Present financial position and performance of the firm
Selected Topics in
Micro and Macro Finance
October 2007
Financial Management in
Agriculture and Food System
LESE 306
Korea University
John B. Penson, Jr.
Regents Professor and Stiles Professor of Agriculture
Department of Agricultural Economics
Texas A&M University
Selected Topics in
Micro and Macro Finance
Table of Contents
Part I: Financial Statement Overview
A. Basic Structure of the Financial Statements 4
B. Some Terms to Know 8
C. Key Financial Indicators 13
D. Financial Strength and Performance of the Firm 19
Part II: Growth of the Firm
A. Economic Climate for Growth 22
B. An Economic Growth Model 25
Part III: Valuing Investment Projects
A. Time Value of Money 30
B. Capital Budgeting Methods 37
C. Overview of Capital Budgeting Information Needs 44
D. Specific Applications of Net Present Value Method 52
E. Other Capital Budgeting Applications 55
Part IV: Valuation of Externalities
A. Historical Assessments 60
B. Forecasting Market Prices and Probability 64
Part V: Adjustments for Risk
A. Exposure to Business Risk 68
B. Risk/Return Preferences 70
C. Exposure to Financial Risk 77
D. Portfolio Effect 79
E. Optimal Capital Structure 83
F. Ranking Potential Projects and the Capital Budget 86
Part VI: Selected Macroeconomic Topics
Agriculture and the global economy 88
B. Economic activity and policy responses 93
C. Big 5 macro variables and agriculture 101
D. Impact on income statement and balance sheet 104
E. Capacity expansion analysis 105
F. Factors influencing monthly price fluctuations 110
G. Cash market versus forward pricing alternatives 113
Appendix I
Present value interest (PIF) factors 117
Equal payment present value interest (EPIF) factors 118
Appendix II
Simple macroeconomic model of an open economy 119
Part I: Financial Statement Overview
There are a minimum of three financial statements comprising the financial accounting system of all businesses: the balance sheet, the income statement, and the cash flow statement. An understanding of the basic designs of these statements and the interrelationships between them is essential to this course. Other financial statements, such as the statement of change in owner equity, while not addressed in this course, basically feed off of these statements.
A. Basic Structure of the Financial Statements
The Balance sheet
The balance sheet is a statement of a firm’s assets and claims on these assets. The basic structure of the statement is illustrated in Figure 1 below. The
Figure 1 – Structure of the Balance Sheet.
left-hand side of the balance sheet lists the firm’s assets in order of their asset liquidity, or ability to be converted to cash quickly without disrupting
the ongoing operations of the firm. Naturally, therefore, cash would be the first asset found in the balance sheet along with other current assets, or assets like time deposits, accounts receivable, and inventories of unsold production that can presumably be converted to cash quickly without disrupting the firm’s operations. Intermediate term assets require more time to convert to cash and can disrupt the ongoing nature of the firm’s operations. Examples include machinery, equipment, and trucks and other motor vehicles. Long term assets are the least liquid assets of all assets owned by the firm, and include such assets as buildings and land.
There are at least two types of balance sheets: a book value balance sheet and a current market value balance sheet. A book value balance sheet reflects the basis the firm has it its intermediate and long term assets, or their original purchase price less accumulated depreciation. A current market value balance sheet, on the other hand, reflects the market value of intermediate and long term assets as of the date of the balance sheet.
The Income Statement
The firm’s income statement provides a record of revenue from the firm’s operations and the expenses associated with generating this revenue. The
Figure 2 – Structure of the Income Statement.
general structure of this financial statement is illustrated in Figure 2. The bottom line of this financial statement is a measure of net income (either net cash income or total net income), which represents a measure of the firm’s profitability during the current year. We will examine a number of other measures of profitability later in this booklet.
The income statement can take on at least two forms: a cash income statement like the one illustrated in Figure 2 above where revenue and expenses are registered when payment is either received or expenses paid, and an accrual statement where revenue and expenses are measured when they are incurred. This is the difference, for example, between cash receipts from product sales versus the value of production when accounting for the gross revenue of the firm in the current year.
The Cash Flow Statement
The cash flow statement provides a record of the sources and uses of cash for the firm during the year. A key element of this statement is the cash position of the firm, or the difference between cash available minus cash
Figure 3 – Structure of the Cash Flow Statement.
required to meet uses of cash by the firm. A negative cash position requires
the firm borrow to cover the negative balance at a minimum. Cash available includes the firm’s beginning cash balance carried over from the previous year’s balance sheet plus cash received from the sale of current production.
The cash flow statement can also take at least three forms: an annual cash flow statement, a quarterly cash flow statement, and a monthly cash flow statement. The latter format is extremely useful in establishing a line of credit with the firm’s creditors for the current year.
Additional detail on all three financial statements will be provided in Power Point slide shows presented during lectures in this course.
B. Some Terms to Know
Before proceeding with key topics in the financial analysis of an agricultural firm, we need to refresh your memory on key terminology and introduce you to others.
Liquidity
Liquidity is generally defined as the ability to generate cash quickly to meet claims on the business without disrupting the ongoing operations of the business. There are three forms of liquidity: (1) asset liquidity, (2) credit liquidity and (3) cash flow liquidity.
Asset liquidity – the ability to convert assets to cash quickly to meet claims on the business without disrupting the ongoing operations of the business. If the firm can convert its current assets to cash, retire its current liabilities and have cash left over, the firm is said to be liquid. Assets that can be converted to cash quickly are referred to as current assets, and include financial instruments (cash itself, stocks and bonds and accounts and notes receivable) as well as non-financial assets like unsold production inventories and inventories of purchased inputs. Assets on a firm’s balance sheet are typically ordered in terms of their liquidity, which explains why cash appears first in the asset column of the balance sheet and land appears last. While land may be sold quickly, its sale will affect the ongoing nature of the firm’s business operations.
Credit liquidity – the ability to borrow cash quickly to meet claims on the business without disrupting the ongoing operations of the business. This concept is related to the firm’s unused credit reserves, or the maximum amount of credit lenders will typically extend to a firm based upon its debt/equity structure.
Cash flow liquidity – the first two measures of liquidity are typically based upon balance sheet evaluations at a specific point in time. Cash flow liquidity on the other hand is a periodic measure of liquidity, usually monthly. The firm’s monthly cash flow statement reflects the cash flows available and cash flows required during the month. If the ensuing difference between these totals, or net cash balance, is positive, the firm is liquid during that month.
Solvency
Solvency refers to the ability of the firm to convert all its assets to cash, retire all of its liabilities and have cash left over. Note the emphasis on all assets and all liabilities as opposed to current assets and current liabilities when we discussed the concept of asset liquidity described above.
Profitability
Profitability refers to the ability of a firm to generate a level of revenue that exceeds its total costs of production. The bottom line of the firm’s income statement reflects the net income of the firm, one of many measures of profitability. Care must be taken when assessing profitability over time to account for changes in the purchasing power of money. Nominal net income refers to net income not been adjusted for inflation while real net income reflects adjustments for changes in the purchasing power of the firm’s profits from one year to the next.
Economic Efficiency
Economic efficiency refers to the ability of the firm to use its resources to achieve a desired result with little or no wasted effort or expense. This differs from technical efficiency or productivity (i.e., yield per unit of input use) that does not take prices or unit costs of production into account.
Debt Repayment Capacity
Debt repayment capacity refers to the ability of the firm to meet its scheduled term debt and capital lease payments and have cash left over. This concept is closely linked to the concept of unused credit reserves in the literature, or the difference between the maximum capacity to borrow as viewed by lenders and the amount of borrowed capital the firm has already undertaken.
Present vs. Future Value
Present value refers to the value today of a sum of money or stream on payments to be received in the future, discounted back to the present using an appropriate required rate of return. Future value, on the other hand, refers to the value at a specific future date of a sum of money or stream of payments (perhaps to an annuity).
Risk
The possible variation associated with the expected return or value measured by the standard deviation or coefficient of variation. There are many forms of risk, including price risk, interest rate risk, yield risk, political risk, relationship risk, etc.). This collectively is often referred to as business risk, or the relative dispersion or variability in the firm’s expected earnings.
This differs then from financial risk, or the potential loss in equity associated with the use of leverage (debt and equity capital). Finally, risk can also be classified as systematic risk (or non-diversifiable risk) and unsystematic risk (the portion of the variation in investment returns that can be eliminated by diversification).
Physical vs. Financial Capital
Physical capital refers to those assets on the firm’s balance sheet that are not in the form of financial instruments. Examples include inventories of unsold production, machinery and equipment, inventories of production inputs, trucks, buildings and land. Financial capital, on the other hand, typically refers to financial instruments (cash, stocks and bonds) as well as the equity capital the owners firm have invested in the firm.
Explicit vs. Implicit Costs
Explicit costs are those expenses such as wages paid to hired labor where a cash payment to others is required. Implicit costs on the other hand are those expenses such as depreciation or opportunity costs that do not involve the payment of money.
Variable vs. Fixed Expenses
Variable expenses are those expenses that vary with the level of production. Fuel and fertilizer expenses are two examples. Fixed expenses, on the other hand, are those expenses that do not vary with the level of production. A property tax payment, which must be paid even if output of the firm is zero, is a form of fixed expenses.
Optimal Capital Structure
This refers to the capital structure that minimizes the firm’s composite cost of capital for a given amount of debt and equity capital. This will be influenced by the relative cost of financing capital acquisition opportunities with debt and equity capital.
Capital Budgeting
The decision making process associated with investment in fixed assets (machinery and equipment, land and buildings. There are various forms of capital budgeting methods available, including the payback period method, the profitability index method, the internal rate of return method and the net present value method. Each will be covered in depth later in this booklet.
Financial Analysis
Financial analysis refers to the assessment of a firm’s financial condition or well being. Its objectives are to determine the firm’s financial strengths and identify its weaknesses. The primary tool in financial analysis is financial ratios or indicators of liquidity, solvency, profitability, economic efficiency, and debt repayment capacity.
More terms and jargon
More terms and jargon will be added as we proceed thru this booklet. Various methods used in financial analysis will also be introduced, including structural econometric business forecasting. We will define these terms as they are introduced.
C. Key Financial Indicators
Measures of Asset Liquidity
There are several approaches to measuring the liquidity of a firm depending on whether you are talking about asset liquidity, credit liquidity or cash flow liquidity.
Two common measures of asset liquidity are the current ratio and the level of working capital. The current ratio is given by:
1) Current ratio = current assets ( current liabilities
where current assets are those assets that can be converted to cash within the year and current liabilities are those liabilities that are due within the year. If this ratio is greater than 1.0, the firm is said to be liquid, or able to retire all its current liabilities with its current assets and have cash left over. Studies have shown, however, that the firm can be liquid and still fail. Other ratios (acid test ratio and cash ratio) represent variations of equation (1), where specific categories of current assets are excluded from the numerator.
The level of working capital is given by:
2) Working capital = current assets – current liabilities
If the level of working capital is positive, then the firm has sufficient current assets to cover all its current liabilities and still have cash left over. This term is sometimes referred to as net working capital. The ratio of working capital to total assets can also be used to reflect the “asset fixity” or lack of assets liquidity exhibited by the firm.
Measures of Solvency
There are numerous approaches to measuring the solvency of the firm. They all involve balance sheet data and are transformations of one another. These measures include the debt ratio, the net worth ratio, the asset ratio and the leverage ratio. Each is defined as follows:
3) Debt ratio = total debt or liabilities ( total assets
4) Net worth ratio = total net worth or equity ( total assets
5) Asset ratio = total assets ( total debt or liabilities
6) Leverage ratio = total debt or liabilities ( total net worth or equity
The debt ratio typically should not exceed 0.50. Beyond that point, the suppliers of capital to you business own more of the firm than you do. We typically see financial stress to begin building beyond this point. A debt ratio of 0.50 translates into a leverage ratio of 1.0. It also suggests that the net worth or equity ratio should not typically fall below 0.50. The graph to the right clearly shows that firms with debt ratios of greater than 0.50 have a much greater chance of failure than firms with a debt ratio below this level. This is particularly true in economic environments characterized by rising market rates of interest.
Measures of Profitability
In addition to the level of net income, which we said earlier is a measure of profitability, other commonly used measures exist. These include the rate of return on assets (ROA) and the rate of return on equity capital (ROE). Another measure is the net or operating profit margin. These measures can be defined as follows:
7) Rate of return on assets = (net income + interest expense) ( total assets
8) Rate of return on equity = net income ( total equity or net worth
9) Net profit margin = (EBIT – tax) ( total revenue
where EBIT represents earnings before interest and tax payments, or net income minus interest and tax payments.
These measures of profitability should all be positive, with higher values being preferred to values approaching zero.
