Chapter 5 Notes Page 1

Transcription

1 Chapter 5 Notes Page 1 COST BEHAVIOR When dealing with costs, it helps for you to determine what drives the cost in question. A Cost Driver (also called Cost Base) is an activity that is associated with, or related to, changes in the cost in question. For example, assume that Redford, Inc. produces classic, wood baseball bats. The cost of the wood materials used in the production of these baseball bats is closely associated with the number of bats produced. In this case, the number of units is a good Cost Driver for the material cost in Redford s production of bats. In fact, the number of units produced is often used as the Cost Driver for costs. For purposes of this discussion, we will assume that the Cost Driver is the number of units produced, but Cost Drivers actually can be a number of different things. For example, the cost of custodial services is driven by the amount of floor space maintained. Thus, floor space is often used to allocate custodial costs between plant departments. Another popular Cost Driver is the number of labor hours that are used to make a product. Variable Costs change as their associated Cost Drivers change. On the other hand, some costs do not change regardless of changes in production or other Cost Drivers. These costs are referred to as Fixed Costs. Fixed Costs A graphical depiction of a Fixed Cost is shown below. Regardless of the number of units produced (or changes in other typical Cost Drivers), the Fixed Cost remains unchanged. Graphical Representation of a Fixed Cost

2 Chapter 5 Notes Page 2 This definition of Fixed Costs is possible because two assumptions have been made. First, the time horizon being discussed is relatively short. Second, the range of production in question is limited. The time horizon is important. In making decisions, we typically assume a relatively short time horizon. For example, we might be considering a time horizon of one year. Over that year, our rent is likely to remain unchanged regardless of a change in the level of our production. Over a long enough time horizon, however, all costs become variable. For example, with a longer time horizon, our monthly rental expense is likely to increase due to the expiration of our lease, an expansion of our operations, and/or the need to increase and modernize our production capacity. In addition to the time horizon, we also typically assume that we will be operating within a range of activity (Relevant Range). For example, assume that our baseball bat manufacturer normally produces and sells between 10,000 and 20,000 bats in a year, and it has the capacity to make up to 40,000 bats in its current facilities. In this case, it is unlikely that it will be called upon to make more than 40,000 bats, and it is unlikely that its Fixed Costs will change within this Relevant Range of production (10,000 20,000 bats). Without the assumption regarding Relevant Range, the Fixed Cost function could become a step cost. It remains fixed for a given range of production (or other Cost Driver), but it eventually changes. A graphical depiction of the cost to rent a factory appears below: Graphical Representation of a Step Cost As long as you assume that the Relevant Range of production (the circle) can be handled by two factories, then factory rent is a Fixed Cost.

3 Chapter 5 Notes Page 3 Variable Costs As noted above, Variable Costs change with changes in their associated Cost Drivers. A graphical depiction of a Variable Cost appears below. Graphical Representation of a Variable Cost The linear representation of a Variable Cost with a constant slope is again an over simplification of the behavior of Variable Costs. For example, the existence of Economies Of Scale would make a Variable Cost function appear more like the following: Graphical Depiction of a Non-Linear Variable Cost When only the Relevant Range is considered, the non-linear cost (within the circle) appears more linear. This fact makes the assumption that the Variable Cost function is linear becomes more realistic.

4 Chapter 5 Notes Page 4 Mixed Costs Mixed Costs are partially fixed and partially variable. For example, if the cost of renting a car is $50 a day plus 20 a mile, then the rental cost has a fixed component (daily charge) and a variable component (mileage charge). Mixed costs can be represented as follows: Graphical Representation of a Mixed Cost Total Costs are a Mixed Cost because they include both Fixed Costs and Variable Costs. If a cost can be divided into a fixed component and a variable component, then: Total Cost = Variable Cost + Fixed Cost Since Variable Costs are driven by the number of units produced (or some other Cost Driver), Variable Cost is made up of two components, the Variable Cost per unit, V, and the number of units produced (or other Cost Driver), x. Fixed Costs can be represented by the single variable, F. Units produced (x) are not included because Fixed Costs do not change with the number of units produced. Using these variables, we can represent the linear cost function as follows: Total Cost = Vx + F As you will recall (remember High School?), the formula for a line is y = mx + b The y-axis is the total cost in question. The slope of the line is the Variable Cost per unit (V), and the y-intercept, b, is the total Fixed Costs (F).

