Bond Pricing

1
Bond Pricing

• Fundamentals

2
Valuation

• What determines the price of a bond?
• Contract features coupon, face value (FV),
  maturity
• Risk-free interest rates in the economy (US
  treasury yield curve)
• Credit risk premium (risk premium required for
  default risk of firm).

3
Terminology and Conventions

• Face Value (FV) is usually 1000.
• Coupon quoted as of FV and usually paid
  semi-annually. Thus, a 5 coupon on a face value
  of 1000 implies that the bond pays 25 every 6
  months.
• 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
  price and held it to maturity.
• The convention in the bond market is that the YTM
  is quoted in nominal annual terms, which is twice
  the equivalent semi-annual yield.

4
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 to
  maturity, then what would be the yield to
  maturity (or the return) that I earn?

5
Calculation of the Price, knowing the YTM

• If I know what return (YTM) I require from the
  bond, how should I price it?
• Answer Discount the cash flows at a discount
  rate that is equal to the YTM.

6
An Example (1/2)

• What should be the price of a Treasury Bill with
  a 5 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) 5.82.
• Answer 25.625/(1 5.82/2)1
  25.625/(15.82/2)2 1025.625/(15.82/2)3
  990.15.
• Note
• Semi-annual Coupon (1/2) (5.125)(1000)
  25.625
• Discount factor (0.0582/2) for a six month
  period.

7
An Example (2/2)

• The price is usually quoted as a of the face
  value, and expressed in 32nds.
• Thus, the price of 990.15 will be written as
  99.05, which should be read as (99 5/32) of
  FV.
• Note that 99 5/32 is 99.15 (approximately).

8
Calculation of YTM when you know the price of the
bond

• This is exactly the opposite the previous
  question - the complication is that we cannot
  algebraically solve for the YTM. So the YTM has
  to be calculated by trial and error.

9
An Example

• 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?
• Answer 4.43

10
Conventions in the Bond Market

• Previously, we considered an example where the
  first coupon date was exactly 6 months from
  today.
• But how do we price a bond if the first coupon
  date is not exactly 6 months from today?
• The answer depends on the day count conventions
  of the bond market - and it differs depending on
  whether the bond is a US treasury bond or a
  corporate bond.

11
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
  number of days is a fraction of a normal period.
• There are two principal day-count conventions
  actual/actual and 30/360.
• Act/Act The actual number of days between coupon
  payments are used to calculate the length of the
  period between coupons. Eg. US Treasury
• 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
• Act/360 Combination of the above.

12
Corporate vs. Treasury bonds

• The treasury market uses a day count convention
  of act/act, while the corporate bond market
  usually uses a convention of 30/360.

13
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
  Treasury quotes.
• http//online.wsj.com/mdc/public/page/2_3020-treas
  ury.html?modmdc_bnd_pglnk

14
Prices of Treasury Issues (WSJ, 11/23/2007)
Treasury Issues
15
Computing the Price from YTM

• Let us consider our earlier question of how to
  compute the price of a Treasury, knowing the
  yield to maturity.
• Consider the 4.625 coupon Treasury security
  maturing on 11/15/2009. The yield of this
  security is 3.05.
• Given this yield, what would be the market quote
  for this Treasury security?

16
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)
• Maturity 11/15/2009.
• Face Value 100 (as we quote prices in , we do
  not need to write 1000)
• 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 1004.625/2
  2.31.

17
Computations (2/4)

• Number of days between coupon payments is equal
  to the actual number of days between the last
  coupon date (11/15/2007) and the next coupon date
  (5/15/2008) gt 182 days.
• Number of days to next coupon date of 5/15/2008
  from settlement date of 11/27/07 gt 170
• Length of period from settlement to next coupon
  date (actual number of days to next coupon /
  actual number of days between coupons)
  170/1820.934.
• Yield to Maturity 3.05 (given)
• PV of bond 2.31/(10.0305/2)0.934
  2.31/(10.0305/2)1.934 2.31/(10.0305/2)2.934
  102.31/(10.0305/2)3.934 103.14
• Thus, the value of the bond is 103.14.

18
Computations (3/4)

• The present value of 103.14 is called the
  dirty price of the bond. However, the bond is
  not quoted at this price.
• The convention is that the bond is quoted in
  terms 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.
• Accrued interest (accrued days)/(actual days
  between last and next coupon date) x (coupon
  amount) (170/182) x 2.31 0.15

19
Computations (4/4)

• Clean price Dirty price accrued interest
  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 gt
  0.98 31/32
• Quoted Price 102 0.31 10231.
• Thus, this is why the WSJ quotes the price of the
  Treasury bond as 10231, even though its true
  value is 103.14.

20
Finding the YTM

• Similarly, we can set up a spreadsheet to compute
  the yield to maturity, given the price of the
  bond.

21
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 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.
• 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.