A model of a lessor

O M E G A The Int. JI of Mgmt ScL Vol 10, No 4, pp 413 to 431, 1982

0305-0483/82,040413-19~t03.00/0 Copyright ~ 1982 Pergamon Press Ltd

Printed tn Great Britain. All rights reserved

A M o d e l of a Lessor R FLAVELL GR

SALKIN

Imperial College, London, UK (Received January 1981) The lease-or-buy decision has been widely considered. This paper examines the reverse side of the situation, namely from the lessor's point of view. Which leases should be accepted or offered, how should they be funded, what decisions should be made if taxable capacity runs out, what is the impact of gearing restrictions or lessee defaults? These are just some of the questions examined below. The paper describes the operations of a lessor, its problems and possible interactions. Based upon this description, a computer simulation model is described and some example results

presented.

INTRODUCTION CONSIDERING ONLY financial leases, t the growth of the leasing industry in the UK over the last ten years has been very large indeed, from £20M of new business in 1967 to £2360M in 1980 [1]. Leasing thus represents a significant proportion (an eighth in 1980) of the new investment funded externally to the company taking place in the UK economy. The business is not restricted to the private sector, a number of major transactions have occurred in the public sector. At the present time in Britain, leasing combines a number of attractions as a financing medium for manufacturing companies. Many companies pay little or no corporation tax as a result of stock relief. If they therefore undertake the purchase of capital equipment themselves, they may not be able to absorb the capital allowances. Secondly, when an asset is purchased it appears on the balance sheet of the company; similarly, if the purchase has been t A broad definition of a financial lease, as opposed to an operating or maintenance lease, would include a noncancellable fixed term based upon the economic life of the asset and rentals that recovered the initial capital investment but do not generally include any form of operating expenses. 413

financed by loans to the company, the liabilities would also have to be displayed. As a result, all other things being equal, the company's published gearing ratio has deteriorated. If however, the equipment is obtained under a leasing arrangement, neither the asset nor the liability aspects of the arrangement appear on the balance sheet. For a company that is already very highly geared, this is obviously an additional attraction. This situation may be shortly changed however if the recommendations of the Accounting Standards Committee that leased assets should appear on the balance sheet with an off-setting loan on the liabilities side are adopted. Finally, because the lessor company retains the ownership of the asset, and hence is entitled to the ensuing capital allowances itself, the rentals charged can be extremely attractive. It is unlikely that most lessor companies would be able to absorb these capital allowances in their own taxable profits. In the UK, the lessors have tended to be subsidiaries of banks and finance houses which generate considerable profits. Under group relief provisions, these profits can be used to absorb the allowances. The funds needed to support the lessor's operations are also more easily obtained if the

Flavell. 5alkin--A Model of a Lessor

414

parent is a bank. However, most banks place an informal limit on the gearing ratio of their leasing subsidiaries so that the maintenance of the 'right' capital structure is a factor when considering new business. The gearing restriction is perhaps not as important as the provision of allowances but it does involve the interaction of accounting conventions with practical economic considerations. The equity part of the gearing ratio will contain the net profit figure for the year as well as undistributed profits brought forward from previous years. This raises the interesting question of just how the profit from a lease can be measured. If the lease is completed within a single accounting period, the 'actual profit' is P = total rentals - depreciation of asset interest on capital - administrative costs -

ignoring, for simplicity, taxation effects. If the lease extends over more than one period, the 'accounting profit' in the first year depends upon the method adopted for dividing the total 'profit' over the life of the lease. There are three main areas for discussion: first, the depreciation policy has an obvious and immediate impact; this is discussed in much greater length below and in Appendix 1. Second, the determination of the rate charged for capital tied up in the lease is complex; three alternatives come to mind: (i) the 'cost of capital', which for an independent company would be some weighted average of equity capital (capital plus retained earnings) and fixed term loans, (ii) a rate 'handed down' from a parent, (iii) an 'opportunity cost of funds', based on an averaging of LIBOR over the period concerned. Third, how uncertainty, especially in the form of the lessee defaulting, would be taken into account; one suggestion might be to use a risk-adjusted interest rate, another is discussed below. If steady profits have been earned over a number of years, the method employed for taking profits over the life of leasing contracts would average out. But if the new business taken on fluctuates from year to year, there would be no consistency in results and over the short period of a year, the reported profits would depend upon the accounting method used. This would have an impact on the equity figure presented in the balance sheet and thus on the company's ability to raise fresh funds.

The whole argument on the accounting treatment of leases has tended to concentrate on individual leases. The treatment of the 'book' of leases in the total context of the leasing company has been ignored, which is hardly surprising considering the difficulties involved, and consequently the pattern of business being written is also ignored. For example, in a cyclical economy, taking profits early using an individual method when business is increasing will amplify the cycles considerably. Present management might well recognise the underlying variation and make provisions to reduce its impact, but this can only be done by adopting a book approach. It would be desirable to examine the different accounting conventions in the context of a working leasing company, subject to different interest rate structures, different business situations in terms of both quality and quantity of business available, different amounts of tax-shelter profits, and so on. The various profits accrued under each convention could be examined, say, looking at average profit and the variation over the horizon period, to identify the 'best' convention. 'Best' here is, of course, highly dependent upon the various priorities inside the company; perhaps an ordering of the conventions might even be possible. Although the discussion to date has centred upon the accounting conventions, the aim of this paper is to describe a simulation model of a leasing company. The model incorporates all the typical decision variables that management uses, and has been 'worked" under a wide variety of different environments. The objectives of creating such a model should be obvious. Apart from investigating the impact of different accounting conventions as discussed above, it would enable a lessor to examine, for example, the effect of different assumptions about future yield curves on his business and more specifically on his future profits, the effect of different gearing assumptions on his ability to withstand lessee defaults, the effect of going for a few large secure 'tickets" rather than many small leases with the concomitant impact of reducing future flexibility, the effect of adopting an aggressive accounting scheme that takes profits early and so allows rapid growth through internal funding against undesirable future changes in the environment.. The list of effects that may be

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line and reverse-sum-of-the-digits methods are similarly explained. The final method is called the actuarial method and it operates in a slightly different fashion. Using equation (1) above, it first calculates that value of i, which would, after the final tax payment relating to the lease, result in a net investment of zero. This value is termed the gross earnings rate (GER) and gross profits are taken by the application of this rate on the periodic net investment in the lease. The actuarial method is essentially a form of internal rate of return calculations, with the G E R being compared to the ACCOUNTING METHODS hurdle rate of borrowing, and as such may be At the heart of any model of a lessor regarded as the 'theoretically' correct method company must be an accounting method. of calculations because of its internal consistThere is, at the moment, a wide variety of dif- ency and lack of arbitrary apportionment ferent accounting methods employed by lessors scheme. See Appendix 1 for a brief example of but this variety is decreasing due to the activi- these methods. ties of the various professional accounting and Actually, the above discussion is too simplisleasing bodies and the fiscal authorities. Each tic for practical use. Invariably there is in force method is based upon the same set of account- a dual rate, i.e. one interest rate for borrowing ing calculations, i.e. and a second (lower) rate for lending when the net investment is negative. But the nature of N I t = N I , _ ~ - R, + 7], + i , N l , _ : (1) the calculations is not altered, except perhaps for the actuarial method where the G E R is when NI, is the net investment in the particular only applied at rests with a positive net investlease by the lessor at the end of time period t, ment. Because of the problems of forecasting R, is the rental received during this period, T, is suitable interest rates over the term of the the tax payment that has been made (treating lease, many lessors are now adopting a variable allowances, etc., as negative payments) and i, is rate system. This is a system in which the lessor an interest rate chargeable on the monies tied ! may vary the rental charges at relatively short up in the investment. (Note that this simplified notice due to specific changes in some base equation assumes equal borrowing and invest- rate. The system is however very unpopular ment rates of interest, which is, of course, not with lessees as it removes one of the major generally true in practice--see below.) attractions of leasing, namely a deterministic The variations in the methods arise from the rental scheme and hence fixed cash liabilities. different ways profits are taken on the lease, The lessors are basically passing some uncerand hence the ways in which the depreciation tainty in their business down to the lessees allowance is determined. For example, the without any real recompense. investment period method (IPM) favoured by The technical variations between the alternathe ELA takes profits in direct relationship tive accounting methods has already been disfrom the inception of the lease to the time cussed, namely that they take profits at differwhen the net investment first becomes nega- ent rates. The I P M and actuarial method take tive; thus, the I P M apportions gross earnings profits very rapidly, over the investment period (i.e. gross of interest) over the accounting and slightly longer respectively, whilst the periods in the ratio of the net investment at other (hire purchase) methods take profits each rest to the total net investment at all rests more slowly and over the entire term of the in the investment period. Alternatively, the lease. Considering these three methods alone, sum-of-the-digits method (primarily used in the sum-of-the-digits takes profits most rapidly, hire purchase accounting) distributes the earn- then the straight-line and finally the reverseings arithmetically in proportion to the outsum-of-the-digits (see Appendix 1). In any disstanding term at any given date. The straight cussion on the applicability of these methods

