Financial Management:
Assignment #2
Boima S, Kamara University of Liberia Graduate School/MBA 12/02/2021
Question 1: True or false? Explain. (a) Longer-maturity bonds necessarily have a longer
duration. (b) The longer a bond’s duration, the lower its volatility? (e) other things equal,
the lower the bond coupon, the higher its volatility? (d) If interest rates rise, bond
durations also rise?
Solution:
a. Longer-maturity bonds necessarily have longer durations
Duration depends on the coupon as well as the maturity of the bond. Therefore,
this statement is false.
b. The longer a bond’s duration, the lower its volatility
Yield to maturity of a bond and volatility of the bond is proportional to the
duration. So, the expression is false.
c. Other things equal, the lower the bond coupon, the higher its volatility.
This is true because a low coupon rate means longer duration and therefore higher
volatility.
d. If interest rates rise, bond durations rise also.
This is a false statement as a higher rate decreases the relative present value of
principal repayments.
Question 2: Calculate the durations and volatilities (modified durations) of securities A,
B, and C. Their cash flows are shown below. The interest rate on 8%:
Table 2: Question 11
Period 1
40
20
10
Period 2
40
20
10
Period 3
40
120
110
Solution:
There are three securities with the same time maturity (3periods) but different cash flows.
The interest rate for all the security is 8%.
The duration is the time required for the bond to make inflow equal to the market price of
the bond/security on a weighted basis. The original version of the duration is Macaulay
duration.
The formula of the duration:
𝐃𝐮𝐫𝐚𝐭𝐢𝐨𝐧 = [
𝟏 ∗ 𝐏𝐕 𝐨𝐟 𝐂𝟏
𝟐 ∗ 𝐏𝐕 𝐨𝐟 𝐂𝟐
𝟑 ∗ 𝐏𝐕 𝐨𝐟 𝐂𝟑
]+[
]+[
]
𝐓𝐨𝐭𝐚𝐥 𝐏𝐕
𝐓𝐨𝐭𝐚𝐥 𝐏𝐕
𝐓𝐨𝐭𝐚𝐥 𝐏𝐕
Duration of Security A
Particulars
Period
Cash flow
Present value at 8%
Total Present value
Fraction of the total PV
Duration
Duration of Security B
Particulars
Period
Cash flow
Present value at 8%
Total Present value
Fraction of the total PV
Duration
Duration of Security C
Particulars
Period
Cash flow
Present value at 8%
Total Present value
Fraction of the total PV
Duration
Period 1
Period 2
1
40
37.04
2
40
34.29
0.36
0.67
Period 1
Period 2
1
20
18.52
2
20
17.15
0.14
0.26
Period 1
Period 2
1
10
9.26
2
10
8.57
0.09
0.16
Period 3
3
40
31.75
103.08
0.92
1.95
Period 3
3
120
95.26
130.93
2.18
2.59
Period 3
3
110
87.32
105.15
2.49
2.74
Volatility is modified duration. The modified duration finds the change in the price of the
bond with a change in the interest rates.
The formula of the volatility or modified duration is:
𝑽𝒐𝒍𝒂𝒕𝒊𝒍𝒊𝒕𝒚 =
𝑫𝒖𝒓𝒂𝒕𝒊𝒐𝒏
𝟏 + 𝒀𝒊𝒆𝒍𝒅
Security A:
Volatility = 1.949/1+0.08
= 1.806
The volatility of security A is 1.80%.
Security B:
Volatility = 2.59/1+0.08
= 2.398
The volatility of security B is 2.398%.
Security C:
Volatility = 1.949/1+0.08
= 2.537
The volatility of security C is 2.537%.
Question 3: A 10-year U.S. Treasury bond with a face value of $10, 000 pays a coupon
of 5.5% (2.75% of face value every six months). The semiannually compounded interest
rate is 5.2% (a six-month discount rate of 2.6%). What is the PV of the bond? Generate a
graph or table showing how the bonds PV changes for semiannually compounded interest
rates between 1% and 15%.
