Bond Pricing
Bond Pricing
Valuation
• What determines the price of a bond?
Contract features: coupon face value (FV) – Contract features: coupon, face value (FV),
maturity
Risk free interest rates in the economy (US – Risk-free interest rates in the economy (US
treasury yield curve)
– Credit risk premium (risk premium required forCredit risk premium (risk premium required for default risk of firm).
Terminology and Conventions
Terminology and Conventions
• Face Value (FV) : is usually $1000.( ) y $
• Coupon: quoted as % of FV and usually paid semi-annually. Thus, a 5% coupon on a face
l $ i li h h b d $
value of $1000 implies that the bond pays $25 every 6 months.
• Price is usually expressed as a % of FV • Price is usually expressed as a % of FV.
• Yield to maturity (YTM): the return you earn on the bond, if you bought the bond at its current e bo d, you boug e bo d s cu e price and held it to maturity.
– The convention in the bond market is that the YTM is
t d i i l l t hi h i t i th
quoted in nominal annual terms, which is twice the equivalent semi-annual yield.
Two Simple Questions
•
If I know the yield to maturity (or the holding-period return), how much should I pay for it? • If I buy a bond at some price, P, and hold it tomaturity, then what would be the yield to maturity (or the return) that I earn?
Calculation of the Price knowing the
Calculation of the Price, knowing the
YTM
• If I know what return (YTM) I require from
the bond, how should I price it?
,
p
• Answer: Discount the cash flows at a
discount rate that is equal to the YTM
discount rate that is equal to the YTM.
An Example (1/2)
p
(
)
• What should be the price of a Treasury Bill with a 5 1/8 % coupon maturing in 1 5 years when the first of 1/8 % coupon, maturing in 1.5 years, when the first of the semi-annual coupons is due exactly 6 months
from today? The bond is priced to yield (YTM) from today? The bond is priced to yield (YTM) 5.82%. • Answer: 25.625/(1+ 5.82%/2)^1 + ( ) 25.625/(1+5.82%/2)^2 + 1025.625/(1+5.82%/2)^3 = $990.15. • Note – Semi-annual Coupon = (1/2) (5.125%)(1000)= $25.625 – Discount factor = (0.0582/2) for a six month period.
An Example (2/2)
p
(
)
• The price is usually quoted as a % of the face
value, and expressed in 32’nds.
,
p
• Thus, the price of $990.15 will be written as
99 05 which should be read as (99 + 5/32) %
99.05, which should be read as (99 + 5/32) %
of FV.
Note that 99 + 5/32 is 99 15 (approximately) – Note that 99 + 5/32 is 99.15 (approximately).
Calculation of YTM when you know the
Calculation of YTM when you know the
price of the bond
• This is exactly the opposite the previous
question - the complication is that we
q
p
cannot algebraically solve for the YTM. So
the YTM has to be calculated by trial and
y
error.
An Example
An Example
• What is the YTM of a Treasury Bill with a
What is the YTM of a Treasury Bill with a
5 1/8 % coupon, maturing in exactly 1.5
years if it is priced at 101 00?
years if it is priced at 101.00?
• Answer: 4.43%
Conventions in the Bond Market
• Previously, we considered an example
h
h fi
d
l 6
where the first coupon date was exactly 6
months from today.
B t h
d
i
b
d if th fi t
• But how do we price a bond if the first
coupon date is not exactly 6 months from
today?
today?
• The answer depends on the day count
conventions of the bond market and it
conventions of the bond market - and it
differs depending on whether the bond is a
US treasury bond or a corporate bond.
Day Count Conventions
• Day Count Conventions specify how to count the number of days between two dates, and how to calculate the size of an interest period when the calculate the size of an interest period when the number of days is a fraction of a normal period. • There are two principal day-count conventions: p p y
actual/actual and 30/360.
• Act/Act: The actual number of days between coupon t d t l l t th l th f th
payments are used to calculate the length of the period between coupons. Eg. US Treasury
• 30/360: The number of days between coupons is • 30/360: The number of days between coupons is
computed under the assumption that each month has 30 days and that the year has 360 days. Eg. US
corporate bond market
C
t
T
b
d
Corporate vs. Treasury bonds
• The treasury market uses a day count
• The treasury market uses a day count
convention of act/act, while the corporate
bond market usually uses a convention of
bond market usually uses a convention of
30/360.
