Transcript MBA_621_Zietlow_Chapter_4 - John Zietlow

Chapter 4
Stock And Bond Valuation
Professor John Zietlow
MBA 621
Spring 2006
Valuation Fundamentals
• Value of any financial asset is the PV of future cash flows
– Bonds: PV of promised interest & principal payments
– Stocks: PV of all future dividends
– Patents, trademarks: PV of future royalties
• Valuation is the process linking risk & return
– Output of process is asset’s expected market price
• A key input is the required [expected] return on an asset
– Defined as the return an arms-length investor would require
for an asset of equivalent risk
– Debt securities: risk-free rate plus risk premium(s)
• Required return for stocks found using CAPM or other asset
pricing model
– Beta determines risk premium: higher beta, higher reqd return
The Basic Valuation Model
• Can express price of any asset at time 0, P0, mathematically
as Equation 4.1:
P0 =
CF 1
CF 2
CF n
+
+
.
.
.
+
(1+ r )1 (1+ r )2
(1+ r )n
(Eq.4.1)
• Where:
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)
Illustration Of Simple Asset Valuation
•
•
Assume you are offered a security that promises to make four
$2,000 payments at the end of years 1-4.
If the appropriate discount rate for securities of this risk is 2%,
what price should you pay for this security (same as asking what
is its present value)?
P0 =
=
$ 2 ,000 $2,000 $2,000 $2,000
+
+
+
2
3
4
( 1.02 ) ( 1.02 ) ( 1.02 ) ( 1.02 )
$2,000
$2,000
$1,000
$1,000
+
+
+
( 1.02 ) ( 1.0404 ) ( 1.0612 ) ( 1.0824 )
 $1,960.78 + $1,922.34 + $1,884.66 + $1,847.75
 $7,615.53
•
Security would be worth $7,615.53 ($7,615.46 calculator) each.
Illustration Of Bond Valuation Using U.S.
Treasury Securities
• The simplest debt instruments to value are U.S. Treasury
securities since there is no default risk.
• Instead of r, the discount rate to use, rf, is the pure cost of
borrowing.
• Assume you are asked to value two Treasury securities,
when rf is 1.75 percent (r = 0.0175):
– A (pure discount) Treasury bill with a $1,000 face value that
matures in three months, and
– A 1.75% coupon rate Treasury note, also with a $1,000 face
value, that matures in three years.
• For the T-Bill, three months is one-quarter year (n=0.25)
• For 3-year bond, n = 3
Illustration Of Bond Valuation Using U.S.
Treasury Securities (Continued)
• The 3-month T-Bill pays no interest; return comes from
difference between purchase price and maturity value.
• 3-year T-Note makes two end-of-year $17.5 coupon
payments (CF1=CF2=$17.5), plus end-of-year 3 payment of
interest plus principal (CF3 = $1,017.5)
• Can value both with variation of Equation 4.1:
$1,000
$1,000
CF0.25
=
=
= $995.67
0.25
0.25
1.0043466
1 + r 
1.0175
$17.5
$17.5
$1,017.5
+
+
PT  Note =
2
1.0175 1.0175 1.01753
PT Bill =
= $17.2+ $16.9+ $965.9= $1,000
Bond Valuation Fundamentals
• Most U.S. corporate bonds:
– Pay interest at a fixed coupon interest rate
– Have an initial maturity of 10 to 30 years, and
– Have a par value (also called face or principal value) of
$1,000 that must be repaid at maturity.
• The Sun Company, on January 3, 2004, issues a 5 percent
coupon interest rate, 10-year bond with a $1,000 par value
– Assume annual interest payments for simplicity
– Will value later assuming semi-annual coupon payments
• Investors in Sun Company’s bond thus receive the
contractual right to:
– $50 coupon interest (C) paid at the end of each year and
– The $1,000 par value (Par) at the end of the tenth year.
Bond Valuation Fundamentals (Continued)
• Assume required return, r, also equal to 5%
• The price of Sun Company’s bond, P0, making ten (n=10)
annual coupon interest payments (C = $50), plus returning
$1,000 principal (Par) at end of year 10, determined as:
P0 =
+
$50
$50
$50
$50
$50
$50
+
+
+
+
+
(1.05 ) (1.05 )2 (1.05 )3 (1.05 )4 (1.05 )5 (1.05 )6
