Question
A borrower has secured a 30 year, $150,000 loan at 7% with monthly payments. Fifteen years later, an investor wants to purchase the loan from the lender. If market interest rates are 5%, what would the investor be willing to pay for the loan?
Answers
GPT-4o mini
To determine what an investor would be willing to pay for the loan, we first need to find out the remaining balance on the loan after 15 years, and then we can calculate the present value of the remaining payments using the market interest rate of 5%.
### Step 1: Calculate Monthly Payment for the Original Loan
For a loan of $150,000 at 7% annual interest (0.07/12 monthly), with a term of 30 years (360 months), we use the formula for the monthly payment \( M \):
\[
M = P \frac{r(1+r)^n}{(1+r)^n-1}
\]
where:
- \( P = 150,000 \) (loan principal)
- \( r = \frac{0.07}{12} = 0.00583333 \) (monthly interest rate)
- \( n = 360 \) (total number of payments)
Calculating \( M \):
\[
M = 150,000 \frac{0.00583333(1 + 0.00583333)^{360}}{(1 + 0.00583333)^{360} - 1}
\]
Calculating \( (1 + 0.00583333)^{360} \):
\[
(1 + 0.00583333)^{360} \approx 7.612255
\]
Now plug this back into the payment formula:
\[
M = 150,000 \frac{0.00583333 \cdot 7.612255}{7.612255 - 1}
\]
\[
M = 150,000 \frac{0.044409}{6.612255} \approx 1,033.14
\]
The monthly payment \( M \approx 1,033.14 \).
### Step 2: Calculate Remaining Balance After 15 Years
To find the remaining balance after 15 years (180 payments), we can use the formula for the remaining balance \( B \):
\[
B = P \frac{(1 + r)^n - (1 + r)^p}{(1 + r)^n - 1}
\]
where:
- \( P = 150,000 \) (original loan amount)
- \( r = 0.00583333 \) (monthly interest rate)
- \( n = 360 \) (total payments)
- \( p = 180 \) (payments made)
Calculating \( B \):
\[
(1 + r)^{180} \approx (1.00583333)^{180} \approx 2.89828
\]
Now, plug values into the balance formula:
\[
B = 150,000 \frac{7.612255 - 2.89828}{6.612255}
\]
\[
B = 150,000 \frac{4.713975}{6.612255}
\]
\[
B \approx 150,000 \cdot 0.71225 \approx 106,837.50
\]
### Step 3: Calculate Remaining Payments
After 15 years, 180 payments remain (180 monthly payments of approximately \( 1,033.14 \)).
### Step 4: Present Value of Remaining Payments
To find out how much the investor would be willing to pay for the remaining payments at a market interest rate of 5% (0.05/12 = 0.00416667 per month), we compute the present value of an annuity:
\[
PV = M \frac{1 - (1 + r)^{-n}}{r}
\]
where:
- \( M \approx 1,033.14 \)
- \( r = 0.00416667 \)
- \( n = 180 \)
Calculating \( PV \):
\[
PV = 1,033.14 \frac{1 - (1 + 0.00416667)^{-180}}{0.00416667}
\]
Calculating \( (1 + r)^{-n} \):
\[
(1 + 0.00416667)^{-180} \approx 0.406619
\]
Now plug this back into the present value formula:
\[
PV = 1,033.14 \frac{1 - 0.406619}{0.00416667}
\]
\[
PV = 1,033.14 \cdot 142.26 \approx 147,318.68
\]
### Conclusion
The present value of the remaining payments, and hence the amount an investor would be willing to pay for the loan, is approximately **$147,318.68**.
### Step 1: Calculate Monthly Payment for the Original Loan
For a loan of $150,000 at 7% annual interest (0.07/12 monthly), with a term of 30 years (360 months), we use the formula for the monthly payment \( M \):
\[
M = P \frac{r(1+r)^n}{(1+r)^n-1}
\]
where:
- \( P = 150,000 \) (loan principal)
- \( r = \frac{0.07}{12} = 0.00583333 \) (monthly interest rate)
- \( n = 360 \) (total number of payments)
Calculating \( M \):
\[
M = 150,000 \frac{0.00583333(1 + 0.00583333)^{360}}{(1 + 0.00583333)^{360} - 1}
\]
Calculating \( (1 + 0.00583333)^{360} \):
\[
(1 + 0.00583333)^{360} \approx 7.612255
\]
Now plug this back into the payment formula:
\[
M = 150,000 \frac{0.00583333 \cdot 7.612255}{7.612255 - 1}
\]
\[
M = 150,000 \frac{0.044409}{6.612255} \approx 1,033.14
\]
The monthly payment \( M \approx 1,033.14 \).
### Step 2: Calculate Remaining Balance After 15 Years
To find the remaining balance after 15 years (180 payments), we can use the formula for the remaining balance \( B \):
\[
B = P \frac{(1 + r)^n - (1 + r)^p}{(1 + r)^n - 1}
\]
where:
- \( P = 150,000 \) (original loan amount)
- \( r = 0.00583333 \) (monthly interest rate)
- \( n = 360 \) (total payments)
- \( p = 180 \) (payments made)
Calculating \( B \):
\[
(1 + r)^{180} \approx (1.00583333)^{180} \approx 2.89828
\]
Now, plug values into the balance formula:
\[
B = 150,000 \frac{7.612255 - 2.89828}{6.612255}
\]
\[
B = 150,000 \frac{4.713975}{6.612255}
\]
\[
B \approx 150,000 \cdot 0.71225 \approx 106,837.50
\]
### Step 3: Calculate Remaining Payments
After 15 years, 180 payments remain (180 monthly payments of approximately \( 1,033.14 \)).
### Step 4: Present Value of Remaining Payments
To find out how much the investor would be willing to pay for the remaining payments at a market interest rate of 5% (0.05/12 = 0.00416667 per month), we compute the present value of an annuity:
\[
PV = M \frac{1 - (1 + r)^{-n}}{r}
\]
where:
- \( M \approx 1,033.14 \)
- \( r = 0.00416667 \)
- \( n = 180 \)
Calculating \( PV \):
\[
PV = 1,033.14 \frac{1 - (1 + 0.00416667)^{-180}}{0.00416667}
\]
Calculating \( (1 + r)^{-n} \):
\[
(1 + 0.00416667)^{-180} \approx 0.406619
\]
Now plug this back into the present value formula:
\[
PV = 1,033.14 \frac{1 - 0.406619}{0.00416667}
\]
\[
PV = 1,033.14 \cdot 142.26 \approx 147,318.68
\]
### Conclusion
The present value of the remaining payments, and hence the amount an investor would be willing to pay for the loan, is approximately **$147,318.68**.