Reading 46

Equity Investments · Equity Valuation: Concepts and Basic Tools

MODULE 46.1: DIVIDENDS, SPLITS, AND REPURCHASES

LOS 46.a

Evaluate whether a security, given its current market price and a value estimate, is overvalued, fairly valued, or undervalued by the market.

Recall from the reading on Market Efficiency that intrinsic value or fundamental value is defined as the rational value investors would place on the asset if they had full knowledge of the asset's characteristics. Analysts use valuation models to estimate the intrinsic values of stocks and compare them to the stocks' market prices to determine whether individual stocks are overvalued, undervalued, or fairly valued. In doing valuation analysis for stocks, analysts are assuming that some stocks' prices deviate significantly from their intrinsic values.

To the extent that market prices deviate from intrinsic values, analysts who can estimate a stock's intrinsic value better than the market can earn abnormal profits if the stock's market price moves toward its intrinsic value over time. There are several things to consider, however, in deciding whether to invest based on differences between market prices and estimated intrinsic values.

The larger the percentage difference between market prices and estimated values, the more likely the investor is to take a position based on the estimate of intrinsic value. Small differences are to be expected.
The more confident the investor is about the appropriateness of the valuation model used, the more likely the investor is to take an investment position in a stock that is identified as overvalued or undervalued.
The more confident the investor is about the estimated inputs used in the valuation model, the more likely the investor is to take an investment position. Analysts must also consider the sensitivity of a model value to each of its inputs. If a small change in one input would shift the estimated value to the market price, the analyst must be quite sure of that input to act on the difference.
Even when market prices sometimes deviate from intrinsic values, market prices must be treated as fairly reliable indications of intrinsic value. Investors must consider why a stock is mispriced. Investors may be more confident about value estimates that differ from market prices when few analysts follow a particular security.
Finally, to take a position in a mispriced stock, an investor must believe that the market price will actually move toward (and not away from) its intrinsic value, and that it will do so to a significant extent within the investment time horizon.

中文翻譯

內在價值（intrinsic value）＝若投資人完全瞭解資產特性時應給予的合理價值。分析師以估值模型估計內在價值，與市價比較，判斷個股高估、低估或合理。能比市場更準確估計內在價值的分析師，當市價向內在價值回歸時可獲超額報酬。

判斷是否依估值差異建倉，需考量：

市價與估值差距百分比愈大，愈可能依估值建倉。
對所用估值模型愈有信心，愈可能進場。
對輸入參數愈有信心愈可能進場；同時須注意對各輸入的敏感度──若某參數小幅變動就讓估值等於市價，須對該參數極有把握。
市價仍應視為相對可靠的內在價值指標；要思考為何被誤訂價。追蹤分析師少的個股，估值差異可信度較高。
必須相信市價會在投資時間範圍內向內在價值收斂（非偏離）且幅度足夠才值得進場。

LOS 46.b

Describe major categories of equity valuation models.

Analysts use a variety of models to estimate the value of equities. Usually, an analyst will use more than one model with several different sets of inputs to determine a range of possible stock values.

In discounted cash flow models (or present value models), a stock's value is estimated as the present value of cash distributed to shareholders (dividend discount models) or the present value of cash available to shareholders after the firm meets its necessary capital expenditures and working capital expenses (free cash flow to equity models).

There are two basic types of multiplier models (or market multiple models):

The first uses the ratio of stock price to fundamentals such as earnings, sales, book value, or cash flow per share. The price-to-earnings (P/E) ratio is most frequently used.
The second uses the ratio of enterprise value to either EBITDA (earnings before interest, taxes, depreciation, and amortization) or revenue. Enterprise value is the market value of all the firm's outstanding securities minus cash and short-term investments. Common stock value can be estimated by subtracting the value of liabilities and preferred stock from enterprise value.

In asset-based models, the intrinsic value of common stock is estimated as total asset value minus liabilities and preferred stock. Analysts typically adjust the book values of the firm's assets and liabilities to their fair values when applying this model.

中文翻譯

分析師通常會用多種模型、多組假設得出價值區間。

折現現金流模型（DCF / 現值模型）：以分配給股東的現金折現（股利折現模型 DDM），或以滿足資本支出與營運資金後可分配給股東的現金折現（自由現金流給股東 FCFE 模型）。
乘數模型（market multiple）：（1）股價對盈餘／銷售額／帳面價值／現金流之比（P/E、P/S、P/B、P/CF）；（2）企業價值（EV）對 EBITDA 或營收之比，EV＝全部證券市值 − 現金與短期投資。
資產基礎模型：股票內在價值＝資產 − 負債 − 特別股，通常以公允價值調整帳面值。

LOS 46.c

Describe regular cash dividends, extra dividends, stock dividends, stock splits, reverse stock splits, and share repurchases.

Cash dividends are payments made to shareholders in cash. They may be regularly scheduled dividends or one-time special dividends.

Regular dividends occur when a company pays out a portion of profits on a consistent schedule (e.g., quarterly). A long-term record of stable or increasing dividends is widely viewed as a sign of financial stability.
Special dividends are used when favorable circumstances allow a one-time cash payment in addition to any regular dividends. Many cyclical firms (e.g., automakers) use a special dividend to share profits with shareholders when times are good but maintain flexibility to conserve cash when profits are poor. Other names: extra dividends and irregular dividends.

Stock dividends are dividends paid out in new shares of stock rather than cash. There will be more shares outstanding, but each one will be worth less. Total shareholders' equity remains unchanged. Stock dividends are commonly expressed as a percentage — a 20% stock dividend means every shareholder gets 20% more stock.

Stock splits divide each existing share into multiple shares. There are now more shares, but the price of each share will drop correspondingly, so there is no change in the owner's wealth. Splits are expressed as a ratio. In a 3-for-1 stock split, each old share becomes three new shares. Stock splits are currently more common than stock dividends.

Reverse stock splits are the opposite of stock splits. After a reverse split, there are fewer shares outstanding but a higher stock price. Because these factors offset, shareholder wealth is unchanged.

A share repurchase is a transaction in which a company buys outstanding shares of its own common stock. Share repurchases are an alternative to cash dividends as a way of distributing cash, and they have the same effect on shareholders' wealth as cash dividends of the same size. A company might repurchase shares to support their price, to signal that management believes the shares are undervalued, or to offset an increase in outstanding shares from the exercise of employee stock options. In countries that tax capital gains at lower rates than dividends, shareholders may prefer share repurchases to dividend payments.

中文翻譯

現金股利：以現金支付，可分為定期或一次性。

定期股利：固定排程（如每季）發放，長期穩定／成長之股利紀錄被視為財務穩健。
特別／額外／不規則股利：景氣循環產業（如汽車業）景氣好時加發，景氣差時保留現金。

股票股利：以新股代替現金。股數增加但每股價值下降；股東權益總額不變。20% 股票股利 = 多 20% 股數。

股票分割（splits）：每股拆成多股。股數增多、股價成比例下降，股東財富不變。3-for-1 即 1 股拆 3 股。目前比股票股利更常見。

反向分割（reverse split）：股數減少、股價上升，財富仍不變。

股票回購（share repurchase）：公司買回自家股票，等同現金股利之分配方式，對股東財富效果相同。動機：支撐股價、示意管理層認為被低估、抵銷員工選擇權行使造成的股數增加。若資本利得稅率低於股利稅率，股東會偏好回購而非股利。

LOS 46.d

Describe dividend payment chronology.

The dates relevant to dividend payments are shown in Figure 46.1.

