real estate ch 4 slution (14 edn)

Solutions to Questions - Chapter 4Fixed Rate Mortgage Loans

Question 4-1What are the major differences between the CAM, and CPM 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 are determined first by computing a constant amount of each monthly payment to be applied to principal. Interest is then computed on the monthly loan balance and added to the monthly amount of amortization to determine the total monthly payment. 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.CAM - lenders recognized that in a growing economy, borrowers could partially repay the loan over time, as opposed to reducing the loan balance in fixed monthly amounts.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.

When both loans are originated at the same rate of interest, the yield to the lender will be the same regardless of when the loans are repaid (ie, early or at maturity).

Question 4-2Define amortization.Amortization is the process of loan repayment over time.

Question 4-3Why 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 4-4What 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 4-5Does 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 4-6Why 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 4-7What 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.

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Question 4-8Does 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 4-9What 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 4-10What 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 4-11When 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 4-12Why 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 4-13As 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 lower rate of interest than a CAM.

Question 4-15What is negative amortization? Why does it occur with a GPM? What happens to the mortgage balance of a GPM over

time?

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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.

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Solutions to Problems - Chapter 4Fixed Rate Mortgage Loans

Problem 4-1A borrower makes a fully amortizing CPM mortgage loan.

Principal = $125,000Interest = 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 = $125,000Term = 20 yearsMonthly amortization = Principal divided by term of loan in monthsMonthly 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 $124,479.172 $124,497.17 $1,141.06 $520.83 $1,661.89 $123,958.333 $123,958.33 $1,136.28 $520.83 $1,657.12 $123,437.504 $123,437.50 $1,131.51 $520.83 $1,652.34 $122,916.675 $122,916.67 $1,126.74 $520.83 $1,647.57 $122,395.836 $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 300I = 9%/12 or .75PV = $80,000FV = 0

Solve for payment:PMT = -$671.36

(b) Month 1:interest payment: $80,000 x (9%/12) = $600principal payment: $671.36 - $600 = $71.36

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(c) Entire Period:total payment:

$671.36 x 300 = $201,408total principal payment: $80,000total interest payments:

$201,408 - $80,000 = $121,408

(d) Outstanding loan balance if repaid at end of ten years = $66,191.67Solution:

N = 25x12 or 300I = 9%/12 or 0.75PMT = $671.36FV = 0

Solve for loan balance:PV = $66,191.67

(e) Trough ten years:total payments:

$671.36 x 120 = $80,563.20total principal payment (principal reduction):

$80,000-66,191.67*= $13,808.33*PV of loan at the end of year 10total 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 251I = 9%/12 or 0.75PMT = $671.36FV = 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)= $568.45principal payment:

$671.36 - $586.45 = $102.91

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Problem 4-3(a) Monthly payment = $733.76

Solution:N = 30x12 or 360I = 8%/12 or 0.67PV = -$100,000FV = 0

Solve for payment:PMT = $733.76

(b) Quarterly Payment = $2,2024.81Solution:

N = 30x4 or 120I = 8%/4 or 2PV = -$100,000FV = 0

Solve for payment:PMT = $2,2024.81

(c) Annual Payment = $8,882.74Solution:

N = 30I = 8%PV = -$100,000FV = 0

Solve for payment:PMT = $8,882.74

(d) Weekly Payment = $169.23Solution:

N = 52x30 or 1,560I = 8%/52 or 0.019PV = -$100,000FV = 0


Problem 4-4Monthly:

total principal payment: $100,000total interest payment:

($733.76 x 360) - $100,000 = $164,153.60Quarterly:


($2,204.81 x 120)-$100,000= $164,577.20Annually:


($8,882.74 x 30) - $100,000= $166,482.20Weekly:


($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.

