White Papers: The Mathematics Of Gross-Up

OneSource Relocation
July 17, 2002

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WHAT'S ALL THE FUSS ABOUT?
Calculating gross-up on employee relocation expenses is one of the most complicated issues in the relocation industry, but it does not need to be. Both corporations and third party service providers treat the matter of gross-up as if it were not an exact science, when in fact it should be a simple matter of mathematics. Questions such as: What is the tax on tax? and,

How many gross-ups are appropriate? should never even be subjects for discussion.

WHAT IS GROSS-UP?
The entire reason for gross-up is to make an employee whole when he or she is reimbursed for relocation expenses and that reimbursement is then subjected to standard payroll tax withholdings. The sole objective is that the employee's net reimbursement after tax be equal to the expenses incurred in the relocation. In a perfect world, employees would never receive a shortfall or a surplus on the monies received for reimbursement after the various taxes are assessed.

THE COMMON APPROACH:
Some corporations keep their accounting as simple as possible by making one gross-up calculation. They may gross-up for Federal Tax only, say 28%, and merely add the 28% to the amount reimbursed. Others may include State and Local income tax and FICA, but again just add the aggregate tax rate to the amount reimbursed. Then there are those companies who realize that merely adding the tax amount to the reimbursement does not make the employee whole after tax. Here is where the tax on tax multiple gross-up concept comes in. Granted, it is an admirable attempt to get closer to the right answer, but it still falls short and is cumbersome to boot.

THE MATHEMATICS OF MULTIPLE GROSS-UPS:
For the purpose of this discussion, assume employee John Doe earns $100,000 per year placing him, after all deductions against household income, in the 28% Federal Income Tax bracket with 6% State Tax and no Local Income Tax. At this taxable income level he would be assessed the Medicare Tax of 1.45% on incremental income. Therefore, John's aggregate tax rate is 34.45%. Assume his reimbursable relocation expenses total $10,000. Here is how many companies calculate John Doe's reimbursement:

Relo Expenses =                                                                   $10,000
ADD:
1st Gross-up       = $10,000 x 0.3545 =                                       3,545
2nd Gross-up       = $10,000 x 0.3545 x 0.3545 =1,257
3rd Gross-up       = $10,000 x 0.3545 x 0.3545 x 0.3545                  446
4th Gross-up       = $10,000 x 0.3545 x 0.3545 x 0.3545 x 0.3545    158
Total Reimb:                                                                          $15,406

The mathematical equation for this operation is
(where R = John's aggregate Tax Rate of 35.45%):

Total Reimbursement
= $10,000 x (R0 + R1 + R2 + R3 + R4)
= $10,000 x (1 + 0.3545 + 0.1257 + 0.0446 + 0157)
= $10,000 x 1.5406
= $15,406

If this calculation were taken to infinity, which carries the tax on tax concept to finality with an infinite number of gross-ups, the equation is:

Total Reimbursement = $10,000 x (R0 + R1 + R2 + R3 +…Rn)

Where n = infinity

A SIMPLE EQUATION FOR A ONE TIME GROSS-UP:
The mathematical term for the above equation is geometric sequence. The sum of this infinite geometric sequence, where 0<R<1, is:

 

                        SUM      =       $ 10,000    

                                                   1 – R

 

                                       =       $ 10,000__

                                                1 – 0.3545

 

                                       =       $ 10,000

                                                   0.6455

 

                                       =       $ 15,492

 

Note that the above equation yields a reimbursement amount of $15,492, which is approximated by the 4 gross-ups at $15,406. However, it is much easier, and exact instead of approximate, to just divide the reimbursed expense by 1 minus the Tax Rate.

ANOTHER WAY TO LOOK AT IT:
This equation can also be arrived at using linear algebra. The word equation is:

Relocation Expenses must equal Total Reimbursement less Tax on Total Reimbursement

WHERE:         X = Total Reimbursement
                     Y = Relocation Expenses = $10,000
                     R = Employee Tax Rate

EQUATION:   Y = X - XR

          $10,000 = X (1 - R)

SOLVE FOR X:         X = $ 10,000
                                        1 – R

OTHER CONSIDERATIONS:
Of course there are other considerations in determining a gross-up policy in addition to the method of calculation. Those include determining which expenses are to be grossed-up, developing a methodology for determining individual employee tax brackets, and deciding on whether to gross-up per check, per period or at year end, to name a few. However, the use of the following equation will take the mystery and misery out of the actual gross-up calculation: