How to calculate an additional gross amount required to achieve a net income in a multiple rate band tax regime?

Question

I'm working on a computer algorithm for working out how much a net (post-tax) increase will cost a person in gross terms. I'm using the UK tax band system but of course the algorithm could generalise on any band ranges/rates.

I am using an example to attempt to clarify my own thinking, which is an individual who receives £1,200 per month after tax. I want to know what the increase will cost them if they want to receive £4,800 per month after tax (an increase after tax of £3,600).

The tax bands are as follows (all per month):

Name       Range £       Rate
Tax free      0- 916      0%
Basic       916-3583     20%
Higher     3583-99999    40%

For the sake of simplicity I have dropped off an extra real-world band after Higher.

I think I have the algorithm worked out, which is broadly this: for each tax band, work out if it would be fully consumed, and if so add the cost of the band plus the tax that would be applied, to running totals. I have a way to calculate the tax on a gross amount, using all bands in the system, so this bit should be easy. Tax bands are gross, so I can just calculate the net income using the top of a band, and if this is not enough to satisfy the need, the whole band must be required.

Using this I have calculated that the large jump in income will completely consume the Basic band, where £533.33 will be levied.

Where I am stuck is working out how much of the Higher band will be required in order to produce the net income result required. I know what the answer is, but I can't spot how to determine it algorithmically. Here are my workings out:

The gross income actually required to produce the desired net income is £6,500 per month. At 40% within the above ranges, this results in an additional tax levy of £1,166.67.

Thus the total increase in gross salary for the additional amount is the sum of:

£3,600.00    Additional amount required
£  533.33    Basic tax amount
£1,166.67    Higher tax amount
£5,300.00    GRAND TOTAL (GROSS ADDITIONAL AMOUNT)

I have verified these figures using an external tax calculator so I believe they are correct.

I wondered if my confusion stems from double-counting the first portion of tax in the Basic band, since the range 916-1200 has already been taxed and written off. However I don't think this value is involved in the totals table above, so that might be a red herring.

Can anyone see where I am getting unstuck?

Edit: one issue I can see is that £1,200 is net, whereas all the other figures are gross. This figure produces a gross figure of around £1,270.80, which produces a Basic band tax of £70.83 on a taxable sum of £354.13. However, I don't understand things well enough to know if this is actually related to my problem. I think this is OK, since if a person receives an amount net of tax, they are interested in how much they wish to earn gross over the existing income.

Answer 1

The solution to this problem was a bit easier than I thought; since the algorithm uses a standard taxation algorithm to discover whether bands can be fully consumed, the remaining calculation only needs to take one more band into account. The loop can then terminate, since no more bands need to be tested.

In my example, the loop consumed the first two bands fully. This is done by examining the upper bound of the band in each case and applying a forward tax calculation (gross to net) to see if this would hit the required net target. Both failed this test, since they did not reach £4800.

We therefore have accrued the following incomes and taxations:

Band          Max        Net income   Tax
Tax free      £  916     £  916       £  0
Basic         £3,583     £2,667     £533.33
         CURRENT TOTAL   £3,049.6   £533.33

The problem therefore boils down to working out what gross increase is required to get from £3,049.6 to £4,800 in a 40% bracket. The gap comes out as:

  £4,800 - £3,049.6
= £1750.4

Apply 40% reverse taxation to get the gross:

= £1750.4 * (1 / (1-0.4))
= £2917.33

The tax due in this band is therefore 40% of that, or £1,166.67.

We can finally expand the table with a new partially-consumed band, applying these tax amounts:

Band          Tax
Tax free      £    0
Basic         £  533.33
Additional    £1,166.67
GRAND TOTAL   £1,700

Thus the gross target income is (£4,800 + £1,700), which is the £6,500 I am after. The cost after £1200 is (£6,500 - £1,200) = £5,300.