Cost-volume-profit analysis incorporating the cost of capital.

Cost-volume-profit (CVP) analysis is a mathematical representation of the economics of producing a product. The relationships between a product's revenue and cost functions expressed within the CVP model are used to evaluate the financial implications of a wide range of strategic and operational decisions. For example, CVP analysis is employed to assess the financial implications of product mix, pricing, and product and process improvement decisions. Perhaps equally important, CVP analysis facilitates measuring the sensitivity of a product's profitability to variations in one or more of its underlying parameters. Finally, CVP analysis may be used to determine the trade-offs in profitability and risk from alternative product design and production possibilities. In effect, CVP is a quantitative model for developing much of the financial information relevant for evaluating resource allocation decisions.

Despite its widespread application, CVP analysis is frequently criticized for its use of simplifying assumptions, such as deterministic and linear cost and revenue functions. Additionally, CVP is disparaged for its focus on a single product and its single-period analysis. However, as noted by Guidry et al.: "Non-linear and stochastic CVP models involving multistage, multi-product, multivariate, or multi-period frameworks are all possible, although a single model embracing all of these extensions would seem a radical departure from the whole point of CVP analysis, its basic simplicity" (1998: 75). Horngren et al. (2000) note that firms across a variety of industries have found the simple CVP model to be helpful in both strategic and long-run planning decisions. Furthermore, a survey of management accounting practices indicates that CVP analysis is one of the most widely used techniques (Garg et al., 2003). However, Horngren et al. (2000) warn that, in situations where revenue and cost are not adequately represented by the simplifying assumption of CVP analysis, managers should consider more sophisticated approaches to financial analysis.

An implicit assumption, and one that is frequently overlooked in evaluating the use of CVP analysis, involves its treatment of the cost of capital. CVP analysis, like other managerial accounting techniques and models, uses accounting profitability as the primary decision criterion for evaluating resource allocation decisions. CVP analysis, like other managerial accounting techniques, ignores the cost of capital and treats it as if it were zero. However, the opportunity cost of the funds invested in the assets used to manufacture a product is a cost the same as the cost of operating resources, such as direct material, labor, and overhead. The failure of CVP analysis to incorporate the cost of capital into a product's cost function can lead to underestimating a product's cost, while overstating its profitability. For products that require a significant investment of capital, ignoring the opportunity cost of invested funds may lead to accepting products whose rate of return is less than the firm's cost of capital. In effect, traditional CVP analysis encourages managers to select products that destroy, rather than create, economic value for the firm. Finally, using an accounting measure of profitability creates a bias to employ capital relative to operating resources because the cost of capital is not reflected in a product's cost like those of operational resources. Therefore, product designers and developers may employ investment funds beyond the point where the marginal benefit of the last dollar of capital used is equal to its marginal cost.

The purpose of this article is to illustrate how the cost of capital may be incorporated into CVP analysis. It develops the mathematical relationship between a product's discounted operating income after taxes less the cost of capital and the product's price, costs, invested funds, and sales quantity. From this relationship, the sales quantity needed to earn a rate of return equal to the firm's cost of capital may be estimated. Incorporating the cost of capital into the CVP model enables managers to determine the value added (destroyed) for a given level of sales. Finally, the article illustrates how the modified CVP model facilitates evaluating the economic implications of alternative investment and cost structures and process improvement of the activities used to manufacture a product.

The remainder of this article is organized as follows. The next section discusses the different approaches to the economic analysis of a product. The following section discusses how the cost of capital may be incorporated into the measurement of a product's cost. The subsequent section derives a mathematical expression for calculating the discounted value of a product's operating income after taxes less the cost of capital used to earn the operating profit. In the next section, a numerical example is used to illustrate how CVP analysis may be developed from the present value of a product's operating income after taxes less the cost of capital. The subsequent section examines the use of the CVP model for assessing the economics of a program of process improvement. The final section presents the summary and conclusions of the paper.

