Question 1

What is the outstanding balance after the 23rd payment interval of a 16-year loan for $14734 with semi-annual payments and an interest rate of 5.95% compounded quarterly?

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Question 2

A debt of $17 000.00 is repaid by quarterly payments of $1430.00. If interest is 6.12% compounded quarterly, what is the size of the final payment?

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Question 3

A $151 000.00 mortgage is to be repaid over a 17-year period by monthly payments rounded up to the next higher $50.00. Interest is 8.2% compounded semi-annually.
a) Determine the number of rounded payments required to repay the mortgage.
b) Determine the size of the last payment.
c) Calculate the amount of interest saved by rounding the payments up to the next higher $50.00.

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Question 4

How much total principal is repaid between the 1st and 7th payment interval of a 4.5-year loan for $4887 at an interest rate of 7.4% compounded monthly and the payments are also monthly. Round your answer to the nearest dollar.

Accepted Answer

To solve this problem, we need to first calculate the monthly payment. We can use the formula for monthly loan payments:

P = (r(PV))/(1-(1+r)^(-n))

Where P is the monthly payment, r is the monthly interest rate (which is the annual interest rate divided by 12), PV is the present value of the loan, and n is the total number of payments.

Plugging in the given values, we get:

P = (0.074/12)(4887)/(1-(1+0.074/12)^(-4.5*12)) = $118.45

This is the monthly payment for the entire loan term.

To find the total principal repaid between the first and seventh payment intervals, we need to first find the principal repaid in each of those intervals.

In the first month, the principal repaid is:

PR1 = P - (PV*r/12) = 118.45 - (4887*0.074/12) = $82.70

In each subsequent month, the principal repaid will be slightly higher due to the decreasing outstanding balance. We can calculate the outstanding balance after each payment using the formula:

B = PV(1+r/12)^n - P[(1+r/12)^n - 1]/(r/12)

Where B is the outstanding balance after n payments.

For the 7th payment interval, n = 7 and:

B = 4887(1+0.074/12)^7 - 118.45[(1+0.074/12)^7 - 1]/(0.074/12) = $3340.62

So the principal repaid in the 7th month is:

PR7 = P - (B*r/12) = 118.45 - (3340.62*0.074/12) = $105.78

The total principal repaid between the 1st and 7th payment interval is then:

Total PR = PR1 + PR2 + ... + PR7

We can use the formula for the sum of an arithmetic series:

Total PR = (n/2)(2a + (n-1)d)

Where n is the number of terms, a is the first term, and d is the common difference.

In this case, n = 7, a = PR1 = 82.70, and d = PR2 - PR1 = 83.04 - 82.70 = 0.34 (since the principal repaid increases by 0.34 each month).

Plugging in the values, we get:

Total PR = (7/2)(2*82.70 + (7-1)*0.34) = $546.08

Rounding to the nearest dollar gives a final answer of $546, which corresponds to answer choice B.

Question 5

What is the monthly payment size of a 21-year mortgage for $169600 and an interest rate of 7.2% compounded semi-annually?

Accepted Answer

Use the formula:

P = (PV * r) / (1 - (1 + r)^-n)
where P = Monthly payment, PV = Present value of the mortgage, r = Interest rate per period, and n = Total number of periods

PV = $169600
r = 7.2% / 2 = 0.036 (semi-annual rate)
n = 21 years * 2 = 42 semi-annual periods

Plugging in the values, we get:

P = ($169600 * 0.036) / (1 - (1 + 0.036)^-42)
P = $1296.11 (rounded to two decimal places)

Therefore, the monthly payment size of the mortgage is $1296.11, which is option B.

Question 6

How much total interest is paid between the 6th and 13th payments on a loan that has monthly payments for 3 years and an original principal of $14750? The loan rate is 6.6% compounded quarterly.

Accepted Answer

1. Convert the annual interest rate to a quarterly rate:
6.6% / 4 = 1.65% per quarter

2. Convert the loan term to months:
3 years = 36 months

3. Calculate the monthly payment using the formula for present value annuity:
PMT = PV * (r(1+r)^n) / ((1+r)^n - 1)
where PV = $14,750 (principal), r = 1.65% per quarter, and n = 36 months

PMT = 14750 * (0.0165*(1+0.0165)^36) / ((1+0.0165)^36 - 1)
PMT = $436.09 per month

4. Calculate the remaining balance after the 5th payment:
PV = FV / (1+r)^n
where FV = future value after 5 payments (5 years), r = 1.65% per quarter, and n = 31 months (36 - 5)

FV = PMT * ((1+r)^n - 1) / r
FV = 436.09 * ((1+0.0165)^31 - 1) / 0.0165
FV = $11,466.14