Figure 46.1: Dividend Payment Chronology

Declaration date	Ex-dividend date	Holder-of-record date	Payment date
August 25	September 15	September 17	September 30

Declaration date. The date the board of directors approves payment of a dividend, specifying the per-share dividend amount, the record date, and the payment date.
Ex-dividend date. The first day on which a share purchaser will not receive the next dividend. The ex-dividend date is one or two business days before the holder-of-record date, depending on the settlement period for stock purchases. If you buy on or after the ex-dividend date, you will not receive the dividend.
Holder-of-record date (record date). The date on which all owners of shares become entitled to receive the dividend payment.
Payment date. The date dividend checks are mailed to, or payment is made electronically to, holders of record.

On the ex-dividend date, the share price will decrease from the previous day's closing price by approximately the amount of the dividend, in the absence of other factors affecting the stock price. Consider shares trading at $25 on the day prior to the ex-dividend date that will pay a $1 dividend. Purchasing on the day prior gives the owner the share and the $1 dividend on payment date. Purchasing on the ex-dividend date gives only the share; the dividend goes to the seller.

中文翻譯

股利支付時程（範例日期：8/25 → 9/15 → 9/17 → 9/30）：

宣告日（declaration date）：董事會通過股利，公布金額、記錄日、支付日。
除息日（ex-dividend date）：買進當日不能領到下次股利的第一天；落在記錄日之前1～2 個營業日（視交割週期）。
記錄日（holder-of-record date）：登記為股東者可領股利之日。
支付日（payment date）：實際支付（郵寄／電子轉帳）股利之日。

除息日當日，股價約下降一筆股利金額。例：股價 $25、股利 $1，前一日買 → 取得股票＋$1 股利；除息日當日買 → 只取得股票，股利歸賣方。

Module Quiz 46.1

1. An analyst estimates a value of $45 for a stock with a market price of $50. The analyst is most likely to conclude that a stock is overvalued if:

A. few analysts follow the stock and the analyst has less confidence in his model inputs.
B. few analysts follow the stock and the analyst is confident in his model inputs.
C. many analysts follow the stock and the analyst is confident in his model inputs.

B — If the analyst is more confident of his input values, he is more likely to conclude that the security is overvalued. The market price is more likely to be correct for a security followed by many analysts and less likely correct when few analysts follow the security. (LOS 46.a)

2. A valuation model based on free cash flow to equity is most likely to be a(n):

A. multiplier model.
B. asset-based model.
C. present value model.

C — One example of a present value model is valuation based on the present value of future cash flows available to equity holders. (LOS 46.b)

3. A company is evaluating the likely effects on its share price of declaring a 50% stock dividend or a 3-for-2 stock split. Other things equal, which of these will result in a lower share price?

A. 3-for-2 stock split.
B. 50% stock dividend.
C. Both should have the same effect.

C — Both a 50% stock dividend and a 3-for-2 stock split will increase the number of shares by 50%, while neither will affect the value of the company. Therefore, the decrease in the share price should be the same in either case. (LOS 46.c)

4. The first date on which the purchaser of a stock will not receive a dividend that has been declared is the:

A. declaration date.
B. ex-dividend date.
C. holder-of-record date.

B — The chronology of a dividend payout is declaration date, ex-dividend date, holder-of-record date, and payment date. The ex-dividend date is the cutoff date for receiving the dividend: stocks purchased on or after the ex-dividend date will not receive the dividend. (LOS 46.d)

MODULE 46.2: DIVIDEND DISCOUNT MODELS

LOS 46.e

Explain the rationale for using present value models to value equity, and describe the dividend discount and free-cash-flow-to-equity models.

The dividend discount model (DDM) is based on the rationale that the intrinsic value of a stock is the present value of its future dividends.

The most general form of the model is:

\[ V_0 = \sum_{t=1}^{\infty} \frac{D_t}{(1+k_e)^t} \]

where:
$V_0$ = current stock value
$D_t$ = dividend at time $t$
$k_e$ = required rate of return on common equity

One-year holding period DDM. For a holding period of one year, the value of the stock today is the present value of any dividends during the year plus the present value of the expected price of the stock at the end of the year (its terminal value):

\[ V_0 = \frac{D_1}{(1+k_e)} + \frac{P_1}{(1+k_e)} \]

Example: One-period DDM valuation

Calculate the value of a stock that paid a $1 dividend last year, if next year's dividend will be 5% higher and the stock will sell for $13.45 at year-end. The required return is 13.2%.

Answer

The next dividend is the current dividend increased by the estimated growth rate:

$ D_1 = D_0 \times (1 + g) = \$1.00 \times 1.05 = \$1.05 $

The present values of expected future cash flows:

\[ \text{PV of dividend} = \frac{1.05}{1.132} = \$0.93 \qquad \text{PV of year-end price} = \frac{13.45}{1.132} = \$11.88 \]

Stock value = $\$0.93 + \$11.88 = \$12.81$

Multiple-year holding period DDM. With a multiple-year holding period, simply sum the present values of estimated dividends over the holding period plus the estimated terminal value. For a two-year holding period:

\[ V_0 = \frac{D_1}{(1+k_e)} + \frac{D_2}{(1+k_e)^2} + \frac{P_2}{(1+k_e)^2} \]

教授提醒

It is useful to think of the subscript $t$ on dividends $(D_t)$ and prices $(P_t)$ as the end of period $t$. For example, in the preceding equation, $P_2$ is the price at the end of Year 2 — think of it as the selling price of a share, immediately after $D_2$ is received.

中文提醒

把 $D_t$ 與 $P_t$ 的下標 $t$ 想成第 $t$ 期期末。例如 $P_2$ 即第 2 年年末的價格——可想像為剛領完 $D_2$ 後立即賣出的價格。

Example: Multiple-period DDM valuation

A stock recently paid a dividend of $1.50 which is expected to grow at 8% per year. The required rate of return is 12%. Calculate the value of this stock assuming that it will be priced at $51.00 three years from now.

Answer

Find the PV of the future dividends:

$ D_1 = 1.50(1.08) = \$1.62 \quad D_2 = 1.50(1.08)^2 = \$1.75 \quad D_3 = 1.50(1.08)^3 = \$1.89 $

\[ \text{PV of dividends} = \frac{1.62}{1.12} + \frac{1.75}{1.12^2} + \frac{1.89}{1.12^3} = \$4.19 \]

Find the PV of the future price:

\[ \frac{51.00}{1.12^3} = \$36.30 \]

Current value = $\$4.19 + \$36.30 = \$40.49$.

The most general form of the DDM uses an infinite holding period because a corporation has an indefinite life. In an infinite-period DDM, the present value of all expected future dividends is calculated and there is no explicit terminal value for the stock. In practice, a terminal value can be calculated at a time in the future after which the growth rate of dividends is expected to be constant.

Free cash flow to equity (FCFE) is often used in discounted cash flow models instead of dividends because it represents the potential amount of cash that could be paid out to common shareholders. FCFE reflects the firm's capacity to pay dividends and is also useful for firms that do not currently pay dividends.

FCFE is the cash remaining after a firm meets all of its debt obligations and provides for the capital expenditures necessary to maintain existing assets and to support assumed growth. It is often calculated as:

\[ \text{FCFE} = \text{NI} + \text{Depreciation} - \Delta\text{WC} - \text{FCInv} - \text{Debt repayments} + \text{New debt issues} \]

FCFE can also be calculated as:

\[ \text{FCFE} = \text{CFO} - \text{FCInv} + \text{Net borrowing} \]

where net borrowing = amount borrowed − amount repaid (assumed available to shareholders). Fixed capital investment must be subtracted because the firm must invest in assets to sustain itself.

