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Free cash flow models, terminal values and the timing of asset replacements.

1. Introduction

A widely used method for valuing companies involves discounting of "free cash flows" up to some horizon along with a "terminal value". The latter is generally obtained by applying some multiple to expected earnings at that point, or invoking a discounted cash flow approach with simple assumptions concerning the expected growth in cash flows from that point (Cornell, 1993; Koller et al., 2005; Brealey and Myers, 2003; Copeland et al., 2005). Implicit in such processes is the assumption that expenditures for replacing fixed assets are expected to grow at a constant rate beyond the horizon point. This process is subject to at least two major concerns. The first is the failure to recognise that the timing of asset replacements is stochastic, and this has been addressed (Mauer and Ott, 1995; Dobbs, 2004). The second issue is the failure to explicitly consider the age profile of a firm's assets at the valuation horizon; this age profile affects the timing of future replacements and therefore affects the present value of these expenditures. Explicit consideration of this age profile may rebut the assumption that replacements expenditures are expected to grow at a constant rate even when the timing of replacements is deterministic. Furthermore, the valuation difference resulting from this second issue could be very substantial.

This paper seeks to investigate this second issue. To focus upon the second issue, we retain the standard assumption that the timing of asset replacements is deterministic. Section 2 develops the relevant theory. Section 3 considers a stylised example, and finds that the firm's value could be overestimated by more than 100% through failure to explicitly consider this age profile issue. Section 4 then considers a real-world example, and finds that similarly large overestimates are possible. Section 5 concludes.

2. Theory

Let [F.sup.u.sub.t] denote unlevered cash flow from operations net of replacement and new investment for year t, k denote the weighted average cost of capital and [V.sub.T] the value of the firm in T years. The current value of the firm is then as follows:

[V.sub.0] = [T.summation over (t=1)] E([F.sup.u.sub.t]/[(1 + k).sup.t] + E([V.sub.T]/ [(1 + k).sup.T]

The terminal value [V.sub.T] is generally obtained by applying some multiple to earnings at that point, or invoking a discounted cash flow (DCF) approach with simple assumptions concerning the expected growth in free cash flows from that point. In respect of DCF approaches, Cornell (1993, Ch. 6) adopts a constant growth rate of 6% and T = 7, Koller et al. (2005, pp. 101-111) adopt a constant growth rate of 4% and T = 10, Brealey and Myers (2003, Ch. 4) adopt a constant growth rate of 6% and T = 6, and Copeland et al. (2005, Ch. 14) adopt a constant growth rate of 5% and T = 10. We also adopt the DCF approach, and decompose the terminal value into the net present value arising from new investment after the terminal year ([V.sup.N.sub.T]) and the present value of the cash flows arising from investments made prior to that point. (1) In respect of the latter cash flows, it would seem to be appropriate to assume an expected growth rate in free cash flows from year T+1 equal to the expected inflation rate in some price index (outputs and/or inputs), denoted i. (2) To overcome the problem that the free cash flow in year T+1 incorporates the replacement investment in that year, and the level of the latter may be untypical, some normalised measure of replacement investment in year T+1 typically substitutes for actual replacement investment in that year. So, defining [W.sub.T+1] as the unlevered post-company tax cash flow in year T+1 from operations before any deduction for replacement investment, [NR.sub.T+1] as normalised replacement investment for year T+1, and defining

[X.sub.T+1] = [W.sub.T+1] - [NR.sub.T+1] (2)

the expected value of the firm in year T is then as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

Substitution of equation (3) into equation (1) yields the following result:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)

The usual measure for [NR.sub.T+1] is accounting depreciation in that year ([AD.sub.T+1]) (3). This will be "correct" if the firm is in "steady-state" (i.e., equal physical replacement each year) and replacement costs have not changed over the period since acquisition of the assets that are included within [AD.sub.T+1]. (4) However, replacement costs will in general have grown over that period and failure to recognise this implies that accounting depreciation will understate replacement costs. Accordingly, Cornell (1993, p. 159) argues that [AD.sub.T+1] should be adjusted as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)

where [N.sub.A] is the accounting life of the firm's asset and [i.sub.H] is the inflation rate in replacement costs over the period in question. (5) However, the model continues to assume that steady-state prevails.

In assuming that the firm is in steady-state, both of these approaches to replacement investment ignore the actual timing of replacement investment, and we now seek to properly account for that. To begin, suppose that the firm has one fixed asset in place at time T, for which the tax saving arising from tax depreciation in year T+1 will be [TS.sub.T+1], the asset is expected to be replaced M years subsequent to year T at a cost (net of any salvage value) of [C.sub.M+T], and it has an economic life of N years. (6) The year T+1 cash flow from operations [W.sub.T+1] includes the tax saving [TS.sub.T+1]. So, define [Z.sub.T+1] as [W.sub.T+1] stripped of this tax savings, i.e.,

[Z.sub.T+1] = [W.sub.T+1] - [TS.sub.T+1] (6)

Defining [i.sub.z] as the expected inflation rate in [Z.sub.t], the specification for E([V.sub.T]) shown in equation (3) is replaced by the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The first term on the right hand side here is the net present value of all new investment beyond year T. The second term is the present value of all cash flows after year T arising from investments in place at time T except those relating to the replacement of the fixed asset in place at time T and the tax savings arising from its tax depreciation. The third term is the present value of the tax savings arising from tax depreciation on this fixed asset from year T+1 until the end of its economic life. The fourth term is the present value of the net replacement cost of the asset in N years. The fifth term is the present value of the tax savings arising from tax depreciation on this replacement asset, over its economic life. The last two terms have counterparts at intervals of N years out to infinity. The depreciation tax saving arising in year t of this asset's life will be some proportion of the cost of the asset, and is denoted [P.sub.t]. So, the last equation can be written as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Letting [i.sub.F] denote expected rate of increase in future replacement costs and [C.sub.0] denote the current replacement cost of the asset (new) net of salvage value (7), the last equation can be written as follows: (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

All terms beyond the first three on the right hand side are now a geometric progression out to infinity. So, the last equation can be written as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)

Substitution of equation (7) into equation (1) yields the following current value for the firm.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)

This formula assumes the presence of only one fixed asset at time T. However, there will in general be multiple assets (j = 1, 2, ... S). For asset j, let the current replacement cost net of salvage value be denoted [C.sub.j0], the term to first replacement after time T be denoted [M.sub.j], the economic life be denoted [N.sub.j], and the tax savings from tax depreciation (in year t of the asset's life) as a proportion of the asset cost be denoted [P.sub.jt]. The current value of the firm is then as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)

The first term on the right hand side here is the present value of the flee cash flows for the next T years. The second term is the net present value of all new investment beyond year T. The third term is the present value of the operating cash flows after year T arising from investments in place at the end of year T, prior to any deductions for replacement investment and their associated tax effects. The fourth term is the present value of the depreciation tax savings from all assets in place at the end of year T until their first replacement. The last term is the present value of all replacement expenditures on these assets along with all subsequent depreciation tax savings.

In the face of many minor assets, a simplification of equation (9) would be to restrict the third and fourth terms to the major fixed assets (j = 1, 2, ... Q), and therefore leave the allowance for depreciation net of the tax effect on the remaining assets within the third term. So, the current value of the firm would be estimated as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (10)

where [Y.sub.T+1] is the unlevered cash flow from operations in year T+1 less "normalised replacement investment" for the minor assets less the depreciation tax savings for the major assets. These valuation formulas in equations (8) and (9) can now be compared with equation (4). Although equations (9) and (10) seem formidable relative to equation (4), all of the additional parameters are either observable or capable of estimation by at least corporate insiders.

3. A Stylised Example

To explore the extent of variations between equations (4) and (9), consider the following stylised example. A firm currently has unlevered cash flow from operations of [W.sub.0] = $100m, which is expected to grow at 2% per annum indefinitely. The firm has one fixed asset, which requires replacement every 20 years, and no further investment is anticipated. The current cost of replacing this asset (new) is [C.sub.0] = $1100m, this cost has grown at 2% per annum since the last replacement, and is expected to grow at this rate indefinitely. Also, the asset's salvage value will be zero, the firm's weighted average cost of capital is k = 10%, the corporate tax rate is 33%, and straight line depreciation over the 20 year economic life of the asset is used by both the firm and the tax authorities.

Suppose that T = 5 and M = 1, i.e., the first replacement of the asset will arise one year after the valuation horizon of T = 5. So, the last replacement was 14 years ago. Given [C.sub.0] = $1100m and a historic inflation rate here of 2% per annum, the historical cost of the asset must have been $833.7m, which in turn implies [AD.sub.6] = $41.7m and [TS.sub.6] = $13.8m. Following equations (2), (3) and (4) with [NR.sub.6] = [AD.sub.6]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

By contrast, invoking equations (6), (7) and (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

So, equation (4) yields an estimate of the value of the firm that is higher than that from equation (8) by 135%.

Now suppose instead that T = 5 and M = 20, i.e., the asset is replaced immediately before the valuation horizon in five years, at a cost of $1214m, leading to [AD.sub.6] = $60.7m and [TS.sub.6] = $20m. Following equations (2), (3) and (4) with [NR.sub.6] = [AD.sub.6], the result is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

By contrast, invoking equations (6), (7) and (8), the result is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Equation (4) now yields an estimate of the value of the firm that is lower than that from equation (8) by 83%.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Table 1 presents results for a range of values for M and T. It also presents results arising from the use of both [NR.sub.T+1] = [AD.sub.T+1] and equation (5) under which [NR.sub.T+1] = [AD.sub.T+1] (1.22). The principal conclusions that can be drawn from these results are as follows. Firstly, the use of Cornell's equation (5) does not substantially alter the results. Secondly, higher values for T generally give rise to smaller valuation differences. This is to be expected, because T corresponds to the period in which the conventional valuation model generates the same valuation of cash flows as that from equation (8). Thirdly, across the range of possible values for M, the range of valuation differences is dramatic; even when T = 10, the valuation differences range from 55% over to 27% under when using [NR.sub.10] = [AD.sub.10] and from 48% over to 39% under when using equation (5). (9)

4. A Real World Example

4.1 Introduction

Having illustrated the potential for a significant divergence between the results from equations (4) and (9) using a stylised example, we now consider a more realistic case with T = 5 years. (10) This involves Telecom New Zealand, which is the largest company listed on the New Zealand Stock Exchange. The most recently available Financial Statements are for the year ended 30.6.2006, and reveal cash flows from operations prior to any deduction for replacement or new investment ([W.sub.0]) of approximately 2000m. (11) Purchases of fixed assets averaged $700m per year over the previous five years, with figures ranging from $607m to $828m. (12) These expenditures on fixed assets include replacement investment, but they include some new investment as well. (13) Furthermore, new investment can take the form of purchasing stakes in other companies, and Telecom's net investments of this type in the last five years have averaged $14m per year, but with considerable variation. So, total investment has averaged $714m per year over the previous five years, and with considerable variation.

All of this information is historic, and gives only limited insight into its anticipated expenditures for replacement and new investment over the next T = 5 years. However, our concern lies in comparing the outcomes from equations (4) and (9), and projected expenditures for replacement investment over the next five years are common to both approaches. So, in the interests of simplification, we assume replacement expenditures of [R.sub.t] = $400m per year. For the same reason, we assume no new investment at any future point. Consistent with assuming no new investment, we project [W.sub.t] to grow at the CPI inflation rate from the current figure of [W.sub.0] = $2000m.(14) We estimate this inflation rate at .02, consistent with recent long-run forecasts (The Treasury, 2006). The expected free cash flows are then as shown in Table 2. We also invoke an estimate of .02 for the expected inflation rate i (in free cash flows after year 5). Finally, we assume a discount rate of k =. 10.

4.2 Conventional Valuation

We start with a valuation of Telecom in accordance with the conventional approach. Following equations (2), (3) and (4) with [NR.sub.6] = [AD.sub.6], the current value of the firm is estimated as follows: (15)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

To express this outcome in terms of the firm's share value on 30.6.2006, we deduct the book value of debt at 30.6.2006 of $3498m, and divide the residue by the number of ordinary shares at that time of 1961m, to yield a share value of $7.98. (16) Had equation (5) been invoked, we would require an estimate for the average accounting life of the firm's assets ([N.sub.A]) and an inflation rate on the replacement costs of assets included in [AD.sub.6] over the period from their acquisition until year 6 ([i.sub.H]). In respect of [N.sub.A], the estimate is 21 years, (17) which implies that [i.sub.H] spans the period from 15 years ago until six years into the future. In respect of [i.sub.H], for illustrative purposes here, we adopt an estimate of .02. (18) With these parameter values, [NR.sub.6] = [AD.sub.6](1.23). Consequently [V.sub.0] = $18,174m and therefore the share value is $7.48 rather than $7.98.

4.3 Valuation with Recognition of Replacement Timing

We now turn to equation (9). This requires expected inflation rates [i.sub.z] (for operating cash flows exclusive of replacement costs) and [i.sub.F] (for replacement costs); for illustrative purposes, we adopt estimates for each of these rates of .02, consistent with the estimate for i. Equation (9) also requires identification of the firm's assets in five years along with their residual depreciation tax savings, current replacement cost (new), residual economic life, the economic life of the replacements, and the time profile of the depreciation tax savings on these replacements. The set of such assets will be differentiated by both their purchase dates and their economic lives. Although an analyst might have such information in respect of the firm's major assets, we lack it. Nevertheless, it is still possible to implement equation (9) by drawing upon information in Telecom's Financial Statements. To do so, it is necessary to make some assumption about the distribution of economic lives across its assets, and we start by assuming that each of the firm's assets has the same economic life. (19)

In respect of the economic life of its assets, Telecom has depreciating fixed assets (at 30.6.2006) with an aggregate historic cost of $11,000m, aggregate depreciated historic cost of $3200m, depreciation of $538m, and it uses the straight-line depreciation methodology. (20) It follows that the average accounting life of the firm's assets is $11,000m/$538m = 21 years. In respect of this process, the Institute of Chartered Accountants (NZ IAS 16, page 21) requires that depreciation be allocated over the "useful" life of the asset. This implies that the average economic life of the firm's assets would also be 21 years, and we ascribe this term to each of its assets.

In respect of the current replacement cost of Telecom's assets, this is not disclosed in its Financial Statements (which are prepared on the Historic Cost rather than a Replacement Cost basis). However, having invoked an estimate for [i.sub.H] (of .02) in implementing equation (5), this implies that the current replacement costs of Telecom's assets would be equal to their historic costs compounded at .02 since the time of purchase.

In respect of the distribution of the depreciation tax savings over the life of Telecom's assets, Inland Revenue allows taxpayers a choice of straight-line or DV depreciation (Inland Revenue Department, 2007). In respect of straight-line depreciation, the allowed rate on telecommunications equipment with a useful life closest to the average life of 21 years referred to above, being 20 years for "cabling", is .066 (Inland Revenue Department, 2007, page 43). The DV rate is .09, which implies a lower present value for the tax savings. So, we invoke the straight-line rate of .066, which implies depreciation of the assets over 15 years. (21)

We now seek to estimate the age profile of Telecom's assets in five years time. To do this, we start by estimating the age profile of its currently held assets. Letting the historic costs of these assets, and their purchase dates relative to the current point in time, be denoted as follows

[HC.sub.-1], [HC.sub.-2], [HC.sub.-3], ....

then any age profile that is invoked must satisfy the dual requirements that the aggregate historic cost is $11,000m and the aggregate current depreciated historic cost is $3200m. Since all assets are assumed to have an economic life of 21 years, this implies the following:

[HC.sub.-1] + [HC.sub.-2] + .... + [HC.sub.-20] = $11,000m (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (12)

In addition, the age profile must be consistent with the replacement expenditures that are expected over the next five years. In respect of the latter assets, the assumption that $400m per year will be expended, along with [i.sub.H] = .02 and an economic life of 21 years for all assets, implies that the historic cost (HC) of each of these five sets of assets must then be $400m discounted at 2% for 21 years, i.e., $264m. The current depreciated historic cost (DHC) of each set then follows from this historic cost along with the date of purchase. The results for these five sets of assets are shown in the first five rows of Table 3.