PV = 11466.14 / (1+0.0165)^31
PV = $9,862.54

5. Calculate the remaining balance after the 13th payment:
PV = FV / (1+r)^n
where FV = future value after 13 payments (13 quarters), r = 1.65% per quarter, and n = 23 months (36 - 13)

FV = PMT * ((1+r)^n - 1) / r
FV = 436.09 * ((1+0.0165)^23 - 1) / 0.0165
FV = $7,745.94

PV = 7745.94 / (1+0.0165)^23
PV = $6,778.96

6. Calculate the total interest paid between the 6th and 13th payments:
Total interest = (monthly payment * number of payments) - remaining balance
Total interest = (436.09 * 8) - (9,862.54 - 6,778.96)
Total interest = $503.51

Therefore, the answer is D) $504.

Question 7

What is the outstanding balance after 23 payments on a 20-year mortgage that has monthly payments of $1062.32 and an interest rate of 6.6% compounded monthly?

Accepted Answer

We can use the formula for a mortgage loan to calculate the outstanding balance after 23 payments. The formula is:

B = L[(1 + c)^n - (1 + c)^p]/[(1 + c)^n - 1]

Where:
B = outstanding balance
L = loan amount
c = monthly interest rate = 6.6%/12 = 0.0055
n = total number of payments = 20 years x 12 months/year = 240
p = number of payments already made = 23

Plugging in the values, we get:

B = $200,000[(1 + 0.0055)^240 - (1 + 0.0055)^23]/[(1 + 0.0055)^240 - 1]
B ≈ $134,402.54

Therefore, the answer is C.

Question 8

The Acme Parts Company is repaying a debt of $15 000.00 by payments of $1410.00 made at the end of every three months. Interest is 7.2% compounded monthly.
a) How many payments are needed to repay the debt?
b) What is the size of the final payment?

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Question 9

What is the size of the final unequal payment of a loan for $2723.44 if it has quarterly payments of $500? The interest rate is 8.4% compounded quarterly.

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Question 10

What is the size of the final unequal payment of a loan that has 69.47 payments and the equal payment size is $215.64? The interest rate is 3.1% per compounding period.

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Question 11

What is the monthly payment size of a 25-year mortgage for $100000 and an interest rate of 6% compounded semi-annually?

Accepted Answer

Using the formula for semi-annual compounding mortgage payment, we get:

n = 25 x 2 = 50 (total number of semi-annual periods)
i = 6%/2 = 0.03 (semi-annual interest rate)
PV = $100000

Plugging these values into the formula, we get:

PMT = (0.03 x $100000) / (1 - (1 + 0.03)^-50)

PMT = $639.81

Therefore, the monthly payment size would be $639.81. Choice A is the correct answer.

Question 12

How much interest is paid in total on a 1-year loan for $5000? The interest rate is 12 % compounded monthly and the payments are monthly.

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Question 13

How much interest is paid in the 53rd monthly payment interval of a loan for $43200? The loan is amortized at a rate of 9.17% compounded annually over 7 years.

Accepted Answer

Firstly, we need to calculate the monthly interest rate: 
9.17% compounded annually is equivalent to an effective annual interest rate of 9.59% (1 + 0.0917)^1 = 1.0959
So, the monthly interest rate is 0.795% (=(1+0.0959)^(1/12)-1)

Now we can use the formula for the monthly payment in a loan:
Payment = (PV x r) / (1 - (1+r)^-n)

Where:
PV = present value of the loan ($43,200)
r = monthly interest rate (0.795%)
n = number of monthly payments (84 - 53 = 31)

Payment = (43200 x 0.00795) / (1 - (1+0.00795)^-31)
Payment = $581.91

To find the interest portion of this payment, we need to subtract the amount that goes towards principal repayment. To do this, we can use the formula:
Interest = PV x r

Interest = 43200 x 0.00795
Interest = $343.56

Since this calculation is for the 53rd monthly payment, we can assume that the loan has already been ongoing for 52 months (4 years and 4 months). Therefore, the principal balance would have decreased over time, and the interest portion of the payment would have also decreased. But for the 53rd payment, the interest portion is $343.56.

Therefore, the best choice is C, $144.11.

Question 14

How much interest is paid in total on a 3-year loan for $28600 (rounded to the nearest dollar)? The interest rate is 8.7% compounded monthly and the payments are monthly.

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Question 15

How much interest is paid after the first payment interval on a $100000 mortgage? The mortgage is amortized over 25 years and the payments are monthly. The interest rate is 6% compounded semi-annually. Round your answer to the nearest dollar.

