Although evidence was cited in the previous section that the numerical impact of interactions between the investment and financing decisions may be less intractable than that suggested by many
authors, the existence of this interaction where there are significant degrees of market imperfection* remains unquestioned. In any rigorous treatment of the theory of valuation of the firm the interaction needs to be treated explicitly.
* The most significant arguments to the contrary embodied in the work of Modigliani and Miller (58) specifically assume perfect market conditions.
A mathematical programming framework affords a potentially very powerful analytical tool for this. The advantage of math
ematical programming models in this area is their representation of the economic value of the firm as the objective function and their explicit treatment of market imperfections as constraints.
Many interesting and economically meaningful deductions can be made from these models by use of the Kuhn-Tucker conditions* for optimality.
The problem of valuation of the firm, within or without the context of mathematical programming, is a core problem of financial theory. The purpose of this part of the thesis is not to tackle directly any of the fundamental issues but to show the contribution that mathematical programming can make to exploring the consequences and logical consistencies of a particular formulation.
This contribution will be discussed more extensively in chapter four. In this section the background material and the nature of one particular problem will be discussed - the horizon truncation problem.
Of necessity any linear programming model of the firm must have a finite horizon. The properties required of this finite horizon focus precisely on the substance of chapter four - ti>e conceptual problems arising from the interactions of capital market imperfections and the impact on the valuation formula used for the objective function. This aspect is best seen in a
historical context and once again the work of Weingartner provides the most
* Simple expositions of the Kuhn-Tucker conditions can be found in standard operational research texts such as Hillier and Lieberman (67)
20
appropriate vehicle.
As we have seen his original approach of maximizing the net present value of the project set was subject to Baumol-Quandt's criticism of inconsistency. Their suggested way out of the
paradox of maximizing the utility of withdrawals from the firm by a model of the form*
Max l U t Wt (1.4.1)
Subject to - l c ^ Xj*Wt S F t (1.4.2)
was rejected by Weingartner because of the problems of specification of an appropriate utility function.^" Instead he resorted to a horizon valuation model.____________________________________________________
* The notation is the same as in section 1.2
^ As mentioned, later writers such as Myers (72) have identified with the relative utility of total funds in time period t and thus with interest rates exogenously determined by the capital market. Thus Myers rewrites the model in the form
M a x U fc |
ft +
1
°tj X j](1.4.3)
-
1
°t Ft +I ]
°t Ctj X j U . 4.4)subject to £
' Gtj V F t (1.4.5)
He argues that in a certain world, investors facing a prevailing interest rate K will all adjust their portfolios so that the following conditions hold
(1.4.6)
Defining Uq * 1 means that the firm can use the observed rate K to infer the marginal utilities required b/ the Baumol and Quandt formulation
Thus Weingartrier's reformulated model was n
Max Z - Ï c x + j-1 3 3
V - w T T
»•t. - l + vx - wa
(1.4.7)
(1.4.8)
j Ctj X j + (1+rL ,Wt-l “ (1+rB ,Vt-l " wt * V t S ? t
t - 1, H - 1 (1.4.9)
Wt S Bt t-1,. (1.4.10)
0 i x^ £ 1 Vj (1.4.11)
V wt * ° - Vt (1.4.12)
with the additional notation borrowing in period t.
lending in period t.
is the interest rate on lending, is the interest rate on borrowing, is a limit on the borrowing in t.
The scalar quantity c r e p r e s e n t i n g the post horizon value of cash flows is given by
00 i t
1
ctj xj / n (1+iT)t-H C3
y
t-h+i t (1.4.13)This approach gives rise to three important questions.
In what sense is the pursuit of optimal wealth at some future time compatible with maximization of the value of the firm now?
What is the significance of and the determinants of the choice of horizon?
What is the appropriate post-horizon valuation procedure?
