Lease Terminology

1. Purchase option (formerly, bargain purchase option)

Suppose you lease a car for three years. Normally, after three years the car is returned to the leasing company, which sells it on the used-car market. The lessee can instead be given a right of first purchase — "before you return it, you may buy that car if you wish." This is the bargain purchase option.

Why is this right needed? A step-by-step view

1. An operating lease ends with a return When the lease term ends, the lessee returns the asset to the leasing company, which disposes of it by selling it on the secondary market.
2. The leasing company has already received most of its money The leasing company has been receiving monthly lease payments. At signing, an amount is set for what the asset is expected to be worth at the end of the lease. This is called the guaranteed RV (Residual Value). The leasing company can recover its investment even if it receives only the guaranteed RV when it disposes of the asset.
3. The lessee may want to buy the asset For example, the lessee may want to keep using a well-maintained car. The right that lets the lessee take over the asset at a low (bargain) price is the bargain purchase option.
4. The problem: the lessee may not buy A right granted at the end of the lease is a choice: the lessee may buy or not. If the used-market price has by then fallen below the guaranteed RV, the lessee has no reason to buy at the higher price, and the burden and loss of selling fall on the leasing company.
5. The solution: fix it at signing If the lease contract also states that "at the end of the lease, the lessee will certainly take over the asset at this price," the leasing company no longer bears disposal risk. With no risk, the guaranteed RV can be set at or above market value, which can make the monthly lease payment as low as possible.
6. In that case, it is effectively "buying the asset in installments" The lessee is using the lease as a financing tool — borrowing money to buy the asset. That is why the former Korean accounting standard (K-GAAP No. 19) required a lease to be classified and accounted for as a finance lease when exercise of the bargain purchase option was certain.

2. Residual value (Residual Value, RV)

For example, if you expect that a machine bought for $22,000 could be sold used for $3,700 after five years of use, its residual value is $3,700. Even a worthless piece of scrap can be sold as scrap metal; that amount is called scrap value or residual value.

Whether the amount is guaranteed

Points to note

• When the asset passes to the lessee: If there is a transfer-of-ownership clause, or the lessee is reasonably certain to exercise a purchase option, the leasing company cannot sell the asset after the lease ends. The residual value in this case is therefore not included in the money the leasing company will receive (future cash flows).
• If the value falls short of the guaranteed amount: When the actual value is lower than the guaranteed residual value, the lessee must pay the difference to the leasing company. That is why the previous standard included the guaranteed residual value in minimum lease payments, while the current standard includes "the amount expected to be payable" in the lease liability.
• Tax law basis: Tax law treats the residual value of tangible assets (machinery, vehicles, etc.) as up to 10% of the acquisition cost, and treats intangible assets (software, etc.) as having no residual value. This may differ from the value estimated for accounting.

3. Fair value (Fair Value)

Think of it as the price that would arise when neither party is rushed or forced and both are well informed about the circumstances. In leasing, it is used as of the date the lease is executed to set the price of the leased asset.

How the price is determined in lease practice

• When the leasing company is the manufacturer or dealer of the asset: the normal selling price, taking discounts into account
• When it is not the manufacturer or dealer
  • New asset: the acquisition cost stated in the lease contract (the price the leasing company actually paid)
  • Used asset: the market price if a market exists; otherwise the appraised value from a reputable appraisal firm

4. Lease payments (former term: minimum lease payments)

The previous standard called this "minimum lease payments." It meant the least the lessee had to pay, and only the portions that had to be paid were counted. In the current standard the name has changed to lease payments, and its composition has changed somewhat.

Items included in lease payments under the current standard (from the lessee's perspective)

1. Fixed lease payments A fixed amount each month. Any incentive the leasing company agrees to give is deducted.
2. Variable lease payments that depend on an index or rate For example, lease payments that rise with the consumer price index. They are calculated using the index at the commencement date.
3. Amounts expected to be payable under residual value guarantees The shortfall described above.
4. The exercise price of a purchase option Included only if the lessee is reasonably certain to buy.
5. Penalties for terminating the lease Included if the lease term was determined on the basis that the lessee will terminate.

