Lease versus Buy: Method

Lease versus Buy: Net Advantage to Leasing, Lease-Purchase Analysis, Vehicle Lease Arithmetic, Lessor Yield and Lease Accounting

Conventions

One currency throughout. Rates are entered in percent per year and are nominal: with m payments a year (1 or 12) the rate per period is the yearly rate divided by m. A cash flow is positive when the party analysed receives it. A tax effect is taken in the period of the item that causes it: a deductible payment X costs X(1 − T) at the tax rate T, and the depreciation allowed for a year is spread evenly over that year's periods. The comparison runs over the lease term, at the end of which the purchased asset is valued at its residual value.

Net advantage to leasing

The two alternatives give the user the same asset for the same term, so only their cash flows differ. Buying: the price and purchase costs at the start (or a down payment followed by loan payments, with the tax saved on the interest), the tax saved by depreciation T D_t, the owner's maintenance, insurance and property tax after tax, and at the end the residual value S less selling costs and less the tax T(S − book value) on the gain (a loss saves tax). Leasing: the payments L_t(1 − T), in advance or in arrears and stepped up each year by the escalation rate, fees at signing, a deposit paid and returned, any capital the lessee spends on the lessor's asset with its write-off over the term, the costs the lessee carries under the lease, an end-of-lease fee and, when the purchase option is exercised, the option price paid and the residual value kept, taxed on S − option price.

A lease commits the user to fixed payments as a loan does and so displaces borrowing. The flows of both alternatives are therefore discounted at the after-tax cost of borrowing, r_b(1 − T) (Myers, Dill and Bautista 1976; Brealey, Myers and Allen), and

NAL = PV(cost of buying) − PV(cost of leasing).

A positive net advantage favours the lease. Because the residual value is an estimate while the payments are contractual, a higher discount rate may be entered for it. When the purchase loan carries the borrowing rate, its payments and interest tax savings discounted at r_b(1 − T) are worth exactly the amount borrowed, so buying with the loan and buying for cash cost the same in present value and the down payment and loan term do not matter; the tool reports this, and the test script checks it. A loan at another rate, or another discount rate, breaks the identity and the loan then counts.

Equivalent loan. Leasing instead of buying for cash releases F₀ at the start (the cost less the first lease outlays) and takes on the later incremental outflows. Their present value at the discount rate is the loan those same outflows could service, the equivalent loan, and NAL = F₀ − equivalent loan. The rate at which the incremental flows have zero present value is the after-tax cost of the financing the lease provides; leasing wins when it is below the after-tax borrowing rate. When the incremental flows change sign more than once, more than one such rate can exist and the tool lists what it finds and says to rely on the net advantage.

Break-even values. The net advantage is linear in the lease payment and in the residual value, so each break-even is exact from two evaluations: L* = L − NAL/(∂NAL/∂L) and likewise S*. The break-even tax rate is found by scanning 0 to 60 % and bisecting each sign change, the discount rate following the tax rate when it is the after-tax borrowing rate. A taxpayer in a low bracket gains little from depreciation and interest deductions; a lessor in a high bracket can use them and share the benefit through the payment, which is the classical tax reason for leasing.

Lease-purchase analysis without taxes (OMB Circular A-94)

Section 13 of the Circular requires that a lease of a capital asset be justified against purchase by comparing "the net discounted present value of the life-cycle cost of leasing with the full costs of buying or constructing an identical asset": the purchase price at fair market value, plus the present value of the ancillary costs of owning, less the residual value at the end of the period, discounted at the Treasury borrowing rate of comparable maturity (nominal rates for nominal payments, real rates for constant-dollar payments; Section 8.c.2 and Appendix C). The normal taxes of the lessor are not deducted, and renewal options are added to the term. The tool's second analysis is that comparison: the same engine with a zero tax rate and the entered discount rate. Payments spread evenly over the year take the mid-year factor (1 + d)^{−(t − 0.5)}. Its default is Example 1 of the DoD Contract Pricing Reference Guides (volume 2, chapter 9): three payments of 10,000 in advance against a price of 29,000 with a salvage value of 2,000; the guide's text names 5.4 % but its factors (0.9470, 0.8968, 0.8492) are those of 5.6 %, which the tool uses, giving 28,437 and 27,302 against the guide's 28,438 and 27,302 from four-figure factors.

Premises: rent or own

The same comparison with monthly rent stepped up yearly, a deposit, tenant improvements written off over the lease term, a mortgage, yearly ownership costs with growth, straight-line depreciation of the building share of the cost over 39 or 27.5 years (land is not depreciated; the mid-month convention is not applied) and a terminal value entered or grown from the price at an appreciation rate, less selling costs. The gain on sale is taxed at the one entered rate; the separate capital gain and recapture rates are not modelled.