$50
$50
$50
$1,050
+
+
+
= $1,000.00
7
8
9
10
(1.05 ) (1.05 ) (1.05 ) (1.05 )
• So this bond would be selling at par value of $1,000
Bond Valuation Fundamentals (Continued)
• Bond’s value has two separable parts:
(1) PV of stream of annual interest payments, t=1 to t=10
(2) PV of principal repayment at end of year 10.
• Can thus also value bond as the PV of an annuity plus the
PV of a single cash flow using PVFA and PVF from tables.
P0 = C x (PVFA5%,10yr) + Par x (PVF5%,10yr)
= $50 (7.7220) + $1,000 (0.6139) = $1,000.00
• Bonds with a few cash flows can be valued with Eq 4.1; for
bonds with many cash flows, use PVFA/PVF factors,
calculator or Excel.
Bond Values If Required Return Is Not Equal
To The Coupon Rate
• Whenever the required return on a bond (r) differs from its
coupon interest rate, the bond's value will differ from its
par, or face, value.
– Will only sell at par if r = coupon rate
• When r is greater than the coupon interest rate, P0 will be
less than par value, and the bond will sell at a discount.
– For Sun, if r >5%, P0 will be less than $1,000
• When r is below the coupon interest rate, P0 will be greater
than par, and the bond will sell at a premium.
– For Sun, if r <5%, P0 will be greater than $1,000
• Exercise: Value Sun Company, 10-year, 5% coupon rate
bond if required return, r =6% and again if r = 4%.
Bond Values If Required Return Is Not Equal
To The Coupon Rate (Continued)
• Value Sun Company bond if r = 6%
P0 = $50 x (PVFA6%,10yr) + Par x (PVF6%,10yr)
= $50 (7.3601) + $1,000 (0.5584) = $926.405 approx
• Bond sells at a discount of $1,000 - $926.405 = $73.595
• Value Sun Company bond if r = 4%
P0 = C x (PVFA4%,10yr) + Par x (PVF4%,10yr)
= $50 (8.1109) + $1,000 (0.6756) = $1081.145 approx
• Bond sells at a premium of $1,081.45 - $1,000 = $81.45
• Premiums & discounts change systematically as r changes
Bond Value & Required Return, Sun Company’s 5 % Coupon Rate,
10-year, $1,000 Par, January 1, 2004 Issue Paying Annual Interest
Market Value of Bond P0 ($)
1,200
1,100
1,081
Premium
Par
1,000
Discount
926
900
800
0
1
2
3
4
5
Required Return, r (%)
6
7
8
The Dynamics Of Bond Valuation Changes
For Different Times To Maturity
• Whenever r is different from the coupon interest rate, the
time to maturity affects bond value
– even if the required return remains constant until maturity.
• The shorter is n, the less responsive is P0 to changes in r.
Assume r falls from 5% to 4%
– For n=8 years, P0 rises from $1,000 to $1,067.33, or 6.73%
– For n=3 years, P0 rises from $1,000 to $1,027.75, or 2.775%
• Same relationship if r rises from 5% to 6%, though
percentage declines in price less than increases (maximum
decline is 100%, increase unlimited)
– For n=8 years, P0 falls from $1,000 to $937.89, or 6.21%
– For n=3 years, P0 falls from $1,000 to $973.25, or 2.675%
• Even if r doesn’t change, premiums and discounts will
decline towards par as bond nears maturity.
Relation Between Time to Maturity, Required Return & Bond Value,
Sun Company’s 5%, 10-year, $1,000 Par Issue Paying Annual Interest
Market Value of Bond P0 ($)
1,100
Premium Bond, Required Return, r = 4%
1,081
1,067.3
1,050
1,027.75
Par-Value Bond, Required Return, r = 5%
1,000
M
950
Discount Bond, Required Return, r = 6%
926
900
10
9
8
7
6
5
4
3
2
Time to maturity (years)
1
0
Relationship Between Bond Prices & Yields, Bonds Of
Differing Current Maturities But Same 6.5% Coupon Rates
Bond Prices and Yields
Bond Price
$2,000
$1,500
$1,000
$500
$0
1
2-year bond
2
3
4
5
10-year bond
6
7
8
9 10 11 12 13 14 15
Yield to maturity, %
Semi-Annual Bond Interest Payments
• Most bonds pay interest semi-annually rather than annually
• Can easily modify basic valuation formula; divide both
coupon payment (C) and discount rate (r) by 2, as in Eq 4.3:
C
C
C
C
 1,000
2 
2
2
Pr ice 

