Present Financial Position And Performance Of The Firm
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Handout #3
Agricultural Economics 489/689 Handout #3
Spring Semester 2008 John B. Penson, Jr. 2
A. Time Value of Money
The concept of the time value of money is based upon the economic fact that $1 today is worth more than the promise of $1 at some future date because of its current earnings potential. Other reasons for preferring payment today may include your personal preference to spend this dollar now on a consumer good rather than postponing consumption until later. Or perhaps the promise of a payment at a future date carries with it less than complete certainty that the payment will be received.
We will focus on the economics of the time value of money. The time value of money can be viewed either within the context of present value of future sums, or future value of present sums. One is the opposite of the other.
We will confine our discussion to the present value of future sums or stream of income given our interest in capital budgeting.
Present value of a future sum
Letting FVN represent the value of a payment to be received N periods from now. We want to know what the present value of that future payment is today. To find this value, we must discount FVN back N periods by the rate of return (R) we could have received as our next best opportunity, or:
N (33) PV = FVN/(1+R) or
(34) PV = FVN(PIFR,N) where PIFR,N is the present value interest factor for interest rate R and N periods found in Appendix Table 1 and PV is the present value of a sum FVN received N periods from now.1
For example, what is the present value of $500 to be received 10 years from today if the discount rate is 6 percent?
(35) PV = $500/(1+.06)10 = $500[1/(1.791)] = $500(.558) = $279
1 N The PIFR,N interest factors in Appendix Table 1 are based upon the formula 1/(1+R) . The EPIF interest factors used starting in equation (36) are based upon the formula (1- PIFR,N)) ÷ R. 3
Thus, the present value of the $500 to be received in 10 years is $279.
Present value of an equal periodic stream
Suppose instead of receiving $500 in a single payment 10 years from now, you were offered the opportunity to receive annual payments of $50 over the next 10 years. Since these payments are of equal value over the 10-year period and the discount rate is the same over time, we can use the following approach to calculating the present value of this stream of payments:
(36) PV = NCFE(EPIFR,N) = $50(7.360) = $368 where EPIFR,N represents the equal payment-present value interest factor found in the equal payment interest factor Appendix Table 2. The value of EPIF.06,10 is 7.360, which gives us a present value of $368. Thus, the present value of a stream of $50 annual payments over a 10 year period ($368) is greater than the present value of a single payment of $500 received 10 years from now ($279). Why? Because of the time value of money received earlier in the 1- year period.
We could have arrived at the same value taking the longer approach of calculating the present value of each annual payment and then adding the payments together, or:
(37) PV = NCF1 (PIF.06,1) + NCF2 (PIF.06,2) + …. + NCF10 (PIF.06,10) = $50(.943) + $50(.890) + …. + $50(.558) = $368 which is the same as:
2 10 (38) PV = NCF1 [1/(1+R)] + NCF2 [1/(1+R) ] + …. + NCF10 [1/(1+R) ] = $50(1/(1+.06)] + $50[1/(1+.06)2] + …. + $50[1/(1+.06)10] = $368
Remember the assumption in equation (36) is that both the size of the annual payments and the annual discount rate chosen are identical in each year.
Present value of an unequal periodic stream
Suppose that, instead of receiving an equal annual stream of $50 payments, you received the $500 in two installments: $250 after 5 years and $250 after 10 years. Equation (37) no longer is applicable in this case. We can instead use the approaches outlined in general form in equations (37) and (38) as follows: 4
(39) PV = NCF5 (PIF.06,5) + NCF10 (PIF.06,10) = $250(.747) + $250(.558) = $186.75 + $139.50 = $326.25 or
5 10 (40) PV = NCF5 [1/(1+R) ] + NCF10 [1/(1+R) ] = $250[1/(1+.06)5] + $250[1/(1+.06)10] = $326.25
This present value is less than the present value of the steady stream of $50 annual payments ($368 given by equation (37) or (38)) since less is received earlier in the period, but more than the single payment received 10 years from now ($279 given by equation (35)).
