Which of the following equations can be used in certain situations to calculate the payback period?

2.3.2 Payback Period

PBP is defined by calculating the time needed (usually expressed in years) to recover an investment. Thus, a break-even point of investment is determined. The PBP of a certain safety investment is a possible determinant of whether to proceed with the safety project, because longer PBPs are typically not desirable for some companies. It should be noted that PBP ignores any benefits that occur after the determined time period and does not measure profitability. Moreover, neither time value of money nor opportunity costs are taken into account in the concept. PBP may be calculated as the cost of safety investment divided by the annual benefit inflows.

It is worth noting that PBP calculation uses cash flows, not the net income. PBP simply computes how fast a company will recover its cash investment.

Payback Period (PBP)

Payback period is widely used when long-term cash flows are difficult to forecast, because no information is required beyond the break-even point. It may be used for preliminary evaluation or as a project screening device for high risk projects in times of uncertainty. Payback period is usually measured as the time from the start of production to recovery of the capital investment. The payback period is the time taken for the cumulative net cash flow from start-up of the plant to equal the depreciable fixed capital investment (CFC – S). It is the value of t that satisfies the equation

(9.2)∑t=0t=(PBP) CCF=(CFC−S)

where CCF = net annual cash flow

CFC = fixed capital cost

S = salvage value

Figure 9-1 shows the cumulative cash flow diagram for a project. The PBP is the time that elapses from the start of the project, A, to the break-even point, E, where the rising part of the curve passes the zero cash position line. The PBP thus measures the time required for the cumulative project investment and other expenditure to be balanced by the cumulative income.

Payback Period

Payback period (PBP) is widely used when long-term cash flows, that is, over a period of several years, are difficult to forecast, since no information is required beyond the breakeven point. It may be used for preliminary evaluation or as a project-screening device for high-risk projects in times of financial uncertainty. Payback period is usually measured as the time from the start of production to recovery of the capital investment. The payback period is the time taken for the cumulative net cash flow from the start-up of the plant to equal the depreciable fixed capital investment (CFC–S). It is the value of t that satisfies the equation

(2-27)∑t=0t =(PBP)CCF=(CFC−S)

where

CCF = net annual cash flow

CFC = fixed capital cost

s = salvage value.

Figure 2-4 shows the cumulative cash flow diagram for a project. The PBP is the time that elapses from the start of the project A, to the breakeven point E, where the rising part of the curve passes the zero cash position line. The PBP thus measures the time required for the cumulative project investment and other expenditure to be balanced by the cumulative income.

EXAMPLE 2-4

Consider the following cash flow:

Here, the cumulative cash flow at the end of the second year is $3000 + $4000 = $7000, which is less than the initial investment, but the cumulative cash flow at the end of the third year is $3000 + $4000 + $5000 = $12,000, which is more than the initial investment. The payback period is thus between two and three years, assuming that the cash flow of year 3 is received uniformly throughout the year. The payback period is calculated as follows:

In this example, 0.2 is the fraction of year number 3 that it will take to recover $1000. Adding this fraction to the two years during which $7000 is recovered yields a payback period of 2 + 0.2 = 2.2 years.

The payback period method of evaluating investments has a number of flaws and is inferior to other methods. A major disadvantage is that after the payback period, all the cash flows are completely ignored. It also ignores the timing of the cash flows within the payback period.

22.6.1 Payback period method

Payback period is defined as the number of years required to recover the original cash investment. In other words, it is the period of time at the end of which a machine, facility, or other investment has produced sufficient net revenue to recover its investment costs. This is further explained in Chapter 16.

If P is the payback period in number of years, Ci and Cs are the initial cash investment and the final scrap value at the end of the period, respectively, and Ca is the average annual cash flow, then:

P=Ci−CsCa

If the payback period calculated as above is less than the minimum acceptable then the decision should be to procure the new equipment.

6.4.1.4.2 Payback time

The payback period is an indicator used to assess the short- and long-term benefits of the proposed energy systems if any. Logically, energy systems with shorter payback periods are more economically favorable than those with longer payback periods. Thus, shorter payback period is associated with higher sustainability. The payback time refers to the time it takes in order for the project to recover all invested amounts and is usually expressed in years. Payback method does not take into account the time value of money different from the previous indicator (net present value or benefit-cost ratio). The calculation of the payback time is simple. The following equation is used to determine the payback period:

(6.27)PBT=P…PCF

where P… represents the total project investment in ($) and PCF represents the periodic cash flow in ($/year). Table 6.3 shows the judgment criteria set to obtain the PBT score. Shorter payback time is advantageous and more attractive.

Table 6.3. Scorecard for payback time.

7.2.5 Simple payback period

The SPP is the number of years in which the initial investment is recovered. The SPP is the ratio of initial investment to annual saving. Mathematically it can be written as [13]

(7.1)∑n=0n=nsp (Bn−Cn)=0

In an Energy Conservation Option usually the annual money saving is only due to energy savings and hence it is the product of the energy saved and the price of energy. But in the case of unequal cash inflows the PB period can be found out by adding up the cash inflows until the total is equal to initial cash outlay. This is the simplest and easiest way to understand but it does not give us the real picture as it does not consider ther time value of money or the cash flows occurring after PB period. There is no clear-cut rule regarding a minimum SPP to accept the project. So the decision to accept or reject is highly subjective.

