Chapter 2: Valuation of Bonds & Stocks
I.
Bonds
A.
Bond Price
Bond’s price =present value of interest payments + present value of par value
Example (semi-annual coupon payments)
A bond pays $80 annual interest (coupon) payments for 10 years, and then pays the
par (face, maturity) value at the end of the 10th year. What is its value (price) if the
annual yield-to-maturity is 10 percent and coupons are paid semi-annually?
0
1
2
20 = N
/-----------------/------------------/-----------/ /--------------/
40
40
40 + 1,000
i = .05
C
C
C+M
Coupon (C):
Par Value (M):
Maturity (N)
Yield (i):
Annual coupon payment divided by 2
Always $1,000
The number of years until the bond expires times 2
The “yield-to-maturity” or the discount rate divided by 2
Texas BAII+ Keys:
FV
PMT
N
I/Y
=
=
=
=
$1,000
$40
20
5
Par value
semi-annual interest payment
Number of periods
semi-annual yield on the bond
Solve for…
PV
=
-$875.38
Value of the bond
Note:
Note:
Note:
The coupon rate (CR) is: $80/$1000 = 8.00%
The current yield (CY) is: $80/$875.38 = 9.14%
The yield-to-maturity (YTM) is: 5 x 2 =10.00%
Rules to memorize:
If price = par, then CR = CY = yield to maturity. Bond is selling “at par”
If price < par, then CR < CY< yield to maturity. Bond is selling “at a discount”
If price > par, then CR >CY > yield to maturity. Bond is selling “at a premium”
Fin 336: Chapter 2 - Valuation of Bonds and Stocks
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Pure Discount Bond – Zero Coupon Bond
No coupon payments are made. Bond is always sold at a discount from par
Solve for…
FV
N
I/Y
=
=
=
$1,000
20
5
Par value
Number of periods
semi-annual yield on the bond
PV
=
-$376.89
Value of the bond
Yield on a Discount Bond:
[Par Value/Price] 1/N -1
$1,000/$376.891/20 – 1 =
.05
The Reinvestment Rate Assumption
When we compute a bond’s “yield-to-maturity” we are implicitly assuming that all the cash
flows from the bond are reinvested at that “yield-to-maturity”. For example, a 2-year, semiannual coupon bond with an 8 percent coupon rate and selling for $964.54 has a “yield-tomaturity” that is calculated to be: 5 percent.
0
1
2
3
4
/-----------------/------------------/-----------------/------------------/
40
40
40
40 + 1,000
i = .05
/
/
/---------------- 42
/
/---------------------------------- 44.1
/---------------------------------------------------- 46.305
Terminal Value = 1,172.405
If we reinvest all coupon payments at 5 percent, we have a Terminal Value of $1,172.405
Since the price is $964.54 we can also say that the present value of these cash flows is: $964.54.
PV (1+r) N = FV
PV = $964.54
FV = $1,172.405
N=4
Solve for r:
r = 5%
This is proof that the “yield-to-maturity” assumes a reinvestment rate.
If we reinvest all coupon payments at 4 percent, we have a Terminal Value of
$1,169.86, and our effective yield-to-maturity is only 4.94 percent, not the 5
percent that we started with!
Fin 336: Chapter 2 - Valuation of Bonds and Stocks
2
B.
The Yield Curve &Term Structure of Interest Rates
The yield curve shows the interest rate that exists for pure discount bonds of different
maturities.
The yield curve can be viewed as the 1-year interest (discount) rates for any one year
period in the future.
Example:
Time
0-1
1-2
2-3
Expected one-year interest rate (%)
8.0
9.6
10.0
The market expects one-year rates to rise from 8 percent this year to 10 percent in 2 years.
An investor can invest $100 in one year discount bonds and roll it over each year.
The future value will be:
$100(1.08) (1.096) (1.10)
=
130.20
The average rate of return is:
(130.20/100)1/3 -1
=
.092
The yield curve will show 9.2 percent as the rate for a pure discount bond with a 3-year
maturity. The 2-year pure discount bond will have an 8.8 percent rate
Yield
1
2
3
4
Time
Fin 336: Chapter 2 - Valuation of Bonds and Stocks
3
C.
Bond Pricing & the Yield Curve
Since the yield curve shows the interest rate that exists for pure discount
bonds of different maturities, and a coupon bond has cash flows of different
maturities, then the correct procedure for pricing a coupon bond is to use the
yield curve to price each of the cash flows (coupons and face value)
separately, using the pure discount bond discount rate that is appropriate for
that time period.
Example for a 3-year coupon bond (annual coupons): Coupon = $80
Time
1
2
3
3
Payment
$80
$80
$80
$1,000
interest rate (%)
8.0
8.8
9.2
9.2
Discount factor
.926
.845
.768
.768
TOTAL =
PV
$74.07
$67.58
$61.44
$768
$971.09
The yield to maturity on this bond:
FV = $1000
PV = -$971.09
Pmt = $80
N=3
Compute I/Y = 9.145%
Bonds with the same maturity but different coupon rates will have different YTM
The proper measure of a bond’s return is…..the YTM.
