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FINM7403 Bond Part
创建日期：2024-05-23 15:16:04
所属分类：金融Finance
标签： FINM7403
Bond Part

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FINM7403 Bond Part II Tutorial Solutions
BKM Chapter 15 problem sets, Q4, 5, 6, 9*, 13, 15, 16*.
Question 4
Suggested Solutions:
The liquidity theory holds that investors demand a premium to compensate them for interest rate exposure and
the premium increases with maturity. Add this premium to a flat curve and the result is an upward sloping yield
curve.
Question 5
Suggested Solutions:
The pure expectations theory, also referred to as the unbiased expectations theory, purports that forward rates
are solely a function of expected future spot rates. Under the pure expectations theory, a yield curve that is
upward (downward) sloping, means that short-term rates are expected to rise (fall). A flat yield curve implies
that the market expects short-term rates to remain constant.
Question 6
Suggested Solutions:
The yield curve slopes upward because short-term rates are lower than long-term rates. Since market rates are
determined by supply and demand, it follows that investors (demand side) expect rates to be higher in the future
than in the near-term.

Question 9
Suggested Solutions:
First calculate the short rates
1
2
2
3
3
4
4
( ) 5.00%
1.06( ) 7.01%
1.05
1.065( ) 7.51%
1.05 1.0701
1.07( ) 8.51%
1.05 1.0701 1.0751
E r
E r
E r
E r
=
= =
= =
×
= =
× ×
In one year, the market’s expectation of the yield curve is:
Bond Years to Maturity YTM(%)
B 1 7.01%
C 2 7.26%
D 3 7.67%

Question 13
Suggested Solutions:
Year
Forward
Rate
PV of $1 received at period end
1 5% $1/1.05 = $0.9524
2 7 1/(1.05×1.07) = $0.8901
3 8 1/(1.05×1.07×1.08) = $0.8241
a. Price = ($60 × 0.9524) + ($60 × 0.8901) + ($1,060 × 0.8241) = $984.14
b. To find the yield to maturity, solve for y in the following equation:
$984.10 = [$60 × Annuity factor (y, 3)] + [$1,000 × PV factor (y, 3)]
This can be solved using a financial calculator to show that y = 6.60%:
PV = -$984.10; N = 3; FV = $1,000; PMT = $60. Solve for I = 6.60%.
c.
Period Payment Received at End of Period: Will Grow by a Factor of: To a Future Value of:
1 $60.00 1.07 × 1.08 $69.34
2 60.00 1.08 64.80
3 1,060.00 1.00 1,060.00
$1,194.14
$984.10 × (1 + y realized)3 = $1,194.14
1 + y realized = 0666.1
10.984$
14.194,1$ 3/1
=




 ⇒ y realized = 6.66%
Alternatively, PV = -$984.10; N = 3; FV = $1,194.14; PMT = $0. Solve for I = 6.66%.

d. Next year, the price of the bond will be:
[$60 × Annuity factor (7%, 2)] + [$1,000 × PV factor (7%, 2)] = $981.92
Therefore, there will be a capital loss equal to: $984.10 – $981.92 = $2.18
The holding period return is: %88.50588.0
10.984$
)18.2$(60$
==
−+
Question 15
Suggested Solutions:
The price of the coupon bond, based on its yield to maturity, is:
[$120 × Annuity factor (5.8%, 2)] + [$1,000 × PV factor (5.8%, 2)] = $1,113.99
If the coupons were stripped and sold separately as zeros, then, based on the yield to maturity of zeros with
maturities of one and two years, respectively, the coupon payments could be sold separately for:
08.111,1$
06.1
120,1$
05.1
120$
2 =+
The arbitrage strategy is to buy zeros with face values of $120 and $1,120, and respective maturities of one year
and two years, and simultaneously sell the coupon bond. The profit equals $2.91 on each bond.
Question 16
Suggested Solutions:
a. The one-year zero-coupon bond has a yield to maturity of 6%, as shown below:
1
$100$94.34
1 y
=
+
⇒y1 = 0.06000 = 6.000%
The yield on the two-year zero is 8.472%, as shown below:
2
2
$100$84.99
(1 )y
=
+
⇒y2 = 0.08472 = 8.472%
b. The price of the coupon bond is: 51.106$
)08472.1(
112$
06.1
12$
2 =+

