### FI4000 Fundamentals of Valuation

```FINC4101
Investment Analysis
Instructor: Dr. Leng Ling
Topic: Bond Pricing and Yields
1
Learning objectives
1.
2.
3.
4.
5.
6.
7.
8.
9.
Compute the price of a zero-coupon bond.
Compute the price of a fixed coupon bond.
Describe the price-yield relationship of bonds.
Distinguish between a bond’s flat price and its invoice
price.
Compute different measures of bond returns.
Calculate how bond prices will change over time for a
given interest rate projection.
Recognize default/ credit risk as a source of risk for
bonds.
Identify the determinants of bond safety and rating.
Understand how default risk can affect yield to maturity.
2
Concept Map
Foreign
Exchange
Portfolio
Theory
Asset
Pricing
FI4000
Equity
Derivatives
Market
Efficiency
Fixed Income
3
Types of fixed-income securities
Fixed-income security / bond:
 A security that obligates the issuer to
make specified payments to the holder
over a period of time.
 We focus on the pricing of two types of
fixed-income securities:
1. Fixed-coupon bond (FCB)
2. Zero-coupon bond (ZCB)
4
Fixed-coupon bond (FCB) 1

Firm pays a fixed amount of interest (‘coupon payment’)
to the investor every period until bond matures.

At maturity, firm pays face value of the bond to investor.

Face value also called par value. Unless otherwise
stated, always assume face value to be \$1000.

Period: can be year, half-year (6 months), quarter (3
months).

Coupon rate: annual coupon payment as a fraction of
face value.
5
fixed-coupon bond: Example
A firm issues an 8% 30-year bond with annual
coupon payments. Par value is \$1,000.
What does the above tell us?







8%: the coupon rate.
Multiply coupon rate by par value to get annual coupon
payment. Coupon= 8% x 1,000 = \$80.
Maturity = 30 years.
Coupon of \$80 is paid annually, i.e., period=annual.
At maturity (end of 30 years), firm will pay \$1000 to
investor.
What happens if coupon is paid semi-annually?
Quarterly?
6
Fixed-coupon bond (FCB) 2
FCB gives you a stream of fixed payments plus
a single payment (face value) at maturity.
 This cash flow stream is just an annuity plus a
single cash flow at maturity.
 Therefore, we calculate the price of a FCB by
finding the PV of the annuity and the single
payment, using an appropriate interest rate.
 We use the financial calculator to compute the
price of the FCB.

7
Fixed-coupon bond (FCB) 3
Reminder:
 The “interest rate” used to find the PV of a
bond is also known as:
•
•
•
•
Yield-to-maturity (“YTM” or “yield” for short)
Discount rate
Required rate of return
Cost of debt
8
Fixed-coupon bond (FCB) 4
Price of the FCB, PFCB
Fixed periodic
coupon
Number of
periods to
maturity
Face value
N
C
F
PFCB  

t
N
1  rd 
t 1 1  rd 
Yield to maturity (in decimals)
9
Find FCB price

Consider an 8%, 30-year coupon bond that pays
coupons semi-annually. Compute the bond’s
price if the yield to maturity is a) 6%, b) 8%, c)
10%.

If YTM = 6%, verify that bond price = \$1,276.76
If YTM = 8%, verify that bond price = \$1000.
If YTM = 10%, verify that bond price = \$810.71


10
Inverse relationship between
price & interest rate
 Notice
that as YTM (interest rate)
increases, bond price decreases.
Conversely, as YTM decreases, bond
price increases.
 This is the inverse relationship between
bond price and yield to maturity (interest
rate).
 This is a crucial general rule in bond
pricing based on time value of money.
