ADVANCED MANAGEMENT ACCOUNTING Lecture 2

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ADVANCED MANAGEMENT ACCOUNTINGLecture (2)

• Cost Estimation and Behaviour II

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Last Lecture Summary

• Cost classifications for predicting cost
  behaviour (i.e. how a certain cost will behave in
  response to a change in activity)
• Variable cost a cost that varies, in total, in
  direct proportion to changes in the level of
  activity.
• Fixed cost a cost that remains constant, in
  total, regardless of changes in the level of
  activity.
• Mixed cost a cost that contains both variable
  and fixed cost elements.

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Last Lecture Summary (cont)

• How does management go about actually estimating
  the fixed and variable components of a mixed
  cost? There are five methods
• The account analysis (inspection of the
  accounts),
• The engineering approach,
• The high and low method,
• The scatter graph (graphical) method, and
• The least-squares regression method (todays
  topic).

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The least-squares regression method

• This method determines mathematically the
  regression line of best fit (i.e. it uses
  mathematical formulas to fit the regression
  line).
• It is a more objective and precise approach to
  estimating the regression line than the scatter
  graph method (the later fits the regression line
  by visual inspection).
• Unlike the high-low method, the least-squares
  regression method takes all of the data into
  account when estimating the cost formula.

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Definition of Terms

• A regression equation (a regression line when
  plotted on a graph) identifies an estimated
  relationship between a dependent (i.e. cost Y)
  and one or more independent variables (i.e. an
  activity measure or cost driver) based on past
  observations.
• Simple regression when the regression equation
  includes a dependent variable and only one
  independent variable.
• Multiple regression when the regression equation
  includes a dependent variable and two or more
  independent variables.

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Definition of Terms (cont)

• Types of Relationships between dependent and
  independent variables
• ??Direct vs. Inverse
• Direct -X and Y increase together
• Inverse -X and Y have opposite directions
• ??Linear vs. Curvilinear
• Linear -Straight line best describes the
  relationship between X and Y
• Curvilinear -Curved line best describes the
  relationship between X and Y

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Possible Relationships Between X and Y in Scatter
Diagrams
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Simple Linear Regression

• Simple -only one independent or predictor
  variable (X)
• Linear -the mathematical relation between X and Y
  is in the form
• Y a bX

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Simple Linear Regression Equation
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Linear Equations
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Estimating the Linear Equation Using the
Least-Squares Method (LSM)

• Looks at differences between actual values (Y)
  and predicted values (Y). Best fit tries to
  make these small
• But positive differences offset negative
• LSRM minimizes the sum of the squared differences
  (or errors)

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Least Squares Method Graphically
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Coefficient Equations
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Computation Table
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Computation Table (cont)

• Where
• X the level of activity (independent variable)
• Y the total mixed cost (dependent variable)
• a the total fixed cost (the vertical intercept
  of the line)
• b the variable cost per unit of activity (the
  slope of the line)
• n number of observations
• S sum across all n observations

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See Drury (2004)

• Example on Page (1044) -
• Exhibit 24.1 Figure 24.3

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The Example from Drury (2004)
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Solution
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Also, See Seal et al. (2006)

• Example on Page (183) -
• Solution on Pages (pp. 193 194)

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The Example from Seal et al. (2006)
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Solution
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Test of Reliability

• To see how reliable potential cost drivers (e.g.
  machine hours, direct labour hours, units of
  output, or number of production runs) are in
  predicting the dependent variable (the total
  mixed cost), three tests of reliability can be
  applied
• The coefficient of determination,
• The standard error of the estimate, and
• The standard error of the coefficient

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The Coefficient of Determination (r2)

• The R-Square is a general measure of the
  usefulness of the regression model. It measures
  the extent, or strength, of the association
  between two variables (X,Y).
• It indicates how much of the fluctuation in the
  dependent variable is produced by its
  relationship with the independent variable (s).
• An R-Square of 1.00 indicates that 100 of the
  variation in the dependent variable is explained
  by the independent variable (s).
• Conversely, an R-Square of 0.0 indicates that
  none of the variation in the dependent variable
  is explained by the independent variable (s).

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r2--Perfect CorrelationAn Example (r21)
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r2--No CorrelationAn Example (r20)
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r2 Computation
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r2-Example -Drury (2004), P. 1044 Solution,
Appendix 24.1
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Coefficient of Correlation (r)
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Various r Values
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r-Example -Drury (2004), P. 1044 Solution,
Appendix 24.1
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Standard Error of Estimate (se)

• r2 gives us an indication of the reliability of
  the estimate of total cost but it does not give
  us an indication of the absolute size of the
  probable deviations from the regression line.
• This is important because the least-squares line
  is calculated from sample data and other samples
  would probably result in different estimates.
• se measures the reliability of the regression
  line.
• It measures the variability, or scatter of the
  observed values around the regression line.

