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(math361)[2008](s)midterm~ma_yxf^sol_10510.pdf
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Midterm for Math 361
Quantitative Methods for Fixed-Income Securities
April 14, 2008


Solutions:
1
1.1 (2) A spot rate is the yield of a zero-coupon bond. A spot-rate curve is the curve formed by spot rates of various maturities.
1.2 (2) A forward rate is the arbitrage free interest rate for a half-year period in the future. A forward-rate curve is the curve formed by forward rates of various maturities.
1.3 (2) A par yield is a coupon rate of a par bond. A par-yield curve is the curve formed by par yields of various maturities.
. 2Tc/2 1 .
1.4 (2) Py = .
() F +.
j 2T
j=1 (1+ y/ 2) (1+ y/ 2)
..
2T .
. c/2 1
1.5 (2) Py() = F.[1 rj j + rT ( )/2] 2T .
. j=1 + .( /2)/2] [1 + . .
.. . 2T .
c/2 1


1.6 (2) Py = .
() F +.
j 2T. j=1 .
rk( /2)/2) (1+ ( /2)/2)
(1+ rk
. .
. k=1 k=1 .
.2T+1
. . .
(1+ rT( .1/ 2)/ 2 )


1.7 (2) rT = 2 ..
() . 1
.2T
.(1+ rT )
.( )/2 .
..
1/ 2 T
. 2T .
.. ..
.() ( rj .1.
1.8 (2) rT = 2 .. 1+ ( /2)/2 ). .
.. j=1 ..

.. 1. ()
dT
1.9 (2) () = T
.
yT 2
dt()

2
t=1
2t ri()


1.10 (2) Substitute dt() = 1+ into the last equation.
( 2 ).1
i=1
2 (4) Solve y by trial and errorfrom
. 2Tc/2 1 .
=
PF. j+ + 2T+ .
. j 0 (1+ y/ 2) (1+ y/ 2)
=.
1

or
F .cc . 1 . 1 .
P =+ 1.+
. . 2T . 2T .
(1+y / 2) .2 y . (1+y / 2) . (1+y / 2) .
with
F =100
c =0.055
T =3
=(31+28 +31) /181 =0.4972
c
AI = F =1.367
2 181 P =110 +1.367 =111.367.
3.
3.1 (4) Differentiating bond price formula
.c . 1 . 1 .
() =F 1 .
Py .. +
. 2T . 2T
y . (1+y / 2) . (1+y / 2) .
.
w. r. t. y we get
dP . c . 1 .. c . T .
F . 1 .+. +.
=. 2 .. 2T . 1. 2T 1
dy . y . (1+y / 2) .. y .(1+y / 2) .
We calculate DV01s using .1 dP
DV 01 =
10,000 dy For 5.5s of 04/07/2008, we have DV 01 =0.027312.
For 5.0s of 04/07/2006, we have DV 01 =0.009637.
3.2 (2) Modified durations are calculated according to
1 dP

D =.
Pdy
For 5.5s of 04/14/2011, we have D =2.7312. For 5.0s of 04/14/2009, we have D =0.9637.

3.3 (2) We should short sell
2



D5.5 2.7312
== 2.8340
D5.0 0.9637 units of 5s 04/14/2009 to form a duration neutral portfolio. =========================== END ===========================
3