CH4M.DOC 01/08/2019
CHAPTER OBJECTIVES - CHAPTER 4
Fixed Rate Mortgage Loans
The student who has successfully completed this chapter should be able to perform the following:
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Calculate loan payment, loan balances, and interest charges on constant payment, constant amortization, and graduated payment mortgages.
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Calculate the borrower’s effective cost of borrowing or the lender’s effective yield, factoring in the effects of origination fees, discount points, early repayment, and prepayment penalties.
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Understand the difference between the effective cost of borrowing and an annual percentage rate (APR).
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Define the truth-in-lending requirements as they relates to the APR of a loan (i.e. rounding standards) and also calculate this rate for all of the above mortgage types.
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Calculate the amount of discount points or origination fees to be charged on a loan in order to reach the lender’s desired yield.
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Solutions to Questions - Chapter 4
Fixed Rate Mortgage Loans
Question 4-1
What are the major differences between the CAM, CPM, and GPM loans? What are the advantages to borrowers and risks to lenders for each? What elements do each of the loans have in common?
CAM - Constant Amortization Mortgage - Payments on constant amortization mortgages were determined first by computing a constant amount of each monthly payment to be applied to principal. Interest was then computed on the monthly loan balance and added to the monthly amount of amortization. The total monthly payment was determined by adding the constant amount of monthly amortization to interest on the outstanding loan balance.
CPM - Constant Payment Mortgage - This payment pattern simply means that a level, or constant, monthly payment is calculated on an original loan amount at a fixed rate of interest for a given term.
GPM - Graduated Payment Mortgage - The objective is to provide for a series of mortgage payments that are lower in the initial years of the loan than they would be with a standard mortgage loan. GPM payments then gradually increase at a predetermined rate as borrower incomes are expected to rise over time.
CAM - lenders recognized that in a growing economy, borrowers could partially repay the loan over time, as opposed to being left to their own devices to reduce the loan balance when the term of the loan ended.
CPM - At the end of the term of the mortgage loan, the original loan amount or principal is completely repaid and the lender has earned a fixed rate of interest on the monthly loan balance. However the amount of amortization varies each month.
GPM - The payment pattern offsets the tilt effect to some extent, hence reducing the burden faced by households when meeting mortgage payments from current income in an inflationary environment.
Amortization
Question 4-2
Define amortization.
Amortization is the process of loan repayment over time.
Question 4-3
Why do the monthly payments in the beginning months of a CPM loan contain a higher proportion of interest than principal repayment?
The reason for such a high interest component in each monthly payment is that the lender earns an annual percentage return on the outstanding monthly loan balance. Because the loan is being repaid over a long period of time, the loan balance is reduced only very slightly at first and monthly interest charges are correspondingly high.
Question 4-4
What are loan closing costs? How can they be categorized? Which of the categories influence borrowing costs and why?
Closing costs are incurred in many types of real estate financing, including residential property, income property, construction, and land development loans.
Categories include: statutory costs, third party charges, and additional finance charges.
Closing costs that do affect the cost of borrowing are additional finance charges levied by the lender. These charges constitute additional income to the lender and as a result must be included as a part of the cost of borrowing. Lenders refer to these additional charges as loan fees.
Question 4-5
Does repaying a loan early ever affect the actual or true interest cost to the borrower?
When loan fees are charged and the loan is paid off before maturity, the effective interest cost of the loan increases even further than when the loan is repaid at maturity.
Question 4-6
Why do lenders charge origination fees, especially loan discount fees?
Lenders usually charge these costs to borrowers when the loan is made, or “closed”, rather than charging higher interest rates. They do this because if the loan is repaid soon after closing, the additional interest earned by the lender as of the repayment date may not be enough to offset the fixed costs of loan origination.
Question 4-7
What is the connection between the Truth-in-Lending Act and the annual percentage rate (APR)?
Truth-in-Lending Act - the lender must disclose to the borrower the annual percentage rate being charged on the loan.
The APR reflects origination fees and discount points and treats them as additional income or yield to the lender regardless of what costs the fees are intended to cover.
Question 4-8
Does the annual percentage rate always equal the effective borrowing cost?
The annual percentage rate under truth-in-lending requirements never takes into account early repayment of loans. The APR calculation takes into account origination fees, but always assumes the loan is paid off at maturity.
Question 4-9
What is meant by a real rate of interest?