11
The Inverse Relationship Between
Bond Prices and Yields
12
Figure 10.1 Prices/Yields of U.S. Treasury Bonds
on Aug 15, 2011
13
Figure 2.3 Prices/Yields of U.S. Treasury Notes
page 32
Minimum tick size: ½ of 1/32
Special case: ¼ of 1/32 =1/128
Bid: ( 112+30/32)% of face value
100:18.75
100:14
112:30
107:9.5
…
37/64+1/128
28/64=14/32
60/64=30/32
19/64=9.5/32
14
Treasury Bonds and Notes
 Treasury
Notes are issued with maturities
between 1 and 10 years.
 Treasury bonds are issued with maturities
ranging from 10 to 30 years.
 Domination: \$100, \$1000.
 Both make semiannual coupon payments.
 Bid and ask are quoted as a percentage of
par value.
15
Invoice price, Flat price,
Accrued interest
when you buy or sell a bond between coupon
payment dates, the invoice price must
incorporate accrued interest.
Invoice price = quoted price + Accrued interest
(price quoted in the financial press)

For a semi-annual payment bond, accrued
interest between two coupon payment dates =
annualcouponpaym ent dayssin ce last couponpaym ent

2
daysseparatingcouponpaym ents
16
Example 10.1
 Suppose
the coupon rate is 8%, face
value is \$1000 and coupon is paid
semiannually. 30 days have passed since
the last coupon payment. The quoted price
is \$990. What is the invoice price?
990+(1000*8%/2)*(30/182)
=990+6.59
=996.59
17
Exercise

Suppose the coupon rate is 10%, face value is
\$1000 and coupon is paid semiannually. 125
days have passed since the last coupon
payment. The Ask quote is 100:11. What is the
invoice price that the investor has to pay?
18
Price of a maturing 8% Treasury bond
What is the quoted price one day before the
maturity? \$1000.
 A purchaser will receive 1040 on the following
day.
 So he should be willing to pay an invoice price of
\$1040.
 40 is the accrued interest since the previous
coupon payment.

19
Corporate Bonds Figure 10.2
20
Find YTM, Coupon rate
(suppose the bond is issued today)
1)A \$1,000 par value bond sells for \$863.05. It matures in
20 years, has a 10 percent coupon rate, and pays
interest semi-annually. What is the bond’s yield to
maturity on a per annum basis (to 2 decimal places)?
Verify that YTM = 11.80%
2) ABC Inc. just issued a twenty-year semi-annual coupon
bond at a price of \$787.39. The face value of the bond is
\$1,000, and the YTM is 9%. What is the annual coupon
rate (in percent, to 2 decimal places)?
Verify that annual coupon rate = 6.69%
What happens if bond pays coupon annually? Quarterly?
21
Zero-coupon bond (ZCB) 1

Zero coupon rate, no coupon paid during bond’s life.
Bond holder receives one payment at maturity, the face
value (usually \$1000).

Price of a ZCB, PZCB

F = face value of the bond
PZCB 
F
1  r

ZCB N
d
N = number of
periods to maturity
Yield to maturity (in decimals)
22
Zero-coupon bond (ZCB) 2
 As
long as interest rates are positive, the
price of a ZCB must be less than its face
value.
 Why? With positive interest rates, the
present value of the face value (i.e., the
price) has to be less than the face value.
23
These problems are just basic TVM problems where you receive a
single cash flow in the future.
ZCB Problems
1) Find the price of a ZCB with 20 years to maturity,
par value of \$1000 and a yield to maturity of 15%
p.a. Assume annual compounding.
N=20, I/Y=15, FV=1000, PMT=0. Price = \$61.10
2) XYZ Corp.’s ZCB has a market price of \$ 354. The
bond has 16 years to maturity and its face value is
\$1000. What is the yield to maturity for the ZCB.
Assume annual compounding.
PV=-354, FV=1000, N=16, PMT=0.
YTM = 6.71% p.a.
What if semi-annual compounding?
24
Treasury Bills (T-bills)
Chapter 2, page 27





Maturities: 4,13,26, 52 weeks.