5. Initial direct costs (Initial Direct Costs)

Put simply, these are "fees and costs spent directly to win one contract." They fall into two broad groups.

• Costs paid to third parties in negotiating and executing the contract (e.g., brokerage fees, legal advisory fees)
• Costs arising directly from specific internal activities performed because of this contract

As has long been the case, where the lessor is a manufacturer or dealer, its costs are generally not included in initial direct costs. Under the current standard, the lessee includes them in the cost of the right-of-use asset, and the lessor includes them in the investment in a finance lease (net investment in the lease) and in the cost of an operating lease asset.

6. Implicit interest rate (IRR: Internal Rate of Return)

If you invest 100 and receive 110 a year later, the rate of return is 10%. IRR generalizes this to cases where the money is received in several installments. In lease calculations, it is used as the reference interest rate when computing lease payments.

As a formula

Fair value + initial direct costs = present value of (lease payments + unguaranteed residual value)

• Left side: the money the leasing company spent (asset price + contract costs)
• Right side: the money it will receive in future (lease payments, unguaranteed residual value), converted to present value
• The discount rate that makes both sides equal is the implicit interest rate. It is the same as the discount rate at which the present values of cash inflows and cash outflows sum to NPV (net present value) = 0.
• However, the costs of a manufacturer or dealer lessor are excluded from initial direct costs.

Worked example: an auto lease

Using the auto lease below as an example, we first calculate the monthly lease payment and then find the IRR of that cash flow.

• Vehicle price: $37,000
• Lease term: 36 months
• Annual interest rate: 5.5% (monthly rate = 5.5% ÷ 12 ≈ 0.4583%)
• Annual insurance: $0, annual automobile tax: $0, monthly fee: $0
• Residual value at maturity: $14,800

Step 1. Calculate the monthly lease payment

The lease payment is calculated by dividing "the vehicle price minus the present value of the residual value received at maturity" so that it is repaid over 36 months. Here i is the monthly rate and n is the lease term in months.

Lease payment = (Vehicle price − Residual ÷ (1+i)ⁿ) ÷ [(1 − (1+i)⁻ⁿ) ÷ i]

Applying the conditions above gives a monthly lease payment of about $738.

Check it with the lease payment calculator ↗

Check it yourself in Excel or Google Sheets (PMT function)

In Excel and Google Sheets, the PMT function calculates the same lease payment. Enter the formula below into a cell as is.

=PMT(5.5%/12, 36, -37000, 14800, 0)

The result is 738.18…, the same as the roughly $738 calculated above.

Step 2. Build the cash flow

From the leasing company's perspective, the cash outflows and inflows are as follows.

Step 3. Find the IRR

This is the equation for finding the monthly IRR (r) at which the sum of the present values of the money received equals the $37,000 investment.

37,000 = 738/(1+r) + 738/(1+r)² + … + 738/(1+r)³⁵ + 15,538/(1+r)³⁶

• The denominator (1+r)t discounts money received t months from now to its present value. The further in the future, the more it is discounted.
• You can also treat the investment as −37,000 and say "all terms add up to 0." It is the same equation. This is NPV = 0.

The monthly IRR that satisfies this equation is about 0.4583%, and multiplying by 12 to annualize gives 5.50% per year.

Implementing it in a program (trial and error)

IRR cannot be solved directly with a single formula, so Excel also works by trying values and getting closer to the answer.

1. Choose an initial value Start with an arbitrary IRR (e.g., 50%).
2. Plug it in Using that IRR, calculate the present value of the money to be received and compare it with the investment.
3. Adjust the value If the difference is large, raise or lower the IRR and calculate again. Repeat until the difference becomes very small.
4. Limit the iterations Set a maximum number of iterations (e.g., 100) to prevent endless looping.

