Final answer:
To find the present value of the lease option for a printer with an 8.4% nominal rate compounded semi-annually, the effective annual rate must first be determined, and then converted to a monthly rate. The present value of the monthly payments, treated as an annuity due, and the present value of the lump sum payment at the end of 6 years are both calculated and summed up.
Step-by-step explanation:
To calculate the present value of the lease option for office equipment, we must consider the present value of an annuity due as the payments are at the beginning of each period, and then add it to the present value of the lump sum payment at the end of 6 years. With an interest rate of 8.4% compounded semi-annually, the effective monthly rate needs to be determined.
First, the effective annual rate (EAR) is calculated from the nominal rate (8.4% compounded semi-annually) using the formula EAR = (1 + r/n)^(n) - 1, where r is the nominal annual rate and n is the number of compounding periods per year. In this case, EAR = (1 + 0.084/2)^(2) - 1 = 0.08568 or 8.568%. Next, we convert the EAR to an effective monthly rate (EMR) by dividing it by 12, since there are 12 months in a year. So, EMR = 8.568% / 12 which is approximately 0.714% per month.
To find the present value of the lease option, we discount the annuity due of $90 per month and the $1,750 final payment:
- The present value of an annuity due can be calculated with the formula PV of an annuity due = P * [(1 - (1 + i)^(-n))/i] * (1 + i), where P is the payment, i is the EMR, and n is the total number of payments.
- The present value of the $1,750 future value is calculated using PV of lump sum = FV / (1 + i)^n.
Once the present values are calculated separately, they are summed up to get the total present value of the lease option.