Rounding shares to whole numbers

D

dannyt41

Board Regular
Joined
Jul 20, 2014
Messages
86
Hi everyone,

This is potentially more of a math question, but I'm hoping there is an excel formula that could also help.

I'm trying to divide a number of shares by percentages with 14 decimals. The end number of shares needs to be a whole number, however whenever I round I am always left with a remainder. Example -




The example above gives a difference of 3. What is the best way to approach this type of rounding? Is there a formula I can use to ensure the rounded total is 33,557?

Many thanks,
Daniel
 

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My approach is as follows. But I think shg has a different approach that randomizes the off-by-one errors, which are unavoidable. I never can remember his approach.

Caveat: Because I was unable to copy-and-paste from your posted image, my percentages and prorated share amounts differ slightly from yours. However, the results are close.

[TABLE="class: grid, width: 500"]
<tbody>[TR]
[TH][/TH]
[TH]A
[/TH]
[TH]B
[/TH]
[TH]C
[/TH]
[TH]D
[/TH]
[/TR]
[TR]
[TD="align: center"]1
[/TD]
[TD="align: right"]Total Shares
[/TD]
[TD="align: right"]33557
[/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[/TR]
[TR]
[TD="align: center"]2
[/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[/TR]
[TR]
[TD="align: center"]3
[/TD]
[TD="align: right"]
Percentage
[/TD]
[TD="align: right"]Prorated
Shares
[/TD]
[TD="align: right"]
Round
[/TD]
[TD="align: right"]Cuml
Round
[/TD]
[/TR]
[TR]
[TD="align: center"]4
[/TD]
[TD="align: right"]4.95849212790250%
[/TD]
[TD="align: right"]1663.921203360240
[/TD]
[TD="align: right"]1664
[/TD]
[TD="align: right"]1664
[/TD]
[/TR]
[TR]
[TD="align: center"]5
[/TD]
[TD="align: right"]3.30567304246942%
[/TD]
[TD="align: right"]1109.284702861460
[/TD]
[TD="align: right"]1109
[/TD]
[TD="align: right"]1109
[/TD]
[/TR]
[TR]
[TD="align: center"]6
[/TD]
[TD="align: right"]6.61133625561081%
[/TD]
[TD="align: right"]2218.566107295320
[/TD]
[TD="align: right"]2219
[/TD]
[TD="align: right"]2219
[/TD]
[/TR]
[TR]
[TD="align: center"]7
[/TD]
[TD="align: right"]6.61134204650717%
[/TD]
[TD="align: right"]2218.568050546410
[/TD]
[TD="align: right"]2219
[/TD]
[TD="align: right"]2218
[/TD]
[/TR]
[TR]
[TD="align: center"]8
[/TD]
[TD="align: right"]2.47926594092860%
[/TD]
[TD="align: right"]831.967271797410
[/TD]
[TD="align: right"]832
[/TD]
[TD="align: right"]832
[/TD]
[/TR]
[TR]
[TD="align: center"]9
[/TD]
[TD="align: right"]9.91699622035434%
[/TD]
[TD="align: right"]3327.846421664310
[/TD]
[TD="align: right"]3328
[/TD]
[TD="align: right"]3328
[/TD]
[/TR]
[TR]
[TD="align: center"]10
