1. finance
2. investing
3. stocks

Valuing Securities
Stocks and Bonds
Bond Cash Flows, Prices, and Yields

Bond Terminology

Face Value


Coupon Rate


Notional amount used to compute the interest payments
Determines the amount of each coupon payment, expressed as an
annual percentage rate
Coupon Payment
CPN 
Coupon Rate  Face Value
Number of Coupon Payments per Year
Zero-Coupon Bonds

Zero-Coupon Bond (Zero)

Does not make coupon payments

Always (almost always) sells at a discount (a price lower than
face value), so they are also called pure discount bonds

Treasury Bills are U.S. government zero-coupon bonds with
a maturity of up to one year.

“Strips”
Zero-Coupon Bonds

Suppose that a one-year, risk-free, zero-coupon bond with
a $100,000 face value has an initial price of $97,323.60.
The cash flows would be:

0
1
-$97,323.60
$100,000
Although the bond pays no “coupon,” your compensation is
the difference between the initial price and the face value.
Zero-Coupon Bonds


The difference between the price and the face value (here
$2,676.40) is the compensation you receive for buying
and holding the bond for a year.
The compensation is commonly talked about as a return
on the initial investment:
($100,000  $97,323.60) $2,676.40
R

 0.0275  2.75%
$97323.60
$97323.60

This is the return you would receive from investing in
(buying) the bond and holding it till maturity.
Zero-Coupon Bonds

Yield to Maturity

The single discount rate that sets the present value of the remaining
promised bond payments equal to the current market price of the
bond.

For a Zero-Coupon bond:
FV
P 
(1  YTM n ) n

For a zero, this is also the actual annualized return (actual spot rate).
Zero-Coupon Bonds

Yield to Maturity

For the risk free one-year zero coupon bond presented above:
97,323.60 
1  YTM1

100,000

97,323.60
 1.0275
Thus, the YTM is 2.75%.


100,000
(1  YTM 1 )
In bond markets the standard presumption is that a “period” is six months.
Why must this equal the one year risk free rate?
Example



Suppose the following three zero-coupon bonds are
trading at the prices shown.
Each has a face value of $1,000 and the time to maturity
shown in the table.
What is the yield to maturity for each bond?
Maturity
1 Year
2 Years
3 years
Price
980.39
942.60
901.94
Example

The general formula is found by re-arranging the present
value equation:
FV
P
(1  YTM n ) n


1
 FV 
 YTM n  
 1
 P 
n
The specific solutions in our example are:
1 Year – YTM
$1,000/$980.39 -1
2.0%
2 Years – YTM
($1,000/$942.60)1/2 - 1
3.0%
3 Years – YTM
($1,000/$901.94)1/3 - 1
3.5%
This is how the spot rates on the yield curve are found.
Coupon Bonds

Yield to Maturity

The YTM is the single discount rate that equates the present
value of the bond’s remaining cash flows to its current price.

Yield to Maturity of a Coupon Bond

1 
1
FV
P  CPN 

1 
N 
y 
(1  y ) 
(1  y) N

For a coupon bond the YTM is an artificial yield intrinsic to the
bond itself. An IRR for the bond.
YTM – Example

On 9/1/95, PG&E bonds with a maturity date of
3/01/25 and a coupon rate of 7.25% were selling
for 92.847% of par, or $928.47 per $1,000 of face
value. What is their YTM?

Semiannual coupon payment = 0.0725*$1,000/2 =
$36.25.
Number of semiannual periods to maturity
= 30*2 – 1 = 59.

YTM - Example
 1

1
1000
$928.47 = $36.25


59
59 
r
/
2
(
r
/
2
)(
1

r
/
2
)
(1
+
r
/
2
)





r/2 can only be found by trial and error. However, calculators
and spread sheets have algorithms to speed up the search.
Searching reveals that r/2 = 3.939% or a stated annual rate
(YTM) of r = 7.878%.
This is (sort of) an effective annual rate of:
(1.03939)  1  8.03%
2
Example



Suppose that now the yield to maturity for the PG&E
bonds has changed to 7% (on a stated annual basis, APR,
with semi-annual compounding).
What must be the current price of the bond?
Note that the bond itself has not changed (assume for
simplicity that the change occurs instantaneously) only its
pricing.
Example

We find the price using the present value of the bond
payments discounted at the YTM.
 1

1
1000
Price = $36.25


59
59 
.
07
/
2
(.
07
/
2
)(
1

.
07
/
2
)
(1
+
.
07
/
2
)


 1

1
1000
$1031.02  $36.25


59
59 
.
07
/
2
(.
07
/
2
)(
1

.
07
/
2
)
(1
+
.
07
/
2
)



Does it make sense that the price changed like this?
Discounts and Premiums

If a coupon bond trades at a discount, an investor will earn a
return both from receiving the coupons and from receiving a
face value that exceeds the price paid for the bond.