D.
Implications of a Non-flat Yield Curve


E.
An upward sloping yield curve indicates rising, short-term interest rates
A downward sloping yield curve indicates falling, short-term interest
rates
Uses of the Yield Curve




Predict economic growth, recessions, stock market direction:
Help estimate future investment costs
Gap management
Inflation forecasting
Fin 336: Chapter 2 - Valuation of Bonds and Stocks
4
F. Yield Spread and Credit Risk
Corporate Bond Credit Spreads
Fin 336: Chapter 2 - Valuation of Bonds and Stocks
5
Practice Problems: (Assume semi-annual coupons for all problems)
1.
A zero coupon bond promises to pay you $5,000 at the end of 10 years.
What would the price of the “zero” be today if the yield to maturity is 14
percent per year?
2.
A $1,000 par value bond is issued with an 8 percent coupon rate and a 10
year maturity. Assuming semi- annual coupon payments, if the bond is
priced to yield 6.75 percent per year, what is its price?
3.
A $1,000 bond is issued at par with a 15 year maturity and a coupon rate of
8 percent. What is its annual yield to maturity?
4.
Three years later the price has fallen from $1,000 to $844. What is the new
annual yield to maturity?
5.
A $1,000 bond is issued at par with an 8 percent coupon rate. In three years
the yield to maturity on the bond is 6.75 percent and the bond is selling for
$1067.96. What was the original maturity of the bond?
Fin 336: Chapter 2 - Valuation of Bonds and Stocks
6
II.
Stock Valuation: Dividend Discount Model (DDM) Approach
-
A stock’s price is the present value of all future cash flows.
r is the discount rate. Sometimes k is used as the discount rate variable.
0
1
2
|-------------------|-------------------|-----/ /--------------|
D1
D2
Dinfinity
$1/(1+r)1
<----------------|
|
|
2
$1/(1+r)
<-------------------------------------|
|
$1/(1 + r)infinity <-----------------------------------------------------------|
Price=Sum of present values to infinity
P0
=
D1
+
1
(1 + r)
D2 + ... + Dinfinity
(1 + r)2
(1 + r)infinity
D
=
Dividends are paid at the end of each quarter. Dividend estimates can be obtained
for various future time periods from various sources (company announcements,
Value Line, I/B/E/S).
r
=
Required rate of return (and discount rate) for this stock.
There is an inverse relations ship between the discount rate and price.
Finding the appropriate r takes experience, analysis, insight, and guesswork.
For a one year time period, the investor’s expected rate of return can be estimated as follows:
You buy the stock for $50 and sell it for $60. During the year you receive $2 in dividends.
P0
P1
Dt
=
=
=
Price you pay for the stock today.
Price you sell the stock for at the end of the year.
The dividends paid during the year.
Total Return =
Dt + (P1- P0)
P0
$2 + (60 – 50)
50
= .24 = 24%
Dividend yield = 2/50 = .04 = 4%
Capital gain
=
10/50 =
.20 = 20%
The dividend is called “income” and taxes are paid at the ordinary income tax rate
The capital gain tax is paid at a lower rate.
Fin 336: Chapter 2 - Valuation of Bonds and Stocks
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A.
Deriving the Dividend Discount Model (DDM)
Given the following:
Total Return =
Dt + (P1- P0)
P0
We can get:
P0
=
Dt + P1
1+r
and
P1
=
Dt + P2
1+r
By continually substituting for Pt , we finally get:
P0
B.
=
D1
+
(1 + r)1
D2 + ... + Dinfinity
(1 + r)2
(1 + r)infinity
Simplifying Assumptions for the Dividend Discount Model (DDM)
Video 1 (8.07) explanation
1.
Video 2 (8:37) explanation
Zero Growth Assumption:
EQ1 reduces to:
P0
2.
EQ1
=
If dividends are constant (zero growth)
(See Link for the Proof)
D
r
Example:
Zema Corporation issues 8 percent preferred stock for $25 per share.
What dividend will Zema be paying (forever) on this preferred stock?
$25
D/.08
=
The dividend will be $2.
Constant Growth Assumption:
If dividends grow at a constant rate
EQ1 reduces to:
P0
=
D1
r-g
Example:
Zema Corporation’s common stock just paid a dividend of $2 and this
dividend is expected to grow at 3 percent forever. The stock is currently
selling for $10. What is the investor’s required rate of return?
P0
=
D1
r-g
$10
=
$2.06/(r- .03)
Rearrange for r:
r
=
D1/Po + g
r
=
=
2.06/10 + .03
.236
Fin 336: Chapter 2 - Valuation of Bonds and Stocks
8
3.
Variable Growth Assumption:
-
Dividends grow at different rates over different periods.
-
Typically at accelerated rates in the early years, then at a constant rate after the company
reaches maturity.
-
Solve by discounting dividends individually during variable growth period, and then use
the constant growth formula.
Example:
IBM just paid a $1.00 dividend. You expect it to grow at 15% per year for 2
years, then at 5% forever. Your required rate of return is 8%. What would you
pay for this stock?
The answer is $41.88
C.