[/TD]
[TD="align: right"]6.61346292107559%
[/TD]
[TD="align: right"]2219.279752425330
[/TD]
[TD="align: right"]2219
[/TD]
[TD="align: right"]2219
[/TD]
[/TR]
[TR]
[TD="align: center"]11
[/TD]
[TD="align: right"]1.65283419699388%
[/TD]
[TD="align: right"]554.641571485235
[/TD]
[TD="align: right"]555
[/TD]
[TD="align: right"]555
[/TD]
[/TR]
[TR]
[TD="align: center"]12
[/TD]
[TD="align: right"]1.65308763318103%
[/TD]
[TD="align: right"]554.726617066559
[/TD]
[TD="align: right"]555
[/TD]
[TD="align: right"]555
[/TD]
[/TR]
[TR]
[TD="align: center"]13
[/TD]
[TD="align: right"]1.65284313954456%
[/TD]
[TD="align: right"]554.644572336969
[/TD]
[TD="align: right"]555
[/TD]
[TD="align: right"]554
[/TD]
[/TR]
[TR]
[TD="align: center"]14
[/TD]
[TD="align: right"]4.95944478963657%
[/TD]
[TD="align: right"]1664.240888058340
[/TD]
[TD="align: right"]1664
[/TD]
[TD="align: right"]1665
[/TD]
[/TR]
[TR]
[TD="align: center"]15
[/TD]
[TD="align: right"]6.61133256465374%
[/TD]
[TD="align: right"]2218.564868720850
[/TD]
[TD="align: right"]2219
[/TD]
[TD="align: right"]2218
[/TD]
[/TR]
[TR]
[TD="align: center"]16
[/TD]
[TD="align: right"]1.65316589599464%
[/TD]
[TD="align: right"]554.752879718920
[/TD]
[TD="align: right"]555
[/TD]
[TD="align: right"]555
[/TD]
[/TR]
[TR]
[TD="align: center"]17
[/TD]
[TD="align: right"]1.32227119770626%
[/TD]
[TD="align: right"]443.714545814291
[/TD]
[TD="align: right"]444
[/TD]
[TD="align: right"]444
[/TD]
[/TR]
[TR]
[TD="align: center"]18
[/TD]
[TD="align: right"]1.98337237546647%
[/TD]
[TD="align: right"]665.560268035282
[/TD]
[TD="align: right"]666
[/TD]
[TD="align: right"]665
[/TD]
[/TR]
[TR]
[TD="align: center"]19
[/TD]
[TD="align: right"]0.82644310004856%
[/TD]
[TD="align: right"]277.329511083295
[/TD]
[TD="align: right"]277
[/TD]
[TD="align: right"]278
[/TD]
[/TR]
[TR]
[TD="align: center"]20
[/TD]
[TD="align: right"]23.13963399287930%
[/TD]
[TD="align: right"]7764.966978990490
[/TD]
[TD="align: right"]7765
[/TD]
[TD="align: right"]7765
[/TD]
[/TR]
[TR]
[TD="align: center"]21
[/TD]
[TD="align: right"]9.91700597806600%
[/TD]
[TD="align: right"]3327.849696059610
[/TD]
[TD="align: right"]3328
[/TD]
[TD="align: right"]3327
[/TD]
[/TR]
[TR]
[TD="align: center"]22
[/TD]
[TD="align: right"]4.13199658098062%
[/TD]
[TD="align: right"]1386.574092679660
[/TD]
[TD="align: right"]1387
[/TD]
[TD="align: right"]1387
[/TD]
[/TR]
[TR]
[TD="align: center"]23
[/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[/TR]
[TR]
[TD="align: center"]24
[/TD]
[TD="align: right"]100.00000000000000%
[/TD]
[TD="align: right"]33557.000000000000
[/TD]
[TD="align: right"]33560
[/TD]
[TD="align: right"]33557
[/TD]
[/TR]
</tbody>[/TABLE]

