Now that they have accumulated a deposit of 55,000 Jack and Jill take out a housing loan to purchase a home. The house costs $755,000. It is to be repaid in equal monthly instalments over a term of 30 years. The interest rate quoted by the bank is an effective annual rate is 7.5%pa. Jack has lost the paperwork showing the annual nominal rate (j12 ) with monthly compounding.
i. How much is the monthly repayment?
ii. How much do Jack and Jill owe the bank immediately before making the 130th repayment?
iii. After making the 150th repayment Jack and Jill receive an amount of $50,000, which they use to reduce their loan. The bank allows them to make the same monthly payment. How much will the term of the loan be reduced by if the interest rate remains the same?
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To find the answers to these questions, we need to use the formula for the monthly repayment of a loan, which is:
\[M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1}\]
Where:
- M is the monthly repayment
- P is the principal amount (the amount of the loan)
- r is the monthly interest rate
- n is the number of payments
For the interest rate, the bank has provided an effective annual rate of 7.5%. We need to convert this to a monthly rate. To do this, we use the formula:
\[r = (1 + i)^{(1/t)} - 1\]
Where:
- i is the annual rate
- t is the number of compounding periods per year
In this case, we don't have the information about the number of compounding periods per year (t), which is necessary to convert the rate. Therefore, we need to make an assumption about the compounding periods. Let's assume it is compounded monthly (t = 12).
Now, let's solve the questions step by step:
i. To find the monthly repayment amount, we use the formula mentioned earlier, using the principal amount (P = $755,000), the monthly interest rate, and the number of payments (n = 30 years * 12 months).
ii. To find the amount owed before the 130th repayment, we need to calculate the remaining principal after 129 repayments. To do this, we calculate the remaining principal at the end of the 129th month using the formula mentioned earlier, replacing the number of payments (n) with 129.
iii. To find the term reduction, we need to calculate the new number of payments (n) after reducing the loan principal by $50,000. We use the formula mentioned earlier, replacing the principal amount (P) with ($755,000 - $50,000), the remaining principal at the end of the 149th month, and solve for the number of payments (n).
Let's plug in the values and solve the equations to find the answers:
i. Monthly repayment amount:
Using P = $755,000, r (monthly interest rate) = ((1 + 0.075)^(1/12)) - 1, n = 30 years * 12 months, we can calculate the monthly repayment (M).
ii. Amount owed before the 130th repayment:
Using P = $755,000, r (monthly interest rate) = ((1 + 0.075)^(1/12)) - 1, n = 129, we can calculate the remaining principal.
iii. Term reduction:
Using P = $705,000, r (monthly interest rate) = ((1 + 0.075)^(1/12)) - 1, remaining principal after 149th payment, we can calculate the new number of payments (n) to find the term reduction.
To solve these questions, we can use the formula for calculating the equal monthly loan repayment:
Monthly payment = (Principal * Monthly interest rate) / (1 - (1 + Monthly interest rate)^(-number of months))
First, we need to calculate the annual nominal rate with monthly compounding (j12).
Step 1: Calculate the monthly interest rate (r).
r = (1 + j12)^(1/12) - 1
Step 2: Calculate the number of months (n).
n = number of years * 12
Step 3: Calculate the monthly repayment using the formula.
i. How much is the monthly repayment?
Monthly repayment = (Principal * Monthly interest rate) / (1 - (1 + Monthly interest rate)^(-number of months))
ii. How much do Jack and Jill owe the bank immediately before making the 130th repayment?
To find this, we need to calculate the remaining principal after 130 monthly repayments have been made. We can use the formula:
Remaining principal = Principal * (1 + Monthly interest rate)^number of months - (Monthly payment * ((1 + Monthly interest rate)^number of months - 1) / Monthly interest rate)
iii. After making the 150th repayment, Jack and Jill receive an amount of $50,000, which they use to reduce their loan. The bank allows them to make the same monthly payment. How much will the term of the loan be reduced by if the interest rate remains the same?
To find this, we need to calculate the new outstanding principal after deducting the $50,000 from the remaining principal. Then, we can use the same loan repayment formula to calculate the reduced loan term.
Now, let's calculate the answers step by step.
Given:
Deposit: $55,000
House cost: $755,000
Term: 30 years
Effective annual interest rate: 7.5% (Unknown j12)
Step 1: Calculate the monthly interest rate (r).
r = (1 + j12)^(1/12) - 1
We don't know the value of j12, so we'll proceed with calculating other values and come back to this later.
Step 2: Calculate the number of months (n).
n = number of years * 12
n = 30 years * 12 months/year
n = 360 months
Now, let's calculate the monthly repayment (i).
i. How much is the monthly repayment?
Monthly repayment = (Principal * Monthly interest rate) / (1 - (1 + Monthly interest rate)^(-number of months))
Here, the principal is the house cost minus the deposit.
Principal = House cost - Deposit
Principal = $755,000 - $55,000
Principal = $700,000
Now, we can calculate the monthly repayment using the above formula with the calculated values for Principal, r, and n.
Monthly repayment = ($700,000 * r) / (1 - (1 + r)^(-360))
Let's calculate the monthly repayment value.
Continue the calculation for ii. and iii. using the obtained monthly repayment value. We'll revisit the calculation for j12 later.