Restating the general form of the DDM in terms of FCFE:

\[ V_0 = \sum_{t=1}^{\infty} \frac{\text{FCFE}_t}{(1+k_e)^t} \]

Estimating the Required Return for Equity. The capital asset pricing model (CAPM) provides an estimate of the required rate of return for security $i$ as a function of its systematic risk $\beta_i$, the risk-free rate $R_f$, and the expected return on the market $E(R_{mkt})$:

\[ k_i = R_f + \beta_i [E(R_{mkt}) - R_f] \]

There is some controversy over whether the CAPM is the best model for the required return on equity. Different analysts will likely use different inputs, so there is no single correct number.

For firms with publicly traded debt, analysts often estimate the required return on common equity by adding a risk premium to the firm's current bond yield. If the firm does not have publicly traded debt, an analyst can add a larger risk premium to a government bond yield.

中文翻譯

股利折現模型（DDM）：股票內在價值＝未來股利之現值。最一般形式：$V_0 = \sum D_t / (1+k_e)^t$。

單期 DDM：$V_0 = (D_1 + P_1) / (1+k_e)$。多期：將各期股利現值與終端價現值加總。

例 1：$D_0=\$1$、g = 5%、$P_1=\$13.45$、$k_e=13.2\%$ → $V_0 = 1.05/1.132 + 13.45/1.132 = 0.93+11.88 = \$12.81$。

例 2：$D_0=\$1.5$、g = 8%、$k_e=12\%$、$P_3=\$51$ → 股利現值 \$4.19 + $P_3$ 現值 \$36.30 = $V_0 = \$40.49$。

無限期 DDM：實務上常設定某期後股利固定成長以計算終端值。

FCFE（自由現金流給股東）＝公司滿足債務與資本支出後可分配給股東的現金，反映「支付股利之能力」，對不發股利的公司也適用。

$\text{FCFE} = \text{NI} + 折舊 - \Delta WC - \text{FCInv} -$ 還債 + 新增舉債；或 $= \text{CFO} - \text{FCInv} + 淨舉債$。代入 DDM 即得 FCFE 估值。

權益必要報酬：CAPM $k_i = R_f + \beta_i[E(R_{mkt})-R_f]$；亦可在公司債殖利率上加風險溢酬，無公開債者可在政府公債上加更大溢酬。

LOS 46.g

Calculate the intrinsic value of a non-callable, non-convertible preferred stock.

教授提醒

At the end of this reading we will address the LOS that concerns advantages and disadvantages of each category of valuation model.

中文提醒

各類估值模型的優缺點（LOS 46.f）會放在本 Reading 末尾統一說明。

Preferred stock pays a dividend that is usually fixed and usually has an indefinite maturity. When the dividend is fixed and the stream of dividends is infinite, the infinite-period dividend discount model reduces to a simple ratio:

\[ \text{Preferred stock value} = \sum_{t=1}^{\infty} \frac{D_p}{(1+k_p)^t} = \frac{D_p}{k_p} \]

Example: Preferred stock valuation

A company's $100 par preferred stock pays a $5.00 annual dividend and has a required return of 8%. Calculate the value of the preferred stock.

Answer

\[ V = \frac{D_p}{k_p} = \frac{\$5.00}{0.08} = \$62.50 \]

中文翻譯

特別股股利通常固定且永續，無限期 DDM 簡化為：

\[ V = \frac{D_p}{k_p} \]

例：面額 $100、股利 $5、必要報酬 8% → $V = 5/0.08 = \$62.50$。

LOS 46.h

Calculate and interpret the intrinsic value of an equity security based on the Gordon (constant) growth dividend discount model or a two-stage dividend discount model, as appropriate.

The Gordon growth model (or constant growth model) assumes the annual growth rate of dividends, $g_c$, is constant. Hence, $D_1 = D_0(1+g_c)$, $D_2 = D_0(1+g_c)^2$, and so on. The extended equation gives:

\[ V_0 = \frac{D_0(1+g_c)}{(1+k_e)} + \frac{D_0(1+g_c)^2}{(1+k_e)^2} + \cdots + \frac{D_0(1+g_c)^n}{(1+k_e)^n} \]

When the growth rate of dividends is constant, this simplifies to the Gordon (constant) growth model:

\[ V_0 = \frac{D_0(1+g_c)}{k_e - g_c} = \frac{D_1}{k_e - g_c} \]

教授提醒

In much of the finance literature, you will see this model referred to as the constant growth DDM, infinite period DDM, or the Gordon growth model. Whatever you call it, memorize $D_1$ over $(k - g)$. Note that our valuation model for preferred stock is the same as the constant growth model with no growth ($g = 0$).

中文提醒

本模型常被稱為固定成長 DDM、無限期 DDM 或 Gordon 成長模型，名稱不同但公式相同──牢記 $D_1/(k-g)$。特別股估值即為 $g=0$ 的特例。

The assumptions of the Gordon growth model are:

Dividends are the appropriate measure of shareholder wealth.
The constant dividend growth rate, $g_c$, and required return, $k_e$, are never expected to change.
$k_e$ must be greater than $g_c$. If not, the math will not work.

If any one of these assumptions is not met, the model is not appropriate.

Example: Gordon growth model valuation

Calculate the value of a stock that paid a $1.50 dividend last year, if dividends are expected to grow at 8% forever and the required return on equity is 12%.

Answer

$ D_1 = D_0(1+g_c) = 1.50(1.08) = \$1.62 $

\[ V_0 = \frac{D_1}{k_e - g_c} = \frac{1.62}{0.12 - 0.08} = \$40.50 \]

教授提醒

When doing stock valuation problems on the exam, watch for words like "forever," "infinitely," "indefinitely," "for the foreseeable future," etc. — these signal the Gordon growth model. Also watch for "just paid" or "recently paid" (refers to $D_0$) versus "will pay" or "is expected to pay" (refers to $D_1$).

中文提醒

考試遇到「永遠」「無限期」「可預見未來」等字 → 用 Gordon 成長模型。「剛發／最近發」＝$D_0$；「預計發／將發」＝$D_1$。

The example shows that the stock's value is determined by the relationship between $k_e$ and $g_c$:

As the difference between $k_e$ and $g_c$ widens, the value of the stock falls.
As the difference narrows, the value of the stock rises.
Small changes in the difference between $k_e$ and $g_c$ can cause large changes in the stock's value.

Because the estimated stock value is very sensitive to the denominator, an analyst should calculate several different value estimates using a range of required returns and growth rates.

An analyst can also use the Gordon growth model to determine how much of the estimated stock value is due to dividend growth. To do this, assume the growth rate is zero, calculate that value, then subtract it from the value estimated using a positive growth rate.

Example: Amount of estimated stock value due to dividend growth

Using the data from the previous example, calculate how much of the estimated stock value is due to dividend growth.

Answer

The estimated stock value with a growth rate of zero:

\[ V_0 = \frac{D}{k} = \frac{1.50}{0.12} = \$12.50 \]

Amount of value due to dividend growth: $\$40.50 - \$12.50 = \$28.00$.

Estimating the Growth Rate in Dividends

To estimate the growth rate, the analyst can use three methods:

Use the historical growth in dividends for the firm.
Use the median industry dividend growth rate.
Estimate the sustainable growth rate.

The sustainable growth rate is the rate at which equity, earnings, and dividends can continue to grow indefinitely, assuming ROE is constant, the dividend payout ratio is constant, and no new equity is sold:

\[ g = (1 - \text{dividend payout ratio}) \times \text{ROE} = b \times \text{ROE} \]

The quantity $(1 - \text{payout})$ is called the retention rate $b$ — the proportion of net income retained (not paid out as dividends).