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A dual analysis of this model provides a foundation for the answers to these questions, the dual of the cash balance equation gives the marginal value of an extra £l of earnings and thus the
ratio of the duals in successive periods gives the interperiod discount rate at which projects ought to.be screening. In effect the dual is the opportunity cost of capital. The relationship between pfc, the lending rate and the borrowing rate and the dual on the debt capacity (X^) is
It should be emphasised that these duals are outputs from the optimum linear programming solutions. Thus where the firm is lending, the left hand inequality becomes an equality and p t = (1+r^)Pt+i J where the firm is borrowing with spare debt capacity pfc = (1+r^) Pt+i an<^
where the firm is borrowing upto its limit Pt = (1+r^) + X^_.
Thus the opportunity cost of capital may be the lending rate, the borrowing rate or the marginal productivity of capital. The
is the value of the cash flow from project j valued at ii. Hence VJU is a generalization* of the net terminal value concept.
In answer to the first of these problems, Weingartner concluded that where borrowing and lending rates were equal with r^ *= rB
* See Weingartner (74) p.164 et seg. Page numbers refer to the 1974 edition, though the 1962 reference will be given where the historical context of the work is important.
(1.4.16)
marginal value of project j is given by p.. =
and borrowing is unrestricted^ then maximazation of the terminal value
t The inequality then implies pfc » (l+r)pt+1 or pfc *» (l+r)H t with p^ = 6j - J ct;j(l+r)H
t=l
value. However, this set of assumptions implies perfect capital markets and under such conditions a linear programming formulation of the capital investment problem is unnecessary. In conditions of capital rationing Weingartner concluded that maximization of the net terminal value was not equal to maximization of the net present value of the project set.
The dual analysis reveals also the difficulty associated with a specification of a suitable valuation function for post horizon cash flows. As Weingartner states
"The rate taken to be appropriate in computing the horizon values Oj is the lending-borrowing rate used in the models.
However, this rate is not the proper one if there are effective limits on borrowing."
Thus Weingartner admits that while the correct discount rate is effectively incorporated into the valuation in the pre-horizon period it is not clear which of the borrowing, lending or marginal re.tes is the correct one in the post horizon period. Weingartner does provide a clear discussion of the requirements of an horizon, though
little further guidance as to how one might determine such an horizon. Thus he states
"In order to unhook the infinite chains of actions and their consequences in the model of the firmb investment decisions, we seek a point in time such that the decisions which call for implementation before this date will be exactly the same, whether or not events past that moment are treated explicitly or implicitly (and hence partially ignored). More concretely, and in terms of our model,
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we seek a value of H such that the set of accepted projects having outlays or revenues in year H or sooner are exactly the same whether the model makes use of an infinite horizon or a horizon set at H.
In dynamic Models in general such a horizon does not necessarily exist or there may be many of them. If there are several the earliest having this property may be designated as the preferred one."
The discussion gives rise to a definition of a suitable horizon valuation - which shall be termed the fundamental horizon valuation principle.
The horizon valuation is a satisfactory valuation model if , for all optimal feasible solutions^the set of pre-horizon decisions with respect to that horizon would be uraltered for any other choice of horizon.
The existence of such a horizon will be discussed in Chapter Four.
Weingartner's approach to the horizon truncation problem implies that the horizon is an intrinsic property of the model and its determinants are found from within the model. The alternative
approach is to regard the horizon as a function of the firms planning.
Such an approach is exemplified by Chambers (67) in his paper 'The allocation of funds subject to restrictions on reported results') he states that
"the horizon is chosen as a date beyond which opportunities cannot be predicted with any confidence, no information is lost by ignoring interactions between projects after that date, or assuming that funds sure reinvested at the standard rate. In this approach in which the aim was to develop a model to
assist management with planning it was convenient to adopt the same planning horizon”.
While this may not be totally satisfactory from a theoretical viewpoint it may well prove necessary in practice.
In the later paper (71) on 'The joint porblem of investment and finance', he adopts a terminal valuation approach since
"This allows the marginal cost of capital in each year upto a planning horizon to be determined within the model”...
