Goal of the Lecture:

Understand how to properly value a stock or bond.

Stock and Bond Stock and Bond ValuationValuationHow to Get a “k” to Discount Cash Flows - Two Methods

I. Required Return on a Stock (k) - CAPM (Capital

Asset Pricing Model) or Other Securities

k = rf + B(rm - rf)

where rf = Risk-Free Rate

B = Beta (Risk Measure)

rm = Expected Market Return

Example: k = .05 + 2(.12 - .05) = .19

II. Risk Premium Approach

kb = Rb = rreal + rrisk + E(I)

= Productivity Growth + Risk Premium +

Expected Inflation

= 2 - 4% + 0 - 10% + 2 - 6% (Estimates)

Stock and Bond Stock and Bond ValuationValuation

III. Bond Valuation

Bond = Annuity Plus Single Par Payment

(often Semi-Annual Payments)

a. Par Value -face value, maturity value -

usually $1000

b. Coupon Interest Rate - stated as a % of Par -

10% coupon on $1000 par => coupon = $100.

c. Maturity - length of time until Par value is paid off

Bond ValuationBond Valuation““A Bond is an Annuity Plus a Single Face Value Payment”A Bond is an Annuity Plus a Single Face Value Payment”

Annual CouponAnnual Coupon

I. Bond Valuation

General Formula

B0 = I + M

= [= PV(k, n, PMT, FV)]

B0= Bond Price

I = Interest (coupon) Payment

M = Par Value

II. Example: Suppose a bond offers a 10% coupon, on $1000 par, for 3 years, and the expected

inflation rate is 2%, the real rate is 3% and the bond’s risk is 1%. What is its price?

B0 = $100 + $1000

= $100(2.673) + $1000(.84)

= [= PV(0.06, 3, -100, -1000)]

= $1107

1

1( ) k n1 1

1k k k n

( )

3)06.01(

1

3)06.01(06.0

1

06.0

1

QUESTION: If the company only agrees to pay $1000 at maturity, won’t those who buy this bond lose $107 at maturity?

QUESTION: Would you buy this bond? Why? - greater coupon than par bonds.

A par bond would cost $1000 but only pays a $60 coupon. The present value of the difference in coupons (100 - 60)(2.673) = 107 which is the difference in price between this bond and a par bond.

Alternatively, a bond that offered a 2% coupon when rates are 6% will have a price of B0 = [= PV(0.06, 3, -20, -1000)]= 893 or $107 less than the par bond.

Bond ValuationBond Valuation““A Bond is an Annuity Plus a Single Face Value Payment”A Bond is an Annuity Plus a Single Face Value Payment”

Semi-Annual CouponSemi-Annual Coupon

I. Adjustments For Semi-Annual Coupon Bonds

a. n = the number of semi-annual payments

(maturity x 2)

b. k = one-half the bond’s annual yield

c. I = one-half the bond’s annual coupon

II. Example:

Suppose a bond pays 10% coupon, semi-annually,

has 10 years until maturity and has a required

return (or Yield to Maturity) of 8%. What is its price?

B0 = $50 + $1000

= $50(13.59) + $1000(.456)

= [= PV(0.04, 20, -50, -1000)]

= $1135.5

QUESTION: Consider two identical bonds except that one pays an annual coupon and the other a semi-annual coupon. Which should have the higher price?

ANSWER: The semi-annual bond.

20)04.01(04.0

1

04.0

120)04.01(

1

YIELD TO MATURITY - The return one can expect on an investment in a bond if the bond pays all its coupons and par and yields do not change after you purchase the bond.

PROBLEM: Suppose you observe a bond in the market with a price of $803 that pays a coupon of 10% till maturity in 5 years. What is its implied yield to maturity?

[=Rate(5, 100, -803, 1000)] = 16%

Yield to MaturityYield to Maturity

Stock ValuationStock Valuation““Valuation is Based Upon Expected Dividend Flow and theValuation is Based Upon Expected Dividend Flow and the

Future Expected Market Value of the Stock”Future Expected Market Value of the Stock”

I. Common Stock Valuation

General Formula

P0 = E(D1)/(1+ke) + E(D2)/(1+ke)2 + ... +

E(Dn)/(1+ke)n + E(Pn)/(1+ke)n

where E means expectation, Dt means dividend at time t,

P means stock price, and ke is the cost of equity.

II. Example:

Suppose a firm pays $4 in Dividends, which will increase by $1 in each of the next 3 years and we expect the price to be $30 at the end of the 3rd

year. Assume stock beta = 1, the risk free rate is 10%, and the expected market return is 15%. What is the stock price?

Ke = .10 + 1(.15 - .10)

P0 = $4/(1.15) + $5/(1.15)2 + $6/(1.15)3 +$30/(1.15)3

= $4(.87) + $5(.756) + $6(.658) + $30(.658)

= [=NPV(0.15, 4, 5, 36)]

= $30.94

Stock Valuation - Constant GrowthStock Valuation - Constant Growth““Valuation is Based Upon Expected Dividends That Grow at Valuation is Based Upon Expected Dividends That Grow at a Constant Rate For Ever”a Constant Rate For Ever”

Constant Growth Discounted Cash Flow Model, or DCF.

