fixed income

Chapter 1
Prices, Discount Factors, and
Arbitrage

STARTING WITH COUPON BONDS
• Three aspects: In May 2010 the U.S. Treasury sold a bond with
– a coupon rate of 2 % and
– a maturity date of May 31, 2015
– a payment frequency of two a year, six months apart s of May 31, 2015”
• This bond is called “
Coupon rate

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Coupon frequency, “s” is for “semi‐annual”
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maturity

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Cash Flow of the Bond
• The unit for bond purchasing is $1,000.
• Suppose that an investor purchases $1m face value of the bond, i.e., 1,000 units.
• The the coupon payment is calculated according to

Year fraction

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Coupon rate

Face value

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Cash Flow of the Bond, cont’d
Table 1.1: Cash Flows of the U.S. 2 ⁄ from books of May 31, 2015

Date
"11/30/2010"
"5/31/2011"
"11/30/2011"
"5/31/2012"
"11/30/2012"
"5/31/2013"
"11/30/2013"
"5/31/2014"
"11/30/2014"
"5/31/2015"
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Coupon
Payment
$10, 625
$10, 625
$10, 625
$10, 625
$10, 625
$10, 625
$10, 625
$10, 625
$10, 625
$10, 625
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Principal
Payment

$1, 000, 000
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Government Bonds
• US Treasury
• Exchange Fund Bills & Notes Fixings (Hong
Kong)

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Mid‐Prices of Selected Bonds

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DISCOUNT FACTORS
• The discount factor for a particular term gives the value today, or the present value, of one unit of currency to be received at the end of that term.
• Denote the discount factor for t years by d(t).
• Let d(.5) =0.99925, then the PV of $1 to be received in six months is 99.925 cents.
• The present value of the cash flow $1,050,000 in six month is
.
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Bootstrapping the Discount Factors
• A coupon bond can be considered as a portfolio of zero‐coupon bonds.
• Given the prices of coupon bonds of consecutive maturities with six months gap, one can bootstrap the prices of zero‐coupon bonds 9/5/2013

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