PRODUCT MIX DECISIONS AND CVP ANALYSIS

The selection of which products to produce, which to abandon, and which to postpone is one of the most critical decisions confronting a firm's management. The products selected from the product mix decision determine the revenue, profit, and cash flow of the firm's operations. Perhaps equally important, the products selected determine, in part, the firm's competitive position vis-a-vis its competitors. The profit and cash flow from the products selected currently provide the funds required to develop and produce products in the future. A final, but frequently overlooked, aspect of product mix decisions involves the investment in long-term assets used to manufacture a product. The investments in these assets, once committed, are frequently difficult and/or costly to reverse. Therefore, once a product enters production, the firm may find it difficult to avoid economic losses.

CVP analysis is generally implemented with financial data taken from the firm's accounting system. Financial data is readily available, as well as congruent, with the accounting profit objective inherent in the use of CVP analysis. The financial data needed for CVP may be taken from either a traditional cost accounting or an activity-based costing (ABC) system. Traditional cost accounting systems allocate overhead to products based on one or more volume-based measures of activity. However, products consume overhead resources based on batch-, product-, facility-, and complexity-, as well as volume- or unit-level, activities. Consequently, traditional cost accounting systems can systematically misallocate overhead to products. Kaplan and Cooper (1998) assert that it is not unusual for firms using a traditional cost accounting system to find that 20% of their products earn 300% of their profit, while the remaining 80% either break even or incur a loss and collectively lose 200% of the firm's profitability. CVP analysis based on data from a traditional cost accounting system may be expected to lead to similar distortions in modeling a product's cost. Consequently, the CVP model developed in this article is based on ABC, rather than a traditional cost accounting system.

CVP analysis is used to measure the economic characteristics of manufacturing a proposed product. Based on accounting data, the CVP model is used to determine the sales quantity needed to break even, as well as the sales quantity required to earn a desired profit or profit margin. Managers then compare a product's expected sales with the sales quantities required to break even and/or earn a target profit margin to determine whether the product should be produced.

Cost-volume-profit, like all financial models, is based on a set of simplifying assumptions that reduce the complexity of a resource allocation decision to make decision making more tractable. To understand a financial model and its usefulness, its assumptions and their role in a decision must be understood. For example, CVP is a one-period model of a product's profitability, although the product may have an economic life of several years. CVP analyses treat a product's life as a single period, or evaluate a single period of its life, such as a year, and extrapolate the one-period's result over the product's life. Managers can use either approach for evaluating a product's profitability with CVP analysis and for making the product mix decision.

A second assumption of the CVP model is that the opportunity cost of the funds invested in capital assets used to manufacture a product is zero. In cases where a product will require an investment in new plant and equipment, a product has to recover its operating cost, as well as the cost of capital for the funds invested in the assets used to create the products, for the firm to be as well-off after manufacturing the product as it was before the product's production. The product mix and the acquisition of the assets needed for its production are frequently evaluated independently of each other and with conceptually different decision models. However, these are interrelated decisions. Incorporating the cost of capital into a product's cost enables product mix decisions to be congruent with capital budgeting decisions (Kee, 2004). Conversely, when existing capital assets will be used to manufacture a product, failure to consider the cost of capital is implicitly assuming that these assets have an abandonment value of zero. Existing assets can generally be sold or used elsewhere in the firm's operation. Consequently, using them to produce a product incurs an opportunity cost the same as that for the acquisition of new assets. Thus, the cost of capital should be incorporated into a product's cost whether new or existing assets are used to manufacture a proposed product.

