The Cost of Capital, and a Note on Capitalization


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1 The Cost of Capital, and a Note on Capitalization Prepared by Kerry Krutilla 8all rights reserved Introduction Often in class we have presented a diagram like this: Table 1 B B1 B2 B3 C Co This kind of picture is useful to develop the mechanical aspects of discounting, but some important detail is left out. In this handout, I will selectively bring out some of these issues. Specifically, I want to discuss the concept of the cost of capital, largely from an individual investor perspective. The more complete social perspective will be deferred for later. Cost of Capital We have often referred to Co in Table 1 as the Ainitial year investment or the Ainvestment in year zero. This language usage is loose. In real life, Co probably refers to a combination of expenses on labor, energy, and capital. For example, the initial financial cost of a solar system includes both installation costs (labor, energy) and the capital cost of the system itself. Expenditures on labor and fuel do represent financial costs in the sense that labor time and energy are they are used up in year zero to install the system. But the Acapital is not really a cost in year zero: the financial cost of the capital is matched by the asset value of the capital, and the asset value of the capital does not begin to depreciate until the project begins (period 1 outward). Thus, the actual cost of the capital shows up in the outyears when the capital asset begins to be used (although the present value of these costs can certainly be stated in year zero terms). This reality is reflected in tax laws. An investor is not allowed to deduct their financial expenditure on capital in year zero. Rather, they deduct the costs of capital depreciation over a future period of time. 1 Fundamentally, there are two costs of capital. First, if the investor had not purchased the capital asset, the investment funds could have earned the default rate of return. Hence, the rate of return on investment funds in the next best alternative use is an opportunity cost of capital in this use. Secondly, if the capital in question is a depreciable asset, it begins to physically degradel with use, and its asset value declines over time. Depreciation, in value terms, is the second 1 The depreciation schedule in tax laws is usually not very close to the actual schedule over which capital value declines.
2 fundamental cost of capital. In Table 1, there is an implicit assumption that the asset value of capital declines to zero by the end of the project horizon, because there is no terminal value for capital at the end of the project horizon. Table 2 indicates a profile for a project with lessthancomplete capital depreciation: Table 2 B= B1 B2 B3 C3 C Co Here, the positive figure C3, shows up. This is the salvage value or value that could be realized by selling what remains of the capital asset at the end of the project. reflects a physical process but, as always, we are concerned with the economic implication B how the value of capital changes over time as a result. The asset value of capital is determined by its value outside the project. Initially, in year zero, the investor pays the market price for capital. The market price for capital is the value of the capital in the market outside the project B what the investor has to pay to bid this capital out of another economic sector. At the end of the project horizon, C3 in Table 2 refers the sale value of the capital to some buyer outside the project. This buyer=s willingness to pay determines the asset value of the capital. The value of the capital in the project at the end of the project horizon is itself zero, or very close to zero, because future returns beyond the project horizon are not positive, on net. The depreciation pathway from the start of the investment to the end of the investment is not clearly specified in Tables 1 and 2. If we ignore the complication of income taxation, and are not considering the project horizon itself as a choice variable, then the particular depreciation pathway from beginning of the project=s life to the end does not matter to the assessment. If the netpresent value of the projects in Tables 1 and 2 are positive, it makes sense to deploy the capital in the project, and how that capital would be valued elsewhere during the project=s life is not relevant. If NPV is less than zero, then the capital should not be deployed into the project in the first place. On the assumption again that NPV is positive, depreciation could even be ignored if the investor had to sell the investment early B say at the end of year 1 B given a couple of conditions. First, that the endofyear 1 net present value of future returns is positive; second, that the returns could be reflected in the sale price of the investment itself (more on this below). In this case, another investor would buy the project and keep the capital in the project. As such, the value of the capital outside the project during the project=s life is not relevant. It only becomes relevant in the last period of the project B the end point of the depreciation pathway B when the possibility of salvage value arises (as in Table 2) We now turn to a more focused look at the two fundamental capital cost components: rate of return and depreciation. A good way to illustrate these concepts is by looking at an auto dealer=s decision to sell a car, versus the decision to lease a car for a certain period.
