separating mixed costs
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METHODS FOR SEPARATING
MIXED COSTS
Linear Cost RelationshipAssume that a firm leases a photocopier. The lease
agreement calls for a lease payment of P3,000 paid at the beginning of each month. The firm is responsible for paying for the operating costs, which average P.02 per copy and cover the costs of toner, paper and maintenance.
Resources acquired in ADVANCE OF USAGE P3,000.
Resources acquired AS NEEDED AND USED P.02
Y = a + bxAssume 100,000 copies have been processed,
Y = P3,000 + (.02)(100,000)Total Cost will be P5,000.
Linear Cost RelationshipY = a + bx
where:Y = Total activity cost (the dependent variable)a = Fixed cost component (the intercept parameter)b = Variable cost per unit (the slope parameter)x = Measure of activity output (the independent
variable
Linear Cost Relationship
. . . . . .
TOTAL COST LINEY = a + bx
FIXED COSTa [intercept]
VARIABLE COSTb [slope]
ACTIVITYindependent
Methods of Separating Mixed Costs1. HIGH-LOW METHOD – preselects the two
points that will be used to compute the parameters “a” and “b”.
b = Change in costChange in activity
a = Y - bx
Problems
b = P3.00
b = P29,000 – P17,0007,000 - 3000
a = P29,000 – P3.00(7,000)= P8,000
a = P17,000 – P3.00(3,000)= P8,000
Methods of Separating Mixed Costs2. SCATTERGRAPH – “visual fit analysis” plots
the observation on a graph and draws conclusion on the relationships depicted by such observation.
b =Y1 – Y2 X1 – X2
a = Y - bx
X1X2
Y1
Y2
X1X2
Y1
Y2
JanuaryFebruary
MarchAprilMay
June July
AugustSeptember OctoberJanuary
July
Problems
b = P4.08
b =Y1 – Y2
X1 – X2
a = Y2 – bX2
a = P18,000 – P4.08(3,800)= P2,496
b = P28,000 – P18,0006,250 – 3,800
= P28,000 – P4.08(6,250)= P2,500
a = Y1 – bX1
Comparison of High-Low Method and Scattergraph Method
Since the two methods can produce significantly different cost formulas, the question of which method is the best naturally arises. Ideally, the method that is objective, and at the same time produces the best fitting line is needed.
High-Low Scattergraph b [slope] 3.00 4.08 a [intercept] 8,000 2,500.00
Methods of Separating Mixed Costs3. LEAST-SQUARE METHOD – identifies the best
fitting line by computing the line with least sum of squared deviations.
EQUATIONAL DERIVATIONEquation 1 Y = a + bX Equation 2 ΣY = na + bΣx Equation 3 ΣXY = Σxa + bΣX2
..
.
..
.
The deviation is the difference between the predicted and actual cost, which is shown by the distance from the line.
Thus, we are looking with the LEAST SUM of deviations.
Since there are negative and positive deviations, SQUARING the deviations avoids the cancellation problem caused by mix of positive and negative deviations.
Problem 13EQUATIONAL DERIVATION
Equation 1 Y = a + bX Equation 2 ΣY = na + bΣx Equation 3 ΣXY = Σxa + bΣX2
Problem 13ΣY = na + bΣx 1,500,000 = 6a + b4,200
Σxy = Σxa + bΣx2 1,107,000,000 = 4,200a + b3,220,000
Equation 2: 1,107,000,000 = 4,200a + b3,220,000Equation 1: (1,050,000,000) = (4,200a) (b2,940,000)
57,000,000 = b 280,000
b = 204
Equation 2: 1,107,000,000 = 4,200a + (204)(3,220,000) 1,107,000,000 = 4,200a + 656,880,000 450,120,000 = 4,200a
a = 107,171.43
Y =107,171.43 + 204x
Reliability of Cost Formula
• Coefficient of correlation, r• Coefficient of determination, r2
• Standard variance or confidence interval
The least square method provides the BEST FITTING LING but it doesn’t answer the GOODNESS OF FIT or the degree of association between cost and activity output.
Goodness-of-Fit Measures
Coefficient of Correlation, r – reflects the relationship between two variables, the dependent variable, y and the independent variable, x.
It is quite likely that a significant percentage of the total variability in cost is explained by our activity output.
Coefficient of Correlation, r
x y (x-x) (x-x)2 (y-y) (y-y)2 (x-x)(y-y)
800 270,000 100 10,000 20,000 400,000,000 2,000,000
500 200,000 (200) 40,000 (50,000) 2,500,000,000 10,000,000
1,000 310,000 300 90,000 60,000 3,600,000,000 18,000,000
400 190,000 (300) 90,000 (60,000) 3,600,000,000 18,000,000
600 240,000 (100) 10,000 (10,000) 100,000,000 1,000,000
900 290,000 200 40,000 40,000 1,600,000,000 8,000,000
4,200 1,500,000 280,000 11,800,000,000 57,000,000
Coefficient of Correlation, r
r = 57,000,000 (280,000)(11,800,000)
r = 0.9916
This means that there is a positive correlation between units produced, x and total cost, y.A coefficient-of-correlation value close to zero indicates no correlation.
Coefficient of Correlation, r
Machine hours Utility CostsThere is a positive correlation r approaches +1
Coefficient of Correlation, r
Hours of Safety Training
Industrial Accidents
There is a negative correlation r approaches -1
Coefficient of Correlation, r
Hair Length Accounting GradeThere is a no correlation r approaches 0
Goodness-of-Fit MeasuresCoefficient of Determination, r2 – reflects the percentage of variability of in the dependent variable, y explained by an independent variable, x.
r2 = ( r ) ( r )
Coefficient of Determination, r2
r2 = 0.98 or 98%
This means that changes in total cost, y can be explained by changes in activity measure, x by 98%.
Goodness-of-Fit MeasuresStandard Variance (Confidence Interval) – arises because the predicted Y value, Y’ is based on samples and are treated using statistical sampling techniques.
SV = (t-value) (s') 1 +1
+(X - X)2
n Σ (X - X)2
Confidence Interval
Predicted ValueUpper Limit
Lower Limit
+ Normal Deviation- Normal DeviationCoCONFIDENCE INTERVAL
Confidence IntervalAssume a plan of 1,100 units and confidence interval of ± 10,200.
Y =107,171.43 + 204x
Y =107,171.43 + 204(1,100)
Y' = 331,571.43Upper Limit = 341,771.43Lowe Limit = 321,371.43