separating mixed costs

additional topic

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