Denomination: 100, 10000
Income from T-bill is taxable at the federal level, not
state or local level.
The cash flow pattern looks like a ZCB. Issued at a
discount from par value, return the par value at maturity.
The discount from the par (face) value is annualized
based on a 360-day year and reported as a percentage
of face value.
25
Figure 2.1
ask : 0.07%  ( 245 / 360)  0.047639%, \$10,000  (1  0.047639%)  \$9,995.236
bid : 0.085%  ( 245 / 360)  0.057847%, \$10,000  (1  0.057847%)  \$9,994.215
10,000 / 9,995.236  1.000477, HPR : 0.000477  0.0477%
Annual R : HPR  (365 / 245)  0.0711%
26
Figure 2.2 (8th edition)
bid : 0.0120  (161 / 360)  0.00536667, \$10,000  (1  0.00536667)  \$9,946.33
ask : 0.0119  (161 / 360)  0.00532194, \$10,000  (1  0.00532194)  \$9,946.78
HPR : (10,000  9,946.78) / 9,946.78  0.00535048
Annual R : HPR  (365 / 161)  0.01212996
27
Example of T-bill
A T-bill with face value \$10,000 and 87 days
to maturity is selling at a bank discount ask
yield of 3.4%, what is the price of the bill?
87 days
Bank discount of 87 days: 0.034 x
= 0.008217
360 days
Price: \$10,000 x (1 – 0.008217) = \$9,917.83
28
Measures of return
 We
measure the rate of return from
investing in a bond in several ways:
1. Yield to maturity (YTM)
2. Current yield (CY)
3. Yield to call (YTC)
4. Realized compound yield (RCY)
5. Holding period return (HPR)
29
Yield to maturity
The discount rate that makes the present value of
a bond’s payments (coupons & par value) equal
to its price.
 Interpretation: it is the compound rate of return
that will be earned over a bond’s life if
 It is bought now and held until maturity
 All coupons are reinvested at the same YTM.

30
Annualizing YTM (1)
If coupons are paid semi-annually, then the YTM
we get from the financial calculator is a sixmonth YTM.
 We can convert this six-month YTM to an annual
YTM using



Simple interest => bond equivalent yield to maturity
(or bond equivalent yield for short)
OR
Compound interest => effective annual yield to
maturity (or effective annual yield for short)
31
Annualizing YTM (2)
 In
general, if a coupon bond pays coupons
m times a year, then:
Bond equivalent yield
= periodic YTM x m
Effective annual yield
= (1 + periodic YTM)m – 1
Stated in decimals
Note: periodic YTM is the value of I/Y you get from the financial calculator.
32
Annualizing YTM (3)


A 20-year maturity bond with par value
\$1000 makes semi-annual coupon
payments at a coupon rate of 8%. Find
the bond equivalent and effective annual
yield to maturity of the bond if the bond
price is \$950.
Verify that


Bond equivalent yield = 8.53%
Effective annual yield = 8.71%
33
Annualizing YTM (4)
 Treasury
bonds paying an 8% coupon rate
with semi-annual payments currently sell
at par value. What coupon rate would they
have to pay in order to sell at par if they
paid their coupons annually ?
34
Practice 6 (1)
Chapter10:
4,5,6,14,15,17,22, 27
35
Homework 6
1.
2.
You are a US. Treasury bond dealer who trades a
4.75%, 3year, semi-annual coupon bond. Your required
YTM is 0.92826126%. How should you quote your
Asked price in percentage of par value as shown in
Figure 10.1?
Suppose today is Oct 23, 2013. A bond with a 10%
coupon paid semiannually every Feb 15 and Aug15 is
listed as selling at an ask price of 102:11. if you buy the
bond from a dealer today, what price will you pay for it?
the coupon period has 182 days.
36
Bond price, coupon rate & YTM (1)
A
\$1,000 par value bond has coupon rate
of 5% and the coupon is paid semiannually. The bond matures in 20 years
and has a yield to maturity of 10%.
