Question
26) A house is sold with an assumable $156,000 below-market loan at 8.5% for a remaining termof 15 years. Current rates are 9.75% for 15 year mortgages. If the house sold for $240,000, what is the cash-equivalent value of the house.
Answers
GPT-4o mini
To determine the cash-equivalent value of the house, we need to calculate the present value of the below-market loan and the cash portion required to purchase the house at the market interest rate.
### Step 1: Calculate the Monthly Payment for the Below-Market Loan
The below-market loan of $156,000 has an interest rate of 8.5% and a term of 15 years.
1. **Convert annual interest rate to a monthly interest rate:**
\[
r = \frac{8.5\%}{12} = \frac{0.085}{12} \approx 0.0070833
\]
2. **Calculate the number of payments (months):**
\[
n = 15 \times 12 = 180
\]
3. **Use the formula for monthly payment (PMT):**
\[
PMT = P \frac{r(1 + r)^n}{(1 + r)^n - 1}
\]
where \( P \) is the principal amount ($156,000).
Substituting values:
\[
PMT = 156000 \frac{0.0070833(1 + 0.0070833)^{180}}{(1 + 0.0070833)^{180} - 1}
\]
First, calculate \( (1 + 0.0070833)^{180} \):
\[
(1 + 0.0070833)^{180} \approx 3.4990
\]
Now substituting into the payment formula:
\[
PMT = 156000 \frac{0.0070833 \times 3.4990}{3.4990 - 1} \approx 156000 \frac{0.02474}{2.4990} \approx 156000 \times 0.009896 = 1545.64
\]
### Step 2: Calculate the Present Value of the Monthly Payments
Now we'll calculate the present value of the monthly payments using the market interest rate of 9.75%.
1. **Convert the market annual interest rate to a monthly interest rate:**
\[
r_{market} = \frac{9.75\%}{12} = \frac{0.0975}{12} \approx 0.008125
\]
2. **Calculate the present value (PV) of an annuity:**
The present value of the annuity can be calculated using the formula:
\[
PV = PMT \frac{1 - (1 + r_{market})^{-n}}{r_{market}}
\]
Substitute in the known values:
\[
PV = 1545.64 \frac{1 - (1 + 0.008125)^{-180}}{0.008125}
\]
First calculate \( (1 + 0.008125)^{-180} \):
\[
(1 + 0.008125)^{-180} \approx 0.2380
\]
Now substituting into the present value formula:
\[
PV = 1545.64 \frac{1 - 0.2380}{0.008125} \approx 1545.64 \frac{0.7620}{0.008125} \approx 1545.64 \times 93.706 \approx 144,547.14
\]
### Step 3: Calculate Cash-Equivalent Value of the House
The cash-equivalent value of the house is the market price minus the present value of the below-market loan:
1. **House sold for:** $240,000
2. **Cash-equivalent value:**
\[
Cash-equivalent value = Selling Price - Present Value of the Loan
\]
\[
Cash-equivalent value = 240,000 - 144,547.14 \approx 95,452.86
\]
### Conclusion
The cash-equivalent value of the house is approximately **$95,452.86**.
### Step 1: Calculate the Monthly Payment for the Below-Market Loan
The below-market loan of $156,000 has an interest rate of 8.5% and a term of 15 years.
1. **Convert annual interest rate to a monthly interest rate:**
\[
r = \frac{8.5\%}{12} = \frac{0.085}{12} \approx 0.0070833
\]
2. **Calculate the number of payments (months):**
\[
n = 15 \times 12 = 180
\]
3. **Use the formula for monthly payment (PMT):**
\[
PMT = P \frac{r(1 + r)^n}{(1 + r)^n - 1}
\]
where \( P \) is the principal amount ($156,000).
Substituting values:
\[
PMT = 156000 \frac{0.0070833(1 + 0.0070833)^{180}}{(1 + 0.0070833)^{180} - 1}
\]
First, calculate \( (1 + 0.0070833)^{180} \):
\[
(1 + 0.0070833)^{180} \approx 3.4990
\]
Now substituting into the payment formula:
\[
PMT = 156000 \frac{0.0070833 \times 3.4990}{3.4990 - 1} \approx 156000 \frac{0.02474}{2.4990} \approx 156000 \times 0.009896 = 1545.64
\]
### Step 2: Calculate the Present Value of the Monthly Payments
Now we'll calculate the present value of the monthly payments using the market interest rate of 9.75%.
1. **Convert the market annual interest rate to a monthly interest rate:**
\[
r_{market} = \frac{9.75\%}{12} = \frac{0.0975}{12} \approx 0.008125
\]
2. **Calculate the present value (PV) of an annuity:**
The present value of the annuity can be calculated using the formula:
\[
PV = PMT \frac{1 - (1 + r_{market})^{-n}}{r_{market}}
\]
Substitute in the known values:
\[
PV = 1545.64 \frac{1 - (1 + 0.008125)^{-180}}{0.008125}
\]
First calculate \( (1 + 0.008125)^{-180} \):
\[
(1 + 0.008125)^{-180} \approx 0.2380
\]
Now substituting into the present value formula:
\[
PV = 1545.64 \frac{1 - 0.2380}{0.008125} \approx 1545.64 \frac{0.7620}{0.008125} \approx 1545.64 \times 93.706 \approx 144,547.14
\]
### Step 3: Calculate Cash-Equivalent Value of the House
The cash-equivalent value of the house is the market price minus the present value of the below-market loan:
1. **House sold for:** $240,000
2. **Cash-equivalent value:**
\[
Cash-equivalent value = Selling Price - Present Value of the Loan
\]
\[
Cash-equivalent value = 240,000 - 144,547.14 \approx 95,452.86
\]
### Conclusion
The cash-equivalent value of the house is approximately **$95,452.86**.