He suggests that at the horizon the net value of post horizon cash flows (NPVH)
"takes no account of any prospects for reinvesting some or part of the capital at more than the marginal rate of return.
In fact managers would normally expect to be able to invest substantial sums after the horizon at better than marginal rates, and this expectation would normally be shared by shareholders. It would seem to follow that NPVH understates the true value of funds available after the horizon.”
Chambers recognises that the interactive nature of post horizon decisions may affect the opportunity cost of cash flows and hence the valuation.
The investment valuation method implicit with both Weingartner and Chambers models can be represented by Figure 1.4.1.
FIGURE 1.4.1.
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In order to avoid problems associated with the Baumol and Quandt paradox the horizon in used artificially to separate out the constraint set and the valuation flows. The reduced cost associated with a
project decision produced by a conventional linear programming analysis is a generalised net present value which is equal to the post-horizon cash flow contribution less the use of capital and debt capacity valued at their opportunity rates in the pre-horizon period. It should be noted that in both models the post horizon cash flow valuation is approximated by using a pre-determined average discount rate and the debt capacity effects are totally ignored.
It can be seen that neither of these models can satisfy the fundamental horizon principle. The implications'and'limitations of such models will be discussed more extensively in chapter .four, while the next section will introduce the idea of using a mathematical programming framework for the evaluation of a particular financing instrument - a financial lease.
1,5 A mathematical programming framework for Lease* evaluation.
A financial lease is a noncanceliable contractual commitment on the part of the lessee to make a series of payments to a lessor for the use of an asset. The lessee acquires most of the economic benefits resulting from the use of the asset though the lessor retains title to it. The payments made by the lessee to the lessor are such as to reimburse the lessor for the assets and the financing costs associated with the assetc^plus any administration costs and to give him a return on his financial investment. Hence the decision to lease a piece of equipment is at one and the same time the decision to acquire that same piece of equipment. The contractual nature of
* In the ensuing discussion it is assumed that a lease refers to a financial lease rather than an operating lease.
a lease repayment schedule means that the firm is undertaking a form of debt financing while simulaneously it is acquiring an asset which will alter the future cash revenues patterns of the organisation.
Thus by its very nature the lease contract is a prime example of an investment and financing instrument.
It would appear that the most suitable method for the evaluation i& to xnclude it within a mathematical programming
model of the firm in which all the available investment and financing opportunities are considered simultaneously. While such an approach obviously offers a mechanism for integrating the lease decision into a formal planning system, the analytical framework afforded by mathematical programming theory can make a major contribution to the development of appropriate valuation formulae. The requisitive
analysis is carried out in Chapter Five. In this section the relevant background and survey of some of the approaches suggested in the financial literature will be discussed.
The initial work of Vancil(63) was followed by a lull but more recently the attention of academics has refocussed on the lease-buy problem as is evidenced by a spate* of papers purporting to solve the
lease-buy decisions. This revival in interest in the evaluation of financial leases would appear to stem in part from its increasing prominence in the planned financing structure of U.K. firms.
As Fawthrop and Terry (76) point out:
"The growing prominence in the U.K. capital market is made
clear by a recent estimate from the Equipment Leasing Association which suggests that the industry now provides equipment with an initial cost of approximately £1,000 million."
* Of the 75 articles cited by ferry C76) the majority of these have been published between 1973-1976.
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While the numerous writing of academics has resulted in little consensus as to the correct method of analysis of a lease. A common but by no means universally accepted approach is to compare the merits of lease financing with that of debt financing via a discounted
present value method of the cash flows resulting from these alternatives.