examined is virtually endless and it is not the aim of this paper to attempt to cover or even discuss many of them. Suffice it to say that the paper describes a model that will enable such effects as identified by the lessor to be examined easily and rapidly at a computer terminal. Some example results are provided to indicate the impact that changing a single parameter can have on the performance measures of a lessor and a small simulation is described in some detail.

416

Flavell. Salkin--A Model of a Lessor

to lease accounting, two opposing forces must be borne in mind. The first is that a lease will typically suffer from higher interest charges at the beginning and thus the argument is for a method that will approximately match these charges and release proportionally higher gross earnings early in the life of the lease. The second is the effect of default on the lessor who, if he has taken the profits early, has to fund the depreciation account from another source. Such defaults can have a catastrophic effect on the lessor if sufficient provision has not been made for the eventuality. Indeed, whilst lessors dealing with relatively small leases may well use the IPM, lessors dealing with a small number of big leases tend to adopt a method which takes profits much more slowly and may even use an 'individual" method, taking a view at each accounting date as to how much profit should be released. LOAN CAPACITY In this section, we will make the assumption that the lessor has no activities other than leasing and therefore the financing of new assets must be done either from retained earnings or from loans. If the lessor is independent of any other organisation, then these loans must be raised on the market; if on the other hand the lessor is a subsidiary of an institution, then such loans may be raised on the parent. The ability of a lessor to raise money on the market is of course based on his financial position and in particular his ability to service and repay the loan on time. The information required by the term lenders is much and varied but would all be aimed to ascertain this ability of the lessor. As a simple surrogate for this information, we suggest the use of the gearing ratio which would generate an upper bound on the lessor's borrowings. The loan capacity to raise monies on the market is therefore determined by the retained earnings (equity) that are available for investment in new assets. It may well be argued that lenders would attempt to look behind any declared figures and through any accounting convention to the actual cash flows themselves, thereby nullifying much of the above discussion. This is indeed true, but are the lenders very much the wiser about the likelihood of the lessor defaulting'?

We would suggest not, and use the gearing ratio as an easily understood proxy for more complex risk measures. Actually. a much more likely scenario is that the lessor is a subsidiary, and that gearing is used as an internal control mechanism over all subsidiaries. In practice, active restrictions on borrowing through gearing tend to be measures of last resort to prevent overstretching. As discussed above, the split of the rental payments into depreciation and ultimately, retained earnings for any individual lease depends entirely upon the accounting method adopted. It is self-evident that, if the earnings are taken early, the loan capacity of the c o m p a n y is increased in the proportion of the gearing ratio and the absolute size of the business may be rapidly enlarged. The corollary to this action is that the depreciation account, and hence any loans that were raised to finance the purchase of the original asset, is satisfied only by the later payments. The immediate consequence is that the lessor would be very vulnerable to a default by the lessee, because of his geared position and commitment of his equity if the rate of new business has slowed down. This shows probably the main reason why the I P M is unpopular amongst some less o r s - t h a t it is not conservative. A further consequence may well be that of exposure. At a point in time, the lessor possesses uncommitted earnings on the strength of which he raises a term loan(s). Such a loan(s) is highly unlikely to match, in both size and term, the cash flow generated by an individual lease. One could think of a number of reasons for this mismatch: e.g. the money required by the lease is too small to be raised economically; the pattern of flows from the lease is not acceptable to a lender, or the term of the lease is non-standard. Actually, of course, it is theoretically possible to raise the precise loan required but the interest charges would certainly reflect its individuality (and perhaps the liquidity problems of the lender!). Thus, in general, the cash inflows from the leases do not match the cash outflows to the loan repayments. This mismatch may have serious consequences for the lessor. If he is exposed or under-covered, i.e. the cash inflows are not sufficient to meet all the loan repayments so that a fresh loan has to be raised (or part of the existing loan rolled over), then he is highly vul-

Omeqa, Vol. 10. ,%'0. 4

nerable to rises in the interest rates. A converse argument may be applied when he is overcovered. The exposure position can only be assessed first, by looking at the entire portfolio of leases and its associated term loans, second, by taking a view on the future movements of interest rates, and third, after deciding upon the value of time. This last point implies that an exposed position in the future may be of 'less" concern than a similar position "now' because it may be possible to cover oneself in the meantime; in this case, how much is 'less'? A different view of the mismatching of loans and leases can be obtained via duration. The lessor can consider the duration of his leases and the duration of his loans and the differences in the respective durations can be related to his planning period. If these are equal then the lessor will be immunised against small changes in interest rates and the lessor has a benchmark for measuring his exposure. For if the net duration of assets and liabilities is not equal to the planning period he is acquiring an exposed position. If more complex forms of duration (i.e. other than Macauley's definition) are used in the calculations it is possible for the lessor to immunise himself against more complex rate movements.

217

IO 9

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TAXABLE C A P A C I T Y

2

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The fundamental rationale for leasing, namely the ability to transfer taxable capacity between two parties to their mutual benefit, has already been noted above. In this section we wish to examine the problems of a lessor when he has less than an infinite amount of capacity available. In this situation of course, the basic problem is to build up a portfolio that may make 'best' use of a scarce resource. To identify the position precisely, since 1972 the capital cost of a lease has been an allowable expense against taxable profits. If insufficient profits (capacity) were available at the initial accounting date, then allowances could be spread over succeeding years, according to an agreed formula and provided that the capacity was then available. Naturally, because of the time value of money, spreading the allowances either increases the rentals and makes the lease less competitive, or decreases the profitability of the lease, or some combination thereof. If one moves from looking at an individual 0~[- I0 4--F

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(a) Zero dilution; (b) 16.67'11odilutiot~; tion: (d) /~P~ dilution.