A real rate of interest is an interest rate expressed in terms of real goods and is equal to the nominal rate less the expected rate of inflation.
Question 4-10
What is a risk premium in the context of mortgage lending?
A reason for loan discount fees is that lenders believe that they can better price the loan to the risk they take. The risk for some individual borrowers is slightly higher than others and these loans may require more time and expense to process and control.
Question 4-11
When mortgage lenders establish interest rates through competition, an expected inflation premium is said to be part of the interest rate. What does this mean?
The uncertainty of future economic factors, including the supply of savings, demand for housing and future levels of inflation, directly affects interest rates. However, interest rates at a given point in time can only reflect the market consensus of what these factors are expected to be. To be competitive, a lender can only charge an interest rate that reflects what the market expects inflation to be even if he expects inflation to more.
Question 4-12
Why do monthly mortgage payments increase so sharply during periods of inflation? What does the tilt effect have to do with this?
In order to receive the full interest necessary to leave enough for a real return and risk premium over the life of the loan, more “real dollars” must be collected in the early years of the loan (payments collected toward the end of the life of the mortgage will be worth much less in purchasing power.)
Tilting - the real payment stream in the early years have to make up for the loss in purchasing power in later years.
Question 4-13
As inflation increases, the impact of the tilt effect is said to become even more burdensome on borrowers. Why is this so?
With the general rate of inflation and growth in the economy, borrower incomes will grow gradually or on a year-by-year basis. However, as expected inflation increases, lenders must build estimates of the full increase into current interest rates “up front” or when the loan is made. This causes a dramatic increase in required real monthly payments relative to the borrower’s current real income.
Question 4-14
A mortgage loan is made to Mr. Jones for $30,000 at 10 percent interest for 20 years. If Mr. Jones has a choice between a CPM and a CAM, which one would result in his paying a greater amount of total interest over the life of the mortgage? Would one of these mortgages be likely to have a higher interest rate than the other? Explain your answer.
A CPM loan reduces the principal balance more slowly; as a result, if Mr. Jones chooses a CPM, he will pay a greater amount of interest over the life of the loan. A CPM may have a lower interest rate. The initial monthly payments for a CPM are considerably less than those of a CAM, and borrowers are more apt to repay the loan. If an economy were experiencing real economic growth with relatively stable prices, increases in income and property values would reduce borrower default risk associated with a CPM loan. Additionally, lenders receive a greater portion of their return (interest earned) early with a CPM. By decreasing default risk and the effects of default, a CPM may have a lower rate of interest than a CAM.
Question 4-15
A borrower makes a GPM mortgage loan. It is originated for $50,000 and carries a 10 percent rate of interest for 30 years. If the borrower decides to prepay the loan after 10 years, would he be paying a higher yield, lower yield, or the same yield as the contract rate originally agreed on? How would this yield compare with that on a CPM or CAM made on the same terms?
If there are no origination fees and no prepayment penalties, the yield on a GPM loan is equal to the contract rate of interest. This is true whether or not the GPM loan, like the CAM and CPM loans, is repaid before maturity.
Question 4-16
Would a lender be likely to originate a GPM at the same rate of interest as a standard CPM loan?
No, the interest rate starts at a lower rate and then is increased gradually.
Question 4-17
What is negative amortization? Why does it occur with a GPM? What happens to the mortgage balance of a GPM over time?
No amortization of principal occurs until payments increase in later periods. The loan balance increases during the first few years after origination, because the initial GPM payments are lower than the monthly interest requirements at a given rate. Once the loan payments are more than the monthly interest payments, the balance begins to decline until it reaches zero at maturity.
Solutions to Problems - Chapter 4
Fixed Rate Mortgage Loans
Problem 4-1
A borrower makes a fully amortizing CPM mortgage loan.
Principal=$125,000
Interest=11.00%
Term=20 years
CPM Payment:
The monthly payment for a CPM is found using the following formula:
Monthly payment=Principal x (MLC, 11%, 20 years)
MLC, 11%, 20 yrs = .0103219 (from appendix B)
Payment=$125,000 x .0103219
=$1,290.24
Calculator solution: PV = -$125,000; i = 11/12%; n = 20x12; FV=0; solve for PMT.
PMT = $1,290.24 (slight difference due to rounding.)