3 Suppose the dealer can sell the car for $13000 in year zero, and invest this money in a secure investment that yields 5%. This option is described in three equivalent ways in the following three tables: Table 3 B 15,054 Table 4 B1 2,054 B2 13,000 Table 5 B B2 13,000 Table 3 shows that the dealer forgoes $13,000 in year 0, in exchange for a cumulated sum of $15,054 at the end of year 3 (13,000 X (1.05) 3 = 15, 054. Table 4 simply breaks out the cumulated sum into the cumulated rate of return component (2,054) and the original 13,000. The layperson=s language to describe this situation might be something like: AI can put 13,000 in the bank for three years, and then withdraw this sum of money plus the accumulated Table 5 breaks out the cumulated return of 2,054 into its annuity stream equivalent. This is just 5% of 13,000 B the rate of return. The annuity stream can be produced by the following logic: the investor could put 13,000 in the bank which would grow to 13,650 by the end of period 1. Removing the payment 650 again gives 13,000 which again grows to 13,650 at the end of period 2. Pulling off another 650 payment yields 13,000 which a grows to 13,650 by end of period 3, at which point, the investor can remove both the 650 and the 13,000. (Alternatively, the 650 could
4 be produced by taking Crn from Table A3 for 5% and three years, and dividing 2,054 by this factor). The form of Table 5 looks like that of Table 2. We can visualize the investment in Table 5 as a financial investment that yields a rate of return of 5% per period (650 per period) without any capital depreciation, in the sense that the $13,000 shows up at the end of year 3 as a terminal value. For the moment, compare the dealers= investment in Table 5 to an alternative financial investment that yields 7% per period: Table 6: 7% investment B B2 13,000 Subtracting Table 5 from Table 6 we have: Table 7 B B2 C 0 Table 7 shows that the net difference between the two investments reduces to the net difference in the rate of return accumulations. That is, for investments without capital depreciation, the net comparison reduces entirely to a comparison of the rates of return accumulations occurring beyond year 0. Therefore, we can state the net difference of the superior investment option in Table 6 as: Net = Gross Rate of Return of 7%  Rate of Return Opportunity Cost of 5% It is thus accurate to say that the opportunity cost of the better investment is the opportunity cost of the foregone rate of return on the investment alternative (5%) beyond year zero. This is obviously not the same thing as saying that the cost of the better investment is $13,000 in year zero. This point gets us back to the looseness of the language that describes investment costs in terms of year zero financial outlays. We now return to the assumption that 5% is what the car dealer can make if the car is sold at the money invested (again Tables 3Table 5). Alternatively, the dealer can lease (or rent) the car. In the lease option, the car dealer allows the customer to lease the car from some fixed time period.
5 We will assume the lease period is three years. At the end of this period, the customer has the option of returning the car, or buying the car at a stated price. The stated buyback price is the dealer=s estimate of the car=s asset value at the end of the lease period. That is, the amount the dealer believes the car can be sold for immediately after the lease period expires. We will assume that value is $8,000. Assuming way minor frictions and transactions costs, the dealer should be basically indifferent whether the customer decides to return or buy the car. If the customer returns the car, the dealer has a physical asset he or she views as worth $8,000. If the customer buys the car, the dealer has a financial asset worth $8, Here=s how the lease option looks from the car dealer=s perspective: Table 8 B1 L L L B2 8,000 The dealer gives up the $13,000 he or she would have realized from the car sale, in exchange for three lease payments, L, and 8,000 in asset value at the end of period 3. The question is: how much should the lease payments be? To get some intuition, consider the investment without the lease payments: Table 9 2 The dealer should estimate the car=s asset value with reasonable accuracy. They obviously have no incentive to understate the value (overestimate depreciation), i.e., to agree to accept a sale price for the car less than the sale price they could receive from another buyer. The dealer may have some incentive to overstate the value (underestimate depreciation), increasing their profit margin somewhat if the customer decides to buy the car. But since the dealer does not know whether or not the customer will buy the car this incentive should not be too strong, particularly since the dealer would just assume sell the car to this customer as another (sparing the transactions costs of handling another sale.) In this problem, we will we assume that the stated buyback price is an accurate reflection of the car=s actual market worth.
