Chapter 13

Stock Valuation

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Topics Addressed

─ Valuation of FIsBased on ch. 13 and a summary of the valuation model topics in various other sources (Damodaran; Berk and DeMarzo; Brealey, Myers, and Allen).

Valuation methods- Dividend discount method- Flow-to-equity method- Method of comparables (method of multiples)

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Computing the Price of Common Stock

• Valuing common stock is, in principle, no different from valuing debt securities:─ determine the cash flows in each period ─ discount them to the present

• We will review three different methods for valuing stock, each with its advantages and drawbacks.

• The choice of the methods discussed in this course is tailored to be relatively more suitable for financial institutions than non-financial firms in general.

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The Dividend Discount Model

• The discount rate is the cost of equity financing of the firm.

• It is obtained from the CAPM regression based on past stock returns, e.g., daily returns during the previous six months or monthly returns from the last five years. First, we run a regression to estimate the firm’s beta using past returns,

𝑟𝑟 − 𝑟𝑟𝑓𝑓 = 𝛼𝛼 + 𝛽𝛽 𝑅𝑅𝑚𝑚 − 𝑟𝑟𝑓𝑓 + 𝜀𝜀• Then we put the alpha and beta estimates in the

CAPM equation with the current market and risk-free returns, and calculate 𝑟𝑟𝑒𝑒.

....)1()1(1 32321

0 ++++++=er

D

erD

erD

P

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The Dividend Discount Model:Constant Growth

• Recall:• A growing perpetuity is an investment offering a growing

stream of cash flows in perpetuity. • Gordon’s growth model (a special case of the Dividend

Discount Model) assumes a constant discount rate and a constant dividend growth rate g.

gerD

ergD

ergD

erD

P −=+++

+++

++= 11110 ....)1(

)1()1(

)1(1 3

2

2

where P0 is the stock price now (at EOY0), and 𝐷𝐷𝑖𝑖 are the dividends per share in year i, 𝐷𝐷𝑖𝑖 = 𝐷𝐷𝑖𝑖−1 ∗ (1 + 𝑔𝑔)

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Example. Constant dividend growth

• Citigroup is in stable growth and its current earnings are $13.993 billion will grow 5% in perpetuity. The dividend payout ratio will be 56.4%. The beta for the stock based on its past returns is 1.00. With these inputs, the risk free rate of 5.1% and market risk premium of 4%, value Citigroup’s equity.

• Solution. • Cost of equity for Citigroup = 5.1% + 1.00 (4%) = 9.1%• The dividend today 𝐷𝐷0 = payout ratio * earnings =

0.564*13.993 =$7.892 billion• Value of Citigroup’s equity

𝐸𝐸 = 𝐷𝐷1𝑟𝑟−𝑔𝑔

= $7.892*1.05 / (0.091 - 0.05) = $202.113 billion

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Deriving the Growing Annuity Formula

• Start with $1 cash flow at the end of year 1, growing at rate g. What is the price of a 3-year growing annuity?

Year 3 breaks the infinite timeline Line 1 into two parts:

Line 2: Infinite stream starting from Y4, whose PV in Y3 is

PV3 = (1+g)3/(r-g), brought to Y0 by further discounting by (1+r)3.

Line 3: PV of CFs from year 1 to year 3.

at Year 0

(1+g)3

(1+g)3

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Example. Changing dividend growth rates

• A bank’s current dividend is $3.50. It will grow 7% per year in years 1-4. Then the bank will increase it to $7.51 in year 5, after which it will grow at 2.3% per year in perpetuity. The cost of capital is 11.5%. What should the current stock price be?

Solution.• We forecast a firm’s dividends to be 𝐷𝐷𝑖𝑖 = 𝐷𝐷𝑖𝑖−1 ∗ (1 + 𝑔𝑔) in years

1-4: D1=3.50*1.07=3.745, D2=$4.007, D3=$4.288, D4=$4.588.

