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Unformatted text preview: Fixed Income Securities
Fixed
and Markets
Chapter 2
Bond Prices, Spot Rates, and
Forward Rates Effective Annual Rates
and Compounding
Quoted as “5% per annum
Quoted
compounded semiannually”, or “5%
compounded semiannually”.
Means paying 2.5% every six months.
Means Semiannual
Compounding
Investing $100 at an annual rate of
Investing
5% compounded semiannually for six
months generates
$100 X (1+0.05/2)=$102.50
Investing $100 at the same rate for
Investing
one year instead generates
$100 X (1+0.05/2) X (1+0.05/2) = $105.0625 In general
In
Investing x at an annual rate of r
Investing
compounded semiannually for T years
generates r
x(1 + ) 2T
2 Semiannually compounded
holding period return
What is the semiannually compounded
What
return from investing x for T years and
having w at the end? ⎡w 1
⎤
r = 2 ⎢( ) 2T − 1⎥
⎣x
⎦ Spot Rates
The spot rate is the rate on a spot
The
loan.
It can vary across different maturities.
It
The tyear spot rate is denoted (t).
(t).
The
Unless otherwise specified, assume
Unless
semiannual compounding frequency. ⎡ 1 21t ⎤
r = 2 ⎢(
) − 1⎥
⎣ d (t )
⎦ d (t ) = 1
ˆ
r (t ) 2t
(1 +
)
2 Term Structure, downwardsloping or
Term
inverted, upwardsloping Term structure of spot rates
A fiveyear zerocoupon bond and a
fiveyear
tenyear zerocoupon bond are
discounted using different rates
Coupon bond: each payment must be
Coupon
discounted at a different rate,
according to the term of the payment. From Chapter 1, the
14.25s of Feb. 15, 2002
108+31.5/32=7.125 d(.5) + 107.125 d(1)
108+31.5/32=7.125 Using
Using d (t ) = 108+31.5/32= 1
ˆ
r (t ) 2t
(1 +
)
2 7.125
107.125
+
ˆ
ˆ
r(.5)
r(1) 2
(1+
)
(1+
)
2
2 Forward Rates
Forward Forward loan and forward rates
A forward loan is an agreement made
forward
to lend money at some future date.
The rate of interest on a forward loan,
The
specified at the time of the agreement
as opposed to the time of the loan, is
called a forward rate. Forward rates
Forward
Define r(t) to be the semiannually
Define
compounded rate earned on a six
month loan ( t  0.5 ) years forward. r(.5)= (.5)=5.008%
r(.5)= ˆ
⎛ r (.5) ⎞ ⎛ r (1) ⎞ ⎛ r (1) ⎞
⎜1 +
⎟ × ⎜1 +
⎟ = ⎜1 +
⎟
2⎠⎝
2⎠ ⎝
2⎠
⎝ 2 ˆ
⎛ r (.5) ⎞ ⎛ r (1) ⎞ ⎛ r (1.5) ⎞ ⎛ r (1.5) ⎞
⎜1 +
⎟ × ⎜1 +
⎟ × ⎜1 +
⎟ = ⎜1 +
⎟
2⎠⎝
2⎠⎝
2⎠⎝
2⎠
⎝ ˆ
⎛ r (.5) ⎞
⎛ r (t ) ⎞ ⎛ r (t ) ⎞
⎜1 +
⎟ ×⋯ × ⎜1 +
⎟ = ⎜1 +
⎟
2⎠
2⎠ ⎝
2⎠
⎝
⎝ 2t Spot sloping upward, forward above spot
Spot
Spot sloping downward, forward below spot
Spot 3 If r(2.5) is above
If
(2). ˆ
⎛ r (t − .5) ⎞
⎜1 +
⎟
2⎠
⎝ 2 t −1 (2), then (2.5)>
(2), ˆ
⎛ r (t ) ⎞ ⎛ r (t ) ⎞
× ⎜1 +
⎟ = ⎜1 +
⎟
2⎠ ⎝
2⎠
⎝ 4 ˆ
ˆ
⎛ r (2) ⎞ ⎛ r (2.5) ⎞ ⎛ r (2.5) ⎞
⎜1 +
⎟ × ⎜1 +
⎟ = ⎜1 +
⎟
2⎠ ⎝
2⎠⎝
2⎠
⎝ 2t 5 Bond prices can be expressed in
terms of forward rates
108+31.5/32=7.125 d(.5) + 107.125 d(1)
108+31.5/32=7.125 108+31.5/32= 108+31.5/32= 7.125
107.125
+
ˆ
ˆ
r(.5)
r(1) 2
(1+
)
(1+
)
2
2 7.125
107.125
+
r(.5) ⎞
r(.5) ⎞ ⎛ r(1) ⎞
⎛
⎛
⎜ 1+
⎟
⎜1+
⎟ ⎜ 1+
⎟
2⎠
2 ⎠⎝
2⎠
⎝
⎝ Maturity and Bond Price
When are bonds of longer maturity
When
worth more than bonds of shorter
maturity, and when is the reverse
true?
Consider five imaginary 4.875%
Consider
coupon bonds with terms from six
months to 2.5 years. As of Feb. 15,
2001, which bond would have the
greatest price? ...
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 Fall '08
 ANDERSON
 Compounding

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