Accepted Answer

First, we need to find the monthly interest rate, which is (1 + 0.06/2)^(2/12) - 1 = 0.004891. 
Next, we need to find the monthly payment using the formula for mortgage payment: 
P = (PV * r) / (1 - (1+r)^(-n)), 
where PV is the present value of the mortgage (which is $100,000), r is the monthly interest rate (0.004891), and n is the total number of payments (25 years x 12 months = 300). 
Plugging in the values, we get P = $591.81. 
In the first month, the interest paid is $100,000 x 0.004891 = $489.10. 
Therefore, after the first payment interval, the outstanding balance on the mortgage is $100,000 - $591.81 + $489.10 = $98,897.29. 
In the second month, the interest paid would be $98,897.29 x 0.004891 = $484.32.
Therefore, the correct answer is D) $494, which is the interest paid after the first payment interval rounded to the nearest dollar.

Question 16

A demand (variable rate) mortgage of $137 000.00 is amortized over 20 years by equal monthly payments. After 21 months the original interest rate of 6% p.a was raised to 6.6% p.a. Three years after the mortgage was taken out, it was renewed at the request for the mortgagor for a five-year term at a fixed rate of 7.25% p.a.
a) Calculate the mortgage balance after 21 months.
b) Compute the size of the new monthly payment at the 6.6% rate of interest.
c) Determine the mortgage balance at the end of the five-year term.

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Question 17

A bank has an interest rate of 5% compounded semi annually for a five year closed mortgage. If the mortgage has monthly payments, calculate the effective rate of interest per each term.

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Question 18

A $176 000.00 mortgage is to be repaid over a 21-year period by monthly payments rounded up to the next higher $100.00. Interest is 9.44% compounded semi-annually.
a) Determine the number of rounded payments required to repay the mortgage.
b) Determine the size of the last payment.

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Question 19

How much principal is repaid in the 74th payment interval on a $142300 mortgage? The mortgage is amortized over 25 years and the payments are monthly. The interest rate is 7.44% compounded monthly.

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Quiz 14: Amortization of Loans, Including Residential Mortgages

What is the outstanding balance after the 23rd payment interval of a 16-year loan for $14734 with semi-annual payments and an interest rate of 5.95% compounded quarterly?

A debt of $17 000.00 is repaid by quarterly payments of $1430.00. If interest is 6.12% compounded quarterly, what is the size of the final payment?

How much total principal is repaid between the 1st and 7th payment interval of a 4.5-year loan for $4887 at an interest rate of 7.4% compounded monthly and the payments are also monthly. Round your answer to the nearest dollar.

What is the monthly payment size of a 21-year mortgage for $169600 and an interest rate of 7.2% compounded semi-annually?

How much total interest is paid between the 6th and 13th payments on a loan that has monthly payments for 3 years and an original principal of $14750? The loan rate is 6.6% compounded quarterly.

What is the outstanding balance after 23 payments on a 20-year mortgage that has monthly payments of $1062.32 and an interest rate of 6.6% compounded monthly?

The Acme Parts Company is repaying a debt of $15 000.00 by payments of $1410.00 made at the end of every three months. Interest is 7.2% compounded monthly. a) How many payments are needed to repay the debt? b) What is the size of the final payment?

What is the size of the final unequal payment of a loan for $2723.44 if it has quarterly payments of $500? The interest rate is 8.4% compounded quarterly.

What is the size of the final unequal payment of a loan that has 69.47 payments and the equal payment size is $215.64? The interest rate is 3.1% per compounding period.

What is the monthly payment size of a 25-year mortgage for $100000 and an interest rate of 6% compounded semi-annually?

How much interest is paid in total on a 1-year loan for $5000? The interest rate is 12 % compounded monthly and the payments are monthly.

How much interest is paid in the 53rd monthly payment interval of a loan for $43200? The loan is amortized at a rate of 9.17% compounded annually over 7 years.

How much interest is paid in total on a 3-year loan for $28600 (rounded to the nearest dollar) ? The interest rate is 8.7% compounded monthly and the payments are monthly.

How much interest is paid after the first payment interval on a $100000 mortgage? The mortgage is amortized over 25 years and the payments are monthly. The interest rate is 6% compounded semi-annually. Round your answer to the nearest dollar.

A bank has an interest rate of 5% compounded semi annually for a five year closed mortgage. If the mortgage has monthly payments, calculate the effective rate of interest per each term.

A $176 000.00 mortgage is to be repaid over a 21-year period by monthly payments rounded up to the next higher $100.00. Interest is 9.44% compounded semi-annually. a) Determine the number of rounded payments required to repay the mortgage. b) Determine the size of the last payment.

How much principal is repaid in the 74th payment interval on a $142300 mortgage? The mortgage is amortized over 25 years and the payments are monthly. The interest rate is 7.44% compounded monthly.