Example: Sustainable growth rate

Green, Inc. is expected to pay dividends equal to 25% of earnings. Green's ROE is 21%. Calculate and interpret its sustainable growth rate.

Answer

$ g = (1 - 0.25) \times 21\% = 15.75\% $

With long-run economic growth typically in the single digits, it is unlikely that a firm could sustain 15.75% growth forever. The analyst should also examine industry growth and the firm's historical growth to determine whether the estimate is reasonable.

Some firms do not currently pay dividends but are expected to begin paying them at some point in the future — perhaps because the firm is in financial distress, or because the return it can earn by reinvesting cash exceeds what stockholders could earn elsewhere. For such firms, an analyst must estimate the amount and timing of the first dividend to use the Gordon growth model. Because these parameters are highly uncertain, the estimate should be checked against estimates from other models.

Example: A firm with no current dividend

A firm currently pays no dividend but is expected to pay a dividend at the end of Year 4. Year 4 earnings are expected to be $1.64, and the firm will maintain a dividend payout ratio of 50%. Assuming a constant growth rate of 5% and a required rate of return of 10%, estimate the current value of this stock.

Answer

First, find the value of the stock at the end of Year 3 ($P_3$ = PV of dividends Years 4 → ∞, calculated one period before the first dividend):

$ D_4 = (\text{payout})(E_4) = (0.5)(1.64) = \$0.82 $

\[ V_3 = \frac{D_4}{k_e - g_c} = \frac{0.82}{0.10 - 0.05} = \$16.40 \]

Second, calculate $V_0$:

\[ V_0 = \frac{16.40}{1.10^3} = \$12.32 \]

Multistage Dividend Growth Models

A firm may temporarily experience a growth rate that exceeds the required rate of return, but no firm can maintain this relationship indefinitely. A firm with extremely high growth will attract competition, and its growth rate will eventually fall. We must assume the firm will return to a more sustainable rate of growth at some point in the future.

One way to value a dividend-paying firm experiencing temporarily high growth is to add the present values of dividends expected during the high-growth period to the present value of the constant-growth value at the end of the high-growth period. This is the multistage dividend discount model:

\[ V_0 = \frac{D_1}{(1+k_e)} + \frac{D_2}{(1+k_e)^2} + \cdots + \frac{D_n}{(1+k_e)^n} + \frac{P_n}{(1+k_e)^n} \]

where:

\[ P_n = \frac{D_{n+1}}{k_e - g_c} \]

is the terminal stock value, assuming dividends at $t = n+1$ and beyond grow at a constant rate $g_c$.

Steps in using the multistage model:

Determine the discount rate, $k_e$.
Project the size and duration of the high initial dividend growth rate, $g^*$.
Estimate dividends during the high-growth period.
Estimate the constant growth rate at the end of the high-growth period, $g_c$.
Estimate the first dividend of the constant growth period.
Use this dividend to calculate the stock value at the end of the high-growth period.
Add the PVs of all dividends during the high-growth period to the PV of the value of the stock at the end of the high-growth period.

Example: Multistage growth

Consider a stock with dividends that are expected to grow at 15% per year for two years, after which they are expected to grow at 5% per year, indefinitely. The last dividend paid was $1.00 and $k_e = 11\%$. Calculate the value of this stock using the multistage growth model.

Answer

Calculate the dividends over the high-growth period:

$ D_1 = D_0(1+g^*) = 1.00(1.15) = \$1.15 $

$ D_2 = D_1(1+g^*) = 1.15(1.15) = \$1.32 $

Calculate the first dividend of the constant-growth period:

$ D_3 = D_2(1+g_c) = 1.32 \times 1.05 = \$1.386 $

Use the constant growth model to get $P_2$, a value for all the (infinite) dividends expected from time = 3 onward:

\[ P_2 = \frac{D_3}{k_e - g_c} = \frac{1.386}{0.11 - 0.05} = \$23.10 \]

Sum the present values of $D_1$, $D_2$, and $P_2$:

\[ V_0 = \frac{1.15}{1.11} + \frac{1.32 + 23.10}{1.11^2} = \$20.86 \]

中文翻譯

Gordon 成長模型（固定成長 DDM）：股利以固定速率 $g_c$ 永續成長：$V_0 = D_1 /(k_e - g_c)$，其中 $D_1 = D_0(1+g_c)$。

假設：股利為適切財富指標；$g_c$ 與 $k_e$ 永不變動；$k_e > g_c$。三者任一不成立則不適用。

例：$D_0=\$1.5$、g = 8%、$k_e = 12\%$ → $V_0 = 1.62/0.04 = \$40.50$。

因 $k_e - g_c$ 在分母，價值對其極敏感──差距愈大價值愈低、差距愈小價值愈高、微幅變動造成大差異。建議用區間估算多組情境。

分解成長貢獻：先設 g = 0 求得 $V_0=D/k=\$12.50$；剩下 $\$40.50-\$12.50=\$28$ 即為股利成長帶來的價值。

成長率估計：歷史股利成長／同業中位數／永續成長率 $g = b \times \text{ROE}$，其中 b＝盈餘保留率＝1−派息率。

例：派息率 25%、ROE 21% → g = 15.75%（須與經濟增長／同業比對是否合理）。

目前不發股利之公司：須估計首次股利的金額與時點才能套用 Gordon。例：第 4 年盈餘 $1.64、派息率 50% → $D_4=\$0.82$；$V_3 = 0.82/0.05 = \$16.40$；$V_0 = 16.40/1.10^3 = \$12.32$。

多階段 DDM：暫時超額成長 → 然後回到永續成長率。$V_0 = \sum D_t /(1+k_e)^t + P_n/(1+k_e)^n$，其中 $P_n = D_{n+1}/(k_e-g_c)$。

例：g\*=15%（2 年），之後 g=5%，$D_0=\$1$、$k_e=11\%$。$D_1=1.15、D_2=1.32、D_3=1.386$。$P_2 = 1.386/0.06 = \$23.10$。$V_0 = 1.15/1.11 + (1.32+23.10)/1.11^2 = \$20.86$。

LOS 46.i

Identify characteristics of companies for which the constant growth or a multistage dividend discount model is appropriate.

The Gordon growth model uses a single constant growth rate of dividends and is most appropriate for valuing stable and mature, non-cyclical, dividend-paying firms.

For dividend-paying firms with dividends that are expected to grow rapidly, slowly, or erratically over some period, followed by constant dividend growth, some form of the multistage growth model should be employed. The important points are that dividends must be estimable and must grow at a constant rate after some initial period, so that the constant growth model can determine the terminal value of the stock. Multistage models can be applied to:

A firm with high current growth that will drop to a stable rate in the future.
A firm temporarily losing market share and growing slowly or shrinking, as long as growth is expected to stabilize at a constant rate at some point in the future.

One variant assumes three stages: growth, transition, and maturity. A 3-stage model is suitable for firms with an initial high growth rate, followed by a lower growth rate during a transition period, followed by the constant growth rate in the long run — such as a young firm still in the high growth phase.

When a firm does not pay dividends, estimates of dividend payments years in the future are highly speculative. In this case, valuation based on FCFE is appropriate as long as growth rates of earnings can be estimated. In other cases, valuation based on price multiples may be more appropriate.