Code: 
Formulas:
D4:  =ROUND(A4*B1,0)
D5:  =ROUND(SUM($A$4:A5)*$B$1 - SUM($D$4:D4), 0)

Copy D5 into D6:D22
 
Last edited:
Upvote 0
joeu2004 said:
My approach is as follows. But I think shg has a different approach that randomizes the off-by-one errors, which are unavoidable. I never can remember his approach.

Caveat: Because I was unable to copy-and-paste from your posted image, my percentages and prorated share amounts differ slightly from yours. However, the results are close.

[TABLE="class: grid, width: 500"]
<tbody>[TR]
[TH][/TH]
[TH]A[/TH]
[TH]B[/TH]
[TH]C[/TH]
[TH]D[/TH]
[/TR]
[TR]
[TD="align: center"]1[/TD]
[TD="align: right"]Total Shares[/TD]
[TD="align: right"]33557[/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[/TR]
[TR]
[TD="align: center"]2[/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[/TR]
[TR]
[TD="align: center"]3[/TD]
[TD="align: right"]
Percentage[/TD]
[TD="align: right"]Prorated
Shares[/TD]
[TD="align: right"]
Round[/TD]
[TD="align: right"]Cuml
Round[/TD]
[/TR]
[TR]
[TD="align: center"]4[/TD]
[TD="align: right"]4.95849212790250%[/TD]
[TD="align: right"]1663.921203360240[/TD]
[TD="align: right"]1664[/TD]
[TD="align: right"]1664[/TD]
[/TR]
[TR]
[TD="align: center"]5[/TD]
[TD="align: right"]3.30567304246942%[/TD]
[TD="align: right"]1109.284702861460[/TD]
[TD="align: right"]1109[/TD]
[TD="align: right"]1109[/TD]
[/TR]
[TR]
[TD="align: center"]6[/TD]
[TD="align: right"]6.61133625561081%[/TD]
[TD="align: right"]2218.566107295320[/TD]
[TD="align: right"]2219[/TD]
[TD="align: right"]2219[/TD]
[/TR]
[TR]
[TD="align: center"]7[/TD]
[TD="align: right"]6.61134204650717%[/TD]
[TD="align: right"]2218.568050546410[/TD]
[TD="align: right"]2219[/TD]
[TD="align: right"]2218[/TD]
[/TR]
[TR]
[TD="align: center"]8[/TD]
[TD="align: right"]2.47926594092860%[/TD]
[TD="align: right"]831.967271797410[/TD]
[TD="align: right"]832[/TD]
[TD="align: right"]832[/TD]
[/TR]
[TR]
[TD="align: center"]9[/TD]
[TD="align: right"]9.91699622035434%[/TD]
[TD="align: right"]3327.846421664310[/TD]
[TD="align: right"]3328[/TD]
[TD="align: right"]3328[/TD]
[/TR]
[TR]
[TD="align: center"]10[/TD]
[TD="align: right"]6.61346292107559%[/TD]
[TD="align: right"]2219.279752425330[/TD]
[TD="align: right"]2219[/TD]
[TD="align: right"]2219[/TD]
[/TR]
[TR]
[TD="align: center"]11[/TD]
[TD="align: right"]1.65283419699388%[/TD]