Icl 20~i;

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lease to a consideration of a portfolio, then the decision problems of the lessor are increased enormously. Figure la shows a typical situation; there are a dozen leases, each of which requires one unit of capacity, and vet there are

-ll 8

Flacell. Salkin--A Model of a Lessor

only ten units available. Now the various alternatives open to the lessor may be partly explained by the term dilution. Dilution implies that the initial 100,°Jo take-up of the capital allowances by the first ten leases (it is assumed that the ordering of the leases is chronological) is reduced by a certain percentage and that the outstanding allowances spread if possible. For example, a dilution of 16.67~o (one-sixth) implies that, if each of the twelve leases took up 83.33~o (five-sixths) of the capital allowances initially, a total of only ten units of capacity would be required (see Fig. lb; the shaded area is the superimposition of Fig. la). Now, if a dilution of 20~o were permitted, then this implies that the last ten leases would only require eight units of capacity between them and hence the first two leases could be totally undiluted. Conversely, if a dilution of only 10~ were allowed, this would only release 1.2 units of capacity for spreading on a future date, and would require a total of 10.8 units. Thus, one of the leases would have to be dropped, leaving a total requirement of 11 units which, at 10~ dilution, implies an immediate take-up of only 9.9 units. The spare 0.1 unit may be given to the first lease, as described above. Notice that in fact the percentage of dilution is a continuum; 0~o dilution implies that the lessor closes his book when he runs out of capacity, whereas at the other extreme, 100~o dilution implies non-tax-based leasing. Perhaps a common maximum dilution is 10~, with this amount then being spread over successive years. Lessors do have alternative strategies that they can adopt as the taxable capacity becomes scarce. The most practical alternative, ignoring such things as off-shore or cross-border leasing, is the 'purchase' of taxable capacity from a third-party company; this is frequently called 'management leasing'. Here the lessor acts as a middle man, arranging all the documentation for the purchase of an asset by a company and its subsequent leasing, through the lessor, to the lessee. The 'purchase' price of the capacity is negotiable but it is generally between 15 and 20~o of the size of the capacity. The selection of the third-party company is, of course, complicated; parameters include mature profits structure and yet not too large (so that it would not be tempted to start leasing in its own right!). Cross-border or cross-currency leasing is a

possible growth area, but it is fraught with additional difficulties such as taxation structures and currency movements. So far in this section we have looked at the various alternatives open to a lessor as he starts to run out of taxable capacity. There is, however, an associated forecasting problem. A lessor, accepting business throughout a particular financial year, raises capital allowances that are to be taken at the appropriate accounting date after the end of the year. Simultaneously, the lessor (and his parent institution) is earning taxable profits and hence generating the capacity out of which the allowances are to be taken. The true (or at least published) capacity is only known after the end of the financial year, and yet the processes have to go on in parallel. Thus, forecasts of this capacity have to be made throughout the year, and the lessor must attempt to match his business with these forecasts. A further source of uncertainty, i.e. over and above the general business risk, is that the end-of-year windowdressing of the accounts that is usually performed can make a mockery of the forecasts. Hence the lessor could find himself either desperately overstretched or with a relatively empty book at the year-end without this state of affairs being a true reflection of his business acumen. This problem may disappear to some extent, at least for the larger leasing companies, because lessors are generating such large capacities themselves that they are unable to use up their parents' capacities as well. DEFAULTS Defaults or bad debts will occur in any form of business in which credit is given. Leasing is, not unnaturally, no exception and defaults can have a widespread and serious effect on the performance of the lessor. Imagine the situation when a lessee does not pay the rentals. The lessor may be able to recover the asset and either sell or release it, generally at a considerable discount. The effect of this is to remove a steady cash inflow and replace it by either a single inflow (from the sale) or a much reduced income stream (from the release). On the other side of the business, the loan repayments and interest charges still have to be met. Thus the business has been relatively undercovered not only to changes in the interest rate but also to

Omega, Vol. 10, No. 4

the existing rate itself. A further complication depends on the accounting method used; if the profits have been taken early, and then the lessee defaults, provision has to be made to complete the depreciation account. Because the default must ultimately reduce the profitability of the business, the loan capacity (and hence future growth) is effected through the gearing ratio. Some lessors, namely those with a small number of large leases on the book, only operate with prime clients so that defaults are unknown to them. But for other lessors, those acting more as finance companies with a large number of relatively small leases, the problem is potentially more acute and active attempts have to be made to reduce the percentage of defaults. A M O D E L O F A LESSOR The major aim of this paper is to describe the construction of a simulation model of a lessor, incorporating all the features described above plus some minor ones. Figure 2 is an outline flowchart of the model and will be used as the basis for the description. Before starting on the details however, it is probably worthwhile to discuss the principal methodology behind the model. Appendix 2 describes the theoretical development of the principal relationships contained in the model. The simulation assumes that an existing lessor has available to it a stream of future leases. TABLE I. INFORMATION NEEDED ON EACH LEASE AND LOAN

For each lease:

cost of lease acquisition date length of primary rental period overall probability of default status of lease, i.e. loss-leader or not number of first type of instalment months between each instalment size of each instalment number of instalments needed as initial down payment number of second type of instalment months between each instalment size of each instalment salvage value at end of primary period For each loan:

size of loan starting date term repayment schedule interest rate, i.e. either fixed or a margin above LIBOR

419

TABLE 2. KEY EXOGENOUSLY PROVIDED PARAMETERS

Accounting method Tax capacity option Exposure limit Maximum acceptable gearing ratio

Depending upon its current and perceived future portfolios of leases and loans, the lessor has to decide which of the new leases it wishes to accept. This is done by calculating the future expected profile of the company on the basis of no new business and then selected new leases are added, together with the concomitant new loans, to arrive at a desirable new profile. Because many of the calculations are complex, the selection is done on the basis of partial enumeration rather than with the aid of any sophisticated optimisation technique. Once the selection has been made for some future period, the accounting procedure is run with the new leases and loans coming on-line automatically. I. Inputs to the model

There are three sources of information. The first is the existing status of the firm, with details of the existing book of leases, the outstanding loans, the outstanding capital equity estimates of the taxable capacity available from other operations, maximum acceptable gearing ratio, liquidity requirements and so on. See Table 1 for a list of the precise information stored on each lease and loan. The other two are the new business available to the lessor, in the form of a long stream of possible leases plus an estimate of the future state of the money-market, and some exogenously controlled parameters set to control to the lessor's operations. These parameters are specified in Table 2. Other information also provided includes negotiation costs as a function of size of lease (as suggested by a leasing company), fixed overheads, recoverable percentage value of asset if sold after repossession, minimum size of any loan plus feasible increment, lag between year end and tax due, proportion of capital allowance that may be spread over future years and so on. 2. Initial analysis o f the new business

The new leases are first sorted into chronological order of their starting dates: this is

Flavell, 5alkin--A Model of a Lessor

420

Read

in STATUS Of f i r m

Read

in new business

Read

in available

Request

under o.=fer

loans and yield curves

inter-active

information

q T

2 Arrange order

new business

Calculate business

negotiation

Calculate default each rental

3

in chronological

Raise predetermined remove old ones

cost of new

probabilities

Repay

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instalments

Raise new Ioan~ if necessary existing business

on

Calculate Calculate expected actuarial JAR) of each lease

term loans and

existing

to cover

loan profile

return Perform monthly existing book

Calculate estimated cost of capital and reject all business with MR < this cost (except loss leaders). The leases remaining form a feasible set

Collect

calculations

on

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Calculate

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Calculate allowances due [modified by dilution if necessary) Calculate apportion

interest payable and to individual [eases

Store periodic Adjust equity

net investment base

4

Looking ahead a short time, identify the portfolio of new business and concomitant new loans that, in combination with the existing business, satisfies the exposure constraint and maximises the net profit

When the end of an accounting year is reached calculate the required taxable capacity for all the accepted leases in the year. If the capacity is insufficient, modify the computations on the basis of the option selected

NO

the required period?

t'YES • End of selection Start accounting Produce a monthly of business Produce account

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FIG. 2.

f!nal STATUS Of firm

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Flowchart of the leasing model.

simply to aid searches in the future. The negotiation costs, if not provided with the lease, are estimated; these costs may be added to the initial costs as it is an accepted principle by the

ICA that such costs may be off-set against earnings, or spread according to some formula. From the overall default probability associated with each lease, a default probability on