INCORPORATING THE COST OF CAPITAL INTO A PRODUCT'S COST

As noted earlier, a deficiency of managerial accounting in general and CVP analysis in particular involves its failure to include the cost of capital as an expense. Alfred Marshall, an English economist in the 1800s, asserted that a firm does not earn a profit until its operating income after taxes exceeds the cost of capital used to generate the operating income. A firm's operating profit after taxes less the cost of capital used to generate the profit measures its economic income. In the 1990s, Stewart (1991) proposed a similar concept he referred to as economic value added (EVA) to evaluate a firm's performance. Economic value added is a registered trademark of Stern Stewart and Company. Stewart (1991) asserts that a positive (negative) EVA increases (decreases) a firm's stock market performance. That is, a firm's stock price will increase (decrease) when it earns a rate of return higher (lower) than its cost of capital (Stewart, 1991). Therefore, when a firm's economic income is positive, it creates economic value for the firm's stockholders. Conversely, when the firm's economic income is negative, it destroys economic value. Studies of the effect of economic income, or EVA, upon a firm's profitability and stock price returns have been somewhat mixed. Hogan and Lewis (2005), in a study of firms adopting economic income for management compensation, found that firms that were well-suited for the use of economic income were more profitable relative to firms that were equally suited for the use of economic income, but chose not to adopt it. In a study of the association between economic income and stock price returns, Chen and Dodd (1997) found that stock price returns were more highly associated with economic income than accounting income. In a similar study, Biddle et al. (1997) found stock price returns were more highly associated with accounting earnings than with economic income. Additional empirical and theoretical studies of the relationship between economic income and stock price returns are provided by Paulo (2002), Chen and Dodd (2002), and Ferguson et al. (2005).

Like accounting income, economic income is a periodic measure of performance. However, unlike accounting income, economic income in different time periods is not additive. The time value of money is implicit in the cost of capital. Consequently, to evaluate the economic income of a product over multiple periods, each period's economic income must be discounted to when production of the product will begin. Hartman (2000) and Shrieves and Wachowicz (2001) provide mathematical proofs that, when the cost of capital is treated as an expense, the discounted value of an investment's economic income is equivalent to its discounted cash flows. In effect, the discounted value of an investment's economic income is equal to its NPV. Hartman (2000) and Shrieves and Wachowicz (2001) indicate their findings are invariant across the different depreciation methods used to compute economic income.

To incorporate the cost of capital into the CVP model, the opportunity cost of the funds invested in capital assets must be traced to the products where the assets are used to manufacture the firm's products. This may be accomplished by tracing the firm's assets to production-related activities. The book value of assets that can be used to produce multiple products is divided by an activity's practical capacity to measure the investment per unit of service, or cost driver, for the activity. The cost of capital is then traced to products by multiplying the quantity of an activity's cost driver used to manufacture each product times the investment per unit of service and the cost of capital rate. Conversely, for assets that can be used to produce only one product, an activity's total investment in these assets is multiplied times the cost of capital rate and charged to the product. In effect, the cost of capital is charged to products for assets that can be used to produce multiple products as a unit-level cost, while those that can be used to manufacture a single product, as a product-level cost. However, unlike overhead-related cost, the cost of capital is subtracted from operating income after taxes to determine a product's economic income. Incorporating the cost of capital as an expense transforms ABC from a system for measuring a cost object's accounting income to measuring its economic income. In the next section, these concepts will be used to develop a CVP model incorporating the cost of capital.

INCORPORATING THE COST OF CAPITAL INTO CVP ANALYSIS

The traditional CVP model is developed by specifying the mathematical relationship between a product's accounting profit and its sales quantity, price, and costs. The resulting equation is then manipulated to measure a product's financial attributes, such as its breakeven sales quantity or the sales required to earn a given profit or profit margin. A CVP model incorporating the cost of capital may be developed in a similar manner. However, when the cost of capital is charged to a product as an expense, the difference between the product's revenue and expenses is its economic income. Unlike accounting profitability, economic income over a product's life must be discounted to when production of the product will begin. Therefore, CVP analysis incorporating the cost of capital is based on an equation of the relationship between a product's discounted economic income and its sales quantity, price, costs, investments, and cost of capital. To develop this relationship, the following notation will be used:

i = period index, i = 1, 2, ..., N,

j = activity index, j = 1, 2 ..., M,

[C.sub.u,j] = cost driver rate for unit-level activity j,

[C.sub.p,j] = cost driver rate for product-level activity] j,

[I.sub.u,j] = investment per unit of output for unit-level activity j,

[I.sub.p,j] = investment in product-level activity j,

N = economic life of a product,

M = number of activities,

NPV = net present value,

[P.sub.i] = price per unit in period i,

[PV.sub.N,r] = present value of an annuity of $1 for N periods and a discount rate of r,

[Q.sub.i] = quantity of a product produced and sold in period i,

r = cost of capital, and

t = effective tax rate.