This gives rise to two particular measures of the cost of a lease which will prove of great value in our analysis. They are the interest rate on the lease and the after tax cost of the lease. The interest rate implied in a lease is just that rate of interest which when applied to the outstanding capital on the lease is such that the lease repayments meet both capital and interest. In order to make precise this definition and to facilitate the subsequent discussions it is convenient to write down the algebraic expressions for the lease-buy decision from the point of view of the lessee, using the following notations:
T *• Marginal tax rate on corporate net income H = Length of the lease contract
Hence we have the lease interest rate i defined by the equation la
and the after tax cost of the lease r defined by the equation ii
Zn general, academics* tend to reject such measures as internal rates of return in favour of net present value methods, though in this particular case under the most rigorous analysis the former measure provides a very good decision parameter.
Mao's analysis(69) exemplifies the more usual net present value approach. The discounted cost of a lease financing i^i
while the corresponding cost of debt financing is
In the first expression only»the lease payments are allowable against tax while in tiie second expression both depreciation charges and interest charges are allowable against tax.- Hence from this analysis it can be seen that the value of the lease-buy decision is:
So far nothing has been said about the appropriate discount rate K to use and this remains the centre of much of the controversy about lease analysis.
Mao suggests that K is the firms marginal investment return;
an assumption which would imply that the lease is being considered under some state of capital rationing. The use of the marginal
H P (1—T) + b T
V t_________ t (1.S.6)
- wo (1.5.9)
* See Van Horne (77) p. 88
30
investment return as the appropriate discount rate is
subject to much dispute.* Other writers such as Vancil adopt an average cost of capital discount rate. Vancil»recognising that other sources of money are available^argues that it is desirable to eliminate the differences in the amounts of financing when comparing specific proposals. Since leasing provides more financing than debt the company Will nave more fixed charges under the lease plan than under the debt plan. These higher fixed charges may prompt investors to discount earnings (or dividends) at different rates. Vancil's (61) approach is to compare leasing with borrowing only after the difference in the amounts of funds provided have been removed. At a particular time t, of a lease repayment Pt , rwt represents the imputed interest expense while the remaining P - rwfc represents repayment of the principal. In order to remove the difference in the amount of
financing provided by leasing and borrowing the Dasic Interest approach focuses on the tax savings associated with the non-interest portion of the lease payments. Hence the cost of leasing under this approach is given by the difference between the price of the assets and the present value of the tax savings associated with the non-interest portion of the lease payments. This is given by the expression:
For the purpose of comparison, the present value of the alternative which is that of debt financing is just given by:
Ao
H W fcT
t£j_ (1+K) t (1.5.11)
* See Bower (73)
have been eliminated already and do not appear in the expression.
Leasing here is viewed as an alternative to debt. One of the difficulties of such an approach is that of comparing diftering amounts of debt financing and loan repayment schedules.
Using a variation of Vancil's algorithm, Bower, Herringer and Williamson(66) specifically tackle this problem by assuming that the loan payment schedule is the same configuration as the lease repayment schedule to 'wash out' this difference. The remaining details of their approach is of less interest to this brief survey than their choice of discount rate - both Vancil and
Bower, Herringer and Williamson chose the’ weighted average cost of capital.
It can be argued that conceptually it is wrong to use the cost of capital in making decisions between methods of financing.
The cash flows under consideration are contractually fixed or are associated with tax savings and involve very little risk.
It thus seems erroneous to use a cost of capital, which emobodies a risk premium for the firm as a whole. The counter-arguments of Vancil and BHW is that investors and creditors, in their valuation of the firm, recognise the difference in tax savings between the two methods. Because both investors and creditors determine the overall cost of capital the average cost of capital is the appropriate rate. A cynic might well remark that the debt rate is avoided because discounting at a debt rate would in general cause leasing to be sufficiently unattractive and that neither of these algorithms would yield results which would explain its popularity. The use of the average cost of capital gives rise to one further problem. Where there is significant portion of
lease finance,which will be usually more expensive than debt finance j
32
then this fact ought to be reflected in the cost of capital rate.
Thus the discount rate used in the above algorithms is dependent upon
Thus the discount rate used in the above algorithms is dependent upon