In respect of the remaining assets, deduction of the aggregate HC and DHC of the assets to be replaced within the next five years from the figures of $11,000m and $3200m reveals that the aggregate HC of the remaining assets must be $9680m and their aggregate DHC must be $3011m. Since the economic lives of all of these assets is 21 years, then the average residual life of these assets (R) must satisfy the following equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

It follows that R is approximately 7 years, and therefore the average period since purchase of the assets must be approximately 14 years. Thus, for illustrative purposes, we will act as if these remaining assets were purchased 13, 14 and 15 years ago. Since there are only two equations to satisfy in the form of (11) and (12), and therefore only two unknowns, we arbitrarily let [HC.sub.-14] = $3000, which implies a current DHC of $1000m as shown in Table 3. The remaining assets (costing [HC.sub.-13] and [HC.sub.-15]) must then have an aggregate HC and DHC of $6680m and $2011 m respectively, i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Simultaneously solving these two equations yields the solution shown in Table 3. We have therefore specified an age profile for Telecom's currently held assets that is consistent with equations (11) and (12) and the replacement expenditures expected over the next five years. (22)

We now turn to the situation in five years. The last three sets of assets in Table 3, along with the five sets of replacements over the next five years, form the complete set of assets in five years, and significant characteristics of them are shown in Table 4. The purchase dates of the assets are shown in the first column of Table 4. The historic costs (HC) of the assets are as described above, and are shown in the second column of Table 4. The residual lives of the assets in five years (M) follow from the purchase dates along with the economic life of 21 years, and appear in the third column. The values for [AD.sub.6] are simply equal to HC/21, and appear in the fourth column. The values for [TS.sub.6] in the fifth column are simply 33% of the tax depreciation, and the latter follows from the purchase date along with the straight-line rate of .066 over 15 years. For each set of assets, the aggregation of these tax savings over the remaining life of the asset (M), using a discount rate of 10%, is shown in the sixth column. As discussed earlier, the current replacement costs of the assets new ([C.sub.0]) reflect the values for HC along with inflation of [i.sub.H] = .02 per annum since purchase, and this is shown in the seventh column of the table. For example, for the assets purchased 15 years ago, at a cost of $5604m, the current replacement cost (new) is $5604m compounded at 2% for 15 years, to yield [C.sub.0] = $7542m.

We now substitute these parameter estimates into equation (9). We start with E([Z.sub.6]) as defined in equation (6), and draw upon the relevant data from Tables 2 and 4, to yield the following result:

E([Z.sub.6]) = E([W.sub.6]) - [TS.sub.6] = $2208m - $45m = $2163m

In respect of the present value of the depreciation tax savings over 15 years, expressed as a proportion of its historic cost, this is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Finally, we require the last term in equation (9) for each of the eight sets of assets in Table 4. For the assets purchased 15 years ago, the calculation is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

This result along with the results for the other seven sets of assets are shown in the last column of Table 4, and their sum over all eight sets of assets is -$13,541m. Substitution of these results into equation (9) yields the following valuation of the firm:

E([V.sub.5]) = [$2163m/.10 - .02] - $13,541m = $13,497m

[V.sub.0] = [$1600m/1.0] + [$1640m/[(1.10).sup.2]] + [$1681m/[(1.10).sup.3]] + [$1722m/[(1.10).sup.4]] + [$1764m+$13,497m]/[(1.10).sup.5]] = $14,725m

To express the result on a per share basis, we again deduct the book value of debt of $3498m, and divide the residue by the number of ordinary shares of 1961m, to yield a share value of $5.72. By contrast, the estimate from equation (4) is $7.98, or $7.48 if Cornell's adjustment is invoked, and these latter figures are 31%-40% larger than from equation (9). (23) The dramatic variation is attributable to three factors. Firstly, in respect of asset replacements after the valuation horizon, the present value of the replacement costs net of the tax savings from depreciation are a large proportion of the value of the firm. (24) Secondly, as shown in Table 4, the overwhelming proportion of the assets held by the firm in five years time appears to require replacement within a short period. Thirdly, equation (4) fails to recognise this fact and implicitly assumes that replacement conforms to a steady-state pattern, i.e., 1/21st part per year with a consequent average term until replacement of 10.5 years.

4.4 Alternative Assumptions

The results from equation (9) shown above are predicated upon all of Telecom's assets having the same economic life (21 years). An extreme alternative to this would be to assume that its assets comprised two types, one with a "low" economic life and the other with a "high" economic life. Inland Revenue (2007, page 43) provides estimated economic lives for a range of telecommunications equipment ranging from 5-50 years, with virtually all in the range 5-25 years. Thus, we invoke figures of 10 and 30 years and seek to construct an age profile for Telecom's currently held assets that is consistent with this "Bernoulli" distribution of economic lives. The age profile must also satisfy the requirement that the aggregate HC of the assets is $11,000m and the aggregate DHC is $3200m. Table 5 presents such a profile, and was derived in a similar fashion to that in Table 3. Table 6 then shows characteristics of the assets that are expected to be held in five years, with the underlying calculations paralleling those for Table 4. (25) Using the results in Table 6, the resulting share valuation for Telecom is $5.40 rather than the figure of $5.72 obtained in the previous section; the difference is less than 6%. Thus, across the range of plausible possibilities for the distribution of the economic lives of Telecom's assets, there is no significant variation in the share value.

