Why is sum of parts not equal total? (weighting issue)

[Airlike]

Dear forum,

I am looking at this problem all day long and still cannot figure out the 'behaviour pattern' of it. Basically I am looking at a sub-set (Unit 2) which makes sense when looking at it isolated (change in ratio leading to 60), but as soon as I consider the change in ratio from month 1 to month 2 from both units (unit 1 and unit 2) the total does not show the sum of it parts (unit 1 + unit 2). I would expect that the yellow highlighted numbers are exactly the same. What am I missing? I would like to attach an Excel spreadsheet which would make it more convencience to understand the issue but I believe I am not allowed to attach such document types.

Hopefully someone can help me out of my misery... :-(

 

[JeffM]

I’d need to understand better what you want to explain before I could advise you, but I think your intuitive expectations are wrong.

When we think about changes in something that involves multiplication or division, simple explanations fail due to interaction effects. A very simple example.

[MATH]5 * 9 = 45. [/MATH]
Now imagine that the 5 increases by 1% and the 9 increases by 2%. The result does not change by 3%.

[MATH]5.05 * 9.18 = 46.359[/MATH]
The percentage change in the result is 3.02%, which is somewhat greater than the sum of the individual changes. Most people find this contrary to intuition. The 0.02% change is an interaction effect.

It is usually quite easy to solve the analytic problem, but how to address the problem of presentation about something that is contrary to intuition is frequently a challenge.

 

[Airlike]

Dear Jeff, thank you very much for taking time and trying to help me.

I guess sometimes it helps to have a fictive background story:
Lets asume Unit 1and Unit 2 are two bank branches. The two bank branches are asked in month 0 to reduce their deposit resp. cash holding ratios and the effort will be reviewed after 1 month by the director of the bank (responsible for branch 1 and branch 2). The bank director is wondering why branch 2 is reporting cash reduction of 60 while headquarters only report cash reduction of 57 in their total report for branch 1 and branch 2.

Now, if you only look at the Total Report from headquarters (which the director will receive) you could assume that the deposit ratio of 70% is like a industry average, hence if you do ‘nothing’, you always have 70% in cash deposits - also when assets are increasing. So if you do ‘nothing’, you would expect to have 70% of total assets of 110 in cash in month 1 as well (which would be 77 in cash), but now the director checks the cash levels in month 1 and sees that they are actually only at 20. So he argues there was a (relative or fictive) decrease of 57 (77-20) and is very happy as the lower the cash-ratio the better (arguing that cash holdings to not generate revenues for the bank but only invested non-cash assets).

Branch 2 looks exactly the same way at their assets and come to the conclusion that they reduced 60 in cash against the local market assumption saying that assets are usually 100% made out of cash. So when they analyze the change in deposit-ratio from month 0 to month 1 they see the cash ratio reduced to 0%, arguing they reduced 60 cash against the market/ industry average of 100%.

The director reports 57 in reduction, branch manager of branch 1 reports no change and the local branch manager of branch 2 shows 60 in reduction against their local industry averages. So you would somehow expect that the two reports of the two branches add up to the total report but clearly this is not the case. So how can the reports fit together?

Please find an Excel spreadsheet better explaining the mechanism under the following link: Calculation Example

 

[JeffM]

I do not need fictive stories about banks. I was a senior bank executive for twenty-five years and a bank director for another ten.

I am quite familiar with the mathematical phenomenon you are talking about.

Now two different things are changing in your example, asset sizes and cash/asset ratios. Moreover, you are ignoring a third difference: you are assuming that the required cash levels at the two branches ought to differ due to local market conditions (a very sensible assumption). The moment multiple things are changing or differ, you get interaction effects. If changes are small, you can ignore interaction effects without making serious errors. But there will be some error if interaction effects are ignored. And those errors will get serious if changes are relatively large (as in your example). Finally, your example is quite artificial. In real life, a branch cannot operate without cash if only to load the ATM. Nor is it at all usual for two branches to have identical asset levels. And it is unheard of for a branch to have exactly identical assets at the end of two different months. So in real life, things will be considerably more complex to present than in your example.

Because you are assuming that the two branches are initially the same size but differ in their required cash asset ratios, you would be correct that
you could simply average the improvements in the cash/asset ratios IF THE ASSET SIZES DID NOT CHANGE.

