## Profitability terms and Formulas for calculation

### Net present value (NPV)

The net present value is determined using the internationally renowned net present value method. It adds all incomes and expenditures of a facility, discounted by the interest rate. The cash flows of the individual years are discounted or written down to the present day with the internal rate of return (which is expressing the cost of capital). The result is the so-called present value of the payments. The net capital value expresses what would have to be paid today as a lump sum for the purchase and the upkeep of the facility for the period under observation, netted and without the cost of capital, and vice versa what the income would be as of today. This single amount is a possible form to express the profitability of a facility. It can be judged by its absolute height or compared with alternative investments.

**Formulas:**

One-off receipts:

Receipts year 0 = one-off subsidies + one-off tax relief + miscellaneous one-off receipts + loan

\(B_{0} = B_{For,e} + B_{div,e} + B_{k}\)

Running receipts:

Yearly receipts = Energy sales * energy sales price * energy price increase + running subsidies + running miscellaneous receipts + thermal energy sales * energy price increase

\(B_{j} = E_{teg}*B_{En}*\left( 1 + e_{En} \right)^{j} + B_{For,\ c} + B_{div,\ \ c} + B_{En,\ \ therm\ }*\left( 1 + e_{En} \right)^{j}\)

One-off expenses:

Plant costs = Investment costs for construction

\(I_{0} = \sum_{k = 1}^{Number\ comp.}I_{k}\)

One-off expenses = Plant costs + loan repayment + miscellaneous one-off expenses

\(A_{0} = I_{0} + A_{k} + A_{div,e}\)

Running expenses:

Replacement rate = Period under consideration / lifetime

\(i_{m,k} = \left\lfloor \frac{n_{BP} – 1}{{\ n}_{tot,k}} \right\rfloor\) [1]

Replacement decision on whether a component must be replaced (0 = no, 1 = yes)

\(a_{k,j} = left{ begin{matrix} 0, (j – 1) mod n_{tot,k} neq 0 \ 1, (j – 1) mod n_{tot,k} = 0 \ end{matrix} right. \) [2]

Replacement investment expenses per year = Investment * replacement decision * price change replacement

\(I_{m,j} = \sum_{k = 1}^{Number\ Comp.}I_{k}*a_{k,j}*\left( 1 + e_{m} \right)^{j}\)

Yearly expense = Maintenance costs + fees + replacement of components per year + energy bought * energy price * energy price increase + maximum energy demand * price per kilowatt * energy price increase + miscellaneous expenses

\(A_{j} = A_{OM} + A_{G}\ + I_{m,j} + {(E}_{eaux} + E_{Par})*A_{En}*{(1 + e_{En})}^{j} + max{(E}_{teg})*A_{P}*{(1 + e_{En})}^{j} + A_{div,c}\)

Remaining value:

Remaining value = Investment expenses * price change replacement * linear depreciation

\(W = (\ \sum_{k = 1}^{Number\ comp.}I_{k}*(1 + {e_{m})}^{i_{m,k}*n_{tot,k}}*\frac{\left( i_{m,k} + 1 \right)*n_{tot,k} – n_{BP}}{n_{tot,k}})\)

Net present value:

\(NPV = \ B_{0} – A_{0} + W*\left( 1 + i_{i} \right)^{{- n}_{BP}}*\left( 1 + i_{r} \right)^{{- n}_{BP}}\ + \sum_{j = 1}^{n_{BP}}{(\left( B_{j} – A_{j} \right)*\left( 1 + i_{i} \right)^{- j} – A_{kr,j})*\left( 1 + i_{r} \right)^{- j}}\)

The loan interest costs \(A_{kr}\) are not indexed with inflation. More information can be found in the chapter „Profitability terms”.

## Interest rates

### Effective interest rate / Interest rate for cost of capital

The internal rate of return is used to make future cash flows, occurring at different times in the future, comparable when it comes to interest. Future income and expenses are discounted or written down using this internal rate of return to the present day. The internal rate of return expresses the cost of capital or, in projects with a net income surplus, the expected return on capital.

*Internal rate of return (IRR)*

The “internal interest rate”-method, internationally known as the method of effective interest or Internal Rate of Return (IRR), expresses the profitability of an investment as the mathematically correct determined yield of all cash flows of the investment in the period under observation. This yield can be compared in its absolute height or in comparison to alternative investments. As opposed to the net capital value method, which has a fixed interest rate (cost of capital rate) used to determine the net present value of all future cash flows, the IRR expresses the interest rate at which the net capital value equals zero. An IRR can sensibly only be calculated if during the period under observation the sum of all income is bigger than the sum of all expenses – only then one can talk of a return on capital.

The IRR makes it possible to calculate a (theoretical) average yearly return for an investment with changing and irregular incomes. If the IRR is negative, it means that the sum of the financial income is smaller than the capital invested.

