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Estimating the call option values embedded in closed-end automobile leases.


There has been a great deal of research in the area of options and option pricing. (See Martin et al., 1988, for an excellent review of empirical tests of option pricing models.) Previous research suggests that the right to buy or sell an asset for a predetermined price at some time in the future has an explicit value (Tucker, 1991). Indeed, options are actively traded in order to maximize an investor's return (Kolb, 1991). While the trading is active, it is at best an infrequent occurrence for the average consumer or small business.

20 percent of new automobiles are leased

However, the average consumer or small business does purchase an automobile from time to time. In recent years more and more households are deciding to lease an automobile rather than purchase it. The number of cars leased by individuals has increased from the one million vehicles leased in 1975 to over 2.5 million in 1992 (Woodruff, 1991). On average 20 percent of the new automobiles today are leased. For certain luxury models leasing accounts for up to half of the cars on the road (Consumer Reports, 1993).

Auto leasing in an option context

One advantage of automobile leasing is the option to purchase the car at a predetermined price (referred to as the car's residual value) at the termination of the lease. This right to purchase the car at a predetermined price at the termination of the lease is analogous to a call option. The fact that the car may be purchased at this "strike price" only at the end of the lease makes the automobile lease an excellent example of a European option (Tucker, 1991).

The call option value in a closed-end automobile lease is transferred to the lessee for no explicit additional charge. The monthly payment is calculated to cover only the estimated decline in the value of the car. The present value of the lease payments does not include a charge for the right to purchase the car at the termination of the lease. The lessee does not have an opportunity to make a lower lease payment by foregoing the opportunity to purchase the car at the termination of the lease. As with any option, this right to buy at a predetermined price could be quite valuable even if the lessee does not desire to purchase the car at the end of the lease period. In the options market, an option trader need not buy or sell the commodity to capitalize on a commodity option, rather, the option itself is sold (Brigham and Gapenski, 1993).

However, the exercise of the option implied in an automobile lease is not as simple. Consider a car that costs $16,000, and at the end of the 36-month lease the lessee may buy it for $7,000. If the car has a market value of only $6,000 and the lessee turns the car in at the end of the lease, the option is worthless. However, if the car's market value is $8,500, it would be profitable to buy the car for $7,000 and then resell it at its market value. The option (as the right to buy) in this case is worth the present value of $1,500 in three years. The significant difference between a closed-end lease transaction and a car purchased with the intention to sell it in two or three years is who assumes the risk of the fluctuation in the cat's value? If the lessee had purchased the car in the example above with the intention of selling it in three years, the fluctuations in the car's value would be the purchaser's risk. The car may be worth $6,000, or it may be worth $8,500. In the case of the lease, the lessee profits if the cat's value is $8,500, and loses nothing if the value is less than $6,000. There is no risk to the lessee. More surprisingly, there has been no charge by the lessor for assuming this risk.

Research methodology

A judgment sample of national closed-end automobile lease programs on 1995 model cars was collected from advertisements found in The Wall Street Journal, the Washington Post and the Atlanta Constitution. To be included in the sample, the lessee had to have the option of purchasing the automobile at a specified residual value at the end of the lease. In addition, the model was required to have been marketed for at least as long as the lease program to establish a historical depreciation pattern. Several lease programs were deleted from the sample because they were new models and lacked a depreciation history. One car, the Nissan Pathfinder, had two separate lease programs in effect simultaneously. While the model was the same, one lease was for 24 months and the other lasted 36 months. This difference was assumed to be sufficient to differentiate those cars and treat them as separate models. Thus, 41 lease programs were included in the final sample, 12 programs by domestic car makers and 29 from foreign producers. The length of the lease programs ranged from 12 months (eight programs) to 48 months (two programs). The modal lease program was 36 months (18 programs).

Estimated minimum value of option

The estimated minimum value of the option implied by the lease is based on the pricing formula for European call options (Chance, 1992):

[P.sub.MIN] = Max[O, S - X/[(1+r).sup.n]] (1)


S = the asset's current value X = the option's exercise price n = time until option may be exercised and r = the discount rate.

Thus, the minimum price of a European option is either zero or the difference between the asset's value and the present value of the strike price, whichever is greater (Chance, 1992).