The graph to the right illustrates the path in profitability taken by firms that fail versus firms that succeed. The linkage between this graph and the previous debt ratio and liquidity graphs should be emphasized. In periods of rising market interest rates, firms using high levels of leverage experience a much higher level of interest expense and hence lower levels of profitability. This can lead to the use of the firm’s existing liquidity or more carryover debt to cover claims on the business normally covered by current net income, which makes the firm less liquid as well as increase its level of indebtedness.
Measures of Economic Efficiency
Like the other financial indicators, there are a variety of measures of economic efficiency used by financial analysts. These include a number of expense ratios (interest expense ratio, variable expense ratio, depreciation expense ratio) as well as several turnover ratios (total asset turnover and fixed asset turnover). These ratios are defined as follows:
10) Interest expense ratio = interest expenses ( total revenue
11) Variable expense ratio = total variable expenses ( total revenue
12) Depreciation expense ratio = depreciation expenses ( total revenue
13) Total asset turnover ratio = total revenue ( total assets
14) Fixed asset turnover ratio = total revenue ( fixed assets
The interest expense ratio is illustrative of the impact that the use of leverage has upon the firm’s profitability. The summation of the interest expense ratio, the variable expense ratio and the depreciation ratio represents the net profit margin ratio before tax. These ratios ideally should reflect stability in operations. One thing they clearly illustrative for firms that are price takers in both input and product markets is the effect that rising unit input costs relative to unit product prices have on the profitability of the firm; i.e., the effects of a price-cost squeeze.
Measures of Debt Repayment Capacity
Finally, measures of debt repayment capacity include term debt and capital lease coverage ratio, times interest earned ratio, and debt burden ratio to name a few. They are calculated as follows:
15) Term debt and capital
lease coverage ratio
16) Times interest earned = (EBIT–taxes) ( scheduled interest payments
17) Debt burden ratio = total debt outstanding ( net income
The financial indicators in equations (15) and (16) should be greater than one. This would indicate the firm has, at minimum, the capability to service their commitments as scheduled out of the firm’s operations. Obviously the greater these ratios are, the greater the comfort zone for the lender.
Equation (17) indicated the number of periods needed to retire total debt outstanding if net income was used entirely for this purpose. If the net income statistic used comes from an annual income statement, then this ratio would reflect the number of years necessary to retire the firm’s entire debt. While this may reflect the eventual application of net income, it does indicate to a lender the firm’s ability to retire debt from operations.
The graph illustrated to the right shows that as the firm’s net cash income declines and becomes negative, the grater chance of ultimate failure. The firm exhibits an increasing inability to service existing debt commitments out of its current operations, thereby being forced to carry over debt to continue operations, assuming lenders are willing to do so.
A final point needs to be made. These measures of liquidity, solvency, profitability, efficiency and debt repayment capacity should be used jointly when analyzing the trends in the firm’s economic performance and financial position. The asset liquidity ratio, for example, was shown for the firms that failed to exceed what is normally required to be considered liquid.
D. Financial Strength and Performance of the Firm
Financial analysis was defined earlier as “the assessment of a firm’s financial condition or well being”. Its objectives were said to be “determine the firm’s financial strengths and identify its weaknesses”. A number of financial indicators were defined and interpreted.
Nothing however was said about the basis for comparison other than a specific level so satisfy the definition of liquidity, etc.
Often the most accurate or reveling assessment of a firm’s financial strength and performance involves assessing the trends in a number of ratios for the firm over time and the deviations from the financial achievements of other similar or “like kind” firms. This is called historical financial analysis and comparative financial analysis, respectively.
Historical Analysis
Historical financial analysis involves comparing the firm’s current performance with its performance in previous years, and identifying the underlying reasons for deviations from expectations. This involves computing the above measures for liquidity, solvency, profitability, economic efficiency and debt repayment capacity (one measure from each group should do) for the latest available year and comparing the values of each measure with say the Olympic average over the last five years (drop the high and low when computing the average).
Understanding why any undesirable deviations in liquidity, solvency, profitability and other categories of financial indicators occurred during the most recent period may help formulate strategies that prevent further occurrences. There may very well be a good explanation tied to one-time events beyond the control of the producer. Conducting a comparative financial analysis with similar firms will help confirm this conclusion
Comparative Analysis
As the name suggests, comparative financial analysis involves comparing the financial strength and performance of the firm with that of similar firms. The current weak performance of the firm may not be a one-time event beyond the control of the producer, but rather something similar firms did not experience, at least not to the same degree.[1]
The study by W. H. Beavers again showed that the financial ratios of firms that subsequently fail are different from those firms that survived.[2] Beavers tracked the current ratio, debt ratio, rate of return on assets, and the reciprocal of the debt burden ratio described earlier in this booklet (equations (1), (3), (7) and (17) respectively).
Catching undesirable differences from other firms in its peer group early enough to minimize or eliminate their long-term effects is paramount to the long run success of the firm. The gap between the successful firms and the firms that failed in the Beavers study was not that great in the initial year. The growth in the use of borrowed funds to cash flow operations in subsequent years, however, eventually led to sharp differences between the successful and failing firms as additional interest expenses grew and ROA declined.
Pro forma Analysis
Pro forma analysis refers to the process of forming expectations about future trends in returns and risk confronting the firm as it makes important investment and financing decisions. Much more will be said about this topic as we move through this booklet.
Summary
The bottom line is that completion of financial statements to meet external reporting requirements only to then store them in a filing cabinet ignores a rich source of information on assessing the financial health of the business. Furthermore, calculation of the various measures described in equations (1) thru (17) in the previous section of this booklet does not necessarily help matters much. The issue is whether or not the firm’s performance improved over last year’s results or the Olympic average over the last 5 years. And, if not, why? This requires the use of historical financial analysis. In addition, comparative financial analysis can tell us whether a down trend in these various performance indicators is something unique to the firm, or if it is being experienced by other “like kind” firms as well.
Part II: Growth of the Firm
A. Economic Climate for Growth
The firm faces a number of decisions with regards to input choices and product choices over time. Let’s begin with short run, with the existing firm having a given amount of land and capital, and a given level of management resources. Let’s further assume for the moment that the firm produces a single product. While we assume conditions of perfect competition, imperfect competitors follow most but not all decision rules discussed here.
Economics of Business Expansion
Up to this point we have been assessing the allocation of current resources to maximize profits. An important question pertaining to investment expenditures and planned growth of the firm is how large the firm should be and the combination of resources to use in expansion. These are two separate issues.
Figure 4 – Economies of Size.
How large should the firm be? Many industries have been the subject of studies focusing on the economies of size. Firms often benefit from growth due to increases in efficiency as well as perhaps being able to buy in bulk and hence pay a lower price for inputs. Let’s discuss the previous graph to better understand the issues involved.
At a current market price of P, firm size #1 would not be covering its cost of production (i.e., SAC > P at Q1). Of the options in this graph, the firm needs to operate at a minimum of Q2 units of output. At size #2, the firm would be earning an average profit of P – SAC2 at Q2.
Should the firm expand to size #4, earning an average profit of P – SAC4? The answer to this question is a qualified “No” in an industry where there are no barriers to entry or exit. Why? Suppose as other firms entered, supply increased shifting the market supply curve to the right and depressing the price of the product to PLR. Firm size #4 would face the prospect of downsizing its operations since it is no longer covering its costs of production at Q4 with PLR. This brings up the problem of asset fixity faced by many businesses, or the inability to sell specialized fixed assets in periods of economic decline.
Suppose the firm was raising cotton and had recently purchased a $300,000 cotton picker. Should cotton prices fall sharply (like from P to PLR above) and the firm wanted to flex to another commodity, it might find it hard to dispose of the cotton picker in a weakened secondary farm machinery market. A cotton picker has no use to growers of other commodities or to non-farm sectors because of its specialized nature. Furthermore, other cotton farmers may well be attempting to sell high cost capital items at the same time.
Firm size #3 is the only size depicted in this graph that is positioned to remain a viable business should the price fall to PLR. It is operating at the minimum point on the long run average cost (LAC) curve, where PLR = LAC = MC3. The LAC curve, which is an envelope of a series of short run average cost (SAC) curves, is known as the long run planning curve. Up to the minimum point on the LAC curve, the firm can benefit from increasing economies of size. After that point, the firm would experience decreasing economies of size. Previous studies for many firms show a third range of the LAC curve exhibiting constant returns to size, where the LAC curve is relatively flat before decreasing returns occur.
Capital Variable in the Long Run
A large segment of a firm’s balance sheet is concentrated in fixed assets in the short run. In agriculture, the firm’s tillable land base, its milking capacity, its feedlot capacity, etc. requires capital expansion if more production is to be forthcoming. Other forms of fixed capital such as harvesting equipment may involve the decision of whether to increase capital or labor, and if so, how much?
Figure 5 – Economics of Resource Expansion.
Suppose the firm depicted above wanted to double its output from 10 units to 20 units. You will recall that an isoquant indicates how different combinations of inputs can produce an identical amount of output, and that the point of tangency between the iso-cost line and the isoquant indicates the profit maximizing level of input use. The firm is currently producing 10 units of output using one unit of capital and 5 units of labor (point A) in the above graph. Given the goal of doubling its output, the firm faces the decision of how much to expand its use of capital and labor.
B. An Economic Growth Model
Before digging into the analytics underlying specific investment analysis models, we think it is beneficial to gain an understanding the how internal and external factors influence the growth of firm equity. An important lesson learned here is the notion that the use of debt capital is a “double edged sword”.
Let the following symbols represent specific variables:
r Rate of return on assets
i Rate of interest on debt capital
D Beginning debt outstanding
E Beginning net worth or equity
A Beginning total assets
tY Income tax rate
w Rate of withdrawals from income
Y Net income
Net income before taxes would be given by:
(18) Y = [r(D + E) – i(D)]
while the level of retained earnings or change in equity would be given by:
(19) Y = [r(D + E) – i(D)](1 – tY)(1 – w)
Rearranging terms, we see that
(20) Y = [(r – i)D + rE] (1 – tY)(1 – w)
Dividing both sides of this equation by the beginning level of equity, the rate of return on equity capital would be given by:
(21) ROE = [(r – i)L + r](1 – tY)(1 – w)
where L is the debt-to-equity ratio, or L = D/E.
Equation (21) gives us a simple yet comprehensive economic model that allows us to examine the effects of internally and externally imposed constraints on growth.
Internally Imposed Constraints on Growth
Internally imposed constrains on growth refers to decisions made by the producer that affect the annual growth in equity. Assume the following values for the variables in the growth model:
r 10% or .10
i 7% or.07
D $0
A $100,000
tY 25% or .25
w 0% or 0.0
Given these values, the rate of return on equity capital or ROE would be:
22) ROE = [(.10 – .07)0 + .10](1 – .25)(1 – 0)
= [.10](.75)(1.0)
= .075 or 7.5%
If the firm withdrew 50 percent of after-tax income for other uses, the firm’s ROE would be equal instead to:
ROE = [(.10 – .07)0 + .10](1 – .25)(1 – .50)
= [.10](.75)(.50)
= .0375 or 3.75%
The firm above internally rationed its use of debt capital to avoid exposure to financial risk. If the firm instead had borrowed $100,000, giving it a debt-to-asset ratio of 0.50 or a debt-to-equity ratio of 1.00, its ROE would be:
ROE = [(.10 – .07)1.0 + .10](1 – .25)(1 – .50)
= [.03 + .10](.75)(.50)
= .04875 or 4.875%
which is higher than the ROE given previously. Thus, the internal rationing of the use of debt capital and the decision to withdraw equity from the firm for other uses such as family living expenses affects the rate of return and economic growth achieved by the firm.
Externally Imposed Constraints on Growth
Externally imposed constrains on growth refers to external policies or events occurring in the economy that an individual consumer has little or no control over, but that affect the annual growth in equity. Let’s assume that the firm is characterized by the following values:
r 10% or .10
i 7% or.07
D $50,000
A $100,000
tY 25% or .25
w 40% or 0.40
The firm’s ROE under these conditions would be:
23) ROE = [(.10 – .07)1.0 + .10](1 – .25)(1 – .40)
= [.03 + .10](.75)(.60)
= .0585 or 5.85%
If the firm must pay a one percentage point risk premium for its term loans, which results in a 8% cost of debt capital, the ROE would fall to:
ROE = [(.10 – .08)1.0 + .10](1 – .25)(1 – .40)
= [.02 + .10](.75)(.60)
= .054 or 5.4%
which is 0.45 percentage points lower than earned by others who do not have to pay this risk premium of term loans.
If lenders not only charge a one percentage point risk premium but also limited the firm to a 75 percent debt-to-equity ratio (an example of external credit rationing), the firm’s rate of return on equity or ROE would be:
(24) ROE = [(.10 – .08)0.75 + .10](1 – .25)(1 – .40)
= [.015 + .10](.75)(.60)
= .05175 or 5.175%
which is .675 percentage points lower than the ROE given by equation (24) that excluded the risk premium and credit rationing.
