Solutions to Questions – Chapter 6
Mortgages: Additional Concepts, Analysis, and Applications
Question 6-1
What are the primary considerations that should be made when refinancing?
Question 6-2
What factors must be considered when deciding whether to refinance a loan after interest rates have declined?
Question 6-3
Why might the market value of a loan differ from its outstanding balance?
Question 6-4
Why might a borrower be willing to pay a higher price for a home with an assumable loan?
Question 6-5
What is a buydown loan? What parties are usually involved in this kind of loan?
Question 6-6
Why might a wraparound lender provide a wraparound loan at a lower rate than a new first mortgage?
Question 6-7
Assuming the borrower is in no danger of default, under what conditions might a lender be willing to accept a
lesser amount from a borrower than the outstanding balance of a loan and still consider the loan paid in full?
Question 6-8
Under what conditions might a home with an assumable loan sell for more than comparable homes with no
assumable loans available?
Question 6-9
What is meant by the incremental cost of borrowing additional funds?
Question 6-10
Is the incremental cost of borrowing additional funds affected significantly by early repayment of the loan?
Solutions to Problems – Chapter 6
Residential Financial Analysis
INTRODUCTION
The following solutions were obtained using an HP 12C financial calculator. Answers may differ slightly due to rounding or
use of the financial tables to approximate the answers. As pointed out in the chapter, there is often more than one way of
approaching the solution to the problems in this chapter. Thus “alternative solutions” are shown were appropriate.
Problem 6-1
(a)
Alternative Interest Rate Loan Term Loan Amount Monthly Payments
(b)
Alternative Loan Amount Points Net Proceeds Monthly Payments
90% Loan $90,000 $1,800 $88,200 $724.70
(c)
We now need the loan balance after 5 years.
Problem 6-2
(a)
For this problem we need to know the effective cost of the $180,000 loan at 9% combined with the $40,000 loan at 13%
Loan Amount Interest Rate Loan Term Monthly Payments
$180,000 9% 20 yrs.. $1619.51
(b) REV
(c)
Assuming the loan is held for the full term (to compare with Part a:)
(c) IRR(CF1, CF2, ….CFn)
CFj nj
© 2019 McGraw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No
reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education.
1,619.51 n = 12
1,619.51 n = 12
1,619.51 n = 12
1,619.51 n = 12
1,619.51 n = 12
1,619.51 n = 12
1,619.51 n = 12
Solve for the IRR:
= 0.79% x 12 = 9.49% (annual rate, compounded monthly)
Note that the payment of $1,619.51 is first discounted as a 10 year annuity (years 11 to 20) and further discounted as a lump
sum for 10 years to recognized the fact that the annuity does not start until year 10. When calculating the IRR in excel input
the monthly payment (annuity) in each cell for each period as opposed to one lump annual payment amount.
Solving for the cost we obtain 9.49%. This is less than 9.5% rate for the single $220,000 loan. Thus, the combined loans are
preferred.
Assuming the loan is held for 5 years (to compare with Part b):
We now need the loan balance after 5 years.
Loan Amount Interest Rate Loan Term Monthly Payments Loan Balance
$180,000 9% 20 yrs.. $1619.51 $159,672.68
40,000 13% 10 yrs.. 597.24 26,248.89
Combined $220,000 $2,216.75 $185,921.57
i(n,PV,PMT,FV)
n = 60
PMT = $2,216.75
PV = -$220,000
FV = $185,921.57
Solve for the annual interest rate:
i = 9.67%
We now obtain 9.67%. This is greater than the 9.5% rate for a single loan.
Problem 6-3
Preliminary calculation:
The existing loan is for $95,000 at a 11% interest rate for 30 years (monthly payments). The monthly payment is $904.71.
The balance of the loan after 5 years is $92,306.41.
Alternate solution:
Amount of new loan $92,306.00
(b)
For a 5-year holding period we must also consider the balance of the old and new loan after 5 years. We have:
© 2019 McGraw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No
reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education.
The negative return tells you this is a bad investment if the new loan is paid off so quickly. The reason for the negative
return, is that you pay for the refinancing up-front, but do not benefit from the lower monthly payments on the new financing
for a period of time long enough to cover and/or justify the cost of refinancing.
Alternative solution:
The effective cost is now as follows:
Problem 6-4
(a)
(b)
To calculate the market value assuming the loan is repaid after 5 additional years, we have:
Problem 6-5
(a)
Alternative 1: Purchase of $150,000 home:
Note to instructors:
It is informative to calculate exactly how much more the borrower could pay for alternative 2. This is found by discounting
the payment savings at 10.5%. We have:
(b)
With the homeowner providing the second mortgage for the additional $20,000 at 9% (purchase money mortgage) we have:
(c)
Alternative 2: Purchase of $160,000 home:
Problem 6-6
Amount
Payment
Term
$150,000
$1,800.25
15 yrs.
