Please help solve equation
jasine2
problem_set_4_q1.docxDue Date: Tuesday, April 29, 2014 Spring 2014
1. (10 points) You obtained the following information:
|
Bond |
Face value |
Coupon rate |
Coupon Payments |
Maturity (years) |
YTM1 |
Zero-Coupon Bond |
$1,000 |
- |
- |
0.5 |
3.25% |
Zero-Coupon Bond |
$1,000 |
- |
- |
1 |
3.50% |
Zero-Coupon Bond |
$1,000 |
- |
- |
1.5 |
3.75% |
Zero-Coupon Bond |
$1,000 |
- |
- |
2 |
3.90% |
US Treasury Note |
$1,000 |
3.00% |
Semiannual |
2 |
? |
US Treasury Note |
$1,000 |
3.50% |
Semiannual |
5 |
4.00% |
|
US Treasury Note |
$1,000 |
3.75% |
Semiannual |
10 |
4.25% |
1 YTMs are annualized rates that have been annualized by multiplying the semiannual rates by two.
(a) (1 point) Determine the price of the 5-year and 10-year US Treasury notes. Please round your answers to the nearest cent (e.g., $1.23).
Price of 5-year US Treasury Note
Price of 10-year US Treasury Note
(b) (1 point) Suppose that interest rates increase 1%. Find the percentage change in the price of the 5-year and 10-year US Treasury notes. Note that you need to re-compute the prices of the two notes with the higher interest rates. Please round your answers to the nearest basis point (e.g., 1.23%).
Price of 5-year US Treasury Note (with YTM =5.00%)
Price of 10-year US Treasury Note (with YTM =5.25%)
Percentage Change in Price of 5-year Treasury Note
Percentage Change in Price of 10-year Treasury Note
(c) (2 points) Compute the duration of the 5-year and 10-year US Treasury notes. Please report the duration in semesters (e.g., 9.8765 semesters).
Duration of 5-year US Treasury Note (using Po = $979.59 from part a)
Duration = 10.97533259 Semesters
|
t |
ct |
ct/(1-y*)^t |
[ct/(1-y*)^t]/p |
t[ct/(1-y*)^t]/p |
|
1 |
17.5 |
17.46506986 |
0.017828959 |
0.017828959 |
|
2 |
17.5 |
17.43020944 |
0.017793372 |
0.035586744 |
|
3 |
17.5 |
17.3954186 |
0.017757856 |
0.053273569 |
|
4 |
17.5 |
17.36069721 |
0.017722412 |
0.070889647 |
|
5 |
17.5 |
17.32604512 |
0.017687038 |
0.088435188 |
|
6 |
17.5 |
17.2914622 |
0.017651734 |
0.105910405 |
|
7 |
17.5 |
17.2569483 |
0.017616501 |
0.123315508 |
|
8 |
17.5 |
17.22250329 |
0.017581338 |
0.140650707 |
|
9 |
17.5 |
17.18812704 |
0.017546246 |
0.157916213 |
|
10 |
1017.5 |
997.3720708 |
1.018152565 |
10.18152565 |
|
|
|
|
D |
10.97533259 |
Duration of 10-year US Treasury Note (using Po = $961.25 from part a)
Duration = 16.75242514 Semesters
|
t |
ct |
ct/(1-y*)^t |
[ct/(1-y*)^t]/p |
t[ct/(1-y*)^t]/p |
|
1 |
18.75 |
18.35985312 |
0.019099977 |
0.019099977 |
|
2 |
18.75 |
17.97782435 |
0.018702548 |
0.037405096 |
|
3 |
18.75 |
17.60374478 |
0.018313389 |
0.054940166 |
|
4 |
18.75 |
17.23744899 |
0.017932327 |
0.071729307 |
|
5 |
18.75 |
16.87877502 |
0.017559194 |
0.087795969 |
|
6 |
18.75 |
16.52756428 |
0.017193825 |
0.10316295 |
|
7 |
18.75 |
16.18366147 |
0.016836059 |
0.117852411 |
|
8 |
18.75 |
15.84691454 |
0.016485737 |
0.131885895 |
|
9 |
18.75 |
15.51717458 |
0.016142704 |
0.145284339 |
|
10 |
18.75 |
15.19429579 |
0.01580681 |
0.158068097 |
|
11 |
18.75 |
14.87813541 |
0.015477904 |
0.170256946 |
|
12 |
18.75 |
14.56855365 |
0.015155843 |
0.181870111 |
|
13 |
18.75 |
14.26541361 |
0.014840482 |
0.19292627 |
|
14 |
18.75 |
13.96858126 |
0.014531684 |
0.203443576 |
|
15 |
18.75 |
13.67792534 |
0.014229311 |
0.213439667 |
|
16 |
18.75 |
13.39331735 |
0.01393323 |
0.22293168 |
|
17 |
18.75 |
13.11463143 |
0.01364331 |
0.231936265 |
|
18 |
18.75 |
12.84174436 |
0.013359422 |
0.240469595 |
|
19 |
18.75 |
12.57453549 |
0.013081441 |
0.248547385 |
|
20 |
1018.75 |
669.0001743 |
0.695968972 |
13.91937944 |
|
|
|
|
D |
16.75242514 |
(d) (2 points) Using duration, find the approximate percentage change in the price of 5-year and 10-year US Treasury notes arising from interest rates increasing 1%. Please round your answers to the nearest basis point (e.g., 1.23%).