Another external constraint on economic growth is imposition of income taxes. If the effective income tax rate is 35 percent rather than 25 percent, staying with the parameters in equation (24), we see the value of ROE would be:
(25) ROE = [(.10 – .08)0.75 + .10](1 – .35)(1 – .40)
= [.015 + .10](.65)(.60)
= .04485 or 4.485%
which is now .915 percentage points lower than the value given earlier in equation (23).
Financial Risk Associated With Revenue Variability
The use of debt capital can be seen as a two-edge sword. As long as the rate of return on assets (r) is greater than the cost of debt capital (i), or r > i., the use of debt capital contributes to the growth of the firm’s equity. The firm however is exposed to financial risk when borrowing if events lead to the situation where the rate of return on assets (r) is less than the interest rate on debt capital (i), or r < i.
Let’s assume that the rate of return on assets fell from 10% to 2% in a given year while the interest on outstanding debt was 8 percent. Holding all other conditions, we see that the firm’s ROE would fall to:
(26) ROE = [(.02 – .08)0.75 + .02](1 – 0)(1 – 0)
= [(-.06).75 + .02](1.00)(1.00)
= -.025 or –2.5%
The value of the income tax rate would be zero due to the negative net farm income. The rate of withdrawal would also be zero for the same reason (which explains the two “1.00” appearing in this equation.
Thus, while the firm’s ROA was 2 percent, the rate of growth in equity would be negative. The more leveraged the firm is, the greater a low ROA will have upon the firm. For example, assume the firm in this case had a leverage ratio of 2.0 rather than .75. Prove to yourself that the value of ROE would be – 10%.
More will be said about business and financial risk as we begin discussing evaluating and ranking investment opportunities later in this booklet.
Part III. Valuing Investment Projects
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:
27) PV = FVN/(1+R)N
or
28) PV = FVN(PIFR,N)
where PIFR,N is the present value interest factor for interest rate R and N periods in the tables distributed in class and PV is the present value of a sum FVN received N periods from now.[3]
For example, what is the present value of $500 to be received 10 years from today if the discount rate is 6 percent?
29) 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:
30) 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 table. 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 10- 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:
31) 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:
32) 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 (30) 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 (30) no longer is applicable in this case. We can instead use a modified form of the approaches outlined in equations (31) and (32) as follows:
33) PV = NCF5 (PIF.06,5) + NCF10 (PIF.06,10)
= $250(.747) + $250(.558)
= $186.75 + $139.50
= $326.25
or
34) 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 (31) or (32)) since less is received earlier in the period, but more than the single payment received 10 years from now ($279 given by equation (29)).
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 distributed in class rests on this assumption. That means that equations (31) and (32) are not applicable if this assumption does not hold. Let’s relax this assumption be restating equation (32) as follows:
35) 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 right-hand side of equation (35) will take the form:
= $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 land annually and the annual discount rate is 6 percent. The present or “capitalized” value of this tract of land can be approximated as follows:
36) 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:
37) PV = $100/.05
= $2,000
We will use this concept later when evaluating two mutually exclusive investment projects with unequal lives.
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
38) $6,000 = PI(EPIF.15,4)
$6,000 = PI(2.855)
so
(39) 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:
40) 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 (40), the principal and interest payments would be:
41) 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:
42) I = $1,000(.08)
= $80.00
The principal portion of this payment in the first year would therefore be:
43) 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
Equation (40) 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:
44) LOAN = PI(EPIFR,N)
The last two options require solving for the equal payment present value interest factor, or:
45) (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) tables distributed in class.
Future Value
While the focus here has been on the present value of a future sum (equations 27 and 28) and future streams of annual sums (equations 30 through 32), some applications may require reversing the timing perspective of a decision. This can be done by solving for the future value (FV) rather than present value (PV) in equations 27 and 28 and dividing by rather than multiplying by the annual interest rate factors in the subsequent equations. For the purposes of this course, our focus will remain on bringing future values back to the present, or present value analysis.
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, 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.
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
46) 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 (46) as follows:
47) 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 (46) and (47) as being very similar to equations (31) and (32) from a discounting of a 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). We will address the issue of budget constraints later in this booklet when ranking projects.
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 4 below:
Table 3 - Determination of the NPV 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
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 (46) by instead using
48) 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 (48) instead of equation (46) 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 (48), we see that the net present value for project A is equal to:
(49) 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 (46) and (47) should be seen as a special case of the present value formulas presented in equations (31) and (32). We shall limit the examples studied for the moment to those which permit us to use equations (46) and (47); that is, we shall assume a constant discount rate over the economic life of the project.
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., IRRR, 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
50) NPV = NCF1(PIFR,1) + NCF2(PIFR,2) + …. + NCFN(PIFR,N) – C ( 0
which is nothing more than equation (46) set equal to zero. If the net cash revenue flows are identical in each year of the investment project, we can instead use
51) NPV = NCFE(EPIFR,N) – C ( 0
which is nothing more than the equation (48) set equal to zero. Thus, all that remains is to find that value of R in these equations which results in a NPV equal to zero.
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 (51), or
(52) $3,000(EPIFR,5) - $10,000 = 0
Solving equation (52) for the interest factor (EPIFR,5), we see that
(53) (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) tables. 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 (50) rather than equation (51) 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 (50). 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 (50). 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), 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 (50) 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
Year 1 net cash flow +10,000
Year 2 net cash flow - 10,000
Year 3 net cash flow +12,000
This pattern of cash flows over a two year period has two sign reversals; from +$10,000 to -$10,000 and then from -$10,000 to +$12,000. So there can be as many as two positive IRRs that will result in a NPV of zero. Which solution is correct? Neither solution is valid! Neither provides any insight to the true project returns. Thus 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.
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).
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 expanding the firm’s resources by making an investment. 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 for Year 1.
| |Before new investment |After new |Net change |
|Item: | |investment | |
|1. Cash receipts |$25,000 |$30,000 |$5,000 |
|2. Cash operating expenses |-15,000 |-18,000 |-3,000 |
|3. Depreciation |-3,000 |-4,000 |-1,000 |
|4. Tax deductible expenses (2+3) |18,000 |22,000 |4,000 |
|5. Taxable income (1 – 4) |7,000 |8,000 |1,000 |
|6. Income tax payments (5 times 25%) |1,750 |2,000 |250 |
|7. Net income after taxes (1 – 4 – 6) |5,250 |6,000 |750 |
|8. Net cash flow (7 + 3) |8,250 |10,000 |1,750 |
Table 4 shows that the additional cash receipts generated by the investment project is $5,000 in year 1. 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.
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. Finally, we have to add back in the depreciation expenses used to compute tax deductible expenses (a non-cash flow) to measure the annual net cash flows associated with the project. Table 4 above indicates that this annual net cash flow in year 1 would be equal to $1,750.
This calculation must be made for each year covered by the capital budgeting decision. If we assume the annual net cash flows are identical over the entire economic life of the project, however, we can use equation (51) when computing the net present value of the project.
Economic and Service Lives
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:
(54) NPV = NCFE(EPIFR,N) – C
= $4,500(EPIF0.05,3) - $10,000
= $4,500(2.723) - $10,000
= $2,254
based upon applying equation (48) presented earlier. Note this is approximately $736 less that the net present value reported for project A in Table 3 ($2,990 - $2,254).
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 (48) 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 in equation (49) to show that:
(55) AeA = NPVA ÷ EPIF0.05,5
= $2,990 ÷ 4.329
= $691
Similarly, the equivalent level annuity for project C is equal to:
(56) AeC = NPVC ÷ EPIF0.05,3
= $2,254 ÷ 2.723
= $828
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 ends. 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 perpetuity. For project A, this means we must compute:
(57) NPV(A = AeA ÷ RA
where NPV(A 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:
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
NVPAA = $600(3.791) - $2,000 NVPBB = $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
NVPAA = $600(6.145) - $2,000 - $2,100(0.621) NVPBB = $375(6.145) - $2,000
= $382.80 = $304.37
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.
The ranking given by the analysis in Table 6 is identical to the ranking suggested by the equivalent annual annuity approach. The value of AeAA would be $63.43 ($274.60/EPIF.05,5) while the value of AeBB would be $39.42 ($304.37/EPIF.05,10).
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 (46), (47) and (48). This value should be the net cash purchase price after installation and 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 is 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 (46) to include this discounted terminal value as follows:
58) NPV = NCF1(PIFR,1) + NCF2(PIFR,2) + …. + NCFN(PIFR,N) – C + T(PIFR,N)
Similarly, equation (47) can be modified to include the discounted terminal value as follows:
59) NPV = NCF1[1/(1+R)] + NCF2[1/(1+R)2] + …. + NCFN[1/(1+R)N] – 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 (48) by instead using
60) NPV = NCFE(EPIFR,N) – C + T(PIFR,N)
Let’s return to the example contained in Table 3. 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.
The net present value of project B now becomes
61) 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 on project A in Table 3, let’s assume you expect to sell the assets acquired under project A for $7,000 at the end of the second year of the project. Using equation (60) above, we see that:
62) 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 (62) is to find the terminal value that results in a net present value of zero. We can rearrange equation (60) to read:
63) T = [C – NCFE(EPIFR,N)] ÷ (PIFR,N)
= [$10,000 – $3,000(EPIF0.05,2)] ÷ (PIF0.05,2)
= [$10,000 – $3,000(1,859)](1.103)
= $4,878
Thus, the 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.
We will initially assume that this discount rate is identical over the entire economic life of the project and that it reflects the investor’s true opportunity cost associated with similar projects. We will relax this assumption later in this booklet when we focus more intensively on accounting for business and financial risk.
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 the following equation:
64) 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 is quite small, 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 depreciable assets are disposed of at the end of the project. One exception to this rule is permanent buildings associated with real estate.
Purchase of Real Estate
The purchase of real estate assets (land and accompanying buildings) 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 real estate with the characteristics of equation (63) is given by:
65) 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:
66) T = C ÷ PIFG,N
The variable G represents the rate of appreciation in real estate 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.
Maximum Bid Price for Real Estate
When participating in an auction or simply offering to purchase real estate in a one-on-one negotiation, it is important to know what the maximum value you can afford to pay for a tract of real estate. This value can be found by merely rearranging the terms in equation (65) to solve for the original purchase price 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. Let’s assume that no additional equipment is required to operate this additional land. Further 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. A final set of assumptions is needed: (a) comparable tracts of land in your area are currently selling for $1,000 (i.e., V0 = $1,000), (b) land is expected to appreciate at a seven percent annual rate over the next 20 years, (c) your capital gains will be taxed at a 25 percent tax rate, and (d) that your discount rate is 5 percent.
Given these assumptions, the present value of the future economic benefits from ownership of this land per acre would be: [4]
67) PV = NCFE(EPIFR,N) + {V0 ÷ PIFG,N}(PIFR,N)
– [tCG({V0 ÷ PIFG,N} – V0)](PIFR,N)
= $75(EPIF0.05,20) +{$1,000 ÷ PIF0.07,20}(PIF0.05,20)
– [0.25({$1,000 ÷ PIF0.07,20} – $1,000)](PIF0.05,20)
= $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
You can justify making this investment as long as the net present value of this project is greater than zero (NPV = PV – C = $0). This suggests that the maximum you can bid for this tract of land on a per acre basis, or C, is also $2,125, or that:
68) $0 = PV – C
= $2,125 – C
Transposing C to the left-hand side, we see that C, the maximum bid price, is equal to $2,125.
If you had ignored the capital gains component when determining how much to bid, you would have incorrectly concluded that you could not justify bidding the current market value of land (V0) since the present value of the annual net cash flows ($935) is less than $1,000. Obviously you do not have to bid $2,125 if land is currently going for comparable values in the area. The point is that you could pay more and still come out ahead.
It should be emphasized that, if the current value of land in the area (V0), the rate of land appreciation (G), the expected net cash flows (NCFE) or your aversion to risk captured in the discount rate (R) change, you have to compute a new maximum bid price. You will notice the valuation approach presented in equation (67) is a far more complex valuation tool than the simple capitalization formula given in equation (36) back on page 31. The added complexity, however, is necessary since the firm will not hold this land in perpetuity.
In summary, it is important to test the sensitivity of this bid price assuming different rates of appreciation of real estate, different market factors influencing expected future net cash flows, and different aversions to risk.
E. Other Capital Budgeting Applications
There are a number of other applications of the time value of money and capital budgeting. The section focuses on two such applications: (1) the optimal age to replace machinery and equipment and (2) the relative merits of purchasing machinery and equipment with a new loan versus acquiring these assets with a capital lease.
Asset replacement decision
The decision to replace an aging major piece of equipment in the firm’s production operations also represents an application of capital budgeting. The decision to replace a machine is somewhat different from the decision to expand because the cash flows from the old machine must also be considered.
Let’s examine the example above where a firm is considering replacing an old machine with a new one costing $12,000. The price received from the sale of the old machine which has a salvage value of zero is $1,000. The salvage value of the new machine at the end of year 5 is $2,000 which is subject to depreciation recapture. Let’s further assume the firm is in the 40 percent income tax bracket.
