Rates Interest and Bond
Valuation
Outline
Bonds and Bond Valuation
More about Bond Features
Term Structure of Interest Rate
Inflation and Interest Rates
Bond Ratings
Some Different Types of Bonds
Bond Markets
Bond Definitions
Bond is a debt security, a class of financial asset
Bond issuer is borrower and has to pay to the bond
holder
+ Coupon payments: Periodic interests (periodic
equal payments)
+ The Principal or Par Value or Face Value
Coupon rate: similar to interest rate which is the
percentage of the principal (usually semi-annually)
Maturity date: The date that the bond finishes and
paid off
Yield or Yield to maturity: The rate required in the
market on a bond (Market Discount Rate)
Bond Valuation and PV Model
Bond is a financial asset which is bought and sold
Why buys bond ? Why do you want to own bond ?
A reminder: Financial Asset generates future
income. Bond is a typical example
Owner of bond owns the future cash flows. What are
they ?
Asset Pricing: How much is this asset valued?
Market Value ?
If bond market value = PV of future cash flows
Bond Value = PV of coupons + PV of par
Bond Value = PV of annuity + PV of lump sum
The Bond Pricing Equation
If we know discount rate r then we have the
formula
If this is market value, r is the key in pricing. We
call it Yield to Maturity (YTM) or Yield
1
1 (1 + r) t
FV
Bond Value = C
+
t
r
(1 + r)
SINGLE
PV OF
ANNUITY, PAYMENT PER
PERIOD =C, t periods
FUTURE CASH
FLOW
Valuing a Discount Bond with
Annual Coupons
Consider a bond with a coupon rate of 10% and
annual coupons. The par value is $1,000, and the
bond has 5 years to maturity. The yield to maturity
is 11%. What is the value of the bond?
Using the formula:
B = PV of annuity + PV of lump sum
B = 100[1 1/(1.11)5] / .11 + 1,000 / (1.11)5
B = 369.59 + 593.45 = 963.04
Price of the bond is lower than its face value,
so called discount bond
Valuing a Premium Bond with
Annual Coupons
Suppose you are reviewing a bond that has a
10% annual coupon and a face value of $1000.
There are 20 years to maturity, and the yield to
maturity is 8%. What is the price of this bond?
Using the formula:
B = PV of annuity + PV of lump sum
B = 100[1 1/(1.08)20] / .08 + 1000 / (1.08)20
B = 981.81 + 214.55 = 1196.36
Price of the bond is higher than its face value, so
called premium bond
Graphical Relationship Between
Price and Yield-to-maturity (YTM)
1500
1400
Bond Price
1300
1200
1100
1000
900
800
700
600
0%
2%
4%
6%
8%
10%
12%
14%
Yield-to-maturity (YTM)
Coupon rate = 8% with annual coupons; Par value = $1,000; Maturity = 10 years
Bond Prices: Relationship
Between Coupon and Yield
If YTM = coupon rate, then par value = bond
price
If YTM > coupon rate, then par value > bond
price
Why? The discount provides yield above coupon rate
Price below par value, called a discount bond
If YTM < coupon rate, then par value < bond
price
Why? Higher coupon rate causes value above par
Price above par value, called a premium bond
Bond Prices: Relationship
Between Coupon and Yield
So, coupon is like interest payment, specified in
the contract, think of it as a contract rate
Yield is related to market price, the key to
pricing. It is the rate but this is the market rate
In addition, price and return are important
concepts investors care about. Whats the notion
of return you should think about in the case of
bond ?
Yield could be seen as Return. Why ?
Example 7.1 Practice of Bond
Pricing
Find present values based on the payment
period
In practice, bond makes coupon payment twice
per year. Then apply to the following example:
For a bond with par of $1000 and a coupon rate
of 14%, APRs yield of 16% and 7 years to
maturity
How many coupon payments are there?
What is the semiannual coupon payment?
What is the semiannual yield?
B = 70[1 1/(1.08)14] / .08 + 1,000 / (1.08)14 = 917.56
Interest Rate Risk and Yield
Yield is the market rate, the key to bond pricing
Interest rate risk here implies the sensitivity of
bonds price to interest rate change
Price Risk
Change in price due to changes in market interest rates. Why ?
Long-term bonds have more price risk than short-term bonds
Low coupon rate bonds have more price risk than high coupon rate
bonds
Reinvestment Rate Risk
Uncertainty concerning rates at which cash flows can be reinvested
Short-term bonds have more reinvestment rate risk than long-term
bonds
High coupon rate bonds have more reinvestment rate risk than low
coupon rate bonds
Figure 7.2
Computing Yield to Maturity
Yield to Maturity (YTM) is the rate implied by the
current bond price
Finding the YTM requires trial and error if you do
not have a financial calculator and is similar to
the process for finding r with an annuity
If you have a financial calculator, enter N, PV,
PMT, and FV, note the sign convention (PMT and
FV need to have the same sign, PV the opposite
sign)
YTM with Annual Coupons
Consider a bond with a 10% annual coupon rate,
15 years to maturity and a par value of $1,000.