中文翻譯

Gordon 模型適用：穩定、成熟、非景氣循環、有發股利之公司。

多階段 DDM適用：股利前期非固定（高速、緩慢、起伏）、之後回到固定成長者。可用於：

當前高速成長、未來回到穩定成長者；
市占暫時下滑成長緩慢／衰退，但預期未來穩定者。

三階段模型（成長 → 轉換 → 成熟）適合仍在高成長期的年輕公司。

不發股利之公司若無法可靠估計未來股利金額／時點 → 改用 FCFE 模型（只要可估計盈餘成長率）；其他情況可用價格乘數法。

Module Quiz 46.2

1. The constant growth model requires which of the following?

A. $g < k$.
B. $g > k$.
C. $g \approx k$.

A — For the constant growth model, the constant growth rate ($g$) must be less than the required rate of return ($k$). (LOS 46.e)

2. What would an investor be willing to pay for a share of preferred stock that pays an annual $7 dividend if the required return is 7.75%?

A. $77.50.
B. $87.50.
C. $90.32.

C — The share value is $7.0 / 0.0775 = \$90.32$. (LOS 46.g)

3. An analyst estimates that a stock will pay a $2 dividend next year and that it will sell for $40 at year-end. If the required rate of return is 15%, what is the value of the stock?

A. $33.54.
B. $36.52.
C. $43.95.

B — $(\$40 + \$2) / 1.15 = \$36.52$. (LOS 46.h)

4. What is the intrinsic value of a company's stock if dividends are expected to grow at 5%, the most recent dividend was $1, and investors' required rate of return for this stock is 10%?

A. $20.00.
B. $21.00.
C. $22.05.

B — Using the constant growth model, $\$1(1.05) / (0.10 - 0.05) = \$21.00$. (LOS 46.h)

5. Assume that a stock is expected to pay dividends at the end of Year 1 and Year 2 of $1.25 and $1.56, respectively. Dividends are expected to grow at a 5% rate thereafter. Assuming that $k_e$ is 11%, the value of the stock is closest to:

A. $22.30.
B. $23.42.
C. $24.55.

C — $D_3 = D_2(1+g) = 1.56 \times 1.05 = \$1.638$. $P_2 = D_3/(k_e - g_c) = 1.638/(0.11-0.05) = \$27.30$. $V_0 = 1.25/1.11 + (1.56+27.30)/1.11^2 = \$24.55$. (LOS 46.h)

6. An analyst feels that Brown Company's earnings and dividends will grow at 25% for two years, after which growth will fall to a constant rate of 6%. If the projected discount rate is 10%, and Brown's most recently paid dividend was $1, the value of Brown's stock using the multistage dividend discount model is closest to:

A. $31.25.
B. $33.54.
C. $36.65.

C — $D_1 = 1.00(1.25) = \$1.25$; $D_2 = 1.25(1.25) = \$1.5625$; $D_3 = 1.5625 \times 1.06 = \$1.6563$. $P_2 = D_3/(k_e - g_c) = 1.6563/0.04 = \$41.41$. $V_0 = 1.25/1.10 + (1.5625 + 41.41)/1.10^2 = \$36.65$. (LOS 46.h)

7. Which of the following firms would most likely be appropriately valued using the constant growth DDM?

A. An auto manufacturer.
B. A producer of bread and snack foods.
C. A biotechnology firm in existence for two years.

B — The constant growth DDM assumes that the dividend growth rate is constant. The most likely choice here is the bread and snack producer. Auto manufacturers are more likely to be cyclical than to experience constant growth. A biotechnology firm in existence for two years is unlikely to pay a dividend, and if it does, dividend growth is unlikely to be constant. (LOS 46.i)

MODULE 46.3: RELATIVE VALUATION MEASURES

LOS 46.j

Explain the rationale for using price multiples to value equity, how the price to earnings multiple relates to fundamentals, and the use of multiples based on comparables.

Because the dividend discount model is very sensitive to its inputs, many investors rely on other methods. In a price multiple approach, an analyst compares a stock's price multiple to a benchmark value based on an index, industry group, or peer group. Common multiples: P/E, P/CF, P/S, P/B.

Price multiples are widely used and readily available. They are easily calculated and can be used in time series and cross-sectional comparisons. Many ratios have been shown to be useful for predicting stock returns, with low multiples associated with higher future returns.

A critique of price multiples is that they reflect only the past because historical (trailing) data are often used in the denominator. Many practitioners use forward (leading or prospective) values in the denominator instead. The use of projected values can result in much different ratios. An analyst should use price multiple calculations consistently across firms.

When we compare a price multiple, such as P/E, for a firm to those of other firms based on market prices, we are using price multiples based on comparables. By contrast, price multiples based on fundamentals tell us what a multiple should be based on a valuation model and therefore do not depend on the current market prices of other companies to establish value.

中文翻譯

因 DDM 對輸入極敏感，許多投資人改用價格乘數法：將個股乘數與指數／同業／同質公司比較。常用：P/E、P/CF、P/S、P/B。

優點：易得、易算、可時間序列與橫斷比較；某些乘數（低值對應高未來報酬）可預測報酬。

缺點：用歷史（trailing）分母只反映過去；改用前瞻（leading）分母可改善但結果可能差異大；同公司間應一致計算。

同類比較（comparables）：與其他公司市價乘數比；基本面乘數（justified）：依估值模型推算，不依賴他人市價。

LOS 46.k

Calculate and interpret the following multiples: price to earnings, price to an estimate of operating cash flow, price to sales, and price to book value.

Price multiples used for valuation include:

Price-earnings (P/E) ratio: stock price ÷ earnings per share. Most widely used.
Price-sales (P/S) ratio: stock price ÷ sales per share.
Price-book value (P/B) ratio: stock price ÷ book value of equity per share.
Price-cash flow (P/CF) ratio: stock price ÷ cash flow per share, where cash flow may be operating cash flow or free cash flow.

Other industry-specific multiples can be used. For example, in the cable television industry, market capitalization is compared to the number of subscribers.

Multiples Based on Fundamentals

Consider the Gordon growth model:

\[ P_0 = \frac{D_1}{k_e - g} \]

Dividing both sides by next year's projected earnings, $E_1$:

\[ \frac{P_0}{E_1} = \frac{D_1 / E_1}{k_e - g} \]

which is the leading P/E for this stock if it is valued in the market according to the constant growth DDM.

This P/E based on fundamentals is also referred to as a justified P/E — "justified" because, with correct inputs for $D_1, E_1, k_e$, and $g$, the equation provides a P/E ratio based on the present value of future cash flows. We refer to this as a leading P/E because it is based on expected earnings next period, not on actual earnings for the previous period (which would produce a lagging or trailing P/E).

This approach makes clear that the firm's P/E ratio is a function of:

$D_1 / E_1$ = expected dividend payout ratio.
$k$ = required rate of return on the stock.
$g$ = expected constant growth rate of dividends.

Example: P/E based on fundamentals

A firm has an expected dividend payout ratio of 30%, a required rate of return of 13%, and an expected dividend growth rate of 6%. Calculate the firm's fundamental (justified) leading P/E ratio.

Answer

\[ \text{Expected P/E ratio} = \frac{0.3}{0.13 - 0.06} = 4.3 \]

The justified P/E serves as a benchmark. If the firm's actual P/E (based on market price and expected earnings) was 8, the stock would be considered overvalued; if the actual P/E was 2, the stock would be undervalued. P/E ratios based on fundamentals are very sensitive to inputs (especially the $k - g$ denominator), so the analyst should use several sets of inputs to indicate a range.

Other things equal, the P/E ratio increases with: (1) a higher dividend payout rate, (2) a higher growth rate, or (3) a lower required rate of return. So if the subject firm has a higher dividend payout, higher growth, and lower required return than peers, a higher P/E may be justified.