100,000
1,100.25
15 yrs. (remaining)
$50,000
$700.25
$700.25 x (MPVIFA, ?%, 15 yrs..) = $50,000
Problem 6-7
(a)
Alternative solution:
Payments on a loan for $100,000 at 9.5%: $873.70
Present value of the saving discounted at 9.5%:
(b)
The balance of the $100,000 loan (9%, 25 yrs.) after 10 years is $82,739.23. We now discount the payments on the $100,000
loan which are $839.20 and the balance after 10 years which is 82,739.23. Both are discounted at the market rate of 9.5%.
Alternative solution:
The difference in payments for a $100,000 loan at 9% and $100,000 at 9.5% is $34.50 (same as alternative solution to part a.)
We must also consider the difference in loan balances after 10 years.
Problem 6-8 Amount of
(a) Reduction Payment will be Months
Monthly payment reduction during the first year (50% of $726.96): $363.48 $363.48 12
(b)
Based on the results from (a), the buydown loan is worth $6,005.66 in present value terms. We would expect the home to
Problem 6-9
(a) Step 1, Calculate the dollar monthly difference between the two financing options.
Find present value of the payments at the market rate of 8%
This is the market (cash equivalent) value of the loan.
© 2019 McGraw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No
reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education.
The buyer made a cash down payment of $60,000.
Cash equivalent value of loan $131,675.49
Cash down payment 60,000.00
Cash equivalent value of property $191,675.49
(b) If it is assumed that the buyer only expected to benefit from the favorable financing for five years:
Problem 6-10
Question: A borrower is making a choice between a mortgage with monthly payment or bi-weekly payments; the loan will
be $200,000@6% interest for 20 years.
A) How would you analyze these alternatives?
B) What if the bi-weekly loan was available for 5.75%? How would your answer change?
Problem 6A-1
(a)
Loan $100,000, 10% interest, 15 yrs (monthly payments)
Monthly payment $1,074.61
Before-tax
After-tax
Value
Month
Payment
Interest
Principal
Balance
of Deduction
AT Payment
0
-100000
-100000
1
1074.61
833.33
241.27
99,759
250.00
824.61
2
1074.61
831.32
243.28
99,515
249.40
825.21
3
1074.61
829.30
245.31
99,270
248.79
825.82
4
1074.61
827.25
247.35
99,023
248.18
826.43
5
1074.61
825.19
249.42
98,773
247.56
827.05
6
1074.61
823.11
251.49
98,522
246.93
827.67
7
1074.61
821.02
253.59
98,268
246.30
828.30
8
1074.61
818.90
255.70
98,013
245.67
828.93
9
1074.61
816.77
257.83
97,755
245.03
829.57
10
1074.61
814.62
259.98
97,495
244.39
830.22
11
1074.61
812.46
262.15
97,233
243.74
830.87
12
1074.61
810.27
264.33
96,968
243.08
831.52
13
1074.61
808.07
266.54
96,702
242.42
832.18
14
1074.61
805.85
268.76
96,433
241.75
832.85
15
1074.61
803.61
271.00
96,162
241.08
833.52
16
1074.61
801.35
273.26
95,889
240.40
834.20
17
1074.61
799.07
275.53
95,613
239.72
834.88
18
1074.61
796.78
277.83
95,335
239.03
835.57
19
1074.61
794.46
280.14
95,055
238.34
836.27
20
1074.61
792.13
282.48
94,773
237.64
836.97
21
1074.61
789.77
284.83
94,488
236.93
837.67
22
1074.61
787.40
287.21
94,201
236.22
838.39
23
1074.61
785.01
289.60
93,911
235.50
839.10
24
1074.61
782.59
292.01
93,619
234.78
839.83
25
1074.61
780.16
294.45
93,325
234.05
840.56
26
1074.61
777.71
296.90
93,028
233.31
841.29
27
1074.61
775.23
299.37
92,728
232.57
842.04
28
1074.61
772.74
301.87
92,427
231.82
842.78
29
1074.61
770.22
304.38
92,122
231.07
843.54
30
1074.61
767.68
306.92
91,815
230.31
844.30
31
1074.61
765.13
309.48
91,506
229.54
845.07
32
1074.61
762.55
312.06
91,194
228.76
845.84
33
1074.61
759.95
314.66
90,879
227.98
846.62
34
1074.61
757.33
317.28
90,562
227.20
847.41
35
1074.61
754.68
319.92
90,242
226.40
848.20
Balance 36
90993.83
752.02