If a coupon bond trades at a premium it will earn a return
from receiving the coupons but this return will be diminished
by receiving a face value less than the price paid for the bond.


If a bond trades at a discount, its yield to maturity will exceed its
coupon rate.
If a bond trades at a premium, its coupon rate will exceed its yield to
maturity.
Most coupon bonds have a coupon rate set so that the bonds
will initially trade at, or very close to, par.
Interest Rate Changes and Bond Prices

The sensitivity of a bond’s price to changes in interest
rates is measured by the bond’s duration.

The prices of bonds with high durations are highly sensitive to
interest rate changes.

The prices of bonds with low durations are less sensitive to
interest rate changes.

A bond’s term or time to maturity is a major determinant of
duration. All else equal, the longer the term the more sensitive
to changes in the interest rates is bond’s price.

However, that is not the whole story:
Interest Rate Changes and Bond Prices



Duration is essentially a weighted average of the times to
receipt of cash flows from the bond.
The face value is usually a comparatively very large cash
flow making the maturity date a very important “time to
receipt” of cash.
However if the coupons are large they will also play an
important role in lowering the duration of a long-term
coupon bond.
The Yield Curve and Bond Arbitrage

Using the Law of One Price and the yields of default-free
zero-coupon bonds, one can determine the price and
yield of any other default-free bond.

The yield curve provides sufficient information
to evaluate (price) all such bonds.
Replicating a Coupon Bond

Replicating a two-year $1,000 bond that pays a 5%
coupon using four zero-coupon bonds:
time
0
1/2
1
1½
2
coupon
bond
- Price
$25
$25
$25
$1,025
6 month
zero
-Price
$25
1 year
zero
-Price
18 month
zero
-Price
2 year
zero
-Price
portfolio
- sum of
prices
$25
$25
$1,025
$25
$25
$25
$1,025
Replicating a Coupon Bond

Yields and Prices (per $100 Face) for Zero Coupon Bonds
Maturity
6 months
1 year
1 ½ years
2 years
YTM
2.0%
3.00%
3.50%
4.00%
Price
$99.01
$97.07
$94.93
$92.38

Again these are the spot rates on the yield curve.
Replicating a Coupon Bond
Zero
Face Value Needed
Cost/Price
6 month
$25
$99.01/4 = $24.75
1 year
$25
$97.07/4 = $24.27
18 month
$25
$94.93/4 = $23.73
2 year
$1,025
$92.38x41/4 = $946.90
Total Cost $1,019.65

By the Law of One Price, the three-year coupon bond must
trade for a price of $1,019.65.
Value a Coupon Bond Using Zero-Coupon
Yields

The price of a coupon bond must equal the present value
of its coupon payments and its face value discounted at
the relevant spot rate.

Price of a Coupon Bond
PV  PV (Bond Cash Flows)
CPN
CPN



2
1  YTM 1
(1  YTM 2 )
CPN  FV

(1  YTM n ) n
25
25
25
25  1000
P 



 $1,019.65
2
3
4
1.01 1.015
1.0175
1.02

Where the YTMn‘s are the yields to maturity of an n-period zero.
Coupon Bond Yield to Maturity

Knowing the price we can of course also find the yield to
maturity of the coupon bond.
25
25
25
25  1000
P  $1, 019.65 



2
3
(1  y)
(1  y)
(1  y)
(1  y) 4
25
25
25
25  1000
P 



 $1, 019.65
2
3
4
1.0198
1.0198
1.0198
1.0198

Annualized this is 3.968%, compare this to the zerocoupon yields.
Stock Valuation: An Example

3M is expected to pay a dividend of $1.92 per share in
the coming year.

You expect the stock price to be $85 per share at the
end of the year.

Investments with equivalent risk have an expected return
of 11% (the cost of equity capital, rE).

What is the most you would pay today for
3M stock?