 ....  2
r
r
r
r
(1  )1 (1  ) 2 (1  ) 3
(1  ) 2 n
2
2
2
2
( Eq.4.3)
• In Eq 4.3, C is the annual coupon payment, so C/2 is the
semi-annual payment.
• r is the annual required return, so r/2 is the semi-annual
discount rate.
• n is the number of years, so there are 2n semi-annual
payments.
Valuing A Bond With Semi-Annual Bond
Interest Payments
• Value a T-Bond with a par value of $1,000 that matures in
exactly 2 years and pays a 4% coupon if r = 4.4% per year.
• Insert known variables into Equation 4.3: C = $40, so C/2 =
$20, r = 0.044, so r/2 = 0.022, n = 2, so 2n = 4:
$40
$40
$40
$40
 1,000
2
2
2
P0 


 2
1
2
3
4
 0.044  0.044  0.044  0.044
1 
 1 
 1 
 1 

2
2
2
2

 
 
 


$20
$20
$20
$1,020




2
3
4
(1.022) (1.022) (1.022) (1.022)
 $19.57  $19.15  $18.74  $934.97  $992.43
The Importance And Calculation Of Yield To
Maturity
• Yield to Maturity (YTM) is the 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 (P0 = Par) will always
equal the coupon interest rate.
– When P0  Par, the YTM will differ from the coupon rate.
• YTM is the discount rate that equates the PV of a bond’s
cash flows with its price. If P0, CFs, n known, can find YTM
– Use T-Bond with n=2 years, 2n=4, C/2=$20, P0=$992.43
$20
$20
$20
$1,020
$992.43 