Thus far we have assumed a single valued discount rate over the 10-year life of this analysis. If an investor is potentially exposed to unique degrees of risk exposure over the economic life of the investment project, we need to account for this when calculated the present value.
Present value with unequal discount rates
All the equations involving calculation of the present value of a future stream thus far has assumed identical discount rates (i.e., R1 = R2 = … = RN). The use of the present value interest factor tables in the back of all financial management textbooks rest on this assumption. That means that equations (36), (37) and (38) are not applicable if this assumption does not hold. Let’s relax this assumption be restating equation (38) as follows:
(41) PV = NCF1 [1/(1+R1)] + NCF2 [1/{(1+R1)(1+R2)}] + …. + NCFN [1/{(1+R1)(1+R2)…(1+RN)}]
If the discount rate increases by one-half a percentage point each year for reasons we will explore later, the present value following equation (41) will be:
= $50(1/(1+.06)] + $50[1/{(1+.06)(1+.065)}] + …. + $50[1/{(1+.06)(1.065)…(1+.105)}]
We have covered a number of variations in the calculation of the present value of a future sum or stream of cash flows over time. There are several popular applications of these concepts we can explore before proceeding with the topic of capital budgeting. 5
Present value of infinitely lived periodic stream
Several examples come to mind. One is a perpetuity or an annuity that continues forever. Another is the expected cash rent received from an infinitely lived asset like land. Assume you can charge a cash rent of $50 per acre for farm land annually and the annual discount rate is 6 percent. The present or “capitalized” value of this farm land is:
(42) PV = NCFE ÷ RE = $50/.06 = $833.33 per acre.
In another example, the present value of a $100 perpetuity discounted back to the present at 5 percent is:
(43) PV = $100/.05 = $2,000
Amortized loans
The procedure for solving for an annuity payment when the discount rate, number of payments and present value are known can also be used to determine the level of payments associated with paying off a loan in equal installments over time. For example, suppose a company wanted to purchase a piece of machinery. To do this, it borrows $6,000 to be repaid in 4 equal payments at the end of each of the next four years. The interest rate to be paid to the lender is 15 percent on the outstanding portion of the loan. What we don’t know is the value of this payment. Given the information, we know that
(44) $6,000 = PI(EPIF.15,4) $6,000 = PI(2.855) so
(45) PI = $6,000/2.855 = $2,101.58
Thus, the annual principal and interest payment for this $6,000 4-year loan carrying an interest rate of 15 percent is $2,101.58.
We can state this problem in terms of the PI payment as follows:
(46) PI = LOAN/(EPIFR,N)
We can calculate the separate principal and interest payments for this loan that is needed to measure interest expenses for taxable income purposes. Let’s assume a $1,000 loan with annual payments over a 5-year period at an interest rate of 8 percent. Using equation (46), the principal and interest payments would be: 6
(47) PI = $1,000/(EPIF.08,5) = $1,000/(3.993) = $250.46 or $250.46 annually starting at the end of the first year. The interest portion of this payment in the first year would be equal to:
(48) I = $1,000(.08) = $80.00
The principal portion of this payment in the first year would be:
(49) P = $250.46 - $80.00 = $170.46 which means the interest payment in year two would be based upon $829.54 rather than $1,000. The entire loan repayment schedule would be:
Table 1 – Amortization table for $1,000 loan at 8% for 5 years. Year P I PI Balance 1 $170.46 $80.00 $250.46 $829.54 2 184.10 66.36 250.46 645.45 3 198.82 51.64 250.46 446.63 4 214.72 35.73 250.46 231.90 5 231.90 18.55 250.46 0.00
The concept of present value and discounting shown above was shown to have many applications.
Equation (46) can be twisted in any of four ways. First, you can solve for the level of the principal and interest payment or PI as we did above given the interest rate R, number of payments N and loan amount (LOAN). Second, you can solve for the level of the loan that is associated with a given payment PI, interest rate R and number of payments N, or:
(50) LOAN = PI(EPIFR,N)
The last two options require solving for the equal payment present value interest factor, or:
(51) (EPIFR,N) = LOAN/PI and then finding the corresponding values of R (if N is known) or N (if R is known) in the equal payment interest factor (EPIFR,N) Appendix table. 7
B. Capital Budgeting Methods
Capital budgeting involves the analysis of the additional net cash flows associated with investment projects over their entire economic life. The objective of capital budgeting, simply put, is to determine if the net benefits from making the investment is positive or negative. The following discussion describes for capital budgeting methods presented in the following order: payback period method, profitability index method, internal rate of return method, and net present value method.