Compute the current price of this bond.
FV=1000, PMT =25, I/Y=5, N=40. CPT, then PV.
PV = -571.02. Thus, price = \$571.02 < par value
37
Bond price, coupon rate & YTM (2)
Go back to the bond in the last problem.
 Suppose annual coupon rate = 10%.
 Verify that price = \$1000 = par value
Suppose annual coupon rate = 12%
 Verify that price = \$1,171.59 > par value.

It turns out that the following property is true.
38
Bond price, coupon rate & YTM (3)
Coupon rate < YTM Price < face value Bond is selling
at a discount
Coupon rate = YTM Price = face value Bond is selling
at par
Coupon rate > YTM Price > face value Bond is selling
39
Apply what we learnt

A 10-year annual coupon bond was issued four years
ago at par. Since then the bond’s yield to maturity
decreased from 9% to 7%. Which of the following
statements is true about the current market price of the
bond?
A.
The bond is selling at a discount
The bond is selling at par
The bond is selling at a premium
The bond is selling at book value
Insufficient information
B.
C.
D.
E.
40
Try one more

One year ago Pell Inc. sold 20-year, \$1,000 par value,
annual coupon bonds at a price of \$931.54 per bond. At
that time the market rate (i.e., yield to maturity) was 9
percent. Today the market rate is 9.5 percent; therefore
the bonds are currently selling:
A.
at a discount.
at par.
above the market price.
not enough information.
B.
C.
D.
E.
41
Current yield (1)
 Annual
coupon payment divided by bond
price.
 Measures return from coupon payments.
 Shortcomings:
Ignores capital gains or losses from bond sale.
 Ignores income from reinvestment of coupon
payments.

42
Current yield (2)


Consider a 6%, 15-year, semi-annual
payment bond with a par value of \$1000.
Compute the current yield if the yield to
maturity is: (a) 7%, (b) 6%, (c) 5%
Verify that
(a) Price = \$908.0398, current yield = 6.61%
(b) Price = \$1000, current yield = 6%
(c) Price = \$1104.6515, current yield = 5.43%
43
Current yield (3)
Observe that…
If bond is selling at…
Coupon
rate (%)
Current
yield (%)
YTM (%)
Discount
Price = 908.0398
6
6.61
7
Par
Price = 1000
6
6
6
Price = 1104.6515
6
5.43
5
44
In general, we have the following relationship
between coupon rate, current yield, & YTM
If bond is selling at… Then we know that…
Discount
(below par)
Coupon rate < Current yield < YTM
Par
Coupon rate = Current yield = YTM
(above par)
Coupon rate > Current yield > YTM
45
Quick review
A
bond has a current yield of 9% and a
yield to maturity of 10%. Is the bond
selling above or below par value?
 Is
the coupon rate of the bond more or
less than 9%?
46
Yield to call (1)

Applicable only to callable bonds.

What’s a callable bond?


Bond that may be repurchased by the issuer at a specified price
(“call price”) before the maturity date.

Call period: The period of time during which the issuer can
repurchase the bond.

Time until call: The period of time before the issuer can start
Motive: if interest rates fall, issuer can repurchase the
bonds and issue new bonds at lower coupon rate. This
lowers interest payments.
47
Yield to call (2)

Yield to call is calculated just like the yield to
maturity, except:


Time until call replaces time to maturity
Call price replaces par value
Fixed periodic coupon
N
PFCB
C
Call Pr ice


t
N
(1  rFC )
t 1 (1  rFC )
Time until call
Yield to call
48
Yield to call (3)




A 20-year maturity bond with par value \$1000 makes
semi-annual coupon payments at a coupon rate of 8%.
The bond is currently selling for \$1,150 and is callable in
10 years at a call price of \$1,100.
Compute the bond equivalent yield to call and the
effective annual yield to call.
What is N? What is FV?