Examples
• Consider the following prices and yields of
Treasury securities, taken from the Wall Street Journal. If you access the online version of WSJ, then go to Markets Data Center for the US
Treas r q otes Treasury quotes.
• http://online.wsj.com/mdc/public/page/2_30
20
h
l?
d
d
b d
l k
Prices of Treasury Issues
Prices of Treasury Issues
(WSJ, 11/23/2007)
Treasury Issues
Maturity Coupon Bid Ask Yield
11/15/2009 4.625 102:30 102:31 3.05
11/15/2014 4.25 102:28 102:29 3.76
2/15/2031 5.375 113:13 113:14 4.44
Trade Date 11/23/2007
Computing the Price from YTM
• Let us consider our earlier question of how
to compute the price of a Treasury, knowing
p
p
y,
g
the yield to maturity.
• Consider the 4 625% coupon Treasury
Consider the 4.625% coupon Treasury
security maturing on 11/15/2009. The yield
of this security is 3 05%
of this security is 3.05%.
• Given this yield, what would be the market
quote for this Treasury security?
Computations (1/4)
• Refer to the spreadsheet. Explanation is provided below.
• Trade date: Friday, 11/23/2007
• Settlement day: 2nd business day (Tuesday, 11/27/2007)
11/27/2007)
• Maturity = 11/15/2009.
• Face Value = 100 (as we quote prices in % we do • Face Value = 100 (as we quote prices in %, we do
not need to write 1000)
• Last coupon was paid on 11/15/2007.Last coupon was paid on 11/15/2007.
• Remaining coupon dates are: 5/15/2008, 11/15/2008, 5/15/2009, 11/15/2009.
• The amount of coupon payment =100*4.625%/2 = $2.31.
C
t ti
(2/4)
Computations (2/4)
• Number of days between coupon payments is equal to the
l b f d b h l d
actual number of days between the last coupon date
(11/15/2007) and the next coupon date (5/15/2008) => 182 days.
N b f d t t d t f 5/15/2008 f
• Number of days to next coupon date of 5/15/2008 from settlement date of 11/27/07 => 170
• Length of period from settlement to next coupon date
( t l b f d t t / t l b f
(actual number of days to next coupon / actual number of days between coupons) = 170/182=0.934.
• Yield to Maturity = 3.05% (given)
PV f b d 2 31/(1 0 0305/2)^0 934
• PV of bond = 2.31/(1+0.0305/2)^0.934 +
2.31/(1+0.0305/2)^1.934 + 2.31/(1+0.0305/2)^2.934 +102.31/(1+0.0305/2)^3.934 = 103.14
Computations (3/4)
p
(
)
• The present value of $103.14 is called the ‘dirty’ price of the bond However the bond is not quoted price of the bond. However, the bond is not quoted at this price.
• The convention is that the bond is quoted in terms q of its ‘clean’ price.
– Clean Price = Dirty Price – Accrued Interest. • As the bond was bought 12 days after the last
coupon date of 11/15/2007, the bond is said to have accrued 12 days of interest
have accrued 12 days of interest.
• Accrued interest = (accrued days)/(actual days between last and next coupon date) x (coupon p ) ( p amount) = (170/182) x 2.31 = $0.15
Computations (4/4)
p
(
)
• Clean price = Dirty price – accrued interest
$103 14
0 15
$102 98
= $103.14 – 0.15 = $102.98.
• The price is quoted in 32nds, so we have to
convert $0.98 into 32nds by multiplying by
32 = > 0.98 = 31/32
• Quoted Price = 102 + 0.31 = 102:31.
• Thus this is why the WSJ quotes the price
Thus, this is why the WSJ quotes the price
of the Treasury bond as 102:31, even
though its true value is 103 14
though its true value is 103.14.
Finding the YTM
• Similarly, we can set up a spreadsheet to
compute the yield to maturity, given the
p
y
y, g
price of the bond.
Risk of Bonds
Risk of Bonds
• What caused the value of a bond to change? There are two main sources of risk:
• 1. Interest rate risk: This is the risk that the yield (th i k f US T t ) ill h
curve (the risk-free US Treasury rates) will change. • 2. Default risk (or credit risk): This is the risk that
the corporation will default on the bond the corporation will default on the bond.
• There are other sources of risk for certain bonds – in particular, if you invest in mortgage-backed securities, there is “pre-payment” risk – the risk that the loan on real estate will be pre-paid before maturity
