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Handout #3
Agricultural Economics 432
Part I - Financial Analysis
Handout #3
Spring Semester 2007
John B. Penson, Jr.
Agricultural Economics 432
Part I - Financial Analysis
Outline
Topic 1: Review of Financial Analysis Concepts
- Introduction to Terminology
- Key Financial Indicators to Track
- Other Variables to Track
- Financial Strength and Performance of the Firm
Topic 2: Growth of the Firm
- Economic Climate for Growth
- An Economic Growth Model
Topic 3: Valuing Investment Opportunities
- Time Value of Money (p. 22)
- Capital Budgeting Methods p (p. 27)
- Overview of Capital Budgeting Information Needs (p. 33)
- Specific Applications of Net Present Value Method (p. 39)
Topic 3. Valuing Investment Opportunities
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:
(33)PV = FVN/(1+R)N
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
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:
(38)PV = NCF1 [1/(1+R)] + NCF2 [1/(1+R)2] + …. + NCF10 [1/(1+R)10]
= $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:
(39)PV = NCF5 (PIF.06,5) + NCF10 (PIF.06,10)
= $250(.747) + $250(.558)
= $186.75 + $139.50
= $326.25
or
(40)PV = NCF5 [1/(1+R)5] + NCF10 [1/(1+R)10]
= $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.
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:
(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.
YearPIPIBalance
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.
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 AProject 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.
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<R). Thus, we would accept all projects where IRR>R, and we would rank these projects according to the size of their 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. 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
(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:
(57)NPV = NCF1[1/(1+R)] + NCF2[1/(1+R)2] + …. + NCFN[1/(1+R)N] - 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