Vehicle lease

The federal disclosure (Regulation M, 12 CFR 1013, model form A-2) builds the payment in steps: gross capitalized cost (agreed value and amounts capitalized) less the capitalized cost reduction is the adjusted capitalized cost; less the residual value it is the depreciation and any amortized amounts; plus the rent charge it is the total of base monthly payments, which divided by the number of payments is the base monthly payment; plus the monthly sales or use tax it is the total monthly payment. The total of payments is the amount due at signing plus the monthly payments plus other charges, less the first monthly payment and the refundable deposit, which would otherwise be counted twice. The default is the sample form of the Federal Reserve's Keys to Vehicle Leasing (22,300; 3,500; 18,800; 12,350; 6,450; 5,295; 11,745; 48; 244.69; 12.25; 256.94; 4,241.94; 16,368.12), reproduced to the cent.

Lessors commonly state the rent charge through a money factor: rent charge = (adjusted capitalized cost + residual value) × money factor × n. The sum in parentheses is twice the average balance of a lease that falls in a straight line from the adjusted capitalized cost to the residual value, so the money factor is a monthly rate on the average balance divided by two: money factor = (yearly rate / 12) / 2, that is, yearly rate in percent = money factor × 2400. This is a trade convention and an approximation. Federal law prescribes no lease rate, and the Federal Reserve guide says of the money factor that it "is not a lease rate and cannot be converted to a lease rate by moving the decimal point". The tool therefore also solves the constant-yield (actuarial) rate i of the base payments P in advance, adjusted capitalized cost = P(1 + i)(1 − vⁿ)/i + residual × vⁿ. For the sample form the rent charge corresponds to a money factor of 0.003541, 8.499 % by the 2400 rule, against a constant-yield rate of 8.497 %.

The lease or loan comparison runs the net-advantage engine monthly without taxes: lease payments in advance, the amounts at signing and the disposition fee against the down payment, the loan payments, the payoff of the balance at the end of the lease term and the value of the vehicle then, at the household's own discount rate.

Lessor

The rate implicit in a lease is the rate per period i at which the payments and the residual value return the lessor's investment I (cost or fair value plus initial direct costs): I = L a + R vⁿ, with a = (1 − vⁿ)/i for payments in arrears and (1 + i) times that in advance. The payment for a target yield is L = (I − R vⁿ)/a at the target rate per period. The schedule earns i on the outstanding net investment each period (the constant yield method), which runs from I down to R. After tax the lessor's flows are −I, the payments L(1 − T), the depreciation tax savings and the residual value less tax on R − book value. They are the lessee's incremental flows with the sign reversed when both have the same tax rate, depreciation and residual value, so that the lessor's net present value at r_b(1 − T) is −NAL: the tool's test script checks this symmetry. The default is the 36-month lease of the Federal Reserve guide (24,000 falling to 12,000 with payments of 488.53 by the constant yield method), whose implicit rate is 10.08 % with payments in arrears.

Lease accounting (lessee, ASC 842 in summary)

The description follows the staff guidance of the Office of the Comptroller of the Currency (Bank Accounting Advisory Series, Topic 3B); the standard itself, FASB Accounting Standards Codification Topic 842, is cited only. At commencement the lessee records a lease liability at the present value of the lease payments not yet paid, discounted at the rate implicit in the lease or, when that is not readily determinable, at its incremental borrowing rate, and a right-of-use asset equal to the liability plus lease payments made before commencement and initial direct costs, less incentives received. Payments here include the periodic payments with their fixed escalation, the price of a purchase option the lessee is reasonably certain to exercise and the amount probable of being owed under a residual value guarantee.

A lease is a finance lease when any one of five criteria is met: ownership transfers by the end of the term; a purchase option is reasonably certain to be exercised; the term is the major part of the remaining economic life; the present value of the payments and of any residual value guaranteed by the lessee is substantially all of the fair value; or the asset is so specialized that it has no alternative use to the lessor. The standard sets no percentages. The guidance gives "75 percent or more" and "90 percent or more" as examples, the thresholds of the earlier standard that practice has kept; the tool applies them and prints the measured ratios so that another threshold can be judged. OMB Circular A-11 (Appendix B) uses the same two numbers as fixed tests for the budget scoring of federal leases.

The liability is carried the same way in both classes: interest at the lease rate on the opening balance, less the payment. Finance lease: the expense is that interest plus straight-line amortization of the right-of-use asset over the lease term (over the economic life when the lessee will keep the asset), so it is front-loaded. Operating lease: a single lease cost, (total payments + initial direct costs + prepaid rent − incentives)/n, each period; the right-of-use asset is reduced by the single cost less the interest accrued, which brings asset and liability to zero together. Variable payments, remeasurement, modification, impairment, short-term leases and the lessor's accounting are outside the summary.

Limits

Taxes are simplified: one rate, deductions used when they arise, the gain on sale at the ordinary rate, the full last-year MACRS allowance (the disposal-year half-year rule reduces the deduction and the gain by the same amount), no limits on luxury vehicles or lease inclusion amounts (IRS Publication 463), no alternative minimum tax, and no test of whether the contract is a true lease for tax purposes. The equipment and premises cases loaded by the Analysis selector are illustrative; the lease-purchase, vehicle and lessor cases are the published examples named above.