Equation (1) requires four variables to estimate the minimum value of a European option, the asset's current value, the strike price, the discount rate and the time until the option's expiration. In the context of a closed-end automobile lease, the strike price and the time variable are readily known. The exercise price is the residual value that is specified in the lease agreement. Similarly, the time variable is the specified length of the lease.

The appropriate discount rate should reflect the risk to the lessee that purchase option in the lease agreement will not be honored. Such an abrogation of the contract assumes that the leasing firm (normally the automobile manufacturer) will be bankrupt at the time the lease expires. Therefore, an appropriate estimate for r in equation (1) is the required return on debt with a maturity similar to the lease agreement, since both the lease and debt represent contractual obligations. Bond yields for high quality corporate bonds with maturities of one to ten years were examined in the Yield Comparisons section of The Wall Street Journal. For the week of 25 June 1995, the average yield was 6.5 percent. This was the estimate of r used in equation (1). The determination of an appropriate estimate for asset's current value (variable S in equation (1)) is not as readily apparent. Normally, an asset's current value is readily known. Furthermore, any dramatic changes in the asset's future cash flow are reflected in this current price, S. In the case of a leased car, however, such is not the case. The car's current value and its value at the end of the lease differ because of the fact that physical assets have a limited life and depreciate with usage. In an automobile lease, the present asset and the asset you have the option to purchase at the end of the lease are not the same. The car currently is brand new. The car you have an option to buy will be 24 or 36 months old. Thus, the question becomes, "How much will the car be worth at the end of the lease?" At the heart of this question is the car's capital depreciation rate. Therefore, the value that should be compared to the residual price is not the car's current price but an estimate of the car's value at the termination of the lease, or [S.sup.*].

However, the car's future value is unknown. The car's purchase price is clearly an inappropriate estimate of its value at the end of the lease because auto values typically depreciate dramatically. The best indication of a car's future depreciation is the model's recent depreciation pattern. This study assumes that the model's depreciation pattern for the last n years (where n is the term of the lease) will be reflected in the subsequent years. For instance, assume that a potential lessee is considering a three-year lease on a car with a manufacturer's suggested retail price (MSRP) of $15,000. Using the NADA Official Used Car Guide (sometimes referred to as the Blue Book), it is determined that three years ago the MSRP on this model was $11,750, and the car is currently worth $6,500, or 55.3 percent ($6,500/$11,750) of the original list price. Therefore, the expected market price of the car is 55.3 per cent of the original $15,000, or $8,298 at the end of the lease.

Two values for used cars

(The NADA Official Used Car Guide lists two values for used cars, a retail value and a trade-in value. Retail values reflect the average of the model's selling prices while trade-in values indicate the average value allowed on a new car when the present car is traded. In every case, retail prices are higher than average trade-in values, making the expected future value of the car higher if retail prices are used as a proxy for future value. Rather than attempting to determine whether the lessee would receive a price closer to the retail or the trade-in value if the car was sold privately at the end of the lease, an average of these two figures is used.)

Since the estimated value of [S.sup.*] is in the future, its value should be discounted to the present. The discount rate for the car's estimated value, k, depends on the particular auto model's variability around the mean value. The NADA Official Used Car Guide reports the mean, but not the standard deviation, of the cars' prices. However, the interest rates charged on used car loans is the discount rate the market considers appropriate for the risks associated with used car transactions. Therefore, this study will use the current market interest rate for used cars as the discount rate, k, for expected used car prices, [S.sup.*]. A national sample of financial institutions for June 1995, found the loan rate on used cars to be approximately 14.25 percent (O'Connell, 1995).

Therefore, the equation to estimate the minimum option value embedded in a closed-end automobile lease is:

[P.sub.MIN] = Max[0, [S.sup.*]/[(1 + k).sup.n] - X/[(1 + r).sup.n]] (2)


[S.sup.*] = the automobile's estimated value at the end of the lease

X = the lease-end purchase option price, or residual value

n = term of lease

k = the interest rate on used car's loans, 14.25 percent and

r = the short-term corporate bond rate, 6.5 percent.

Empirical results

Estimating equation (2) yielded surprising results. Twenty-four of the option values in the sample were positive. The option values ranged from $5,032 for the Nissan Pathfinder (36 month lease) to $384 on a Chevrolet Lumina. The residual value exceeded the estimated market value in the other 17 leases, making the option value zero. For the Mitsubishi Montero LS, the residual value was $5,015 greater than the estimated market value at the end of the lease.