Therefore: yield to maturity for the coupon bond = 8.333%
[On a financial calculator, enter: n = 2; PV = –106.51; FV = 100; PMT = 12]
c.
2 2
2
2
1
(1 ) (1.08472)1 1 0.1100 11.00%
1 1.06
yf
y
+
= − = − = =
+
d. Expected price =
$112
1.11
=$100.90
(Note that next year, the coupon bond will have one payment left.)
Expected holding period return = %00.60600.0
51.106$
)51.106$90.100($12$
==
−+
This holding period return is the same as the return on the one-year zero.
e. If there is a liquidity premium, then: E(r2) < f 2
E(Price) =
2
$112 $100.90
1 ( )E r
>
+
E(HPR) > 6%

CFA Problems 2, 3, and 5*.
Problem 2
Suggested Solutions:
d. Investors bid up the price of short term securities and force yields to be relatively low, while doing just the
opposite at the long end of the term structure. Therefore, they must be compensated more for giving up liquidity
in the long term.
Problem 3
Suggested Solutions:
a. (1+y4 )4 = (1+ y3 )3 (1 + f 4 )
(1.055)4 = (1.05)3 (1 + f 4 )
1.2388 = 1.1576 (1 + f 4 ) ⇒ f 4 = 0.0701 = 7.01%
b. The conditions would be those that underlie the expectations theory of the term structure: risk neutral market
participants who are willing to substitute among maturities solely on the basis of yield differentials. This
behaviour would rule out liquidity or term premia relating to risk.
c. Under the expectations hypothesis, lower implied forward rates would indicate lower expected future spot
rates for the corresponding period. Since the lower expected future rates embodied in the term structure are
nominal rates, either lower expected future real rates or lower expected future inflation rates would be consistent
with the specified change in the observed (implied) forward rate.

Problem 5
Suggested Solutions:
The present value of each bond’s payments can be derived by discounting each cash flow by the appropriate rate
from the spot interest rate (i.e., the pure yield) curve:
Bond A: 53.98$
11.1
110$
08.1
10$
05.1
10$PV 32 =++=
Bond B: 36.88$
11.1
106$
08.1
6$
05.1
6$PV 32 =++=
Bond A sells for $0.13 (i.e., 0.13% of par value) less than the present value of its stripped payments. Bond B
sells for $0.02 less than the present value of its stripped payments. Bond A is more attractively priced.

BKM Chapter 16 problem sets, Q1, 2, 3, 4*, 5, 7, 9, 12*, 14*.
Question 1
Suggested Solutions:
While it is true that short-term rates are more volatile than long-term rates, the longer duration of the longer-
term bonds makes their prices and their rates of return more volatile. The higher duration magnifies the
sensitivity to interest-rate changes.
Question 2
Suggested Solutions:
Duration can be thought of as a weighted average of the maturities of the cash flows paid to holders of the
perpetuity, where the weight for each cash flow is equal to the present value of that cash flow divided by the
total present value of all cash flows. For cash flows in the distant future, present value approaches zero (i.e., the
weight becomes very small) so that these distant cash flows have little impact and, eventually, virtually no
impact on the weighted average.
Question 3
Suggested Solutions:
The percentage change in the bond’s price is:
7.194 0.005 0.0327 3.27%,
1 1.10
D y
y
− ×∆ = − × = − = −
+
or a 3.27% decline
Question 4
Suggested Solutions:
a. YTM = 6%
(1) (2) (3) (4) (5)
Time until Payment
(Years)
Cash Flow PV of CF (Discount Rate = 6%) Weight Column (1) × Column (4)
1 60.00 56.60 0.0566 0.0566

