Chapter 9 - Selected Quantitative Problems & Solutions
Question 4
A 2-year $1,000 par zero-coupon bond is currently priced at $819.00. A 2-year $1,000
annuity is currently priced at $1,712.52. If you want to invest $10,000 in one of the two
securities, which is a better buy? You can assume
a. the pure expectations theory of interest rates holds,
b. neither bond has any default risk, maturity premium, or liquidity premium, and
c. you can purchase partial bonds.
Solution: With PV  $819; FV  $1,000; PMT  0; and N  2, the yield to maturity on
the two-year zero-coupon bonds is 10.5% for the two-year annuities. PV 
$1,712.52; PMT  0;
FV  $2,000; and N  2 gives a yield to maturity of 8.07%. The zero-coupon
bonds are
the better buy.
Question 5
Consider the following cash flows. All market interest rates are 12%
Year
0
1
2
3
4
Cash Flow
160
170
180
230
a.
What price would you pay for these cash flows? What total wealth do you expect
after 2½ years if you sell the rights to the remaining cash flows? Assume interest
rates remain constant.
b. What is the duration of these cash flows?
c. Immediately after buying these cash flows, all market interest rates drop to 11%.
What is the impact on your total wealth after 2½ years?
Solution:
a.
Price 
160 170
180
230



 $552.67
2
3
1.12 1.12 1.12 1.124
Expected Wealth  160  (1.12)1.5  170  (1.12)5 
180
230

 $733.69
5
1.12 1.121.5
160
170
180
230
(1) 
(2) 
(3) 
2
3
4
1.12
1.12
1.12
1.12
b. Duration 
 2.50
552.67
180
230
c. Expected Wealth  160  (1.11)1.5  170  (1.11).5 