Managerial implications

Having identified the option values in the closed-end auto leases in the sample, it is important to identify the source of this value. Theoretically, a lessee should pay only for the value of the car that is consumed over the life of the lease. Therefore, the present value of the lease payments (plus the down payment at the lease's inception, if any) and the present value of the specified residual at the end of the lease should sum to the car's current value. The present value of the payments and residual is often referred to as the "capitalized cost" in a lease. (This capitalized cost amount is analogous to the selling price.)

The existence of positive option values implies that the discounted residual value exceeds the discounted value of the car's expected market price. This means that the lease payments were too high and the lessee has paid for more than the amount the car depreciated. Unfortunately for the lessee, the only means available to recover this overpayment is to exercise the option, buy the car for the residual value and sell it at the higher market value. This recovered difference is the option value.

Leased cars are often returned to the lessor at the termination of the lease despite the existence of significant option value. One reason for this seemingly irrational behavior is that the lessee is unable to sell the car at the estimated market value (the average of the retail and trade-in prices) but rather faces a market where the car commands a price closer to the average trade-in value. This lower than expected market price may reduce the spread between the market value and the residual to an amount insufficient to compensate the lessee's time required to sell the car privately.

Since trade-in values on cars are averages based on model year and miles, leased cars offered privately should fall into a "better than average" category, or else the lessee would have returned the car to the lessor at the [TABULAR DATA FOR TABLE I OMITTED] termination of the lease (Akerlof, 1970). However, auto buyers do place a premium on buying a used car from an established dealer, with a business reputation, as opposed to the private seller. With the dealer, you do receive some type of warranty, even if it is implied only by the dealer's reputation. With the private seller, it is the most basic situation of caveat emptor (Rosenman and Wilson, 1991).

Having explained the positive option value, how can a residual value greater than the expected market value be explained? This is a situation where the car has depreciated to a level less than the residual value. The present value of the lessee's monthly payments is less than the decline in the car's value. This does not necessarily imply irrational behavior (not value maximizing) or a mistake by the leasing company in pricing the lease. A more likely explanation is that the advantageous lease terms were offered on a model the automobile company was anxious to sell. To stimulate sales, an automobile manufacturer could reduce the price or offer more attractive terms in lieu of a price reduction. Oligopoly theory suggests, and recent history indicates, that auto companies are reluctant to reduce prices (Scherer, 1990). However, rebates, reduced financing charges and special prices on optional equipment packages are a common means of non-price competition among automobile producers. Offering advantageous lease terms is merely another means of competing on a non-price basis that allows the manufacturer's suggested retail price to remain the same.


In option markets, high option values correspond with option contracts most likely to be exercised. This would suggest that only those cars with high option values should be leased, especially since the option is transferred to the lessee at no explicit cost. However, the option value of a leased car results from its value at the end of the lease being more than the residual value. This implies that the lessee actually bought the option by paying too much over the course of the lease, since the lease payments should have covered only the decline in the car's value. The only way to recoup this overpayment is to buy the car for the residual value and then resell it. However, exercising the option to buy the car in order to recoup your option value entails the significant transaction costs associated with a private sale. If your goal is to pay no more during the lease than the amount the car depreciated, then the cars to lease are those with an option value of zero.

An additional application of these results is that a potential lessee may want to negotiate the residual value with the dealer. If the residual value is a smaller percentage of the MSRP than the used car depreciation values currently exhibit, then the lessee should challenge the lessor's depreciation assumptions on that particular model. By negotiating the residual value to reflect more closely the expected value at the end of the lease, the potential lessee avoids having to "exercise the call option" (buy and then resell the car) in order to recover any overpayment.

The author gratefully acknowledges the support received from Dean Carl Gooding and the College of Business Administration's Summer Research Grant Program at Georgia Southern University


Akerlof, G.A. (1970), "The market for 'lemons': quality, uncertainty and the market mechanism", The Quarterly Journal of Economics, Vol. 84, pp. 488-500.

Brigham, E.F. and Gapenski, L.C. (1993), Intermediate Financial Management, Dryden Press, Chicago, IL.

Chance, D.M. (1992), An Introduction to Options and Futures, Dryden Press, Chicago, IL. Consumer Reports (1993), "Should you lease?", Vol. 58 No. 4, April, pp. 204-6.

Kolb, R.W. (1991), Options: An Introduction, Kolb Publishing, Miami, FL.

