We already know the PV (that is, the lease amount) and the FV (the residual value), and we want to solve the above equation for the monthly payment amount. Therefore, we have the following formula as our starting point: The principle of value additivity states that the present value (lease amount) is equal to the present value of the monthly payments (an annuity) plus the present value of the residual value (a lump sum). Note that I have compressed the time line for space considerations. In this case, the cash flows of the lease resemble those on a bond, as can be seen in the picture below: Our example lease has a present value of $3,500, a residual value of $1,000, and a monthly payment of $121.71 (which we solve for below). The lease cash flows are an annuity (the monthly payment) and a lump sum (the residual value) at the end of the lease. If we assume that the lease does not call for any advance payments, then calculating the regular monthly payment is straightforward. Then, when the lease is signed, you would pay $600 (2 payments) and the first regular payment of $300 would be due in one month. For example, suppose that a lease calls for a $300 monthly payment with 2 advance payments. The number of advance payments could be 0, 1, 2, or more. Advance Payments Sometimes the lease terms call for a number of payments to be paid in advance, when the lease is signed. In this example, the lease has a built-in call option with a strike price equal to the residual value. If the lessee chooses not to exercise that option, then the vehicle will be turned over to the lessor. At that time, the lessee has the right, but usually not the obligation, to purchase the vehicle for $15,000. For example, an auto lease may specify a residual value of $15,000 when the lease is up in 3 years. Often, as is the case for auto loans, the lessee has the option of purchasing the asset for this price when the lease is up. Residual Value This is the assumed value of the asset at the end of the life of the lease. It is the present value of the future payments on the lease, including the residual value. Definitions Lease Amount This is the presumed value of the asset being leased, at the time that the lease is signed. Just be sure that N is the number of periods and i is the interest rate per period. The formulas presented here can be for any other payment frequency. Throughout this tutorial, I will assume that the lease payments are made monthly (12 times per year). However, with a little bit of algebra, we can solve this problem as well. This complicates matters a bit, because you have to know the regular payment to calculate the advance payment amount. The up front payment is some multiple of the regular monthly payment amount. Some leases call for up front payments (also called advance payments) at the time that the lease is signed. In its basic form, calculating the payment on a lease contract is quite straightforward, and if you have worked through the time value of money tutorials on the present value of lump sums and annuities then you will be able to follow this tutorial with no difficulty. The lease contract will specify the payment terms and other details, such as the residual value of the property at the end of the lease term. Are you a student? Did you know that Amazon is offering 6 months of Amazon Prime - free two-day shipping, free movies, and other benefits - to students? Click here to learn moreĪ lease is a contract, between the lessor and lessee, for the use of equipment or other property for a fixed amount of time.
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