In practice, other things are not equal. An increase in the dividend payout ratio will reduce the firm's sustainable growth rate. While higher dividends increase firm value, a lower growth rate decreases firm value. This is referred to as the dividend displacement of earnings. The net effect on firm value of increasing the dividend payout ratio is ambiguous. Firms cannot continually increase their P/Es or market values by raising the payout ratio — otherwise all firms would have 100% payout.

教授提醒

Watch for the wording "other things equal" or "other variables unchanged" in any exam questions about the effect of changing one variable.

中文提醒

注意題目是否有「其他條件不變」字眼──實務上派息率上升會壓低永續成長率，效果不能單看。

Example: Fundamental P/E ratio comparison

Holt Industries makes decorative items. Compare Holt and the industry:

	Holt Industries	Industry Average
Dividend payout ratio	25%	16%
Sales growth	7.5%	3.9%
Total debt to equity	113%	68%

Which of these factors suggest a higher fundamental P/E ratio for Holt?

Answer

The higher dividend payout ratio supports Holt having a higher P/E than the industry.
Higher sales growth suggests Holt can increase dividends faster, supporting a higher P/E.
The higher level of debt, however, indicates higher risk and a higher required return on equity, supporting a lower P/E for Holt.

Multiples Based on Comparables

Valuation based on price multiple comparables (comps) uses a price multiple to evaluate whether an asset is properly valued relative to a benchmark. Common benchmarks include the stock's historical average (a time-series comparison) or similar stocks and industry averages (a cross-sectional comparison).

The economic principle guiding this method is the law of one price: two identical assets should sell at the same price, or two comparable assets should have approximately the same multiple.

Cautions:

Comparables must really be comparable — different size, industry, or growth makes multiples non-comparable.
P/E ratios for cyclical firms are complicated due to sensitivity to economic conditions; the P/S ratio may be favored because sales are less volatile than earnings (due to operating and financial leverage).

Disadvantages of comparables: (1) a stock may appear overvalued by the comparable method but undervalued by the fundamental method, or vice versa; (2) different accounting methods produce non-comparable multiples, especially internationally; (3) cyclicality affects multiples at given points in time.

Example: Valuation using comparables

The following figures are for Renee's Bakery. All figures except the stock price are in millions.

Fiscal Year-End	20X3	20X2	20X1
Total stockholders' equity	$55.60	$54.10	$52.60
Net revenues	$77.30	$73.60	$70.80
Net income	$3.20	$1.10	$0.40
Net cash flow from operations	$17.90	$15.20	$12.20
Stock price	$11.40	$14.40	$12.05
Shares outstanding	4.476	3.994	3.823

Calculate Renee's lagging P/E, P/CF, P/S, and P/B ratios and judge whether the firm is undervalued or overvalued using these industry averages for 20X3:

Lagging Industry Ratios	20X3
Price-to-earnings	8.6
Price-to-cash flow	4.6
Price-to-sales	1.4
Price-to-book value	3.6

Answer

For example, the 20X3 P/S ratio: sales per share = $77.30/4.476 = 17.27$; P/S = $11.40/17.27 = 0.7$.

	20X3	20X2	20X1
P/E	15.9	52.3	115.2
P/CF	2.9	3.8	3.8
P/S	0.7	0.8	0.7
P/B	0.9	1.1	0.9

Compared to industry averages for 20X3, Renee's multiples are lower in all cases except P/E. Cross-sectional evidence suggests Renee's Bakery is undervalued. The high P/E warrants investigation — Renee's may have a higher P/E because earnings are depressed by high depreciation, interest expense, or taxes; the price-EBITDA ratio would provide an alternative unaffected by these expenses.

On a time-series basis, the ratios trend downward — also suggesting current undervaluation. Three-year averages: P/E 61.1, P/CF 3.5, P/S 0.7, P/B 1.0. The current P/E, P/CF, and P/B are below their 3-year averages, again suggesting Renee's may be currently undervalued (though systematic factors may also have lowered market-wide P/E).

中文翻譯

常用價格乘數：P/E、P/S、P/B、P/CF（cash flow 可定義為 CFO 或 FCF）；產業特殊：例有線電視業以「市值／訂戶數」。

基本面 P/E（justified P/E）：由 Gordon 模型 $P_0=D_1/(k-g)$ 同除 $E_1$ 得：

\[ P_0/E_1 = (D_1/E_1)/(k_e - g) \]

此為leading P/E（前瞻），用本期實際盈餘則為 trailing P/E。決定要素：派息率、必要報酬率 k、成長率 g。

例：派息 30%、k=13%、g=6% → P/E = 0.3/0.07 = 4.3。實際 8 → 高估；實際 2 → 低估。

其他不變下：派息率↑、g↑、k↓ → P/E↑。但實務中三者互動：派息↑會壓低永續成長率（股利擠出（dividend displacement of earnings）），淨效果不確定。

Holt 比較：派息 25% > 16%（→ 高 P/E）、銷售增長 7.5% > 3.9%（→ 高 P/E）、債務比 113% > 68%（→ 高風險、k↑、低 P/E）。

同類比較（comps）：依一價法則，類似資產應有類似乘數。需注意：規模／行業／成長率須相當；景氣循環公司之 P/E 較難用，可改用 P/S（營收波動較小）。

缺點：comp 與 fundamental 結論可能相反；會計差異（國際間）；景氣影響大。

Renee's Bakery 範例：除 P/E 外其他乘數皆低於業界平均 → cross-section 看是低估；高 P/E 可能因折舊／利息／稅高，可改用 P/EBITDA 檢視。時間序列也呈下降 → 亦顯示低估（但市場整體 P/E 也可能下降）。

LOS 46.l

Describe enterprise value multiples and their use in estimating equity value.

Enterprise value (EV) measures total company value. EV can be viewed as what it would cost to acquire the firm:

\[ \text{EV} = \text{MV of common and preferred stock} + \text{MV of debt} - \text{cash and short-term investments} \]

Cash and short-term investments are subtracted because an acquirer's cost would be decreased by the target's liquid assets — although the acquirer assumes the firm's debt, it also receives the firm's cash. Enterprise value is appropriate when comparing firms with significant differences in capital structure.

EBITDA (earnings before interest, taxes, depreciation, and amortization) is the most frequently used denominator for EV multiples; operating income can also be used. Because the numerator represents total company value, it should be compared to earnings of both debt and equity owners.

Advantage of using EBITDA instead of net income: EBITDA is usually positive even when earnings are not. When net income is negative, value multiples based on earnings are meaningless.
Disadvantage of EBITDA: it often includes non-cash revenues and expenses.

A potential problem with EV is that the market value of debt is often not available. The analyst can use market values of similar bonds or use book values. Book value, however, may not be a good estimate of market value if conditions have changed significantly since issuance.

Example: Calculating EV/EBITDA multiples

Daniel, Inc. is a manufacturer of small refrigerators and other appliances. The following figures are from Daniel's most recent financial statements except for the market value of long-term debt, which has been estimated from financial market data.

Stock price	$40.00
Shares outstanding	200,000
Market value of long-term debt	$600,000
Book value of long-term debt	$900,000
Book value of total debt	$2,100,000
Cash and marketable securities	$250,000
EBITDA	$1,000,000

Calculate the EV/EBITDA multiple.

Answer

Estimate market value of short-term debt: $\$2{,}100{,}000 - \$900{,}000 = \$1{,}200{,}000$ (book value of short-term debt; assume market value ≈ book value).

Market value of total debt: $\$600{,}000 + \$1{,}200{,}000 = \$1{,}800{,}000$.

Market value of equity: $\$40 \times 200{,}000 = \$8{,}000{,}000$.