Payback Period Method
The purpose of the payback period method is simply to find the number of years it would take for an investment to pay for itself. Suppose you are considering two mutually exclusive projects. Both cost $10,000 and have an economic life of 5 years. Further assume that the net cash flows generated by these two investment opportunities (project A and project B) are represented by the net cash flows below:
Table 2 – Net cash flows for two projects. Year Project A Project B 1 $3,000 $2,000 2 3,000 3,000 3 3,000 5,000 4 3,000 2,000 5 3,000 1,000 Total 15,000 13,000
Finally, assume that the terminal value (the market value of any assets acquired in by the project) at the end of the 5th year in both instances is equal to zero. We will tackle that issue later in this course.
Based on this information, the payback period or length of time required to recover your initial investment of $10,000 is 4 years for project A and 3 years for project B. That is, 4 years would elapse before you would accumulate enough net cash flows from project A to “pay back” the initial $10,000 as opposed to just 3 years for project B.
If we were to rank these projects according to the length of their payback period, we would prefer project B over project A.
The payback period method is computationally easy to use. It also provides a measure of the project’s liquidity. However, it fails to consider the timing of the net cash flows generated by a project both before the payback period has been reached as well as afterward. It largely ignores the time value of money. Finally, there is no objective decision rule associated with the method. That is, we are not maximizing profits, minimizing costs or attempting to satisfy some other objective. 8
Internal Rate of Return Method
The present value discussion thus far was based upon assuming a particular discount rate. One can ask the question of how much higher or lower this rate would have to be before the net present value of these projects would fall to zero. This information is provided by another capital budgeting technique incorporating the time value of money concept: the internal rate of return method.
The internal rate of return for an investment project is defined as that discount rate in equations that equate the present value of the annual net cash flows with the project’s net capital outlay. For discount rates lower than the internal rate of return (i.e., IRR>R), the net present value of the project will be positive. Conversely, the net present value of an investment project will be negative if the discount rate is higher than the internal rate of return (i.e., IRR
In computing the internal rate of return, therefore, we must find that value of R which results in a net present value equal to zero, or
(52) NPV = NCF1(PIFR,1) + NCF2(PIFR,2) + …. + NCFN(PIFR,N) – C 0 which is nothing more than equation (56) set equal to zero. If the net cash revenue flows are identical in each year of the investment project, we can instead use
(53) NPV = NCFE(EPIFR,N) – C 0 which is nothing more than equation (58) set equal to zero. Thus, all that remains is to find that value of R in these equations which results in NPV = 0.
Suppose that you wanted to know the internal rate of return for project A and project B described above. For project A, the internal rate of return can be found by substituting the values for YE and C into equation (53), or
(54) $3,000(EPIFR,5) - $10,000 = 0
Solving equation (54) for the interest factor (EPIFR,5), we see that
(55) (EPIFR,5) = $10,000/$3,000 = 3.333
The internal rate of return is then found by locating that value of R associated with the equal payment present value interest factor (EPIFR,N) of 3.333 for N = 5 in the equal payment interest factor (EPIFR,N) Appendix table. Doing this, we find a value of R that is approximately equal to 15 percent. This would suggest that your opportunity of return (R) would have to exceed 15 percent before you should consider not investing in project A. 9
For project B, where equation (52) rather than equation (53) must be used, the value of R yielding a net present value of zero must be found by trial and error.
The search procedure is begun by selecting the value of R you think most closely approximates the true value of R into equation (52). If the resulting solution for the net present value is greater than zero, you have underestimated the true value of R and must try a higher value in equation (52). If the solution value, however, was less than zero, you have overestimated the true value of R and must try a lower value in subsequent attempts.