Approximate % Change Price of 5-year US Treasury Note
Approximate % Change Price of 10-year US Treasury Note
(e) (1 point) How does the approximate percentage change in price obtained in part (d) compare with the percentage change in price obtained in part (b). Why? Hint: you may want to recognize: (i) the convex relation between a bond price and its YTM; and (ii) the fact that duration provides a linear approximation to this relation.
The approximate percentage change in price overestimates the price change of the 5-year and 10-year Treasury Notes compared with the percentage change obtained in part b. Using the approximate percentage change using durations, the linear approximation overestimates the price as interest rates increases estimating a -5.49% and –8.38% change in the 5-year and 10-year Treasury Note respectively. However, the true relation between bond prices and interest rates is convex and using part (b) the percentage change in price is -4.02% and -7.52% in the 5-year and 10-year Treasury Note respectively.
(f) (3 points) Suppose that there are no arbitrage opportunities. Find the price and the YTM of the 2-year US Treasury Note. Hint: in determining the price of the Treasury note you may want to replicate the Treasury note with a portfolio of zero-coupon bonds following the approach covered in class.
|
Zero-coupon maturity |
YTM |
T |
Value |
|
0.5 |
0.0325 |
1 |
984.0098401 |
|
1 |
0.035 |
2 |
965.8977718 |
|
1.5 |
0.0375 |
3 |
945.7952635 |
|
2 |
0.039 |
4 |
925.6591098 |
Price of 2-year Treasury Note = $982.98
= (0.015*984) + (0.015*965.9) + (0.015*945.8) + (1.015*925.66)
= 982.98
YTM of 2-year Treasury Note =
Solve for y:
Y =
1
P 0 =18.75× 1− 1
(1+ 0.02125)19
0.02125
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
+ 1000
(1+ 0.02125)19 ≈ 961.25
P
0
=18.75´
1-
1
(1+0.02125)
19
0.02125
é
ë
ê
ê
ê
ê
ù
û
ú
ú
ú
ú
+
1000
(1+0.02125)
19
»961.25
P 0 =17.5× 1− 1
(1+ 0.025)9
0.025
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
+ 1000
(1+ 0.025)9 ≈ 940.22
P
0
=17.5´
1-
1
(1+0.025)
9
0.025
é
ë
ê
ê
ê
ê
ù
û
ú
ú
ú
ú
+
1000
(1+0.025)
9
»940.22
P 0 =18.75× 1− 1
(1+ 0.02625)19
0.02625
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
+ 1000
(1+ 0.02625)19 ≈ 888.92
P
0
=18.75´
1-
1
(1+0.02625)
19
0.02625
é
ë
ê
ê
ê
ê
ù
û
ú
ú
ú
ú
+
1000
(1+0.02625)
19
»888.92
Δ%P 0 = P1 P0
−1
= $940.22 $979.59
−1
Δ%P 0 = -0.04019 = - 4.02%
D%P
0
=
P1
P0
-1
=
$940.22
$979.59
-1
D%P
0
= -0.04019 = - 4.02%
Δ%P 0 = P1 P0
−1
= $888.92 $961.25
−1
Δ%P 0 = -0.0752 = -7.52%
D%P
0
=
P1
P0
-1
=
$888.92
$961.25
-1
D%P
0
= -0.0752 = -7.52%
Δ%P 0 ≈ − D
1+ y* ×Δy*
≈ − 5.49 1
×1%
≈ −5.49%
D%P
0
»-
D
1+y*
´Dy*
»-
5.49
1
´1%
»-5.49%
Δ%P 0 ≈ − D
1+ y* ×Δy*
≈ − 8.38 1
×1%
≈ −8.38%
D%P
0
»-
D
1+y*
´Dy*
»-
8.38
1
´1%
»-8.38%
V 0 = 982.98 =15×
1− 1
(1+ y 2
)4
y 2
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥
+ 1000
(1+ y 2
)4
V
0
=982.98=15´
1-
1
(1+
y
2
)
4
y
2
é
ë
ê
ê
ê
ê
ê
ê
ù
û
ú
ú
ú
ú
ú
ú
+
1000
(1+
y
2
)
4
P 0 =17.5× 1− 1
(1+ 0.02)9
0.02
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
+ 1000
(1+ 0.02)9 ≈ 979.59
P
0
=17.5´
1-
1
(1+0.02)
9
0.02
é
ë
ê
ê
ê
ê
ù
û
ú
ú
ú
ú
+
1000
(1+0.02)
9
»979.59