What dividend yield and capital gain rate would
you expect at this price?
Example

Solution: the law of one price implies that to value a security we must
determine the discounted expected cash flow associated owning it.
Div1  P1
$1.92  $85
P0 

 $78.31
(1  rE )
(1 .11)
Div1
$1.92
Dividend Yield 

 2.45%
P0
$78.31
P1  P0
$85.00  $78.31
Capital Gains Yield 

 8.54%
P0
$78.31

Total Return = 2.45% + 8.54% = 11% (but for rounding) so at this price the
investor is receiving an appropriate return.
A Multi-Year Investor

What is the price if we plan on holding the stock for two
years?
Div1
Div2  P2
P0 

2
1  rE
(1  rE )

Again the price must be the present value of the future
cash flows or one side (i.e. either the buyer or the seller)
will refuse to trade.
A Multi-Year Investor

What is the price if we plan on holding the stock for N years?
P0 

Div1
Div2


2
1  rE
(1  rE )

DivN
PN

(1  rE ) N
(1  rE ) N
This is known as the Dividend Discount Model.

Note that the above equation holds for any horizon. Thus all investors
(with the same beliefs) will attach the same value to the stock,
independent of their investment horizons.
Div3
Div1
Div2
P0 



2
3
1  rE
(1  rE )
(1  rE )




n 1
Divn
(1  rE ) n
The price of any stock is equal to the present value of all of the expected
future dividends it will pay.
The Dividend-Discount Model

Constant Dividend Growth

The simplest forecast for the firm’s future dividends states that they
will grow at a constant rate, g, forever.
Div1
P0 
,
rE  g

rE
Div1

 g
P0
The value of the firm depends on the current dividend level, the cost
of equity, and the growth rate. The expected return is from dividend
yield and the expected capital gain (g by assumption).
Example

Problem

AT&T plans to pay $1.44 per share in dividends in the coming
year.

Its equity cost of capital is 8%.

Dividends are expected to grow by 4% per year
in the future.

Estimate the value of AT&T’s stock.
Example

Solution
Div1
$1.44
P0 

 $36.00
rE  g .08  .04

Note that its price after one year is expected to be:
P1  P0
Div2 $1.44(1.04)
P1 

 $37.44 
 0.04  4%
rE  g 0.08  0.04
P0

The expected capital gain equals the dividend growth rate,
g.
Changing Growth Rates

We cannot use the constant dividend growth model to
value a stock if the growth rate is not constant.

For example, young firms often have very high initial earnings
growth rates. During this period of high growth, these firms
often retain 100% of their earnings to exploit profitable
investment opportunities.

As they mature, their growth slows. At some point, their
earnings exceed their investment needs and they begin to pay
dividends.
Changing Growth Rates
PN

DivN  1

rE  g
Dividend-Discount Model with Constant Long-Term Growth
Div1
Div2
P0 


2
1  rE
(1  rE )
 DivN  1 
DivN
1




N
(1  rE )
(1  rE ) N  rE  g 
Example

Batesco Inc. just paid a dividend of $1. The dividends
of Batesco are expected to grow by 50% next year
(time 1) and 25% the year after that (year 2).
Subsequently, Batesco’s dividends are expected to
grow at 6% in perpetuity.

The proper discount rate for Batesco is 13%.

What is the fair price for a share of Batesco stock?
Example cont…





First, determine the dividends. Draw the timeline!
D0 = $1
g1 = 50%
D1 = $1(1.50) = $1.50
g2 = 25%
D2 = $1.50(1.25) = $1.875
g3 = 6%
D3 = $1.875(1.06) = $1.9875
0 g =50% 1
1
1.50
g2=25%
2 g =6%
3
1.875
3
1.9875
g4=6%
4
......
2.107
Example cont…

Supernormal growth period:
Ps =

D 1 + D 2 = 1.50 + 1.875 = $2.796
(1+ r E ) (1+ r E )2 (1.13) (1.13 )2
Constant growth period. Value at time 2:
Pc =

Discount Pc to time 0 and add to Ps:
P0 = P s +

D 3 = 1.9875 = $28.393
r E - g 3 0.13 - 0.06
28.393
Pc
=
2.796
+
= $25.03
2
2
(1+ r E )
(1.13 )
What if supernormal growth lasted 8 yrs at 50%?
Limitations of the Dividend-Discount Model

There is a tremendous amount of uncertainty associated
with forecasting a firm’s dividend growth rate and its
future dividends (particularly those many periods from
now).