Solution:
Data:
Face Value = $10,000.00
Coupon rate = 5.5% (2.75% every 6months)
Semiannual Interest rate = 5.20%
6-months discount rate =
2.60%
Period =
10 * 2 = 20
𝟏 − (𝟏 + 𝐫)−𝐧
𝐅𝐚𝐜𝐞 𝐕𝐚𝐥𝐮𝐞
]+[
]
(𝟏 + 𝐫)𝐧
𝐫
𝐂𝐨𝐮𝐩𝐨𝐧 (𝐂) = 𝐂𝐨𝐮𝐩𝐨𝐧 𝐫𝐚𝐭𝐞 ∗ 𝐅𝐚𝐜𝐞 𝐯𝐚𝐥𝐮𝐞
𝐏𝐕 𝐨𝐟 𝐁𝐨𝐧𝐝 = 𝐂 [
1 − (1 + 0.026)−20
$10,000
PV of Bond = $275 [
]+[
]
(1 + 0.026)20
0.026
PV of Bond = $275[15.4429] + $5,984.84
PV of Bond = $4,246.79 + $5,984.84
𝐏𝐕 𝐨𝐟 𝐁𝐨𝐧𝐝 = $𝟏𝟎, 𝟐𝟑𝟏. 𝟔𝟑
Table showing how the bonds PV changes for semiannually compounded interest rates
between 1% and 15%.
Interest Rate Semiannual Rate
1%
0.005
2%
0.010
3%
0.015
4%
0.020
5%
0.025
6%
0.030
7%
0.035
8%
0.040
9%
0.045
10%
0.050
11%
0.055
12%
0.060
13%
0.065
14%
0.070
15%
0.075
PV of Interest PV of Face Value
5,221.54
9,050.63
4,962.53
8,195.44
4,721.38
7,424.70
4,496.64
6,729.71
4,287.02
6,102.71
4,091.31
5,536.76
3,908.41
5,025.66
3,737.34
4,563.87
3,577.18
4,146.43
3,427.11
3,768.89
3,286.36
3,427.29
3,154.23
3,118.05
3,030.09
2,837.97
2,913.35
2,584.19
2,803.49
2,354.13
PV of bond
14,272.17
13,157.97
12,146.08
11,226.36
10,389.73
9,628.06
8,934.07
8,301.21
7,723.61
7,196.00
6,713.64
6,272.28
5,868.06
5,497.54
5,157.62
Question 4: A six-year government bond makes annual coupon payments of 5% and
offers a yield of 3% annually compounded. Suppose that one year later the bond still
yields 3%. What return has the bondholder earned over the 12-month period? Now
suppose that the bond yields 2% at the end of the year. What return would the bondholder
earn in this case?
Solution:
Computation of return the bondholder earned over the 12-months period at 3% yield:
Data:
Face Value = $1,000.00 (say)
Coupon rate = 5%
Yield =
3% annually
Period =
6yrs
𝐂𝐨𝐮𝐩𝐨𝐧 (𝐂) = 𝟓% ∗ $𝟏, 𝟎𝟎𝟎. 𝟎𝟎
𝐂𝐨𝐮𝐩𝐨𝐧 (𝐂) = $𝟓𝟎
1 − (1 + 0.03)−6
$1,000
PV of Bond = $50 [
]+[
]
(1 + 0.03)6
0.03
PV of Bond = $50[5.4172] + $837.48
PV of Bond = $270.86 + $5,984.84
𝐏𝐕 𝐨𝐟 𝐁𝐨𝐧𝐝 = $𝟏, 𝟏𝟎𝟖. 𝟑𝟒