Enterprise value: $\$1{,}800{,}000 + \$8{,}000{,}000 - \$250{,}000 = \$9{,}550{,}000$.

\[ \text{EV/EBITDA} = \frac{\$9{,}550{,}000}{\$1{,}000{,}000} = 9.6 \]

If competitor or industry average EV/EBITDA is above 9.6, Daniel is relatively undervalued; if below, relatively overvalued.

中文翻譯

企業價值 EV＝（普通股＋特別股市值）＋債務市值 − 現金及短期投資。可視為併購目標公司之總成本：併購方雖承擔債務但同時取得現金。EV 適用於資本結構差異大的跨公司比較。

分母最常用 EBITDA（亦可用營業利益）；EBITDA 涵蓋全部投資人（債權＋股權）報酬。

優：EBITDA 通常為正（淨利可能為負，使盈餘乘數無意義）。
缺：常含非現金收支。

限制：債務市值常不可得，可改以同類債券估或用帳面值，但帳面值未必準確。

Daniel 例：股權市值 $40×200,000 = $8M；債務市值 = $600K（長期）+ $1.2M（短期，書值）= $1.8M；EV = 8M + 1.8M − 0.25M = $9.55M；EV/EBITDA = 9.55/1 = 9.6。產業平均高於 9.6 → 相對低估；低於 → 相對高估。

LOS 46.m

Describe asset-based valuation models and their use in estimating equity value.

Our third category of valuation model is asset-based models, based on the idea that equity value = market or fair value of assets − market or fair value of liabilities. Because market values of firm assets are usually difficult to obtain, the analyst typically starts with the balance sheet. In most cases, market values are not equal to book values. Possible approaches: depreciated values, inflation-adjusted depreciated values, or estimated replacement values.

Asset-based models are problematic for firms with a large amount of intangible assets (on or off the balance sheet). Lost current owner talents and customer relationships are difficult to measure. Analysts often consider asset-based values as floor or minimum values when significant intangibles (e.g., business reputation) are involved. Supplement with a forward-looking valuation such as DCF.

Asset-based models are most reliable when:

The firm has primarily tangible short-term assets.
Assets have ready market values (e.g., financial or natural resource firms).
The firm will cease to operate and is being liquidated.

Asset-based models are often used to value private companies but may be increasingly useful for public firms as they move toward fair value reporting.

Example: Using an asset-based model for a public firm

Williams Optical is a publicly traded firm. An analyst estimates that the market value of net fixed assets is 120% of book value. Liability and short-term asset market values are assumed to equal their book values. The firm has 2,000 shares outstanding.

Cash	$10,000	Accounts payable	$5,000
Accounts receivable	$20,000	Notes payable	$30,000
Inventories	$50,000	Term loans	$45,000
Net fixed assets	$120,000	Common stockholders' equity	$120,000
Total assets	$200,000	Total liabilities and equity	$200,000

Answer

Market value of assets (adjusting fixed assets):

\[ \$10{,}000 + \$20{,}000 + \$50{,}000 + \$120{,}000(1.20) = \$224{,}000 \]

Market value of liabilities: $\$5{,}000 + \$30{,}000 + \$45{,}000 = \$80{,}000$.

Adjusted equity value: $\$224{,}000 - \$80{,}000 = \$144{,}000$.

Per share: $\$144{,}000 / 2{,}000 = \$72$.

中文翻譯

資產基礎模型：股票價值 = 資產公允價 − 負債公允價。多自資產負債表起步；公允值通常≠帳面值，可用折舊後值、通膨調整後折舊值或重置成本估計。

對無形資產多的公司不適用（人才、客戶關係難估），常作下限值並輔以 DCF。

最適合：（1）以有形短期資產為主；（2）資產有現成市價（金融、天然資源業）；（3）公司清算中。

常用於私人公司，公開公司逐漸採公允價值表達後，亦愈具參考價值。

Williams Optical 例：固定資產調整為帳面 120% → $10+20+50+120\times1.2 = \$224\text{K}$；負債 $5+30+45=\$80\text{K}$；權益＝\$144K；每股＝144K/2K = $72。

LOS 46.f

Explain advantages and disadvantages of each category of valuation model.

Discounted Cash Flow Models

Advantages:

Based on the fundamental concept of discounted present value, well grounded in finance theory.
Widely accepted in the analyst community.
FCFE model is useful for firms that currently do not pay dividends.
Gordon growth model is useful for stable, mature, noncyclical firms.
Multistage models can be used for firms with nonconstant growth.

Disadvantages:

Inputs must be estimated/forecast.
Value estimates are very sensitive to input values.
For the Gordon growth model specifically: very sensitive to the $k - g$ denominator; required return must exceed growth rate; $k_e$ and $g$ must remain constant; firm must pay dividends.

Comparable Valuation Using Price Multiples

Advantages:

Some price multiples are useful for predicting stock returns.
Widely used by analysts.
Readily available.
Can be used in time series and cross-sectional comparisons.
EV/EBITDA multiples are useful when comparing firm values independent of capital structure or when earnings are negative and P/E cannot be used.

Disadvantages:

Lagging price multiples reflect the past.
May not be comparable across firms with different size, products, or growth.
Multiples for cyclical firms are greatly affected by economic conditions; P/S may be more appropriate for cyclical firms than P/E.
A stock may appear overvalued by comparables but undervalued by fundamentals (or vice versa).
Different accounting methods produce non-comparable multiples (especially internationally).
Negative denominator in a price multiple results in a meaningless ratio. P/E is especially susceptible.
Potential problem with EV/EBITDA: market value of debt is often not available.

Price Multiple Valuations Based on Fundamentals

Advantages: based on theoretically sound valuation models; correspond to widely accepted value metrics.

Disadvantage: very sensitive to inputs (especially the $k - g$ denominator).

Asset-Based Models

Advantages:

Can provide floor values.
Most reliable when the firm has primarily tangible short-term assets, assets with ready market values, or when the firm is being liquidated.
Increasingly useful for valuing public firms that report fair values.

Disadvantages:

Market values are often difficult to obtain.
Market values are usually different from book values.
Inaccurate when a firm has a high proportion of intangible assets or future cash flows not reflected in asset values.
Assets can be difficult to value during periods of hyperinflation.

中文翻譯

DCF 模型
優：理論基礎強、業界廣為使用；FCFE 適用不發股利之公司；Gordon 適用穩定／成熟／非循環公司；多階段模型適用非固定成長者。
缺：輸入需預測；對輸入極敏感。Gordon 限制：對 $k-g$ 極敏感、k>g、k 與 g 永久不變、必須發股利。

價格乘數（comparables）
優：某些可預測報酬；廣用、易得；可作時間序列與橫斷比較；EV/EBITDA 適用資本結構差異大或淨利為負之比較。
缺：trailing 反映過去；公司大小／產品／成長不同時不可比；景氣循環影響大（P/E 尤甚，可改 P/S）；comp 與 fundamental 結論可能相反；會計差異使國際間不可比；分母為負使乘數無意義（P/E 易發生）；EV/EBITDA 之債務市值不易取得。

基本面乘數（justified）
優：理論完整、與通用估值對應。
缺：對輸入極敏感（特別是 $k-g$）。

資產基礎模型
優：可提供下限值；最適合有形短期資產為主、資產有市價或公司在清算者；公允價值報導擴大其應用。
缺：市值難取得且異於帳面值；高無形資產／未反映未來現金流時失準；惡性通膨時資產價值難估。

Module Quiz 46.3

1. Which of the following is least likely a rationale for using price multiples?

A. Price multiples are easily calculated.
B. The fundamental P/E ratio is insensitive to its inputs.
C. The use of forward values in the divisor provides an incorporation of the future.