Estimating the rate of growth (g) for a stock
1.
The Retention Ratio Approach:
g
=
ROE x retention ratio = ROE x (1-DPO)
*
ROE is return on equity investment for each investor
*
If each investor reinvests half that return, their investment should grow by half that
amount.
Example:
Imagine a company worth $100,000 with a single investor
The $100,000 investment returns 20% per year and pays it all out
If the investor reinvests half in the first year, the company is now worth $110,000
2.
The Dividend Discount Model Approach
P
=
D
r-g
constant growth
Rearrange and solve for g:
g
=
r
-
D/ P
Growth is the additional return that the stock is providing above and beyond the yield on the
dividends the company is currently paying. That is, growth is the promise of future increases in
dividends.
Fin 336: Chapter 2 - Valuation of Bonds and Stocks
9
III.
The Capital Asset Pricing Model (CAPM)
A.
Diversification Benefits
Video (8:13) explanation
As the number of stocks in a portfolio increases, the average total risk (SD) of the portfolio
decreases, up to a point. There are limits to the benefits of diversification
Total risk
=
systematic risk + unsystematic risk
Standard deviation = Beta risk + other
(“other” is the diversifiable risk)
(From: The Dangers of Over-Diversifying Your Portfolio, 2/10/2010)
In the real world, few securities are negatively correlated
The benefits of diversification can be achieved by combining securities that have low
correlations
Total risk
=
=
standard deviation
Systematic risk + unsystematic risk
Systematic risk (market risk):
Non-diversifiable!
Unsystematic risk (firm-specific risk):
Diversifiable!!
Thus: Total risk is not the risk we should be concerned with.
Only systematic risk is the risk that we should require a return for.
Fin 336: Chapter 2 - Valuation of Bonds and Stocks 10
B.
Measuring Systematic Risk: Beta
Beta measures the systematic risk of a stock (or firm) and tells how a stock’s return moves with
the market’s return. The definition of beta is:
Betai =
Cov(ri,rm)/Var(rm)
ri
rm
a stock’s returns
the market’s returns
=
=
In practice, beta is calculated by regressing returns for specific stock onto the returns for the
market (see below). The S&P500 Index is the universe of stocks that is often used to represent (be
a proxy for) the market. It is common practice to regress 60 months (5 years) worth of returns
for a stock onto the market in order to calculate the stock’s beta.
The slope of the regression line is beta.
RIBM =
α + βIBM RS&P + e
Y
a + bX + e
=
(Watch: How to calculate beta, 9:48)
Formula for a straight line
In EXCEL, the output looks like:
Intercept
X Variable 1
Coefficients Standard Error
-0.00034
0.000389
1.134888
0.061777
RIBM
t Stat
-0.885
18.37072
P-value
0.396935491
4.91706E-09
The Characteristic Line
RS&P500
Fin 336: Chapter 2 - Valuation of Bonds and Stocks 11
C.
Using Beta to Compute Return
E(r)
=
RF
+
RP
The General formula
The expected return for any security (such as a stock) is equal to the risk-free rate
plus a risk premium.
The market’s risk premium is: RPm= E(rm)- RF
The stock’s risk premium is:
E(ri) =
RF
+
RPi = [E(rm)- RF]Betai
[E(rm)- RF]Betai
The Security Market Line (SML)
The SML equation from the CAPM is a theoretical relationship between a stock’s
expected return and the risk premium for that stock.
E(r)
|
|
|
|
RF |
|________________________
Beta
The Security Market Line (SML)
If the stock has the same beta as the market, its return should be the same as the
market’s return.
Please watch: The Security Market Line (SML) vs the Capital Market
Line(CML) 7:43
Portfolio Beta:
The beta for any portfolio is simply the weighted average of the betas for all the
stocks in the portfolio, with the weights the percentage of the total portfolio that
the stock represents.
Fin 336: Chapter 2 - Valuation of Bonds and Stocks 12
D.
Common Stock Issues
1.
Cash Dividend or Share Repurchase
-
Dividends are taxed as ordinary income
Share repurchases are not
Dividends are forced on shareholders
A share repurchase is elective
Tax laws prevent corporations from executing share repurchases solely for the purpose of
allowing shareholders to avoid paying taxes on dividends.
What does dividend policy say about a company?
Dividend yield
=
Annual dividend / Price
Dividend payout rate
=
Annual dividend / EPS
-
A high dividend yield is good, if it can be sustained
A high payout rate may not be sustainable
Firms with high dividend yields and low payout rates send the signal that they are earning
enough to even pay higher dividends, or that they can easily sustain their current payout.
They would be promising companies to invest in….if the price is right.
A good company is not always a good investment (if it’s overvalued).
2.
Dividend and Signals
Dividends should be paid if earnings cannot be reinvested profitably
Some dividends are necessary to please investors, regardless of investment opportunities
Raising dividends:
Can be used by managers to signal future growth prospects
A false signal is too expensive for management
Dividends reduce agency problems
Dividend paying firms declined in the 90s…..more should pay now!
Fin 336: Chapter 2 - Valuation of Bonds and Stocks 13