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(usually normalised) subjectively determined weight. There are two alternative views: the first that the weights should decrease into the future because the company has time to adjust its position, and the second that as the future is increasingly uncertain the c o m p a n y should not take a position. We prefer the first argument and, in the model, use a negative-exponential curve to generate the weights. More complex formulae are also possible (e.g. weighting outflows more heavily than inflows, restricting the m a x i m u m exposure (or cumulative exposure) in any period, etc.) and easily implemented. This section of the model is perhaps the most time-consuming. Selected leases from the offered new business are added to the existing business and the necessary loans raised. The exposure and net profit from this combination is calculated. The selection is raised by (almost) total enumeration until a final new portfolio is found that satisfies the exposure limit and maximises total net profit. Within the model is the idea of selection periods and overlap periods, each of which m a y be endogenously set. To illustrate, suppose that the selection period is set to 6 months and the overlap 3 months: the model would consider its selection of leases in months 1, 4, 7, 10. . . . . and would then look at all leases starting in the next 6 months. For each possible selection, and concomitant loan structure, the effect on the exposure would be calculated over the entire lifetime of the present business plus the new selection. This approach was adopted as a compromise between con3. Estimate existing lease and loan profiles sidering all possible selections which would This section is primarily book-keeping. have been far too time-consuming and not Matured loans are removed and pre- looking ahead at all but only examining those determined and necessary new loans raised. leases starting in the current rest period. This Any loan instalments due are paid and the last method would result in a very inflexible monthly net investments made in the current and short-sighted policy. It is implicitly assumed at this point that the book calculated. most desirable value for the exposure as calcu4. Satisfy the exposure constraint lated by equation (2) is zero. Of course this may not be so in practice because, as discussed The principle of exposure has already been above, it is the .total organisation's exposure discussed but to set a constraint on it implies that is important and a non-zero position may that it must be measurable. There is no single accepted measure; all measures are based upon well be desirable. It would be extremely simple to modify the program accordingly. the following formula

each rental payment is calculated on the basis of a geometrically decreasing series. The thinking behind this is that the more rentals that are successfully paid up, the less likely the lessee is to default. This assumption is also conservative and adopted by many leasing companies. Finally, the expected actuarial return of each new lease is calculated and, as an initial hurdle, the lease rejected if the return does not exceed the estimated cost of capital. The calculation of the cash flows for the AR is detailed in Appendix 1, if a dual method is used then when the lease is in surplus an exogenous current deposit rate is used. The cost of capital is based on a weighted average of the current yield curve and the current cost of equity. The weights change as the model progresses, reflecting the balance of outstanding loans and equity actually drawn down, at any point in time, but the rate used in the lease selection at a single moment depends only on the current book and not on possible future business. As an estimate for the cost of equity, the model takes the current deposit rate: the idea is that this is the certain rate that could be obtained on the equity if it were invested elsewhere. Alternatively, if the lessor is a subsidiary, the deposit rate may be an internal company transfer rate reflecting the current desire for funds elsewhere in the business, and that the equity is a limited source of capital seeking the best place for internal investment. For a loss-leader of course, this hurdle is ignored.

exposure = ~ w, lo, - i,[

(2)

t=O

where o, and i, are the expected cash outflows and inflows respectively in period t and w, is a

5. Taxable capacity At the end of each accounting year, the required taxable capacity is calculated for all the newly accepted leases. If there is insufficient

422

Flarell. Salkin--A Model of a Lessor

capacity available from the parent plus the existing business then one of the options described above has to be adopted and the portfolio modified accordingly. If the full capital allowance cannot be taken up in the first year and the dilution option is adopted, the residue is spread over the succeeding four years according to an exogenous formula.

The ultimate purpose of this model is to provide the opportunity to gauge the effects of different operating policies very quickly. This Would be done by simulating the lessor over a number of years under alternative policies and then comparing the various financial results. We present in this section one such simulation run together with extracts from the results. In this simulation, the lessor was started up from 6. Monthly calculations scratch, offered leases unevenly over about four Sections 2 and 5 are repeated for a pre-speci- years and then allowed to run down again, fled number of years. At the end of the time thereby demonstrating phases of growth, periods, all the desirable new business has been steady business and decline. The lessor initially had equity stake of identified. The final major step is now to start again from the beginning and do all the £I0,000 together with a gearing restriction of accounting calculations, bringing the new 5:1. There was a taxable capacity of £50,000 leases and loans on stream as already identified p.a. available and equal dilution on all the leases was opted for if this capacity were to be and performing all the other routine transexceeded. This is one of the default options, actions. with any diluted allowance spread in a declinVarious reports are routinely produced: ing fashion over the next four years. Annual yield curves were supplied along with a varying - - a monthly cash flow summary, deposit rate. Some fifty leases (all obtained --individual Profit and Loss (P & L) accounts, from Lombard North Central) were offered, with a repossession account if the lease with lifetimes ranging from 2 to 5 years and asset costs from £1000 to £15,000. The investdefaults, ment period method was selected with a maximum permitted discounted exposure of - - a n n u a l company balance sheet. £10,000. Table 3A shows the P & L accounts over Finally, a status report of the final position of the company is produced. This is in the form eight years. Notice that, as expected with the ready to be fed back to the model as an open- t P M , gross earnings ceased near the end of the ing position for the next time period. Thus period as the net investment (in leasing terms) quite long spans of time may be modelled by became negative and this is reflected in the bottom line which shows a loss in years 5 and this method. 6. Table 3B shows the balance sheets, where it can be seen that a single 5-year long-term loan was raised, together with a number of shortSOME E X A M P L E RESULTS term and overnight facilities. The tax reserves In a paper such as this, there is little merit in shown are to be paid 15 months (another parproducing an enormous amount of numerical ameter) after the end of the relevant financial output; it is the model that is of prime import- year--see Table 3c for further details. The ance, any results are inevitably highly selective actual assets acquired rise to a peak and then and data-dependent. In the sections above, the decline as they are salvaged. A minimum cash broad inter-relationships between the key par- level is kept and all other unused monies are ameters were discussed, e.g. the impact of the shown in short-term deposits. The current form accounting convention adopted on the bad of the model is as a money pump, i.e. it pays no debt incurred, etc. The discussion was on a dividends to shareholders or any parent, but it qualitative basis and in this section, we present would be easy to superimpose some policy. some quantitative evidence. First, a simulation Table 3c is an example of the monthly calcuof a lessor is described and then some com- lations that are performed. This particular parative results when the most important par- month was selected to demonstrate also the tax payments being made on revenues received in ameters are changed.

423

Omega, Vol. I0, ,~o, ,~

TABLE 3A. SUMMARYP & L ACCOUNTSFOR SI,~LATION Year Rentals Less depreciation

1

2

3

4

5

13,345.03 24,458.42 40,543.40 39.486.42 8793.80 14,539.38 27,422.55 29,173.68

Gross earnings

4551.23

9919.04 I3,120.85 10,312.74

Less negotiation cost Loan interest Loss on sale of asset

2081.45

1508.70

2198.30

1930.58

3031.75

0 539.20

Net profit before taxation

6

37,885.19 35,990.86

7

8

23,309.14 12,851.49 2348.37 23,162.93 12,851.49 2348.37

1894.33

146.21

0

0

42.24

0

0

0

0

3526.84

2691.68

3814.77

553._')0

0

0

0

0

0

0

1330.(30

0

0

5378.59

7395.70

7578.83

- 1920.44

0

0

- 1736.99

pact of these looked at over the next six years (this period being the length of the longest lease). The figure of 'annual net profit before taxation' accruing from the newly accepted leases is used as a comparative indicator, although of course many other measures could have been taken. Table 4 presents the annual profits accruing under different exposure limits and accounting options. As the exposure limit was increased, it was possible to select a wider variety of leases and therefore receive larger rentals. However it is possible, if the matching is not very good because of the deliberate restrictions in the model to discrete sizes of loans, to receive a larger rental and yet be worse off because of the excessive interest charges being borne. Such

the first year and also the capital allowances claimed. Notice that the first year allowances exactly total £50,000, the capacity available, whilst the balance sheet shows an acquisition of £55,750.83. Dilution of 10.3~ was applied and the remaining allowances spread, as we can see in Table 3D. Here the total assets acquired in year 2 do not exceed the capacity, and so no additional dilution is needed. Finally, Table 3E gives the details of a single lease, namely the one that was bought in Table 3c. This is a short, highly profitable lease of a car. In the second set of runs, an existing lessor has been assumed, with a current profile of loans and leases. For each run, some twenty leases were offered to the company and the im-