5 Chapter 5 Notes Page 5 Deriving the Cost Function From Past Behavior Most books emphasize objective techniques that can be used to derive the cost function from a firm s past cost experience. There are other approaches that also include subjective analysis. One of the benefits of using more subjective techniques is that you can incorporate changes and events that are not reflected in past data. These more subjective techniques include: Engineering Method. With this method, you make a detailed step-by-step analysis of the manufacturing process and the costs involved in each step. From this analysis, you decide how much time and materials are needed. The same analysis decides how much Manufacturing Overhead is needed for the entire manufacturing process, and this Manufacturing Overhead can then be allocated to the production. From your analysis, you estimate the cost function. Conference Method. With this method, different departments and employees give their opinions as to the costs (and the likely Cost Drivers) involved in their operations. A consensus cost function is derived from these opinions. Account Analysis Method. With this method, you examine all of the accounts of the business in order to determine an operation s Variable Costs (and their Cost Drivers) and Fixed Costs. With this information you estimate the firm s cost function. Objective methods to derive a cost function from the past behavior range from a very low tech approach, the High-Low Method, which can be done with simple Geometry and Algebra, to the use of an OLS Regression, which can be done using Excel. Before using either of these approaches, it is important for you to do a scattergraph, in which you plot the past history of the cost in question. The reason that you do the scattergraph is to visually confirm that you are dealing with a linear relationship, and to determine whether there are outlying points that can throw off your analysis. Scattergraph

6 Chapter 5 Notes Page 6 High-Low Method A simple way to estimate a linear cost function from your past costs, is the High-Low Method. As you will recall, only one line runs through two points. Thus, you need only two points to define the cost formula. The High-Low Method says that the two points that define the cost formula are: (i) the highest activity level, and (ii) the lowest activity level. (If either or both of these points were outlying points, you should select others.) High-Low Method Cost Function With the High-Low Method, you first start by estimating the Variable Cost per unit (the slope of the cost formula). As you will recall, the slope of a line is derived from the following formula: V = m = y 1 - y 0 x 1 - x 0 Now that you know the Variable Cost per unit, you can figure out the Fixed Cost. Take your cost function: Total Cost = Vx + F Plug in: (i) The Variable Cost per unit, (ii) the Total Costs for one of the two points (either the high or the low point), and (iii) the activity level associated with the Total Costs; and solve for the Fixed Cost. Example of High-Low Method Assume that Redford, Inc. makes baseball bats, and the highest activity level last year was 40,000 baseball bats and the cost to make those bats during that month was $500,000. The lowest activity level last year was 20,000 baseball bats, and the cost to

7 Chapter 5 Notes Page 7 make these bats during that month was $300,000. If you were to use the High-Low Method, you would calculate the Variable Cost per unit (the slope of the linear cost function): Variable Cost per Unit = m = V = V = V = y 1 - y 0 x 1 - x 0 $500,000 - $300,000 40,000-20,000 $200,000 20,000 $10 per bat Now that you know the Variable Cost per unit, you can figure out the Fixed Cost: Take the generic cost formula (Total Cost = Vx + F), and plug in: (i) the $10 Variable Cost per unit, (ii) the Total Costs for one of the two points that you used to determine the Variable Cost per unit (either the high or the low point), and (iii) the activity level for that point. Then, solve for the Fixed Cost. Either point will give you the same Fixed Cost. Using the High Point: Using the Low Point: Total Cost = Vx + F Total Cost = Vx + F 500,000 = 10 (40,000) + F 300,000 = 10 (20,000) + F 500,000 = 400,000 + F 300,000 = 200,000 + F 500, ,000 = F 300, ,000 = F 100,000 = F 100,000 = F So Redford, Inc. s cost formula is: TC = $100,000 + [$10 x (number of bats)] or TC = 10x + 100,000 [In Peggy Sue Got Married, Peggy Sue goes back in time to her High School Algebra class on the day of an examination. Having forgotten her Algebra, she turns in a blank answer sheet.] TEACHER: What's the meaning of this, Peggy Sue? PEGGY SUE: (patiently) Mr. Snelgrove, I happen to know that in the future, I will never have the slightest use for Algebra. And I speak from experience. Apparently, Peggy Sue never took Cost Accounting.