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Problem 4-5(a) Monthly Payment:

Solution:N = 20x12 or 240I = 8%/12 or 0.67PV = -$100,000FV = 0


(b) Entire Period:total payment:

$836.44 x240 = $200,745.60total principal payment: $100,000total interest payment:

$200,745.60 - 100,000 =$100,745.60

(c) Outstanding loan balance if repaid at end of year eight = $77,272.67Solution:

N = 12x12 or 144I = 8%/12 or 0.67PMT = -$836.44FV = 0

Solve for mortgage balance:PV = $77,272.67

Total interest collected:

total payment + mortgage balance - principal$836.44 x (8x12) + 77,272.67 - 100,000total interest collected = $57,570.91

(d) Step 1, Solve for the loan balance at the end of year 5:N = 15x12 or 180I = 8%/12 or 0.67PMT = -$836.44FV = 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 0.67PMT = -$836.44PV = $82,525.58FV = 0

Solve for maturity:N = 161.37

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5- The new payment would be $788.66

Solution:I = 8%/12 or 0.67N = 15x12 or180PV = $82,525.58FV = 0


Problem 4-6(a) Monthly payment reduction due to principal reduction.

Initial principal = $75,000Interest rate = 10.00%Initial term = 30 yearsInitial 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.

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Alternative SolutionStep 1, Solve for the original monthly payment:

I = 10%/12 or 0.83N = 30x12 or 360PV = -$75,000FV = 0


Step 2, Solve for current balance:I = 10%/12 or 0.83N = 20x12 or 240PV = -$75,000PMT = $658.18

Solve for mortgage balance:FV = $68,203.24

(a) New Monthly Payment = $561.67Solution:

I = 10%/12 or 0.83N = 12x20 or 240PV = $58,203.24*FV = 0


(b) New Loan Maturity = 161 monthsSolution:

I = 10%/12 or 0.83PMT = -$658.18PV = $58,203.24*FV = 0

Solve for maturity:N = 161

*$68,203.24 - 10,000

Problem 4-7The loan will be repaid in 158 months.

Solution:I = 7.5%/12 or 0.625PMT = $1,000PV = $100,000FV = 0

Solve for maturity:N = 157.4226

Problem 4-8 The interest rate on the loan is 12.96%.

Solution:N = 25x12 or 300PMT = -$900PV = $80,000FV = 0

Solve for the annual interest rate:I = 1.08 (x12) or 12.96%

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Problem 4-9(a) Monthly Payments = $656.70

Solution:N = 10x12 or 120I = 9%/12 or 0.75PV = -$60,000FV = $20,000

Solve for monthly payment:PMT = $656.70

(b) Loan balance at the end of year five = $44,409.83Solution:

N = 5x12 or 60I = 9%/12 or 0.75PMT = $656.70FV = $20,000

Solve for the loan balance:PV = -$44,409.83

Problem 4-10(a) Monthly Payments = $800

Solution:N = 10x12 or 120I = 12%/12 or 1PV = -$80,000FV = $80,000

Solve for monthly payments:PMT = $800

(b) Loan balance = $80,000Solution:

N = 12x5 or 60I = 12%/12 or 1PV = -$80,000PMT = $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 60PMT = $800PV = -$80,000FV = $80,000

Solve or the annual yield:I = 1 (x12) or 12

(d) Yield to the lender =12%Solution:

N = 12x10 or 120PMT = $800PV = -$80,000FV = $80,000

Solve or the annual yield:I = 1 (x12) or 12%

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Problem 4-11Monthly Payments = $982.63

Solution:N = 10x12 or 120I = 8%/12 or 0.67PV = $90,000FV = -$20,000

Solve for monthly payments:PMT = $982.63

Yield to the lender =8.41%Solution:

N = 12x10 or 120PMT = $982.63PV = -$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 72I = 8%/12 or 0.67FV = $20,000PMT = $982.63

Solve for the loan balance:PV = -$68,439.23

Step 2, Solve for the yield:Solution:

N = 12x4 or 48PMT = $982.63PV = -$88,200*FV = $68,439.23

Solve or the annual yield:I = .722 (x12) or 8.66%

*-$90,000 x (100-2)% = -$88,200

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Problem 4-12(a) At the end of year ten $94,622.86 will be due:

Solution:N = 12x10 or 120I = 8%/12 or 0.67PV = -$50,000PMT = 0

Solve for loan balance:FV = $94,622.86

(b) Step 1, Solve for loan balance at end of year eightSolution:

N = 8x12 or 96I = 8%/12 or 0.67PV = -$50,000PMT = 0

Solve for loan balance:FV = $94,622.86

Step 2, Solve for the yield:Solution:

N = 8x12 or 96PMT = 0PV = -$50,000FV = $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 96PMT = 0PV = -$49,500*FV = $94,622.86

Solve or the annual yield:I = .68 (x12) or 8.13%

*-$50,000 x (100-1)% = -$49,500

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Problem 4-13(a)

Property value = $105,000Principal = $84,000Interest rate = 12.00%Maturity = 30 yearsLoan 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


$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.


$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.

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Problem 4-14Points required to achieve a yield to 10% for the 25 year loan.

Monthly payments:$95,000 x (MLC, 9%, 25 years) = Monthly payment$95,000 x (MLC, 9%, 25 years) = $797.24

PV of 300 payments of $797.24 discounted at 10% = $87,773.67

Subtracting $87,773.67 from $95,000.00, we get $7,226.33

The loan origination fee should be $7,226.33 if the loan is to be repaid after 25 years and the lender requires a 10% yield.

If the loan is expected to be repaid after 10 years, the loan balance at the end of 10 years must be determined:

Loan balance after 10 years = Present value of 180 payments of $797.24 discounted at 9%.Loan balance after 10 years = 78,602.27

Discounting by the desired yield of 10%:

Present value = $797.24 (MPVIFA, 10%, 10 years) + $78,602.27 (MPVIF, 10%, 10 years)Present value = 89,364.04

Subtracting $89,364.04 from $95,000.00, we get $5,635.96.

The loan origination fee should be $5,635.96 if the loan is to be repaid after 10 years, and the lender requires a yield of 10%.

Problem 4-15(a) In order to find which loan is the better choice after 15 years, the effective interest rate of each loan must be calculated.

Loan A Loan BPrincipal $75,000 $75,000Nominal interest rate 10.00% 11.00%Term (years) 30 30Points 6 2Payment $658.18 $714.24Loan Balance after 15 years $61,248.42 $62,840.44Loan Balance after 5 years $72,430.74 $72,873.48

Loan A$70,500.00 = $658.18 x (MPVIFA, ?%, 180 months) + $61,248.42 x (MPVIFA, ?%, 180 months)Effective interest rate = 10.85%

Loan B$73,500.00 = $714.24 x (MPVIFA,?%, 180 months) + $62,840.44 x (MPVIF, ?%, 180 months)Effective interest rate = 11.29%

Loan A is the better alternative if the loan is repaid after 15 years.

(b) This part is solved the same as (a) except using the assumption that the loan is repaid after 5 years.

Loan A$70,500.00 = $658.18 x (MPVIFA, ?%, 60 months) + $72,430.74 x (MPVIF, ?%, 60 months)Effective interest rate = 11.61%

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Loan B$73,500.00 = $714.24 x (MPVIFA, ?%, 60 months) + $72,873.48 x (MPVIF, ?%, 60 months)Effective interest rate = 11.53%

Loan B is the better alternative if the loan is repaid after 5 years.

Problem 4-16(a) Monthly Payments = $1,382.50

Solution:N = 10x12 or 120I = 11%/12 or 0.92PV = 0FV = -$300,000

Solve for monthly payments:PMT = $1,382.50

(b) The borrower will have monthly payments of $1,382.50 during months 1 to 36 Solve for loan balance at the end of month 36

Solution:N = 36I = 11%/12 or 0.92PV = 0PMT = $1,382.50

Solve for loan balance:FV = -$58,649.97

(c) The borrower will have monthly payments of $626.22 during months 51 to 120 Step 1, Solve for loan balance at the end of month 50

Solution:N = 50I = 11%/12 or 0.92PV = 0PMT = $2,000

Solve for loan balance:FV = -$126,139.10

Step 2, Solve for payments during months 51 to 120Solution:

N = 120-50 or 70I = 11%/12 or 0.92PV = $126,139.10FV = -$300,000

Solve for monthly payments:PMT = $626.22

Problem 4-17Find the balance at the end of 5 years for a fully amortizing $200,000, 10% mortgage with a 25 year amortization schedule:

PV = -200,000 FV = 0i = 10% Solve PMT = $1,817.40n = 300

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Solve for balance at end of 5 years:

i = 10% PMT = $1,817.40n =240 FV = 0

Solve PV = -188,327.38

Problem 4-18 Comprehensive Review Problem

Loan = 100,000, 12% interest, 20 yearsA. Monthly payments if

(1) Fully amortizing:PV = -100,000 n = 240i = 12% Solve PMTs = $1,101.09FV = 0

(2) Partial amortizing:PV = -100,000 n = 240i = 12% Solve PMTs = $1,050.54FV = $50,000

(3) Interest onlyPV = 100,000 n = 240i = 12% Solve PMTs = $1,000.00FV = 100,000

(4) Negative amortization:PV = -100,000 n = 240i = 12% Solve PMTs = $949.46FV = 150,000

B. Loan Balances for A.1. – A.4 after 5 years

A.1 PMTs = 1,101.09 FV = 0i = 12% Solve PV = $91,744.33

A.2 PMTs = 1,050.54 FV = 50,000i = 12% Solve PV = $95,872.16n = 180

A.3 PMTs = 1,000.00 FV = 100,000i = 12% Solve PV = 100,000n = 180

A.4 PMTs = $949.46 FV = 150,000i = 12% Solve PV = 104,127n = 180

C. Interest at the end of month 61 for A.1 – A.4

A.1 $91,744.33 * .01 = $ 917.44A.2 $95,872.16 * .01 = $ 958.72A.3 $100,000.00 * .01 = $1,000.00A.4 $104,127.84 * .01 = $1,041.28

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D. APR* for loans in A.1 – A.4

A.1 PV = -97,000, PMT = 1,101.09, FV = 0, n = 240 Solve i = 12.50A.2 PV = -97,000, PMT = 1,050.54, FV = 50,000, n = 240 Solve i = 12.44A.3 PV = -97,000, PMT = 1,000.00, FV = 100,000, n = 240 Solve i = 11.76A.4 PV = -97,000, PMT = 949.46, FV = 150,000, n = 240 Solve i = 12.375*Solution shown based on calculation – final answers S/B rounded to nearest 1/8%

E.Effective yield if loan prepaid EOY 5A.1 PV = -97,000, PMT = 1,101.09, FV = 91,744.33 n = 60 Solve i = 12.84A.2 PV = -97,000, PMT = 1,050.54, FV = 95,872.16 n = 60 Solve i = 12.83A.3 PV = -97,000, PMT = 1,000.00, FV = 100,000.00 n = 60 Solve i = 12.82A.4 PV = -97,000, PMT = 949.46, FV = 104,127.00 n = 60 Solve i = 12.80

F. Assume monthly payments in A.1 = 0 for 36 mos. What must payments be from yr. 4-17 to fully amortize the loan at the end of 24 mos?Part 1:PV = -100,000 PMT = 0i = 12% Solve FV = 143,076.88n = 36Part 2:PV = -143,076.88 n = 204 Solve PMT = $1,647.12i = 12% FV = 0

G.(1) Total PMTs = (949.46 * 240) + 150,000 = $377,870Principal = 100,000Interest = 277,870

(2) n = 204 FV = 150,000 PMTs = 949.46 i = 12% Solve PV = 102,177 balance

(3) 12% because there are no points

(4) 4 points charged, loan payoff 36 months, what is effective interest rate?

PV = -96,000 PMT = 949.46n = 36 Solve i = 13.62%FV = 102,177

Problem 4-19

The effective cost is now 12.64% versus 12.82%.

Problem 4-20

The loan balance is now $61,680 versus $63,793.

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real estate ch 4 slution (14 edn)

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