This article is based on a model which Deloitte Haskins & Sells (now part of PricewaterhouseCoopers) developed for Woolworths plc with technical assistance from Eudoxus Systems Ltd. It is said that there are two things you cannot escape: death and taxes. Most of us do, however, go through our lives without experiencing Capital Gains Tax (CGT). Such is the ferocious complexity of the tax that this can only be a good thing, unless you are an out-of-work accountant. Companies, however, are not so fortunate. In the ordinary course of their business they acquire fixed assets: premises; plant; equipment, etc. Over time their business will inevitably change and they will sell some assets and buy some new ones. When they do, Capital Gains Tax is payable on any gains which they make. In most cases, companies only sell a handful of assets in a year which are worth more than they paid for them. When this happens, the CGT calculations, although onerous, are straightforward. Where, however, a company sells a large number of assets with capital gains, the problem becomes much more complicated. This is precisely the problem which Woolworths faced. They had owned shops up and down the land, most of them freeholds dating from the first half of the century. They were rationalising their stores, closing small ones, opening new branches in shopping centres and developing new chains such as B&Q and Comet. These transactions gave rise to a huge number of sales with capital gains, and a correspondingly large number of capital investments. Capital Gains Tax In principle, Capital Gains Tax is payable on the real gain in value of the item which has been sold, i.e. the gain after allowing for inflation. In practice there are a number of complications, particularly with older assets, but it suffices to suppose that for each asset we can identify a date when the asset was in effect acquired and its cost at that date. Tax is charged at a flat rate on the real gain, i.e. independent of the magnitude of the gain. The rate may vary from year to year with changes in taxation but had been stable at 30% for many years when this work was carried out. CGT Reliefs We now come to an unusual feature of Capital Gains Tax which distinguishes it from Income Tax. You do not need to pay Capital Gains Tax if you have spent the proceeds from the sale on buying other assets. This is not magnanimity by the Inland Revenue. It simply reflects economic reality. It would stifle economic activity if companies were charged tax every time they sought to replace their old assets with new ones of equal value. This relief from paying CGT is provided in two forms:
Roll-over relief applies where the new assets which are acquired are "non-wasting", e.g. freehold land and buildings, long leases. In this case the new assets can be acquired at any time between one year before and three years after the old asset was sold. The tax liability arising from the sale of the old asset is "rolled into" the new asset and is thereby deferred for so long as the new asset is retained. If the new asset is subsequently sold, the rolled-up CGT becomes payable, but it can be deferred again by buying another non-wasting asset. Hold-over relief is similar to roll-over relief and applies where the new assets which are acquired are "wasting", e.g. short leaseholds, fixtures and fittings. It differs from roll-over relief in that it lasts a maximum of 10 years, i.e. the CGT from the sale of the old asset becomes payable 10 years after the purchase of the wasting asset for which hold-over relief is claimed. Those 10 years' grace are, however, very valuable, both because of the improved cash flow and because the CGT can be deferred indefinitely by "rolling the gain" into non-wasting assets purchased during those 10 years. Opportunities for Optimization We can now begin to see why CGT gives rise to an optimization problem. If two assets are sold, each for £100,000, different amounts of CGT will be payable depending on the effective dates when the assets were acquired and their purchase costs. We may calculate the effective tax rate of the assets as the CGT payable divided by the proceeds realised. If we now buy a non-wasting asset for £100,000, we must obviously claim the roll-over relief which is available. But which asset's proceeds have been "rolled into" the new asset? The answer is: it's up to us to decide. A moment's thought will show that we should choose the asset with the higher effective tax rate. Now multiply this problem many times over, with hundreds of disposals and purchases of non-wasting assets over a number of years. Add in the opportunities provided by hold-over relief where wasting assets have been bought. Finally, complicate matters further by taking account of interest charges, both on unpaid tax liabilities (e.g. where proceeds are held for 3 years before being invested) and on cash within the business. The result is a problem which is genuinely complex. We must allocate the proceeds from disposals into the available purchases of wasting and non-wasting assets in accordance with the rules. There is an enormous number of ways of doing this. Different sets of allocations have different outcomes in terms of the CGT which is payable and the dates when it becomes due. We thus have all the ingredients of a Mathematical Programming problem:
Building the Model A number of issues arose with the model which are typical of practical applications:
If each transaction had been represented individually, the model would have been impracticably large. Disposals in each year were therefore grouped into bands with similar effective tax rates, e.g. effective tax rate between 18% and 20%. The model treated each group of transactions as a homogeneous lump and could allocate any quantity of proceeds from the group as it chose. Another type of aggregation which was used was to collect together all transactions of the same effective tax band which occurred in a year. The time window during which proceeds can be reinvested runs from 1 year before the date of disposal to 3 years after, so this form of aggregation could generate some spurious allocations and fail to make use of others which were actually possible. It might be thought that as CGT is payable on past events, the data should be certain. But this is not so. The opportunities for claiming relief depend largely on what will happen in future years, so estimates are required. Furthermore, the "costs" of assets which have been sold may themselves be estimates which need to be negotiated with Inland Revenue valuers. Another area of uncertainty concerned interest rates and the tax regime. Interest rates are volatile and the values predicted for future years have a significant effect on the objective function. Changes to the tax regime are less predictable but can be as significant. All in all, therefore, the uncertainties in the data meant that any model could only be an aid to decision-making and not the complete solution. Formulating the Problem At first sight, the rules for Capital Gains Tax might appear to require the use of Integer Programming to capture the way in which money can be held for up to 3 years before reinvestment and hold-over relief lasts 10 years. But an entirely linear formulation sufficed. The key concept is illustrated in Figure 1, as it related to proceeds held for reinvestment. Figure 1: Cascade of Proceeds into Investment We define four separate "pots" of money for each effective tax rate band for each year:
As we roll forward from one year to the next, money which has not been invested during the year cascades from one of these pots to the next, e.g. from the "this year's disposals" pot to the "last year's disposals" pot. If the money hasn't been invested within the required 3 years it then cascades out of the final pot and becomes a tax liability. Using the Results The mathematical formulation ensures that the rules about reinvestment are followed (at least within the limits of the aggregation of the data). But note that when reinvestments are made, the model merely identifies the "age" of the money which is used and its effective tax rate band. Both because of this and because of the aggregation of the data, the results could not be used directly in preparing the tax statement for the Inland Revenue. Instead, a post-solution heuristic was used to prepare detailed allocations of individual disposals to individual investments and compute the tax which was due. Thus the Mathematical Programming model played a substantial part in the system but was not the complete solution. Related articles include Data Envelopment Analysis and its Use in Banking To find other articles, refer to the MP in Action page. |
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