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Scatter Around the Regression Line
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Formula to Compute se
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Se -Example -Drury (2004), P. 1044 Solution,
Appendix 24.1
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The standard error of the coefficient (s )
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Sb-Example -Drury (2004), P. 1044 Solution,
Appendix 24.1
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Computer Programs to perform a simple regression
analysis

• SPSS and Performing A Simple Regression Analysis
• Microsoft Excel and Performing A Simple
  Regression Analysis

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Multiple Linear Regression

• The simple least-squares regression analysis is
  based on the assumption that total cost was
  determined by one activity-based variable only
  (only one factor is taken into consideration).
• However, other variables besides activity are
  likely to influence total cost.
• E.g. shipping costs may depend on both the number
  of units shipped and the weight of the units.
• In a situation such as this, multiple regression
  is necessary (where several factors are
  considered in combination).

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Multiple Linear Regression (cont)

• If two independent variables (e.g. machine hours
  and temperature) influence the total cost (e.g.
  the cost of steam generation) and the
  relationship is assumed to be linear, the
  regression equation will be
• ya b1x1 b2x2
• where
• a -represents the total fixed cost.
• b1 represents the regression coefficient for
  machine hours (i.e. the average change in y
  resulting from a unit change in x1, assuming that
  x2 remains constant).
• X1 is the number of machine hours.
• b2 is the regression coefficient for temperature
  (i.e. the average change in y resulting from a
  unit change in x2, assuming that x1remains
  constant).
• X2 represents the number of days per month in
  which the temperature is less than 15ºC.

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Multicollinearity problem

• Multiple regression analysis is based on the
  assumption that the independent variables are not
  correlated with each other.
• When the independent variables are highly
  correlated with each other, it is very difficult
  to separate the effects of each of these
  variables on the dependent variable.
• This condition is called multicollinearity.
• Generally, a coefficient of correlation between
  independent variables greater than 0.70 indicates
  multicollinearity.

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Non-linear regression (the learning-curve-effect)

• Changes in the efficiency of the labour force may
  render past information unsuitable for predicting
  future labour costs.
• A situation like this may occur when workers
  become more familiar with the tasks that they
  perform, so that less labour time is required for
  the production of each unit.
• This phenomenon is known as the
  learning-curve-effect.

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Non-linear regression (the learning-curve-effect)
cont

• The learning curve can be expressed in equation
  form as follows
• Yx axb
• Where
• Yxthe cumulative average time required to
  produce X units.
• a the time required to produce the first unit of
  output.
• X the number of units of output under
  consideration.
• The exponent b is defined as the ratio of the
  logarithm of the learning curve improvement (e.g.
  80) divided by the logarithm of 2.

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Example An application of the 80 learning curve

• The labour hours are required on a sequence of
  six orders where the cumulative number of units
  is doubled for each order.
• If the first unit of output was completed on the
  first order in 2000 hours.
• Required
• calculate the cumulative average time (per unit)
  taken to produce 2, 4, 8, 16 32 units
  respectively, assuming that the average time per
  unit were 80 of the average time per unit of the
  previous cumulative production.

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Solution
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Solution (cont)

• The cumulative average time (per unit) taken to
  produce 2, 4, 8, 16 32 units
• First determine
• Order1 Y1 2000 hours
• Order 2 -Y2 2000 2-0.322 1600 hours
• Order 3 -Y4 2000 4-0.322 1280 hours
• Order 4 -Y8 2000 8-0.322 1024 hours
• Order 5 -Y162000 16-0.322 819 hours
• Order 6 -Y322000 32-0.322 655 hours

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Solution (cont)
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Solution (cont)Graphical method
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Factors to be considered when using past data to
estimate cost functions

• The cost data and activity should be related to
  the same period (e.g. some costs lag behind the
  associated activity wages paid).
• Number of observations (a sufficient number of
  observations must be obtained).
• Accounting policies (do not lead to distorted
  cost functions the allocated costs).
• Adjustments for past changes (any changes of
  circumstances in the future).
• Relevant range (i.e. the range of activity within
  which a particular straight line provide a
  reasonable approximation to the real underlying
  cost function) and non-linear cost functions (see
  next two slides)

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The Linearity Assumption and the Relevant Range
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Fixed Costs and Relevant Range
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Summary

• The least-squares regression method is an
  objective and precise approach to estimating a
  cost function based on the analysis of past data.
  The stages involved in the estimation are
• 1) Select the dependent variable (y) to be
  predicted,
• 2) Select the potential cost drivers (Xs),
• 3) Collect data on the dependent variable and
  cost drivers,
• 4) Plot the observations on a graph,
• 5) Estimate the cost function, and
• 6) Test the reliability of the cost function.

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Workshop (2)

• See Exercises P5-15 P5-16 (Seal et al., 2006)