• The present value of the perpetuity that starts in year 5 is 𝐷𝐷𝐷𝐷𝑉𝑉5

0.115−𝑔𝑔in year 4 dollars. We therefore discount it to EOY0 by

dividing it by 1 + 𝑟𝑟 𝟒𝟒.

456511511

02301150517

11515884

11512884

11510074

11517453

44321 .$..-.

...

..

..

..

=

×++++=

+

×++

++

++

++

= 45

44

33

221

0 )1(1

)1()1()1(1 rr-gDIV

rDIV

rDIV

rDIV

rDIVP

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Errors in Valuation

Although the pricing models are useful, market participants frequently encounter problems in using them. Any of these can have a significant impact on price in the Gordon model.• Problems with Estimating Growth• Problems with Estimating Risk• Problems with Forecasting Dividends

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Errors in Valuation:Dividend growth rates

Table 13.1 Stock Prices for a Security with D0 = $2.00, re = 15%, and Constant Growth Rates as Listed

Growth (%) Stock Price ($)

1 14.43

3 17.17

5 21.00

10 44.00

11 55.50

12 74.67

13 113.00

14 228.00

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Errors in Valuation:Required returns

Table 13.2 Stock Prices for a Security with D0 = $2.00, g = 5%, and Required Returns as Listed

Required Return (%) Stock Price ($)

10 42.00

11 35.00

12 30.00

13 26.25

14 23.33

15 21.00

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Case: The 2007–2009 Financial Crisis and the Stock Market

• The financial crisis, which started in August 2007, was the start of one of the worst bear markets.

• The crisis lowered “g” in the Gordon Growth model - driving down prices and also impacted re - higher uncertainty increases this value, again lowering prices.

• Similarly, the market was overoptimistic about the dot com economy in the 1990s. Dow Jones index:

883,1208.092.

6.154)( 2000 March =−

=−

=gr

DivindexPV

589,8074.092.6.154)( 2002October =

−=

−=

grDivindexPV

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Errors in valuation. The Dividend discount model for Financial Firms

• Constant growth rate– It is a relatively good assumption for large firms, firms

having intense competition from other firms, with few changes in government regulation.

• Cost of equity– The CAPM betas are stable over time for financial firms.

Leverage itself and the effects of variation in leverage on the cost of equity are difficult to estimate for financial firms and are, thus, often ignored, unlike for non-financial firms.

• Forecasting dividends– Financial firms pay a much higher % of earnings as

dividends than non-financial firms. The payout ratio has little variation over time. This speaks if favor of using a dividend discount model for FIs

• Dividends are high because FIs need to invest little in capital expenditures, are relatively more mature, and historically developed a reputation of reliable payers of high dividends, which attracted investors that like dividend payments.

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The Flow-to-Equity Method

• Note that firms do not have to pay dividends. A method that accounts for all (incl., undistributed) earnings is the Flow-to-Equity method.– A valuation method that calculates the free cash flow

available to equity holders after taking into account all payments to and from debt holders.

– These cash flows to equity holders are then discounted using the equity cost of capital.

– The result is what the market value of Equity should be based on our forecasts of future cash flows.

– Finally, the stock price is E divided by the number of shares outstanding, P=E/n.

...)1()1( 2

21

1 ++

++

=E

e

E

e

rFCF

rFCFE

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Calculating the Free Cash Flow to Equity

• Free Cash Flow to Equity (FCFE)– The cash flow that remains after adjusting for interest

payments, as well as debt issuance and repayments.

• The first step in the FTE method is to determine the project’s free cash flow to equity.

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Using the FTE method to value an investment project: Example

• Assume Avco is considering introducing a new line of packaging, the RFX Series.– Avco expects the technology used in these products to

become obsolete after four years. The marketing group expects annual sales of $60 million per year over the next four years for this product line.

– Manufacturing costs and operating expenses are expected to be $25 million and $9 million, respectively, per year.

