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Math  /  Numbers & Operations

QuestionJack and Jill recently negotiated a rate of 4.8%4.8 \% on their mortgage loan that has a 5-year term and 25-year amortization period. What would be their monthly payment on a \$250,000 mortgage? Fixed-rate mortgages are compounded semi-annually.
Select one: a. $1,420\$ 1,420 b. $1,426\$ 1,426 c. $1,311\$ 1,311 d. $1,315\$ 1,315

Studdy Solution

STEP 1

1. The mortgage amount is \$250,000.
2. The annual interest rate is \(4.8\%\).
3. The interest is compounded semi-annually.
4. The amortization period is 25 years.
5. We need to calculate the monthly payment.

STEP 2

1. Convert the annual interest rate to an effective monthly interest rate.
2. Calculate the total number of payments.
3. Use the mortgage payment formula to find the monthly payment.

STEP 3

Convert the annual interest rate to a semi-annual rate, then to an effective monthly interest rate.
The nominal annual interest rate is 4.8%4.8\%, compounded semi-annually. First, calculate the semi-annual interest rate:
Semi-annual rate=4.8%2=2.4%\text{Semi-annual rate} = \frac{4.8\%}{2} = 2.4\%
Convert this to a decimal:
Semi-annual rate (decimal)=2.4100=0.024\text{Semi-annual rate (decimal)} = \frac{2.4}{100} = 0.024
Now, convert the semi-annual rate to an effective monthly rate using the formula for converting a nominal rate compounded nn times per year to an effective monthly rate:
(1+Semi-annual rate)16−1(1 + \text{Semi-annual rate})^{\frac{1}{6}} - 1
Calculate the effective monthly rate:
(1+0.024)16−1≈0.003965(1 + 0.024)^{\frac{1}{6}} - 1 \approx 0.003965

STEP 4

Calculate the total number of payments over the amortization period.
The amortization period is 25 years, with monthly payments:
Total number of payments=25×12=300\text{Total number of payments} = 25 \times 12 = 300

STEP 5

Use the mortgage payment formula to find the monthly payment:
The mortgage payment formula is:
M=Pr(1+r)n(1+r)n−1M = P \frac{r(1 + r)^n}{(1 + r)^n - 1}
Where: - MM is the monthly payment. - PP is the principal amount (\$250,000). - \(r\) is the effective monthly interest rate (\(0.003965\)). - \(n\) is the total number of payments (300).
Substitute the values into the formula:
M=250,000×0.003965(1+0.003965)300(1+0.003965)300−1M = 250,000 \times \frac{0.003965(1 + 0.003965)^{300}}{(1 + 0.003965)^{300} - 1}
Calculate MM:
M≈250,000×0.003965×3.2253.225−1≈1,420M \approx 250,000 \times \frac{0.003965 \times 3.225}{3.225 - 1} \approx 1,420
The monthly payment on the mortgage is approximately:
$1,420\boxed{\$1,420}

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