Transcript FINANCIAL ADMINISTRATION OF THE FIRM FIN 5043--930

Chapter 4
Stock And Bond Valuation
Professor Del Hawley
Finance 634
Fall 2003
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?
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 each.
Illustration Of Simple Asset Valuation
With most debt securities, the cash flows are smooth (equal
amounts at equal time intervals, except possibly the last cash
flow) and so it can be treated like an annuity or an annuity
with a balloon.
PV =
FV =
n =
i =
PMT =
FV = 0
Price or value of the security
Maturity or par value (usually $1000)
Number of remaining interest payments
Market required return per payment period
Periodic interest payment (Coupon Rate x 1000)
n=4
I=2
PMT = 2000
PV = $7,615.53
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, the discount rate to use (rf) is the pure cost of
borrowing.
rf = Real Rate of Interest + Inflation Premium
The Fisher Effect And Expected Inflation
• The relationship between nominal (observed) and real
(inflation-adjusted) interest rates and expected inflation is
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 the real rate equals 3% (a = 0.03) and expected annual
inflation rate equals 2% (i = 0.02), then:
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%
Illustration Of Bond Valuation
Using U.S. Treasury Securities
Assume you are asked to value two Treasury securities,
when rf is 1.75 percent:
– 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.
Illustration Of Bond Valuation Using
U.S. Treasury Securities (Continued)
• The 3-month T-Bill pays no interest; return comes from the
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
CF 0.25
=
=
= $995.68
PT  Bill =
0.25
0.25
1.00434 
1 + r 
1.0175 
$17.5
$17.5
$1,017.5
+
+
PT  Note =
2
1.0175  1.0175  1.0175 3
= $17.2 + $16.9 + $965.9 = $1,000
Illustration Of Bond Valuation Using
U.S. Treasury Securities (Continued)
For the 3-month Treasury Bill:
PV
-995.68
FV
1000
N
1/4
I
1.75
PMT
0
Illustration Of Bond Valuation Using
U.S. Treasury Securities (Continued)
For the 3-year Treasury Note:
PV
-1000
FV
1000
N
3
I
1.75
PMT
17.5
Illustration Of Bond Valuation Using
U.S. Treasury Securities (Continued)
For the 3-year Treasury Note:
FV
n
i
PMT
PV
1000
3
1.75
17.5
($1,000.00)
For Excel: =PV(rate,nper,PMT,FV,Type)
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.
Bond Valuation Fundamentals
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, is 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 )
Bond Valuation Fundamentals (Continued)
PV
-1000
FV
1000
N
10
I
5
PMT
50
For Excel: =PV(rate,nper,PMT,FV,Type)
Bond Valuation Fundamentals (Continued)
A 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.
Therefore, we can also value a bond as the PV of an annuity
plus the PV of a single cash flow.
PV
-386.09
-613.91
FV
0
1000
N
10
10
I
5
5
PMT
50
0
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
• 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)
PV
-926.405
-1081.45
FV
1000
1000
N
10
10
I
6
4
PMT
50
50
At r = 6%, the bond sells at a discount of $1,000 - $926.405 = $73.595
At r = 4%, the 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%
The Dynamics Of Bond Valuation Changes
For Different Times To Maturity
• The same relationship holds if r rises from 5% to 6%,
(though the percentage decline in price is less than the
percentage increase was in the previous example).
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 the bond’s par value as the 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
926
Discount Bond, Required Return,rr = 6%
900
10
9
8
7
6
5
4
3
2
Time to maturity (years)
1
0
Relationship Between Bond Prices & Yields, Bonds Of
Differing Original Maturities But Same Coupon Rates
Bond Prices and Yields
$1,600
10-year bond
$1,400
Bond Price
$1,200
2-year bond
$1,000
$800
$600
$400
$200
$0
1
2
3
4
5
6
7
8
9
10
Yield to maturity, %
11
12
13
14
15
Semi-Annual Bond Interest Payments
• Most bonds pay interest semi-annually rather than annually
• Can easily modify the 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
• N is always the number of PAYMENT PERIODS
• I is always the required return PER PERIOD
( Eq.4.3)
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.
$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
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.
PV
-992.43
FV
1000
N
4
I
2.2
PMT
20
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.
• YTM is the discount rate that equates the PV of a bond’s
cash flows with its price.
• 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.
The Importance And Calculation
Of Yield To Maturity
Suppose you purchase a T-Bond for $875.00 that has 2 years to
maturity and pays its 5% coupon rate in semi-annual payments.
What is the YTM for the bond?
PV
-875.00
FV
1000
N
4
I
6.117
PMT
25
This is the semi-annual yield
on the bond, whereas the
YTM is always stated as an
annual rate. To annualize the
semi-annual yield, simply
multiply it by 2. So, the YTM
on the bond is 12.23%.
The Importance and Calculation
of Yield to Maturity
For the 3-year Treasury Note:
PV
FV
n
PMT
i
-875
1000
4
25
6.12%
For Excel: =RATE(nper,PMT,PV,FV,<Type>,<guess>)
The “Current Yield” is an
Approximation of the YTM
In bond quotes, the Current Yield (Cur Yld) is computed as:
Cur Yld = Annual $ of Interest / Price
It is an approximation of the YTM.
The long the time to maturity and the larger the coupon rate, the
better the approximation.
See http://www.bondpickers.com/?source=gglBondPrice for
detailed price quotes for corporate bonds.
Holding-Period Returns
If you purchase a bond with YTM = 10% and hold it until
maturity, you will earn an average of 10% return on the
bond over its life even though the value of the bond will
fluctuate throughout its life.
