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CHAPTER 7
Bonds and Their Valuation




Key features of bonds
Bond valuation(price)
Measuring yield(return)
Assessing risk
7-1
What is a bond?

A long-term debt instrument in
which a borrower agrees to make
payments of principal and interest,
on specific dates, to the holders of
the bond.
7-2
Key Features of a Bond




Par value – face amount of the bond, which
is paid at maturity (assume $1,000).
Coupon rate – stated interest rate (generally
fixed) paid by the issuer. Multiply by par to
get dollar payment of interest.
Maturity date – years until the bond must be
repaid.
Issue date – when the bond was issued.
7-3
The value of financial assets
0
1
2
r
Value
n
...
CF1
CF2
CFn
CF1
CF2
CFn
Value 

 ... 
1
2
n
(1  r) (1  r)
(1  r)
7-4
Bond valuation model


Rd is the required return of the bond. It is
also the market rate of interest of the bond,
and yield to maturity (YTM) of the bond.
Rd is NOT coupon rate
Vb (bond price) 
coupon payment
coupon payment face value

...


1
N
(1  rd )
(1  rd )
(1  rd ) N
7-5
What is the value of a 10-year, 10% annual
coupon bond, if rd = 10%?
0
1
2
r
Vb = ?
n
...
100
100
100 + 1,000
$100
$100
$1,000
Vb 
 ... 

1
10
(1.10)
(1.10)
(1.10)10
Vb  $90.91  ...  $38.55  $385.54
Vb  $1,000
7-6
Using a financial calculator to value a
bond

This bond has a $1,000 lump sum due at t = 10,
and annual $100 coupon payments beginning at
t = 1 and continuing through t = 10, the price of
the bond can be found by solving for the PV of
these cash flows.
INPUTS
OUTPUT
10
10
N
I/YR
PV
100
1000
PMT
FV
-1000
7-7
An example:
Increasing inflation and rd

Suppose inflation rises by 3%, causing rd =
13%. When rd rises above the coupon rate,
the bond’s value falls below par, and sells at
a discount.
INPUTS
OUTPUT
10
13
N
I/YR
PV
100
1000
PMT
FV
-837.21
7-8
An example:
Decreasing inflation and rd

Suppose inflation falls by 3%, causing rd =
7%. When kd falls below the coupon rate,
the bond’s value rises above par, and sells
at a premium.
INPUTS
OUTPUT
10
7
N
I/YR
PV
100
1000
PMT
FV
-1210.71
7-9
Semi-annual coupon bond

Majority of bonds pay interest semiannually.







Coupon rate=10%/Y
Going (nominal) interest=5%
15 year bond
Solution: P semi annual=$1523.26
How: N=15*2 i=5/2 PMT=$100/2
Compared with annual coupon bond, which
should have a higher price?
P annual=$1518.98
7-10
What is the YTM on a 10-year, 9% annual coupon,
$1,000 par value bond, selling for $887?

Must find the rd that solves this model.
VB 
coupon payment
coupon payment face value

...


1
N
(1  rd )
(1  rd )
(1  rd ) N
$887 
90
90
1,000

...


(1  rd )1
(1  rd )10 (1  rd )10
7-11
Using a financial calculator to find YTM

Solving for I/YR, the YTM of this bond is
10.91%. This bond sells at a discount,
because YTM > coupon rate.
INPUTS
10
N
OUTPUT
I/YR
- 887
90
1000
PV
PMT
FV
10.91
7-12
Find YTM, if the bond price was $1,134.20.

Solving for I/YR, the YTM of this bond is
7.08%. This bond sells at a premium,
because YTM < coupon rate.
INPUTS
10
N
OUTPUT
I/YR
-1134.2
90
1000
PV
PMT
FV
7.08
7-13
Relationship between interest and bond
price (inversely realted)
Interest
Rate
12%
10%
8%
Bond Value
874.50
1,000
1,152.47
7-14
Interest sensitivity and maturity

Annual Coupon rate=10%
% change 1 yr
+4.8% $1,048
rd
5%
10yr
$1,386
$1,000
10%
$1,000
$957
15%
$749
-4.3%
% change
+38.6%
-25.1%
The 10-year bond is more sensitive to interest
rate changes, and hence has more interest rate
risk.
7-15
Interest sensitivity and coupon rate
10 year annual bond
% change
+44%
-28%