Line 6 reflects the decrease in operating costs if the replacement is made due to increased efficiency ($3,000) less the associated increase in taxes associated with this economic gain ($1,200 or .40 x $3,000).
Assuming a discount rate of 12 percent resulting in the present value interest factors on line 17, we see the net present value of the replacement project would be -$522. Thus, we would reject the decision to replace the machine at the present time and continue to use the old machine in the firm’s production operations. The payback period of the project based on the net operating cash flows on line 11would be 4.1 years, reflecting that it would take almost the entire period to recover the total net investment of $11,400 shown on line 5.
Optimal replacement age. A different twist on asset replacement decisions is the determination of the optimal age to replace machinery and equipment. This involves finding the year in the service life of an asset prior to the year where the marginal costs associated with its current use becomes greater than the cost of replacing the asset. The year in which the present value of a stream of future ownership costs is minimized represents the optimal age to replace an aging depreciable asset.
Lease vs. buy
The decision to finance the purchase of an asset or finance the use of an asset for a specific number of years involve determining which of these two alternatives results in the least cost to the firm. This is done by comparing the present value of the net outflow of funds over the life of the two alternatives.
The net advantage to leasing calculation is as follows:
Installed cost of the asset
Less: Investment tax credit retained by the lessor (company providing asset)
Less: Present value of the after-tax lease payments
Less: Present value of the depreciation tax shield
Plus: Present value of after-tax operating costs incurred if owned but not if leased
Less: Present value of the after-tax salvage value
Equals: Net advantage to leasing (NAL)
The installed cost of the asset equals the purchase price plus installation and shipping charges. This forms the basis upon which depreciation and investment tax credit (if allowed) are computed.
The present value of the after-tax lease payments reduces the NAL. These payments are discounted at the firm’s after-tax cost of borrowing rather than the firm’s risk adjusted required rate of return to reflect the fact that lease payments are contractually known in advance and thus not subject to uncertainty.
The present value of the depreciation tax shield reduces the cost of ownership and hence is subtracted when computing the NAL. Since the annual depreciation amounts are also known with relative certainty, they are also discounted at the firm’s after-tax cost of borrowing.
Sometimes there are operating costs incurred if the asset is owned but not if leased. These may include property tax payments, insurance, and some maintenance expenses. If they do exist, they represent a benefit to leasing and thus increase the NAL. Since they too are also known with relative certainty, they are discounted at the firm’s after-tax cost of borrowing.
Finally, if the asset is owned, the owner will receive the after-tax salvage value. This is lost of the assets is instead leased. Thus, the after-tax salvage value reduces the NAL. Since this value is not known with relative certainty, it should be discounted by the firm’s after-tax weight cost of capital which includes a risk premium.
Let’s consider the following example. Suppose a firm is considering leasing an asset that can be purchased for $50,000, including delivery and installation. Alternatively, the asset can be leased for a six-year period at a beginning of the year lease payment of $10,000.
Suppose the firm can borrow the funds to purchase the asset at a rate of 10 percent. If the asset is purchased, it will require insurance and a maintenance contract costing $750 annually. The asset would me depreciated as a five-year asset where the allowable annual rates are 15%, 22%, 21%, 21% and 21%. Assume an investment tax credit rate of 10% exists.
The calculation of the net advantage to leasing in this case example would be as follows:
where:
This case example suggests that the net advantage to leasing is negative, which means the firm would be better off economically if it borrowed and purchased the asset rather than leasing it.
This general procedure can be used to evaluate any lease versus buy decision once it has been determined using standard capital budgeting techniques (i.e., net present value) that an asset should be acquired.
Part IV: Valuation of Externalities
A. Historical Assessments
An important consideration in the financial decision making processes of firms is the valuation of variables over which the firm has little or no control. A perfect example in agriculture is crop yields. Forming knowledgeable expectations about future trends in this and other externalities is imperative to making sound investment and financing decisions. The old GIGO rule (garbage in – garbage out) is a good rule to keep in mind.
Successful firms will find use for all the information available at its disposal, including historical on past product and input prices and productivity (e.g., bushels per acre, gain per pound of feed). Gross revenue or simply revenue for a particular enterprise is given by:
Revenue ( Product price/unit ( output/unit ( number of units used
Historical product and input prices can help explain deviations from historical trends in the financial indicators discussed earlier in equations (1) through (17) given uniform or constant productivity (i.e., output per unit of input). However, future trends and deviations from trend are influenced by changes in productivity, external global market events in both domestic and competitor nations influenced by government policies, financial crises in client nations, and other factors that present new risks and returns.
Yields in crop and livestock production are more local in nature. Understanding long run trends in crop yields and deviations about these trends, for example, can be instrumental to making projections of future revenue flows. Assume the annual yields in wheat production over the last ten years on a farm were as depicted in the scatter diagram below:
where this scatter reflects the following observations:
1996 = 35.6 1997 = 34.1 1998 = 40.2 1999 = 36.5
2000 = 31.8 2001 = 37.7 2002 = 39.1 2003 = 36.4
2004 = 41.2 2005 = 36.8
The average or mean of this time series is 36.94, which was found by using the AVE function in the Excel spreadsheet. The standard deviation for this same time series is equal to 2.80, which was found by using the STD function in the Excel spreadsheet.
These elements of this historical probability distribution can be depicted as shown below:
This information suggests that we can be approximately 70 percent confident that the yield on the firm’s tract of land planted to wheat will range between 34.14 bushels and 39.74 bushels.
The “least squares” line passing through the scatter diagram above can be found by using regression analysis in Excel. Using the SLOPE and INTERCEPT functions in the Excel spreadsheet, we see that the linear time trend over the 1996 – 2005 time period is given by:
69) Yield = - 632.318 + 0.334545(Year)
The percent deviations about this historical time trend can be super-imposed on the forecasted long run yield trend starting in 2006 to assess how sensitive an investment decision is to past weather variability. More will be said about this topic in the next section.
Armed with this information, we can assess the long run yield trend for the firm’s local yields by substituting successive years into equation (69). This equation suggests that the yield for wheat in 2006, for example, would be:
70) Yield = - 632.318 + 0.334545(2006)
= 38.78
or 38.78 bushels per acre. This process can be extended over the life of the investment project to forecast future trends in yields reflecting annual productivity gains due to improved varieties or cultural practices, giving us a projection like that illustrated in the figure below:
We can then superimpose observed yield deviations in the past on this long-term trend to examine the sensitivity of the investment project’s feasibility to known weather patterns in alternative scenario simulations. For example, one simulation of the NPV could be based upon the long term yield trend while a second simulation might be based upon deviations about this trend reflecting a reoccurrence of past weather patterns.
Another approach is to compute a cumulative density function or CDF that illustrates the probability that the yield be less than or equal to a specific yield over the time interval used to estimate the CDF.
B. Forecasting Market Prices and Probability
One of the more important dimensions to financial decision-making is the formation of expected future values of income and cost streams over time. Accountants and economists alike have a term they use for this; it is called pro forma analysis. There are a varied of approaches one can take when conducting pro forma analysis. Some are simple, such as using last year’s price or buying projections from private consultants, while others are more sophisticated, as we will demonstrate. We will start with the simple approaches.
Market Outlook Information Approach
Perhaps the easiest approach to acquiring forward information on commodity and input prices is from government and university sources. Many universities, in conjunction with government agencies, provide 12-24 month outlook materials for producers of major crop and livestock commodities. The information is free, but is often somewhat dated. The Amber Waves publication available at usda. is a prime example of monthly market assessments available to U.S. producers.
More detailed and frequently updated price information is available at a cost from private sources. This can range from market newsletters to contractual consulting arrangements with consulting companies specializing in specific sectors of an economy or global market.
Historical-Based Approaches
Another option is to make your own projections of what future trends in commodity and unit input prices will be over the economic life of an investment project. For example, one can use the naïve model approach, which assumes that
71) Pi = Pi-1
where Pi is the projected price in the ith year and Pi-1 is the price in the previous year. While this approach is used because of its simplicity, recent commodity price trends for wheat easily refute the validity of this approach.
Another historical-based approach is to employ the Olympic average approach. This involves using past prices for 5 or more years, dropping the “high” and the “low” price, and computing the arithmetic average for the remaining observations. This approach will most likely out-perform the naïve model approach, but still ignores probabilistic future events that can lead to sharp departures from historical-based expectations. Other more sophistical time series models require a much longer sample period.
Structural Econometric Simulation
As an alternative to the market outlook information and historical-based approaches, one can estimate market demand-supply relationships for the commodities produced by the firm or employ elasticity estimates from previous studies for these commodities. Let the demand for the ith crop be given by:
where the demand curve will shift with changes in the price of substitutes, consumer income (Y-T), real wealth (W), exchange rate relationships with client nations and other domestic and export demand developments. The supply curve will shift with changes in marginal input costs (MIC), productivity, and other domestic supply and import developments.
Time series data on these variables, combined with the use of regression analysis to estimate the demand and supply relationships in double log form, provides price and income elasticity estimates that are useful in pro forma analysis in absence of simulating estimated demand and supply equations. For example, the reciprocal of the econometrically-estimated own price elasticity of supply for a commodity of say 0.25 gives us the price flexibility for the commodity, or
72) %(P = 4.0(%(Q)
This suggests that a one percent increase in the supply of a commodity coming onto the market will cause a 4 percent drop in the price of the commodity. This begins to help the firm assess the magnitude of future price fluctuations.
More expansive econometric analyses that capture the structure of multi-market relationships can provide an appropriate basis for making long run projections of commodity and unit costs of projections. Simulation of these relationships is referred to as structural pro forma analysis. This involves the assessment of alternative scenarios that result in a distribution of market prices for commodities and inputs to the firm’s operations as it considers investment projects. These scenarios include potential developments in farm programs, weather and disease, macroeconomic policies, foreign trade
policies and global market developments. Much like we did for historical observations for the firm’s yield history, we can compute the standard deviations and coefficients of variation for annual distributions of net cash flows reflecting the effects of each scenario.
Triangular Probability Distributions
We shall assume for the moment that we dealing with three scenarios, a “best case” scenario, a “worst case” scenario and a “most likely” scenario. This results in a triangular normal probability distribution where the two equally distributed tails of this distribution reflect the subjective probabilities associated with the “best case” and “worst case” scenarios. The probabilities assigned to these two tails will likely increase over time, reflecting the increasing uncertainty as we move away from the current period and out over the remainder of the investment project’s economic life.
More sophisticated probability distributions can be developed using programs like Simetar. For example, one can test for the assumption of normality assumed in the triangular probability. Often, multivariate empirical distributions are the most representative of real world relationships. This is beyond the scope of this course.
Part V: Adjustments for Risk
A. Exposure to Business Risk
Expected Future Net Cash Flows
Let’s assume a normal triangular probability distribution for the annual net cash flow in the ith year can be expressed mathematically as follows:
(73) E(NCFi) = Pi,1(NCFi,1) + Pi,2(NCFi,2) + Pi,3(NCFi,3)
where:
E(NCFi) Expected additional net cash flow attributable to the project in the ith year
Pi,1 Probability that “optimistic” economic conditions will occur in the ith year
Pi,2 Probability that “most likely” economic conditions will occur in the ith year
Pi,3 Probability that “pessimistic” economic conditions will occur in the ith year
NCFi,1 Net cash flow if “optimistic” economic conditions occur in the ith year
NCFi,2 Net cash flow if “most likely” economic conditions occur in the ith year
NCFi,3 Net cash flow if “pessimistic” economic conditions occur in the ith year
Given the assumption above, the expected value E(NCF1) or mean of this triangular probability distribution is equal to its “most likely” value, or NCFi,2 given in equation (73) above.
Measurement of Business Risk
There are two traditional measures of business risk, the standard deviation above the mean or expected value and the coefficient of variation. Using our notation above, the standard deviation associated with the net cash flows generated by the project in the ith year is given by:
(74) SD(NCFi) = ( [Pi,1(NCFi,1 - E(NCFi))2 + Pi,3(NCFi,3 - E(NCFi))2]
or
75) SD(NCFi) = ( 2[Pi,1(NCFi,1 - E(NCFi))2]
You will notice several shortcuts taken in equations (74) and (75). First, the deviation between the potential net cash flow associated with the “most likely” scenario and the mean of the probability distribution is absent from equation (74). This term drops out under the normal triangular probability distribution assumed here since these two terms are identical! Second, since both tails of this distribution are identical in absolute terms, we can multiply either one of them by 2.0 and drop the other as shown in equation (75).
While the standard deviation is useful for other reasons, it is not a very good measure of risk is it offers to basis of comparison to the mean of the distribution. We can rectify that by calculating the coefficient of variation as follows:
(76) CV(NCFi) = SD(NCFi) ( E(NCFi)
where CV(NCFi) represents the coefficient of variation for net cash flow in the ith year, or business risk per dollar of expected net cash flow. We will use this statistic as our measure of the firm’s annual exposure to business risk associated with a particular investment project later in this booklet.
