0

Two bank branches are asked in month 0 to reduce their 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. What is the issue?

Balance sheet Branch 1 Month 0 Month 1
Total Assets 50 50
Cash Deposits 20 20
Cash Ratio 40% 40%

Branch 1 reports 0 change in cash ratio and that no cash reduction took place (40% in y0 and 40% in y1)

Balance sheet Branch 2 Month 0 Month 1
Total Assets 50 60
Cash Deposits 50 0
Cash Ratio 100% 0%

Branch 2 reports 100% change in cash ratio and argues that 60 of cash has been reduced (100% - 0% = 100% x 60)

Balance sheet Total Month 0 Month 1
Total Assets 100 110
Cash Deposits 70 20
Cash Ratio 70.0% 18.2%

Headquarters reports cash ratio reduction of 51.8%ppt (70% - 18.2%) and argues that 57 of cash has been reduced (51.8% x 110)

Here a more compact overview of the tables above: spreadsheet view of tables above

Airlike1
  • 1
  • Related: https://math.stackexchange.com/questions/601265/percentage-change-of-ab-different-than-percentage-change-of-a-percentage-ch –  Apr 16 '21 at 06:47
  • Thank you - the reference to this question is indeed most likely talking about the same issue however I am struggeling to follow the arithmetic expressions. How would that be applied on the example above? – Airlike1 Apr 16 '21 at 06:56
  • It is important to keep in mind that percentages are ratios adjusted to parts per hundred (the meaning of per centum in Latin). Branch 1 did nothing to change its cash-holding ratio, while Branch 2 eliminated all of its cash holdings. So the behavior of the two branch managements were totally different (I suspect this is the point of this problem...) So Branch 1 is comparing $ \ 20/50 \ $ to $ \ 20/50 \ , $ while Branch 2 is comparing $ \ 50/50 \ $ to $ \ 0/60 \ $ . HQ is therefore doing something potentially misleading by combining the ratios in the way they did. (continued) –  Apr 16 '21 at 07:12
  • What they are doing is comparing $ \ (20 + 50)/(50+50) \ $ to $ \ (20+0)/(50+60) \ $ , so the bases of these ratios are different from what is being compared for the individual branches. If HQ is concerned with their overall situation, this is fine, but it smoothes over how the reduction was accomplished (one branch did all of the cash reduction). This illustrates why it is always of value to ask how reported statistics are actually computed. (All manner of invidious comparisons get made in business and politics in ways like this.) Was this what you were asking for? –  Apr 16 '21 at 07:17
  • I might mention an old joke based on this sort of "ratio play". A rat cornered by nine cats can take solace in the fact that, averaged over the group, the rat is $ \ 90 \ $ percent cat, while each cat is $ \ 10 \ $ percent rat... –  Apr 16 '21 at 07:20
  • Dear Boojum, many thanks. Yes, I assume HQ should not combine the ratios they way they do but what would be the right approach resp. what needs to be considered/ explained to the director that he can understand the deviation in the two numbers? – Airlike1 Apr 16 '21 at 07:36
  • This is another issue with comparing ratios with different bases. Branch 2 is comparing $ \ 0/60\ $ to $ \ 50/50 \ $ ; the absolute cash reduction is really $ \ 50 \ $ and not $ \ 60 \ $ . The director is comparing $ \ 70/100 \ $ to $ \ 20/110 \ $ , when the absolute reduction is $ \ 50 \ $ , and neither of these ratios has the same base as Branch 2 refers to in the first month, much less the second. So this carelessness about the bases is rather confusing the issue: the director's $ \ 51.8 \ $ percent of $ \ 110 \ $ does not reflect how the reduction actually took place. –  Apr 16 '21 at 08:02
  • If branch 2 would not have done anything, they would have had also 100% cash ratio in month 2, hence 60 cash but they managed to go down to 0 by end of month 2, hence a reduction of 60 I would figure is correct if you look at it from a cash-ratio reduction. – Airlike1 Apr 16 '21 at 08:08
  • But Branch 2 never reduced its cash-holding by $ \ 60 \ $ ; the statement shows that the reduction is $ \ 50 \ $ to $ \ 0 \ $ , while their total assets went from $ \ 50 \ $ to $ \ 60 \ . $ If anything, the director is making it look like more absolute holding was reduced with the figure of $ \ 57 \ $ . The HQ balance sheet also shows the $ \ 50 \ $ absolute cash reduction; the use of ratios to find the absolute change is inaccurate. –  Apr 16 '21 at 08:14
  • Thanks. The intention is not to find the absolute change but the relative reduction, e.g. if it is assumed that in a normal world the cash-ratio is always 50% even when assets increases, then any reduction of that ratio would be a positive change even if the cash volumes are unchanged in absolute terms but total assets increases and hence the cash ratio reduced – Airlike1 Apr 16 '21 at 08:21
  • Just to come back to one statement you made near the end, if Branch 2 had done nothing about their cash holding of $ \ 50 \ $ , but their total assets went up to $ \ 60 \ $ , their cash ratio would go down to $ \ 83.3 \ $ percent, but not because they got rid of any cash, but because their ratio base got larger. By the director's "reasoning", since the ratio went from $ \ 70/100 \ $ to $ \ 70/110 \ $ , he would claim a drop in cash ratio of $ \ 6.36 \ $ percentage points, and so a cash reduction of $ \ 7 \ $ , which is clearly untrue... –  Apr 16 '21 at 08:46
  • Yes, exactly. How does that fit into the issue above? Both, branch 2 and HQ are arguing their reports are correct. – Airlike1 Apr 16 '21 at 08:49
  • The two parties are talking about different things in comparing their ratios, although the director is also using the ratio improperly. It is better to just show the "dollar amounts" in the annual report and let the readers make their own ratios... –  Apr 16 '21 at 08:53
  • Thanks, unfortunately this is not solving my problem as the report is build to show the change as shown in the example above and I need to understand how these two reports can fit together. – Airlike1 Apr 16 '21 at 08:55
  • The two reports fit together in that everyone agrees that the total cash has been reduced by $ \ 50 \ $ . Both parties are using changes in percentages of cash against the wrong bases. Branch 2 reduced their cash by $ \ 100 \ $ percent from the $ \ 50 \ $ of the first month, not the $ \ 60 \ $ of the second. As for HQ, a change in percentage from one month to the next does not tell you how much absolute cash reduction occurred (as my last hypothetical example also shows), because their percentages are based on differing total assets from one month to the next. –  Apr 16 '21 at 09:06
  • Understood. Still wondering how the improvement in cash ratio from 70% to 18.2% from the HQ report then would be translated into an absolute figure if its not 57... – Airlike1 Apr 16 '21 at 10:00
  • The change needs to be reported against the first month's total: $ \ 70/100 \ $ goes to $ \ 20/\mathbf{100} \ $ , so it's a change by $ \ 50 \ $ percentage points and the cash reduction is $ \ .50 · 100 \ = \ 50 \ $ . The change should not be taken against a base of $ \ 110 \ $ since that is not what the change in cash was departing from. –  Apr 16 '21 at 10:26

0 Answers0