As noted earlier, Hartman (2000) and Shrieves and Wachowicz (2001) provide mathematical proofs that discounting an investment economic income is equivalent to its NPV. Using this proof, a product's discounted economic income may be expressed as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)

The first term on the right-hand side of Equation 1 is the present value of a product's revenue less the cost of unit- and product-level activities adjusted to an after-tax basis. Batch-level activities have been included in unit-level activities to simplify the analysis. The second and third terms in Equation 1 are the present value of the cost of capital for the funds invested in the long-term assets required for the unit-and product-level activities used to manufacture a product. The expression (N+ 1 - i)/N in both the second and third terms adjusts for an activity investment in long-term assets as depreciation expenses calculated with the straight-line depreciation method are taken over successive periods. Like the first two terms in Equation 1, the second and third terms are discounted to the beginning of Period 1.

Summing across activity j, Equation 1 may be restated as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)

Each term in Equation 2 is a convergent series. Simplifying these series, Equation 2 may be expressed as:

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

The expression, [1 - (1 + r)[sup.-N]/r, in each term on the right-hand side of Equation 3, is the present value of an annuity of $1 for N periods discounted at an interest rate of r. Replacing the mathematical expression for an annuity with the symbol PVvEquation 3 may be restated as:

NPV=((P-[C.sub.u])Q-[C.sub.p])(1-t)[PV.sub.N,r]-([I.sub.u]Q+[I.sub.p]) [[1/1-[PV.sub.N,r]/N] (4)

As indicated in Equation 4, a product's NPV is the present value of its operating income after taxes, less the cost of capital for the investment in unit- and product-level activities used to earn the profit. The cost of capital for unit- and product-level activities, as indicated in Equation 4, is the original investment less the present value of the depreciation expense taken over the period of time that an activity's assets are used to produce a product. Equation 4 is the foundation from which CVP analysis incorporating the cost of capital may be developed. Managers can manipulate one or more variables in Equation 4 to measure the financial attributes of a product. For example, a product's breakeven sales quantity can be determined by setting the left-hand side of Equation 4 equal to zero and solving for Q. Similarly, for a desired level of profitability, Equation 4 can be solved for the sales quantity required to provide this level of discounted economic profit. In the following section, a numerical example will be used to illustrate the application of the CVP model.

Numerical Example

To illustrate the use of CVP analysis incorporating the cost of capital, consider a firm's management evaluating the economics of manufacturing Product X. Due to competitive pressure, the firm must decide whether to manufacture Product X currently or forego its production. The resource requirements to manufacture Product X are listed in Panel I of Table 1. As indicated in Panel I, Product X requires two and one-half pounds of material and one labor hour. The product uses two machine hours in the assembly activity and one-half hour in the packaging activity. Product X will be produced in production runs of 1,000 units. Each production run requires two hours from the setup activity and 20 orders from the purchasing activity. Finally, Product X requires 600 engineering drawings each period to incorporate new features and technology into the product. Product X's sales price and maximum demand are expected to be $114 and 500,000 units a year, respectively, over its three-year life.

Panel II of Table 1 lists the investments in the equipment and other assets necessary, to manufacture Product X. Each overhead-related activity and its cost driver listed in parentheses are given in column one. The finds that will be invested in each activity and the practical capacity they provide are listed in columns two and three, respectively. Investments of $48,000,000 and $18,000,000 are required for the equipment comprising the assembly and packaging activities, respectively. The equipment for the assembly and packaging activities provides large, discreet quantities of production capacity. However, the assembly equipment is designed specifically to manufacture Product X and cannot be used to produce other products of the firm. Therefore, the operating cost and cost of capital of the equipment for the assembly activity is attributable to only Product X. Conversely, the equipment that will be purchased for the packaging activity can be used to manufacture Product X, as well as other products of the firm. Therefore, the cost of the capacity of the packaging activity is attributable to multiple products and traced to these products based on the quantity of its service used in their production. The remaining activities, set-up and purchasing, are batch-level costs, while engineering is a product-level cost. The last column in Panel II lists the investment per unit of capacity for each activity in year one. For example, the investment per hour for the packaging activity in year one is $72, or the investment of $18,000,000 in packaging divided by the 250,000 hours of capacity that the investment provides. The investment per unit of capacity for the remaining activities can be interpreted in a similar manner. The investment per unit of capacity is given for the assembly activity. However, the product-specific nature of its assets makes it a product-level investment.