322.59
89,919
225.60
90768.23
(b)
Before-tax
After-tax
Value
Month
Payment
Interest
Principal
Balance
of Deduction
AT Payment
0
-95000
-96500
1
1074.61
833.33
241.27
94,759
250.00
824.61
2
1074.61
789.66
284.95
94,474
236.90
837.71
3
1074.61
787.28
287.32
94,186
236.18
838.42
4
1074.61
784.89
289.72
93,897
235.47
839.14
5
1074.61
782.47
292.13
93,605
234.74
839.86
6
1074.61
780.04
294.57
93,310
234.01
840.59
7
1074.61
777.58
297.02
93,013
233.28
841.33
8
1074.61
775.11
299.50
92,714
232.53
842.07
9
1074.61
772.61
301.99
92,412
231.78
842.82
10
1074.61
770.10
304.51
92,107
231.03
843.58
11
1074.61
767.56
307.05
91,800
230.27
844.34
12
1074.61
765.00
309.61
91,490
229.50
845.11
13
1074.61
762.42
312.19
91,178
228.73
845.88
14
1074.61
759.82
314.79
90,863
227.95
846.66
15
1074.61
757.19
317.41
90,546
227.16
847.45
16
1074.61
754.55
320.06
90,226
226.36
848.24
17
1074.61
751.88
322.72
89,903
225.56
849.04
18
1074.61
749.19
325.41
89,578
224.76
849.85
19
1074.61
746.48
328.12
89,250
223.94
850.66
20
1074.61
743.75
330.86
88,919
223.12
851.48
21
1074.61
740.99
333.62
88,585
222.30
852.31
22
1074.61
738.21
336.40
88,249
221.46
853.14
23
1074.61
735.41
339.20
87,910
220.62
853.98
24
1074.61
732.58
342.03
87,568
219.77
854.83
25
1074.61
729.73
344.88
87,223
218.92
855.69
26
1074.61
726.86
347.75
86,875
218.06
856.55
27
1074.61
723.96
350.65
86,524
217.19
857.42
28
1074.61
721.04
353.57
86,171
216.31
858.29
29
1074.61
718.09
356.52
85,814
215.43
859.18
30
1074.61
715.12
359.49
85,455
214.54
860.07
31
1074.61
712.12
362.48
85,092
213.64
860.97
32
1074.61
709.10
365.50
84,727
212.73
861.87
33
1074.61
706.06
368.55
84,358
211.82
862.79
34
1074.61
702.99
371.62
83,987
210.90
863.71
35
1074.61
699.89
374.72
83,612
209.97
864.64
Balance 36
90993.83
696.77
377.84
83,234
209.03
90784.80
Problem 6A -2
Monthly loan schedule
Cumulative
Beginning
Ending
Interest
Cumulative
Cumulative
Deferred
Month
Balance
Payment
Interest
Principal
Balance
Deduction
Deduction
Interest
Interest
1
100,000
500.00
750.00
-250.00
100,250
500.00
500.00
750.00
250.00
2
100,250
500.00
751.88
-251.88
100,502
500.00
1000.00
1501.88
501.88
3
100,502
500.00
753.76
-253.76
100,756
500.00
1500.00
2255.64
755.64
4
100,756
500.00
755.67
-255.67
101,011
500.00
2000.00
3011.31
1011.31
5
101,011
500.00
757.58
-257.58
101,269
500.00
2500.00
3768.89
1268.89
6
101,269
500.00
759.52
-259.52
101,528
500.00
3000.00
4528.41
1528.41
7
101,528
500.00
761.46
-261.46
101,790
500.00
3500.00
5289.87
1789.87
8
101,790
500.00
763.42
-263.42
102,053
500.00
4000.00
6053.29
2053.29
9
102,053
500.00
765.40
-265.40
102,319
500.00
4500.00
6818.69
2318.69
10
102,319
500.00
767.39
-267.39
102,586
500.00
5000.00
7586.08
2586.08
11
102,586
500.00
769.40
-269.40
102,855
500.00
5500.00
8355.48
2855.48
12
102,855
500.00
771.42
-271.42
103,127
500.00
6000.00
9126.90
3126.90
13
103,127
875.20
773.45
101.75
103,025
875.20
6875.20
9900.35
3025.15
14
103,025
875.20
772.69
102.51
102,923
875.20
7750.40
10673.04
2922.63
15
102,923
875.20
771.92
103.28
102,819
875.20
8625.60
11444.96
2819.35
16
102,819
875.20
771.15
104.06
102,715
875.20
9500.80
12216.10
2715.30
17
102,715
875.20
770.36
104.84
102,610
875.20
10376.01
12986.47
2610.46
18
102,610
875.20
769.58
105.62
102,505
875.20
11251.21
13756.05
2504.84
19
102,505
875.20
768.79
106.41
102,398
875.20
12126.41
14524.83
2398.42
20
102,398
875.20
767.99
107.21
102,291
875.20
13001.61
15292.82
2291.21
21
102,291
875.20
767.18
108.02
102,183
875.20
13876.81
16060.00
2183.19
22
102,183
875.20
766.37
108.83
102,074
875.20
14752.01
16826.38
2074.37
23
102,074
875.20
765.56
109.64
101,965
875.20
15627.21
17591.94
1964.72
24
101,965
875.20
764.74
110.47
101,854
875.20
16502.41
18356.67
1854.26