TABLE 3B. SUMMARY BALANCESHEET FOR SIMULATION Year

1

2

3

4

5

6

7

8

Authorised capital Revenue reserves P & L A/c

10,000.00 10,000.00 0 539.20 539.20 5378.59

10,000.00 5917.79 7395.70

10,000.00 13,313.50 7578.83

10,000.00 20,892.33 --1920.44

10,000.00 18,971.89 - 1736.99

10,000.00 17,234.90 O

10,000.00 17,234.90 0

Long-term loans Short-term loans Tax reserves Overnight facilities

20.000.00 20,000.00 8000.00 8431.36 4666.50 14,625.48 3856.72 13,084.77

20,000.00 12,734.23 27,368.1 I 20.045.80

20,000.00 0 35,785.38 0

20,000.00 0 35.411.46 0

0 0 27,742.02 0

0 0 17.516.07 0

0 0 7599.93 0

47,062.42 72,059.41

10,3461.63

86,677.71

84,383.35

54,976.92

44,750.97

34,834.83

55.750.83 95,233.40 8793.80 23.333.17

153,984.23 50,755.73

155.050.09 79,929.40

155,050.09 I15.920.26

137,222.09 122,022.23

111,910.09 I09,561.72

732,53.54 73,253.54

Assets acquired Less depreciation Short-term deposits Cash

0

0

0

11,248.09

44,963.81

39,504.71

42,130.25

34.562.48

105.39

159.18

233.13

308.92

289.72

272.35

272.35

272.35

47,062.42 72.059.41

103.461.63

86,677.71

84,383.35

54,976.92

44.750.97

34,834.83

424

FlarelL Salkin--A .'~todel of a Lessor TABLE

Lease

Cost

Rental

3C. EXAMPLEOF MONTHLYCALCULATIONSFOR SIMULATION Interest payable

Tax payable

First year capital allowance

Written down allowance

13.460.79 6 167.62 2417.90 977.56 3511.16 5834.89 4695.89 3072.60 1525.39 4659.12 3677.08 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

181.65 771.35 407.16 103.85 589.74 -3875.65 -400.92 - 153.63 432.66 863.99 989.26 1664.93 6174.45 5272.46 748.71 1987.23 823.31 2440.33 4367.86 2141.28 3663.68 785.35 11,735.34 6156.78 2567.42 1029.90 3721.76 3508.46

0

58,698.70

L005 L007 L006 L008 L001 L002 L009 L010 L003 L012 L014 L013 L016 L018 L015 LI06 LI08 L110 L112 LlI3 L114 LI15 L205 L207 L206 L208 L201 L202

0 0 0 422.06 0 0 0 0 0 0 0 3006.93 0 173.68 0 111.41 0 0 0 338.97 0 0 0 191.76 0 0 0 0 0 0 0 0 0 0 0 111.41 0 0 0 191.76 0 0 0 0 0 0 0 422.06 0 0 0 0 0 0 6506.00 3006.93

0.49 2.06 1.09 0.28 1.58 - 18.32 - 1.90 - 0.73 1.16 2.31 2.65 4.45 16.52 14.11 2.00 5.32 2.---'I0 6.53 11.69 5.73 9.80 2.10 31.40 16.47 6.87 2.76 9.96 9.39

1145.20 542.54 232.29 110.13 354.21 1299.76 496.94 260.82 135.26 52.15 46.21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Total

6506.00

147.94

4666.50

7976.97

50,000.00

Net investment

The month is month 4, year 3, i.e. 15 months after the end of the first accounting year, and is the month in which tax is paid and capital allowances claimed. Lease 202 starts this month--see Table 3E. Net investment adopts the usual leasing convention of - v e for surplus, + ve for deficit.

is the case at an e x p o s u r e limit of £2500; the c o m p a n y is d o i n g m o r e business a n d yet is paying m u c h m o r e interest t h a n necessary. At yet g reat er e x p o s u r e limit, the c o m p a n y is d o i n g m o r e business a n d a g o o d m a t c h has been achieved. T h e r e is belief i n c o r p o r a t e d in the m o d e l that m o r e business is best a n d so an e x p o s u r e limit w o u l d tend to be used to its full a d v a n t a g e if possible. T u r n i n g to the i n d i v i d u a l a c c o u n t i n g options, first, the N P V is c a l c u l a t e d using a d i s c o u n t rate o f 7%, r o u g h l y the average internal cost o f e q u i t y a s s u m e d o v e r the first six years. N o t i c e that the a c tu a r i a l a n d I P M have high N P V s because, as is clear f r o m the a n n u a l profits, the profits are" t a k e n early an d the final years left to pay off interest c harg es a n d loans. In c o n t r a s t , the m o r e conse r v at i v e m e t h o d s t a k e their profits later and h a v e s o m e w h a t l o w e r N P V . In the p r o d u c t i o n of this table, it was a r r a n g e d that no leases

defaulted. T a b l e 5 sh o w s the same set of leases but n o w with a m u c h higher p r o b a b i l i t y of default, and indeed two leases did default. O b v i o u s l y the N P V for each m e t h o d is far worse, but the m o s t interesting statistic is the c h a n g e in N P V ; the o r d e r of merit of the m e t h o d s is reversed, with the c o n s e r v a t i v e m e t h o d s suffering relatively less. T h e s e tables d e m o n s t r a t e o n e of the c o m p l i c a t i o n s in selecting the a p p r o p r i a t e a c c o u n t i n g m e t h o d and the need for a reserve fund if an aggressive m e t h o d is chosen. Finally, T a b l e 6 was p r o d u c e d by reducing the t a x a b l e c a p a c i t y o f the c o m p a n y d o w n to such a size (£30,000) that all the leases the c o m p a n y w a n t e d to accept with an e x p o s u r e limit o f £5000 r e q u i r e d r a t h e r m o r e capacity. T h e effect o f two o p t i o n s is s h o w n - - t h a t closing the b o o k s results in a smaller portfolio, and, indeed, o n e that is unprofitable, and that

Omeqa, Vol. I0, No..t

-~25

TABLE 3D. MONTHLY CALCULATIONS FOR MONTN 4, YEAR 4

' Rental

Interest payable

First year capital allowance

Tax payable

Written down allowance

Net investment

Lease

Cost

L005 L007 L006 LO~ LO0I L002 LOft) LOIO L003 L012 L014 L013 L016 L018 L015 L106 LI08 LII0 L112 LlI3 Ll14 Ll15 L205 L207 L206 L208 L201 L202 L209 L210 L203 L212 L204 L214 L215

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 422.06 0 0 0 0 173.68 111.41 0 338.97 0 191.76 0 0 0 0 0 111.41 0 191.76 0 0 0 422.06 0 0 0 3006.93 173.68 111.41 0 338.97 0 0 0

- 11.94 -1.65 -0.13 -0.52 -0.17 - 15.93 -9.78 - 5.67 0.64 0.07 1.16 -0.46 2.81 2.54 0.20 0.94 0.29 - 1.62 2.73 0.64 2.67 0.92 29.51 15.96 6.67 2.56 9.69 2.40 11.30 7.85 4.61 14.41 9,03 12.26 2.61

1514.24 719.91 296.97 137.35 429.37 1453.16 961.60 612.03 160.84 566.13 428.70 271.20 744.13 616.70 107.56 234.74 80.8l 227.97 187.06 37.55 44.35 126.60 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 2660.00 8397.95 7125.90 1065.86 2696.00 1090.00 3426.00 5195.00 2660.00 4100.00 1065.86 0 0 0 0 0 0 0 0 0 0 0 0 0