Verify that


Bond equivalent yield to call = 6.64%
Effective annual yield to call = 6.75%
49
Realized compound yield (1)
 Compound
rate of return based on coupon
payments, reinvestment income and sale
price during the holding period.
 Realized compound yield depends on:
Reinvestment rate: interest rate at which
coupon payments are reinvested.
 Holding period
 YTM at the end of holding period

50
Realized Compound Yield (2)

Realized compound yield, y
Length of holding period
1
n
 final proceeds
  1
y  
 purchase price 



Purchase price = what you paid for the bond
Final proceeds = future value of coupon payments and
reinvestment income + sale price
n = length of holding period (could be in years, halfyears, quarters etc).
51
Realized Compound Yield (3)
 Suppose
you buy a 30-year, 7.5%, annual
payment coupon bond for \$980 and plan
to hold it for 20 years. Your forecast is that
the bond’s YTM will be 8% when it is sold
and that the reinvestment rate on the
coupons will be 6%. Compute the annual
realized compound yield.
52
Realized Compound Yield (4)



Five years ago, XYZ Inc issued a 5% semi-annual
payment coupon bond with a maturity of ten years. You
buy the bond now at a price of \$683.94 and plan to hold
it for three years. You forecast that you can invest the
coupon payments at a stated annual rate of 6.25% and
that at the end of three years, the yield will be 7.75%.
What is the bond equivalent realized compound yield?
(i.e., annualize using simple interest)
What is the effective annual realized compound yield?
(i.e., annualize using compound interest)
53
Yield to Maturity vs.
Realized Compound Yield (1)

Consider a 7% annual payment bond with two
years to maturity. The YTM is 8% right now.
Compute the realized compound yield if the
reinvestment rate is (a) 7%, (b) 8%, (c) 9%.

Verify that the realized compound yield is:
7.97% < YTM
8%
= YTM
8.03% > YTM
a)
b)
c)
54
Yield to Maturity vs.
Realized Compound Yield (2)
 In
general, if you hold a bond to maturity,
then:
 RCY < YTM if reinvestment rate < YTM
 RCY = YTM if reinvestment rate = YTM
 RCY > YTM if reinvestment rate > YTM
55
Holding period return (HPR)

Rate of return over a single investment (holding)
period. Has two components:


Price change = ending price – beginning price
Coupon payment
HPR
P rice change + Coupon payment
=
Beginning price
Ending price - Beginning price + Coupon payment
=
Beginning price
56
HPR problem

Suppose you buy a bond of General Electric at a
price of \$990. The bond pays coupons semiannually, has an annual coupon rate of 6%, a
face value of \$1,000 and will mature in six
months’ time. You intend to hold the bond till it
matures.

What is the 6-month HPR?
57
HPR vs. YTM (1)
 When
the YTM stays the same during the
holding period, HPR = YTM.
 If the YTM changes after you bought the
bond, then HPR will be different from the
initial YTM.
If YTM falls after you bought the bond,
HPR > initial YTM.
 If YTM rises after you bought the bond,
HPR < initial YTM.

* This property is true only if the end of the holding period is NOT
the maturity date.
58
HPR vs. YTM (2)

Consider a 30-year bond paying an annual
coupon of \$80 and selling at \$1000. The bond’s
initial YTM is 8%. Suppose you buy the bond
now and hold it for one year. Compute the 1year HPR if YTM at the end of the year is
(a) 8%, (b) 7%, (c) 9%.
Verify that:
a) If year-end YTM is 8%, then HPR = 8%
b) If year-end YTM is 7%, then HPR = 20.28%
c) If year-end YTM is 9%, then HPR = -2.2%
59
Bond prices over time

Suppose the yield to maturity is the same from the time
you purchased a bond to the bond’s maturity.
If you bought the Bond price over time
bond at ….
Price falls and approaches par value.
Par
Price remains at par value till maturity.