B — The fundamental P/E ratio is sensitive to its inputs. It uses the DDM as its framework, and the denominator $k - g$ in both has a large impact on the calculated P/E or stock value. (LOS 46.j)

2. A firm has an expected dividend payout ratio of 60% and an expected future growth rate of 7%. What should the firm's fundamental price-to-earnings (P/E) ratio be if the required rate of return on stocks of this type is 15%?

A. 5.0x.
B. 7.5x.
C. 10.0x.

B — Using the earnings multiplier model, $0.6 / (0.15 - 0.07) = 7.5\text{x}$. (LOS 46.k)

3. Enterprise value is defined as the market value of equity plus:

A. the face value of debt minus cash and short-term investments.
B. the market value of debt minus cash and short-term investments.
C. cash and short-term investments minus the market value of debt.

B — Enterprise value is market value of equity plus market value of debt minus cash and short-term investments. (LOS 46.l)

4. Which of the following firms would most appropriately be valued using an asset-based model?

A. An energy exploration firm in financial distress that owns drilling rights for offshore areas.
B. A paper firm located in a country that is experiencing high inflation.
C. A software firm that invests heavily in research and development and frequently introduces new products.

A — The energy exploration firm would be most appropriately valued using an asset-based model. Its near-term cash flows are likely negative, so a forward-looking model is of limited use. Furthermore, it has valuable assets in the form of drilling rights that likely have a readily determined market value. The paper firm would not be appropriately valued using an asset-based model because high inflation makes the values of a firm's assets more difficult to estimate. An asset-based model would not be appropriate to value the software firm because the firm's value largely consists of internally developed intangible assets. (LOS 46.m)

5. Which type of valuation model is viewed as having the disadvantage of producing results that may not be comparable across firms?

A. Asset-based models.
B. Price multiple models.
C. Discounted cash flow models.

B — Results that may not be comparable across firms are considered a disadvantage of valuation models based on price multiples. (LOS 46.f)

Key Concepts

LOS 46.a — Over/Under/Fairly Valued

An asset is fairly valued if the market price equals its estimated intrinsic value, undervalued if market price is less than estimated value, and overvalued if market price is greater than estimated value.

For security valuation to be profitable, the security must be mispriced now and price must converge to intrinsic value over the investment horizon.

Securities followed by many investors are more likely to be fairly valued than securities neglected by analysts.

LOS 46.b — Categories of Valuation Models

Discounted cash flow models estimate the present value of cash distributed to shareholders (DDM) or the present value of cash available after capex/working capital (FCFE).
Multiplier models compare stock price to earnings, sales, book value, or cash flow; or compare enterprise value to sales or EBITDA.
Asset-based models define stock value as total asset value minus liabilities and preferred stock, on a per-share basis.

LOS 46.c — Dividends, Splits, Repurchases

Regular cash dividends are paid at set intervals. A special dividend is a one-time cash payment.

Stock dividends are additional shares; stock splits divide each share into multiple shares — both decrease per-share value because the total value is unchanged. Each shareholder's ownership percentage is also unchanged.

In a reverse stock split, shares decrease and per-share value increases.

A share repurchase is a purchase by the company of its outstanding shares — an alternative to cash dividends.

LOS 46.d — Dividend Payment Chronology

Declaration date: board approves the dividend.
Ex-dividend date: first day a share trades without the dividend, 1–2 business days before the holder-of-record date. On this day, share value decreases by approximately the dividend amount.
Holder-of-record date: shareholders entitled to the dividend are identified.
Payment date: dividend checks are sent or payment is transferred.

LOS 46.e — DDM and FCFE

The DDM is based on the rationale that a corporation has an indefinite life and a stock's value is the present value of its future cash dividends:

\[ V_0 = \sum_{t=1}^{\infty} \frac{D_t}{(1+k_e)^t} \]

FCFE can be used instead of dividends. FCFE is the cash remaining after a firm meets all of its debt obligations and provides for necessary capital expenditures. FCFE reflects the firm's capacity for dividends and is useful for firms that currently do not pay a dividend. By using FCFE, an analyst does not need to project the amount and timing of future dividends.

LOS 46.f — Pros & Cons of Each Model

DCF: pros — easy to calculate, well accepted, FCFE works for non-dividend firms, Gordon for stable firms, multistage for nonconstant growth. Cons — inputs must be forecast, results are highly input-sensitive (Gordon especially: $k-g$ sensitive, $k_e$>$g$, constants required, must pay dividends).

Price multiples: pros — predictive, widely used, easy/available, time-series & cross-section, EV/EBITDA for capital-structure comparisons or negative earnings. Cons — fundamental P/E sensitive, cross-firm comparability issues, cyclical issues (use P/S instead of P/E), comp vs. fundamental conflicts, accounting differences, negative denominator (esp. P/E), debt market value problem for EV.

Asset-based: pros — floor values, reliable for tangible/short-term/liquidating firms, increasingly useful with fair-value reporting. Cons — market values hard to obtain and ≠ book values, inaccurate with intangibles, hard during hyperinflation.

LOS 46.g — Preferred Stock

Preferred stock typically pays a fixed dividend and does not mature:

\[ \text{Preferred stock value} = \frac{D_p}{k_p} \]

LOS 46.h — Gordon Growth & Multistage Models

The Gordon growth model assumes the growth rate in dividends is constant:

\[ V_0 = \frac{D_1}{k_e - g_c} \]

The sustainable growth rate is the rate at which earnings and dividends can continue to grow indefinitely:

\[ g = b \times \text{ROE} \]

where $b$ = earnings retention rate = 1 − dividend payout ratio.

A firm with high growth followed by constant growth can be valued with a multistage model:

\[ V_0 = \frac{D_1}{(1+k_e)} + \frac{D_2}{(1+k_e)^2} + \cdots + \frac{D_n}{(1+k_e)^n} + \frac{P_n}{(1+k_e)^n} \]

where $P_n = D_{n+1}/(k_e - g_c)$, $g_c$ = constant growth rate, $n$ = number of supernormal-growth periods.

LOS 46.i — When to Use Which DDM

The constant growth model is most appropriate for firms with constant-rate dividend growth — stable and mature firms or noncyclical firms such as utilities and food producers in mature markets.

A 2-stage DDM is most appropriate for a firm with high current growth that will drop to a stable rate, an older firm experiencing temporary high growth, or an older firm with a market share that is decreasing but expected to stabilize.

A 3-stage model is appropriate for a young firm still in a high growth phase.

LOS 46.j — P/E Based on Fundamentals

\[ \frac{P_0}{E_1} = \frac{D_1/E_1}{k - g} \]

If the subject firm has a higher dividend payout ratio, higher growth rate, and lower required return than its peers, it may be justified in having a higher P/E.

Price multiples are widely used by analysts, easily calculated and readily available, and can be used in time-series and cross-sectional comparisons.

LOS 46.k — Price Multiples

P/E = stock price ÷ EPS.
P/S = stock price ÷ sales per share.
P/B = stock price ÷ book value per share.
P/CF = stock price ÷ cash flow per share (CFO or FCF).

LOS 46.l — Enterprise Value

Enterprise value (EV) measures total company value:

\[ \text{EV} = \text{MV of common and preferred} + \text{MV of debt} - \text{cash and short-term investments} \]

EBITDA is frequently used as the denominator because EV represents total company value and EBITDA represents earnings available to all investors.

LOS 46.m — Asset-Based Models

Asset-based models value equity as the market or fair value of assets minus liabilities. These models are most appropriate when a firm's assets are largely tangible and have fair values that can be established easily.