 r   r  2  r 3  r  4
1   1   1   1  
 2  2  2  2
• The YTM can be found by trial and error, calculator or with
spreadsheet program (Excel).
The Fisher Effect And Expected Inflation
• The relationship between nominal (observed) and real
(inflation-adjusted) interest rates and expected inflation
called the Fisher Effect (or Fisher Equation).
• Fisher said the nominal rate (r) is approximately equal to
the 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%
• The true Fisher Effect is multiplicative, rather than additive:
(1+r) = (1+a)(1+i) = (1.03)(1.02) = 1.0506; so r = 5.06%
The Term Structure Of Interest Rates
•
•
•
•
At any point in time, will be a systematic relationship between YTM
and maturity for securities of a given risk
– Usually, yields on long-term securities higher than short-term
– Generally look at risk-free Treasury debt securities
Relationship between yield and maturity called the Term Structure
of Interest Rates
– Graphical depiction called a Yield Curve
Yield curves normally upwards-sloping (long yields > short)
– Can be flat or even inverted during times of financial stress
Won’t cover term structure in depth, but three principal
“expectations” theories explain term structure:
– Pure expectations hypothesis: YC embodies prediction
– Liquidity premium theory: Investors must be paid more to invest L-T
– Preferred habitat hypothesis: Investors prefer maturity zones, so
different supply and demand characteristics in sub-sectors
Yield Curves for US Treasury Securities
16
14
May 1981
Interest Rate %
12
10
8
January 1995
August 1996
6
October 1993
4
2
1
3
5
10
15
Years to Maturity
20
30
Yield Curve, March 23, 2006
From www.cnnfn.com
%
Years to maturity
%
Changes In The Shape And Level Of Treasury
Yield Curve During Early October 1998
5.1
October 9
4.9
October 8
Yield %
4.7
October 2
4.5
4.3
4.1
3.9
3.7
1
5
10
Maturity in Years
30
Equity Valuation
•
•
•
•
As will be discussed in chapter 5, the required return on common
stock is based on its beta, derived from the CAPM
– Valuing CS is the most difficult, both practically & theoretically
– Preferred stock valuation is much easier (the easiest of all)
^
Disequilibrium: Whenever investors feel the expected return, r, is
not equal to the required return, r, prices will react:
– If exp return declines or reqd return rises, stock price will fall
– If exp return rises or reqd return declines, stock price will rise
Asset prices can change for reasons besides their own risk
– Changes in asset’s liquidity, tax status can change price
– Changes in market risk premium can change all asset values
Most dramatic change in market risk: Russian default Fall 98
– Caused required return on all risky assets to rise, price to fall
Bond Risk Premiums, February 97-November 98
600
500
400
High-yield Bond
Yields less yield
on 10-year
Treasurys in
basis points
300
200
100
0
97
98
Preferred Stock Valuation
• PS is an equity security that is expected to pay a fixed
annual dividend over its (assumed infinite) life.
• Preferred stock’s market price, P0, equals next period’s
dividend payment, Dt+1, divided by the discount rate, r,
appropriate for securities of its risk class:
D t 1
P0 =
r
• A share of PS paying a $2.30 per share annual dividend and
with a required return of 11% would thus be worth $20.91:
P0 =
Dt 1 $2.30
=
= $20.91 / share
r
0.11
• Formula can be rearranged to compute required return, if
price and dividend known:
Dt 1
$2.30
r=
=
= 0.11= 11.0%
P0 $20.91
Common Stock Valuation
• Basic formula for valuing a share of stock easy to state; P0
is equal to the present value of the expected stock price at
end of period 1, plus dividends received, as in Eq 4.4:
P0 
P1  D1
(1  r )
(Eq.4.4)
• But how to determine P1? This is the PV of expected stock
price P2, plus dividends. P2 in turn, the PV of P3 plus
dividends, and so on.
• Repeating this logic over and over, find that today’s price
equals PV of the entire dividend stream the stock will pay in
the future, as in Eq 4.5:
D2
D1
D3
D4
D5
P0 




 ....
2
3
4
5
(1  r) (1  r)
(1  r) (1  r)
(1  r)
(Eq.4.5)
The Zero Growth Valuation Model
• To value common stock, must make assumption about
growth of future dividends.
• Simplest approach, the zero growth model, assumes a
constant, non-growing dividend stream:
D1 = D2 = ... = D
• Plugging constant value D into Eq 4.5, valuation formula
reduces to simple equation for a perpetuity:
D
P0 
r
• Assume the dividend of Disco Company is expected to
remain at $1.75/share indefinitely, and the required return
on Disco’s stock is 15%. P0 is determined to be $11.67 as:
D $1.75
P0  
 $11.67
r
0.15
The Constant Growth Valuation Model
• The most widely used simple stock valuation formula, the
constant growth model, assumes dividends will grow at a
constant rate, g, that is less than the required return (g<r).
• If dividends grow at a constant rate forever, can value stock
as a growing perpetuity. Denoting next year’s dividend as D1:
D1
P0 
rg
Eq.4.6
• This is commonly called the Gordon Growth Model, after
Myron Gordon, who popularized model in the 1960s.
• The Gordon Company’s dividends have grown by 7% per
year, reaching $1.40 per share. This growth is expected to
continue, so D1=$1.40 x 1.07=$1.498. If required return is
15%:
D1
$1.498
$1.498
P0 


