A lease is a method of financing the use of an asset and is an agreement between a lessee (who rents the asset), and a lessor (who owns the asset). The lessor is usually a lease company or finance company.

The lessee rents the asset from the lessor in return for a periodic rental payment. The lessee never owns the asset, and at the end of the term it is returned to the lessor or a secondary period of rental is entered into.

If paid at the end of each period, the periodic rental payments are in fact an annuity, and given the terms of the lease, the present value of an annuity formula can be used to calculate the lease payment required to be paid by the lessee.

The lease payment cash flow diagram would look as follows:

Period | 1 | 2 | . . . n |
---|---|---|---|

Cash flow | Pmt ↓ | Pmt ↓ | ..Pmt ↓ |

The present value of these payments is given by the annuity formula as shown below:

PV = Pmt x (1 - 1 / (1 + i)^{n}) / i

**Variables used in the formula**

PV = Present Value

Pmt = Periodic payment

i = Discount rate per period

n = Number of periods

This can be rearranged to solve for the payment (Pmt)

Pmt = PV x i / (1 - 1 / (1 + i)^{n})

If we treat the present value (PV) as the asset value to be financed at the start of period 1 (A), the discount rate (i) as the lease interest rate, and n as the number of lease payments required under the agreement, then the leasing calculation formula can be restated as follows:

Pmt = A x i / (1 - 1 / (1 + i)^{n})

**Variables used in the formula**

A = Asset value to be financed

Pmt = Periodic lease payment

i = Lease interest rate per period

n = Number of lease payments required

## Simple Lease Payment Example

Suppose a business (lessee) wants to lease an asset costing 20,000. The finance company (lessor) offers to lease the asset to the business in return for monthly payments at the end of each month, over a term of 3 years, at a lease interest rate of 6%. The monthly lease payment (Pmt) is calculates as follows:

A = Asset value to be financed = 20,000 i = Lease rate = 6%/12 = 0.50% (monthly) n = Number of lease payments required = 3 x 12 = 36 (months) Pmt = Lease payment Pmt = A x i / (1 - 1 / (1 + i)^{n}) Pmt = 20,000 x 0.50% / (1 - 1 / (1 + 0.50%)^{36}) Pmt = 608.44 monthly lease payment

## Advance Payments

In most situations, the lessee is required to make advance payments under the lease agreement. The advance payments are usually a multiple of the normal lease payment.

Suppose the lessee was required to make a number of payments in advance (a), leaving (n-a) further lease payments to be made at the end of each month.

The cash flow diagram would be as follows:

Period | 1 | 2 | . . . (n-a) |
---|---|---|---|

Advance payment | ↓ a Pmts | ||

Single payment | Pmt ↓ | Pmt ↓ | ..Pmt ↓ |

The present value of these lease payments must be the same as the asset value to be financed (A).

Since the advance payments are made at the start of period 1 (today), then their present value is the same as the amount paid or a x Pmt. The present value of the remaining payments is given as before, by the present value of an annuity formula with (n-a) payments

A = PV of advance payments + PV of periodic payments A = a x Pmt + Pmt x (1 - 1 / (1 + i)^{(n-a)}) / i

Again this can be rearranged to give a formula for Pmt

Pmt = A x i /((a x i) + (1 - 1 / (1 + i)^{(n-a)}))

**Variables used in the formula**

A = Asset value to be financed

Pmt = Periodic lease payment

i = Lease interest rate per period

n = Number of lease payments required by the lease agreement

a = Number of payments made in advance

## Advance Lease Payment Example

Assuming the same figures as in the example above except that two payments are made in advance, then the payment (Pmt) can be calculated using the formula as follows:

A = Asset value to be financed = 20,000 i = Lease interest rate = 6%/12 = 0.50% (monthly) n = Number of lease payments required = 3 x 12 = 36 (months) a = Number of advance payments = 2 Pmt = Lease payment Pmt = A x i /((a x i) + (1 - 1 / (1 + i)^{(n-a)})) Pmt = 20,000 x 0.5% /((2 x 0.5%) + (1 - 1 / (1 + 0.5%)^{(36-2)})) Pmt = 602.49 monthly lease payment

The value of the lease payment is lower as two payments are made in advance.

## Asset Residual Value

In a lease agreement, the lessor buys the asset, leases it to the lessee, and then sells the asset at the end of the term. If the asset has no residual or salvage value at the end of the term, then the cost to the lessor is the original cost of the asset, and this becomes the asset value to be financed (A) under the terms of the lease agreement.

However, if the asset does have a residual value then lessor sells the asset at the end of the lease term and receives a residual value amount (R), as shown in the cash flow diagram below.

Period | 1 | 2 | . . . n |
---|---|---|---|

Cash flow | ↓ Original cost (C) | Residual value (R) ↑ |

The total cost to the lessor is the present value of these two cash flows. As the asset is purchased at the start of period 1 (today), the present value of the original cost of the asset is its cost (C). The present value of the residual value (R) is found using the present value of a lump sum formula.

The effect of the residual value is to lower the overall cost of the asset to the lessor, assuming the benefit of this is passed on to the lessee, then the asset value to be financed (A) under the lease agreement is the same as the overall cost of the asset to the lessor.

A = Cost of asset + PV of residual value A = C + R / (1 + i)^{n}

Substituting this value of A in our calculating lease payments formula above we get the following.

Pmt = A x i /((a x i) + (1 - 1 / (1 + i)^{(n-a)})) Pmt = (C + R/(1 + i)^{n}) x (i/((a x i) + (1 - 1/(1 + i)^{(n-a)})))

**Variables used in the formula**

A = Asset value to be financed

Pmt = Periodic lease payment

i = Lease interest rate per period

n = Number of lease payments required by the lease agreement

a = Number of payments made in advance

C = Original cost of the asset

R = Residual value of the asset (a negative number)

## Asset Residual Value Lease Payment Example

Again, using the same numbers from the example above, suppose the asset has a residual value of 3,000 at the end of the lease term, then the payment can be calculated as follows:

Pmt = (C + R/(1 + i)^{n}) x (i/((a x i) + (1 - 1/(1 + i)^{(n-a)}))) i = Lease interest rate = 0.5% per period n = Number of lease payments required = 36 a = Number of payments made in advance = 2 C = Original cost of the asset = 20,000 R = Residual value of the asset (a negative number) = -3,000 Pmt = Periodic ease payment Pmt = (C + R/(1 + i)^{n}) x (i/((a x i) + (1 - 1/(1 + i)^{(n-a)}))) Pmt = (20000 - 3000/(1 + 0.5%)^{36}) x (0.5%/((2 x 0.5%) + (1 - 1/(1 + 0.5%)^{(36-2)}))) Pmt = 526.97 monthly lease payment

The value of the lease payment is lower as the asset value to be financed is less as a result of the lessor being able to sell the asset for the residual value of 3,000 at the end of the term.