The results for equation (9) shown above also reflect an estimate of the expected rate of appreciation in future replacement costs ([i.sub.F]) and an estimate of the rate of appreciation in the replacement costs of assets included in [AD.sub.6] over the period from their acquisition until year 5 ([i.sub.H]). Any variation from these two parameter estimates will affect the difference in results from equations (4) and (9), and variation in [i.sub.H] is the more significant issue. So, we examine variations in the estimate for [i.sub.H], which will affect the current replacement cost of assets in equation (9) and affect Cornell's depreciation adjustment in equation (5); the results are shown in Table 7. As [i.sub.H] increases, the share value arising from equation (4) declines in so far as Cornell's adjustment is invoked. However, the share value from equation (9) declines more significantly, and therefore the divergence from equation (4) becomes even greater. At [i.sub.H] = .04, the estimate of the share value from equation (4) will then be 111%-145% higher than from equation (9) rather than 31%-40% higher. This suggests that knowledge of the current replacement costs of the firm's assets is very significant information for the purposes of valuing firms. (26) This information will appear in Financial Statements based upon replacement cost asset valuations, and such Statements are therefore preferable for company valuation purposes over those based upon historic cost valuations.

4.5 Comparison of Assumptions

Before concluding, we summarise the assumptions relating to asset replacements that underlie the "conventional" valuation model as shown in equation (4) and the model proposed in equation (9). Both approaches assume that the timing of future asset replacements is known rather than stochastic. Equation (4) further assumes that the firm is in "steady-state" (i.e., equal physical replacement each year). Furthermore, in the absence of Cornell's adjustment shown in equation (5), it is also implicitly assumed that replacement costs have not changed over the period since acquisition of the assets included in [AD.sub.T+1]. By contrast, equation (9) does not require either of these additional assumptions. In particular the model is designed to accommodate variations from steady-state and recognises inflation in replacement costs over the period since acquisition of the assets included in [AD.sub.T+1]. Thus, equation (9) embodies less restrictive assumptions than that of equation (4). However, it requires considerably more information about a firm's fixed assets. Nevertheless, with the exception of estimates for [i.sub.F] and [i.sub.H], which are in any case required to implement equation (4), a set of Financial Statements constructed using Historic Cost is sufficient to implement equation (9) to an acceptable degree of approximation. Furthermore, in the event of Financial Statements prepared upon the Replacement Cost basis being available, an estimate for [i.sub.H] is then redundant. Such Statements are therefore preferable for company valuation purposes over those based upon historic cost valuations.

5. Further Comments

The analysis presented in sections 3 and 4 indicates that the divergence in valuation results between the conventional model and that proposed here is most pronounced when fixed asset replacement costs are relatively large, when they are (at least sometimes) concentrated around a point in time, when such a concentration occurs close to the valuation horizon used in the conventional model, and when the valuation horizon is not far into the future. The first two of these points suggest some characteristics of firms for which the divergence in valuation results would be most pronounced. In respect of fixed asset replacement costs being large, this simply requires a firm to have a high level of fixed assets relative to its revenues. Such a characteristic would be typical of firms that are engaged in manufacturing, but is not unique to such firms; Telecom is a contrary example. In respect of replacement costs (at least sometimes) being concentrated in time, this will occur if a firm's fixed assets are dominated by a single individual asset. Alternatively, the firm might have a large number of individual fixed assets with none involving a significant replacement cost but were all acquired at around the same time and have the same economic life. Alternatively, even if the firm had a large number of individual fixed assets with diverse economic lives and were acquired at different times, the timing of replacements will sometimes coincide; for example, an asset acquired 10 years ago with an economic life of 15 years and one acquired five years ago with an economic life of 10 years will simultaneously require replacement in five years and every 30 years thereafter.

Turning to the third point noted above, involving a concentration of replacements close to the valuation horizon, it might be thought the issue arises only when the replacement concentration occurs shortly after the valuation horizon and that under these circumstances simply shifting the valuation horizon forwards to include this replacement concentration would largely address the issue. However, the effect of doing so would simply be to replace an overvaluation of the firm with an undervaluation and the undervaluation may be even more serious. To illustrate this point, consider the results in Table 1 and suppose that T = 5 and M = 5, i.e., the asset replacement occurs five years after the valuation horizon and equation (4) generates an overvaluation by 36% when Cornell's approach is used and 48% otherwise. If T is increased to 10, then M becomes 20, i.e., the asset is now replaced just before the valuation horizon and consequently the next replacement occurs 20 years later. Consequently, as shown in Table 1, equation (4) now yields an undervaluation of the firm by 39% when Cornell's approach is used and 27% otherwise. In the case of T = 5 and M = 1, the results would have been even more dramatic. In particular, the overvaluation using equation (4) would be 118% using Cornell's approach and 135% otherwise, as shown in Table 1. However, if T is increased to 6, then M becomes 20 and the use of equation (4) then gives rise to an undervaluation of 88% using Cornell's approach and 61% otherwise. (27)

6. Conclusions

This paper analyses the issue of the timing of expenditures in replacing fixed assets within the context of valuing firms using the free cash flow approach. Standard practice amongst both practitioners and academics is to assume a smooth pattern in these expenditures past some future point, and such a pattern is improbable. This paper develops a model that rests upon much less restrictive assumptions, shows that the model is readily amenable to implementation, and that the difference in valuation results could be quite substantial. In implementing the model, it is necessary to estimate the current replacement costs of the firm's existing assets. This information will appear in Financial Statements based upon replacement cost asset valuations, and such Statements are therefore preferable for company valuation purposes over those based upon historic cost valuations.

