Assume a bank loan requires an interest payment of $85 per year and a principal payment of $1,000 at the end of the loan's eight-year life. What would be the present value of this loan if it carried an 8.5 percent interest rate?
1,546
To calculate the present value of the loan, we need to discount the future cash flows using the interest rate. Here are the steps to determine the present value:
Step 1: Find the present value of the interest payments.
The interest payment is $85 per year and the loan has a total of 8 years. We can use the formula for the present value of an annuity to calculate the present value of the interest payments:
PV = Pmt * [(1 - (1 + r)^-n) / r]
Where:
PV = Present Value
Pmt = Payment per period
r = Interest rate
n = Number of periods
In this case, the payment per period (Pmt) is $85, the interest rate (r) is 8.5% (or 0.085 as a decimal), and the number of periods (n) is 8.
PV = $85 * [(1 - (1 + 0.085)^-8) / 0.085]
Step 2: Find the present value of the principal payment.
The principal payment is $1,000, due at the end of the 8-year loan term. To find the present value of this payment, we can use the formula for the present value of a single lump sum:
PV = FV / (1 + r)^n
Where:
PV = Present Value
FV = Future Value
r = Interest rate
n = Number of periods
In this case, the future value (FV) is $1,000, the interest rate (r) is 8.5% (or 0.085 as a decimal), and the number of periods (n) is 8.
PV = $1,000 / (1 + 0.085)^8
Step 3: Calculate the total present value.
To find the total present value of the loan, we need to sum up the present value of the interest payments and the present value of the principal payment:
Total PV = PV of interest payments + PV of principal payment
Simply plug in the values we calculated from Step 1 and Step 2 into this formula to get the answer.
To calculate the present value of the loan, we need to discount the future cash flows using the given interest rate. The present value is the current value of future cash flows.
First, let's calculate the present value of the interest payments. The interest payment is $85 per year for 8 years, so:
PV(interest) = $85 / (1 + 0.085)^1 + $85 / (1 + 0.085)^2 + ... + $85 / (1 + 0.085)^8
Next, let's calculate the present value of the principal payment at the end of the loan. Since it occurs in 8 years, we can calculate its present value as:
PV(principal) = $1,000 / (1 + 0.085)^8
Now, we can calculate the present value of the loan by summing the present values of the interest and principal:
Present Value of the Loan = PV(interest) + PV(principal)
Once you calculate these values, you can add them together to find the present value of the loan.