B. Risk/Return Preferences
Now that we have a measure of the unique annual exposure to business risk, we need to relate that to the firm’s required rate of return, or the discount rate used in assessing the net present value associated with the investment project. To do this, we must first assess the firm’s aversion to business risk. This can be done in the context of a “hurdle” rate, or the minimum rate of return the firm requires for accepting additional risk.
Firm’s respond to exposure to risk. Few are risk neutral when evaluating investment projects unless they inadvertently ignore the risk associated with the expected returns from a project.
This suggests that the risk neutral investor will not require any additional return over the risk-free rate of return. The lowly risk-averse investor will require RRRL,i as a hurdle or required rate of return while the highly risk-averse investor will require RRRH,i. The difference between RF,i and either RRRL,i or RRRH,i represents the business risk premium or additional return for taking additional risks.
Assume you are a consultant discussing an investment project with a client and he has told you that he requires a minimum rate of return of 12% if he is to invest in a project with a risk of 10 cents on the dollar (i.e., a coefficient of variation of 0.10). This response helps you develop what is know as a risk/return preference function. To see this, let’s use the following general form of the risk/return preference function:
(77) RRRi = RF,i + bi(CVi)
where:
RRRi Required rate of return in the ith year
bi Slope of the firm’s risk/return preference curve ((RRRi /((CVi)
For example, if the risk free rate of return (RF,i) is 5%, then we can solve equation (77) for the slope of the risk/return preference curve bi as follows:
(78) bi = (RRRi - RF,i) ÷ CVi
which in our example at the bottom of page 70 would be equal to:
79) bi = (.12 - .05) ÷ .10
= 0.70
Thus the risk/return preference function in this case can be expressed as follows:
(80) RRRi = .05 + 0.70(CVi)
This risk /return preference curve can be displayed graphically as follows:
It is important to note that each year can have a unique required rate of return. Why? There are several reasons: (a) the risk free rate of return (RF,i) can change from one year to the next, (b) the coefficient of variation (CVi) can change from one year to the next, and (c) the slope of the risk/return preference curve can change.
The difference between the required rate of return and the risk free rate of return for an opportunity of equal maturity is known as the business risk premium. This represents the additional rate of return you require over a risk free investment for taking on the business risk involved in the project in the ith year.
These annual values of RRRi represent the discount rates associated with the corresponding annual net cash flows. We can now adjust our net present value model for depreciable assets to account for the presence of business risk as follows:
(81) NPV = E(NCF1)((1+RRR1) + E(NCF2) ( [(1+RRR1)(1+RRR2)] + … + E(NCFn) ( [(1+RRR1)(1+RRR2)…(1+RRRn)] + T÷[(1+RRR1)(1+RRR2)…(1+RRRn)] – C
where:
E(NCF1) Expected additional net cash flow attributable to the project in year 1
1/(1+RRR1) Present value discount factor in year 1 reflecting required rate of return based upon unique risk exposure in year 1
T Expected terminal value of assets acquired
C Initial net outlay for assets acquired
To illustrate, let’s assume the following states of nature facing a firm in year 1 which is considering an investment that will enhance its annual net cash flows:
Table 8 – Elements of Triangular Probability
Distribution.
State of nature: Net cash flow Probability
1. Optimistic $8,382 5.00%
2. Most Likely 7,620 90.00%
3. Pessimistic 6,858 5.00%
We know from our previous discussion that the expected net cash flow in the ith year or E(NCF1) will be $7,620. Let’s prove that to be true using equation (73) as follows:
82) E(NCF1) = 0.05($8,382) + 0.90($7,620) + 0.05($6,858)
= $419.10 + $6,858.00 + $342.90
= $7,620
Using equation (74), we can calculate the standard deviation associated with the annual net cash flows in year 1 of this project as follows:
83) SD(NCFi) = ( [0.05($8,382 - $7,620)2 + 0.05($6,858 - $7,620)2]
= ( $29,032.20 + $29,032.20
= $240.97
We could have also used equation (75) to calculate this standard deviation given the normal nature of our triangular probability distribution and achieved the same solution:
84) SD(NCFi) = ( [2.0(0.05($8,382 - $7,620)2 )
= ( 2.0[$29,032.20]
= $240.97
The next step is to calculate the coefficient for the net cash flows expected in year 1 under this investment project. Using the format outlined in equation (76) we see that the coefficient of variation would be:
85) CV(NCFi) = $240.97 ($7,620
= 0.0316
or approximately 3.2 cents per dollar of expected net cash flow in year 1. Given the specification of the risk/return preference function given in equation (80), we see that the required rate of return in year 1 would be:
(86) RRR1 = .05 + 0.70(0.0316)
= .05 + .022
= .072
or 7.2%. This process is completed for each year in the economic life of the project.
For example, assume the expected value of the net cash flows E(NCFi) over the remaining 3 years of the 4-year economic life of this investment and their corresponding standard deviations are as follows:
Table 9 – Expected Value and Standard Deviation.
Year Expected Standard
Value deviation
1 $ 7,620 $241
2 10,920 488
3 14,220 779
4 14,220 899
The corresponding annual coefficients of variation, business risk premiums and required rates of return using equation (86) would be:
Table 10 – Required return and business risk premium.
Year Coefficient Risk-free Business risk Required
of variation rate of return premium rate of return
1 0.0316 6.89% 2.21% 9.10%
2 0.0447 7.16% 3.13% 10.29%
3 0.0548 7.12% 3.83% 10.95%
4 0.0632 7.26% 4.43% 11.69%
In addition to these annual net cash flows, the firm expects to receive a terminal value of $7,810 when it sells the assets acquired under this project at the end of the 4th year.
The annual required rates of return in Table 10 above are then included in equation (81) when calculating the net present value for this project costing $45,000 as follows:
(87) NPV = $7,620 ( (1+.0910) + $10,920 ( [(1+.0910)(1+.1029)] + … + $14,220 ( [(1+.0910)(1+.1029)…(1+.1169)] + 7,810[(1+.0910)(1+.1029)…(1+.1169)] – $45,000
We can express this calculation in table form to give you a better idea about the individual components of equation (87) as follows:
Table 11 – Use of Risk Adjusted Discount Rates.
(1) (2) (3)
Net Cash Present Value Present Value
Year Flow Interest Factors of NFCi
(i) (NCFi) (1) x (2)
1 $ 7,620 0.9166 $ 6,984
2 10,920 0.8310 9,075
3 14,220 0.7490 10,651
4 14,220 0.6702 9,530
4 7,810 0.6702 5,234
$ 54,790 $41,474
Less initial cost - 45,000
Net present value $ - 3,526
Thus, we would reject this project after adjusting for risk since the net present value is negative.
If we discounted the net cash flows above at the risk-free rate of return (RF,i), we would have calculated a net present value of:
Table 12 – Use of Risk Free Discount Rates.
(1) (2) (3)
Net Cash Present Value Present Value
Year Flow Interest Factors of NFCi
(i) (NCFi) (1) x (2)
1 $ 7,620 0.9355 $ 7,129
2 10,920 0.8753 9,558
3 14,220 0.8167 11,613
4 14,220 0.7625 10,843
4 7,810 0.7625 7,518
$ 54,790 $46,661
Less initial cost - 45,000
Net present value $ 1,662
Using the risk-free discount rate would have led us to conclude that this was an economically feasible investment opportunity!
Finally, how important was it for us to account for the possibility of increasing risk over time rather than use the interest factor calculated for year 1 in Table 11? This table involves using the 9.1% required rate of return reported for year 1 in Table 10 when calculating the interest factors for the subsequent years. The results of this adjustment are reported in Table 13 below:
Table 13 – Use of Constant Risk Discount Rates.
(1) (2) (3)
Net Cash Present Value Present Value
Year Flow Interest Factors of NFCi
(i) (NCFi) (1) x (2)
1 $ 7,620 0.9166 $ 6,984
2 10,920 0.8401 9,174
3 14,220 0.7701 10,951
4 14,220 0.7058 10,036
4 7,810 0.7058 5,512
$ 54,790 $42,657
Less initial cost - 45,000
Net present value $ - 2,343
Thus we still would have concluded that the business risk involved with this project would have made it an infeasible economic opportunity, although the net present value is less negative than that reported in Table 11.
C. Exposure to Financial Risk
The economic growth model presented in equation (24) helped us see the advantages and disadvantages associated with the use of financial leverage to grow the firm. If the rate of return on assets exceeds the rate of interest on debt capital, leverage will contribute to the growth of the firm’s equity. However, if the rate of return on assets is less than the rate of interest on debt capital, leverage will detract from the growth of the firm’s equity. Leverage thus is associated with financial risk. The greater the use of leverage, or greater the debt-to-equity ratio, the greater the potential exposure to loss in equity capital well be.
We can modify the risk/return preference function presented initially in equation (77) to reflect financial risk as follows:
88) RRRi = RF,i + bi(CVi) + ci(Li)
where bi(CVi) represents the business risk premium and ci(Li) represents the financial risk premium. We can visualize the addition of the financial risk premium below:
It can be shown using equation (77) that the required rate of return for a project in the ith year of a project having a risk per dollar of expected net cash flows of 8 cents and risk free rate of return of 5 percent would be:
89) RRRi = .05 + 0.70(.08)
= .106
Adding the financial risk premium to equation (89), we see that:
90) RRRi = .106 + ci(Li)
Let’s now assume that the firm said it would require a rate of return equal to 15 percent given its exposure to business and financial risk if its leverage ratio was 1.0. Given this information we can compute the coefficient in the financial risk premium by transposing terms, or:
91) ci(Li) = .15 - .106
Solving for the coefficient associated with the liquidity variable, we see that
(92) ci = (.15 - .106) ÷ Li
= .044 ÷ 1.0
= .044
which represents the change in the required rate of return for a given change in the firm’s leverage position, or (RRRi/(Li.
With the addition of the financial risk premium, we now assemble the entire risk/return preference function. This function, which includes both the business risk premium and the financial risk premium as well as the risk-free rate of return on assets of similar maturity, takes the form:
(93) RRRi = .05 + .70(CVi) + .044(Li)
This equation suggests that the higher the coefficient of variation associated with expected annual net cash flows over the life of a project or the higher the firm’s debt relative to equity, the greater the “hurdle” or required rate of return a new project will have to “clear” in order to be economically acceptable to the firm’s decision makers.
D. Portfolio Effect
The firm can benefit from diversifying its portfolio of assets and enterprises if certain conditions hold. One of these conditions is that the net cash flows associated with the firm’s existing operations be highly negatively correlated with the net cash flows generated by the new project.
We can illustrate the nature of the path taken for annual net cash flows generated by the firm’s existing assets that are highly negatively correlated returns with the annual net cash flows associated with a new investment project by examining the following figure:
The figure above illustrates the case where the net cash flows generated by the firm’s existing assets are low when the net cash flows from the new project are high, and vice versa. Thus, the peaks of one stream help offset, at least in part, the valleys of the other stream.[5] If the expected net cash flows generated by these two sources are weighted approximately the same, the time path taken by the E(NCFi) will be a relatively flat line parallel to the time axis, reflecting constant net cash flows over time.
When this is the case, the firm’s overall exposure to risk is lowered, allowing us to reduce the required rate of return given by equation (93) due to the investment project’s risk reducing features.
To illustrate how negatively correlated investment projects affect the firm’s exposure to business risk, let’s examine the following situation. Suppose you are considering investment in project C and are concerned about the degree of business risk associated with the project’s expected net cash flows. Let the expected rate of return from project C in the ith year be represented by E(ROAC,i) and the standard deviation of these returns be represented by SD(ROAC,i). Further assume that the expected rate of return from firm’s existing assets in the ith year is represented by E(ROAEX,i) and the standard deviation of these returns be represented by SD(ROAEX,i). Finally, assume that the expected rate of return generated by the new project is highly negatively correlated with the expected rate of returns generated by the firm’s existing assets (see the figure above).
The expected rate of return for the entire portfolio in the ith year after the project is completed would be given by:
94) E(ROAT,i) = WC(E(ROAC,i)) + WEX(E(ROAEX,i))
where WC + WEX = 1.0
The standard deviation in the ith year for the new portfolio of assets would be given by:
95) SD(ROAT,i) = {WC2(SD(ROAC,i))2 + WEX2(SD(ROAEX,i))2
+ [2(WC)(WEX)(()(SD(ROAC,i)) (SD(ROAEX,i))] }1/2
where ( represents the correlation coefficient between the rate of return generated by the firm’s existing assets and a project it is considering. If these rates of return are highly negatively correlated, the value of ( will be at or close to –1.0. If these rates of return are highly positively correlated, the value of ( will be at or close to +1.0. A value of ( equal to zero means these two annual rates of return are uncorrelated.[6]
The sum of the first two terms in equation (95) represents the weighted average variance for the new portfolio while the entire last term represents the covariance associated with the net cash flows from the new project C and the firm’s existing assets.