In Panel III, the cash expenditures and depreciation expenses for each activity are listed in columns two and three, respectively. The depreciation expense was computed using straight-line depreciation on the assets listed in Panel II. The cash expenditures and depreciation expense for each activity are summed and divided by an activity's practical capacity to estimate its cost driver rate. For example, the packaging activity is expected to have cash expenditures of $1,000,000 and an annual depreciation expense of $6,000,000. The packaging activity's operating expenses of $7,000,000 were divided by its practical capacity of 250,000 hours to estimate its cost driver rate of $28 per hour. The cost driver rates for the other activities in Panel III were computed in a similar manner.

In the last panel in Table 1, Panel IV, unit- and product-level costs and investments are listed. The resources used by the assembly activity are a product-level cost, while the cost of the packaging activity is a unit-level cost. The set-up and purchasing activities are batch-related activities. However, to simplify the analysis, their costs and investments are treated as unit-level costs and investments. The last activity, engineering, is required to produce the product independently of the number of units, or batches, manufactured. Accordingly, its cost and investment are a product-level cost, and investment, respectively. Summing across activities in Panel IV, Product X's unit- and product-level costs are $44.70 and $21,440,000, respectively. The unit--and product-level investments required for the equipment needed to manufacture Product X are $40.80 and $51,600,000, respectively.

The breakeven sales quantity is determined by setting the left-hand side of Equation 4 equal to 0 and solving for Q. Solving Equation 4 with Product X's price, unit-, and product-level cost and investment listed in Table 1, and cost of capital of 10% and a tax rate of 20%, results in a breakeven quantity of 393,305 units a year. At this level of sales, Product X is expected to earn a rate of return equal to the firm's cost of capital. Alternatively, the breakeven sales quantity may be viewed as the sales quantity at which each additional unit of Product X sold will create economic value for the firm.

To gain a broader perspective of Product X's economics, Equation 4 was solved for Q ranging from 0 to 500,000 at intervals of 50,000 units. The graph of the resulting set of NPVs with respect to unit sales is provided in Figure I. The vertical and horizontal axes of Figure I represents dollars and unit sales of Product X, respectively. In Figure I, the diagonal line denoted with the diamond symbol is the NPV, while the diagonal line denoted by the square symbol is accounting income. The accounting income line is presented for comparative purposes. In Figure I, Product X's NPV ranges from a minimum value of -$51,480,631 at 0 sales to a maximum value of $13,965,500 at sales of 500,000 units. The NPV crosses the horizontal axis at 393,305 units, or the breakeven point. The distance from the horizontal axis to the NPV line represents the economic value either created or destroyed by Product X for the sales quantity indicated on the horizontal axis.

[FIGURE I OMITTED]

The accounting profit for Product X was computed by setting r in Equation 1 to 0. The denominator of the first term in Equation 1 is equal to 1, while the denominators of the remaining terms are equal to N. However, with the cost of capital equal to 0, the numerators of the second and third terms in Equation 1 reduce to 0. Simplifying the first term in Equation 1, it is equivalent to the first term in Equation 4, with [PV.sub.N,r] equal to three. If the equation for accounting income is set equal to 0 and solved for Q the resulting accounting breakeven quantity is 309,380 units. The accounting breakeven point is the sales quantity necessary to earn a rate of return equal to 0. The additional 83,925 units needed to increase the accounting to the economic breakeven point is the sales quantity needed to increase Product X's rate of return from 0 to the firm's cost of capital. The accounting breakeven point measures the sales quantity necessary to recover a product's explicit costs, which is analogous to the payback capital budgeting technique. The economic breakeven point represents the sales quantity necessary to recover the cost of all resources used in a product's production. The economic breakeven sales quantity is a variation of the NPV capital budgeting technique. However, unlike the NPV method discussed in finance that determines a point value based on discounted cash flows, the NPV model expressed in Equation 4 enables the entire range of NPVs a product may earn to be determined based on accounting, and not cash flow, variables.