387.05 177.34 69.52 28.11 100.96 167.78 135.03 88.35 43.86 133.97 105.73 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

- 2058.01 - 285.18 - 22.29 - 89.26 - 29.41 - 2747.33 - 1686.73 -977.49 192.46 22.53 348.89 - 79.63 847.19 766.33 61.17 282.50 86.98 - 280.15 824.43 194.06 807.58 277.39 8912.67 4820.78 2015.93 772.99 2926.21 724.06 3413.68 2370.91 1391.40 4353.17 2727.25 3701.65 788.74

Total

0

5594.10

96.59

9958.98

39.482.57

1437.7l

35.375.48

TABLE 3E. EXAMPLE OF INFORMATION PROVIDED ON EACH LEASE IN SIMULATION

Acquisition date

Cost

Deposit

First rental

Second rental

Salvage value

(£)

(Months)

(£)

(£)

(£)

(£)

6506

28

I x 3006.93

2 x 3006,93

0

12

Due

Life

IRR

(months) (months) 12

36

(°0) 40.844

Summary information of lease 202 A/c period

Rental

Depreciation

Gross earnings

Negotiation cost

Interest payable

Net profit

3 4 5 6 7 8

3006.93 3006.93 3006.93 0 0 0

2015.42 1736.12 2754.47 0 0 0

991.51 1270.81 252.46 0 0 0

245.43 0 0 0 0 0

134.07 t32.28 -146.87 - 194.47 - 126.91 - 107.45

612.01 1138.53 399.33 194.47 126.9t 107.45

I. 'Due ' is the number of months between rental payments. 2. The deposit is a multiple of primary rental payments. 3. The interest charges assume a re-investment of net profit at a base rate.

Flarell, Salkin--A Model o f a Lessor

426

TABLE 4. ANNL'AL PROFITS UNDER DIFFERENT EXPOSURE CONSTRAINTS Exposure limit (£)

Annual net profit before taxation (at year end, £) ~ 3 4 5 6

Ac option

I

500

I 2 3 4 5

-218 573 - 1193 13 - 2399

2796 3482 601 1034 165

-493 43 809 188 1430

2500

1 2 3 4 5

-557 745 - 1795 -364 -3225

3186 4175 481 1091 -83

5000

1 2 3 4 5

155 1327 - 941 376 -2257

10,000

1 2 3 4 5

195 1415 -985 404 -2374

-1344 -1578 - 363 -770 45

-797 -1157 6 -530 542

Leases accepted

NPV

-42 -146l 42 -33 119

214 608 -174 81 -431

1.2,5,8,9,10

-419 -1643 -1005 143 - 1827 - 1435 823 -291 169 238 -804 -614 1405 220 951

-161 -2-100 14 -146 133

-157 443 -678 -314 - 1005

1,2, 5, 6, 7,9,10

3141 3732 537 1117 -47

-206 148 697 267 1127

-1559 -1633 - 227 -669 215

-1018 -1408 249 -595 1094

-162 -1815 36 -145 219

697 1162 187 513 -140

1,2,5,6.7,8,10

3315 3935 605 1215 - 11

-297 106 749 257 1241

-1658 -1764 -278 -761 206

-1057 -1460 210 -635 1055

-173 -1900 24 -155 201

720 1193 173 512 - 172

1,2,5,6,7,8,9

Code: (1)---investment period method ; (2~---act uarial: (3 ~---straight line; (4)--rule of 78 ; (5.)-----reverse rule of 78.

TABLE 5. THE EFFECT OF HIGH DEFAULTS ON ANNUAL PROFITS

A/c option

1 -2109 - 1452 -2655 -1917 -3393

2 -357 -52 - 1645 -1162 -2129

Annual net profit before taxation (at year end, £) 3 4 5 6 23 502 859 702 1016

-57 -95 889 514 1263

-2 -195 770 255 1285

0 -1207 -718 -893 -543

NPV -2309 -2008 -2468 -2255 -2683

Dccl'ease in NPV from Table 3 3006 3170 2655 2768 2543

Leases accepted

Leases defaulted

1,6,7,8,9,10

1,6

TABLE 6. THE EFFECT OF LIMITED TAX CAPACITY UNDER DIFFERENT ASSUMPTIONS

Tax option

A/c option

l

Annual net profits before taxation (at year end, £) 2 3 4 5 6

NPV

Leases accepted

-542 760 - 1270 -472 -2069

1638 2018 -230 303 -763

22 -59 - 108 -35 -- 182

- 1280 --1193 -- 13 - 399 372

-891 -1222 376 -469 1220

--142 --765 --1500 -357 51 -- 1184 - 124 -927 5 -- 1591

1,5,6,7,8 (total cost = 29,587)

- 1273 -101 -2368 - 1052 -3685

3133 3731 527 1111 -42

-236 117 666 236 1095

- 1605 -1679 --273 -715 169

- 1060 -1450 207 -638 1051

- 151 -1804 42 - 133 -19

1, 2, 5, 6, 7, 8, I0 (capacity purchased = 9519)

-727 -257 --1242 -909 -- 1721

Code of tax option: (l)--close the books; (2)---buy a ddi t i ona l capacity at 15~ premium.

Omega, Vol. 10, .Vo. 4

427

REFERENCE

buying additional capacity produces a large negative flow in year 1 and then the profits are roughly as in Table 5 (the slight difference is due to the larger loans needed to carry the purchase of the capacity). The other two options involving dilution are not shown because these affect after-taxfigures only and as such are less meaningful.

1. Financial Times (198l) 17th June.

RB Flavell. Department of Science, Imperial College, London SW7

ADDRESS FOR CORRESPONDENCE:

Management 2BX, UK

CONCLUSIONS A P P E N D I X 1" DETAILS O F LESSOR ACCOUNTING

In this paper, we have described the key operations of a leasing company, the problems it has to face and some of the inter-relationships between these problems. Based on this description and various discussions with leasing companies in London, a simulation model of such a company has been constructed and this, together with some example results, was discussed. The purpose of the model is to examine the effect of a changing environment and/or changing key parameters on a leasing company and thereby aid in the planning and decisionmaking processes. Work is still continuing on the model, primarily in speeding up some of the heuristic procedures used for the construction of portfolios, but all the essential features are now included.

The reason for including this appendix is that the different methods of lessor accounting are only described in the specialist leasing literature, not the standard finance textbooks and certainly not in any management science context. Yet the implications of the different methods are crucial to an understanding of the lessor model and the accompanying discussion on such aspects as bad debts, loan repayments and so on. Let us start with a simple lease. The lessor purchases an asset for £1000, immediately leases it for five annual payments of £230 each after which the asset becomes valueless. We will also assume for simplicity that all transactions occur at the end of a financial year, capital allowances of 100%~, and the tax at 50~o are received and paid immediately and that the lessor's cost of borrowing is a straight 10~. The gross profit (i.e. before interest charges) on this lease is £150 and it is the way in which the different methods spread this profit over the five years that is of interest. Table AI shows the actual cash flows involved during the lifetime of the lease, together with the net investment annually remaining in the lease. (The first tax payment is due to the receipt of a capital allowance and so it is not strictly speaking a cash flow.) The apportionment of the £150 gross profit for the five different methods discussed in the main text is shown in Table A2 and the net profit after the interest payments (as shown in Table AI) in Table A3.

ACKNOWLEDGEMENTS Our thanks are due to numerous leasing companies with which we have had discussions over the years and in particular to Lombard North Central.