Let’s assume that WC = .20 and WEX = .80 in the first year of the project and that the annual rate of return in this year are expected to be E(ROAC,i) = 10% and E(ROAEX,i) = 8%. Using equation (94), the expected rate of return for the entire portfolio after the investment is made is expected in year 1 to be:
96) E(ROAT,1) = 0.20(0.10) + 0.80(0.08)
= 0.084 or 8.4%
If the values of the corresponding standard deviations are SD(ROAC,i) = 0.02 and SD(ROAEX,i) = 0.03 and the value of the correlation coefficient ( = - 1.0, then the standard deviation for the entire portfolio in year 1 using equation (95) would be:
(97) SD(ROAT,1) = {(0.20)2(0.02)2 + (0.80)2(0.03)2
+ [2(0.20)(0.80)(-1.0)(0.02)(0.03)] } 1/2
= 0.02
which results in a coefficient of variation for the entire portfolio in year 1 after the new investment is made of:
98) CV(ROAT,1) = SD(ROAT,1) ÷ E(ROAT,1)
= 0.02 ÷ 0.084
= 0.238
or 23.8 cents per dollar of expected net returns.
How can we use the total portfolio information given by equations (96) and (97)?
The coefficient of variation or risk per dollar of expected return for the existing portfolio was equal to:
99) CV(ROAEX,1) = SD(ROAEX,1) ÷ E(ROAEX,1)
= 0.03 ÷ 0.08
= 0.375
or 37.5 cents per dollar of expected net returns. Therefore, the incorporation of the new project (project C) into the firm’s total portfolio lowers the risk per dollar of expected return from 37.5 cents to 23.8 cents, a 36 percent reduction! This occurs only because the returns from project C are highly negatively correlated with the returns stemming from the firm’s existing assets.
This would suggest that a reduction in the required rate of return for project C when computing its net present value is justified. One approach would be to lower the business risk premium by 36 percent, leaving the financial risk premium and risk-free rate of return unchanged. Suppose the business risk premium associated with the firm’s existing assets was 5 percent and the risk free rate of return was 10 percent, we can graph the adjustment proposed here in year 1 of the project would be as follows:
[pic]
E. Optimal Capital Structure
Explicit and implicit costs of capital
Thus far we have focused on the required rate of return, mentioning the cost of debt capital only in comparison to the rate of return on assets (ROA) when discussing the rate of growth in equity capital and when discussing a project’s internal rate of return, or IRR. Even then we only addressed the explicit cost of debt capital, or the externally determined rate specified on the mortgage or note.
There implicit costs of debt capital that cause firms to internally ration their use of debt capital that were more or less implied when we discussed the concept of financial risk and the financial risk premium. As the firm reduces its credit liquidity as it uses up its credit reserves, its implicit cost of debt capital rises, causing the total cost of debt capital
to rise as depicted in the graph above. This concept is an important component to analyzing the firm’s weighted average cost of capital and optimal capital structure.
Weighted average cost of capital
The weighted average cost of capital (WACC) employed by the firm is given by the following equation:
(100) WACC = WEQ(rE) + WDT(rD)
where WEQ is the relative importance of equity in the firm’s balance sheet, rE is the cost of equity capital, WDT is the relative importance of debt in the firm’s balance sheet, and rD is the total cost of debt capital. The optimal capital structure of the firm’s balance sheet is given by the least cost combination of debt and equity capital. We can illustrate the point where this occurs is the graph below:
Two features are worth noting in the graph above. The first is the fact that the cost of debt capital is less than the cost of equity capital. How can this be? Think of the cost of debt capital as the minimum opportunity rate of return available to the firm. After all, one of the opportunities available to using the firm’s equity capital is to make loans to others at the going cost of debt capital!
The other feature has to do with the shape of the weighted cost of capital curve and the optimal location on that curve. This curve falls sharply at low debt/equity ratios since the cost of equity capital is higher and carries a higher weight. The optimal location on the weighted average cost of capital curve is at its lowest point. At this point, the firm is minimizing its cost of capital. Any other combination of debt and equity capital would reduce the returns from the firm’s portfolio.
A numerical example corresponding to the graph depicted above is presented below:
Table 14 – Calculation of the Weighted Cost of Capital.
Leverage Source of Unit
ratio capital cost WACC
0.0 Debt 0.04
Equity 0.06 0.060
0.5 Debt 0.04 Equity 0.06 0.053
1.0 Debt 0.04 Equity 0.06 0.050
1.5 Debt 0.05
Equity 0.08 0.062
2.0 Debt 0.06
Equity 0.10 0.074
We see above that the least cost combination of debt and equity capital occurs where the firm achieves a 50-50 balance of debt and equity capital on its balance sheet. At this point we see that the weighted average cost of capital is 5 percent.
F. Ranking Potential Projects and the Capital Budget
An important topic covered in this booklet is the role that the capital budget plays in the selection of economically feasible investments to fund in the current period. Let’s assume the firm is facing the following investment opportunities this year and has a $90,000 capital budget to work with:
Table 15 – Cost and Benefits from Alternative Projects.
(1) (2) (3)
Present Net
Value of Present
Cost of Net Cash Value
Project Project Flows (2) – (1)
A $10,000 $14,500 $4,500
B 24,000 33,120 9,120
C 7,500 8,850 1,350
D 43,000 46,500 3,400
E 5,250 3,360 -1,890
Totaling up the costs of the five projects the firm is considering, we see that this total ($89,750) does not exceed the amount of debt and internal equity capital available this year to the firm ($90,000, which is comprised of $50,000 in debt capital and $40,000 in equity capital). What projects would you advise this firm to invest in?
First, we can throw out project E because it has a negative net present value. This leaves us with $84,500 in projects that have a positive net present value. Should the firm invest in all four projects?
If the firm wants to minimize its cost of financing (i.e., use 50 percent retained earnings and 50 percent debt capital given by the minimum point on its weighted average cost curve in the previous example), the answer is no. For example, if the firm sticks by this least-cost decision rule, it would prefer to spend only $77,000 on new investment projects in the current period, using $38,500 in equity capital and $38,500 in debt capital. The firm in this instance would invest in projects A, B, and D. The firm would hold the unused portion of its available equity capital (i.e., $1,500 or $40,000 - $38,500) in reserve in the short run in the form of a liquid income-earning asset. And it would not borrow additional debt capital it might otherwise employ ($11,500 or $50,000 - $38,500).
If the firm had invested in all four projects costing $84,500, and used up its available debt capital first, the D/E or leverage ratio for these projects would be approximately 1.45 ($50,000/($84,500 - $50,000), which is much higher than its cost minimizing target capital structure of 1.0.
In fact, all other combinations of available debt and equity capital to finance all four projects would also lead to a capital structure other than 1.0.
Other factors such as achieving a greater market share in the short run might well convince a firm to invest in all four projects fitting within its available budget, thus ignoring the optimal capital structure rule. However, a firm bumping up against its maximum leverage ratio rationed externally to it by its lender (and hence more exposed to financial risk) will be more apt to consider this concept when taking on new projects.
Part VI: Selected Macroeconomic Topics
The purpose of this section of the booklet is to familiarize you with the key ways in which events in the domestic and global economies beyond the farm gate influence the profitability and financial strength of a nation’s farmers and ranchers.
A. Agriculture and the Global Economy
The agricultural sector in developed economies benefits from sophisticated delivery system for inputs to production, including the loan funds necessary to purchase inputs on credit. The sector also benefits from a market infra-structure that allows farmers and ranchers an array of opportunities to sell their output.
The food and fiber system is essentially comprised of four key sectors in the economy. These are the farm input supply sector, the farm sector itself, the processing and manufacturing sector, and the wholesale and retail trade sector. These sectors combined utilize natural, human and manufactured scarce resources to deliver finished food and fiber products to consumers (households, governments and foreign entities). The supply of these products is influenced by the cost and availability of loan funds from credit markets as well as the cost and availability of labor, particularly in the fresh fruit and vegetable segment of the farm sector.
The impacts of domestic macroeconomic policy include interest rates, inflation rates, economic growth, unemployment, budget deficits and the national debt, exchange rates and the trade deficit. The impacts of macroeconomic policies of client nations include the growth of their disposable incomes, exchange rates and any barriers to trade. These policies, taken together, fuel the components of gross domestic product (consumption, investment, net exports and government spending) by influencing activity in product, labor and financial markets.
The farm input supply sector, the first in the chain of sectors comprising the food and fiber system, is a major source of new capital embodied technologies and hence productivity in the production of raw agricultural inputs on the nation’s farms and ranches. This sector is typically characterized by a high degree of concentration (small number of firms) with the market power to set prices farmers and ranchers must pay for inputs. These firms have become much more diversified in recent years. John Deere, for example, has globally diversified its market presence in farm machinery and equipment markets, and has increased the relative importance of its construction machinery and home and garden machinery divisions in recent years.
The food processing and fiber manufacturing sector in a nation’s food and fiber system adds value to raw agricultural products (examples: wheat to flour to bread; corn to ethanol to blended gasoline. The diagram on the top of the next page illustrates the linkages between labor markets, farm input markets and financial markets to a corn farmer on the one hand, and the distribution of the corn farmer’s crop once harvest takes place. Fuel use of
of corn as a feedstock for the production of ethanol in the United States, for example, has grown from 5.5 percent in 1997 to roughly 27 percent of the total use of corn ten years later in 2007 (see graph below). This, in turn, has led to a reduction in the amount of the crop available for feed use, which has driven up feed prices to livestock farmers and ranchers. Food use, which includes the use of corn syrup in the manufacturing of candies and other products containing sweeteners, has remained relatively constant. This process is illustrated in the graph below.
Once the processing and manufacturing phase is completed, valued added products containing corn components move through wholesale and retail trade firms to their final consumer. These firms often practice “just in time” inventory strategies to gain efficiencies in storage and handling.
The total portion of the food dollar going to food processors and manufacturers and to wholesale and retail trade firms is 80 cents, or 80 percent. This share is known as the marketing bill.
Commodity and environmental policy of domestic and foreign nations play a role in regulating the use of resources and product safety of the final product going to consumers. Governments also subsidize specific sectors in the food and fiber system in the name of food security and trade competitiveness.
B. Economic Activity and Macro Policy Responses
This section presents a rather simple Keynesian macroeconomic model for an open economy. There are four markets in this model, three of which are obvious (product market, money market and labor market) and one which is less obvious (government bond market). Each is described below before we discuss the impacts of monetary and fiscal policy.
The model expressed in this section of the booklet is the Keynesian IS-LM representation of general equilibrium. The equations underlying these graphs are presented in the appendix to this booklet.
If we plot a series of equilibrium interest rates, we get what is known as the LM curve, where L represents the demand for “liquidity” and M represents the money supply.
Each and every point along the LM curve depicted above represents a equilibrium in the money market.
If we plot these equilibrium gross domestic product levels, we get what is known as the IS curve, where I represents the demand for investment expenditures and S represents the level of savings in the economy, or:
Each and every point along the IS curve depicted above represents an equilibrium in the product market. We can now plot both curves in the same space.
This graph suggests that only one interest rate and one level of gross domestic product satisfies the equilibrium conditions in both markets simultaneously.
How does monetary and fiscal policy affect the IS and LM curves? Expansionary (contractionary) monetary policy will shift the LM curve to the right (left) while expansionary (contractionary) fiscal policy will shift the IS curve to the right (left).
Monetary policy actions shift the LM curve and leave the IS curve alone. Expansionary monetary policy will shift the LM curve to the right, lowering interest rates and stimulating aggregate demand through investment, jobs and consumption. Contractionary monetary policy will have exactly the opposite effects.
A graphical representation of expansionary monetary policy (expansion of the money supply by increasing total reserve or lowering the reserve requirements) is illustrated below. This lowers interest rates in the money market which stimulates aggregate demand.
Contractionary monetary policy actions (contraction of the money supply by decreasing total reserve or raising reserve requirements) would shift the LM curve in the opposite direction. This lowers interest rates in the money market which depresses aggregate demand.
Fiscal policy actions on the other hand shifts the IS curve and leaves the LM curve alone. Expansionary fiscal policy (tax cut or increase in government spending) will shift the IS curve to the right, stimulating aggregate demand through increased after tax income and government spending, but increasing interest rates. Contractionary fiscal policy (tax increase or decrease in government spending) will have exactly the opposite effects.
A graphical representation of expansionary fiscal policy is illustrated below. We see an increase in aggregate demand in the product market but an increase in the nation’s money market.
Contractionary fiscal policy actions, on the other hand would shift the IS curve in the opposite direction, resulting in a reduction in aggregate demand in the product market, but lower interest rates in the money market.