This iterative procedure is continued until the solution for the net present value is approximately equal to zero. In the case of project B, the internal rate of return is approximately equal to 10 percent. Thus, based upon a comparison of these internal rates of return, we would again prefer project A to project B because since IRRA > IRRB.
Although any project can have only one net present value (NPV) and one profitability index value (PFT), a single project under certain circumstances can have more than one internal rate of return (IRR). The reason for this can be traced to the calculation of the IRR. If the initial capital outlay is the only negative value in equation (53) and all of the annual net cash flows are positive, there is no problem. Problems occur when there are sign reversals in the annual cash flow stream. There can be as many solutions for IRR as there are sign reversals. To illustrate, consider the following example:
Annual cash flows Initial outlay - 1,600 Year 1 net cash flow +10,000 Year 2 net cash flow - 10,000
This pattern of cash flows over a two year period has two sign reversals; from -$1,600 to +$10,000 and then from +$10,000 to -$10,000. So there can be as many as two positive IRRs that will result in a NPV of zero. You can prove to yourself that two discount rates (25 percent and 400 percent) result in a NPV of zero. Which solution is correct? Neither solution is valid! Neither provides any insight to the true project returns. This when there is more than one sign reversal in the flows of funds over the project’s economic life, the possibility of multiple IRRs exists, and the normal interpretation of the IRR loses its meaning.
Net Present Value Method
To remedy the deficiencies noted above for the payback period method, we can use a capital budgeting technique that accounts for the present value of the entire stream of net cash flows over the life of the project. One such technique is the net present value method. In the case where the discount rate is expected to remain constant over the entire economic life of the investment project (i.e., R1 = R2 = … = RN), the net present value of an investment project (NPV) is given by
10
(56) NPV = NCF1(PIFR,1) + NCF2(PIFR,2) + …. + NCFN(PIFR,N) - C where C is the initial capital outlay for the assets acquired under the project. Since this outlay is made at the start of the project, no discounting is needed. We can restate equation (56) as follows:
2 N (57) NPV = NCF1[1/(1+R)] + NCF2[1/(1+R) ] + …. + NCFN[1/(1+R) ] - C where NCF1 once again represents the annual net cash flow generated by the project in year 1, NCF2 represents the net cash flow generated by the project in year 2, etc., R is the discount rate and N is the number of years in the life of the project. Finally, C represents the original cash purchase price less any cash discounts (but not the trade-in value of used machinery deducted from the purchase price at the time of the purchase). You should recognize equations (56) and (57) as being very similar to equations (37) and (38) from a discounting net cash flows standpoint.
The net present value of an investment project can be viewed as the “profit” or dollar measure of the amount saved by making this investment now. Given the assumptions of profit maximization and complete certainty, you should accept those projects whose net present values are positive (i.e., NPV > 0). You will be indifferent between whether or not to invest when the net present value equals zero (i.e., NPV = 0), and you should reject all projects whose net present value is negative (i.e., NPV < 0).
Let us assume that you expect a constant discount rate of 5 percent over the 5-year economic lives of two mutually exclusive investment projects: project A and project B. The net present values for both projects are reported in Table 3 below:
Table 3 - Determination of the Net Present Value for Projects A and B.
Project A Project B
(1) (2) (3) (4) (5) (6) Net Cash Present Value Present Value Net Cash Present Value Present value Flow Interest Factors of NFCi Flow Interest Factors of NFCi Year (NCFi) PIF0.05,I (1) x (2) NFCi PIF0.05,I (4) x (5) (i)
1 $ 3,000 0.952 $ 2,856 $ 2,000 0.952 $ 1,904 2 3,000 0.907 2,721 3,000 0.907 2,721 3 3,000 0.864 2,592 5,000 0.864 4,320 4 3,000 0.823 2,469 2,000 0.823 1,646 5 3,000 0.784 2,352 1,000 0.784 784 $ 15,000 $ 12,990 $ 13,000 $ 11,375 Less initial cost - 10,000 Less initial cost - 10,000 Net present value $ 2,990 Net present value $ 1,375 11
This table shows, for example, that project A has a higher net present value than project B, because it generates higher net cash flows in the first year of the project where the interest factor is at its highest and also has a higher cumulative net cash flow over the entire 5-year period. While both projects should be considered since each was found to have a positive net present value, project A should be preferred over or ranked higher than project B.