PV of bond one year later:
1 − (1 + 0.03)−5
$1,000
𝑃𝑉 𝑜𝑓 𝐵𝑜𝑛𝑑 = $50 [
]+[
]
(1 + 0.03)5
0.03
𝑃𝑉 𝑜𝑓 𝐵𝑜𝑛𝑑 = $50[4.5797] + $862.61
𝑃𝑉 𝑜𝑓 𝐵𝑜𝑛𝑑 = $228.99 + $862.61
𝑷𝑽 𝒐𝒇 𝑩𝒐𝒏𝒅 = $𝟏, 𝟎𝟗𝟏. 𝟔𝟎
𝐀𝐧𝐧𝐮𝐚𝐥 𝐢𝐧𝐭𝐞𝐫𝐞𝐬𝐭 + (𝐏𝐕 @ 𝐲𝐫𝟏 − 𝐂𝐮𝐫𝐫𝐞𝐧𝐭 𝐏𝐫𝐢𝐜𝐞
]
𝐂𝐮𝐫𝐫𝐞𝐧𝐭 𝐏𝐫𝐢𝐜𝐞
$50 + ($1,091.6 − $1,108.32
Return to bondholders = [
]
$1,108.34
𝐑𝐞𝐭𝐮𝐫𝐧 𝐭𝐨 𝐛𝐨𝐧𝐝𝐡𝐨𝐥𝐝𝐞𝐫𝐬 = 𝟎. 𝟎𝟐𝟗𝟗 𝒐𝒓 𝟑%
𝐑𝐞𝐭𝐮𝐫𝐧 𝐭𝐨 𝐛𝐨𝐧𝐝𝐡𝐨𝐥𝐝𝐞𝐫𝐬 = [
Computation of return the bondholder earned over the 12-months period at 2% yield:
Data:
Face Value = $1,000.00 (say)
Coupon rate = 5%
Yield =
2% annually
Period =
6yrs
PV of bond one year later @ 2% yield:
1 − (1 + 0.02)−5
$1,000
]+[
]
(1 + 0.02)5
0.02
PV of Bond = $50(4.7135) + $905.73
PV of Bond = $235.68 + $905.73
𝐏𝐕 𝐨𝐟 𝐁𝐨𝐧𝐝 = $𝟏, 𝟏𝟒𝟏. 𝟒𝟏
PV of Bond = $50 [
𝐀𝐧𝐧𝐮𝐚𝐥 𝐢𝐧𝐭𝐞𝐫𝐞𝐬𝐭 + (𝐏𝐕 @ 𝐲𝐫𝟏 − 𝐂𝐮𝐫𝐫𝐞𝐧𝐭 𝐏𝐫𝐢𝐜𝐞
]
𝐂𝐮𝐫𝐫𝐞𝐧𝐭 𝐏𝐫𝐢𝐜𝐞
$50 + ($1,141.41 − $1,108.32
Return to bondholders = [
]
$1,108.34
𝐑𝐞𝐭𝐮𝐫𝐧 𝐭𝐨 𝐛𝐨𝐧𝐝𝐡𝐨𝐥𝐝𝐞𝐫𝐬 = 𝟎. 𝟎𝟕𝟒𝟗 𝐨𝐫 𝟕. 𝟒𝟗%
𝐑𝐞𝐭𝐮𝐫𝐧 𝐭𝐨 𝐛𝐨𝐧𝐝𝐡𝐨𝐥𝐝𝐞𝐫𝐬 = [
Question 5: A 6% six-year bond yields 12% and a 10% six-year bond yields 8%.
Calculate the six-year spot rate. Assume annual coupon payments.
Solution:
Finding the six-year spot rate:
Assuming the face value of the bond is $1,000, then the 6% six-year bond has a coupon
of $60 and the 10% six-year coupon has a coupon of $100.
Current price of the 6% six-year coupon bond with yield 12% as follows:
1 − (1 + 0.12)−6
$1,000
]+[
]
(1 + 0.12)6
0.12
PV of Bond = $50(4.1114) + $506.64
PV of Bond = $246.68 + $506.64
𝐏𝐕 𝐨𝐟 𝐁𝐨𝐧𝐝 = $𝟕𝟓𝟑. 𝟑𝟐
PV of Bond = $60 [
Current price of the 10% six-year coupon bond with yield 8% as follows:
1 − (1 + 0.08)−6
$1,000
PV of Bond = $100 [
]+[
]
(1 + 0.08)6
0.08
PV of Bond = $100(4.6229) + $630.16
PV of Bond = $463.29 + $630.16
𝐏𝐕 𝐨𝐟 𝐁𝐨𝐧𝐝 = $𝟏, 𝟎𝟗𝟐. 𝟒𝟓
The sum of the coupon payment of 2 6%, 6-years bond and 1.2 10%, 6-years bond would
be same in the first five years regardless of the spot rate in these first five years.
