return, the company's intrinsic value is \$10 (\$1 / 0.10). This is a little higher than the table above, where the dividend was only expected for 30 years.

Will the expected intrinsic value of this security be higher next year? Note that since dividends do not change, the expected price next year will remain \$10 unless the required rate of return changes. In this case the expected return is based solely on the dividend to be received each year. The dividend yield (dividend divided by price) would be 10% (\$1 / \$10), the same as the required rate of return.

Example: Mark Mitchell is considering an investment in a regional utility that has paid the same annual dividend for the past 10 years of \$1.25 per share. Mark views this investment as relatively low risk, but lacking in growth. Mark determines that he would be happy to receive a return of 8% per year on such investments. Using the no-growth dividend discount model Mark computes an intrinsic value of \$15.62 (\$1.25 / 0.08). Mark observes that the current market price is \$17.50 and concludes that this is not a satisfactory investment since the market price exceeds his computed intrinsic value.

There are few cases (some utilities and preferred stock issues) in which one can expect a constant, eternal stream of dividends. However, the formula's assumptions can be relaxed to permit a dividend that grows at a constant rate, g:

[V.sub.0] [D.sub.1]/r - g

This is the formula for a constant growth dividend discount model (also known as the Gordon Growth Model), where [D.sub.1] is next year's expected dividend, r is the investor's required rate of return, and g is the rate at which dividends are expected to grow. Another version of this formula starts with this year's dividend rather than next year's expected dividend:

[V.sub.0] = [D.sub.0] (1 + g)/r - g

This formula is mathematically equivalent to the prior formula since next year's dividend is the same as this year's dividend plus one year of growth.

Assume that a company paid a dividend this past year of \$1.00 and that the dividend is expected to grow at a constant rate of 5% per year. The required rate of return is 10%. The intrinsic value of this security would be:

[V.sub.0] = [D.sub.0] (1 + g)/ r - g = 1.00(1.05)/(0.10 - 0.05) = \$21

Note that adding a growth rate of 5% increased the value of the security from \$10 in the no-growth case to \$21 in the growth case. This is due to the fact that, not only is next year's dividend expected to grow at 5%, but so are all future dividends. By year 30, the expected dividend will be \$4.32.

In this case, the expected price of the security will change next year--assuming the required rate of return and expected growth rate do not change. The value at time one ([V.sub.1]) would be:

[V.sub.1] = [D.sub.2]/r - g = 1.1025/(0.10 - 0.05) = \$22.05

Why is the price expected to increase from \$21.00 to \$22.05, an increase of 5%? This is a result of the expected growth rate of 5%. Investors in stocks derive two sources of returns, the dividend yield and the expected growth in share price (appreciation). Rewriting the dividend discount model and using the observed market price, rather than intrinsic value, results in:

r = [D.sub.1]/P + g

Here r is the expected return, [D.sub.1] is next year's expected dividend, P is the current observed market price, and g is the expected growth rate. So, if a security is expected to pay a dividend next year of \$2.00, dividends are expected to grow at 6% per year, and the current price is \$40, the expected return would be 11% (5% dividend yield plus 6% growth).

Example: Evelyn Garcia is considering an investment in a large manufacturing company that paid dividends last year of \$1.95. Evelyn believes that dividends will grow at 4% per year. Evelyn would like to earn an 11% return on this investment. Using the constant growth dividend discount model, Evelyn computes an intrinsic value of \$28.97 [(\$1.95 x (1 + 0.04)) / (0.11 - 0.04)]. Evelyn observes that the current market price is \$26 and concludes that this investment meets her criteria. Based on the current market price, Evelyn expects a return of 11.80% [((\$1.95 x (1 + 0.04)) / \$26) + 0.04].

What if growth is not expected to be constant? The dividend discount model can be extended to permit two (or more) separate growth rates.

