Stock & Bond Valuation

Professor XXXXXCourse Name / Number

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Valuation FundamentalsPresent Value of Future Cash Flows

Link Risk & Return

Expected Return on Assets

Valuation

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The Basic Valuation Model

• P0 = Price of asset at time 0 (today)• CFt = cash flow expected at time t• r = discount rate (reflecting asset’s risk)• n = number of discounting periods (usually

years)

This model can express the price of any asset at t = 0 mathematically.

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Valuation FundamentalsBond Example

Using the P0 equation, the bond would sell at a par value of $1,000.

• Company issues a 5% coupon interest rate, 10-year bond with a $1,000 par value on 01/30/05– Assume annual interest payments

• Investors in company’s bond receive the contractual rights– $50 coupon interest paid at the end of

each year – $1,000 par value at the end of the 10th

year

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P0 < par value

P0 > par value

BondsPremiums & DiscountsWhat happens to bond values if required return is not equal to

the coupon rate?

The bond's value will differ from its par value

R > Coupon Interest Rate

R < Coupon Interest Rate

DISCOUNT =

PREMIUM =

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BondsTime to Maturity

What does this tell you about the relationship between bond prices & yields for bonds with the equal coupon rates, but different maturities?

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BondsSemi-Annual Interest Payments

An example....

Value a T-Bond

Par value = $1,000

Maturity = 2 years

Coupon pay = 4%

r = 4.4% per year

= $992.43

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Yield to Maturity (YTM)

Rate of return investors earn if they buy the bond at P0 and hold it until maturity.

The YTM on a bond selling at par will always equal the coupon interest rate.

YTM is the discount rate that equates the PV of a bond’s cash flows with its price.

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The Fisher Effect And Expected Inflation• The relationship between nominal and real (inflation-

adjusted) interest rates and expected inflation called the Fisher Effect (or Fisher Equation).

• Nominal rate (r) is approximately equal to real rate of interest (a) plus a premium for expected inflation (i).– If real rate equals 3% (a = 0.03) and expected

inflation equals 2% (i = 0.02):r a + i 0.03 + 0.02 0.05 5%

• True Fisher Effect multiplicative, rather than additive:(1+r) = (1+a)(1+i) = (1.03)(1.02) = 1.0506; so r = 5.06%

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Term Structure of Interest Rates• Relationship between yield and maturity is

called the Term Structure of Interest Rates– Graphical depiction is called a Yield Curve– Usually, yields on long-term securities are higher

than on short-term securities– Generally look at risk-free Treasury debt securities

• Yield curves normally upwards-sloping – Long yields > short yields– Can be flat or even inverted during times of

financial stress

What to you think a Yield Curve would look like graphically?

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Yield CurvesU.S. Treasury Securities

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4

6

8

10

12

14

16

5 10 15 20 30

Years to Maturity

Inte

rest

Rat

e %

August 1996October 1993

May 1981

January 1995

1 3

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Valuation FundamentalsPreferred Stock

Preferred stock is an equity security that is expected to pay a fixed annual dividend for its life

rD = P 1t

0 • P0 = Preferred stock’s market

price• Dt+1 = next period’s dividend

payment• r = discount rate

An example: A share of preferred stock pays $2.3 per share annual dividend and with a required return

of 11% share= =r

D = P t0 /90.20$

11.03.2$1

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Valuation FundamentalsCommon Stock

• P0 = Present value of the expected stock price at the end of period 1

• D1 = Dividends received • r = discount rate

Value of a

Share of

Common Stock

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• But how is P1 determined? – This is the PV of expected stock price P2, plus

dividends– P2 is the PV of P3 plus dividends, etc...

• Repeating this logic over and over, you find that today’s price equals PV of the entire dividend stream the stock will pay in the future

Valuation FundamentalsCommon Stock

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Zero Growth Valuation Model• To value common stock, you must make

assumptions about the growth of future dividends

• Zero growth model assumes a constant, non-growing dividend stream:

D1 = D2 = ... = D

• Plugging constant value D into the common stock valuation formula reduces to simple equation for a perpetuity:

rDP 0

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Constant Growth Valuation Model• Assumes dividends will grow at a constant

rate (g) that is less than the required return (r)

• If dividends grow at a constant rate forever, you can value stock as a growing perpetuity, denoting next year’s dividend as D1:

This is commonly called the Gordon Growth Model.

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Variable Growth ModelExample• Estimate the current value of Morris

Industries' common stock, P0 = P2005 • Assume

– The most recent annual dividend payment of Morris Industries was $4 per share

– The firm's financial manager expects that these dividends will increase at an 8% annual rate over the next 3 years

– At the end of the 3 years the firm's mature product line is expected to result in a slowing of the dividend growth rate to 5% per year forever

– The firm's required return, r , is 12%

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Variable Growth ModelValuation Steps #1 & #2• Compute the value of dividends in 2006, 2007, and

2008 as (1+g1)=1.08 times the previous year’s dividendDiv2006= Div2005 x (1+g1) = $4 x 1.08 = $4.32Div2007= Div2006 x (1+g1) = $4.32 x 1.08 = $4.67

Div2008= Div2007 x (1+g1) = $4.67 x 1.08 = $5.04

• Find the PV of these three dividend payments:PV of Div2006= Div2006 (1+r) = $ 4.32 (1.12) = $3.86

PV of Div2007= Div2007 (1+r)2 = $ 4.67 (1.12)2 = $3.72

PV of Div2008= Div2008 (1+r)3 = $ 5.04 (1.12)3 = $3.59

Sum of discounted dividends = $3.86 + $3.72 + $3.59 = $11.17

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• Find the value of the stock at the end of the initial growth period using the constant growth model

• Calculate next period dividend by multiplying D2008 by 1+g2, the lower constant growth rate: D2009 = D2008 x (1+ g2) = $ 5.04 x (1.05) = $5.292

• Then use D2009=$5.292, g =0.05, r =0.12 in Gordon model:

60.7507292.5

2292.5 $ =

0.$ =

0.05 -0.1$ =

g -rD = P

2

92008200

Variable Growth ModelValuation Step #3

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• Find the present value of this stock price by discounting P2008 by (1+r)3

81.53405.1

60.75$)12.1(

60.75$)1( 33

$ = = = r

P =PV 0820

Variable Growth ModelValuation Step #3

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• Add the PV of the initial dividend stream (Step #2) to the PV of stock price at the end of the initial growth period (P2008):

P2005 = $11.17 + $53.81 = $64.98

Variable Growth ModelValuation Step #4

Current (year 2005) stock price

Remember: because future growth rates might change, the variable growth model

allows for changes in the dividend growth rate.

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Common Stock ValuationOther Options• Book value

– Net assets per share available to common stockholders after liabilities are paid in full

• Liquidation value– Actual net amount per share likely to be realized upon

liquidation & payment of liabilities– More realistic than book value, but doesn’t consider firm’s

value as a going concern

• Price / Earnings (P / E) multiples– Reflects the amount investors will pay for each dollar of

earnings per share– P / E multiples differ between & within industries– Especially helpful for privately-held firms

Questions?