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 the loan if the interest rate is 8 percent?
To calculate the present value of the loan, we need to add the present values of the interest payments and the principal payment.
The interest payments are an annuity, as they occur at regular intervals (every year) for a fixed period of time (eight years). To calculate the present value of an annuity, we can use the formula:
PV = PMT * ((1 - (1 + r)^(-n)) / r)
where PV is the present value of the annuity, PMT is the payment per period, r is the interest rate per period, and n is the total number of periods.
In this case, the payment per period (interest payment) is $85, the interest rate per period (annual interest rate) is 8%, and the total number of periods (years) is 8.
Let's calculate the present value of the annuity:
PV_annuity = $85 * ((1 - (1 + 0.08)^(-8)) / 0.08)
PV_annuity = $85 * ((1 - 1.08^(-8)) / 0.08)
PV_annuity = $85 * ((1 - 0.46319392) / 0.08)
PV_annuity = $85 * (0.53680608 / 0.08)
PV_annuity = $85 * 6.710075
PV_annuity ≈ $571.34
Next, we need to calculate the present value of the principal payment. The principal payment is a single lump sum payment occurring at the end of the loan's eight-year life. To calculate the present value of a single lump sum, we can use the formula:
PV = FV / (1 + r)^n
where PV is the present value, FV is the future value, r is the interest rate per period, and n is the number of periods.
In this case, the future value (principal payment) is $1,000, the interest rate per period (annual interest rate) is 8%, and the number of periods (years) is 8.
Let's calculate the present value of the principal payment:
PV_principal = $1,000 / (1 + 0.08)^8
PV_principal = $1,000 / 1.08^8
PV_principal = $1,000 / 1.7186972
PV_principal ≈ $581.46
Finally, to get the present value of the loan, we need to add the present values of the annuity and the principal payment:
Present Value of the Loan = PV_annuity + PV_principal
Present Value of the Loan ≈ $571.34 + $581.46
Present Value of the Loan ≈ $1,152.80
Therefore, the present value of the loan would be approximately $1,152.80.