𝐂𝐨𝐮𝐩𝐨𝐧 𝐩𝐚𝐲𝐦𝐞𝐧𝐭 (𝟐 𝟔% 𝐁𝐨𝐧𝐝𝐬) = 𝟐 ∗ 𝐅𝐚𝐜𝐞 𝐯𝐚𝐥𝐮𝐞 ∗ 𝐂𝐨𝐮𝐩𝐨𝐧 𝐫𝐚𝐭𝐞
Coupon payment (2 6% Bonds) = 2 ∗ $1,000 ∗ 6%
𝐂𝐨𝐮𝐩𝐨𝐧 𝐩𝐚𝐲𝐦𝐞𝐧𝐭 (𝟐 𝟔% 𝐁𝐨𝐧𝐝𝐬) = $𝟏𝟐𝟎
𝐂𝐨𝐮𝐩𝐨𝐧 𝐩𝐚𝐲𝐦𝐞𝐧𝐭 (𝟏. 𝟐 𝟏𝟎% 𝐁𝐨𝐧𝐝𝐬) = 𝟏. 𝟐 ∗ 𝐅𝐚𝐜𝐞 𝐯𝐚𝐥𝐮𝐞 ∗ 𝐂𝐨𝐮𝐩𝐨𝐧 𝐫𝐚𝐭𝐞
Coupon payment (1.2 10% Bonds) = 1.2 ∗ $1,000 ∗ 10%
𝐂𝐨𝐮𝐩𝐨𝐧 𝐩𝐚𝐲𝐦𝐞𝐧𝐭 (𝟏. 𝟐 𝟏𝟎% 𝐁𝐨𝐧𝐝𝐬) = $𝟏𝟐𝟎
Coupons of two 6% coupon bonds are equal to the one and half of the 10% coupon
bonds.
The difference between the prices of 2 6% coupon bonds and 1.2 10% coupon bonds
(cost of portfolio) is as follows:
𝐂𝐨𝐬𝐭 𝐨𝐟 𝐏𝐨𝐫𝐭𝐟𝐨𝐥𝐢𝐨 = [(𝐏𝐕 𝐨𝐟 𝟔% 𝐛𝐨𝐧𝐝 ∗ 𝟐) − (𝐏𝐕 𝐨𝐟 𝟏𝟎% 𝐛𝐨𝐧𝐝 ∗ 𝟏. 𝟐)]
Cost of Portfolio = [($753.32 ∗ 2) − ($1,092.45 ∗ 1.2)]
Cost of Portfolio = [$1,506.64 − $1,310.94]
𝐂𝐨𝐬𝐭 𝐨𝐟 𝐏𝐨𝐫𝐭𝐟𝐨𝐥𝐢𝐨 = $𝟏𝟗𝟓. 𝟕
The difference between the cash flow of the two bonds in year 6 at maturity:
𝐂𝐚𝐬𝐡𝐟𝐥𝐨𝐰 (𝐂𝐅) 𝐨𝐟 𝐏𝐨𝐫𝐭𝐟𝐨𝐥𝐢𝐨
= [(𝐂𝐅 𝐘𝐫 𝟔, 𝟔% 𝐛𝐨𝐧𝐝 ∗ 𝟐) − (𝐂𝐅 𝐘𝐫 𝟔 𝟏𝟎% 𝐛𝐨𝐧𝐝 ∗ 𝟏. 𝟐)]
Cashflow of Portfolio = [($1000 + $60] ∗ 2) − ([$1000 + $100] ∗ 1.2)]
Cashflow of Portfolio = [$2,120 − $1,320]
𝐂𝐚𝐬𝐡𝐟𝐥𝐨𝐰 𝐨𝐟 𝐏𝐨𝐫𝐭𝐟𝐨𝐥𝐢𝐨 = $𝟖𝟎𝟎
The six-year spot rate:
𝐂𝐚𝐬𝐡𝐟𝐥𝐨𝐰 𝐨𝐟 𝐏𝐨𝐫𝐭𝐟𝐨𝐥𝐢𝐨 = 𝐂𝐨𝐬𝐭 𝐨𝐟 𝐏𝐨𝐫𝐭𝐟𝐨𝐥𝐢𝐨 ∗ (𝟏 + 𝐫𝟔 )𝟔
$800 = $195.7 ∗ (1 + r6 )6
$800
$195.7
=
∗ (1 + r6 )6
$195.7
197.7
1
(4.0879)6 = (1 + r6 )
1.2645 = 1 + r6
𝐫𝟔 = 𝟎. 𝟐𝟔𝟒𝟓 𝐨𝐫 𝟐𝟔. 𝟒𝟓%
Question 6: Is the yield on high-coupon bonds more likely to be higher than that on lowcoupon bonds when the term structure is upward-sloping, or when it is downwardsloping? Explain.
