Telephone bill analysis: how to unravel cost behaviour

Introduction: intended audience

This case relates to a stage two cost behaviour analysis problem: for a first year seminar in cost accounting by accounting undergraduates; or a second year management accounting class exercise for business or management studies undergraduates.  The case is also suitable as a cost behaviour analysis case for MBA students.

The data that follow are taken from an actual telephone bill or statement in the United Kingdom some time during early 1999.

The purpose of this page is to take us through a first and then second stage analysis of the bill.  That is, we would like to derive a cost function for the telephone expense.

Unscientific analysis

Firstly, have a look at the data that follow and try to estimate the overall behaviour of the data: are they fixed, variable or semi variable?  Once you have done that, you should attempt to estimate the cost function for the data!

                 

Duration	Cost	Duration	Cost	Duration	Cost
(secs)	(£)	(secs)	(£)	(secs)	(£)
181	0.101	3620	1.514	4	0.042
134	0.074	80	0.088	4	0.042
106	0.059	52	0.057	783	8.824
35	0.042	83	0.092	52	0.147
358	0.074	75	0.083	54	0.153
13	0.042	575	0.644	442	1.253
110	0.061	1540	0.224	228	0.646
40	0.042	38	0.042	150	0.425
83	0.046	696	0.411	182	0.516
85	0.047	1254	14.127	1	0.08
108	0.059	16	0.172	133	0.377
165	0.091	595	6.587	585	1.659
326	0.182	317	3.562	40	0.113
1	0.042	653	7.356	248	0.703

Not easy is it?  Let’s sort out how we should do it properly.

Stage one analysis

The first stage is to take the data in their entirety and assess how the cost is behaving: refer to the page on cost behaviour if you are unsure of what this means.

This case represents an excellent example of why the first thing we should do with any attempt to assess the behaviour of a cost is to plot the data on a graph  

Helpful?  Well, possibly.  There is a section of the data that extend from just under £4 to £14; and there’s another one that seems to extend from about 350 seconds to almost 4,000 seconds.  Then there’s another series that seems to start around 150 seconds and rises to 600 seconds at a cost of around 1.8 pounds. Apart from that, it’s difficult, isn’t it?

Just for the sake of argument, let’s carry on and assume that we can apply the method of Ordinary Least Squares (OLS), or regression analysis, to help us to estimate a cost function.  The following graph contains the basic data and the line of best fit as derived from OLS.  The graph also includes the cost function derived from OLS.

   

The function tells us that we can estimate the telephone cost, Y’, by applying the function

Y’ = 0.6492 + 0.0017x

where x is the duration, in seconds, of the call relating to the cost we would like to estimate or predict.

However, we can see from the graph that even if we carry out all of the calculations now available to us, it won’t be much use because, as both the line of best fit and the r² value, 0.1286, tell us, we haven’t found an especially good model. 

Have we failed in our quest?

We haven’t failed yet.  As we saw just a short while ago, there do seem to be some patterns buried in the data: we found suggestions of at least three different sub series of data.  Our quest now, then is to unravel the data and see if we can derive the cost function we are looking for.

Stage two analysis: disaggregate the data

Take a closer look

In a situation such as this, the accountant has to go back to the invoice, the journals, the technical analysis and find out how he might unravel the telephone bill story.

Imagine the accountant has sent his assistant to the files to take a closer look at the telephone bills and she comes back with some additional information.  The assistant finds that telephone calls can be classified according to whether the calls are invoiced as

o        Local

o        Regional and National

o        International

o        Chargecard

That is, local calls are invoiced at one cost per unit/minute; regional and national calls are invoiced at another cost per unit/minute; and so on.  The assistant has found, therefore, that there are at least four different rates that telephone costs are invoiced at. 

So, what do we do now?  The accountant asked the assistant to sort the telephone cost data according to whether they relate to local, regional/national, international and Chargecard calls. The assistant does this and comes up with the data below.  We should see straight away that the assistant accountant has provided some very useful information for us.  Take a look at the costs per minute that have been derived from the newly sub classified data.  Not perfect; but better.

Local				Regional and National
duration	cost	cost/min		duration	cost	cost/min
181	0.101	0.0335		4	0.042	0.63
134	0.074	0.0331		4	0.042	0.63
106	0.059	0.0334		3620	1.514	0.0251
35	0.042	0.072		80	0.088	0.066
358	0.074	0.0124		52	0.057	0.0658
13	0.042	0.1938		83	0.092	0.0665
110	0.061	0.0333		75	0.083	0.0664
40	0.042	0.063		575	0.644	0.0672
83	0.046	0.0333		1540	0.224	0.0087
85	0.047	0.0332		38	0.042	0.0663
108	0.059	0.0328		696	0.411	0.0354
165	0.091	0.0331
326	0.182	0.0335
1	0.042	2.52
International				BT Chargecard
duration	cost	cost/min		duration	cost	cost/min
1254	14.127	0.6759		52	0.147	0.1696
16	0.172	0.645		54	0.153	0.17
595	6.587	0.6642		442	1.253	0.1701
317	3.562	0.6742		228	0.646	0.17
653	7.356	0.6759		150	0.425	0.17
783	8.824	0.6762		182	0.516	0.1701
				1	0.08	4.8
				133	0.377	0.1701
				585	1.659	0.1702
				40	0.113	0.1695
				248	0.703	0.1701

Let’s go back to basics again and put these on a graph: 4 separate graphs this time.  On each graph, we have the actual data and, by use of OLS analysis, what is called “Linear” that is the OLS based cost prediction for the costs of the telephone calls.

In the case of both International and Chargecard calls, we seem to have found a near perfect model because all we can see on each graph is the cost prediction for each call.  The actual costs are there, trust me!, but as we can see from the r² values, the OLS model is so good that cost prediction virtually exactly equals actual costs in every case.

Y'_{international} = -0.02788 + 0.011276X

r² 1.000

Y'_chargecard = 0.015453 + .002791X

r² 0.998

What about the models for the other calls?

What about local and regional and national calls, however?  Why do they have relatively poor r² values and hence why are their models relatively poor?  Have a look at the charts and the statistics and verify that this viewpoint is true.

Y'_local = 0.034781 + 0.000272X

r² 0.588

Y'_{regional and national} = 0.065861 + 0.000372X

r² 0.837