6 B1 B2 8,000 Without lease payments, the dealer is giving up 13,000 in year zero in exchange for 8,000 in year three. Compare this to the default financial investment depicted in Table 4, which shows the dealer giving up 13,000 in year zero in exchange for 13,000 at the end of year three and a 5% rate of return accumulation of 2,054. If we subtract Table 9 from Table 4 we can see that the default alternative does better than the investment in Table 9 as follows: Table 10 B1 2,054 B2 5,000 Total 7,054 From the year 3 window, the financial investment is better than the lease investment Bwithout any lease payment B by the sum of the rate of return accumulation of 2,054, plus the sum of the asset value that does not depreciate, This situation is graphically depicted in Figure 1. $15,054 is how much 13,000 invested in the bank accumulates to at the end of year 3. On the other hand, $8,000, is how much the investment Alease car without lease accumulates to at the end of year 3. The bank investment is better than the lease arrangement by the sum of the rate of return accumulation, 2054, plus the sum of the avoided depreciation, 5,000, for a total of 7,054. The implication of these numbers is that the stream of lease payments shown in Table 8 must accumulate in value to 7,054 by the end of year three for the lease arrangement to do as well as the alternative Asell the car and invest the money in a nondepreciable financial asset at That is, the lease payment must cover (a) the opportunity cost of the rate of return on the alternative use of capital (2,054) and (b) the depreciation of the asset value of the car (5000). This particular example illustrates a totally general proposition. Since the opportunity cost of capital is in the form of a forgone rate of return accumulation and asset depreciation, the benefit of the capital deployment in the project must cover these costs. The lease payment can be calculated in a number of ways. Way 1. Find the relevant annuity compounding factor in the financial table, Table A.3. It is for 3 years at 5%. Then do this division: 7,054/3.152> about Way 2. From Table 9, discount 8,000 at 5% to the present, subtract from 13,000 to give: Table 11
7 B2 6,912 < 8,000 Net 6,088 Note that 6,088 is the present value of the difference between the bank investment and the lease investment, which can be verified by discounting the future value difference 7,054: 7,054 X.864 = 6, That is, 7,054 is the FV cost of capital in the lease option, while 6,088 is the PV of the cost of capital in the lease option. The next step is to use the reciprocal of discount factor, from table A4, annualize the 6,088, which is about Thus, AWay 1" and AWay 2" are consistent. 4 There is one final point to make. The lease payment stream, about 2338, covers the total cost of the capital in question: its rate of return of return component, and its depreciation component. These two can be separated. Since we already know that the rate of return accumulation on 13,000 is 650, we can subtract that from the lease payment to get the residual depreciation component. We thus have: Table 12 Future Value Accumulation Rate of Return ,054 Depreciation ,000 Total Lease Payment ,054 3 Difference due to rounding error from discount factors rounded to three decimal places. 4 The discrepency being a function of roundoff error.
8 You can verify that the future value of each of these annuities is approximately as shown. 5 As a final step, we can translate this whole set up into percentage terms, by dividing the annuities by 13,000. Thus, we have: Table 13 Rate of Return 5% Depreciation 12% Total Lease Payment 17% Hence, the average return from the lease payment (17%) must be sufficient to cover the rate of return opportunity cost (5%) and the average rate of deprecation (12%). A note on capitalization I have referred to a couple of times and this idea is important enough to discuss further. Suppose we have this kind of investment Table 14 4 B C 100 The person who is considering this investment has a 5% discount rate. At a 5% discount rate, The NPV of this investment is: Suppose this person unexpectedly had to sell the investment at the end of year 2. In this case, the investment would not pass the twoyear Apayback criterion. However, I mentioned before that this isn=t a good argument for the payback criterion if the future value of the investment can be capitalized, i.e., that the investment could be sold at the end of period three for the value of the future revenue stream. Discounting the revenue stream in year 3 and year 4 back to the end of year 2 converts the investment in Table 14 as follows: Table B B C Again, there will be slight differences from roundoff error.
9 where B2 is the sale price of the investment that Acapitalizes the future As such, the NPV of the investment in Table 15 is the same as for the investment in Table 14.
10 Figure 1 PV FV End of Year 3 15, 054 2,054 13,000 13,000 7,054 5,000 8, Year
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