The accounting profit line is at its closest to the NPV line at 0 sales. Thereafter, the accounting income diverges from the NPV as Q increases. The difference between the accounting income and NPV lines in Figure I for a given value of Q is the cost of capital. The increasing divergence of the accounting profit and NPV lines is a result of the additional cost of capital for the increasing usage of investments in unit-level activities as Product X's sales increase.

In traditional CVP analysis, a product's contribution margin is used to measure the rate of change in its profit with respect to a unit change in its sales. The contribution margin enables managers to evaluate the sensitivity of a product's profitability with respect to a change in its sales. A similar concept for the CVP model incorporating the cost of capital may be developed by taking the first derivative of Equation 4 with respect to Q. The resulting rate of change in a product's NPV with respect to a unit change in sales may be expressed as:

dNPV/dQ =(P-[C.sub.u](1-t)[PV.sub.N,r]-[I.sub.u][[1/1-[PV.sub.N,r]/N] (5)

The rate of change in the NPV for a unit change in demand is the slope of the NPV line illustrated in Figure I. Using the data for Product X listed in Table 1, the rate of change in Product X's NPV with respect to Q is $130.89 per unit. Therefore, for every additional unit of Product X sold each year over its three-year life, its discounted economic income will increase by $130.89. For example, if the sales of Product X were expected to increase by 25,000 units annually as a result of a promotional campaign, its NPV would increase by $3,272,250, or 25,000 units, at $130.89 per unit.

IMPROVING A PRODUCT'S PROFITABILITY

The CVP model proposed in this article provides a more detailed representation of a product's revenue, investment, and cost structure relative to that of a traditional model. The relationships reflected in the CVP model incorporating the cost of capital enable a firm's managers to evaluate alternative investment and cost structures to enhance a product's profitability. For example, Product X's economic income is affected by the product-specific nature of the assets purchased for the assembly activity. Whenever production of Product X is less than 500,000 units, the assembly activity will have unused capacity that cannot be used elsewhere in the firm's operations. Consequently, one strategy for improving Product X's profitability is to use more flexible assembly equipment. Suppose assembly equipment costing $57,000,000 could be purchased that can be used to manufacture Product X, as well as other products of the firm. The alternative investment in the assembly activity changes the investment and cost structure of Product X. First, it increases the unit cost of Product X since the alternative investment's annual depreciation expense will be $3,000,000 higher than that of the planned investment. However, the flexibility of the alternative investment in assembly equipment transforms the activity from a product- to a unit-level cost and investment that enables the firm to avoid unused capacity cost.

To evaluate the trade-offs between the planned and alternative investments in assembly equipment, the NPV for producing Product X with each investment is illustrated in Figure II. The vertical and horizontal axes in Figure II represent Product X's NPV and sales quantity, respectively. Sales of Product X are restricted to values ranging from 250,000 to 500,000 units to illustrate clearly the behavior of the two investment structures as the discounted values of their economic incomes intersect. The line denoted with the diamond symbol represents the NPV for manufacturing Product X with the planned investment in the assembly activity, while the line denoted with the square symbol represents the NPV for producing Product X with the alternative investment in the assembly activity.

