## TVM Math

#### Definitions & Other

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A lease is a contract, between the lessor and lessee, for the use of equipment or other property for a fixed amount of time. 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.

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.

Some leases call for up front payments (also called advance payments) at the time that the lease is signed. The up front payment is some multiple of the regular monthly payment amount. This complicates matters a bit, because you have to know the regular payment to calculate the advance payment amount. However, with a little bit of algebra, we can solve this problem as well.

Throughout this tutorial, I will assume that the lease payments are made monthly (12 times per year). The formulas presented here can be for any other payment frequency. Just be sure that N is the number of periods and i is the interest rate per period.

## Definitions

Lease Amount
This is the presumed value of the asset being leased, at the time that the lease is signed. It is the present value of the future payments on the lease, including the residual value.
Residual Value
This is the assumed value of the asset at the end of the life of the lease. 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. For example, an auto lease may specify a residual value of $15,000 when the lease is up in 3 years. At that time, the lessee has the right, but usually not the obligation, to purchase the vehicle for$15,000. If the lessee chooses not to exercise that option, then the vehicle will be turned over to the lessor. In this example, the lease has a built-in call option with a strike price equal to the residual value.
Sometimes the lease terms call for a number of payments to be paid in advance, when the lease is signed. The number of advance payments could be 0, 1, 2, or more. For example, suppose that a lease calls for a $300 monthly payment with 2 advance payments. 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. ## Calculating the Monthly Lease Payment If we assume that the lease does not call for any advance payments, then calculating the regular monthly payment is straightforward. The lease cash flows are an annuity (the monthly payment) and a lump sum (the residual value) at the end of the lease. 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). In this case, the cash flows of the lease resemble those on a bond, as can be seen in the picture below:

Note that I have compressed the time line for space considerations.

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). Therefore, we have the following formula as our starting point:

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. This requires a minor amount of algebra to rearrange the equation, and the result is:

### An Example

Imagine that you are considering an equipment lease (rather than a purchase) of a computer for your office. The lease terms call for a lease amount of $3,500, a residual value of$1,000 and 24 monthly payments. The lease carries an interest rate of 9% per year. How much would your monthly payments be?

Using our formula from above, and converting the annual rate to a monthly rate (0.09/12 = 0.0075) we can calculate the monthly payment amount as follows:

So, the monthly payment would be $121.71. Note that you can easily find this same answer using a financial calculator. Simply enter 24 into N, 9/12 into i, -3500 into PV, and 1000 into FV. Solve for the PMT and you will get the same answer. ## Calculating the Monthly Lease Payment With Advance Payments When a lease calls for advance payments, the calculation gets more complex and it cannot easily be done in a financial calculator without a special program (not all calculators are programmable). In this section, we will derive a formula that will calculate the payment. At first glance, it seems that you would need to know the monthly payment amount before you can calculate the advance payment. But, the presence of the advance payment changes the amount of the monthly payment. So, how can you calculate the monthly payment without knowing the advance payment, and vice versa? It appears to be a tough problem. However, realize that the advance payment is a multiple of the monthly payment, and that simplifies the problem enormously. The advance payment serves to reduce the effective lease amount and also reduces the number of monthly payments to be made by the number of advance payments. This insight makes it relatively simple to solve the problem using a variant of our PV equation above. In the equation below, A is the number of advanced payments: So, we are simply subtracting the number of advance payments from the lease amount (because they are both at period 0), and reducing the number of payments from N to N – A. Now, we simply need to solve the equation for the monthly payment. After some algebraic manipulation, we get the following formula: ### An Example with Advance Payments Let's return to our previous example, but add in some advance payments: Imagine that you are considering an equipment lease (rather than a purchase) of a computer for your office. The lease terms call for a lease amount of$3,500, 3 advance payments due at signing, a residual value of $1,000 and 24 monthly payments. The lease carries an interest rate of 9% per year. How much would your monthly payments be? Using our formula, it is relatively simple to calculate the monthly payment: The result is that you monthly payment would be$119.13. Note that you would make three payments at signing, so you would have to immediately write a check for \$357.40. Your last payment would be made 3 months earlier than if you hadn't made the advance payments.

One final note: If the number of advance payments equals 1, then the problem is greatly simplified because the monthly payment can be treated as an annuity due. This calculation can be done in a financial calculator — just put the calculator into Begin mode. However, if the number of advance payments is 2 or more, then the above formula must be used.

I hope that you have found this tutorial useful. I also have an Excel spreadsheet to calculate lease payments (Excel 2003 version) available.