Typical Payback Periods – Totem versus conventional Boiler plus Mains Electricity

Typical Payback Periods – Totem versus conventional Boiler plus Mains Electricity

Discounted Payback Method (DPB)

The DPB computes the discounted net cost savings of Eq. (8.1) or the cash flow of Eq. (8.2) by applying the cost of capital also called discount rate (r%) of the investor and determines how many years it takes for the sum of the discounted amounts to equal the ICC. By this procedure, the DPB takes the time value of money into consideration. The formula for the DPB is

(8.3)ICC=∑i=1TAnnual energy cost saving−Annual operating cost1+rt

where t is the time of the net cost saving and T is the total number of years the discounted net cost savings will equal the total initial investment cost. For an investor, the equation becomes

(8.4)ICC=∑t=1TCash flow1+rt

Another variation of Eqs. (8.3), (8.4) is

(8.5)ICC≤Discounted savingsor cash flow

Discounted savings are the term within the summation sign (Σ) in Eq. (8.2). To prevent duplication of equations, it is worth mentioning that

(8.6)Netcashsaving=Annual energy cost saving–Annual operating cost=Cashflow

These three terms shall be used interchangeably throughout this work.

If the investment consists of an initial capital outlay (i) and future incremental investments (Δi) such as replacement costs and decommissioning costs, then these incremental investments would be discounted to the base year as follows:

(8.7)ICC=i+∑t=1TΔi1+rt

Eqs. (8.3)–(8.5) are preferred approaches to finding the payback time than Eqs. (8.1), (8.2) as the former takes the time value of money into consideration [33].

The DPB rule is to accept projects whose sum of discounted net cost savings provides a payback period less than or equal to some prespecified number of years. It is the time it takes to breakeven in an economic or financial sense [32].

Payback Period

The payback period of an investment is a measure of how long it takes to break even on the cost of that investment. In other words, how many weeks, months, or years does it take to earn the investment capital aid out for a project or a piece of equipment?

Obviously, those projects with the fastest returns are highly attractive. The technique for determining the payback period again lies within present value; however, instead of solving the present-value equation for the present value, the cost and benefit cash flows are kept separate over time.

First, the project's anticipated benefit and cost are tabulated for each year of the project's lifetime. These values are converted to present values using the present-value equation, with the firm's discount rate plugged in as the discount factor. Finally, the cumulative total of the benefits (at present value) and the cumulative total of the costs (at present value) are compared on a year-by-year basis. At the point in time when the cumulative present value of the benefits starts to exceed the cumulative present value of the costs, the project has reached the payback period. Ranking projects then becomes a matter of selecting those projects with the shortest payback period.

Although this approach is straightforward, there are dangers in selecting P2 projects based on a minimum payback-time standard. For example, because the P2 benefit stream can extend far into the future, discounting makes its payoff period very long. Another danger is that the highest costs and benefits associated with many environmental projects generally are due to catastrophic failure, also an event far into the future. Because the payback period analysis stops when the benefits and costs are equal, the projects with the quickest positive cash flow dominate. Hence, for a P2 project, with a high discount rate, the long-term costs and benefits may be so far into the future that they do not even enter into the analysis. In essence, the importance of life-cycle costing is lost in using the minimum payback-time standard, because it considers costs and benefits only to the point where they balance, instead of considering them over the entire life of the project.

The payback period of an investment is best applied as a screening calculation between options. It gives the investor a first pass at deciding whether a particular investment is worth examining in greater detail or can provide a relative ranking of alternatives in terms of payback options.

12.4.4.1 Payback period index evaluation

The payback period is calculated as the total cost of project life (Sl. no. “c” in Table 12.7) divided by annual savings. Interest rate is assumed to be 10%, inflation rate is assumed to be 6% (Al-Kayiem & Aja, 2016; Sullivan, Wicks, & Koelling, 2009), and project lifetime, N is assumed to be 25 years per system.

Table 12.7. Total project life cost and payback period for different solar PV systems.

Annual savings are estimated from Eq. (12.13) energy yield commodity (experimental Eac values taken from Table 12.3) and energy expense of 8INR/kWh. Solar PV electricity depletion (Jordan & Kurtz, 2013), inverter life cycle, and solar tracker motor repair costs are taken into account for estimating payback time.

Table 12.7 lists the estimated payback time. Payback term is minimal (6.50 years) for DASI and average (8.42 years) for FAMI. MI’s high energy yield leads to higher annual energy savings, but MI’s cost is 87.5% higher than SI, so DAMI and FAMI’s payback period is higher than DASI and FASI systems. Total initial cost of DA-structured systems is about 67% higher than the FA system, but the increase in annual energy yield is about 35% hence the payback period of DASI and DAMI systems is less than FASI and FAMI.