– Developing the product will require upfront R&D and marketing expenses of $6.67 million, together with a $24 million investment in equipment.

• The equipment will be obsolete in four years and will be depreciated via the straight-line method over that period.

– Avco expects no working capital requirements for the project.

– Avco borrows $30.62 million initially and pays off this loan over four years leaving the balances: $23.71, $16.32, $8.43, and 0 at the end of years 1, 2, 3, 4, respectively.

– Avco pays a corporate tax rate of 40%. The cost of equity capital of the firm is 10%, and the interest rate is 6%.

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Expected Free Cash Flow to Equity from Avco’s RFX Project

ngNetBorrowikCapitalIncreaseWrCAPEXonDepreciatiNetIncomeFCF E +−−+=

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Calculating the Free Cash Flow to Equity (cont'd)

Notes:• Depreciation is not a cash expense (only used to calculate the

tax) and is, therefore, added back to the net income.• Expenses increasing the working capital (e.g., buying

inventories) reduce the FCF available for shareholders; selling inventories increases the FCF. Thus, we subtract the change in Net Wrk capital.

• The proceeds from the firm’s net borrowing activity are added in.– These proceeds are positive when the firm issues debt

(intuitively, because the cash available to shareholders is increased) and are negative when the firm reduces its debt by repaying the principal.

𝑁𝑁𝑁𝑁𝑁𝑁 𝐵𝐵𝐵𝐵𝑟𝑟𝑟𝑟𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝑔𝑔𝑡𝑡 = 𝐷𝐷𝑡𝑡 − 𝐷𝐷𝑡𝑡−1Also note that debt repayments take place at the end of the year; therefore, interest each year accrues based on the debt balance at the beginning of the year.

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Valuing Equity Cash Flows

• Because the FCFE represent payments to equity holders, they should be discounted at the project’s equity cost of capital.– The alternatives are the assets cost of capital that includes the cost of

debt financing.

2 3 4

9.98 9.76 9.52 9.27( ) 2.62 $33.25 million1.10 1.10 1.10 1.10

NPV FCFE = + + + + =

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Summary of the Flow-to-Equity Method

1. Determine the free cash flow to equity of the investment.

2. Determine the equity cost of capital.3. Compute the equity value by discounting the free

cash flow to equity using the equity cost of capital.

• Other popular valuation methods (the Weighted Average Cost of Capital (WACC), the Adjusted Present Value (APV) methods) produce the value of assets rather than equity. Then, one can calculate equity: E=A-D. They are less appropriate for financial firms than the FTE method because debt for financial firms is what raw materials are for manufacturing firms, rather than simply a liability. Therefore, debt is hard to define for fin. firms, and its value is hard to measure. Instead of calculating both A and D, one can calculate E directly using the FTE method.

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Method of Comparables (Multiples)

• The method of comparables involves using a price multiple to value an asset or a firm in relation to a benchmark, such the value of another company, peer firms, or the industry average.

• The price–earnings (P/E) ratio is the most common measure for the multiple. Others: P/Sales, P/FCF, P/EBITDA.

• Example. You are asked to value a new wireless phone/internet provider firm. You take an average P/E ratio of the firms in the wireless industry in Canada, which is, say, equal to 10. This is the multiple. By the method of comparables, this multiple should approximate the P/E ratio of the new firm being valued. Given the firm’s earnings of $2 million, an approximation for the value of the firm’s equity is Earnings* Multiple = $2 million * 10 = $20 million. Or, if you have the firm’s EPS, say, at $0.2/share, then the stock price should be $0.2*10=$2/share.

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Method of Comparables (cont’d)

• The economic rationale underlying the method of comparables is solid—the law of one price, that two identical assets should sell at the same price (assuming the market is efficient, i.e., nothing is under- or over-priced).

• Major problems of the method: – everything hinges on the correctness of one number (an

accounting ratio)– subjective estimates of the ratio– accounting practices greatly influence valuation results.