This represents a reversal of the rankings given by the payback method. A profit- maximizing farm operator should consider investing in both of these projects as long as their combined cost is less than or equal to the amount of funds available to finance new projects.
In the case where NCF1 = NCF2 = … NCFN, we can simplify the computational procedure along the lines initially suggested by equation (53) by instead using
(58) NPV = NCFE(EPIFR,N) – C where NCFE represents the equal annual net cash flows generated by the new investment project. For example, we could have used equation (58) instead of equation (57) to compute the net present value for project A. Locating the interest factor (EPIFR,N) in the equal payment interest factor table and substituting this value into equation (58), we see that the net present value for project A is equal to:
(59) NPV = $3,000(4.329) - $10,000 = $2,990 which is identical to the net present value reported for project A in Table 3.
The net present value formula presented in equations (58) and (59) should be seen as a special case of the present value formulas presented in equations (52) and (53). We shall limit the examples studied for the moment to those which permit us to use equations (58) and (59); that is, we shall assume a constant discount rate over the economic life of the project.
Which method should we use?
The payback period method, because of its failure to account for the level of cash flows beyond the payback period, does not in itself represent a desirable capital budgeting method. It is the only method that suggested project B should be ranked higher than project A! The net present value and the internal rate of return methods, which do account for these factors, will provide the same order of ranking for mutually exclusive investment projects in most but not all cases. The possibility of multiple solutions with the internal rate of return method poses a problem. Another issue is that the net present value method discounts the net cash flows at the investor’s desired discount rate while the internal rate of return method assumes that the net cash flows can be reinvested at their internal rate of return (which may not be true). 12
In practice, many analysts will report as many as all four of the statistics discussed in this section. The net present value method, however, remains the appropriate basis for ranking the economic benefits generated by two or more mutually exclusive investment projects. The IRR is reported generally because many are familiar with this statistic and can directly compare it with the cost of debt capital or borrowed funds. The payback period also gives analysts an insight to the liquidity associated with alternative projects.
C. Overview of Capital Budgeting Information Needs
Composition of net cash flows
The net cash flows generated by an investment project represent the net change in the annual net cash flows generated by the firm’s existing resources, or before the investment is made. These annual net cash flows require a forward assessment of all the forces that affect future net cash flows over the economic life of the investment. The following table will help make its measurement clear:
Table 4 – Measuring annual net cash flows. Before new After new Net Item: investment investment change Cash receipts $25,000 $30,000 $5,000 Cash operating expenses -15,000 -18,000 -3,000 Depreciation -3,000 -4,000 -1,000 Tax deductible expenses 18,000 22,000 4,000 Taxable income 7,000 8,000 1,000 Income tax payments 1,750 2,000 250 Net income after taxes 5,250 6,000 750 Net cash flow 8,250 10,000 1,750
The additional cash receipts generated by the investment project are $5,000 annually over the life of the project. The firm’s cash operating expenses (i.e., fuel expenses, hired labor expenses, fertilizer and chemical expenses) are expected to increase $3,000 annually while depreciation expenses are expected to increase by $1,000. This means the firm’s tax deductible expenses will be $4,000 higher than the current level if the investment project is undertaken. Subtracting these tax deductible expenses from cash receipts results in an increase in taxable income of $1,000 annually.
If the tax rate is equation to 25 percent, income tax payments would be $250 higher annually, giving a net income after taxes of $750 annually. Finally, we have to add back in the depreciation expenses used to compute tax deductible expenses (a noncash flow) to 13 measure the annual net cash flows associated with the project. Table 4 above indicates that this annual net cash flow would be equal to $1,750.
Note this will allow us to use equation (58) when computing the net present value of the project. Why? The annual net cash flows are expected to be identical over the entire economic life of the project.