Example: Salazar Ventures, Inc. (SVI) paid a dividend last year of \$1.00. An investor who requires a return of 10% expects SVI's dividend to grow 10% a year for the next three years. Subsequently, the growth rate is expected to fall to 5% in perpetuity. To apply a two-stage model, first estimate the dividends for the next three years, and compute their present value (2):
```   Year       Dividend     Present
Value

1               1.10        1.0000
2               1.21        1.0000
3               1.33        1.0000
```

The total present value of the first three years (stage 1) of dividends is \$3.00. The second stage applies the constant growth dividend discount model to estimate the value of the stock at the end of year 3:

[V.sub.3] = [D.sub.3] (1 + g)/ r - g = 1.33(1.05)/(0.10 - 0.05) = \$27.93

The intrinsic value of SVI's stock at the end of year three is expected to be \$27.93.

To determine the value today, discount the \$27.93 back to the present for three years at the required return of 10%. This results in a present value of \$20.98. Adding the second stage result to the \$3.00 (present value of dividends) from stage one results in a total value today of \$23.98.

Discounted Free Cash Flow Models

What if a company does not pay dividends? Other discounted cash flow models are available, and are often preferred over dividend-based models even when the company pays dividends. One of the most frequently used discounted cash flow models is based on free cash flow to equity rather than dividends. Free cash flow to equity (FCFE) is the company's operating cash flow, less capital investments and debt repayments. While this cash flow is available for distribution as dividends, many companies choose to retain it for future use.

FCFE can be viewed as the cash flow that would be available to an entity that acquired control of the subject company. Even shareholders who buy only a few shares of stock have an interest in the value of a firm on a FCFE basis since the potential exists for some other entity to acquire control of the subject company or for the subject company to begin paying dividends at some future date.

A FCFE model is simply a dividend discount model where the dividend is replaced by FCFE per share:

[V.sub.0] = [FCFE.sub.1]/r - g

The intrinsic value is denoted by [V.sub.0], as before. [FCFE.sub.1] represents next year's forecast of free cash flow to equity. The required rate of return (3) for an equity investment is r and the estimated growth in FCFE is g.

If a company has estimated FCFE of \$3.50 per share next year, is expected to grow its FCFE 3% annually, and requires a return of 12%, the intrinsic value using this model would be \$38.89. As with the dividend discount model, the FCFE model can be expanded for multiple growth stages as in the example above, by replacing dividends with FCFE per share. This method is frequently used to value acquisitions including closely held businesses.

Free cash flow to the firm (FCFF) is another alternative that is often used in valuing firms in leveraged buyouts or where the new owner plans to change the capital structure of the acquired firm. Free cash flow to the firm is the cash flow available to all capital providers, including debt and equity holders. It is therefore a broader concept. When valuing equity using free cash flow to the firm, the value obtained is that of the overall enterprise (debt plus equity), not just the value of the equity (stock).

Companies that are able to generate free cash flow (whether FCFE or FCFF) are often viewed as attractive takeover candidates. Free cash flow is also often discussed in company press releases as one metric to evaluate how well the company is doing.

Capitalized Earnings

The previous two sections presented models based on discounting some measure of future cash flows. Another approach is to discount earnings rather than cash flows. This method is related to the market multiple (Price/Earnings) approach that is discussed later and is very often used for public companies. The capitalized earnings method is frequently used to evaluate real estate investments (including real estate investment trusts). In a capitalized earnings approach, some measure of historic earnings is divided by a capitalization rate (encompassing both the required rate of return for risk and expected growth) to obtain intrinsic value:

[V.sub.0] = Earnings/Capitalization Rate

The capitalization rate can be determined by subtracting the expected growth rate from a required rate of return that compensates for risk, as in the previous discounted cash flow models. The required rate of return, however, is not the same as that used in discounted cash flow models. Cash flows such as dividends and free cash flow are available to be spent currently, whereas investors may be unable to access earnings (for example, the company may reinvest them in inventory or other assets). Another approach, used frequently in real estate, is to develop a capitalization rate by looking at similar real estate companies/projects and determining their implied capitalization rate.

Example: Bill Grant wants to value a real estate project that had net operating earnings of \$100,000 last year. Net operating earnings are defined as revenues minus all operating expenses. Operating expenses exclude income taxes and interest expense. Bill identifies several other real estate projects that sold recently and are similar in size, expected growth and amenities to the subject property:
```               (1) Net                            Effective
Operating         (2) Sales      Capitalization
Property       Income             Price         Rate (1)/(2)

A             \$150,000         \$1,875,000           0.080
B             \$200,000         \$2,350,000           0.085
C             \$180,000         \$2,000,000           0.090
```

On average, the subject companies have a capitalization rate of 0.085 or 8.5%. Bill uses this rate to estimate the intrinsic value of his subject property:

[V.sub.0] = Earnings/ Capitalization Rate = \$100,000/0.085 = \$1,176,471

Care must be taken when using a capitalization of earnings approach since earnings are subject to a great deal of management discretion. Earnings may be overstated (or even understated) and result in an inappropriate estimate of value.