Martin, J.D., Cox, S.H. and MacMinn, R.D. (1988), The Theory of Finance: Evidence and Applications, Dryden Press, Chicago, IL.

NADA Official Used Car Guide (1995), National Automotive Dealers Association, McLean, VA, July.

O'Connell, V. (1995), "The best bank in America", Money, June, p. 126.

Rosenman, R.E. and Wilson, W.W. (1991), "Quality differentials and prices: are cherries lemons?", The Journal of Industrial Economics, Vol. 39, December, pp. 649-58.

Scherer, F.M. (1990), Industrial Market Structure and Economic Performance, 3rd edition, Houghton Mifflin, Boston, MA.

Tucker, A.L. (1991), Financial Futures, Options and Swaps, West Publishing, St. Paul, MN.

Woodruff, D. (1991), "Why auto companies can't win for leasing", Business Week, 20 May, pp. 104-5.

Executive summary and implications for managers and executives

Leasing, honesty and the car salesman

(A precis of the article "Estimating the call option values embedded in closed-end automobile leases".)

Buying a car is, for most people, the second biggest purchase of their life. Only the buying of a family home is usually larger. Over time the market has sought to make the purchase of a car as easy as possible. Moreover, there is a realization that the only way to sustain the car market at the size it is currently is to enable people to buy a car without having the entire cost available today.

As a result a multitude of financing schemes are available from traditional hire purchase, through more involved finance packages to the leasing schemes considered in Professor White's article. Most of these schemes are intended to get the prospective buyer on the road and feeling that a good deal has been struck. As Professor White shows, leasing schemes are no cheaper than hire purchase and, it could be argued, confuse the consumer by presenting them with a possible asset (and associated profit) at the end of the lease period. In reality the best lease is one where the entire risk lies with the owner (i.e. the dealer running the contract) and the lessee does not pay over the difference between purchase price and value at the end of the lease. Professor White finds that this is not usually the case, suggesting that, in many instances, leasing options are unattractive to customers.

So why have these leasing contracts arrived and succeeded? Is this just another indication of the gullibility of consumers or is there a gain over and above the actual asset value? Without examining the details of the leases considered in this article it is difficult to provide an answer. I suspect that part of the answer lies in the re-lease option incorporated in many of these schemes rather than the residual value. If the buyer is happy with the manufacturer the option to convert the car's residual value into a new lease is an attraction.

For some people leasing makes sense. It emerged from the business market where the ability to reduce the capital commitments implicit in a car fleet (or indeed a range of other equipment) made leasing attractive. For the ordinary individual it presents little benefit, costs as much as hire purchase and does not really deliver the benefits at lease end that are implied in the contract. Car dealers are probably not being disingenuous in selling lease contracts and some may convey genuine advantages to buyers, but consumers should be aware that the deal is often less attractive than it seems.

Since, given the price of cars these days, with financing packages essential to the new car business, it is worth asking what steps dealers and manufacturers should take to ensure that consumers are aware of the implications of different financing options. Presenting different options (such as leasing and hire purchase) to consumers, revealing the extent to which the dealer is rewarded for selling a finance package and not penalizing people who arrive with finance from a third party source all stand out as possibilities. There is absolutely no reason other than the history of motor car sales that suggest dealers will deceive buyers to secure their own best deal. For manufacturers one option is to consider changing the dealer relationship or even doing without dealers altogether (as Korean manufacturer Daewoo has done in the UK). Whatever happens, the need to deliver consistent sales volumes so that manufacturers can operate production lines at levels approaching full capacity, will affect the way in which cars are sold. Sometimes this presents an advantage to the consumer by placing a premium on the trade-in. At other times the reverse is true.

Professor White provides a useful insight into one small part of the motor dealers' business. As ever this shows that the schemes on offer for private individuals to lease rather than buy are sometimes less attractive than they might at first seem. Consumer finance can prove a nightmare even when legislation exists forcing providers to be more explicit about the contracts offered. I suspect that leasing, like other approaches before, will be endlessly tinkered with to the extent that comparing contracts between different providers becomes nigh on impossible.

(Supplied by Marketing Consultants for MCB University Press

John B. White is Associate Professor of Finance, Georgia Southern University, Statesboro, Georgia, USA.
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Author:White, John B.
Publication:Journal of Consumer Marketing
Date:Jun 22, 1996
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