TABLE A1. CASH FLOWS INVOLVEDIN LEASE End of year

Payment of asset

Rent received

0 1 2 3 4

1000 -----

230 230 230 230 230

Tax paid -385.00 95.75 100.53 105.56 110.84

Interest

Net investment

-38.50 29.93 18,87 8.32

385.00 289.25 188.7l 83.15 -27.69

TABLE A2. APPORTIONMENT OF GROSS PROFIT BY ALTERNATIVE METHODS

End of year

IPM

Sum-of-digits

Straight line

Reverse-sum of-digits

Actuarial

0 1 2 3 4

0.407 0.306 0.199 0.088 0.0 1.000

0.333 0.267 0.200 0.133 0.067 1.000

0.200 0.200 0.200 0.200 0.200 1.000

0.067 0.133 0.200 0.267 0.333 1.0(I)

0.000 0.386 0.300 0.207 0. 107 1.000

428

Flavell. Salkin--A Model of a Lessor

TABLE A3. NET

PROFIT FROM EACH METHOD

End of year

IPM

Sum-oUdigits

S t r a ~ h t line

0 1 2 3 4

61.04 7.35 0.99 - 5.69 18.32

49.45 1.55 1.07 1.08 1.73

30.00 -8.50 1.07 11.13 21.68

TABLE A4. End of year

Payment of asset

Rent received

0 1 2 3 4

1000 -----

230 230 230 230 230

'A5. T O T A L NET PROFITS IN EACH PERIOD

Year

Individual

Portfolio

1 2 3 4 5 6 7 8 9

6.22 25.26 86.78 18.01 25.28 58.53 64.77 30.86 10.76

16.72 38.18 43.54 64.80 36.13 53.99 72.75 44.08 25.65

10 Total

2.43 328.90

10.05 - 18.55 1,07 21.18 41.63

NOTIONAL CASH FLOWS UNDER BORROWING RATE OF

The apportionment for the IPM is calculated by adding up all the net investments until the first time the lessor has no capital remaining in the lease, and then working out the proportion of net investment outstanding in any year. For example, the total is (385 + 289.25 + . . . ) = 946.11 and the apportionment in the first year 385/946.11, etc. The next three methods are more straightforward but the derivation of the actuarial apportionment needs some explanation. Table A4 shows cash flow calculations for a cost of borrowing of 15.05~o; notice that the final net investment is now effectively zero. (This is, of course, how the interest rate is determined.) The sum of the interest payments is £150 by definition and it is the relative sizes of the interest payments that are used to calculate the apportionment. Because each of the methods declare gross, and hence net, profits at different rates, the selection of a particular method can have a significant impact on an organisation. Two examples: first, borrowing capacity m a y be dependent upon retained net profits so a method that takes profits early, such as the IPM, will enable a c o m p a n y to expand rapidly; second, if a lessee defaults this then has little capital impact on a c o m p a n y that has adopted a scheme for taking profits late such as reverse-sum-of-d!gits. There is a further area of decision for the lessor, namely, should he treat all leases on an individual basis and spread

TABLE

Reverse-sum o~digits

7 .__38 405.82

Tax paid -385.00 86.03 92.50 99.46 106~45

Actuarial 0.00 19.39 16.01 12.20 7.78

15.05~ p.a.

Interest

Net investment

-57.89 44.94 31.07 16.10

385.00 298.97 206.46 106.99 0.04

the profits accordingly or should his entire "book' be treated, in essence, as a single lease'? Table A5 shows the net profits for five leases taken as individuals and then summarised and alternatively taken as a portfolio. (The IPM method was used.) There is a considerable difference in the columns; the portfolio approach gives a more stable profile of earnings and also recovers the total capital invested more rapidly because the surplus monies from early leases can be used to cover the later leases. This appendix has attempted to explain the details of lessor accounting conventions and their impact on the organisation. Obviously, the example is extremely simple but the fundamental principles are true in reality. The portfolio approach as described last is the one adopted in the computer model as it is strongly felt that overall performance is of more importance than individual performance. The decision whether or not to proceed with an individual lease should, at the end of the day, depend upon its marginal impact on the portfolio.

APPENDIX 2" THEORETICAL DEVELOPMENT OF A LEASING MODEL Let Li [si, f/, ai, [ p , , ri,; t = O...nl]l denote the ith lease, i = 1 ... N where s~ is the starting date, f~ is the finishing date, a~ is the cost of the asset, Pit is the rental p a y m e n t at the end of period t after s, r, is the interval between rental payments. Hence, n~

f,=s,+ ~

ri,.

t=o

Let D)Ibj, e), d r, ['k). h),, g),: t = 0 . . . m / I f denote t h e j t h loan, j = 1 ... M where b~ is the starting date, ej is the maturity date,

Omeqa. ~ol. I0..Yo. 4 dj k j, h, y j,

is the initial size of the loan. is the interest rate pertaining over the period t. is the loan repayment at the end of period t. is the interval between loan repayments and or interest revisions.

Assume: 1. The end of the first accounting year is n~ periods from the beginning of the model. 2. Each accounting year is m, periods long. hence 0

<_ n ~ <

m a.

3. There is a constant tax rate of 7, and a capitaI allowance rate of Ca. 4. Tax payments and/or relief are made n, periods after the end of the accounting year in which they are raised. The net investment at the end of period 7-. N l r . is

Nlr=

Notice that here we are also assuming that k j, is defined as the appropriate interest rate over the period of 9,. This could lead to some problems of dimensionaiity which would be overcome by assuming dimensions of interest over a single time period. Again. in the above, cumulative definitions would make the computations easier.

Loan repayment We ~,ill a s s u m e that the loan is repaid at the end of the period and after the calculation of any interest charges. The easiest expression is simply {LRT) = [ D j r - I - Ojr] V T using the above definition as this expression will also cover the raising of the loan. Hence the total I o a n r e p a y m e n t l = ~ [ D j r - i - Ojr] in period T ) jg't

N l r - ~ - rentals received - tax relief - lendings + interest payments on loan + capital repayment + tax payments.

The fendings here may be a special type. i.e. the funding of more assets for leasing. This will only occur if the net investment is positive and so will not be considered just

429

and when T = bj. then D~r_t = 0 and the expression is equal to - D j r =- dj.

Tax payment--only considered if [{T - n,) - n,~] modtrn=) = 0.

yet.

The tax payable for any period is on the total rental received less the total interest paid {assuming of course that this net figure is positive). Based on the same tax period as the tax relief calculations, we get

Rentals The total rental received) ~. {0p i f T < & o r T > f ,. if 3t* s.t. in period T ( R r ) j~ = ~ :o g(j~ t i e

~

Y

-

(Ter) = T,~(R,

si,

e=O

(Computationally, evaluation of this expression is made much easier by (i) ordering all leases s.t. s~ < & . t - - h e n c e only need to search a band (ii) storing cumulative values of r.)

Tax relief---only considered if [(T - , n , ) - n~] m o d (m~) =0.

If we first assume that 100°,o of the allowable tax relief may be taken up (i.e. infinite taxable capacity), then the relief at the end of period T is (TRT) =

(l - T,)C,,T.S', '~ { ; T -otherwise ifn , - m , - n , < s , < T - n , - n ~ . i

Loan calculation Define Dir as the a m o u n t of loan j outstanding at the end of period T. This is calculated from: 0 i f T < b j o r T >_el Io

dj-

-- hi,

if

3t*s.t.

¢=0

~

t'+l

91, -< T -

bi <

e=O

~

91,-

=

t

'"

ki,.D~r- t if ~ 9~, < T -

j=l

ki,.-tDjr-~

for

T--n,-

m~-n,

< e <_ T -

n,-

n~.