5%c
$1,000
rd
5%
10%c
$1,386
$693
10%
$1,000
15%
$749
$498
% change
+38.6%
-25.1%
Low coupon bond has CF concentrated at the end of
maturity from repayment of principal.
The bond with lower coupon rate is more sensitive to
interest rate changes
7-16
Calculation details
N
1Y,10% coupon
bond
10Y, 10% coupon
bond
10Y, 5% coupon
bond
10Y, 10% coupon
bond
I(%)
1
1
1
10
10
10
10
10
10
10
10
10
5
10
15
5
10
15
5
10
15
5
10
15
PV=? PMT FV
-1048
100 1000
-1000
100 1000
-957
100 1000
-1386
100 1000
-1000
100 1000
-749
100 1000
-1000
50 1000
-693
50 1000
-498
50 1000
-1386
100 1000
-1000
100 1000
-749
100 1000
7-17
Summary of Factors that Affect bond Prices
and Price sensitivity when Interest Rates
Change

Interest Rate



Time Remaining to Maturity


negative relation between interest rate changes
and bond price
increasing interest rates correspond to bond price
decrease (at a decreasing rate)
longer time to maturity corresponds to larger price
change for a given interest rate change
Coupon Rate

the lower the coupon rate, the bigger the price
change for a given change in interest rates
7-18
Reinvestment risk


The risk that bondholders have to
reinvest future cash flows (coupon and
principal when expires)at lower interest
rates if general interest level declines
EXAMPLE: Suppose you just won
$500,000 playing the lottery. You
intend to invest the money and live off
the interest.
7-19
Reinvestment Rate Risk Example


You may invest in either a 10-year bond or a series of
ten 1-year bonds. Both 10-year and 1-year bonds
currently yield 10%.
If you choose the 1-year bond strategy:


After Year 1, you receive $50,000 in income and have $500,000
to reinvest. But, if 1-year rates fall to 3%, your annual income
would fall to $15,000.
If you choose the 10-year bond strategy:

You can lock in a 10% interest rate, and $50,000 annual income
for 10 years.
7-20
7-20
Conclusions about Interest Rate and
Reinvestment Rate Risk
Interest rate risk
Reinvestment rate risk
Short-term
AND/OR
High-coupon
Bonds
Low
High
Long-term
AND/OR
Low-coupon
Bonds
High
Low
CONCLUSION: Nothing is riskless!
 Interest risk is a big concern in low
interest environment, and reinvestment
risk is a big concern in high interest
7-21
7-21
environment

Default Risk


If an issuer defaults, investors receive less
than the promised return.
Influenced by the issuer’s financial
strength and the terms of the bond
contract.
7-22
7-22
Evaluating Default Risk:
Bond Ratings
Investment Grade
Junk Bonds
Moody’s
Aaa Aa A Baa
Ba B Caa C
S&P
AAA AA A BBB
BB B CCC C


Bond ratings are designed to reflect the
probability of a bond issue going into
default.
High risk demands high yield. (PIGS 4
are paying high yield 7-23
on their bonds) 7-23
Optional: Bond investing in real world

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Find bond information
Where to buy?
Don’t buy individual bond (unless riskfree T bond)
Risk-free could be illusion, price does
change
Bond ETF (HYG,JNK, BSV, SCPB)
7-24
Factors to consider when investing in
bonds

Yield

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For single bond: current yield( annual
interest payment/current price) and yield
to maturity(consider principal payment).
For bond ETF or mutual fund: Yield
(current yield) and SEC yield (consider ytm
and expense)
Default risk and rating
Interest risk and duration

Duration: the effective maturity of a bond
7-25
(average maturity for a bond ETF)
Optional: practical questions

Is Treasury bond really risk-free?

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Risk free bond price changes
Is it good time to buy bonds now?
Interest changes
 LT vs. ST bond
Should I buy individual high-yield corporate
bond?


7-26
Example of treasure bond

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Yahoo-Finance-Market data-Bond
Comparing LT bond yield with ST yield
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Expectation of interest rate
Default risk?
Interest risk and time to maturity
7-27
Examples of bond portfolio
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HYG (bond ETF)

12-month Yield: the sum of a fund’s total
trailing 12-month interest and dividend
payments divided by the last month’s ending
share price (NAV) plus any capital gains
distributed over the same period.

SEC yield: based on 30-day period ending on the
last day of previous month, reflects the
deduction of the fund’s expenses.
7-28
Examples of bond portfolio
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
Rating(default risk):
Duration(interest risk)
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Duration: Average years to maturity
Longer duration is more sensitive to interest
changes
When interest rate increases from 3% to 4%, a
bond with duration of 4 years will see price
decline of approximate 4*(4%-3%)=4%
7-29
Examples of bond portfolio
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Portfolio holdings
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Weight, maturity and coupon rate
Risk of each individual issue and the benefit of
diversification
Compared with other bond ETF
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BSV (ST T bond+ corporate,yield,grade, duration)
SCPB (corporate A bond)
JNK-high yield (similar to HYG)
7-30