Bond and Stock Valuation
• The market value of the firm is the present value of the cash flows
generated by the firm’s assets:
CFt
PV  
t
t  0 (1  r )
N
• The cash flows generated by the firm’s assets are divided among the
investors who pay for the assets. If these investors include only debt
and equity holders, the market value of the firm can be expressed as:
PVfirm = PVdebt+ PVStock
Bond (Debt) Valuation
• The price of bonds in the market place is the present value
of the cash flows that bondholders have claim to:
N
CFd ,t
t 1
(1  rd )t
PVd  
• These cash flows consist generally of two components,
interest and principal. They are generally divided as
follows:
I
P

t
t 1 (1  rd )
(1  rd ) N
N
PVd  
• That is, interest is paid every period, and the principal is
paid at maturity, when the bond comes due.
Bond Valuation (Continued)
Terms:
– Coupon Payment: the interest paid annually, or semiannually (I).
Typically, these payments are fixed so that the interest paid each
year is the same.
– Principal: the amount borrowed, and repaid at maturity (P).
– Coupon Rate: the annual interest payment divided by the
principle (I/P)
– Current Yield: the annual interest payment divided by the price
(I/PV)
– Capital Gains Yield: the change in price (over one year) divided
by the price at the beginning of the year [(PV1-PV0)/ PV0]
– Yield to Maturity: the return investors expect if they buy the bond
and hold it until it matures. If the market is in equilibrium, the
yield to maturity is also the return investors require given the
bond’s risk (rd).
Bond Valuation (Continued)
• Numerical Example: Suppose a bond with 10 years to maturity has a
coupon rate of 10%, a principal amount of $1,000, and a yield-tomaturity of 10%. Assuming interest is paid annually and the bond is in
equilibrium,
– What is the price of the bond?
100
1,000

?
t
t 1 (1.10)
(1.10)10
10
PVd  
– What is its current yield?
• Current Yield = I/PV =
– What is its expected capital gains yield?
• Capital Gains Yield = [(PV1-PV0)/ PV0] =
Bond Valuation (Continued)
• Suppose now that everything else remains constant, but the yield to
maturity is 12%. What are the price, the current yield, and the
expected capital gains yield?
100
1,000

?
t
10
t 1 (1.12)
(1.12)
10
PVd  
Current Yield = I/PV =
Capital Gains Yield = [(PV1-PV0)/ PV0] =
• What would cause the yield to maturity to be 12% instead of 10%?
Bond Valuation (Continued)
• The yield to maturity, the return investors expect, is linked to the return
investors require, rd.
• The required return, rd , is a function of
– The real rate of return - the return investors require for deferring
consumption (that is, the time value of money)
– The expected rate of inflation - the compensation investors require to
guard against losses in their purchasing power.
– The risk premium - the compensation investors require to accept the
possibility that their return will be lower than what they were promised.
• If rd is 12%, not 10%, one or more of the three components of the
required rate of return must be higher in the second instance than in the
first.
• Why is yield to maturity linked to rd?
Bond Valuation (Continued)
• Suppose the expected rate of return does not equal the required rate of
return. If the bond above should be priced at $887 because the required
rate of return is 12%, but it is priced at $1,000 to give an expected
return of 10%, investors are not being compensated for the risk that
they bare.
– The NPV from buying the bond will be negative (887-1,000), so new
investors will not buy.
– The NPV from selling the bond will be positive (1,000-887), so existing
investors will want to sell.
– The combination of new investors not buying and existing investors
wanting to sell will cause the price of the bond to fall.
– How far? Why?
Bond Valuation (Continued)
Risk Return Relationship
20.00%
18.00%
16.00%
14.00%
Return
12.00%
10.00%
8.00%
6.00%
4.00%
2.00%
0.00%
0.0
0.2
0.4
0.6
0.8
1.0
Risk
1.2
1.4
1.6
1.8
2.0
Bond Valuation (Continued)
• What you see in the Wall Street Journal, May 1, 2000
Bonds
IBM, 63/8 00
IBM, 71/4s,02
Cur Yld
6.4
7.2
Vol
4
60
Close
Net Change
-1/32
9913/16
-1/8
1001/2
• What do you know about IBM’s yield to maturity and expected capital
gains yield based on this information?
Stock Valuation
• The price of stocks in the market place is the present value
of the cash flows that stockholders have claim to:
N
CFs ,t
t 1
(1  rs )t
PVs  
• These cash flows consist generally of two components,
dividends and capital gains. They are generally divided as
follows:
PVs ,0
PVs , N
Div t