Solution:
The term structure of the interest rates represents the relationship between the short and
long-term interest rates. The term structure on interest rate is also called the yield curve.
The yield curve shows how the yield of the interest securities is plotted over the maturity
period.
If the term structure of interest rate is upward sloping, the long-term interest rates will be
higher than the short-term interest rates. It does not mean that investing in long-term
securities is more profitable than investing in short-term securities.
If the term structure of interest rate is upward sloping, the investors expect short-term
interest rates to rise. When the term structure is upward sloping, the investors like to
borrow short with the expectation of increase in interest rates.
If the term structure of the interest rate is downward sloping, the short-term interest rates
will be higher than the longer-term interest rates.
The highest proportion of the cash flows from the higher coupon bonds will be provided
in the early years. Hence, a higher coupon bond will be considered as a shorter bond.
Therefore, when the term structure is downward sloping, the high coupon bonds’ yield
will be higher than the low coupon bonds’ yield.
Question 7: You have estimated spot rates as follows: r1 = 5.00%, r2 = 5.40%, r3 =
5.70%, r4 = 5.90%, and r5 = 6.00%.
a. What are the discount factors for each date (that is, the PV of $1 paid in year t)?
b. Calculate the PV of the following bonds assuming annual coupon»: (i) 5%, two-year
bond (ii) 5%, 5-year bond, and (iii) 10%, 5-year bond
c. Explain intuitively why the yield to maturity on the 10% bond is less than that on the
5% bond
d. What should be the yield to maturity on a five-year zero-coupon bond?
e. Show that the correct yield to maturity on a five-year annuity is 5.75%
f. Explain intuitively why the yield on the five-year bond described in part (b) must lie
between the yield on a five-year zero-coupon bond and a five-year annuity.
Solution:
a. Calculate the discount factor of the spot rate
𝐃𝐢𝐬𝐜𝐨𝐮𝐧𝐭 𝐅𝐚𝐜𝐭𝐨𝐫 =
Year
1
2
3
4
5
Rate
5.00%
5.40%
5.70%
5.90%
6.00%
Calculation
1/(1+5.0%)
1/(1+5.4%)^2
1/(1+5.7%)^3
1/(1+5.9%)^4
1/(1+6.0%)^5
𝟏
(𝟏 + 𝐫)𝐧
Discount Factor
0.9524
0.9002
0.8468
0.7951
0.7473
b. Calculation of PV of the bonds
i)
5%, two-year bonds
Let say the face value is $1,000, then the coupon is $50 ($1,000 * 5%). The
rate is given in part (a).
