# Annuity Method

## Introduction

Compared to other approaches, the annuity method is more suitable for assessing absolute economic efficiency and for comparing various investments with very divergent projected lifetimes. The annuity method is a reliable means of comparing the economic viability of various investment options. It takes into account reinvestments and differences in system mortality.

According to the simplified approach presented here, however, a single cost increase factor is applied for all inputs, i.e. energy, services, spare parts, etc. As we have seen in the course of the past few years, though, the cost of energy and of wares with a close tie-in to the cost of energy (such as chemical fertilizers) has been increasing more rapidly than, say, the national wage index of most countries. Also, discrepancies can always be expected to be particularly pronounced in countries where the state intervenes in the price structure. Thus, if the economic efficiency of a particular system is to be projected with any real degree of accuracy, the price-increase rates for each individual product must be taken into account.

## Calculation

Basically, the annuities method converts the investment into fixed annual costs suitable for direct comparison with the annual benefits.

AN = B - C - I0 CR (i,T) or

AN = R - ANI

where

AN = annuity, i.e. the annual gain, calculated for the first year (year 0)
ANI= annuity of the investment
B = annual benefits (savings and/or returns on investment), calculated for the year 0
C = annual costs, calculated for the year 0
R = annual reflux (R = B - C)
I0 = total initial investment volume, calculated for the year 0
CR = capital recovery factor
i = assumed interest rate (discount rate)
T = projected service life or time required for amortization of the investment

#### Annuity (AN)

The purpose of the annuities calculation is to convert all net payments in connection with an investment project to a series of uniform annual payments - the so-called annuities. Conversion is effected by multiplying the individual payments by the capital recovery factor CF.

AN = ANR - ANI
AnI = I0 CR (i, T)
ANR= R for constant annual benefits

As long as the annuity AN is positive, the project may be regarded as profitable in absolute terms under the postulated conditions. If it is negative, the project must be regarded as unprofitable. The annuity can be equated with the anticipated mean annual profit/loss. It is calculated for the year 0, i.e. the year in which the investment is undertaken.

#### Annual Benefits (B)

The annual benefits comprise the monetarily evaluable returns, savings, etc. yielded by the investment. These may derive from:

• Power generation
Naturally, only the net energy gain can be counted, i.e. the process energy fraction (for agitators, pumps, heating, and any outside energy input) must be subtracted from the total gas yield. If the generated power is sold, the returns are included in the calculation. Any energy used to replace previous outside energy inputs counts as savings.
• The substitution of digested sludge for chemical fertilizers can often yield savings in developing countries, where in the past, much of the material used as substrate has so far not been used as fertilizer. Accurate monetary evaluation is difficult, because the fertilizing effect of digested sludge is substantially influenced by the type of storage, the climate, the techniques employed in spreading the sludge and working it into the soil, etc.
• Savings attributable to the superior properties of digested sludge: These may result from the improved fertilizing effect of the sludge, its hygienization, reduced odor nuisance, and more advantageous handling properties such as reduced viscosity, improved homogeneity, etc. However, it is normally quite difficult to attach a monetary value to such benefits. Legal regulations pertaining, for example, to reducing odors or improving hygiene can be of decisive influence.

#### Annual Costs (C)

The current annual costs are made up of the expenses incurred for:

• maintenance and repair,
• plant operation,
• inspection fees. etc..
• system attendance.

Most such items can only be estimated, whereby 1 - 3 % of the investment volume is generally accepted as rule-of-thumb quota for maintenance and repair. For simple biogas systems in developing countries, the percentage is usually somewhat lower, though it could be even higher for the more complicated types of systems used in industrialized countries.

Operating costs are largely attributable to the depletion of consumables (such as desulfurizer cleaning agents) and to outside energy requirements, e.g. electricity for running agitators and mixers.

Inspection fees usually arise in connection with pressurized biogas systems. (According to German standards, a system is defined as pressurized if it operates on an internal pressure of 1.1 bar, = 0.1 bar gage, or more.)

Expenses in connection with system attendance by the owner-operator himself or by his employees should usually be taken into account, whereby the hourly wage and time expenditure are subject to wide variance.

#### Total Investment Volume (I0)

The total investment volume includes the capital outlay for:

• the digester, including agitating, mixing and heating equipment,
• gas storage and safety provisions,
• gas usage, including integration into existing systems,
• linkage between the biogas system and the farm estate, i.e. Iiquid-manure and gas lines, structural alterations on stabling structures, etc,
• planning, construction supervision, licensing fees, etc.

Reinvestment costs for the replacement of individual components (pumps, floating gas holder, etc.) with service lives that expire prior to the end of the projected system service life T must be included in the total investment volume. For the purposes of this simplified approach, the cost of such reinvestments may be quoted for the year 0:

I = I0 + I1 + I2 + ....

where

I = total investment volume
I0 = initial investment volume
I1, I2, .... = reinvestments

#### Capital Recovery Factor (CR)

CR accounts for the cost of financing a project for which the investment volume has to be raised by way of loans (interest, compound interest) . If the capital outlay is covered by cash funds, CR is used to account for ceasing gain in the form of lost interest and compound interest on assets.

CR is calculated according to the formula:

CR (i,t) = (qt (q-1)) / (qt-1) = ((1+i)t i) / ((1+i)t - 1)

where

q = 1+i
i = assumed interest rate in percent
t = time in years

#### Assumed Interest Rate (i)

The assumed interest rate must be determined with due regard to specific individual conditions. In this context, the assumed interest rate is defined as a real interest rate, i.e. after adjustment for inflation. In the case of cash outlay, the real interest rate would equal the rate of interest that the capital would have borne on the money market. Accordingly, the assumed interest rate is equal to the current mean debt interest rate demanded by the bank for the loan capital, when the entire project is financed with borrowed money. Moreover, money costs in the form of bank service charges, the owner's own administrative overhead, etc. must also be included. Since, however, most projects involve a certain degree of mixed financing, the assumed interest rate will take on a value located somewhere between the debt interest rate and the credit-interest rate, depending on the case situation. (Note: All rates adjusted for inflation!).