Compounding the issue, dividends are discretionary.

Small changes in the assumed dividend growth rate can
lead to large changes in the estimated stock price.

Many firms pay no dividends (especially true recently).
Discounted Free Cash Flow Model

The Discounted Free Cash Flow Model

Determines the value of the firm to all investors, including both
equity and debt holders
Enterprise Value  Market Value of Equity  Debt  Cash

The enterprise value can be interpreted as the net cost of
acquiring control of the firm, buying its equity, taking its excess
cash, paying off all debt, and owning the then unlevered
business or “enterprise”.
Discounted Free Cash Flow Model

Valuing the Enterprise
Unlevered Net Income
Free Cash Flow  EBIT  (1  c )  Depreciation
 Capital Expenditures  Increases in Net Working Capital

Free Cash Flow


Cash flow available to pay both debt holders and
equity holders
Discounted Free Cash Flow Model
V0  PV (Future Free Cash Flow of Firm)
V0  Excess Cash 0  Debt 0
P0 
Shares Outstanding 0
Discounted Free Cash Flow Model

Implementing the Model

Since we are discounting cash flows to both equity holders and
debt holders, the free cash flows should be discounted at the
firm’s weighted average cost of capital, rwacc. If the firm has no
debt, rwacc = rE.
Discounted Free Cash Flow Model

Implementing the Model
V0 

FCF1
FCF2


2
1  rwacc
(1  rwacc )

FCFN
VN

(1  rwacc ) N
(1  rwacc ) N
Often, the terminal value is estimated by assuming a constant
long-run growth rate gFCF for free cash flows beyond year N, so
that:
VN
 (1  g FCF )  FCFN 
FCFN  1

 

rwacc  g FCF
 (rwacc  g FCF ) 
Example: Stock Price from FCF

A proforma forecast of the (simplified) income statement for Clive.com appears
below. As Clive.com is a mature firm it is forecast that future capital expenditures
will be for the replacement of worn equipment. The firm’s experience is that
necessary levels of net working capital are 12% of sales in any year. The firm has
$75 Million in excess cash on hand, $50 Million in outstanding debt, 100 Million
shares of stock outstanding, and faces a 35% tax rate. After 2014 it is expected that
free cash flows for the firm will increase at a constant rate of 3% and due to its risk
and use of debt a WACC of 15% is appropriate.

What is the value of Clive.com’s equity and its share price at the start of 2010?
Proforma Income Statement for Clive.com ($ in Millions)
2009
2010
2011
2012
2013
2014
sales
672.3
739.53 813.483 878.5616 931.2753 977.8391
cogs
235.305 258.8355 284.7191 307.4966 325.9464 342.2437
op exp
268.92 295.812 325.3932 351.4247 372.5101 391.1356
interest
3.5
3.5
3.5
3.5
3.5
3.5
taxes
57.60125 63.48388 69.95476 75.64914 80.26159 84.33592
Net Income 106.9738 117.8986 129.916 140.4913 149.0572 156.6239
Solution

First we need a forecast of future free cash flow.
EBIT
Unlevered NI
NWC
DNWC
Plus Depr
Less Cap Ex
FCF

2010
2011
2012
2013
2014
184.8825 203.3708 219.6404 232.8188 244.4598
120.1736 132.191 142.7663 151.3322 158.8989
80.676 88.7436 97.61796 105.4274 111.753 117.3407
8.0676 8.87436 7.809437 6.325644 5.587652
----------112.106 123.3166 134.9568 145.0066 153.3112
What about the rest?



FCF in 2015 is forecast to be $153.31x1.03 = $157.91
Terminal value is then $157.91/(0.15-.03) = $1,315.92 (end of
2014).
PV of TV is $1,315.92/(1.15)5 = $654.25
Solution

The present value of the free cash flow for the forecast
period is:
$112.11 $123.32 $134.96 $145.01 $153.31
PV 




 $438.60
2
3
4
5
1.15
1.15
1.15
1.15
1.15

Now just add the present values:

PV(forecast period) + PV(Terminal Value)
V0  $438.60  $654.25  $1,092.84

Enterprise value plus Excess Cash less existing Debt = Equity value
Equity Value  $1, 092.84  $75  $50  $1,117,84

Share price is Equity value divided by the number of shares
Share Price  $1,117.84 /100  $11.18