Suppose that you have the option to lease a new car, which you otherwise intend to purchase for $21,000. The lease terms: $3000 down and payments of $304 per month for 48 months, at the beginning of each month. Upon termination, you can purchase the car for an addition payment of $7000 at lease expiration. If your financing rate is 5.6% APR, and you discount the lease-purchase option using that same rate, how much will pay to buy car (in present-value terms) using the lease-purchase option?
Calculation of present value of the lease payments done.
As the annuity payments are done at the beginning of each year, it is called as the annuity due.
So formula for calculating present value of annuity due=C*[(1-(1+i)^-n)/i]*(1+i).
Where c is the annuity payment.
i is the interest rate
n is the no.of payments
Present value of annuities=304*[(1-(1+0.056/12)^-48)/(0.056/12)]*(1+0.056/12)
Bringing the last payment of 7000 to the present value I.e., bringing 7000 paid at the end of 4 years to the present value at the discounting rate of 5.6%
Therefore present value=Future value/(1+interest rate) ^n
Where n is the no.of years
Hence the total present value to be paid=down payment+present value of annuity+present value of lease payment done at the end of 4 years.
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