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Tax By Mathematics

01 August 2001 / Patrick Morris
Issue: 3818 / Categories:

PATRICK MORRIS takes a mathematical approach to the allocation of losses in order to maximise double tax relief.

PATRICK MORRIS takes a mathematical approach to the allocation of losses in order to maximise double tax relief.

The allocation of losses against sources of income in order to maximise double tax relief for a company is usually left to tax software programs. The tax professional's consideration of the allocation is usually a brief look to make sure that the losses are being allocated against the income with the lowest rate of overseas tax first. This article considers the mathematical derivation of the ideal amount to allocate in a 'reasonably' mathematically rigorous manner.

The article discusses the impact of advance corporation tax on the calculations. This is still potentially relevant to many companies, although the shadow advance corporation tax rules may prevent the actual use of any surplus advance corporation tax brought forward against a tax liability.

From the point of view of the legislation, most losses (including trade and non-trade charges, management expenses, losses on unquoted shares) can be set off against any kind of income. The rules about the order of these offsets are not considered here as it makes little difference to the derivation of the formulae.

The main kinds of losses that cannot be offset against any kind of income are as follows:

  • Brought forward trading losses can only be offset against profits of the same trade in general. Therefore if one of the client's trades has an overseas branch (or branches) then the losses can be allocated in order to maximise the double tax relief available.
  • Brought forward capital losses can only be offset against capital gains of the period.
  • Non trading loan deficits offset against section 83(2)(d), Finance Act 1996 can only be offset against non trading income. These losses can therefore only be allocated against non trading income for double tax relief purposes.

The article ignores all of these complexities and assumes that the losses under consideration can be offset against any kind of income. In addition I have assumed that there is only one kind of income source whereas in reality the income sources will be split between many kinds of income (Schedule D, Case I, Schedule A, etc.).


A single United Kingdom company, Double Tax Relief Ltd has n sources of income S1 to Sn. In addition, the company has £L of tax losses which it can offset against the sources of income in any way it sees fit.

Each source of income Si has an amount of overseas tax associated with it Oi such that Oi is greater than or equal to £0.

Thus the taxable profit of the company is P where P is defined as the sum of the income sources Si less the losses to be allocated L.

The United Kingdom tax on this profit P is Tu. This is calculated in the normal way and therefore the average United Kingdom tax rate for the company, Ru, is defined as Tu/P providing P is not zero.

The rate of advance corporation tax is defined as Ra, and in many cases is likely to be 20 per cent.


The derivation of the formulae is based on the three following cases which would be applied to a source of income in turn:


Step 1: Allocate losses to preserve advance corporation tax and double tax relief.
Step 2:
Allocate losses to preserve double tax relief.
Step 3:
Allocate remaining losses.

Thus each source of income is considered for step 1 above. If there are any more losses unallocated then step 2 is undertaken for each source. If there are more losses then step 3 is undertaken.

Step 1: Allocate losses in order to preserve double taxation relief and advance corporation tax

While this may no longer be necessary for most companies now, there are still some companies with advance corporation tax brought forward to be used, so step 1 is still necessary when considering the generic case.

It is necessary here to consider the optimal amount of losses to allocate Li which would be where both the advance corporation tax capacity and the double taxation relief are preserved. This means that losses are allocated so that the full amount of advance corporation tax can be offset on the reduced profits while not wasting any double taxation relief.

Symbolically this is expressed as follows:

(Si – Li) Ra = (Si - Li) Ru – Oi
(Si - Li)(Ra - Ru) = -Oi
Si – Li = -Oi/(Ra – Ru)
Si – Li = Oi/(Ru – Ra)
Li = Si – Oi/(Ru-Ra)

In words this means that the ideal allocation of losses is equal to the taxable amount of the income source less the overseas tax divided by the difference between the rate of United Kingdom corporation tax and the rate of advance corporation tax (or the modern equivalent). By way of example, consider a source of income of £10,000 with overseas tax of £500. The average tax rate in the United Kingdom on the company's profits is 30 per cent and the advance corporation tax rate is 20 per cent. Therefore the ideal amount of losses to allocate to this source of income is as follows:

Li = 10,000 – 500/(0.3 – 0.2) = 5000

When calculating the tax due on this source after the above loss allocation, the outstanding tax liability is £1,000, which is exactly the capacity for advance corporation tax offset on the income after the loss allocation (i.e. advance corporation tax capacity = £5,000 x 20 per cent = £1,000). Therefore providing there is at least £1,000 of advance corporation tax to use (and assuming there is no restriction on its use) there will be no remaining tax liability on this income source.

As an aside we have the requirement that 0 <= Li <= Si (i.e. allocate a loss that is negative or is more than the income cannot be allocated). From this the following further condition is derived:

Li = Si – Oi/(Ru –Ra) and Li >= 0
0 <= Si – Oi/(Ru – Ra)
Issue: 3818 / Categories:
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