# 3.2 General Applications

This section uses simple examples to illustrate the core principles. It also contains a slightly more complex example related to cost recovery in project-based industries, such as oil and gas. The next section covers profit-sharing in private equity, which is another frequent area of application that has specific features which also add some complexity.

#### Example (I): Single Period Tax Calculations

The core principle of waterfalls as an allocation mechanism can be illustrated in the context of income tax calculations. Typically, for personal taxation, the tax rate varies according to the level of income. For example, in the following image, the rate is 5% on the first \$15000 of income, 20% on the next \$20000, 40% on the next \$30000, and 60% thereafter:

That is, a person with an income of \$25000, would pay 5% tax on the first \$15000 (\$750), and 20% tax on the remaining \$10000 (or \$2000), to give \$2750 tax in total, or an average tax rate of 11% (i.e. \$2750/\$25000).

To perform such calculations, one first needs therefore to allocate income to the bands. For example, with an income of \$38000, the first two bands (\$15000 and \$20000) would be full, with \$3000 being in the third band:

The formula required to do this must keep track of the absolute value at which a specific band starts (i.e. the cumulative capacity of all previous bands). For example, in cell E6 (formula shown in cell I6), the amount of income in the second band is based on taking the minimum of:

• The total income less the amount that would have been allocated to earlier bands.
• The capacity of the band.

Once the allocation is done, the tax calculation for each band is a simple multiplication (income in that band, times the tax rate), and the tax is a simple addition of the results:

Whilst straightforward in principle, the set of calculations required can become quite large in more general cases.

For example, if one wishes to create a model with similar calculations in which income will develop over time (to be shown horizontally), then one may first restructure the above calculations to have the blocks underneath each other:

… and then complete the data, such as in the following where the income grows over time (row 2), as do the tax bands (rows4-8), and the tax rates could also change (rows 22-26):

Note that whilst already more complex – simply due the size of the tabular structures – the calculations are relatively straightforward: Essentially each column is independent of the others, so taht the model is mainly a single column that has been copied several times. In more complex cases, there may be relationships required across time (i.e. between the columns), which would complicate the calculations further.

#### Application: Interest Charges Based on Debt Tranches

The same principle can be applied to the calculation of interest charges on a “tranched” loan. This is where a borrower can draw down (i.e. use) from a pre-arranged lending facility, which is structured so that the interest rate increases each time that a borrowing threshold is crossed. The higher interest charges on the higher borrowing are to compensate the lender for the additional risk of lending in these latter tranches, as – in the case of default by the borrower or shortfall in repayment – the lower tranches are to be repaid first.

The nature of the calculations is essentially the same as in the above example, but where the terminology would be modified:

• The taxable income used earlier would be replaced by the loan amount.
• The capacity of each tax band would be replaced by the size of the loan facility for each tranche.
• The tax rates would be replaced by interest rates, and the tax amount by interest charges.

As there is such a direct comparison in principle, this is not illustrated further here.

#### Example (II): General Profit Splitting

In some contexts, the way that profits of a project or investment are to be split between two between two partners depends on the level of profits that arise. For example, in private equity, deals are sourced and arranged by a project sponsor (initiator, or General Partner, “GP”). The GP will put in a relatively small proportion of the equity capital, with the majority of the equity provided by several external partners (Limited Partners, or “LPs”). In addition, the GP will take a higher share of the profits if the project is highly profitable, whilst at lower levels of profitability, the profits will be shared more equally.

For example, the following shows the parameters for a case where the GP provides 10% of the capital, whilst its share of the profits increases from 10% through to 40% depending on the level of profits (defined by three profits hurdles)

(Note that the band size defined is the capacity of the band (not the threshold values, which are the cumulated figures i.e. \$20m, \$50m, and \$60m.)

Then, for any level of profit that arises in the project this could be split across the bands. For example, a profit of \$55, would fil the first two bands, as well as half of the third band:

… and the share of each band could then be calculated for each party according to the split:

In other words, the \$55m profit would be shared into \$7.5 for the GP (a 13.6% share), and \$47.5m for the LPs (86.4% share). Note that the profit split is higehr than the split of capital that was provided (i.e of 10% and 90% respectively).

In practice, the incentive schemes used are more complex than this, as the band sizes are calculated using target returns based on the capital that each party provides, even as the principle of allocation remains the same. An example of a realistic situation is given in the next section.

#### Example (III): Revenue Sharing Agreements with Basic Cost Recovery

In extraction industries (such as oil, gas, or mining), the resources are often owned by governments, whereas the discovery and development of the operational fields are done by the private sector. In such cases, the government may wish to retain a share of the profits, whilst the private investor may agree to invest only if they are able to retain a larger share of the first profits made, and take a smaller share of profits once its investment costs have been recovered.

For example, a project could have a profile for its margin (revenues less operating costs), investment cost, and cumulated investment cost as follows:

… whilst the investor will retain a large percentage (100% in this example) of the maring until its investments have been recovered, and a smaller share (40% in this example) thereafter:

In each year, the investor’s share is one or the other figure depending on whether it has recovered its cumulated investment or not (through its accumulated margin). In the example case, the investor has a 100% share for the first four years (see row 17), which then reduces to 40%, before briefly rising again to 100% in the middle of a second investment phase, and then returning to the 40% level at the end. The percentage to use is determined by whether the investor’s margin, cumulated from the beginning) is more than the investment amount (accumulated from the beginning).

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