8 Chapter 5 Notes Page 8 OLS Regression Analysis Ordinary Least Squares (OLS) Regression analysis also involves estimating linear cost relationships from past data. Rather than merely drawing a line between two points, we use the computer to calculate the line that best fits the data. With OLS Regression, the computer squares the distances between a proposed cost function line and the actual cost data. The computer finds the line that produces the lowest sum of the squared distances. In the figure that appears below, the line in the center produces the lowest sum of these squared distances. You can run a Regression using Excel. With a Regression, you have independent variables and a dependent variable. The dependent variable is the variable whose behavior you are trying to estimate (cost). Independent variables are the explanatory variables (e.g., number of units). If you only have one independent variable, this is called a Simple Regression. If you have more than one independent variable, this is called a Multiple Regression. When you run a Regression using any statistical package (including Excel), the Regression will provide you with: (i) the coefficients that go before the independent variable (e.g. Variable Cost per unit), and (ii) the y-intercept value (for Fixed Cost). With a Simple Regression, this gives you the linear cost function: Total Cost = [Coefficient x Independent Variable] + Y-Intercept Coefficient With a Multiple Regression, you are no longer in a two dimensional world [total cost (y) & 1 independent variable (x) as the two axis]. Instead, you are now in a greater than two dimensional world. The number of dimensions is equal to the number of dependent

9 Chapter 5 Notes Page 9 and of independent variables in your cost function. The cost function given by the Regression reflects the increased number of dimensions. When running a Regression, a key statistic given in the Regression output is the R 2. The R 2 is referred to as the Coefficient of Determination. This statistic gives the Goodness Of The Fit of the model (or the percentage of the dependent variable s behavior that is explained by the model). The higher the R 2, the better the model fits the data. The best R 2 would be That means 100% of the data is explained by the model. When you have a Multiple Regression, then the Adjusted R 2 is looked at because it adjusts the R 2 value for the number of explanatory variables. The Regression output also gives you the statistical significance (t-statistic) for the coefficient(s) of the explanatory variable(s) and the y-intercept. This gives the likelihood that the coefficient or y-intercept is significantly different than zero. Usually, people want these coefficients to be significant at the 99%, 95%, or 90% level. If you ever see a Multiple Regression with a high Adjusted R 2, but low t-statistics for the independent variable coefficients, this tells you that the model is good, but you have chosen related independent variables. Having correlated independent variables is called Multicollinearity. The program is saying that your model is explaining the cost behavior, but the model can not figure out how much each variable is contributing to the explanatory power of the model. If you see this, try dropping one of the independent variables. OLS Regression Analysis Using Excel In order to run an OLS Regression using Excel, you need to have Data Analysis added as a Tool. Click on Tools on the main Menu Bar, and check to see if you have Data Analysis as an option. If not, click on Add Ins and then check the Analysis ToolPak box, and then click OK. You will now have Analysis ToolPak as an option under the Tools Menu item. You need to set up the data so that you can run the Regression. You need to input the cost (the dependent variable) and the independent variables. We will use two independent variables, hours and units:

10 Chapter 5 Notes Page 10 Now, click on Tools on the main Menu Bar, and then click on Data Analysis. The Data Analysis Dialog Box will open, and you will click on Regression. Now, the Regression Dialog Box will open. Click on the Input Y Range Box. Then, highlight the O/H Cost information on the spreadsheet (D2:D13). Next, Click on the Input X Range Box. Then, highlight the information under Hours (B2:B13). Now, click on OK. A new worksheet should open that has the Regression output: The Coefficients column gives you the OLS Regression s formula for your cost function: O/H Costs = [$3.814 x (Number of Hours)] + $219,385 Using this formula, you can estimate that if you had 69,000 hours, then your Manufacturing Overhead would be: ($3.814 x 69,000) + $219,385 = $482,551 We could also construct the 90% Confidence Interval using the formula that we discussed above. You would adjust the Cost Estimate using the Standard Error ($6,620) and the 90% t-statistic for 10 Degrees of Freedom (1.8125): $482,551 ± ( x $6,620) This produces the following Confidence Interval: 90% Confidence Interval: $470,552 to $494,550 Note that the actual cost in July (which had 69,000 hours) was $475,000; and actual cost lies within our projected Confidence Interval.

11 Chapter 5 Notes Page 11 In the Regression Dialogue Box, you will notice that you can specify a Confidence Level. This is not a reference to the Confidence Interval for the Cost Estimates produced by the Regression. This Confidence Interval is a reference to a Confidence Interval for the Regression s estimates for the coefficients (Variable Cost per unit & Fixed Costs). For example, the Regression will tell you that 90% of the time, the Variable Cost per unit will lie between $3.60 and $4.03 and the Fixed Cost will lie between $203,589 and $235,182: You cannot use these figures to produce a Confidence Interval for the Cost Estimate. The R 2 (R square) gives you the Goodness Of The Fit of this model. In real life, 99.07% is superb. The t-statistics that are reported for each coefficient are also great. You can tell this from the P-value that is also reported, which gives the significance level of the t- Statistic. [The Confidence Level related to the t-statistic is 1-(P-value).] We are happy if it is significant to the 90% level (less than.10); the 95% level (less than.05) and the 99% level (less than.01). Here the t-statistic of is significant beyond the 99% level ( ), and the t-statistic of 32.6 is also significant beyond the 99% level ( ). What if you ran an OLS Regression using Units as the independent variable?

12 Chapter 5 Notes Page 12 The OLS Regression s formula for your cost function is now: O/H Costs = [$18.03 x (Number of Units)] + $379,984 Using this formula, we can estimate that if we had 6,000 units, then our Manufacturing Overhead would be: ($18.03 x 6,000) + $379,985 = $488,165 and 90% of the time the actual cost will lie between $472,599 and $503,731: $488,165 ± ( x $8,588) Note that the actual cost in July (which had 6,000 units) was $475,000; and the actual cost lies within our projected Confidence Interval. The R 2 and t-statistics reported for this Regression are also very significant; although the R 2 (98.4%) is slightly lower than that of the preceding regression (99.07%). What if you ran an OLS Regression using both Hours (X1) and Units (X2) as the independent variable? This would be a Multiple Regression because you have more than one independent variable: Does the use of the two variables really improve the model? The R 2 of this Multiple Regression is as high as the Hours Regression. But look at the t-statistics that are reported for each coefficient. The t-statistic for the y-intercept is significant, but not to the level that we saw in the Simple Regressions, and the t-statistics for the coefficients for the two independent variables are not significant at all. What is going on? You have Multicollinearity. The two independent variables are correlated. The model still does a great job explaining cost behavior, but the Regression cannot tell which independent variable is doing the explaining.

13 Chapter 5 Notes Page 13 You should also note that the Multiple Regression s Adjusted R 2 ( ) declined from the Adjusted R 2 produced from the Simple Regression using Hours as the independent variable ( ). This suggests that you do not add additional explanatory power to the model by including Units as an additional independent variable in a Multiple Regression.