P0 = =

where D0 = present dividend paid at time 0D1 = dividend expected at time 1g = constant future growth in dividendsk = required return (discount rate)

Note: Works only for k > g and dividend paying firm

D g

k g0 1( )

D

k g1

5

10 05

5

05100

(. . ) .

5 11

15 10

( . )

(. . )

PROBLEM: Suppose a company will pay a dividend of $5 in one year, has a required return of 10% and dividends grow 5% per year. What is the stock price?

P =

Mixture of Dividends

PROBLEM: Suppose a firm will pay a dividend of $5 per year for 5 years and then increases the dividend by 10% per year thereafter. The firm has k = 15%. What is its price now?

P = 5 +

= 5(3.352) + 110(.497)

= [= PV(0.15, 5, -5, 0)] + [= PV(0.15, 5, 0, -110)] = 71.43

5)15.01(15.0

1

15.0

15)15.01(

1

PROBLEM: Suppose we know the price of the stock in the market is $80, and it pays a dividend of $3 that will grow by 10% per year. What is the return the market requires on the stock?

k =

=

= .14

This version can also be used to assess the performance of the firm. For example, if we know k = 0.14 from the CAPM, then a firm should work to produce a combination of dividend yield and growth to produce at least 14%.

Stock Valuation - Implied Required Returns Stock Valuation - Implied Required Returns

and Growth Ratesand Growth Rates

““Just Manipulate the Constant Growth Formula”Just Manipulate the Constant Growth Formula”

gP

gD

0

0 )1(

3 11

8010

( . ).

D

P1

5

50

PROBLEM: If the market price of a stock is $50, its required return is 15 percent and next year’s dividend is expected to be $5, by what percent must the market expect the company’s dividends to grow.

g = k -

so g = .15 - = .05

PRICE-EARNINGS RATIOSPRICE-EARNINGS RATIOS““PE’s Are Commonly Used to Compare Stocks”PE’s Are Commonly Used to Compare Stocks”

PE Ratio - This is the number of dollars investors are willing to pay for each dollar of a company’s earnings.

You can use the growth model to see why stocks have different price-earnings ratios

P =

=

=> =

where E = earnings per shared = dividend payout ratio =

Clearly, a larger growth rate and payout ratio and a smaller discount rate (k) makes for a larger price earnings ratio.

D

k g1

( )

dE

k g1

( )

P

E

d

k g( )

D

E

PRICING STOCKS USING P/E & NORMAL EARNINGS

• GET INDUSTRY AVERAGE P/E OR SIMILAR COMPANY P/ E

• ESTIMATE NORMAL EARNINGS PER SHARE

- VARIOUS WAYS - ADJUST FOR ACCOUNTING

GIMMICKS

• MULTIPLY P/E TIMES NORMAL EARNINGS

• eg. suppose normal earnings = $3/share

and P/ E of industry = 10

• Price estimate = $3 * 10 = $30

7

6

5

4

3

2

1

0

Earnings Per Share

Year

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95

For price earnings ratios, use secular not cyclical earnings as a measure of normal earnings.

Hewlett-Packard Earnings Per Share

cyclical

secular trend

PRICING STOCKS USING P/S & NORMAL SALES

• WHEN STOCKS HAVE NO EARNINGS, SAY FOR A NEW COMPANY OR A COMPANY IN A NEW INDUSTRY, YOU CANNOT USE THE P/E METHOD, INSTEAD USE PRICE TO SALES (P/S) IN THE WAY THAT YOU USE PRICE EARNINGS RATIOS.

– Get industry average or comparable firm P/S ratio

– Get firm’s sales per share - normalized

– Multiply to get price estimate

• OR USE PRICE TO BOOK (P/B)

• OR USE PRICE TO EBITDA (P/EBITDA)

• OR USE PRICE TO R&D (P/R&D)

Enterprise Value

Private equity funds and investment banks who advise companies on mergers and acquisitions often compute Enterprise Value. They use it to decide how much to pay for a whole company, including paying off minority interest holders to get full ownership of consolidated subsidiaries but liquidating interests in unconsolidated associate companies.

Enterprise Value = Market Value of Common + Market Value of Preferred Equity+ Market Value of Debt + Minority Interest – Market Value of Associate Companies - Cash and Cash Equivalents.

(minority interest is added to account for the value of fully consolidated subsidiaries for which the firm owns less than 100%. The reverse occurs for associate companies consolidated into other firms)

(other possible adjustments include unfunded pension liabilities, executive stock options, other claims against assets)

Enterprise Value is an estimate of the market value of a firm’s operating assets.

Question: Is it possible for a firm to have a negative enterprise value? What would that mean for a buyout? (see biotechs around 2000)