But the asset sizes did change: branch 2 is now a bigger share of the pie. Which means that the improvement in the cash-asset ratio of branch 2 is now more important overall than the failure to improve at branch 1. There is an old saying about not counting apples and oranges. That is what you are trying to do.

One way to present this is to compare assuming everything but one stays the same. That means you end up with multiple comparisons. Branch 2 beat branch 1 in asset growth by 20%. Branch 2 beat branch 1 in reduction of cash-asset ratio by 100%. Overall, assets increased by 10% (10 on a base of 100), and the cash-asset ratio improved from 70/100 to 20/110 (an improvement of just under 52%). As a result of the COMBINED EFFECTS of both of those improvements, the investible cash derived from branch 1 and 2 together jumped from 30 to 90, an improvement of 200%. And 10% + 52% is a lot less than 200%.

In other words, when you are looking at something which depends on multiple things, you can explain quite simply how the individual components change, but the combined effect of those changes will usually not be some simple adding up of the component parts.

You can say in that example, of the 10% increase in assets overall, 100% is attributable to branch 2. Of the 52% improvement in the cash-asset ratio, 100% is attributable to branch 2. The combined effect of the increase in asset size of 10% and the improvement of 52% in the cash asset ratio gave a 200% increase in earning assets from the region.

You can attribute changes in individual metrics to different units, but addition will not let you show the combined effect of all the changes taken together.

 

[Airlike]

Dear Jeff, first and foremost many thanks for taking so much time and coming back with such a detailed answer - very much appreciated. Also very glad that I took an example which is so familiar to you - you absolutely got the issue and what I am trying to understand - trying to use the change in cash-ratio to make an indirect statement about the revenues from increase in earning assets while multiple elements are moving/ changing (total assets and deposits). And yes, I took a very artifical example to highlight the issue as much as possible. Furthermore, I agree also with your saying not to compare apples with oranges. But I am still wondering a bit how to resolve the ficitive argument between the director and branch manager 2, both arguing that their number are correct and here I believe branch manager 2 is forgetting about branch 1’s impact on the regional situation.

Do you have a suggestion how to amend the logic such to get a somewhat “stable report” going forward? Maybe taking the regional report and allocate the total achievement of 57 by some factor/ weighting down to the branches (in this case branch 2 would get the entire achievement as they contributed 100% of the change)?

 

[JeffM]

Based on a lot of experience in presenting numerical arguments to people with very disparate degrees of mathematical knowledge, I always liked a prsentation as follows. I am going to change your example to make it more realistic.

First, I would present the aggregate (just as you did).

Assets: old = 120, new 132, change 10% Obvious.

Earning assets: old = 90 new = 105.6, change approximately 17.33%. Obvious.

Earning asset ratio: old = 90/120 = 75%, new = 105.6/132 = 80%, change = 5%.

And obviously 10% + 5% < 17.33%.

You want everyone to see the numbers in a way that requires only basic arithmetic to understand. You want to be able to humiliate (very politely of course) anyone who is going to sidetrack YOUR presentation. Like chess, a career is not for the kind-hearted, but the rude incur many enemies. One must emulate Iago and smile and smile and be a villain. Whenever anyone asks an idiotic question, always apologize for having been unclear.

Second, you move on to an explanation of what caused the increase in earning assets. Here is a nice way to present it.

If the earning asset ratio had stayed the same (namely 75%), the increase in earning assets would have been
(132 - 120) * 0.75 = 12 * 0.75 = 9. But earning assets increased by 105.6 - 90 = 15.6. So the increase in assets alone account fot
9 /15.6 or about 58% of the increase in earning assets. Show that.

If the assets had stayed the same (namely 120), the increase in earning assets would have been
120 * (0.8 - 0.75) = 120 * 0.05 = 6. So the increase in the earning asset ratio alone accounts for
6/15.6 or about 38% Show that.

Obviously then, the effect of increasing assets and improving the earning asset ratio together (the interaction effect) is about 4%. Show that. Now people understand that the change cannot be neatly divided into two buckets. The ibtuition that it can is wrong. People must be shown that their intuition is wrong.

So, in terms of the aggregate you have fully explained the change 58% + 38% + 4% = 100%.

Moreover, you can now ignore the interaction effect because everyone can see that it is is relatively minor.

It is easy to allocate the change in assets among various branches. When it comes to allocating the improvement in the earning asset ratio, now you do need weighting as your title implied.

Presentation is something not taught in math class, but in terms of persuasion and self-promotion, it is an important art.