Economic and service life
The economic life of an investment project represents the length of time the firm intends to hold the assets acquired. It does not represent the service life (sometimes called the useful life) of the assets, or the amount the amount of time taken before they wear out. For example, suppose that the firm plans to purchase a piece of equipment that normally wears out over a ten-year period but only plans to hold this piece of equipment for three years. Thus, the economic life of the investment is three years while the service life of the equipment is ten years. This distinction is important when accounting for the effects that the terminal value of a project has upon its feasibility.
On another front, two investment projects can also have different or “unequal” service lives. Thus far, as in Table 3, we have assumed that two or projects have identical service lives. When this assumption is not valid, we cannot directly compare the net present values generated by two or more investment projects.
Suppose we were considering investing in one of two projects that provide identical services to the firm. One project has a service life of five years while the other has a service life of three years. While both projects provide identical services in a specific year, one project would require more frequent replacement. An each project must be replaced as it wears out over time in order to maintain the firm’s productive capital stock.
One approach to comparing these two projects is to compute the equivalent level annuity that yields the same net present value if invested at a rate R over a period of N years. Let us assume that, as an alternative to project A which had a five-year economic life in Table 3, the firm is considering project C, which also costs $10,000 but generates equal annual net cash flows (NCFE) of $4,500 over a three-year service life. Continuing to assume that the firm requires a 5 percent discount rate and that the terminal value is equal to zero, the net present value over its original service life is:
(60) NPVC = NCFE(EPIFR,N) – C = $4,500(EPIF0.05,3) - $10,000 = $4,500(2.723) - $10,000 = $2,254 based upon applying equation (58) presented earlier. Note this is approximately $736 less that the net present value reported for project A in Table 3. 14
The equivalent level annuity for an investment project is equal to the present value of NPV if invested at rate R for N periods. You will recall that we said in equation (58) that the present value of an equal annual net cash flow is equal to the value of the net cash flow (NCFE) multiplied by the “equal payment” present value interest factor (EPIFR,N).
Letting AeA represent the equivalent level annuity for project A, we can rearrange terms to show that:
(61) AeA = NPVA ÷ EPIF0.05,5 = $2,990 ÷ 4.329 = $691
Similarly, the equivalent level annuity for project C is equal to:
(62) AeC = NPVC ÷ EPIF0.05,3 = $2,254 ÷ 2.723 = $829
Thus, while project A had a higher net present value during its original service life, project C is preferred after you take into account of how quickly project C wears out. As long as both projects have the same discount rate and can be repeated over time, this approach will lead to the appropriate ranking. If a different discount rate is required for whatever reason, we must take the additional step of converting the equivalent level annuity into a perpetuity. For project A, this means we must compute:
(63) NPVA = AeA ÷ RA
where NPVA represents the net present value of an “infinity-lived” project.
Another approach to evaluating two investment projects with different service lives is to find the shortest common replacement chain. Assume we are considering two projects with different service lives, project AA and project BB. The cash flows for these projects are as follows: 15
Table 5 – Net present values for two projects with unequal lives.
Project AA Project BB
(1) (2) (3) (4) Net Cash Net Net Cash Net Year Flow Investment Flow Investment
0 $2,000 $2,000 1 $ 600 $ 375 2 600 375 3 600 375 4 600 375 5 600 375 6 - 375 7 - 375 8 - 375 9 - 375 10 - 375
NPVAA = $600(3.791) - $2,000 NPVBB = $375(6.145) - $2,000 = $274.60 = $304.37
The net present value of project BB therefore exceeds that of project AA. Note, however, that the service life for project BB is twice as long as project AA’s. If we take the replacement chain approach to account for the differences between their service lives, the shortest common service life would be ten years, the exact length of project BB’s service life. Taking this into account, we can put both projects on an equal footing using the replacement chain approach as follows:
Table 6 – Net present value for two projects with unequal lives after applying the Replacement Chain approach.