2 60.00 53.40 0.0534 0.1068
3 1,060.00 890.00 0.8900 2.6700
Column sums $1,000.00 1.0000 2.8334
Duration = 2.833 years
b. YTM = 10%
(1) (2) (3) (4) (5)
Time until
Payment (Years)
Cash Flow PV of CF (Discount Rate = 10%) Weight Column (1) × Column (4)
1 60.00 54.55 0.0606 0.0606
2 60.00 49.59 0.0551 0.1102
3 1,060.00 796.39 0.8844 2.6532
Column sums $900.53 1.0000 2.8240
Duration = 2.824 years, which is less than the duration at the YTM of 6%.
Question 5
Suggested Solutions:
For a semi-annual 6% coupon bond selling at par, we use the following parameters: coupon = 3% per half-year period,
y = 3%, T = 6 semi-annual periods.
(1) (2) (3) (4) (5)
Time until Payment
(Years)
Cash Flow PV of CF (Discount Rate = 3%) Weight Column (1) × Column (4)
1 $ 30.00 $ 29.13 0.02913 0.02913
2 30.00 28.28 0.02828 0.05656
3
30.00
27.45 0.02745 0.08236
4 30.00 26.65 0.02665 0.10662
5 30.00 25.88 0.02588 0.12939
6 1030.00 862.61 0.86261 5.17565
Column sums $1000.00 1.00000 5.57971
D = 5.5797 half-year periods = 2.7899 years
If the bond’s yield is 10%, use a semi-annual yield of 5% and semi-annual coupon of 3%:
(1) (2) (3) (4) (5)
Time until Payment
(Years)
Cash Flow PV of CF (Discount Rate = 5%) Weight Column (1) × Column (4)
1 $ 30.00 $ 28.57 0.03180 0.03180
2 30.00 27.21 0.03029 0.06057

3
30.00
25.92 0.02884 0.08653
4 30.00 24.68 0.02747 0.10988
5 30.00 23.51 0.02616 0.13081
6 1030.00 768.60 0.85544 5.13265
Column sums $898.49 1.00000 5.55223
D = 5.5522 half-year periods = 2.7761 years
Question 7
Suggested Solutions:
D. Investors tend to purchase longer term bonds when they expect yields to fall so they can capture significant
capital gains, and the lack of a coupon payment ensures the capital gain will be even greater.
Question 9
Suggested Solutions:
a.
(1) (2) (3) (4) (5)
Time until Payment
(Years)
Cash Flow PV of CF (Discount Rate = 10%) Weight Column (1) × Column (4)
1 $10 million $ 9.09 million 0.7857 0.7857
5 4 million 2.48 million 0.2143 1.0715
Column sums $11.57 million 1.0000 1.8572
D = 1.8572 years = required maturity of zero-coupon bond.
b. The market value of the zero must be $11.57 million, the same as the market value of the obligations.
Therefore, the face value must be: $11.57 million × (1.10)1.8572 = $13.81 million

Question 12
Suggested Solutions:
a. PV of the obligation = $10,000 × Annuity factor (8%, 2) = $17,832.65
(1) (2) (3) (4) (5)
Time until Payment
(Years)
Cash Flow PV of CF (Discount Rate = 8%) Weight Column (1) × Column (4)
1 $10,000.00 9,259.259 0.51923 0.51923
2 10,000.00 8,573.388 0.48077 0.96154
Column sums $17,832.647 1.00000 1.48077
D = 1.4808 years
b. A zero-coupon bond maturing in 1.4808 years would immunize the obligation. Since the present value of the
zero-coupon bond must be $17,832.65, the face value (i.e., the future redemption value) must be
$17,832.65 × 1.081.4808 = $19,985.26
c. If the interest rate increases to 9%, the zero-coupon bond would decrease in value to
92.590,17$
09.1
26.985,19$
4808.1 =
The present value of the tuition obligation would decrease to $17,591.11
The net position decreases in value by $0.19
d. If the interest rate decreases to 7%, the zero-coupon bond would increase in value to
99.079,18$
07.1
26.985,19$
4808.1 =
The present value of the tuition obligation would increase to $18,080.18
The net position decreases in value by $0.19
The reason the net position changes at all is that, as the interest rate changes, so does the duration of the stream
of tuition payments.