CAM Payments:
Principal=$125,000
Term=20 years
Monthly amortization=Principal divided by term of loan in months
Monthly amortization=$125,000 / (240 months)
Monthly amortization=$520.83 (Rounded)
Set up the following table similar to Exhibit 4-4 to solve for the initial 6 monthly payments.
(1) / (2) / (3) / (4) / (3) + (4) / (2) -(4)Month / Opening Balance / Interest (11%/12) / Amortization / Monthly Payment / Ending Balance
1 / $125,000.00 / $1,145.83 / $520.83 / $1,666.67 / $124,479.17
2 / $124,497.17 / $1,141.06 / $520.83 / $1,661.89 / $123,958.33
3 / $123,958.33 / $1,136.28 / $520.83 / $1,657.12 / $123,437.50
4 / $123,437.50 / $1,131.51 / $520.83 / $1,652.34 / $122,916.67
5 / $122,916.67 / $1,126.74 / $520.83 / $1,647.57 / $122,395.83
6 / $122,395.83 / $1,121.96 / $520.83 / $1,642.80 / $121,875.00
Problem 4-2
(a) Monthly payment = $671.36
Solution:
N = 25x12 or 300
I = 9%/12 or .75
PV =$80,000
FV=0
Solve for payment:
PMT=-$671.36
(b) Month 1:
interest payment:
$80,000 x (9%/12) = $600
principal payment:
$671.36 - $600 = $71.36
(c) Entire Period:
total payment:
$671.36 x 300= $201,408
total principal payment:$80,000
total interest payments:
$201,408 - $80,000=$121,408
(d) Outstanding loan balance if repaid at end of ten years = $66,191.67
Solution:
N=25x12 or 300
I=9%/12 or 0.75
PMT=$671.36
FV=0
Solve for loan balance:
PV=$66,191.67
(e) Trough ten years:
total payments:
$671.36 x 120 =$80,563.20
total principal payment (principal reduction):
$80,000-66,191.67*=$13,808.33
*PV of loan at the end of year 10
total interest payment:
$80,563.20-13,808.33= $66,191.67
(f) Step 1, Solve for loan balance at the end of month 49:
N= 300/49 or 251
I=9%/12 or 0.75
PMT=$671.36
FV=0
Solve for loan balance:
PV=-$75,793.68
Step 2, Solve for the interest payment at month 50:
interest payment:
$75,793.68x(.09/12)=$568.45
principal payment:
$671.36 - $586.45=$102.91
Problem 4-3
(a) Monthly payment = $733.76
Solution:
N=30x12 or 360
I=8%/12 or 0.67
PV=-$100,000
FV=0
Solve for payment:
PMT=$733.76
(b) Quarterly Payment = $2,2024.81
Solution:
N=30x4 or 120
I=8%/4 or 2
PV=-$100,000
FV=0
Solve for payment:
PMT=$2,2024.81
(c) Annual Payment = $8,882.74
Solution:
N=30
I= 8%
PV=-$100,000
FV= 0
Solve for payment:
PMT=$8,882.74
(d) Weekly Payment = $169.23
Solution:
N=52x30 or 1,560
I=8%/52 or 0.019
PV=-$100,000
FV= 0
Solve for payment:
PMT=$169.23
Problem 4-4
Monthly:
total principal payment:$100,000
total interest payment:
($733.76 x 360) - $100,000 =$164,153.60
Quarterly:
total principal payment:$100,000
total interest payment:
($2,204.81 x 120)-$100,000=$164,577.20
Annually:
total principal payment:$100,000
total interest payment:
($8,882.74 x 30) - $100,000=$166,482.20
Weekly:
total principal payment:$100,000
total interest payment:
($169.23 x 1560)-$100,000 =$163,998.80
The greatest amount of interest payable is with the Annual Payment Plan because you are making payments less frequently. Therefore, reducing you balance less frequently and paying interest on a greater amount each year.