𝐂𝐨𝐮𝐩𝐨𝐧 𝐢𝐧 𝐘𝐞𝐚𝐫 𝟏
𝐂𝐨𝐮𝐩𝐨𝐧 𝐢𝐧 𝐘𝐞𝐚𝐫 𝟐 + 𝐅𝐚𝐜𝐞 𝐕𝐚𝐥𝐮𝐞
]+[
]
𝐧
(𝟏 + 𝐫𝟐 )𝐧
(𝟏 + 𝐫𝟏 )
$50
$50 + $1,000
PV of Bond = [
]+[
]
(1 + 0.054)2
(1 + 0.05)
$50
$50 + $1,000
PV of Bond = [
]+[
]
(1 + 0.054)2
(1 + 0.05)
PV of Bond = $47.62 + $945.17
𝐏𝐕 𝐨𝐟 𝐁𝐨𝐧𝐝 = $𝟗𝟗𝟐. 𝟕𝟗
𝐏𝐕 𝐨𝐟 𝐁𝐨𝐧𝐝 = [
ii)
5%, five-year bonds
$50
$50
$50
$50
]+[
]
+
[
]
+
[
]
(1 + 0.054)2
(1 + 0.057)3
(1 + 0.059)4
(1 + 0.05)
$1050
+[
]
(1 + 0.060)5
PV of Bond = $47.62 + $45.01 + $42.34 + $39.75 + $784.62
𝐏𝐕 𝐨𝐟 𝐁𝐨𝐧𝐝 = $𝟗𝟓𝟗. 𝟑𝟒
PV of Bond = [
iii)
10%, five-year bonds
$100
$100
$100
$100
]+[
]+[
]+[
]
2
3
(1 + 0.054)
(1 + 0.057)
(1 + 0.059)4
(1 + 0.05)
$1,100
+[
]
(1 + 0.060)5
PV of Bond = $95.24 + $90.02 + $84.68 + $79.51 + 821.98
𝐏𝐕 𝐨𝐟 𝐁𝐨𝐧𝐝 = $𝟏, 𝟏𝟕𝟏. 𝟒𝟑
PV of Bond = [
c. Intuitive explanation on why the yield to maturity on the 10% bond is less than that on
the 5% bond:
The yield on 5% bond
Cashflow (pmt)
$
50.00
Face value
$ 1,000.00
PV of the bond
$ 959.34
Number of periods
5
Yield
5.96%
The yield on 10% bond
Cashflow (pmt)
$ 100.00
Face value
$ 1,000.00
PV of the bond
$ 1,171.43
Number of periods
5
Yield
5.94%
The yield on the 10% bond is lower than the yield on the 5% bond. The calculation of
yield depends on the spot rate and the coupon payment. The coupon payment for the 10%
bond is higher as compared with the 5% bond. Hence the yield is lower for a 10% bond.
d. What should be the yield to maturity on a five-year zero-coupon bond?
The YTM on a 5-year zero-coupon bond is the 5-year spot rate. Since it is zero-coupon
bond, no coupon interest will be available. Hence the YTM is 6.0% which is the yield at
the end of the 6th year.
e. Show that the correct yield to maturity on a five-year annuity is 5.75%
The price of a 5-year annuity is calculated as follows:
𝐏𝐕 𝐨𝐟 𝐁𝐨𝐧𝐝 = [
𝐂𝟏
𝐂𝟐
𝐂𝟑
𝐂𝟒
𝐂𝟓
]+[
]+[
]+[
]+[
]
𝟐
𝟑
𝟒
(𝟏 + 𝐫)
(𝟏 + 𝐫)
(𝟏 + 𝐫)
(𝟏 + 𝐫)𝟓
(𝟏 + 𝐫)
$1
$1
$1
$1
$1
]+[
]
+
[
]
+
[
]
+
[
]
(1.054)2
(1.057)3
(1.059)4
(1.05)5
(1.05)
𝐏𝐕 𝐨𝐟 𝐁𝐨𝐧𝐝 = $𝟒, 𝟐𝟒𝟏𝟕
PV of Bond = [
Now, the YTM is calculated as follows:
𝐂𝟏
𝐂𝟐
𝐂𝟑
𝐂𝟒
𝐂𝟓
]+[
]+[
]+[
]+[
]
𝟐
𝟑
𝟒
(𝟏 + 𝐫)
(𝟏 + 𝐫)
(𝟏 + 𝐫)
(𝟏 + 𝐫)𝟓
(𝟏 + 𝐫)
1
1
1
1
1
$4.2417 = [
]+[
]
+
[
]
+
[
]
+
[
]
(1 + r)2
(1 + r)3
(1 + r)4
(1 + r)5
(1 + r)
𝐫 = 𝟎. 𝟎𝟓𝟕𝟓 𝐨𝐫 𝟓. 𝟓𝟕%
𝐏𝐕 𝐨𝐟 𝐁𝐨𝐧𝐝 = [
f. Intuitive explanation on why the yield on the five-year bond described in part (b) must
lie between the yield on a five-year zero-coupon bond and a five-year annuity.
The yield is somewhere between the 5-year bond and the 5-year coupon bond, as their
cash flow lies between the two instruments when interest rates rise