Discount
Price rises and approaches par value.
60
Bonds over Time
61
Quick review (1)

a)
b)
c)
d)
Consider a bond with a 10% coupon rate and
with YTM of 8%. If the bond’s YTM remains
constant, then in one year, will the bond price
be
Higher
Lower
Unchanged
None of the above
62
Quick review (2)
Which of the following statements is correct?
If market rates do not change, the price of a bond selling
at a premium increases over time.
b) If interest rates are greater than zero, it is possible for a
zero-coupon bond to sell at a premium (i.e. for more than
par value).
c) If a bond’s yield to maturity is greater than its coupon rate,
the bond will sell at a discount.
d) If market rates do not change, the price of a bond selling
at a discount decreases over time.
e) All of the above statements are false.
a)
63
Default risk

What is default?


Failure of the bond issuer to make the promised
coupon &/or par value payments to the bond holder.
Default risk

To the bond holder, default risk is the uncertainty in
cash flows arising from the possibility that the issuer
can fail to make promised payments.

With the exception of the U.S. government, all issuers
have default risk.

Some issuers have greater default risk than others.
64
Measuring default risk

Credit rating agencies measure default risk of:




Well-known credit rating agencies:





Large corporate bond issues
Municipal bond issues
International sovereign bond issues
Moody’s
Standard & Poor’s (S&P for short)
Duff & Phelps
Fitch
Each agency assigns a letter grade to reflect
level of default risk.
65
S&P, Moody’s credit rating scheme
66
Bonds fall into one of the two broad groups:
Rated BBB and above by S&P, or
 Rated Baa and above by Moody’s

 Speculative
Rated BB or lower by S&P, or
 Rated Ba or lower by Moody’s or
 Unrated bond

67
Determinants of credit rating


To assign rating, credit rating agencies
look at level and trend of issuer’s financial
ratios.
Key ratios:
1.
2.
3.
4.
5.
Coverage: times-interest-earned
Leverage: debt-to-equity, debt-to-assets
Liquidity: current, quick
Profitability: ROA
Cash flow-to-debt
68
Protection Against Default

Credit Default Swap (CDS)
An insurance policy on the default risk of a corporate
percentage of each \$100 of bond principal). The sellers
compensate the buyers for loss of bond value in the event
of a default.

Bond indenture specifies restrictions on
issuer to protect bondholder against
default.
Sinking fund; Subordination of further debt;
Dividend restrictions; Collateral
69
Yield to Maturity and Default Risk (1)
 Promised/stated


YTM:
YTM you get if issuer makes all promised payments.
Maximum possible YTM.
 Expected
YTM:
YTM you get if you consider the possibility of default.
Consider the following example.


70
Yield to Maturity and Default Risk (2)



A firm issued a 9% semi-annual payment coupon bond
20 years ago. The bond now has 10 years left until its
maturity date, but the firm is having financial difficulties.
Investors believe that the firm will be able to make good
on the remaining interest payments but that at the
maturity date, the firm will be forced into bankruptcy, and
bondholders will receive only 70% of par value. The
bond is selling at \$750.
Calculate the promised (bond equivalent) YTM and the
expected (bond equivalent) YTM.
Verify that the promised YTM is 13.7% and the expected
YTM is 11.6%.
71
Summary
1.
Compute the price of (a) fixed-coupon bond, b) zero-coupon bond.
2.
Bond price is inversely related to interest rates.
3.
Different measures of bond returns:
a)
yield to maturity,
b)
current yield,
c)
yield to call,
d)
realized compound yield,
e)
holding period return.
4.
Relationships between different measures of bond returns.
5.
Evolution of bond prices over time if YTM remains the same.
6.
Credit/ default risk as a source of risk in bond investing.
7.
Determinants of bond safety and rating.
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Practice 6 (2)
Chapter10: 7,8,9,12,16,23,28,32.
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