Project AA Project BB
(1) (2) (3) (4) Net Cash Net Net Cash Net Year Flow Investment Flow Investment
0 $2,000 $2,000 1 $ 600 $ 375 2 600 375 3 600 375 4 600 375 5 600 $2,100 375 6 600 375 7 600 375 8 600 375 9 600 375 10 600 375
NPVAA = $600(6.145) - $2,000 - $2,100(0.621) NPVBB = $375(6.145) - $2,000 = $382.80 = $304.37 16
By replacing the equipment acquire in project AA at the start of the 6th year (i.e., at the end of the 5th year) for $2,100 and repeating its use for another five-year period, we see that NVPAA > NVPBB. This reverses the investment conclusion we would have reached if we did not account for the unequal lives.
Original and terminal value
Information is also needed on the original net capital outlay when the assets are acquired as well as their terminal or market value at the end of the economic life of the investment project. The net capital outlay was represented by C in equations (56), (57) and (58). This value should be the net cash purchase price after any cash discounts.
The terminal value of assets purchased in an investment project must also be accounted for before the net present value of the project can be determined. This value may be nothing more than the salvage value in the case of depreciable assets if the economic life of the investment project coincides with the service life of these assets. If the economic life is less than the service life, the terminal value will play a key role in determining the economic feasibility of the project. This value if what you expect to be able to sell these assets at the end of the project’s life. If land is involved, the terminal value may be considerably higher than its original value at the beginning of the project’s economic life.
Let T represent the terminal value of an asset at the end of the economic life of an investment project. Because the firm receives this value at a specific point in the future, we must discount this value back to the present. We can modify equation (56) to include this discounted terminal value as follows:
(64) NPV = NCF1(PIFR,1) + NCF2(PIFR,2) + …. + NCFN(PIFR,N) – C + T(PIFR,N)
Similarly, equation (57) can be modified to include the discounted terminal value as follows:
2 N (65) NPV = NCF1[1/(1+R)] + NCF2[1/(1+R) ] + …. + NCFN[1/(1+R) ] – C + T[1/(1+R)N]
Finally, in the case where NCF1 = NCF2 = … NCFN, we can simplify the computational procedure along the lines suggested by equation (58) by instead using
(66) NPV = NCFE(EPIFR,N) – C + T(PIFR,N)
Let’s return to the example contained in Table 3 on page 9. The net present value of project A was $2,990 while the net present value of project B was $1,375. Let’s now assume that, while the terminal value of the assets acquired under project A is zero, the terminal value of the assets acquired under project B is $2,500. 17
The net present value of project B now becomes
(67) NPV = $11,375 - $10,000 + $2,500(PIF0.05, 5) = $11,375 - $10,000 +$2,500(0.784) = $3,335 where $11,375 represents the present value of the annual net cash flows over the five- year period illustrated in Table 3. The existence of the discounted terminal value now makes project B’s net present value of $3,335 higher than project A’s net present value of $2,990.
Another application of the terminal value is when you know in advance that the economic life of the investment project will be less than the service life of the assets acquired in the project. Suppose you plan to retire in two years and want to know whether or not it is profitable to invest in a project that normally would have a service life of five years. Focusing again on project A in Table 3 on page 9, let’s assume you expect to sell the assets acquired under project A for a net of $7,000 at the end of the project’s second year. Using equation (66) above, we see that:
(68) NPV = $3,000(EPIF0.05,2) - $10,000 + $7,000(PIF0.05, 2) = $3,000(1.859) - $10,000 +$2,500(0.907) = $1,926
Thus, you would accept this particular project if your discount rate was 5 percent.
An interesting twist on equation (68) is to find the terminal value which results in a net present value of zero. We can rearrange equation (66) to read:
(69) T = [C – NCFE(EPIFR,N)] ÷ (PIFR,N) = [$10,000 – $3,000(EPIF0.05,2)] ÷ (PIF0.05,2) = [$10,000 – $3,000(1.859)] ÷ 0.907 = $4,878
Thus, the net terminal value of the assets acquired under this project would have to be below $4,878 before the firm should consider rejecting investment in this particular project.