[TD="align: right"]554.641571485235[/TD]
[TD="align: right"]555[/TD]
[TD="align: right"]555[/TD]
[/TR]
[TR]
[TD="align: center"]12[/TD]
[TD="align: right"]1.65308763318103%[/TD]
[TD="align: right"]554.726617066559[/TD]
[TD="align: right"]555[/TD]
[TD="align: right"]555[/TD]
[/TR]
[TR]
[TD="align: center"]13[/TD]
[TD="align: right"]1.65284313954456%[/TD]
[TD="align: right"]554.644572336969[/TD]
[TD="align: right"]555[/TD]
[TD="align: right"]554[/TD]
[/TR]
[TR]
[TD="align: center"]14[/TD]
[TD="align: right"]4.95944478963657%[/TD]
[TD="align: right"]1664.240888058340[/TD]
[TD="align: right"]1664[/TD]
[TD="align: right"]1665[/TD]
[/TR]
[TR]
[TD="align: center"]15[/TD]
[TD="align: right"]6.61133256465374%[/TD]
[TD="align: right"]2218.564868720850[/TD]
[TD="align: right"]2219[/TD]
[TD="align: right"]2218[/TD]
[/TR]
[TR]
[TD="align: center"]16[/TD]
[TD="align: right"]1.65316589599464%[/TD]
[TD="align: right"]554.752879718920[/TD]
[TD="align: right"]555[/TD]
[TD="align: right"]555[/TD]
[/TR]
[TR]
[TD="align: center"]17[/TD]
[TD="align: right"]1.32227119770626%[/TD]
[TD="align: right"]443.714545814291[/TD]
[TD="align: right"]444[/TD]
[TD="align: right"]444[/TD]
[/TR]
[TR]
[TD="align: center"]18[/TD]
[TD="align: right"]1.98337237546647%[/TD]
[TD="align: right"]665.560268035282[/TD]
[TD="align: right"]666[/TD]
[TD="align: right"]665[/TD]
[/TR]
[TR]
[TD="align: center"]19[/TD]
[TD="align: right"]0.82644310004856%[/TD]
[TD="align: right"]277.329511083295[/TD]
[TD="align: right"]277[/TD]
[TD="align: right"]278[/TD]
[/TR]
[TR]
[TD="align: center"]20[/TD]
[TD="align: right"]23.13963399287930%[/TD]
[TD="align: right"]7764.966978990490[/TD]
[TD="align: right"]7765[/TD]
[TD="align: right"]7765[/TD]
[/TR]
[TR]
[TD="align: center"]21[/TD]
[TD="align: right"]9.91700597806600%[/TD]
[TD="align: right"]3327.849696059610[/TD]
[TD="align: right"]3328[/TD]
[TD="align: right"]3327[/TD]
[/TR]
[TR]
[TD="align: center"]22[/TD]
[TD="align: right"]4.13199658098062%[/TD]
[TD="align: right"]1386.574092679660[/TD]
[TD="align: right"]1387[/TD]
[TD="align: right"]1387[/TD]
[/TR]
[TR]
[TD="align: center"]23[/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[/TR]
[TR]
[TD="align: center"]24[/TD]
[TD="align: right"]100.00000000000000%[/TD]
[TD="align: right"]33557.000000000000[/TD]
[TD="align: right"]33560[/TD]
[TD="align: right"]33557[/TD]
[/TR]
</tbody>[/TABLE]