MARKET BASED METHODS

A market-based approach to valuation involves estimating the intrinsic value of a security with reference to the value of other similar securities. This is done by comparing the market price, divided by some fundamental value such as earnings for the subject company, to that of similar peer companies. This method is sometimes referred to as a market multiple, price multiple, or guideline company method. Several market multiples are commonly used for this comparison:

* Price/Earnings

* Price/Free Cash Flow

* Price/Sales

* Dividend/Price (Dividend Yield)

Note that the last one listed is evaluated inversely to the others. Customarily, investors like to think in terms of dividend yields rather than price as a multiple of dividends.

Price/Earnings Approach

A price/earnings ratio (P/E) is a stock's price per share divided by earnings per share. It is a measure of relative value. A P/E ratio of 15 indicates that the price is 15 times the earnings per share of the company. A high P/E ratio relative to other companies indicates that the company is expensive relative to earnings.

What is the appropriate P/E for a company? The P/E ratio is directly related to the dividend discount model presented earlier. Let's revisit the no-growth dividend discount model from earlier:

[V.sub.0] = D/r

First, let's replace intrinsic value, [V.sub.0], with the current observed market price, P:

P = D/r

A no-growth company is one in which earnings are paid out as dividends to shareholders rather than being reinvested in the company (either because the company simply chooses to or has no opportunities for growth). Therefore, for a no-growth company, earnings are equal to dividends and:

P = E/r

Where P is the price per share, E is the earnings per share and r is the required rate of return. Now dividing both sides by E results in:

P/E = 1/r

In this modification of the dividend discount model, it can be seen that the P/E ratio for a no-growth company should be equal to one divided by the required rate of return. So, a no-growth company having a required rate of return of 10% (r = 0.10) would have a theoretical P/E ratio of 10 (1 / 0.10). This is also known as a justified or warranted P/E. The intrinsic value of a no-growth company can be assessed by multiplying the justified P/E times the earnings per share:

V = Justified P / E x Earnings Per Share

Say that the no-growth company with a justified P/E of 10 has earnings per share of \$1.20. Its intrinsic value would be \$12.00 (10 x \$1.20).

Now let's take a look at a theoretical or justified P/E for a growth company. The formula derived from the dividend discount model is:

P/E = Payout Ratio (1 + g)/r - g

Where P/E is the current price divided by last year's earnings per share, Payout Ratio is the proportion of earnings paid out as dividends (in decimal form), r is the required rate of return based on risk, and g is the expected growth rate.

A company that pays out 40% (0.40) of its earnings as dividends, is expected to grow at 7% (0.07), and has a required return of 10% (0.10), would have a justified P/E of 14.27 [(0.40 x (1 + 0.07)) / (0.10 - 0.07)]. If last year's earnings were \$1.50, the intrinsic value would be \$21.40 (14.27 x \$1.50).

Since the theoretical P/E formulas are derived directly from the dividend discount model, they will provide the same estimate of intrinsic value as the dividend discount model. Nonetheless, the distinct models are useful in understanding which factors influence P/E ratios.

Two critical factors are risk (which impacts r) and expected growth. A company with higher risk than other companies will have a higher required rate of return and therefore should trade at a lower P/E ratio. On the other hand, a company with a higher growth rate should trade at a higher P/E ratio.

A market multiple or guideline valuation approach uses these concepts to assess the relative value of a subject company. Assume an investor is comparing a company to three similar peer companies (same industry, similar size, etc.). They have the following P/E ratios:
```Company           Price         Earnings      P/E Ratio
Per Share

Subject           \$20.00         \$1.00            20
Company X         \$36.00         \$2.00            18
Company Y         \$24.00         \$1.50            16
Company Z         \$25.20         \$1.80            14
```

Absent any additional information, it appears that the subject company is overvalued since its price relative to earnings is higher than those of the three peer companies. How much overvalued? If it is assumed that the subject company deserves a P/E multiple equal to the median (i.e., middle, which in this case is the same as the average) multiple of the peers (16), then the intrinsic value would be \$16.00 (\$1.00 x 16) and the stock could be considered overvalued by \$4.00 (\$20 - \$16).