Received on February 28, 2007; received in revised form on October 3, 2007; accepted on November 23, 2007.

References

Brealey, R. and Myers, S (2003). Principles of Corporate Finance, 7th edition, McGraw-Hill.

Copeland, T., Weston, J. and Shastri, K (200). Financial Theory and Corporate Policy, 4th edition, Pearson Addison Wesley.

Cornell, B (1993). Corporate Valuation: Tools for Effective Appraisal and Decision Making, Business One Irwin.

Dobbs, I (2004). 'Replacement Investment: Optimal Economic Life Under Uncertainty', Journal of Business Finance and Accounting, 31: 729-757.

Jardine, A. and Tsang, A (2006). Maintenance, Replacement and Reliability: Theory and Applications, Taylor and Francis.

Koller, T., Goedhart, M. and Wessels, D (2005). Valuation: Measuring and Managing the Value of Companies, 4th edition, John Wiley & Sons. Inland Revenue Department (2007). General Depreciation Rates (IR 265), www.ird.govt.nz.

Institute of Chartered Accountants of New Zealand (2005). Applicable Financial Reporting Standards, Institute of Chartered Accountants of New Zealand.

Mauer, D. and Ott, S (1995). 'Investment Under Uncertainty: The Case of Replacement Investment Decisions', Journal of Financial and Quantitative Analysis, 30 (4): 581-605.

Reserve Bank of New Zealand (2006). Consumers Price Index www.rbnz.govt.nz.

The Treasury (2005). Half Year Economic and Fiscal Update 2006 www.treasury.govt.nz.

* The helpful comments of the referees and the Associate Editor, Lyndon Moore, are gratefully acknowledged. The paper has also benefited from comments by Jerry Bowman, Glenn Boyle, and participants at a Victoria University workshop.

(1) New investment is that which expands the productive capacity of the firm whereas replacement investment merely maintains that capacity. In the process of replacing an asset, a firm might acquire an asset with greater productive capacity; in this event, it would be simultaneously replacing an asset and undertaking new investment. Nevertheless, even in this case, the two actions are still conceptually distinct.

(2) If revenues and pre-tax cash operating costs are expected to grow at rate i, post tax cash flows from operations will grow at a lesser rate between successive asset replacements, and experience upward jumps at replacement times, because the tax savings from tax depreciation are not subject to inflation until the time of replacing the asset. However, the valuation overstatement from acting as if post-tax cash flows from operations grow at rate i will be low for low values of i.

(3) Accounting depreciation is implicitly invoked by Koller et al (2005, p. 11 l) and Copeland et al (2005, p. 535) in the course of using NOPLAT (net operating profit less adjusted taxes) to generate the terminal value.

(4) "Correct" means that the value for [NR.sub.T+1] and its projection for subsequent years (at growth rate i) exactly matches the set of expected expenditures for asset replacements.

(5) The model implicitly assumes that the firm has only one class of asset. However, the model can be readily adapted to multiple asset classes with different lives by applying (5) to each individual asset class and then adding up over the results.

(6) There is a considerable literature on the question of determining the "optimal" economic life of an asset (see, for example, Jardine and Tsang, 2006). However, the firm's decision concerning the economic life of its assets is exogenous to the present exercise, which is concerned with the implications for the firm's value of the actual timing of its replacements beyond the valuation horizon (and therefore the age profile of its assets at the valuation horizon).

(7) The use of potentially different expected inflation rates [i.sub.z] and [i.sub.F] is natural once the operating cash flows [Z.sub.t] and replacement costs (net of tax savings from depreciation deductions) are separately acknowledged. By contrast, equation (4) does not separately treat such cash flows and therefore applies a single expected inflation rate i to the total free cash flows. This expected inflation rate i can be interpreted as a composite rate.

(8) This formulation readily admits technological development. For example, if one expects to replace the fixed asset in place at time T with a superior model, the expected inflation rate [i.sub.F] should impound any advantage that the new model has over the existing model in terms of purchase price, and the future cash flows [Z.sub.t] will impound any reduction in operating costs implicit in the new model. In respect of the first point, [i.sub.F] might be below [i.sub.z] on account of this. By contrast, equation (4) faces difficulties in taking account of a situation in which [i.sub.F] < [i.sub.z] because it invokes a composite inflation rate i.

(9) These numbers are peculiar to the stylised example considered here and different numbers would arise in alternative stylised examples.

(10) This term is less than the average figure of 8 years across the (academic) examples noted on page 4. However, the author's experience of practitioners' work has been that five years is typical.

(11) The Statement of Cash Flows reports net cash flows to equity holders from operating activities of $1807m. Adding back the interest payments of $294m, net of a tax deduction at the corporate tax rate of 33%, yields unlevered net cash flow from operations of $2004m. These figures are prior to any deduction for replacement or new investment.

(12) These figures are prior to any deduction for salvage values. Salvage values are not disclosed, but are presumably embodied within "sales of property, plant and equipment" in the Statement of Cash Flows. The latter figures average about 3% of fixed asset purchases over the last five years. Consequently, over that period, salvage values have been no more than 3% of the purchase price of the new assets. This is sufficiently small that we treat salvage values as zero.