The current price is $928.09.
Will the yield be more or less than 10%?
N = 15; PV = 928.09; FV = 1,000; PMT = 100
YTM with Semiannual
Coupons
Suppose a bond with a 10% coupon rate and
semiannual coupons, has a face value of $1,000,
20 years to maturity and is selling for $1,197.93.
Is the YTM more or less than 10%?
What is the semiannual coupon payment?
How many periods are there?
N = 40; PV = 1,197.93; PMT = 50; FV = 1,000;
Then solve for discount rate = 4% (Is this the YTM?)
YTM = 4%*2 = 8% (because we want annual rate
quote similar to APRs)
Current Yield, Capital Gain Yield
and Yield to Maturity
Recall what we did in RETURN calculation
The insight is we want to distinguish between income
generated by asset and the capital gain from long and short
the asset in the market
Dollar return = Income + Capital Gain (dollar)
We want to follow that direction here. Not quite exactly the
same but lets think about 1 period return
Current Yield = annual coupon / price (Income)
Yield to maturity = current yield + capital gains yield (%)
Example: 10% coupon bond, with semiannual coupons, face
value of 1,000, 20 years to maturity, $1,197.93 price
Current yield = 100 / 1,197.93 = .0835 = 8.35%
Price in one year, assuming no change in YTM = 1,193.68
Capital gain yield = (1,193.68 1,197.93) / 1,197.93 = -.0035 = -.35%
YTM = 8.35 - .35 = 8%, which is the same YTM computed earlier
Yield to Maturity and Return
So if we agree Yield is
C
C
FV
similar to return which is Pt = (1 + y ) + .... (1 + y )T t + (1 + y )T t
paramount to investors,
C
C
FV
then think
Pt +1 =
+ ....
+
T t 1
(1 + y )
(1 + y )
(1 + y )T t 1
Current Yield as return
due to income from
One period dollar return = Pt +1 Pt + C
asset
Pt +1 Pt + C C Pt +1 Pt
% Return =
=+
Capital Gain Yield as
Pt
Pt
Pt
return due to the
difference in selling and Can you show that % Return = y ???
buying price
Current
Yield
Capital
Gain
Yield
Bond Pricing Theorems
Bonds of similar risk (and maturity) will be
priced to yield about the same return,
regardless of the coupon rate
If you know the price of one bond, you can
estimate its YTM and use that to find the
price of the second bond
This is a useful concept that can be
transferred to valuing assets other than
bonds
Bond Prices with a
Spreadsheet
There is a specific formula for finding bond
prices on a spreadsheet
PRICE(Settlement,Maturity,Rate,Yld,Redemption,
Frequency,Basis)
Settlement YIELD(Settlement,Maturity,Rate,Pr,Redemption,
Frequency,Basis)
and maturity need to be actual dates
The redemption and Pr need to be input as % of par
value
Click on the Excel icon for an example
Clean vs. Dirty Prices
Clean price: quoted price
Dirty price: price actually paid = quoted price plus
accrued interest
Example: Consider a T-bond with a 4% semiannual
yield and a clean price of $1,282.50:
Number of days since last coupon = 61
Number of days in the coupon period = 184
Accrued interest = (61/184)(.04*1000) = $13.26
Dirty price = $1,282.50 + $13.26 = $1,295.76
So, you would actually pay $ 1,295.76 for the bond
Inflation and Interest Rates
Real rate of interest change in purchasing power
Nominal rate of interest quoted rate of interest,
change in actual number of dollars
The ex ante nominal rate of interest includes our
desired real rate of return plus an adjustment for
expected inflation
The Fisher Effect
The Fisher Effect defines the relationship
between real rates, nominal rates, and inflation
(1 + R) = (1 + r)(1 + h), where
R = nominal rate
r = real rate
h = expected inflation rate
Approximation
R=r+h
Example 7.5
If we require a 10% real return and we
expect inflation to be 8%, what is the
nominal rate?
R = (1.1)(1.08) 1 = .188 = 18.8%
Approximation: R = 10% + 8% = 18%
Because the real return and expected
inflation are relatively high, there is
significant difference between the actual
Fisher Effect and the approximation.
Term Structure of Interest Rates
Term structure is the relationship between time to
maturity and yields, all else equal
It is important to recognize that we pull out the
effect of default risk, different coupons, etc.