For this, the NPV is set as zero. The formula is solved for i_{r}. The solution for i_{r} is found through iteration and equals the IRR.

More information can be found in the chapter „Profitability terms”.

### Payback period

The determination of the payback period is done in a cumulative way. Beginning with the year of the first payment, present values of the yearly net payments are added up, until the sum reaches zero or a positive amount for the first time. The point in time when the NPV is zero or positive for the first time is the payback period.

Depending on the settings the NPV can reach the value zero several times. If this situation occurs because of the entries made, the result will be presented with a remark pointing out that effect. In this case, the payback period is not unequivocal.

### Dynamic payback period

The simple payback period accumulates all paybacks from the time the investment was made nominally, until the (theoretical) balance is back to zero. The dynamic payback period used in the tool accumulates all paybacks from the time the investment was made as present values, until the net capital value is back to zero. In other words, the dynamic payback period expresses the time period an investment needs to reach the interest rate used in the calculation of the net values.

### Annuities

**Formulas:**

\(AF = \frac{\left( 1 + i_{r} \right)^{n_{BP}}*i_{r}}{\left( 1 + i_{r} \right)^{n_{BP}} – 1}\)

Annuity = Capital value * annuity factor (AF)

\(A_{A} = NPV*AF\)

### Inflation

Inflation describes the general and prolonged rise in the prices for goods, equalling at the same time the fall in the purchasing power of money. Inflation is usually expressed by yearly price changes of goods found in a hypothetical basket of goods. How can the expected future inflation incorporated in the cash flows of a simulation in Polysun?

If you expect continued low inflation, we urgently advise you not to inflate future costs and incomes for the following reasons:

The effects on prices of non-inflationary causes like supply and demand in case of energy prices, or the life-cycle-caused price changes in case of electrical apparatuses, are much stronger than the influence of inflation.

At low inflation rates, the economic subjects fall victim to the so-called money illusion. This means that inflation is so low and restricted to limited groups of goods that decisions are made based on nominal prices.

Long-term predictions of the future inflation rate of an economic area are only possible with a considerable amount of uncertainty.

As of 2016, the inflation rate is considered to remain low for the considerable future in the Swiss Franc- and Euro-Area. If expected inflation rates are high (multi-year average of above 5 % per year), it may make sense to inflate future prices and costs in the tool, as that will also mean that the money illusion of the economic subjects will be destroyed and the mechanisms of automatic indexing are starting to grip again. It is for this case that the tool offers the possibility to inflate future incomes and costs.

In the results (as in the reports), all values are shown nominally (including inflation). This might lead to some confusion. This is especially true if, for example, the same amount for yearly maintenance costs is entered manually, and the total for maintenance costs does not equal the multiple of the values entered: inflation is the reason for this deviation. Based on this change, however, it is possible to gauge the effect of inflation.

### Cost of capital

The calculation of the economic advantageousness of investment projects is generally based on the total capital approach, as the economic performance of a facility is not dependent upon how it is financed. The capital cost rate or internal rate of return must therefore be a weighted, average interest rate derived from debt and own capital.[3]

If the economic advantageousness of a partially debt-financed investment is to be calculated, all income and costs related to the debt have to be considered (loan payout = income, loan payback = cost, interest payment = cost). The income and cost of a loan are considered by the tool if a loan is defined (interest rate, loan amount, duration). The net cash flows then only have to be funded by own capital. The internal rate of return would then have to equal the own capital cost rate or – in case of projects with a net surplus of income – the expected interest on own capital.

In case of a loan (like a bank loan), the tool automatically considers the payback and calculates the corresponding cash flows. In the year zero the payout of the loan is considered to be income and the subsequent payback as cost. Both add up to zero. The loan payback payments are shown in the charts depicting the cash flows. The payback and the interest on loans are not subject to inflation.

### Energy production costs

The energy production costs mirror the cost for a usable, provided energy unit. In ist calculation, the calculation method Levelized Cost of Energy (LCOE) is used. It determines the running costs per year under consideration, adds them up, discounts them with internal rate of return for the referenced point in time and adds the one-off costs. The discounted costs are then divided by the usable, provided energy units discounted for the referenced point in time. The LCOE is a common method used in the industry to calculate energy production costs. The formula can be used to calculate the production costs of all kinds of energy, be in thermal or electric.

As both the cost and the amount of energy produced during the period under observation are discounted to a referenced point in time, the comparability of energy production cost is given. The energy production costs provide comparable prices based on costs, not in dependence of feed-in tariffs. The value of the individual amounts of energy will not be determined separately, as it does not have to be.