However, perhaps the subject company does not deserve the same multiple as the peer companies. Why could this be true? Even similar companies have different exposure to risk and growth. If the subject company has a higher growth rate than its peers (or lower risk), it might deserve its higher multiple. A commonly used method of assessing P/E ratios relative to growth is to compute a PEG ratio:

PEG = P/E/g

Here, expected growth is in percentage form (e.g., g = 100 x 10% = 10). Adding expected growth to the subject and peer companies yields:
```Company         P/E Ratio       Expected      PEG Ratio
Growth

Subject             20            10%            2.00
Company X           18             9%            2.00
Company Y           16             8%            2.00
Company Z           14             8%            1.75
```

Adjusting for growth, it can be seen that the subject company has the same P/E relative to growth as companies X and Y. It would appear that its higher P/E is justified by its higher growth expectations. Company Z appears to be the best value in terms of price relative to earnings and growth.

Care must be taken when using the PEG ratio for several reasons. Expected growth is normally only available for short periods of time (five years). Companies that have a lower but more persistent growth rate may be penalized by the model. Also the PEG formula assumes a linear relationship between growth and P/E ratios, which is not strictly correct. Lastly, the PEG ratio does not consider differences in risk between the companies, which is another major factor in justified P/E ratios.

P/E ratios in general are further limited in that earnings can be subject to differences between companies due to accounting methods and other non-operating reasons that can make companies non-comparable. Ideally, peer companies should be in the same industry, of similar size, and use similar accounting methods. Above all, it is imperative to remember that any estimate of future growth is just that--an estimate. No one, even experienced Wall Street analysts can predict the future with certainty.

P/E ratios are not only useful in evaluating companies, but are useful in evaluating the holdings of mutual funds. One metric used to evaluate equity mutual funds is the average (or median) P/E ratio of the underlying stocks.

P/E ratios can also be viewed in an inverse format as E/P. E/P is known as the earnings yield. A company with a P/E of 20 would have an earnings yield of 5%. This is an alternative view of value. Companies with high P/E ratios have low earnings yields and companies with low P/E ratios have high earnings yields.

Price/Free Cash Flow Approach

Another market multiple that can be used to assess intrinsic value is the Price/Free Cash Flow or P/FCFE ratio. A theoretical P/FCFE model can be derived as:

P/FCFE = 1 + g/r-g

Where FCFE is free cash flow to equity for the last year. A company with a required return of 10% and growth rate of 5% would have a justified P/FCFE ratio of 21. Practically speaking, a P/FCFE approach involves comparison of a subject company's ratio to those of similar companies to determine if it appears properly valued. As with the P/E ratios, differences may be justified by differences in risk and growth.

Price/Sales Approach

Price to sales (P/S) is yet another multiple that can be used for a relative valuation. A theoretical P/S ratio is:

P/S = Profit Margin x Payout Ratio x (1 + g)/r - g

Profit margin is a company's earnings divided by sales, payout ratio is the portion of earnings paid out as dividends, g is the expected growth rate, and r is the required rate of return. The practical application is the same as with the P/E ratio where the P/S ratio is compared to those of peer companies.

Dividend Yield Approach

Another valuation metric utilizes price per share and dividends. While a dividend multiple (P/D) could be computed, it is more traditional to express relative dividend valuation in terms of dividend yield (D/P). Companies with high dividend yields would indicate a good value (low price relative to dividends). Some investors have a preference for investing in stocks with high dividend yields, especially in light of favorable tax treatment in recent years. Companies with low dividend yields (or zero) are likely to be growth oriented companies that choose to reinvest cash flow in the business. Other investors often favor growth stocks and prefer to defer receipt (and taxation) of income.