Check it with the lease cash-flow IRR calculator ↗

To reproduce the values in the example above, set annual insurance and annual automobile tax to 0 in the calculator. A vehicle price of $37,000, a 36-month term, a 5.5% annual rate, and a $14,800 residual value at maturity are the same as the defaults.

7. Weighted-average rate

When calculating acquisition cost and executing a lease, if several prepaid assets (funds paid in advance) carry different interest rates, they may be combined into a single reference rate. This is not a simple average; the rates on larger principal amounts count for more.

Weighted-average rate = Σ(prepaid asset principal × rate) ÷ Σ(prepaid asset principal) × 100

Example: With $100,000 at 5% and $300,000 at 3%, the simple average is 4%, but the weighted average is ($100,000 × 5 + $300,000 × 3) ÷ $400,000 = 3.5%.

8. Gross and net investment in the lease

Net investment = present value of (lease payments + unguaranteed residual value) = fair value + initial direct costs

In a finance lease, the leasing company records the net investment in its books as "money to be received (a receivable)." The difference between gross and net investment is the money to be earned as interest in future (unearned finance income).

9. Stipulated damages

If the asset is lost or damaged beyond use during the lease term, or the lessee terminates the contract early, the leasing company does not receive the money it was due. This amount is meant to make up that loss, and is interpreted either as damages for breach of contract fixed in advance or as a risk-bearing amount for loss of or damage to the asset fixed in advance.

Values the system must define

• Base amount: which to use as the base — unrecovered principal (principal not yet received), unrecovered principal and interest, or unexpired lease payments (lease payments for the period yet to come)
• Method: the general method or the weighted method
• Applicable rate: the rate set in the contract or terms and conditions

10. Security deposit

• Amount: Agreed between the parties, but usually set according to the lessee's creditworthiness — less for good credit, more for low credit.
• Return: Returned to the lessee when the lease term ends.
• On early termination: Unpaid lease payments or stipulated damages are first offset against the deposit, and the remainder is returned.

Example: If you place a deposit of $7,400, you get it back after 36 months regardless of the monthly lease payment. If you fail to pay a lease payment partway through, it can be offset against the deposit first.

11. Long-term advance payment

"Advance payment" and "long-term advance payment" are treated as having the same meaning, and this page uses long-term advance payment throughout. The word "long-term" is attached because the lease term is long, so the application continues for more than a year and part of the balance is recorded as a non-current liability.

• Monthly lease payment: It is deducted from the principal used to calculate lease payments, so the monthly lease payment decreases.
• Is it returned? No. It is used up as lease payments over time.
• On early termination: The portion already applied to lease payments is not refunded; the remainder is handled according to the settlement rules.

Example: If $7,400 of a $37,000 vehicle price is paid up front as a long-term advance payment, the monthly lease payment is calculated on the remaining $29,600.

How a long-term advance payment is refunded

When the remaining long-term advance payment must be returned on early termination or at maturity, it is settled in one of the following ways.

• Lump-sum refund: returned all at once
• Equal refund: returned in equal amounts over a period
• Top-Down refund, Bottom-Up refund: returned in installments in a set order (largest amount first / smallest amount first)

Security deposit vs. long-term advance payment

12. Late return charge

If the lessee unilaterally postpones the return, the leasing company suffers losses such as the following.

• The cost of taking out insurance again that is unnecessary for a contract that has already ended
• Additional risk and interest loss from being unable to sell the returned used car

To compensate for this, the charge is calculated and collected for the number of days of delay as follows.

Late return charge = (lease payment ÷ 31) × days late × delay multiplier

Example: With a monthly lease payment of $459, a 5-day delay, and a delay multiplier of 200%, the charge is ($459 ÷ 31) × 5 × 2 = about $148.

13. Trade and insurance terms

Terms that often come up when leasing imported equipment or handling insurance for auto leases.