**Formula:**

Energy production costs = discounted costs / discounted energy produced

\(G = \frac{A_{0} – W*\left( 1 + i_{r} \right)^{{- n}_{BP}}\left( 1 + i_{i} \right)^{{- n}_{BP}}\ + \sum_{j = 1}^{n_{BP}}{\left(A_{j}*\left( 1 + i_{i} \right)^{- j} + A_{kr,j} \right)*\left( 1 + i_{r} \right)^{- j}}}{\sum_{j = 1}^{n_{BP}}{\left( Q_{use,j} + Q_{inv,j} -Q_{Loss\ of\ energetic\ yield\ PV,j} \right)*\left( 1 + i_{r} \right)^{- j}}} \)

In calculating the share oft he combined cost of energy production, the remaining value of the replacement investments is subtracted, interest costs on loans are considered under miscellaneous.

More information can be found in the chapter „Profitability terms”.

### Credit costs

The credit costs (borrowing costs) mirror the interest costs of a loan at the nominal rate (inflation is not factored in). Depending on the payback, the total borrowing costs vary. For the payback of the loan, you can choose from three options:

Option 1: Type annuity with a constant payment. The sum of the payment equals the declining share of interest payments and the growing share of paybacks.

**Formulas:**

\(Annuity\ = \ Credit\ sum\ *\ \frac{(1 + Credit\ interest\ rate)^{Credit\ duration}*Credit\ interest\ rate}{(1 + Credit\ interest\ rate)^{Credit\ duration} – 1}\)

\(Payback\ rate\ year\ 1\ {(T}_{1}) = \ Annuity\ – \ Credit\ sum\ *\ Credit\ interest\ rate\)

\(Payback\ rate\ (T_{t}) = \ T_{1}*{(1 + Credit\ interest\ rate)}^{Credit\ duration\ – \ 1}\)

\(Yearly\ interest\ costs = \ Annuity\ – \ T_{t}\)

Option 2: Constant partial amortization (payback) and decreasing interest payments

**Formulas:**

\(Payback\ = \ \frac{Credit\ sum\ }{Credit\ duration}\)

\(Interest\ costs\ = \ Credit\ interets\ rate\ *(Credit\ sum\ – \ \sum_{}^{}{interest\ payments})\)

Option 3: Fixed loan (bullet loan) with constant, yearly interest payments

**Formulas:**

\(Interest\ costs\ = \ Credit\ interest\ rate\ *Credit\ sum\)

You should choose the option that resembles your own credit repayment conditions most. In general practice, the annuity option is quite common.

### Replacement investments and technical life expectancy

The technical life expectancy’s main purpose is to determine the costs for components having to be replaced for the period under observation, thus making the consideration of a life cycle analysis in the calculation possible. If, for example, one component has a life expectancy of half the period under observation, it is assumed that this component has to be replaced once during the period under observation. The investment costs for this component double accordingly. Is the (remaining) life expectancy in years longer than the (remaining) period under observation, a remaining value remains after the end of the period under observation, which is then deducted. The wear and tear of the components is assumed to be linear. Price changes taking place from the beginning of the period under observation until the time the component has to be replaced are considered with a factor. This factor is meant to reflect the learning curve for purchase prices and can be entered at will. It should be entered after talks with the constructors or picked from a trustworthy source. If the factor is positive, components are considered to increase in price over time and vice versa.

### Automatised sensitivity analysis

The exactitude of the result of a profitability analysis is highly dependent on the reliability of the values entered. Small variations in these values can have huge effects on the result. It is thus recommended to conduct a sensitivity analysis with a view to the final result for those values entered with a large effect. This means the profitability calculation is done with the upper and the lower extreme value of this parameter, while the other parameters stay at their original value. The sensitivity analysis helps you to judge risks and possibilities of a project.

### Cost savings through PV

Energy yield from PV modules is calculated with the current electricity rates in order to see clearly how much does the electricity generated from photovoltaic modules would cost, if it were consumed from electricity grid. This number shows how much money the owner would have to pay without installing a PV system.

Cost savings through PV are calculated in a following way:

\(Q_{inv}(hour) – E_{teg}(hour)*SupplyTariff(hour) + E_{teg}(hour)*FeedInTariff(hour)\)

### More profitable than reference from year

Payback period comparison for PV systems shows in which year the payback is reached in comparison to a reference system. In other words, in which year the system with renewable energy fraction becomes more profitable than the reference system. Dashes appear instead of a number of years, if the new system is less profitable in comparison with the reference one. On the example shown in the figure the system Nr. 3 (red line) is less profitable as the reference (black line), therefore the hyphen is shown. The system Nr. 2 (green line) reaches payback in 9 years.

[1] The rounding function (Floor function) rounds down to the closest integer number.

[2] The mod function delivers the value after the decimal.

[3] Own capital also costs interest, namely the opportunity costs: the income foregone that could have been made if another, alternative investment had been made.