Problem 4-5
(a) Monthly Payment:
Solution:
N=20x12 or 240
I=8%/12 or 0.67
PV=-$100,000
FV=0
Solve for payment:
PMT=$836.44
(b) Entire Period:
total payment:
$836.44 x240 =$200,745.60
total principal payment:$100,000
total interest payment:
$200,745.60 - 100,000 =$100,745.60
(c) Outstanding loan balance if repaid at end of year eight = $77,272.67
Solution:
N=12x12 or 144
I=8%/12 or 0.67
PMT=-$836.44
FV=0
Solve for mortgage balance:
PV=$77,272.67
Total interest collected:
total payment + mortgage balance - principal
$836.44 x (8x12) + 77,272.67 - 100,000
total interest collected = $57,570.91
(d) Step 1, Solve for the loan balance at the end of year 5:
N=15x12 or 180
I=8%/12 or 0.67
PMT=-$836.44
FV=0
Solve for loan balance:
PV=$87,525.58
After reducing the loan by $5,000, the balance is:
$87,525.58 - 5,000=$82,525.58
4-The new loan maturity will be 161 months after the loan is reduced at the end of year 5.
Solution:
I=8%/12 or 0.67
PMT=-$836.44
PV=$82,525.58
FV=0
Solve for maturity:
N=161.37
5-The new payment would be $788.66
Solution:
I=8%/12 or 0.67
N=15x12 or180
PV=$82,525.58
FV=0
Solve for payment:
PMT=-$788.86
Problem 4-6
(a) Monthly payment reduction due to principal reduction.
Initial principal=$75,000
Interest rate=10.00%
Initial term=30 years
Initial monthly payment=Principal x (MLC, 10%, 30 years)
Initial monthly payment=$658.18
Mortgage loan balance after 10 years = PV of 340 payments of $658.18 discounted @10%
Mortgage loan balance after 10 years = $68,203.51
Reducing the mortgage balance by $10,000 leaves a principal balance of $58,203.51. The new payment would be based on 10% interest and a 20 year term.
New monthly payment = New principal x (MLC, 10%, 30 years)
New monthly payment = $561.68
(b) Maturity shortening due to principal reduction.
Initial monthly payment = Principal x (MLC, 10%, 30 years)
Initial monthly payment = $658.18
The new maturity is the time necessary for the original monthly payments of $658.18 to fully amortize the remaining principal balance of $58,203.51.
From a financial calculator, using PV of $58,203.51, an interest rate of 10%, and payments of $658.18, we get a new maturity of 161 months or 13 years and 5 months.
This can be checked by noting that $58,203.51 divided by 4658.18 equals 88.43118. This number corresponds to the MPVIFA at 10% interest with a maturity of 161 months.
Alternative Solution
Step 1, Solve for the original monthly payment:
I=10%/12 or 0.83
N=30x12 or 360
PV=-$75,000
FV=0
Solve for payment:
PMT=$658.18
Step 2, Solve for current balance:
I=10%/12 or 0.83
N=20x12 or 240
PV=-$75,000
PMT=$658.18
Solve for mortgage balance:
FV=$68,203.24
(a) New Monthly Payment = $561.67
Solution:
I=10%/12 or 0.83
N=12x20 or 240
PV=$58,203.24*
FV=0
Solve for payment:
PMT=-$561.67
(b) New Loan Maturity = 161 months
Solution:
I=10%/12 or 0.83
PMT=-$658.18
PV=$58,203.24*
FV=0
Solve for maturity:
N=161
*$68,203.24 - 10,000
Problem 4-7
The loan will be repaid in 158 months.
Solution:
I=7.5%/12 or 0.625
PMT=$1,000
PV=$100,000
FV=0
Solve for maturity:
N=157.4226
Problem 4-8
The interest rate on the loan is 12.96%.
Solution:
N=25x12 or 300
PMT=-$900
PV=$80,000
FV=0
Solve for the annual interest rate:
I=1.08 (x12) or 12.96%
Problem 4-9
(a) Monthly Payments = $656.70
Solution:
N=10x12 or 120
I=9%/12 or 0.75
PV=-$60,000
FV=$20,000
Solve for monthly payment:
PMT=$656.70
(b) Loan balance at the end of year five = $44,409.83
Solution:
N=5x12 or 60
I=9%/12 or 0.75
PMT=$656.70
FV=$20,000
Solve for the loan balance:
PV=-$44,409.83
Problem 4-10
(a) Monthly Payments = $800
Solution:
N=10x12 or 120
I=12%/12 or 1
PV=-$80,000
FV=$80,000
Solve for monthly payments:
PMT=$800
(b) Loan balance = $80,000
Solution:
N=12x5 or 60
I=12%/12 or 1
PV=-$80,000
PMT= $800
Solve for loan balance:
FV=$80,000
You also know the loan balance will be the same as initial loan amount because this is an interest only loan where there is no loan amortization or reduction of principal.