We can determine the general equilibrium price level in the economy (PE) by associating the general equilibrium conditions in the IS-LM graph with a graph of the equilibrium in the product market.
We can see relationship graphically in the two graphs below:
There are three more graphs to add to the graphical macroeconomic model presented above: a money market underlying the LM curve, a labor market graph and a short run Phillips curve. YPOT represents potential GDP.
Beginning with the money market, the demand curve will shift to the right (left) if income increases (decreases). Supply is dictated by the level of total reserves determined by the nation’s central bank as well as the required reserve ratios. The graph for the money market is illustrated below:
[pic]
An increase (decrease) in the money supply and shift the MS curve to the right (left), decreasing (increasing) interest rates. This corresponds to a shift in the LM curve to the right (left) which represents the same change in the interest rates in that graph.
The labor market will be affected by changes in aggregate demand in the product market. This is because meeting the change will require a change in the number of workers employed in the economy.
where WRE represents the equilibrium wage rate, LD represents the demand curve for labor, LS represents the supply curve for labor, LE represents the equilibrium level of employment and LMAX represents the size of the civilian labor force. The demand for labor is derived from the level of activity in the nation’s product market. An increase (decrease) in aggregate demand in the product market will shift the demand curve for labor to the right (left), increasing (decreasing) wage rates in the economy. Finally, one minus the ratio of the level of employment to the size of the civilian labor force gives the unemployment rate, or UR = 1 – LE/LMAX.
Finally, the short run Phillips curve illustrates the relationship between the unemployment rate (UR) and the rate of inflation (INF = %∆P). This graph is illustrated below:
An increase (decrease) in aggregate demand in the product market leading to an increase (decrease) in the demand for labor will lower (rasie) the unemployment rate but increase (decrease) inflation, causing a movement down (up) the Phillips curve.
Inflationary and recessionary gaps
A major objective of macroeconomic policy is to promote economic growth while keeping inflationary pressures under control. One approach to visualizing this objective is to determine whether a recessionary or inflationary gap exists in the nation’s product market.
The graph below illustrates the concepts of recessionary and inflationary gaps in the product market:
Let YFE represent full employment output or the natural rate of employment of labor and capital that promotes economic growth at low inflation. If equilibrium output lies to the left of full employment output (YE < YFE) as illustrated above, a recessionary gap is said to exist in the economy. If equilibrium output lies to the right of full employment (YE > YFE) and is crowding the economy’s potential output, an inflationary gap is said to exist.
Expansionary monetary and/or expansionary fiscal policy can be used to eliminate a recessionary gap in the economy. Contractionary monetary and/or fiscal policy can be used to eliminate an inflationary gap.
C. Big 5 Macro Variables and Agriculture
How do the macroeconomic variables identified in the IS-LM model described in the previous section affect agriculture. There are five major variables that transmit the events in the macroeconomy to agriculture. I refer to them as the “Big 5.” They are:
(1) the rate of interest,
(2) the rate of growth in GDP,
(3) the unemployment rate,
(4) the rate of inflation, and
(5) the foreign currency exchange rate
where the exchange rate is influenced by the relative interest rates and inflation rates (or real rates of interest) between two trading nations. These variables transmit the effects of changes in macroeconomic policy to the nation’s farm sector. These effects are identified in the next section.
We can summarize the effects of expansionary and contractionary monetary and fiscal policy on the “Big 5” variables in the table below.
Table 16 – Impact of Macroeconomic Policy on General Economy.
| |Expansionary |Contractionary |Expansionary |Contractionary |
|“Big 5” variables |Monetary |Monetary Policy |Fiscal |Fiscal |
| |Policy | |Policy |Policy |
|Interest rate |Lower |Higher |Higher |Lower |
|GDP growth rate |Higher |Lower |Higher |Lower |
|Unemployment rate |Lower |Higher |Lower |Higher |
|Inflation rate |Higher |Lower |Higher |Lower |
|Exchange rate |Lower |Higher |Higher |Lower |
The initial affects of the Big 5 variables are as follows:
• Rates of interest – affects farm interest expenses and farm land values
• Rate of growth in GDP – affects the domestic demand for agricultural products (see income elasticity of demand)
• Rate of unemployment – affects the cost of hired labor as well as off-farm income of farm operator families, an important source of internal equity capital for many farmers and ranchers
• Rate of inflation – affects prices for farm inputs in the short run and interest rates over the longer run
• Rate of foreign exchange – affects the value of the dollar relative to client nation currencies and hence the export demand for agricultural products
To understand the effects of activity in the nation’s product market, let’s look at the following set of graphs, keeping corn as our focus:
Expansionary monetary policies, for example, expands GDP (and hence national income). This expands the demand for corn as the demand for food, feed, ethanol and other domestic uses of corn of corn increases. The decline in exchange rate falls ($/unit of foreign currency) will also increase export demand. As shown in the corn market graph above, this pulls up the price of corn.
Individual corn growers are price takers, or have no influence on the market price of corn. The corn rower responds to the higher price signal in the market by planting additional acres to corn (operating where MR=MC) as shown in the last graph above. The growers profit increases (change in average profit (MR – ATC) times the output. The corn grower buys more production inputs from the farm input supply sector and sends more output to processors and manufacturers, who in turn add value and the ending product moves through the wholesale and retail trade sectors and ultimately to consumers, including client nations.
Everything moves in the opposite if contractionary monetary policies are adopted by the nation’s central bank. Lower income levels would shift the market demand curve for corn to the left as GDP falls and unemployment rates rise. The decline in market price sends a signal to producers to plant less corn, which in turn means fewer production inputs are sold by the farm input supply sector and less output is sent to processors and manufacturers. Fewer value added product moves through the wholesale and retail trade sectors and ultimately to consumers, including client nations.
D. Impact on Income Statement and Balance Sheet
Recall the structures of the firm’s income statement and balance sheet discussed in Part I of this booklet (see pages 4 and 5).
To understand the full effects of these variables on the financial strength and economic performance of the farm sector, we need to examine the impact on farm income statements and farm balance sheets.
Table 17 – Impact of Macroeconomic Policy on Farm Sector.
| |Expansionary |Contractionary |Expansionary |Contractionary |
|Farm sector |Monetary |Monetary Policy |Fiscal |Fiscal |
|variables |Policy | |Policy |Policy |
|Farm revenue |Higher |Lower |Higher |Lower |
|Farm expenses |Higher |Lower |Higher |Lower |
|Net farm income |Higher |Lower |Higher |Lower |
|Farm land values |Higher |Lower |Higher |Lower |
|Exports |Higher |Lower |Lower |Higher |
This table illustrates that expansionary monetary and fiscal policies are typically good for farmers and ranchers. The higher interest rates that come with expansionary fiscal policy can lead to lower corn exports. It is often said that the worst thing that can happen to agriculture is for the central bank to get serious about fighting inflation. The second column in the above table clearly illustrates why this is so. Higher interest rates and a contracting economy lead to lower net farm income values in farm income statements and lower equity in farm balance sheets.
E. Capacity Expansion Analysis
There probably is not a harder sector of an economy for forecast market activity than agriculture. Why? The answer is the unique role that weather and disease play from both a domestic and global production perspective. Let’s focus on the demand for farm machinery and equipment. This is relevant to strategic planners at firms like John Deere who need to know the potential volume of sales in the coming years for purposes of scheduling expansion of their manufacturing capacity.
Estimation of the farm machinery market model
Theory of the firm suggests that firms will acquire production inputs up to the point where the marginal value product is equal to the marginal input cost, which implies the desired capital stock of fixed inputs is given by:
(101) K*T = ([E(PT) × E(YT)]/E(CT)
where:
K*T Desired capital stock at end of year t
( Partial elasticity of production for capital in year t
E(PT) Expected product price in year t
E(YT) Expected output in year t
E(CT) Expected implicit rental price of capital in year t
and where production is given by:
(102) YT = A((L*T)((.5K*T+.5KT-1)
with additional variables are defined as follows:
L*T Desired labor use in year t
KT-1 Capital stock at the beginning of year t
α Partial elasticity of production for labor
( Partial elasticity of production for capital
Annual net investment expenditures for depreciable inputs like farm machinery can therefore be defined as follows:
(103) NIT = K*T – KT-1 = IT – DT
where IT represents gross investment or capital expenditures for machinery and DT represents depreciation or replacement investment.
If we want to project the demand for farm machinery, therefore we can estimate the following equation using ordinary least squares regression:
(104) NIT = B0 + B1(RATIOT) + B2(KT-1)
where:
RATIOT E(PT × YT)/E(CT)
KT-1 Existing capital stock
Bi Coefficients to be estimated
Once we have estimated this equation, we can project the level of net investment in farm machinery by inserting our expectations for farm revenue (E(PT × YT)) and the implicit cost of capital E(CT) into the right-hand side and solving for net investment expenditures (NIT).
Gross investment expenditures – or total sales in the industry – is finally found by:
(105) It = NIt + Dt
The graph to the right illustrates the annual trends in machinery and equipment sales to US farmers over a recent 30 year period. This gives us a sufficient historical time period over which we can estimate the statistical model given by equation (104).
Before estimating this equation, however, let’s examine how well a naïve linear time trend explains the annual fluctuations in farm machinery sales over this period. Estimation of this model, with only time as the explanatory variable, results in the following statistical fit:
This model has little or no explanatory ability as evidenced by the low R2 and adjusted R2 circled above. Graphing this estimated equation, we clearly see the inability of the naïve linear time trend model to explain annual sales.
If we instead fit the time series data to the statistical model presented earlier in equation (104), we get a much better forecasting model as shown below:
Graphing this estimated equation presented above, we see visual evidence of the goodness of fit for the model based upon economic theory.
Forecasting future machinery sales
Given the superior econometric results from the model based upon economic theory, the next step is to solve this equation beyond the sample period to forecast future sales activity.
Let’s begin by forecasting machinery sales one year into the future using the recent historical average for the ratio of marginal value product to marginal input cost (RATIOT) as our expected value. A point forecast based on a single set of expectations using the last period’s sales of $9.5 billion results in the following forecast:
(106) NIt = -0.455 + 0.351(RATIOt) + 0.888(NIt-1)
= -0.455 + 0.351(4.19) + 0.888(9.5)
= $9.45
or $9.45 billion in expected machinery sales in the coming year. Given a recent historical standard deviation of $1.49 billion for the RATIO variable, we see that there is a significant probability that sales could be as low as $8.93 billion or as high as $9.97 billion, or:
(107) NIT = -0.455 + 0.351(RATIOT) + 0.888(NIT-1)
= -0.455 + 0.351(4.19 – 1.49) + 0.888(9.5) = $8.93
= -0.455 + 0.351(4.19 + 1.49) + 0.888(9.5) = $9.97
which yields the triangular probability distribution shown to the right. Based upon our expectation using the recent historical value for RATIOT, we can say that while we expect total sales in the industry of $9.45 billion, we also see the potential for sales to be as high as $9.97 billion or as low as $8.93 billion given recent variability in total sales. Stochastic simulation of the forecasting model would identify other probabilities of occurrence.
F. Factors Influencing Monthly Price Fluctuations
This section initially documents a business forecasting model for the U.S. corn market. The general approach taken here can be applied to other agricultural commodities as well. The forecast horizon is assumed to be monthly observations over two crop years for the purposes of our discussion.
Conceptualizing the corn market model
The model described in this section contains elements of the U.S. corn pricing model used to support the MDA Federal CROPCast service provided to approximately 1,500 clients internationally that have an interest in the corn and related markets. Current applications of the model are now focused on the 2006/07 (September 2006 through August 2007) and 2007/08 (September 2007 through August 2008) marketing years.
The underlying concept of the model is the solution of a system of simultaneous equations capturing supply and demand for corn. This requires not only modeling corn specifically but also the demand for corn by-products (i.e., distillers dried grains, gluten meal and corn oil) as well as the demand for corn by ethanol producers.
The graph to the left depicts the traditional downward sloping demand curve for a commodity as
an upward sloping supply curve. The
supply curve will shift to the right as
more acres are planted or as weather
and disease conditions result in
favorable growing and harvesting
conditions. Beginning stocks and imports
also affect supply. The decision to plant corn
depends upon the expected relative profitability of this crop versus other
planting alternatives such as planting soybeans. These expectations are driven or derived from modeling the forwardly-linked markets where corn is used to process value added products like ethanol. For further discussion of these relationships, see pages 65-67 and pages 90-92.
The approach taken is to take acres planted as given by surveys of planting intentions of producers. These acres are then multiplied by estimated yields per acre updated periodically using satellite imagery. A system of estimated demand relationships for corn use are simulated in concert with a model of the ethanol market and corn by-product markets. These models solve for the annual equilibrium corn price by minimizing excess demand.