(13) The latest Financial Statements (year ended 30.6.2006) provide some information on new versus replacement expenditure. In respect of New Zealand fixed lines capex (which comprises 65% of total capex), 70% of this is new rather than replacement expenditure.

(14) Telecom may be subject to increased competition and/or regulatory pressure in the future, and therefore the CPI inflation rate may overstate the expected growth rate in [W.sub.t] in the absence of new investment. However, the rate used here is purely illustrative, and would equally affect both the conventional and proposed models.

(15) The value for [AD.sub.6] of $556m is derived later in implementing equation (9), and is shown in Table 3.

(16) The book value of debt and the number of shares at 30.6.2006 were obtained from the Financial Statements. Also, by way of comparison, the observed share price at this time was $4.04. The variation may be attributable to the models used, to the parameter values used, or to the fact that the valuation conducted here assumed no new investment after year 5.

(17) This corresponds to the average accounting life of Telecom's existing assets, as discussed shortly.

(18) This corresponds to the average CPI inflation rate over the past 15 years (Reserve Bank of New Zealand, 2006). Alternative estimates will be considered shortly.

(19) This is a distinct issue to that of the age profile of the firm's assets at the valuation horizon. For example, even if each of the firm's assets had the same economic life of 21 years, the average residual life of its assets at the valuation horizon could be as low as one year or as high as 21 years. In respect of the assumption that all of the firm's assets have the same economic life, extreme variations from this assumption will be considered later.

(20) The firm's fixed assets are described in Note 17 to the Financial Statements (www.telecom.co.nz), and principally comprise telecommunications equipment. The firms fixed assets include some land, which has been excluded from the above analysis because it does not depreciate.

(21) Using the straight-line rate over 15 years, the resulting present value of the tax savings as a proportion of the asset cost is. 166 (as derived shortly). By contrast, using the DV rate of .09, this present value figure is only. 10.

(22) There are a range of alternative age distributions that satisfy equations (11) and (12) and the expected replacement expenditures over the next five years. However, these alternative distributions do not yield significantly different valuation results, and an example of this will be provided later.

(23) This result reflects one particular age profile for Telecom's existing assets that is consistent with equations (11) and (12) and the expected replacement expenditures over the next five years. Alternative profiles that satisfy these requirements do not lead to significantly different valuation results. For example, suppose that there were no purchases 14 years ago, purchases 13 years ago of $2,575m and purchases 15 years ago of $7,105m. The resulting share value of the firm is $5.74, which is almost identical to the $5.72 derived above.

(24) The present value of these net expenditures is $13,541m discounted for five years, which is $8,407m. This represents 57% of the $14,725m value of the firm.

(25) The only difference is in relation to the tax depreciation on the assets with a 10 year economic life. Inland Revenue (2007, page 43) indicates that telecommunications equipment with a 10 year economic life would be eligible for straight-line depreciation at the rate of 15% per year, and this depreciation process would then be completed in almost seven years. Consequently the present value of the resulting tax savings, as a proportion of the cost of the asset, would be .232 rather than the figure of .166 shown earlier. In respect of the assets with a 30 year life, the closest economic life in the Inland Revenue schedule is still 25 years, and therefore we continue to depreciate these assets for tax purposes at a straight-line rate of .066 over 15 years.

(26) The share prices noted in Table 5 range from $4.00 to $7.98, and this range embraces the observed value of $4.04 on 30.6.2006. Prima facie, this suggests that the market price does reflect the actual timing of replacement expenditures. However, there are alternative explanations. For example, the discount rate used in both equations (4) and (9) may be too low or the expected cash flows [W.sub.t] may be too high. Our purpose here is not to explain the observed share price but to show that value is highly sensitive to the treatment of replacement expenditures and to the current replacement cost of assets.

(27) The last two numbers are not shown in Table 1 but are determined in the same fashion as those in the table.

Martin Lally, School of Economics and Finance, Victoria University of Wellington. Phone 64-4-4635998; Fax 64-4-463-5014; Email: martin.lally.@vuw.ac.nz
Table 1: Valuation front Equation (4) Relative to Equation (8)

 [NR.sub.T+1] = [NR.sub.T+1] =
 [AD.sub.T+1] [AD.sub.T+1] (1.22)

 T = 5 T = 10 T = 5 T = 10

M = 1 135% 55% 118% 48%
M = 5 48% 25% 36% 19%
M = 10 8% 5% -2% -1%
M = 15 -11% -24% -21% -45%
M = 20 -83% -27% -100% -39%

Note: This table shows the valuation of the firm described in
section 3 using equation (4) less that from (8), expressed as a
proportion of the latter. The comparison is conducted for a range
of values for T (the valuation horizon), a range of values for M
(the residual economic life of the asset after T years), and two
methods for determining the normalised replacement cost of the
asset in year [T.sub.+1] when invoking equation (4).

Table 2: Projected Cash Flows for Telecom New Zealand

 Yr 1 Yr 2 Yr 3 Yr 4 Yr 5 Yr 6

E([W.sub.t) 2000 2040 2081 2122 2164 2208
- [R.sub.t] 400 400 400 400 400
=E([F.sup.u.sub.t]) 1600 1640 1681 1722 1764

Note: This table shows expected cash flows from operations for
Telecom over the next six years E([W.sub.t]), expected expenditures
on replacement investment over the next five years [R.sub.t], and
expected free cash flows over the next five years
E([F.sup.u.sub.t]) (all in Sin).