(c) Yield to the lender =12%
Solution:
N=12x5 or 60
PMT=$800
PV=-$80,000
FV=$80,000
Solve or the annual yield:
I=1 (x12) or 12
(d) Yield to the lender =12%
Solution:
N=12x10 or 120
PMT=$800
PV=-$80,000
FV=$80,000
Solve or the annual yield:
I=1 (x12) or 12%
Problem 4-11
Monthly Payments = $982.63
Solution:
N=10x12 or 120
I=8%/12 or 0.67
PV=$90,000
FV=-$20,000
Solve for monthly payments:
PMT=$982.63
Yield to the lender =8.41%
Solution:
N=12x10 or 120
PMT=$982.63
PV=-$88,200*
FV=$20,000
Solve or the annual yield:
I=.7011 x 12 or 8.413%
*-$90,000 x (100-2)% = -$88,200
Step 1, Solve the loan balance if repaid in four years:
Solution:
N=6x12 or 72
I=8%/12 or 0.67
FV=$20,000
PMT=$982.63
Solve for the loan balance:
PV=-$68,439.23
Step 2, Solve for the yield:
Solution:
N=12x4 or 48
PMT=$982.63
PV=-$88,200*
FV=$68,439.23
Solve or the annual yield:
I=.722 (x12) or 8.66%
*-$90,000 x (100-2)% = -$88,200
Problem 4-12
(a) At the end of year ten $94,622.86 will be due:
Solution:
N=12x10 or 120
I=8%/12 or 0.67
PV=-$50,000
PMT= 0
Solve for loan balance:
FV=$94,622.86
(b) Step 1, Solve for loan balance at end of year eight
Solution:
N=8x12 or 96
I=8%/12 or 0.67
PV=-$50,000
PMT= 0
Solve for loan balance:
FV=$94,622.86
Step 2, Solve for the yield:
Solution:
N=8x12 or 96
PMT=0
PV=-$50,000
FV=$94,622.86
Solve or the annual yield:
I=.67 (x12) or 8%
Note: because there were no points, the yield must be the same as the initial interest rate of 8% so no calculations were really necessary.
(c) Yield to lender with one point charged = 8.13%
Solution:
N=8x12 or 96
PMT=0
PV=-$49,500*
FV=$94,622.86
Solve or the annual yield:
I=.68 (x12) or 8.13%
*-$50,000 x (100-1)% = -$49,500
Problem 4-13
(a)
Property value=$105,000
Principal=$84,000
Interest rate=12.00%
Maturity=30 years
Loan origination fee=$3,500
Lender will disburse $84,000.00 less the loan origination fee of $3,500.00 or $80,500.00
(b) Monthly payments would be:
$84,000 x (MLC, 12%, 30 years)=$864.03
The effective interest cost would be:
$864.03 x (MPVIFA, ?%, 30 years)=$80,500
Solving for the interest rate, we get 12.58%
(c) The annual percentage rate (APR) is the same as the interest rate in part (b) rounded to the nearest .125%. Therefore, the APR is 12.625%.
Note to Instructors: APRs are rounded to the nearest 1/8 of a percent.
(d) Assuming the loan payoff occurs after 5 years, determine the mortgage balance:
Mortgage balance = PV of 300 monthly payments of $864.03 discounted at 12.00%
Mortgage balance = $82,037.10
The effective interest cost would be:
$864.03 x (MPVIFA, ?%, 5 years) + $82,037.10 x (MPVIF, ?%, 5 years) = $80,500
Solving for the interest rate, we get 13.15%.
The effective interest rate in this part is different from the APR because the loan origination fee is amortized over a much shorter period (5 years instead of 30 years).
(e) With a prepayment penalty of 2% on the outstanding loan balance of $82,037.10, the penalty would be $1,640.72.
The effective interest cost would be:
$864.03 x (MPVIFA, ?%, 5 years) + $83,677.85 x (MPVIF, ?%, 5 years) = $80,500
Solving for the interest rate, we get 13.44%.
This rate is different from the APR because penalty points are not used in the calculation of the APR.
Problem 4-14
Points required to achieve a yield to 10% for the 25 year loan.
Monthly payments:
$95,000 x (MLC, 9%, 25 years)=Monthly payment