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).

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.

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?

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.

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/(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)

• The price is usually quoted as a % of the face value, and expressed in 32’nds.

• 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).

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.

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%

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.

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.

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.

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-treasury.html?mod=mdc_bnd_pglnk

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 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?

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 =100*4.625%/2 =

$2.31.

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) => 182 days.

• 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 (actual number of days to next coupon / actual number of days between coupons) = 170/182=0.934.

• Yield to Maturity = 3.05% (given)• 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

• Thus, the value of the bond is 103.14.

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

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 = > 0.98 = 31/32

• Quoted Price = 102 + 0.31 = 102:31.

• Thus, this is why the WSJ quotes the price of the Treasury bond as 102:31, even 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 price of the bond.

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.