Code: 
Formulas:
D4:  =ROUND(A4*B1,0)
D5:  =ROUND(SUM($A$4:A5)*$B$1 - SUM($D$4:D4), 0)

Copy D5 into D6:D22
Click to expand...


Thank you very much, the formulas work great.

Is this a fairly standard rounding approach for allocating shares? I can't find many references online to this type of calculation and would like to reference something if possible!

Many thanks,
Daniel
 
Upvote 0
dannyt41 said:
Is this a fairly standard rounding approach for allocating shares?
Click to expand...

Ad hoc, I'm afraid.

I believe there is a "standard" algorithm or genereally accepted method. I'm sure I read a wiki page or other onine article long ago. But I'm having trouble finding it now.

My algorithm can be poorly behaved (or at least non-optimal). Consider this....

For my posted example, several rows round in the opposite direction: row 7, 13, 15, 18 and 21 round down; and row 14 and 19 round up. The sum of squared errors (SSE) is about 4.03.

In contrast, the SSE is lower, about 2.17, if we force only row 6, 15 and 18 to round down, the opposite direction.
 
Upvote 0
Another way:

[Table="width:, class:grid"][tr][td="bgcolor:#C0C0C0"][/td][td="bgcolor:#C0C0C0"]
A​
[/td][td="bgcolor:#C0C0C0"]
B​
[/td][td="bgcolor:#C0C0C0"]
C​
[/td][/tr][tr][td="bgcolor:#C0C0C0"]
1​
[/td][td="bgcolor:#F3F3F3"]
Total Shares​
[/td][td]
33557​
[/td][td="bgcolor:#F3F3F3"]
​
[/td][/tr]
[tr][td="bgcolor:#C0C0C0"]
2​
[/td][td][/td][td][/td][td][/td][/tr]
[tr][td="bgcolor:#C0C0C0"]
3​
[/td][td="bgcolor:#F3F3F3"]
Pct​
[/td][td="bgcolor:#F3F3F3"]
Shares​
[/td][td][/td][/tr]
[tr][td="bgcolor:#C0C0C0"]
4​
[/td][td]
4.95849212790250%​
[/td][td="bgcolor:#E5E5E5"]
1664​
[/td][td]B4: =ROUND(A4 / (A$24 - SUM(A$3:A3)) * (B$1 - SUM(B$3:B3)), 0)[/td][/tr]
[tr][td="bgcolor:#C0C0C0"]
5​
[/td][td]
3.30567304246942%​
[/td][td="bgcolor:#E5E5E5"]
1109​
[/td][td][/td][/tr]
[tr][td="bgcolor:#C0C0C0"]
6​
[/td][td]
6.61133625561081%​
[/td][td="bgcolor:#E5E5E5"]
2219​
[/td][td][/td][/tr]
[tr][td="bgcolor:#C0C0C0"]
7​
[/td][td]
6.61134204650717%​
[/td][td="bgcolor:#E5E5E5"]
2219​
[/td][td][/td][/tr]
[tr][td="bgcolor:#C0C0C0"]
8​
[/td][td]
2.47926594092860%​
[/td][td="bgcolor:#E5E5E5"]
832​
[/td][td][/td][/tr]
[tr][td="bgcolor:#C0C0C0"]
9​
[/td][td]
9.91699622035434%​
[/td][td="bgcolor:#E5E5E5"]
3328​
[/td][td][/td][/tr]
[tr][td="bgcolor:#C0C0C0"]
10​
[/td][td]
6.61346292107559%​
[/td][td="bgcolor:#E5E5E5"]
2219​
[/td][td][/td][/tr]
[tr][td="bgcolor:#C0C0C0"]
11​
[/td][td]
1.65283419699388%​
[/td][td="bgcolor:#E5E5E5"]
555​
[/td][td][/td][/tr]
[tr][td="bgcolor:#C0C0C0"]
12​
[/td][td]
1.65308763318103%​
[/td][td="bgcolor:#E5E5E5"]
555​
[/td][td][/td][/tr]
[tr][td="bgcolor:#C0C0C0"]
13​
[/td][td]
1.65284313954456%​
[/td][td="bgcolor:#E5E5E5"]
555​
[/td][td][/td][/tr]
[tr][td="bgcolor:#C0C0C0"]
14​
[/td][td]
4.95944478963657%​
[/td][td="bgcolor:#E5E5E5"]
1664​
[/td][td][/td][/tr]
[tr][td="bgcolor:#C0C0C0"]
15​
[/td][td]
6.61133256465374%​
[/td][td="bgcolor:#E5E5E5"]
2218​
[/td][td][/td][/tr]
[tr][td="bgcolor:#C0C0C0"]
16​
[/td][td]
1.65316589599464%​
[/td][td="bgcolor:#E5E5E5"]
555​
[/td][td][/td][/tr]
[tr][td="bgcolor:#C0C0C0"]
17​
[/td][td]
1.32227119770626%​
[/td][td="bgcolor:#E5E5E5"]
444​
[/td][td][/td][/tr]
[tr][td="bgcolor:#C0C0C0"]
18​
[/td][td]
1.98337237546647%​
[/td][td="bgcolor:#E5E5E5"]
665​
[/td][td][/td][/tr]
[tr][td="bgcolor:#C0C0C0"]
19​
[/td][td]
0.82644310004856%​
[/td][td="bgcolor:#E5E5E5"]
277​
[/td][td][/td][/tr]
[tr][td="bgcolor:#C0C0C0"]
20​
[/td][td]
23.13963399287930%​
[/td][td="bgcolor:#E5E5E5"]
7765​
[/td][td][/td][/tr]
[tr][td="bgcolor:#C0C0C0"]
21​
[/td][td]
9.91700597806600%​
[/td][td="bgcolor:#E5E5E5"]
3328​
[/td][td][/td][/tr]
[tr][td="bgcolor:#C0C0C0"]
22​
[/td][td]
4.13199658098062%​
[/td][td="bgcolor:#E5E5E5"]
1386​
[/td][td][/td][/tr]
[tr][td="bgcolor:#C0C0C0"]
23​
[/td][td][/td][td="bgcolor:#E5E5E5"][/td][td][/td][/tr]
[tr][td="bgcolor:#C0C0C0"]
24​
[/td][td="bgcolor:#E5E5E5"]
100.00000000000000%​
[/td][td="bgcolor:#E5E5E5"]
33557​
[/td][td][/td][/tr]
[/table]
 
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