BUT, if you sell the bond prior to maturity your average
return will depend on the selling price of the bond,
which will depend on the prevailing level of interest
rates and the relative risk of the bond at the time you
sell it.
This is called “Interest rate risk” or “price risk”. Even
“riskless” treasury bonds have this risk.
Holding-Period Returns
The value of the bond will move inversely with the
prevailing level of interest rates.
Increase in required yield → decrease in value
Decrease in required yield → increase in value
So, if required return on your bond rises while you hold
it your selling price will be lower that you expected and
your holding period return (the average you earn over
the time you hold the bond) will fall. And vice versa.
Valuing A Bond With Semi-Annual
Bond Interest Payments
Suppose you purchase a bond today for $960 that has a
15-year maturity, $1000 face value, 8% coupon rate, and
pays interest semi-annually. The next coupon payment
is due in exactly six months. Also suppose that you sell
the bond four years from now, immediately after the 8th
coupon payment you receive. At the time that you sell
the bond, its required return in the market is 12%. What
is your average annual holding period return on this
bond?
Holding Period Returns
First, find the selling price of the bond in four years.
This is the price that the buyer would pay in
four years.
PV
759.17
FV
1000
The buyer will still value the bond with its
terminal value as a cash flow.
N
22
The buyer will get 11 more years of semiannual payments.
I
6
The buyer will value the bond based on a 12%
required return.
PMT
40
The bond has an 8% coupon rate, and that
won’t change over time.
Holding Period Returns
Next, use the price you just found as the FV, and
determine the rate that you would earn over four years.
PV
-960.00
This is the price you paid when you bought
the bond.
FV
759.17
This is the price that you will sell the bond for
after four years.
N
8
I
1.70
PMT
40
This is the number of semi-annual periods
you will hold the bond.
This is the semi-annual return you earned, so
double that for the annual return of 3.40%.
The bond has an 8% coupon rate, and that
won’t change over time.
The Term Structure Of Interest Rates
•
•
•
•
At any point in time, there will be a systematic relationship between
YTM and maturity for securities of a given risk.
– Usually, yields on long-term securities are higher than the yields on
short-term securities.
The relationship between yield and maturity is called the Term
Structure of Interest Rates.
– The graphical depiction of the term structure is called a Yield Curve.
Yield curves are normally upwards-sloping (long yields > short), but
can be flat or even inverted during times of financial stress
We won’t cover term structure in depth, but three principal
“expectations” theories explain the term structure:
– Pure expectations hypothesis: YC embodies prediction
– Liquidity premium theory: Investors must be paid to invest L-T
– Preferred habitat hypothesis: Investors prefer maturity zones
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, February 12, 2003
From www.cnnfn.com
%
Years to maturity
%
Yield Curve Today
%
For detailed current information on the yield curve, go to
http://www.bondsonline.com/asp/news/yieldcurve.html
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
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
Equity Valuation
•
•
•
•
As you learned in MBA 611, the required return on common stock is
based on its beta coefficient, which is derived from the CAPM
– Valuing common stock is the most difficult, both practically and
theoretically, since nothing (except the current price, is known with
certainty.
– Preferred stock valuation is much easier (the easiest of all)
Whenever investors feel the expected return, rˆ, is not equal to the
required return, r, prices will react:
– If exp return declines or req’d return rises, stock price will fall
– If exp return rises or req’d return declines, stock price will rise
Asset prices can change for reasons besides their own risk
– Changes in the asset’s liquidity or 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
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 $2.3 per share annual dividend and
with a required return of 11% would thus be worth $20.90:
P0 =
Dt 1 $2.3
=
= $20.90 / share
r
0.11
• Formula can be rearranged to compute required return, if
price and dividend known:
Dt 1
$2.3
r=
=
= 0.11= 11.0%
P0 $20.90
Common Stock Valuation
The 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 during the period, as in Eq 4.4:
P0 
P1  D1
(1  r )
(Eq.4.4)
The problem is how to determine P1.
Common Stock Valuation
P1  D1
P0 
(1  r )
(Eq.4.4)
P1 is the PV of expected stock price P2, plus dividends
received during period 1. P2 in turn, the PV of P3 plus
dividends, and so on.
Repeating this logic over and over, you will find that today’s
price equals the 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, we must make an assumption
about the growth rate of future dividends.
• The simplest approach, the zero growth model, assumes a
constant, non-growing dividend stream:
D1 = D2 = ... = D
• Plugging the constant value D into Eq 4.5 reduced the
valuation formula to the simple equation for a perpetuity:
D
P0 
r
The Zero Growth Valuation Model
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%. The next dividend will be one year
from now. 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, we can value
stock as a growing perpetuity. Denoting next year’s dividend
as D1:
P0 
D1
rg
Eq.4.6
• This is commonly called the Gordon Growth Model, after
Myron Gordon, who popularized model in the 1960s.
The Constant Growth Valuation Model
P0 
D1
rg
Eq.4.6
The Gordon Company’s dividends have grown by 7% per
year, reaching $1.40 per share this year. This growth is
expected to continue, so D1=$1.40 x 1.07=$1.50. If the
required return on this stock is 15%, then its market value
should be:
D1
$1.50
$1.50
P0 


 $18.75
r  g 0.15  0.07 0.08
Valuing Common Stock Using The
Variable Growth Model
• Because future growth rates might change, we 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.
• The constant growth rate model can be generalized for two
or more changes in growth rates, but let’s keep it simple for
now.
• For a single change in growth rates, we 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, 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).
– 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 an 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
3
(1  r ) (1.12)
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
– Assumes assets can be sold at book value, so may overestimate realizable value
– Ignores non-balance-sheet assets (value of reputation,
human capital, unrealized gains, etc.)
• 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