Table 3: Characteristics of Telecom's Existing Assets

Purchase
Date HC DHC

-20 $264 $264(1/21) = $13
-19 $264 $264(2/21) = $25
-18 $264 $264(3/21) = $38
-17 $264 $264(4/21) = $50
-16 $264 $264(5/21) = $63
-15 $5604 $5604(6/21) = $1601
-14 $3000 $3000(7/21) = $1000
-13 $1076 $1076(8/21) = $410

 [SIGMA] = $11,000 [SIGMA] = $3,200

Note: This table shows a possible distribution for the purchase
dates, historic costs (HC) and depreciated historic costs (DHC) for
Telecom's existing assets. This distribution is consistent with the
aggregate historic cost of the firm's existing assets, their
depreciated historic cost, the current depreciation and the firm's
use of straight-line depreciation.

Table 4: Characteristics of Telecom's Assets in Five Years Time

Pchd HC M [AD.sub.6] [TS.sub.6]

-15 $5604 1 $267 0
-14 $3000 2 $143 0
-13 $1076 3 $51 0
1 $400 17 $19 $9
2 $400 18 $19 $9
3 $400 19 $19 $9
4 $400 20 $19 $9
5 $400 21 $19 $9

 [SIGMA] = $556 [SIGMA] = $45

 [M.summation over (t=1)]
Pchd [TS.sub.5+1]/[(1.10).sup.t]

-15 [C.sub.0] []
-14
-13 0 $7542 -$8,098
1 0 $3958 -$3,941
2 0 $1392 -$1,285
3 $57 $392 -$69
4 $59 $384 -$55
5 $62 $377 -$42
 $64 $370 -$31
 $66 $362 -$20

 [SIGMA] = -$13,541

Note: This table shows characteristics of Telecom's assets in
five years, arising from the distribution in Table 3 and the
expenditures on asset replacements over the next five years. The
columns show the purchase date of the assets relative to the present,
the historic cost of the assets (HC), the residual economic life in
five years (M), the accounting depreciation in six years time
([AD.sub.6], the tax savings arising from depreciation at that time
([TS.sub.6]), the present value in five years of the depreciation
tax savings over the residual life of the assets, the current
replacement cost of the assets ([C.sub.0]), and the present value
in five years of all subsequent replacement expenditures net of
depreciation tax savings ([]).

Table 5: Characteristics of Telecom's Existing Assets

 Purchase
Life Date HC DHC

30 -23 $2640 $2640(7/30) = $616
30 -22 $5820 $5820(8/30) = $1552
10 -9 $328 $328(1/10) = $33
10 -8 $328 $328(2/10) = $66
10 -7 $328 $328(3/10) = $98
10 -6 $328 $328(4/10) = $131
10 -5 $328 $328(5/10) = $164
10 -4 $900 $900(6/10) = $540
 [SIGMA] = $11,000 [SIGMA] = $3200

Note: This table shows a possible distribution for the purchase
dates, historic costs (HC) and depreciated historic costs (DHC)
for Telecom's existing assets. This distribution is consistent
with the aggregate historic cost of the firm's existing assets,
their depreciated historic cost, the current depreciation and
the firm's use of straight-line depreciation.

Table 6: Characteristics of Telecom's Assets in Five Years Time

Pchd HC M [AD.sub.6] [TS.sub.6]

-23 $2640 2 $88 0
-22 $5820 3 $194 0
-4 $900 1 $90 0
1 $400 6 $40 $20
2 $400 7 $40 $20
3 $400 8 $40 $20
4 $400 9 $40 $20
5 $400 10 $40 $20
 [SIGMA] = $572 [SIGMA] = $100

 [M.summation over (t=1)]
Pchd [TS.sub.5+t]/[(1.10).sup.t] [C.sub.0] [ ]

-23 0 $4163 -$3678
-22 0 $8998 -$7371
-4 0 $974 -$1444
1 $49 $392 -$380
2 $63 $384 -$299
3 $75 $377 -$255
4 $86 $370 -$214
5 $96 $362 -$176
 [SIGMA] = $3,817

Note: This table shows characteristics of Telecom's assets in five
years, arising from the distribution in Table 5 and the expenditures
on asset replacements over the next five years. The columns show the
purchase date of the assets relative to the present, the historic
cost of the assets (HC), the residual economic life in five years
(M), the accounting depreciation in six years time ([AD.sub.6]), the
tax savings arising from depreciation at that time ([TS.sub.6]), the
present value in five years of the depreciation tax savings over the
residual life of the assets, the current replacement cost of the
assets ([C.sub.0]), and the present value in five years of all
subsequent replacement expenditures net of depreciation tax
savings ([]).

Table 7: The Effect of Variations in Historical Inflation Rates

Model [i.sub.H] = 0 [i.sub.H] = .02 [i.sub.H] = .04

Equation (4) with
 [NR.sub.6] =
 [AD.sub.6] $7.98 $7.98 $7.98
Equation (4)
 with (5) $7.98 $7.48 $6.90
Equation (9) $7.06 $5.72 $4.00

Note: This table shows the effect of historical inflation rates in
fixed asset prices ([i.sub.H]) upon the estimates of Telecom's share
value using equation (9). Results for equation (4) are also reported,
with and without the correction in equation (5), and the latter
results are not dependent upon such historical inflation rates.
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Date:Jun 1, 2008
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