Question 14
Suggested Solutions:
a. The duration of the perpetuity is: 1.05/0.05 = 21 years
Call w the weight of the zero-coupon bond. Then
(w × 5) + [(1 – w) × 21] = 10 ⇒ w = 11/16 = 0.6875
Therefore, the portfolio weights would be as follows: 11/16 invested in the zero and 5/16 in the perpetuity.
b. Next year, the zero-coupon bond will have a duration of 4 years and the perpetuity will still have a 21-year
duration. To obtain the target duration of nine years, which is now the duration of the obligation, we again solve
for w:
(w × 4) + [(1 – w) × 21] = 9 ⇒ w = 12/17 = 0.7059
So, the proportion of the portfolio invested in the zero increases to 12/17 and the proportion invested in the
perpetuity falls to 5/17.

CFA Problems 2, 3*, 4a.i. 4.a.ii. only.
Problem 2
Suggested Solutions:
a. Bond price decreases by $80.00, calculated as follows:
10 × 0.01 × 800 = 80.00
b. ½ × 120 × (0.015)2 = 0.0135 = 1.35%
c. 9/1.10 = 8.18
d. (i)
e. (i)
f. (iii)

Problem 3
Suggested Solutions:
a. Modified duration 26.9
08.1
10
YTM1
durationMacaulay
==
+
= years
b. For option-free coupon bonds, modified duration is a better measure of the bond’s sensitivity to changes in
interest rates. Maturity considers only the final cash flow, while modified duration includes other factors, such
as the size and timing of coupon payments, and the level of interest rates (yield to maturity). Modified duration
indicates the approximate percentage change in the bond price for a given change in yield to maturity.
c. i. Modified duration increases as the coupon decreases.
ii. Modified duration decreases as maturity decreases.
d. Convexity measures the curvature of the bond’s price-yield curve. Such curvature means that the duration
rule for bond price change (which is based only on the slope of the curve at the original yield) is only an
approximation. Adding a term to account for the convexity of the bond increases the accuracy of the
approximation. That convexity adjustment is the last term in the following equation:
* 21( ) Convexity ( )
2
P D y y
P
∆  = − ×∆ + × × ∆  
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CIVL2010 ECMT2150 CS 333 CP1406 KIT308 Economics 701 ECON3740 ECE 5759 BFC3340 FINC2012 COMP4500 MATH3871 FIT3080 18-819C Economic ECO202Y5Y PHY100 FINC-780 Econ 100B EC602 IE582 EE450 COMP9517 WRDS 150B ECON 570 LECTURE 2 ECA5101 ECE 4420 ADAD9311 FIT3031 CS 544 MF 703 CS 6140 KIT501 ENGSCI 344 CSC171 IE 332 FINM8007 21S2 COSC 1114 FE545 E471 SIT102 CSC108 MAT301 IAE 101 COP3503 STAT2004 EC 303 CS1026 STAT1203 MATH3171 ISOM5160 IFN657 FIT2014 ACCT2102 DSCI550 STATS 330 ELEC421 COMP3670 QBUS6840 MEES:698B MAST30027 MIET1077 Economics IFT 6758 FIT5122 BUSN5101 CptS 111 HRMT5502 Math2018 PP 312 INFS3200 CZ3005 CSSE1001 SENG 2011 MKTG 553Q BUFN 640 COMP507 ETF5952 DATA 311 ECE4043 BANK3011 finance M3H623544 HW1 INFS 7410 AI6103 STAT3888 INFS3208 ACCT7103 ECOS3013 PSYCH 20B F79MA STAT3006 ENGG952 2X MGSC 7000 ECE4132 MSF 526 MATH 3033 CDS 511 EEEE4120 EEET2465 ECA5103 MATH2070 ELEC3104 BU51019 EEE8147 ENGR20003 COSC 1107 ETF3200 FNCE 435 MSFI 435 ISyE6420 GENG4405 HW ETC5523 IE 7315 MOEC0021 STA 106 ENGG 1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A 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