The NPV lines for the planned and alternative investments in the assembly activity intersect at 432,371 units. If sales of Product X are projected to be less than 432,371 units, then the alternative investment in assembly will lead to a higher NPV. The alternative investment in assembly increases Product X's unit cost. However, at sales less than 432,371, the increased unit cost is more than offset by the reduction in its unused capacity cost. Conversely, if sales are expected to be greater than 432,371 units, the planned investment in assembly will lead to a higher NPV. Therefore, analysis of the uncertainty of Product X's NPV for the planned and alternative investments in assembly can provide additional insights into the risk and return attributes of the different investment and cost structures.

Another avenue for improving a product's profitability involves implementing a program of process improvement. A product's cost can be reduced by decreasing the quantity of an activity's services used in its production and by decreasing the cost of the resources used to provide an activity's service. Since every activity cannot be improved simultaneously, identifying activities with the greatest potential for cost reduction is critical to achieving the potential benefits of a program of process improvement. Since the assembly activity has the highest operating cost, it provides a useful starting point for investigating the benefits of process improvement. Assuming the firm decided to use the original investment in the assembly activity, reducing the quantity of machine hours required to manufacture a unit of Product X will increase the firm's efficiency, but not improve its profitability. The equipment in the assembly activity is product-specific. Therefore, every machine hour saved through process improvement becomes an additional hour of unused capacity.

[FIGURE II OMITTED]

Alternatively, a 1% decrease in the cash-related resources in the assembly activity will reduce its cost driver rate from $20 to $19.96 per machine hour. Since it takes two machine hours to produce a unit of Product X, the cost saving would reduce Product X's cost after taxes by $.064 per unit. Suppose Product X's annual sales are expected to be 425,000 units. Using the data in Table I in Equation 4, the discounted economic income for producing and selling 425,000 units is $4,148,580. A 1% reduction in the assembly activity's cash-related resources would result in an annual cost savings of $27,200. The present value of this cost savings over Product X life would be $67,642. The $67,642 discounted cost savings is 1.63% of X's NPV prior to the program of process improvement, or $67,642/ $4,148,580. Therefore, each 1% reduction in the cash-related resources in the assembly activity increases Product X's NPV, based on sales of 425,000 units, by 1.63%.

Unlike the assembly activity, the investment in the packaging activity can be used to produce Product X, as well as other products of the firm. Consequently, every hour saved in packaging may be able to be used productively elsewhere in the firm's operations and thereby reduce Product X's cost. For example, a 1% decrease in packaging time for Product X will lower packaging time by .005 hours. The effect of reducing packaging time by .005 hours per unit reduces the packaging cost from $14 to $13.86 per unit, or $.112 per unit after taxes. The present value of a $. 112 per unit cost reduction for annual sales of 425,000 units over Product X's three-year life is $118,374. This cost savings represents a 2.85% increase in Product X's NPV of $4,148,580 prior to the cost reduction. Conversely, decreasing the cash expenditures of the packaging activity by 1% reduces its cost driver rate from $28 to $27.96 per hour. Since it takes one-half hour in the packaging department, a 1% reduction in the packaging's cash-related resources decreases Product X's cost by $.016 per unit after taxes. The cost savings of $.016 per unit based on production of 425,000 units over Product X's three-year life increases its present value by $16,911, or .41%, relative to its NPV prior to process improvement. Using the CVP model, the change in Product X's discounted economic income for other process improvements can also be assessed. Based on the cost savings from the process improvement of each activity and the difficulty of its implementation, a firm's managers can prioritize activities for implementing a program of process improvement.

SUMMARY AND CONCLUSIONS

Cost-volume-profit (CVP) analysis is a mathematical model of the economics of producing a planned product. The relationship between a product's revenue and cost functions expressed in the CVP model is used to evaluate the financial consequences of a wide rage of strategic and operational decisions. Like other managerial accounting techniques, CVP analysis ignores the opportunity cost of the funds used to manufacture a product and treats the cost of capital as if it were zero. The failure of CVP analysis to incorporate the cost of capital can lead managers to accept products whose rate of return is less than the firm's cost of capital. In effect, traditional CVP analysis can lead managers to produce products that destroy, rather than add, economic value to the firm.

