Fixed Income Securities
and Markets
Chapter 1
Prices, Discount Factors, and
Arbitrage
Overview
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Bond Prices and Bond Features
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Discount Factors
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Arbitrage
Bond Basics
Face
Value,
Coupons (annual, semiannual, …)
Zero-Coupon Bond
Par
Value
Maturity
Coupon Rate = Annual Coupon / Face Value
1
Example
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A treasury bond with a $1,000,000
face value, a coupon rate of 2.125%
and a maturity date of May 31, 2015.
Annual coupon payment is
$1,000,000 X 2.125% = $21,250
$21,250 / 2 = $10,625
An actual coupon payment of $10,625
every six months
Example
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A treasury bond with a $10,000 face
value, a coupon rate of 5.25% and a
maturity date of Aug. 15, 2003.
Annual coupon payment is
$10,000 X 5.25% = $525
$525 / 2 = $262.50
An actual coupon payment of $262.50
every six months
2
T+1 settle
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Delivery or settlement one day after a
transaction.
Treasury bonds have T+1 settle.
Most spot foreign exchange
transactions have T+2 settle.
3
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Numbers after the hyphens denote
32nds, often called ticks. "+" half tick.
(101+12.75/32)%=101.3984%
Discount Factors
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The discount factor for a particular
term gives the value today (present
value) of one unit of currency to be
received at the end of that term.
The discount factor for t years is
written as d(t).
If d(.5)=.97557, it means
$105 to be received 6 months later is
worth .97557X$105=$102.43 now
Extracting discount factors
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100.55=(100+1.25/2) d(.5)
D(.5)=.99925
104.513=(4.875/2) d(.5) +
(100+4.875/2) d(1)
D(1)=.99648
4
The law of one price
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Absent confounding factors (liquidity,
taxes, credit risk, etc.), two securities
(or portfolio of securities) with exactly
the same cash flows should sell for the
same price.
Discount factors extracted from one
set of bonds may be used to price any
other bond with cash flows on the
same set of dates.
5
Coupon 0.75%
Nov. 30, 2011
Theoretical
Price = ?
Arbitrage and the law of one price
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Arbitrage: a trade that generates or
that might generate profits without
any risk.
A violation of the law of one price
implies the existence of an arbitrage
opportunity.
6
Deriving the replicating portfolio
Treasury Strips
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Zero-coupon bonds issued by the US
Treasury are called STRIPS
STRIPS (Separate trading of registered
interest and principal securities)
Coupon or interest STRIPS, called
TINT, INT, or C-STRIPS
Principal STRIPS, called TP, P, or PSTRIPS
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8
Accrued Interest
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On June 1, 2010, consider the 3.625s
of Aug. 15, 2019 (Feb. 15 and Aug. 15
coupon payments)
If Investor B buys $10,000 face value
of this bond from investor S,
Accrued Interest
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75 days between June 1, 2010 and
Aug. 15, 2010
181 days between Feb. 15, 2010 and
Aug. 15, 2010
Thus B should receive only
(75/181)X$181.25 = $ 75.10 of the
coupon payment.
S gets the rest: $106.15
9
Flat price and full price
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Quoted or Flat price: 102-26
Accrued Interest: $106.15 per $10,000
face value or $1.0615 per $100 face
Full price:
102+26/32+1.0615=103.8740
Invoice price on $10,000 face amount:
$10,387.40 (actual amount paid)
Flat Price + Accrued Interest =
PV (future cash flows)
With an accrued interest convention, if
yield does not change then the quoted
price of a bond does not fall as a
result of a coupon payment.
Full price does fall.
PB and PA: quoted prices of a bond
right before and right after a
coupon payment of c/2.
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PB+c/2=c/2+PV(cash flows after the
next coupon)
PB=PV(cash flows after the next
coupon)
PA+0=PV(cash flows after the next
coupon)
PA=PB
10