The next step is to distribute this annual price projection for corn over a series of 12 monthly prices constituting a marketing year. This was accomplished by using clustering analysis of historical monthly price fluctuations annually over a historical period. Two approaches were taken to analyze these trends: Agglomerative Hierarchical Analysis and K-Means
Analysis. The results from the K-Means Analysis, which essentially represents a disaggregation of the Agglomerative Hierarchical Analysis, are depicted in the graph above. Cluster 5 represents only three years out of the 1987-2005 time period; three unusual years where there were major disruptions to the corn market. This graph clearly shows, however, relatively little correlation among clusters of monthly price fluctuations. Applying these clusters to forecasting monthly price fluctuations into the future requires identifying the unique characteristics of each cluster of years (low beginning stocks, weak yields, high planted acres, low global production, etc.), and then matching these characteristics with what is known or expected for the next 12 to 24 months.
The graph illustrated above shows the monthly price forecast for the 2006/07 marketing year, which ended in August 2007, and 2007/08 marketing year made by MDA Federal’s CROPCast service. A weighted average of cluster 3 and cluster 5 was used in the 2006/07 forecast. Stochastic simulation, where yields forecasted by MDA Federal were treated as random variables in the corn market simulation model, resulted in the forecasted price and associated standard deviation. The actual monthly price exceeded the range of potential price trends in only one month (February), and was extremely accurate in forecasting both the direction and levels of prices beginning in April 2007. Their forecast hit the actual price in August ($3.17) right on the nose. What value does this forecast have for corn farmers and commodity traders? That is the subject of the final section of this booklet.
G. Cash Market Versus Forward Pricing Alternatives
Corn farmers have several options in the marketing their production during the marketing year. First, they can rely entirely on the local cash market by selling their entire crop at harvest time. Second, they can hedge at least a portion of their crop by selling a futures market for a certain date after harvest and then buying an equal offsetting contract. Third, they can sign a production contract with a local ethanol processing plant for part of the crop that specifies the price at delivery. And fourth, they can store their crop in either on-farm storage facilities or at a local elevator, and then sell into the market each month to achieve an average price over the year.
An Example of the Short Hedging Process
Suppose a corn farmer decides to explore a variety of marketing alternatives, including futures, rather than settle for whatever the local elevator is willing to pay at harvest time. The farmer's ultimate goal is to improve his bottom line.
Assume the farmer estimates it will cost $2 to produce a bushel of corn. Further assume the farmer sees a December futures price on the Chicago Board of Trade (CBOT) where a profit can be made. He can hedge a portion of the crop by selling a December corn futures contract or contracts on the Chicago Board of Trade (CBOT).[7]
For example, suppose CBOT December futures in early May hits $2.60 a bushel. Using a service like CROPCast provided by MDA Federal, the farmer feels this is the best time to lock in a selling price. The farmer contacts his broker and sells a CBOT December corn futures contract. Let’s assume the Midwest experienced a bumper crop year, causing corn yields to be well above normal and the cash market price of corn to fall to $1.90 a bushel at harvest time. The farmer offsets his futures position by purchasing a CBOT December corn futures contract or contracts.
The end result of the farmer's hedging activities can be summarized as follows:
Table 18 – Benefits From Hedging When Cash Price Falls.
|Cash Market |Futures Market |
|May: Plans to sell 5,000 bushels of corn; sells a CBOT December corn |Sells a CBOT December corn futures contract @ $2.60/bu |
|October: Sells 5,000 bushels of corn in the cash market @ $1.90/bu |Buys a CBOT December corn futures contract @ $1.90/bu |
|Sales price of corn |$1.90/bu |
|Plus futures gain ($2.60 - $1.90) |$0.70/bu |
|Net sale price |$2.60/bu |
By using the CBOT corn futures market to mitigate his exposure to price risk, the farmer increased his net sales price from $1.90 to $2.60 a bushel. Better yet, the final sale was 60 cents a bushel higher than his production expenses of $2.00 a bushel.[8] Relying totally on the local cash market at harvest time would have resulted in a loss of 10 cents a bushel ($1.90 sales price minus production costs of $2.00).
Suppose instead that yields were lower than normal and the cash market price rose to $3.50 instead of falling to $1.90 a bushel. The following example shows the net result of this occurrence. The farmer’s net sales price
Table 19 – Benefits From Hedging When Cash Price Rises.
|Cash Market |Futures Market |
|May: Plans to sell 5,000 bushels of corn; sells a CBOT December corn |Sells a CBOT December corn futures contract @ $2.60/bu |
|October: Sells 5,000 bushels of corn in the cash market @ $3.50/bu |Buys a CBOT December corn futures contract @ $3.50/bu |
|Sales price of corn |$3.50/bu |
|Plus futures gain ($2.60 - $3.50) |- $0.90/bu |
|Net sale price |$2.40/bu |
in this instance would be $2.40 a bushel. While he missed out on the prospect of realizing a substantial profit if he had not hedged his crop, the farmer did lock in a profit of 40 cents a bushel over his production costs of $2.00 a bushel. In this case, average pricing likely would have resulted in a greater profit as a risk mitigation marketing strategy.
We can summarize the advantages of hedging a storable commodity like corn with the following table. Relying on a marketing strategy based solely
Table 20 – Summary of Benefits From Hedging.
|Advantages and Disadvantages of a Selling Hedge with Futures |
|Advantages |Disadvantages |
|1. Reduces risk of price declines |1. Gains from price increases are limited |
|2. Could make it easier to obtain credit |2. Risk that actual basis will differ from projection |
|3. Aids in management decisions by stabilizing income in a crop|3. Year-to-year income fluctuations may not be reduced with |
|year |hedging |
|4. Easier to cancel than a forward contract |4. Contract quantity is standardized and may not match cash |
| |quantity |
| |5. Futures position requires a margin deposit and margin calls |
| |are possible |
on marketing the crop on the local cash market is an extremely risky proposition in today’s global demand-driven market given recent history of price volatility.
Appendix I
Present Value Interest Factor Tables
Appendix II
Simple Macroeconomic model of an Open Economy
Money Market:
(A.1) MD = L/P = n – g(i) + k(Y)
(A.2) MS = M/P = (1/rrm)(TR)/P
(A.3) MD = MS
where:
MD demand for real cash balances
MS real money supply
n autonomous demand for money
g slope of the demand curve
k shift coefficient for income
rrM reserve requirements for commercial bank deposits
TR total reserve requirements ratio
P general price level (GDP price deflator)
Y real national income
The equilibrium interest rates in the money market for given levels of national income is given by:
(A.4) i* = [n + k(Y) –(1/ rrM (TR/P))] (g
Product Market:
The expenditure and revenue equations for can be expressed as follow:
(A.5) C = a + b(Y – T)
(A.6) I = j – f(i)
(A.7) G = G*
(A.8) Y = C + I + G + XN
(A.9) T = h + t(Y)
where:
C Planned real consumption expenditures
a Autonomous consumption
b Marginal propensity to consume
Y Real national income and product
T Real government revenue
I Real planned investment expenditures
j Autonomous investment
G Real government expenditures (* denotes fixed spending)
h Tax base
t Marginal tax rate
The equilibrium levels of aggregate demand in the product market for different levels of interest rates is given by:
(A.10) Y* = [a – b(h) + j + G*]/(1 – b +b(t))- [f/ (1 – b +b(t))] i
The Aggregate Supply Curve:
Let’s assume the aggregate supply curve for the economy takes the following form:
Let’s assume for now that this economy is currently in the Keynesian or “depression” range of the aggregate supply curve. The general price level in this economy has a base period of 2007. Hence the general price index is currently at 1.0
The Labor Market:
It is assumed that there is an unlimited supply of labor available to meet demand in this economy given the state of the economy. That is, the labor supply curve is perfectly elastic over the “Keynesian range” of the aggregate supply curve. This means that wage rates are fixed in real terms as well.
The labor demand curve for this economy is given by:
(A.11) LD = s(WR) + v(Y)
where WR represents the real wage rate. Given equation (A.11) and the size of the civilian labor force, we can calculate the unemployment rate.
Real Exchange Rate:
The real exchange rate is positively related to the interest rates, or
(A.12) ER = m[(iUS)/(iROW)]
where:
ER Real exchange rate
m Exchange rate sensitivity to change in the real interest rate
Net Exports with Flexible Exchange Rates:
The next step is to allow for a dependence of net exports (XN) on income and a marginal propensity to import as follows:
(A.13) XN = aX – d(Y) – u(ER)
where:
aX Autonomous net exports
d Marginal propensity to import
u Trade sensitivity to flexible exchange rates
In light of equation (A.12), we can re-write the general linear form of the net export equation as follows:
(A.14) XN = aX – d(Y) – u(m(i))
General Equilibrium Conditions:
We can solve for the general equilibrium conditions based upon the money and product market conditions expressed in equations (A.4) and (A.10). These equations take the following form:
(A.15) YE = [A – (f + um)(n)/g + (f + um)(M/P)/g]/(z + (f + um)(k)/g)
(A.16) iE = [k(A) + z(n – M/P]/(g(z) + (f + um)(k))
where:
(A.17) A = a – b(h) + j + G* + aX
(A.18) Z = 1 – b + b(t) + d
-----------------------
[1] References to similar or “like kind” firms refers to comparisons with firms of similar size producing the same products in the same geographical area for the same market structure.
[2] W. H. Beavers, “Financial Ratios and Predictors of Failure”, in Empirical Research in Accounting: Selected Studies, Supplement to Journal of Accounting Research, Volume 66:77-111.
[3] The PIFR,N interest factors in the Appendix are based upon the formula 1/(1+R)N. The EPIF interest factors used starting in equation (30) are based upon the formula (1- PIFR,N)) ÷ R.
[4] We have simplified the algebra here by assuming the expected terminal value is based upon the value of comparables in the area and not the asking price set by the seller. That is, V0 does not necessarily equal C.
[5] Highly positively correlated returns would have the opposite effect. They would fluctuate in an identical fashion with the flows generated by the firm’s existing operations. Thus, they do not reduce the firm’s overall exposure to risk; they increase it by putting more “eggs into the same basket”.
[6] For a further discussion of this topic when more than two investment projects are being considered, see Van Horne, Financial Management and Policy, Chapter 3.
[7] The standard contract size for a CBOT corn futures contract equals 5,000 bushels. This example focuses on a short hedge, which protects the corn farmer from falling prices.
[8] Note the farm only contracted for a part of his expected production. This is to mitigate the possibility of poor yields that would require him to buy corn at the time the futures contract matures to fulfill the terms of the contract. The decision on what portion of the expected crop to hedge should be based upon historical yield estimates for the acres planted to this crop. Please note that basis (spread between futures and cash market prices) as well as margin requirements, brokerage commissions and storage costs were omitted from this example.
-----------------------
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
= (EBIT – taxes) ( term debt and capital lease payments
[pic]
[pic]
[pic]
Q1
PBP=3
PBP=4
[pic]
[pic]
[pic]
Demand
Supply
QD = f(P, Y-T, W, ...)
QS = f(P, MIC, …)
QD = QS
Solve for PE and QE
PE
QE
[pic]
Required
Rate of
return
Highly risk
averse
RRRH,i
Coefficient of variation
Lowly risk averse
RRRL,i
RF,i
Risk neutral
CVi
Slope equal to 0.70
RRRi =.12
Business Risk
Premium = .07
RF,i =.05
Required
Rate of
return
}
0.10
Coefficient of variation
.9166 = 1/(1+.0910)
Ignores changing risk free rates and increasing risk over time
Required
Rate of
return
CVi
Coefficient of variation
RRRi
Financial risk premium
RRRi
Reflects+ ci(Li)
RF,i
Business risk premium
Highly Negatively Correlated Net Cash Flows
Annual fluctuations of net cash flows from existing assets
NCFi
Annual fluctuations of net cash flows from new project
Time
Reduction in RRR1 due to a reduction in risk from the portfolio effect associated with the new investment.
.150
.132
.100
.238 .375
Percent
Total cost
Implicit cost
12%
9%
Explicit cost
75%
Use of credit capacity
$/unit
Cost of equity capital
Weighted average cost of capital
Cost of debt capital
1.0
D/E ratio
ii3
i2
i1
LM
Y1 Y2 Y3
IS
ii3
i2
i1
Y3 Y2 Y1
IS
LM
iE
YE
IS
LM
LM*
Expansionary
Monetary policy
(i
(Y
IS*
Expansionary
Fiscal policy
(Y
LM
IS
(i
iE
LM
IS
YE
AS
AD
PE
YE YPOT
iE
MS
M
MD
LD
LS
WRE
LE LMAX
UR
INF
AS
AD
PE
YE YPOT
YFE
$8.93 $9.45 $9.97
Quantity
QE
[pic][9]0129Price of corn
D
S
PE
-- 2008/09 --
-- 2007/08 --
AS
AD
P
1.0
YE
Y
Profitability Index:
Project A = 4,500 ÷ 10,000 = .45
Project B = 9,120 ÷ 24,000 = .38
Project C = 1,350 ÷ 7,500 = .18
Project D = 3,400 ÷ 43,000 = .08
................
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