The above calculations are independent of any accounting convention. However, before one can discuss gearing and exposure relationships, the profitability of a lease must be considered and this is dependent upon the convention selected. Once the convention is chosen, the profitability of a given stream may be assessed and hence the equity base calculated. Given a gearing ratio, this provides debt capacity which may be used to fund the given stream. There are also two other conditions that must be considered, i.e. exposure and taxable capacity: the first occurs in the selection of loans to match t h e extant leases and the second is really an absolute constraint provided by external conditions. The investment period method (for example) operates in the following manner: Given the net investment at the end of each period T. N i t then t. Identify T* s.t. N l r . < 0 for the first time. 2. Assuming no > 0, calculate

r=O

interest payment If we assume that interest is payable at the end of the period and is paid in arrears, i.e. the interest on Djr-~ is paid at the end of period 7. then the total interest paid in period T{lr) 0 if T - 1 < bj or T - 1 ~. ei

~

- I~)

e

t* e=O

t=l

and

GRx=V,_',NI,-n.+(K

-2)m~
t

<_ rain [ T * . n . + (K - 1)re.I*,. For K greater than that defined above. ~ R x = 0. 3. Then calculate the gross earnings from

bl

r=O

if 'Y" 9i, = T -

GRt = ~ Nl,

bi.

-130

Flarell. Salkin--A Model of a Lessor

where

c o m p a n y is open to in period T is simply

'3 -~ V (~RK.

E X r = ~ a,(OL, - IC,)

a, > 0

t=r

The expression in the square brackets is simply the gross profit accruing from each lease, ignoring all money charges and taxes. 4. To calculate the profits from the method, let T i t be the interest paid in the accounting year K (calculated as in (2)). but V K under consideration). The profit is

where a, is a weighting factor. This formulation uses the convention that exposure is positive, overcovering negative. The determination of a, is subjective but there are two major and conflicting theories. (i)

NPx = GRx - T l r V K. Gearing ratios and debt capacity

We will now assume that the c o m p a n y has established an overall gearing ratio as one criterion for control, i.e. debt/equity < G at all times. Thus. if Er is the equity holding at the end of year K. then the total capacity of the c o m p a n y is (1 + G ) E r and the debt capacity G ' E r . The capacity of the c o m p a n y at the end of each accounting year for the succeeding one is obviously dependent upon the equity holding; if no dividends are paid then

The effect of being exposed in the long-term future is less than in the short-term because the c o m p a n y will have time to make adjustments to cover itself. Hence at should be a monotonically decreasing function of t.

(ii) BecauSe of increasing uncertainty in the future, it is important to reduce the possibility of a loss by being increasingly closely matched. Hence a, should be a monotonically increasing function of t. Personally, our view is for the former with a, = e r - ' as an example.

Er = EK- • + NPK + ~ part of the depreciation charge and (l + G)Er is the m a x i m u m capacity in the (K + l)th year (ignoring all overheads, etc.). When a new lease is considered, we will a s s u m e that it is financed in the ratio of G parts debt to one part equity, thus ensuring that both sides are used up at an equal rate. Similarly the depreciation charge is broken up in the same rate. Funding restrictions

With the m a x i m u m debt capacity in any accounting year being defined by GEt there is obviously a limit on the a m o u n t of business that can be performed. Assuming certainty, we have max~{Dj,: t

n, + ( K - 2)m,, < t < n , + ( K - l)m°}

Defaults and bad debts

Obviously with the undertaking of any business venture, there are associated risks. An important problem for the lessor is that of the lessee defaulting on the rental payments. Consider a lease Lt with payments p,, t = O. 1. . . . . n~. Theoretically it is possible to associate with each payment a probability pi', that payment will not be forthcoming and hence that the lessee has defaulted. Let us first consider the effect of the default. Obviously the rentals will have stopped, quite possibly before the cost of the asset has been recovered. To calculate the outstanding value of the asset can be done in a number of ways: perhaps the simplest is to suppose that the ith lease defaults in period T, where s~ < T < f~. The rentals that r-, would have been paid = )-" p, whereas the total rentals ,s0

1

that should have been paid = ~ p,. On the basis of a < G'Er-I

i.e. m a x i m u m debt occurred in period K is limited by equity base at the end of period K - 1.

:=0

straight line diminution of the asset's value, the outstanding value is

Exposure constraints

Exposure occurs when the loans that have been raised do not exactly match the leases that have been undertaken. If the loans are shorter than the leases, so that the loans would need to be rolled over. the c o m p a n y is 'exposed" to the danger that a higher interest charge may have to be paid. Conversely, if the loans are longer than the leases, the c o m p a n y is committed to pay the loans off at possibly high interest charges and with a reduced flexibility and competitiveness. Needless to say, the c o m p a n y m a y take a view whether to be exposed or overcovered depending on the forecasted m o v e m e n t of interest rates. The problems of exposure are of course magnified if the possibility of lessee default is introduced because this severely upsets the pattern of cash flows. The calculation of exposure is complex and to a large extent subjective. Let ICr = Rr + T R r - T P r - l r be the income received in period T after the payment of

ignoring

taxes and interest charges. Similarly. OLr =

V, =

~ D jr is the

if a proportion {say ,8) is recoverable by resale of the asset, then ,sV~r is a cash inflow at this point in time. The expected cash flow from a lease, ignoring constants such as loan repayments and interest charges as well as tax charges, is simply ~.

t-t

t=0

k=0

[ I (l - p ' ~ ) f p ' , , s v , ,

+ 1 - p',p,,]

with no discounting. An approximation to this expression is

~

(1 -

p',)p.

t=o

~

second-order

terms

and

setting

8 = 1 and

PJk. In practice it is relatively easy to take into

k=t+l

outstanding loan in period 7". T h u s the exposure that the

account defaults: merely adjust the rental stream to ,SV~,

Omec ". Vol. I0. No. 4 and set the remaining rentals to zero. The interaction between default and exposure should be noted. Finite taxable capacity Above, it was assumed that 1003o of the allowable tax relief would be taken up in the first year possible. If there is finite taxable capacity, then this may not be possible and the relief may have to be spread over a number of years. At the present time in the UK, the lessor is allowed to take as little as 25°,; of the relief in the first year and spread the remainder over the next three years (or the lifetime of the lease, whichever is the shorter). If L~ is the number of accounting periods that the ith lease spans, then define a~, >_ 0, t = I, 2. . . . rain [4, L~] such that ~ a,, = 1 t

and o'il -> 0.25.Li is calculated as follows:

+ Is, - n~] mod(m~)I/m~ and 1 ~ + 1 if[f~ - no] mod(ma) 4= 0 ( + l if[si n . ] m o d ( m o ) 4= 1

and then TRr =

(l

-

T,)C~

N {0 i,ai i f O < S , + n , + n , + m ~ otherwise

Y'z

~_---

The choice of a is to a large extent arbitrar? for the lessor, but the Inland Revenue do not encourage frequent fluctuations. As mentioned previously, the taxable capacit? of the lessor is to a large extent exogenous because it is primarily provided by other subsidiaries in the group. T R r is the tax relief claimed in period T only if sufficient capacity exists: if TCr is the absolute capacity available and if T C r < T R r for any period even after all possible spreading, then this can have considerable impact on the profitability and perhaps even more important the competitiveness of the lessor. The main body of the text describes, in a situation such as this. what are the alternative courses of action, i.e. dilution, undertaking management leasing (buying capacity), closing the books or some combination thereof. Conclusion

Li = ',(f, - si) - If/ - ha] mod(m~)

Li = L ~ -

'~',1

-

T <_t = u .

In this appendix, we have attempted to construct the main theoretical equations used in the leasing model. Minor points such as overheads, negotiation costs, etc., have been omitted in order to maintain clarity. Having rejected all new leases with an internal rate of return, before considering interest charges, less than some historic gross earnings rate, all possible portfolios of existing and new business likely to occur within some (exogenoust look-ahead period are considered. All portfolios that violate any of the constraints are again rejected and the more profitable remaining one chosen. There are many parameters that one can manipulate here: the look-ahead period obviously, but also. because it is impractical to trace each portfolio through to its final conclusion, the analysis period for each portfolio.