t
t 1 (1  rs )
(1  rs ) N
N
Stock Valuation (Continued)
• What are the differences between bond and stock cash flows?
– Interest vs Dividends
• Interest is paid before dividends.
• Interest is generally fixed ; dividends are variable.
• Interest is a contractual obligation; dividends are discretionary.
– Principal vs. Future Stock Prices
• Principal is contractually binding to the firm; future stock prices are not.
• In liquidation, claims to both principal and interest must be satisfied before
payments can be made to stockholders
• What do these differences imply about potential differences between rs
and rd?
Stock Valuation (Continued)
Risk Return Relationship
20.00%
18.00%
16.00%
14.00%
Return
12.00%
10.00%
8.00%
6.00%
4.00%
2.00%
0.00%
0.0
0.2
0.4
0.6
0.8
1.0
Risk
1.2
1.4
1.6
1.8
2.0
Stock Valuation (Continued)
• If, for simplicity, we assume that dividends grow forever at a constant
rate, g, and that that rate is lower than the required rate of return on
the stock, rs, then the present value of the dividends and future stock
price can be expressed as
Div1
PVs , 0 
rs  g
• This says that the price of the stock today equals the expected dividend
one year from today (Div1) divided by the difference between the
required rate of return and the constant growth rate (rs-g)
• Under these same assumptions, the required return on the stock could
be estimated as
Div1
rs 
g
PVs , 0
Stock Valuation (Continued)
• Another useful way to think about the price of a stock is as
PVs , 0
EPS1

 NPVGO
rs
• The first component, EPS1/rs, is the price of the stock if equity cash
flows (or earnings) remain constant forever. The second component is
the expected NPV from future growth opportunities.
• What determines whether NPVGO is positive or negative?
Stock Valuation (Continued)
• By setting the two stock pricing relationships equal to each other and
recognizing that (Div1/EPS1) equals 1-b, where b is the firm’s retention
ratio, and g is the ROE*b, we can express NPVGO as
NPVGO  EPS1
b * ( ROE  rs )
rs * (rs  g )
• The above relationship tells us that NPVGO will be positive so long as
the ROE on the investment exceeds the required rate of return,rs
Stock Valuation (Continued)
• Numerical Example: Suppose a firm’s expected earnings per share
next year and every year thereafter are $2.50 and that the required
return on the firm’s stock is 15%. What is the price of the firm’s
stock?
• PV,s,0 = EPS1/rs = 2.50/.15 = $16.67
• Now suppose the firm discovers an opportunity to invest in new assets
that produce additional earnings annually forever of $.20 per share.
The investment will be funded by a one-time dividend reduction of
$1.00 per share. What is the value of this investment to current
shareholders if investors can make it only once?
• NPV1 = -$1 + [$.20/.15] = $.33
Stock Valuation (Continued)
• If the firm could retain 40% of its eps in year 2 to make an investment
that costs $1.08 and the ROE on this investment would also be 20%,
what would be the value of this investment to shareholders?
• NPV2 = -$1.08 + .2*1.08/.15 = .36
• What would happen if the firm could make these investments
indefinitely by retaining 40% of its earnings and producing ROEs of
20%?
b * ( ROE  r )
NPVGO  EPS1
s
rs * (rs  g )
 .4 * (.20  .15) 
 $2.50 * 
  $4.76
.
15
*
(.
15

.
08
)


• What would be the price of the stock?
• EPS1/rs + NPVGO = $2.5/.15 + $4.76 = $21.43
Stock Valuation (Continued)
• How does that coincide with the earlier model
PVs ,0 
Div1
$1.50

 $21.43
rs  g (.15  .4 * .20)
Stock Valuation (Continued)
Assumptions
ROE
Retention Ratio
Payout Ratio
Required Return
Growth
Expected Earnings without Investment
Time
Earnings with No New Investment
Present Value of Earnings
Amount Retained
Additional Earnings from Investment
PV of Additional Earnings
NPV of Additional Earnings
Total Earnings
Amount Retained
Additional Earnings from Investment
PV of Additional Earnings
NPV of Additional Earnings
Total Earnings
Amount Retained
Additional Earnings from Investment
PV of Additional Earnings
NPV of Additional Earnings
$
20.00%
40.00%
60.00%
15.00%
8.00%
2.50
0
$
$
1
2.50 $
$
(1.00)
$
1.33
4
2.50 $
5
2.50 $
6 ….
2.50 $
2.50
….
$
2.50
0.20
$
0.20
$
0.20
$
0.20
$
0.20
$
0.20
$
0.20
$
2.50
2.70 $
(1.08)
$
1.44
2.70
$
2.70
$
2.70
$
2.70
$
2.70
$
2.70
0.22
$
0.22
$
0.22
$
0.22
$
0.22
$
0.22
2.92 $
(1.17)
$
1.56
2.92
$
2.92
$
2.92
$
2.92
$
2.92
0.23
$
0.23
$
0.23
$
0.23
$
0.23
0.29
$
$
$
$
3
2.50 $
16.67
$
$
2
2.50 $
0.27
$
2.50
$
2.70
$
$
$
$
0.26