Discount rate
The selection of an appropriate discount rate when calculating the net present value of an investment project involves finding that rate which reflects the after-tax rate of return the firm requires to cover the opportunity cost of not undertaking its next best alternative of a similar maturity and risk exposure. 18
We will initially assume that this discount rate is identical over the entire economic life of the project. We will relax this assumption later in this course when we focus more intensively on accounting for business and financial risk.
Stopped here for 1st Hour Exam
D. Specific Applications of Net Present Value Method
Purchase of depreciable assets
The purchase of depreciable assets required that we account for their cumulative depreciation when determining the net present value of the asset. Let’s continue to assume for the moment that the annual net cash flows generated by the project are identical as are the annual discount rates. This allows us to use equation (70), or:
(70) NPV = NCFE(EPIFR,N) – C + T(PIFR,N)
The uniqueness of this decision is in the valuation of the terminal value or T. In the case of depreciable assets, the market value of assets at the end of the economic life of the investment project, particularly when the economic and service lives are identical. At this point, the terminal value will be represented by the salvage value from depreciating the asset over time for tax purposes. There is typically no capital gain to be taxed when assets are disposed of at the end of the project.
Purchase of land
The purchase of land is unique in that you have to account for capital gains income when determining the net present value of the asset. Recall from our earlier discussion that there are several deficiencies associated with simple capitalization formula which involved treating the value of land as a perpetuity or “infinitely lived” asset. The net present value of an investment in land with the characteristics of equation (66) is given by:
(71) NPV = NCFE(EPIFR,N) – C + T(PIFR,N) – [tCG(T – C)](PIFR,N) where tCG is the capital gains tax rate, and where the terminal value T is given by:
(72) T = C ÷ PIFG,N
The variable G represents the rate of appreciation in land values expected over the investment project. It is often the cash in agriculture where the after-tax value of the terminal value often exceeds the present value of the annual net cash flows associated with the operations of the firm. 19
Maximum bid price for land
When participating in an auction or simply offering to purchase land in a one-on-one negotiation, it is important to know what the maximum value you can afford to pay for a tract of land. This value can be found by merely rearranging the terms in equation (71) to solve for the original purchase price C which results in a net present value of zero!
To illustrate, suppose you are considering the purchase of some additional land that is expected to increase your firm’s annual net cash flows by $75 per acre. We shall assume that no additional equipment is required to operate this additional land. Assume you plan to retire in 20 years and are interested in knowing the maximum price you can justify paying now from an economic standpoint. If comparable tracts of land are selling for $1,000 in your area, that land is expected to appreciate at a seven percent annual rate over the next 20 years, you are in the 25 percent tax bracket and that your discount rate is five percent, the present value (PV) of this land per acre would be:
(73) PV = NCFE(EPIFR,N) + {C ÷ PIFG,N}(PIFR,N) – [tCG({C ÷ PIFG,N} – C)](PIFR,N) = $75(12.462) + {$1,000 ÷ 0.258}(0.377) – [0.25({$1,000 ÷ 0.258} – $1,000)](0.377) = $935 + $1,461 – [0.25($2,876)](0.377) = $935 + $1,461 – $271 = $2,125 where EPIF0.05,20 = 12.462, PIFG,N = 0.258, and PIFR,N = 0.377.
This implies an NPV of $1,125, or $2,125 minus the initial purchase price of $1,000. Thus, you could afford to pay up to $2,125 an acre for this land given the foregoing assumptions before the NPV fell to zero. If you had ignored the capital gains component, you would have incorrectly concluded that you could not afford this additional acreage since the present value of the annual net cash flows ($935) is less than the current sales price ($1,000). You don’t have to pay this much if land is currently selling for $1,000. The point is that you could pay more than $1,000 and still come out ahead.
It should be emphasized that, if the current sales price C or the rate of land appreciation G change, you have to compute a new maximum bid price. You will notice the valuation approach presented in equation (73) is a far more complex valuation tool than the simple capitalization formula given by equation (42) on page 4. The added complexity, however, is necessary since the firm will not hold this land in perpetuity, and capital gains income is owed in the year of the sale. Finally, it is important to test the sensitivity of this bid price assuming different rates of appreciation of land as well as those market factors influencing expected future net cash flows.