 $733.74
.5
1.11 1.111.5
Since you are holding the cash flows for their duration, you are essentially immunized
from interest rate changes (in this simplistic example).
Question 6
The yield on a corporate bond is 10% and it is currently selling at par. The marginal tax
rate is 20%. A par value municipal bond with a coupon rate of 8.50% is available. Which
security is a better buy?
Solution: The equivalent tax-free rate  taxable interest rate  (1  marginal tax rate).
In this case, 0.10  (1  0.20)  8%. The corporate bond offers a lower aftertax yield given the marginal tax rate, so the municipal bond is a better buy.
Chapter 11 - Selected Quantitative Problems & Solutions
Question 1 (Useful)
Compute the required monthly payment on a $80,000 30-year, fixed-rate mortgage with a
nominal interest rate of 5.80%. How much of the payment goes toward principal and interest
during the first year?
Solution: The monthly mortgage payment is computed as:
N  360; I  5.8/12; PV  80,000; FV  0
Compute PMT; PMT  $469.40
The amortization schedule is as follows:
Month
1
2
3
4
5
6
7
8
9
10
11
12
Total
Beginning
Balance
$80,000.00
$79,917.26
$79,834.13
$79,750.59
$79,666.65
$79,582.30
$79,497.55
$79,412.38
$79,326.81
$79,240.82
$79,154.41
$79,067.59
Payment
Interest
Paid
Principal
Paid
Ending
Balance
$ 469.40
$ 469.40
$ 469.40
$ 469.40
$ 469.40
$ 469.40
$ 469.40
$ 469.40
$ 469.40
$ 469.40
$ 469.40
$ 469.40
$5,632.83
$ 386.67
$ 386.27
$ 385.86
$ 385.46
$ 385.06
$ 384.65
$ 384.24
$ 383.83
$ 383.41
$ 383.00
$ 382.58
$ 382.16
$4,613.18
$ 82.74
$ 83.14
$ 83.54
$ 83.94
$ 84.35
$ 84.75
$ 85.16
$ 85.58
$ 85.99
$ 86.41
$ 86.82
$ 87.24
$1,019.65
$79,917.26
$79,834.13
$79,750.59
$79,666.65
$79,582.30
$79,497.55
$79,412.38
$79,326.81
$79,240.82
$79,154.41
$79,067.59
$78,980.35
Question 2
Compute the face value of a 30-year, fixed-rate mortgage with a monthly payment of $1,100,
assuming a nominal interest rate of 9%. If the mortgage requires 5% down, what is the
maximum house price?
Solution: The PV of the payments is:
N  360; I  9/12; PV  1100; FV  0
Compute PV; PV  136,710
The maximum house price is 136,710/0.95  $143,905
Question 3 (Useful)
Consider a 30-year, fixed-rate mortgage for $100,000 at a nominal rate of 9%. If the borrower
wants to pay off the remaining balance on the mortgage after making the 12th payment,
what is the remaining balance on the mortgage?
Solution: The monthly mortgage payment is computed as:
N  360; I  9/12; PV  100,000; FV  0
Compute PMT; PMT  $804.62
The amortization schedule is as follows:
Month
1
2
3
4
5
6
7
8
9
10
11
12
Beginning
Balance
Payment
Interest
Paid
Principal
Paid
Ending
Balance
$100,000.00
$ 99,945.38
$ 99,890.35
$ 99,834.91
$ 99,779.05
$ 99,722.77
$ 99,666.07
$ 99,608.95
$ 99,551.40
$ 99,493.41
$ 99,434.99
$ 99,376.13
$804.62
$804.62
$804.62
$804.62
$804.62
$804.62
$804.62
$804.62
$804.62
$804.62
$804.62
$804.62
$750.00
$749.59
$749.18
$748.76
$748.34
$747.92
$747.50
$747.07
$746.64
$746.20
$745.76
$745.32
$54.62
$55.03
$55.44
$55.86
$56.28
$56.70
$57.12
$57.55
$57.98
$58.42
$58.86
$59.30
$99,945.38
$99,890.35
$99,834.91
$99,779.05
$99,722.77
$99,666.07
$99,608.95
$99,551.40
$99,493.41
$99,434.99
$99,376.13
$99,316.84
Just after making the 12th payment, the borrower must pay $99,317 to pay off the loan.
Question 4
Consider a 30-year, fixed-rate mortgage for $100,000 at a nominal rate of 9%. If the borrower
pays an additional $100 with each payment, how fast will the mortgage be paid off?
Solution: The monthly mortgage payment is computed as:
N  360; I  9/12; PV  100,000; FV  0
Compute PMT; PMT  $804.62
The borrower is sending in $904.62 each month. To determine when the loan
will be retired:
PMT  904.62; I  9/12; PV  100,000; FV  0
Compute N; N  237, or after 19.75 years.
Question 5
Consider a 30-year, fixed-rate mortgage for $100,000 at a nominal rate of 9%. A S&L issues
this mortgage on April 1 and retains the mortgage in its portfolio. However, by April 2,
mortgage rates have increased to a 9.5% nominal rate. By how much has the value of the
mortgage fallen?
Solution: The monthly mortgage payment is computed as:
N  360; I  9/12; PV  100,000; FV  0
Compute PMT; PMT  $804.62
In a 9.5% market, the mortgage is worth:
N  360; I  9.5/12; PMT  $804.62; FV  0
Compute PV; PV  $95,691.10
The value of the mortgage has fallen by about $4,300, or 4.3%
Question 6 (Very Useful – No financial calculator required!)
Consider a 30-year, fixed-rate mortgage for $80,000 at a nominal rate of 5.8%. Refer to the
amortization schedule below.
a) How much of the payment goes toward principal and interest during the first year?
b) If the borrower wants to pay off the remaining balance on the mortgage after making the
12th payment, what is the remaining balance on the mortgage?
c) Calculate the Interest Paid for Month 13 if the principle paid is $87.66 and the ending
balance of month 13 is $78,892.68.
The amortization schedule is as follows:
Month
Beginning
Balance
Payment
Interest
Paid
Principal
Paid
1
2
3
4
5
6
7
8
9
10
11
12
$80,000.00
$79,917.26
$79,834.13
$79,750.59
$79,666.65
$79,582.30
$79,497.55
$79,412.38
$79,326.81
$79,240.82
$79,154.41
$79,067.59
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
469.40
469.40
469.40
469.40
469.40
469.40
469.40
469.40
469.40
469.40
469.40
469.40
386.67
386.27
385.86
385.46
385.06
384.65
384.24
383.83
383.41
383.00
382.58
382.16
82.74
83.14
83.54
83.94
84.35
84.75
85.16
85.58
85.99
86.41
86.82
87.24
Ending
Balance
$79,917.26
$79,834.13
$79,750.59
$79,666.65
$79,582.30
$79,497.55
$79,412.38
$79,326.81
$79,240.82
$79,154.41
$79,067.59
$78,980.35
Answer:
a) Principal Paid for 12 months = $1,019.65, Interest Paid for 12 months = $4,613.18
b) $78,980.35
c) $381.74