# Calculations with return on capital

In addition to the calculations described above without return on capital, investors usually expect payments of interest for their investment of the order of usual paid rates for comparable investments. Starting with the payment A0

in the initial year, the capital cn after a time of n years, with the interest rate ir, can be estimated using compound interest formulae. With q = 1 + ir, the expression becomes:

For an initial capital of A0 = €6500 (following the example of the photovoltaic system in the section headed 'Costs for a photovoltaic system', p237) invested with an interest rate of ir = 6 per cent = 0.06 over a time of n = 25 years, the capital after 25 years including interest becomes c25 = €6500 • (1 + 0.06)25 = €27,897

Later payments yield interest as well, though over a lower term. If the investor pays another €1500 after 10 years (for a replacement inverter), the term reduces to 15 years. The capital after 25 years would then be:

c25 = A0 • q25 + A10 • q15 = €6500 • 1.0625 + €1500 • 1.0615 = €31,492

The following formula describes this in general:

If all payments Ai in the different years i are the same, the geometric progression provides the following formula:

For later payments it is necessary to calculate the additional capital that would have to be invested at the beginning with the interest rate ir so that the capital at the end of the investment period is the same.

The payment Ai in the year i can be discounted to the initial year:

Thus, the respective initial capital for a payment of €1500 after 10 years for the example above becomes:

In other words, if 6 per cent interest is paid on €838 over 25 years, the final capital is the same as if €1500 was invested after 10 years, bearing 6 per cent interest over 15 years.

The following equation provides the discounting of several such payments at different times:

If the payments Ai in the different years i are all the same, it becomes:

W-DV

Then, the capital after n years can be calculated again with the equation for compound interest:

When investing in technical systems for energy conversion, there is likely to be little capital left that can be repaid after the end of its operating life in n years. On the contrary, most end-of-life systems are in a poor state of repair and therefore worthless. Selling the converted energy of the system yields income for the repayments of the invested capital during the operating time. Hence, it is now possible to calculate the price at which a unit of energy must be sold so that the investor gets the required rate of interest.

The sales return also yields interest. If the investor gets a repayment as early as the beginning of the first operating year, he can reinvest this amount with payments of interest over the whole operating time. The interest period decreases for later repayments. The initial capital c0 is calculated as described above with the initial payment A0 and the payments Ai in the following years discounted to the initial year. These payments are compared with the income B-r They must also be discounted to the initial year. For simplification, payments and income within a year are treated as if they were made at the end of the year. For an operating period of n years the calculations become:

NPV is called the net present value; it must be greater than or equal to zero if the investment is not to yield losses.

In the following, it is assumed that the income B at the end of each year is the same. The size of the required annual income can be estimated if the equation of the net present value is solved for B and the NPV is set to zero. With the annuity factor, a, and:

Table 6.5 Annuity Factors a for Various Interest Rates ir and Interest Periods n

Interest rate (discount rate) ir = q - 1

Table 6.5 Annuity Factors a for Various Interest Rates ir and Interest Periods n

Interest rate (discount rate) ir = q - 1

 n 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 10 0.1056 0.1113 0.1172 0.1233 0.1295 0.1359 0.1424 0.1490 0.1558 0 1627 15 0.0721 0.0778 0.0838 0.0899 0.0963 0.1030 0.1098 0.1168 0.1241 0 1315 20 0.0554 0.0612 0.0672 0.0736 0.0802 0.0872 0.0944 0.1019 0.1095 0 1175 25 0.0454 0.0512 0.0574 0.0640 0.0710 0.0782 0.0858 0.0937 0.1018 0 1102 30 0.0387 0.0446 0.0510 0.0578 0.0651 0.0726 0.0806 0.0888 0.0973 0 The annuity factor a is: Table 6.5 shows the annuity factors a for different interest periods n and interest rates ir. The annuity factor is sometimes simply called annuity and the interest rate is also known as the discount rate. With the annuity factor a and the annual generated amount of energy Ea the levelled cost for one unit of energy cE is easy to calculate: The level of the interest rate depends on the risk of the investment. Risks associated with renewable energy systems can include the overestimation of the available renewable energy resource such as solar irradiation or wind energy at one location, unforeseen technical troubles or changing legal conditions. Since these risks are usually higher than those of a typical bank account, the discount rates are also higher. However, it is nearly impossible to make reliable calculations for the level of the discount rate. In the end, the market with all its moods and the subjective feelings of investors define the discount rate. The calculations above can be extended by applying price increases on operating and maintenance costs or fuel costs for conventional energy systems. However, statements on price increases over long investment periods are very difficult to make. Unforeseen events such as crises in oil-producing regions or shortages of fossil energy resources can change future fuel or raw material prices significantly. Therefore, statements on future fuel prices or operating and maintenance costs are not given here; instead the final sections give a critical overview of conventional interest calculations. 