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Unformatted text preview: Fixed Income Securities
and Markets
Chapter 5
OneFactor Measures of Price
OneSensitivity How much price changes
as interest rates change
Initial + shifted rate curve
Initial
computing prices is
computing
straightforward
Defining changes in interest rates
Defining Many different possible
changes in interest rates
Parallel shift in curve
Parallel
Each spot rate moves in some
Each
proportion to its maturity
Inverse proportion to maturity
Inverse 1 Interest Rate Factor
An interest rate factor is a random
An
variable that impacts interest rates in
some way.
One factor driving all interest rates
One
that factor itself is an interest rate.
that
Two or more factors.
Two
Factors are not interest rates.
Factors Measures of Price Sensitivity
Ch. 5: very generally, onefactor
Ch.
onemeasures of price sensitivity
Ch. 6: the special case of parallel yield
Ch.
shifts
Ch. 7: multifactor formulations.
Ch.
multiCh. 8: to model interest rate changes
Ch.
empirically. DV01
P(y): pricerate function of a fixedP(y): pricefixedincome security, where y is an interest
rate factor.
For example: the term structure of
For
interest rates is flat at 5% and the
rates move up and down in parallel. 2 In this 5% flat environment
As of Feb 15, 2001
As
US Treasury bond: 5% coupon,
US
maturity Feb. 15, 2011
Oneyear European call option struck
Oneat par on the 5s of Feb 15, 2011
This option gives its owner the right to
This
purchase some face amount of the
bond after exactly one year at par. Price expressed as a percent of face value Price expressed as a percent of face value 3 DV01: dollar value of an '01 (i.e., .01%) The change in the value (price) of a
The
fixed income security for a onebasis
onepoint decline in rates. DV 01 ≡ − ΔP
10, 000 × Δy (ΔP)/(Δy) ==> Slope of the line ==>
Use points relatively close ==> In the
limit, the slope of the line tangent to
the pricerate curve at the desired rate
pricelevel ==> derivative DV 01 ≡ − 1 dP ( y )
10, 000 dy 4 DV01 at 4%
DV01
− ΔP
8.0866 − 8.2148
=−
= .0641
10, 000 × Δy
10, 000 × (4.01% − 3.99%) This chapter's definition: very general.
This A hedging example, part I:
Hedging a call option
If a market maker sells $100 million
If
face value of the call option and rates
are at 5%, how might the market
maker hedge interest rate exposure by
trading in the underlying bond?
Buy or sell the bond?
Buy 5 .0779
.0369
= 100, 000, 000 ×
100
100
.0369
= $47,370, 000
F = 100, 000, 000 ×
.0779
.0369
= $36,900
100, 000, 000 ×
100
.0779
= $36,901
$47,370, 000 ×
100
− FA × DV 01A
FB =
DV 01B
F× Duration
Duration measures the percentage
Duration
change in the value of a security for a
unit change in rates (10,000 basis
points). D≡− 1 ΔP
P Δy 1 dP
P dy
(108.0901 − 108.2615) /108.1757
D=−
= 7.92
4.01% − 3.99%
ΔP
= − DΔy
P
ΔP = (−7.92 × .0001) × 108.1757 = −0.0857
D≡− 6 Duration Terminology
Effective duration:
Effective
duration computed for any assumed
change in the term structure of
interest rates.
Macaulay duration or modified
Macaulay
duration:
interest rate sensitivity with respect to
a change in yieldtomaturity.
yield to Convexity 7 Convexity
Convexity: how interest rate sensitivity
Convexity:
changes with rates 1 d 2P
C=
P dy 2 100 − 100.0780
= −779.8264
5% − 4.99%
Δ 2 P −779.0901 + 779.8264
=
= 7363
Δy 2
5.005% − 4.995%
1 Δ 2 P 7363
C=
=
= 73.63
P Δy 2
100 8 Measuring the price
sensitivity of portfolios
P = ∑ Pi
dP
dP
=∑ i
dy
dy
1 dP
1 dPi
=∑
10, 000 dy
10, 000 dy
DV 01 = ∑ DV 01i 1
P
1
−
P
− 1 dPi
dP
= ∑−
dy
P dy
P 1 dPi
dP
= ∑− i
dy
P Pi dy Pi
Di
P
P
C = ∑ i Ci
P
D=∑ A hedging example:
the negative convexity of
callable bonds
A callable bond is a bond that the
callable
issuer may repurchase or call at some
fixed set of prices on some fixed set of
dates
Price = (price of bond)  (value of
Price
embedded option) 9 At 5%, callable bond price =
At
1003.0501=96.9499
100DV01=0.07790.0369=0.0410
DV01=0.0779Convexity: 223=
Convexity:
103.15%X73.633.15%X9503.33
103.15%X73.63 Firstorder approximation
First ΔP
≈ − DΔy
P Estimating price changes and returns
with DV01, duration, and convexity
1 d 2P 2
dP
P ( y + Δy ) ≈ P( y ) +
Δy +
Δy
2 dy 2
dy 1 1 d 2P 2
ΔP 1 dP
≈
Δy + ×
Δy
2 P dy 2
P
P dy
1
ΔP
≈ − DΔy + C Δy 2
2
P
%ΔP = −120.82 × 0.0025 + (1/ 2)9503.3302 × 0.00252
= −0.30205 + 0.02970
= −27.235%
2.2194 = 3.0501× (1 − 27.235%) 10 11 ...
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 Fall '08
 ANDERSON
 Interest Rates, Interest, Interest Rate

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