 $18.73
r  g 0.15  0.07
0.08
Valuing Common Stock Using The Variable
Growth Model
• Because future growth rates might change, need to
consider a variable growth rate model that allows for a
change in the dividend growth rate.
• Let g1 = the initial, higher growth rate and g2 = the lower,
subsequent growth rate, and assume a single shift in
growth rates from g1 to g2.
• Model can be generalized for two or more changes in
growth rates, but keep simple now.
• For a single change in growth rates, can use four-step
valuation procedure:
Valuing Common Stock Using The Variable
Growth Model (Continued)
• Step 1: Find the value of the dividends at the end of each
year, Dt, during the initial high-growth phase.
• Step 2: Find the PV of the dividends during this highgrowth phase, and sum the discounted cash flows.
• Step 3: Using the Gordon growth model, (a) find the value
of the stock at the end of the high-growth phase using the
next period’s dividend (after one year’s growth at g2).
– (b) Then compute PV of this price by discounting back to
time 0.
• Step 4: Determine the value of the stock today (P0) by
adding the PV of the stock price computed in step 3 to the
sum of the discounted dividend payments from step 2.
An Example Of Stock Valuation Using The
Variable Growth Model
• Estimate the current (end-of-2003) value of Morris
Industries' common stock, P0 = P2003 , using the four-step
procedure presented above, and assuming the following:
– The most recent (2003) annual dividend payment of Morris
Industries was $4 per share.
– The firm's financial manager expects that these dividends
will increase at a 8 percent annual rate, g1 , over the next
three years (2004, 2005, and 2006).
– At the end of the three years (end of 2006) the firm's mature
product line is expected to result in a slowing of the dividend
growth rate to 5 percent per year forever (noted as g2).
– The firm's required return, r , is 12 percent.
An Example Of Stock Valuation Using The
Variable Growth Model (Continued)
• Step 1: Compute the value of dividends in 2004, 2005, and
2006 as (1+g1)=1.08 times the previous year’s dividend:
Div2004= Div2003 x (1+g1) = $4 x 1.08 = $4.32
Div2005= Div2004 x (1+g1) = $4.32 x 1.08 = $4.67
Div2006= Div2005 x (1+g1) = $4.67 x 1.08 = $5.04
• Step 2: Find the PV of these three dividend payments:
PV of Div2004= Div2004  (1+r) = $ 4.32  (1.12) = $3.86
PV of Div2005= Div2005  (1+r)2 = $ 4.67  (1.12)2 = $3.72
PV of Div2006= Div2006  (1+r)3 = $ 5.04  (1.12)3 = $3.59
Sum of discounted dividends = $3.86 + $3.72 + $3.59 = $11.17
An Example Of Stock Valuation Using The
Variable Growth Model (Continued)
• Step 3: Find the value of the stock at the end of the initial
growth period (P2006) using constant growth model.
• To do this, calculate next period dividend by multiplying
D2006 by 1+g2, the lower constant growth rate:
D2007 = D2006 x (1+ g2) = $ 5.04 x (1.05) = $5.292
• Then use D2007=$5.292, g =0.05, r =0.12 in Gordon model:
D2007 = $5.292 = $5.292= $75.60
=
P2006
r - g 2 0.12- 0.05 0.07
• Next, find the PV of this stock price by discounting P2006 by
(1+r)3.
$75.60 $75.60
PV = P2006 3 =
=
= $53.81
(1  r )
(1.12)3 1.405
An Example Of Stock Valuation Using The
Variable Growth Model (Continued)
• Step 4: Finally, add the PV of the initial dividend stream
(found in Step 2) to the PV of stock price at the end of the
initial growth period (P2006):
P2003 = $11.17 + $53.81 = $64.98
• The current (end-of-year 2003) stock price is thus $64.98
per share.
Other Approaches To Common Stock
Valuation
•
•
•
Book value: simply the net assets per share available to common
stockholders after liabilities (and PS) paid in full; equals total
common equity on the balance sheet
– Assumes assets can be sold at book value, so may over-estimate
realizable value
Liquidation value: is the 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 (think of how many
shares will issue as go public, multiply that by P/E for industry to
get market value for all new shares)