Leasing a car (or vehicle) is much like renting a vehicle. The lessee (person leasing the car) makes a series of equal-sized payments (lease payments), at regular intervals over the course of a fixed time period (lease period). These payments will be smaller than loan payments. This is because at the end of the lease, the lessee either must pay an additional amount called the residual value to purchase the car or they must return the car to the car dealership. To calculate any of the above values, we enter the following in for each value:

Leases are examples of annuities due. We will examine this term in the section below.

See the sections below for key formulas, tips and examples related to leases and annuities due.

Ordinary Annuities vs Annuities Due

When calculating an ordinary annuity (an annuity where the payments occur at the end of each payment interval), you do not likely need to change anything in your calculator. By default, your calculator will be set to END mode (which is the setting you need for ordinary annuities).

It is also possible that payments occur at the beginning of each interval. In this case, you need to “tell” your calculator that the payments are occurring at the beginning of the payment interval. You do this by setting the calculator to BGN mode. This type of annuity is called an annuity due.

How Do We SWITCH between Bgn and end on the BAII Plus?

How Do We Set the BAII Plus to BGN?

• Hit [2ND] followed by [PMT] on your calculator (this gets you into the BGN/END menu).
• You will see END displayed on the screen. To change this to BGN, click on [2ND] and [ENTER]
• You will now see BGN displayed on the screen as well as in the top right of the screen.
• Hit [2ND] followed by [CPT] to exit the BGN/END menu.
• Be careful!  Ordinary annuities are more common that annuities due so once you have completed your annuity due question, turn off BGN (ie: set your calculator back to END).

How Do We Set the BAII Plus to END?

• Hit [2ND] followed by [PMT] on your calculator.
• You will see BGN displayed on the screen. To change this to END, click on [2ND] and [ENTER].
• You will now see END displayed on the screen and you will see BGN disappear from the top right of the calculator screen.
• Hit [2ND] followed by [CPT] to exit the BGN/END menu.

Now, you can practice these steps in the exercise below.

Calculating the Rate Charged & Down Payments on leases

When leasing a car, it can be required to make a down payment. If a down payment is made, we deduct that from the selling price to determine the amount leased:

[latex]\textrm{Amount Leased (PV)} = \textrm{Selling Price} - \textrm{Down Payment}[/latex]

If we instead need the selling price, we re-shuffle the above equation to give:

[latex]\textrm{Selling Price} = \textrm{Amount Leased (PV)} + \textrm{Down Payment}[/latex]

Let us now look at Raj’s lease. He is hoping to lease a Tesla Model X.

EXAMPLE 5.4.1

Raj can lease a Tesla Model X car for three years. From the Tesla website, he finds the following pricing information:

• Selling Price: $110,880
• Term: 36 months
• Required Down Payment: $7,500
• Monthly payment size: $1,702, the first monthly payment is due the day the car is purchased
• Amount owed (residual value) in 3 years to purchase the Model X: $50,568.84

What effective interest rate is Tesla charging on the 3-year lease?

Cost of Financing for Leases

We also call the interest charged on leases the cost of financing. The same interest formula is used:

[latex]\begin{align*} \textrm{Cost of Financing} &= \textrm{Money Out} - \textrm{Money In} = \textrm{\$ OUT} - \textrm{\$ IN} \end{align*}[/latex]

In the case of car leases, we consider the selling price of the car, PV to be “money in.” We consider the regular payments (PMT) and the residual value (FV) to be ‘money out’:

This gives us the following equation for cost of financing for leases:

[latex]\begin{align*} \textrm{Cost of Financing} &= \textrm{\$ OUT} - \textrm{\$ IN}\\ &=( \textrm{Lease Payments}+ \textrm{Residual Value}) - \textrm{Amount Leased}\\ &= ( \textrm{PMT}×\textrm{N}+\textrm{FV})-\textrm{PV} \end{align*}[/latex]

Key Takeaways

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