In this article, traditional CVP analysis has been expanded to incorporate the cost of capital. The cost of capital is traced to products, like the cost of overhead-related resources, using the principles of activity-based costing. However, unlike the cost of overhead-related resources, the opportunity cost of invested funds is deducted from a product's operating income after taxes to measure its economic income. When a product's economic income over its life is discounted to when production will begin, it is equivalent to a product's NPV (see Hartman (2000) and Shrieves and Wachowicz (2001)). The discounted economic income of a product models the interrelationships of the revenue, cost, and cost of funds used in a product's production. It thereby enables managers to perform CVP analysis that incorporates the cost of capital used in manufacturing a proposed product. As demonstrated in this article, the CVP model based on the discounted economic income of a product enables managers to compute a product's breakeven sales quantity, to measure a product's profitability over the range of its sales, and to determine the rate of change in its profitability. The CVP model also facilitates measuring the tradeoffs in alternative investment and cost structures, as well as estimating the impact upon a product's profitability from a program of process improvement.

The CVP model incorporating the cost of capital proposed in this article is more complex and costly to develop than the traditional CVP model. As noted by Guidry et al. (1998), one of the reasons traditional CVP analysis has survived is its simplicity. However, Kaplan and Cooper (1998) indicate that firms using traditional cost accounting systems frequently find that 80% of their products either break even or incur a loss. Consequently, simplicity is not a desirable characteristic when either a cost system or managerial techniques, such as CVP analysis, fail to reflect the economics of producing a product. Therefore, managers and managerial accountants must consider how the CVP model incorporating the cost of capital would change their product mix decisions and the payoffs relative to the increased cost and complexity of the model.

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Robert Kee

Joe Lane Professor of Accounting

University of Alabama Table 1 Investment, Cost, and Operating Data Panel I: Product X Resource Requirements Direct Material (Lbs @ $5/Lb) 2.5 Lbs /Unit Direct Labor (DLH @ $15/DLH) 1 DLH/Unit Assembly (MH) 2 MH/Unit Packaging (Hours) .5 Hour/Unit Set-up (Hours) 2 Hours/Batch Purchasing (Orders) 20 Orders/Batch Engineering (Drawings) 600 Drawings Batch Size 1,000 Maximum Annual Demand (Units) 500,000 Useful Life 3 Years Price $114.00 Panel II: Required Investment

Product X

Invested Practical Investment Per Activity (Cost Driver): Funds Capacity Unit of Capacity Assembly (MH) $48,000,000 1,000,000 $ 48 Packaging (Hours) $18,000,000 250,000 $ 72 Set-up (Hours) $ 1,200,000 1,000 $1,200 Purchasing (Orders) $ 1,200,000 10,000 $ 120 Engineering (Drawings) $ 3,600,000 600 $6,000

$72,000,000 Panel III: Cost Driver Rates

Cash Depreciation Practical Cost Activity: Expenditures Expense * Capacity Driver Rate Assembly (MH) $4,000,000 $16,000,000 1,000,000 $ 20.00 Packaging $1,000,000 $ 6,000,000 250,000 $ 28.00

(Hours) Set-up (Hours) $ 200,000 $ 400,000 1,000 $ 600.00 Purchasing $ 600,000 $ 400,000 10,000 $ 100.00

(Orders) Engineering $ 240,000 $ 1,200,000 600 $2,400.00

(Drawings)

$6,040,000 $24,000,000 Panel IV: Unit- and Product-level Operating and Investment Data

Unit- Product- Product-

level Unit-level level level

Cost Investment Cost Investment Direct Material (Lbs) $12.50 -- -- -- Direct Labor (DLH) $15.00 -- -- -- Assembly (MH) -- -- $20,000,000 $48,000,000 Packaging (Hours) $14.00 $36.00 -- -- Set-up (Hours) $ 1.20 $ 2.40 -- -- Purchasing (Orders) $ 2.00 $ 2.40 -- -- Engineering -- -- $ 1,440,000 $ 3,600,000 (Drawings)

$44.70 $40.80 $21,440,000 $51,600,000 * Straight-line depreciation is used. DLH and MH are direct labor hours and machine hours, respectively.