ASSET BASED METHODS

A less frequently used approach is to assess the value of a company based on the book value of its assets. Book value is the amount reported on the company's financial statements. A balance sheet shows the assets a company owns and the claims against those assets.
```Alpha Beta, Inc.
Balance Sheet
As of December 31, 2006

Cash                                                 \$ 500,000
Equipment                                            2,000,000
Total Assets                          \$2,500,000
Liabilities                           \$1,000,000
Stockholders Equity      1,500,000
Total Liabilities
And Equity                            \$2,500,000
```

Book value (assets minus liabilities) for this firm is \$1,500,000. This is also known as stockholders equity--the amount that stockholders have invested over time in the company. If the company has 50,000 total shares outstanding (owned by stockholders), book value per share is \$30 (\$1,500,000 / 50,000). Most companies sell for more than their book value for a couple of reasons; many of the assets are reported on the balance sheet based at what the company paid for them, which may not reflect their current value. Additionally, most companies are more valuable as a going concern than the sum of their assets minus liabilities. If the value of assets can be determined, the book value can be adjusted. Alternatively, the book value per share can be used to compute a price to book value ratio (P/B) and a relative valuation can be performed as in the market approach presented in the previous section.

WHERE CAN I FIND OUT MORE ABOUT IT?

1. John Stowe, Thomas Robinson, Jerald Pinto, and Dennis McLeavey, Analysis of Equity Investments: Valuation (Association for Investment Management and Research, 2002).

2. Shannon P. Pratt, Valuing a Business, 4th Edition (New York, NY: McGraw Hill, 2000).

3. Z. Christopher Mercer, "Adjusting Capitalization Rates for the Differences Between Net Income and Net Free Cash Flow," Business Valuation Review (December 1992, Vol. 11, No. 4, 201-207).

Question--What is the difference between a stock's price (market value), intrinsic value, and book value?

Answer--A stock's market value is the observed price trading on the stock market. The intrinsic value is the perceived or underlying value of the stock determined by using a valuation method. Book value is simply the difference between a company's assets and liabilities without regard to its value as an ongoing business.

Question--What are the factors that determine value in a dividend discount model?

Answer--The three factors used to determine intrinsic value in a dividend discount model are the expected dividend, the required return, and expected growth.

Question--What factors impact a company's P/E ratio?

Answer--A P/E ratio is primarily influenced by risk and growth. Higher (lower) risk relative to other companies should result in a lower (higher) P/E ratio. Higher (lower) expected growth relative to other companies should result in a higher (lower) P/E ratio.

Question--If the information in the table is correct and using the constant growth dividend discount model, what rate of return (r) does the market expect (require) for Verizon? Based on this information what would Verizon's justified P/E multiple be?

Answer--The following information relates to shares of telecom services provider Verizon, Inc. as of July 13, 2006:
```Annual dividend (last twelve months)      \$1.62
Earnings (last twelve months)             \$2.57
Payout ratio (5-year average)               63%
Share price                              \$31.60
Estimated 5-year growth rate               2.9%
```

The expected return (r) would equal next year's expected dividend (D1), divided by the price (P), plus the expected long term growth rate (g): 8.18% [((1.62 x (1 + 0.029)) / 31.60) + 0.029].

The justified P/E multiple is found by multiplying the payout ratio by one plus the growth rate (g), then dividing the total by the difference between the required return (r) and the growth rate (g): 12.27 [((0.63 x (1 + 0.029)) / (0.0818 - 0.029)].

Question--Verizon's closest peer company is AT&T, which has a trailing P/E multiple of 17.5 and a payout ratio of 85%. Assuming both will have similar growth rates, what does this say about the market's beliefs for AT&T relative to Verizon (refer to last question)?

Answer--An important factor for P/E ratios, besides growth, is relative risk. If the market is awarding AT&T a higher P/E ratio than Verizon and they have similar growth prospects, it may view AT&T as less risky.

CHAPTER ENDNOTES

(1.) How much greater? By making [V.sub.0]= \$9 and solving for r, an investor would obtain an expected return of 14.33%. This is known as an internal rate of return, which makes the present value of the future cash flows equal to the initial investment. See Chapter 33, "Measuring Investment Return."

(2.) Forecasted dividends are rounded to two decimal places.

(3.) Methods of assessing an appropriate required return on equity are presented in Chapter 36, "Asset Pricing Models."