Yield curve graphical representation of the term
structure
Normal upward-sloping; long-term yields are
higher than short-term yields
Inverted downward-sloping; long-term yields
are lower than short-term yields
Figure 7.6 UpwardSloping Yield Curve
Figure 7.6 DownwardSloping Yield Curve
Figure 7.7
Insert new Figure 7.7 here
Factors Affecting Bond Yields
Default risk premium remember bond
ratings
Taxability premium municipal versus taxable
Liquidity premium bonds that have more
frequent trading will generally have lower
required returns
Anything else that affects the risk of the cash
flows to the bondholders will affect the
required returns
Differences Between
Debt and Equity
Debt
Not an ownership interest
Creditors do not have
voting rights
Interest is considered a
cost of doing business and
is tax deductible
Creditors have legal
recourse if interest or
principal payments are
missed
Excess debt can lead to
financial distress and
bankruptcy
Equity
Ownership interest
Common stockholders vote
for the board of directors and
other issues
Dividends are not considered
a cost of doing business and
are not tax deductible
Dividends are not a liability of
the firm, and stockholders
have no legal recourse if
dividends are not paid
An all equity firm can not go
bankrupt merely due to debt
since it has no debt
The Bond Indenture
Contract between the company and the
bondholders that includes
The basic terms of the bonds
The total amount of bonds issued
A description of property used as security, if
applicable
Sinking fund provisions
Call provisions
Details of protective covenants
Bond Classifications
Registered vs. Bearer Forms
Security
Collateral secured by financial securities
Mortgage secured by real property, normally
land or buildings
Debentures unsecured
Notes unsecured debt with original maturity
less than 10 years
Seniority
Bond Characteristics and
Required Returns
The coupon rate depends on the risk
characteristics of the bond when issued
Which bonds will have the higher coupon,
all else equal?
Secured debt versus a debenture
Subordinated debenture versus senior debt
A bond with a sinking fund versus one without
A callable bond versus a non-callable bond
Bond Ratings Investment Quality
High Grade
Moodys Aaa and S&P AAA capacity to pay is
extremely strong
Moodys Aa and S&P AA capacity to pay is
very strong
Medium Grade
Moodys A and S&P A capacity to pay is
strong, but more susceptible to changes in
circumstances
Moodys Baa and S&P BBB capacity to pay is
adequate, adverse conditions will have more
impact on the firms ability to pay
Bond Ratings - Speculative
Low Grade
Moodys Ba and B
S&P BB and B
Considered possible that the capacity to pay will
degenerate.
Very Low Grade
Moodys C (and below) and S&P C (and below)
income bonds with no interest being paid, or
in default with principal and interest in arrears
Government Bonds
Treasury Securities
Federal government debt
T-bills pure discount bonds with original maturity
of one year or less
T-notes coupon debt with original maturity
between one and ten years
T-bonds coupon debt with original maturity greater
than ten years
Municipal Securities
Debt of state and local governments
Varying degrees of default risk, rated similar to
corporate debt
Interest received is tax-exempt at the federal level
Example 7.4
A taxable bond has a yield of 8%, and a
municipal bond has a yield of 6%
If you are in a 40% tax bracket, which bond do
you prefer?
8%(1 - .4) = 4.8%
The after-tax return on the corporate bond
is 4.8%, compared to a 6% return on the
municipal
At what tax rate would you be indifferent
between the two bonds?
8%(1 T) = 6%
T = 25%
Zero Coupon Bonds
Make no periodic interest payments (coupon rate
= 0%)
The entire yield-to-maturity comes from the
difference between the purchase price and the
par value
Cannot sell for more than par value
Sometimes called zeroes, deep discount bonds,
or original issue discount bonds (OIDs)
Treasury Bills and principal-only Treasury strips
are good examples of zeroes
Floating-Rate Bonds
Coupon rate floats depending on some index
value
Examples adjustable rate mortgages and
inflation-linked Treasuries
There is less price risk with floating rate bonds
The coupon floats, so it is less likely to differ
substantially from the yield-to-maturity
Coupons may have a collar the rate cannot go
above a specified ceiling or below a specified
floor
Other Bond Types
Disaster bonds
Income bonds
Convertible bonds
Put bonds
There are many other types of provisions
that can be added to a bond and many
bonds have several provisions it is
important to recognize how these
provisions affect required returns
Bond Markets
Primarily over-the-counter transactions with
dealers connected electronically
Extremely large number of bond issues, but
generally low daily volume in single issues
Makes getting up-to-date prices difficult,
particularly on small company or municipal
issues
Treasury securities are an exception
Treasury Quotations
Highlighted quote in Figure 7.4
8 Nov 21 136.29 136.30 5 4.36
What is the coupon rate on the bond?
When does the bond mature?
What is the bid price? What does this mean?
What is the ask price? What does this mean?
How much did the price change from the
previous day?
What is the yield based on the ask price?
Comprehensive Problem
What is the price of a $1,000 par value bond
with a 6% coupon rate paid semiannually, if the
bond is priced to yield 5% and it has 9 years to
maturity?
What would be the price of the bond if the yield
rose to 7%.
What is the current yield on the bond if the YTM
is 7%?
End of Slide