# COMPOUNDING FACTORS

Two types of payment factors are considered in this section. These payment factors are useful in the economic evaluation of various energy audit projects. Without loss of generality, the interest period is assumed in the remainder of this chapter to be one year. Moreover, a nominal discount rate *d* is used throughout this chapter. This discount rate is an effective interest rate that includes the effects of several parameters such as inflation and taxation discussed above.

## Single Payment

In this case, an initial payment is made to implement a project by borrowing an amount of money *P.* If this sum of money earns interest at a discount rate *d,* then the

FIGURE 3.2 Cash flow diagram for single payment.

value of the payment *P* after *N* years is provided by Eq. (3.16). The ratio *F/P* is often called the single payment compound amount factor (SPCA). The *SPCA* factor is a function of *i* and *N* and is defined as:

Using the cash flow diagram of Figure 3.1, the single payment represents the case where C_{0} = *P,* C, = • • • = C,_{v} -1=0, and *C _{N} = F,* as illustrated in Figure 3.2.

The inverse ratio *P/F* allows us to determine the value of the cash flow *P* needed to attain a given amount of cash flow *F* after *N* years. The ratio *PIF* is called the single payment present worth (SPPW) factor and is equal to:

## Uniform-Series Payment

In the vast majority of energy retrofit projects, the economic benefits are estimated annually and are obtained after a significant initial investment. It is hoped that during the lifetime of the project, the sum of all the annual benefits can surpass the initial investment.

Consider then an amount of money *P* that represents the initial investment, and a receipt of an amount *A* that is made each year and represents the cost savings due to the retrofit project. To simplify the analysis, the amount *A* is assumed to be the same for all the years during the lifetime of the project. Therefore, the cash flows—using Figure 3.1—are C_{0} = *P,* C, = = C_{lV} = A, as depicted in Figure 3.3.

To correlate between *P* and A, we note that for any year *k,* the present worth *P _{k}* of the receipt A can be determined using Eq. (3.17):

By summing up the present worth values for all the annual receipts A, the result should equal the cash flow *P:*

FIGURE 3.3 Cash flow diagram for uniform-series payment.

The sum can be rearranged to obtain a geometric series that can be evaluated as shown below:

The ratio *А/P* is called the uniform-series capital recovery factor (USCR). This *USCR* factor can be determined as a function of both *d* and *N*

The uniform-series present worth factor (USPW), which allows us to determine the value of *P* knowing the amount Л, is the ratio *P/А* and can be expressed as follows:

The various compounding factors are estimated in Example 3.4 for specific values of discount rate and life-cycle.

Example 3.4

Find the various compounding factors for *N =* 10 years and c/= 5 percent.

SOLUTION

The values of the compounding factors for *d* = 0.05 and *N =* 10 years are summarized below:

Compounding Factor |
Equation Used |
Value |

SPCA |
Eq. (3.16) |
1.629 |

SPPW |
Eq. (3.17) |
0.613 |

USCR |
Eq. (3.21) |
0.129 |

USPW |
Eq. (3.22) |
7.740 |

# ECONOMIC EVALUATION METHODS AMONG ALTERNATIVES

To evaluate the cost-effectiveness of energy retrofit projects, several evaluation tools can be considered. The basic concept of all these tools is to compare among the alternatives the net cash flow that results during the entire lifetime of the project. As discussed earlier, a simple addition of all the cash flows such as those represented in Figure 3.1 is not possible. However, by using compound factors discussed in Section 3.3, “conversion” of the cash flows from one period to another is feasible. This section provides a brief description of the common evaluation methods used in engineering projects.

## Net Present Worth

The basic principle of this method is to evaluate the present wortli of the cash flows that occur during the lifetime of the project. Referring to the cash flow diagram of Figure 3.1, the sum of all the present worth of the cash flows can be obtained by using the *SPPW* factor defined in Eq. (3.17):

Note that the initial cash flow is negative (a capital cost for the project), whereas the cash flows for the other years are generally positive (revenues).

In the particular but common case of a project with constant annual revenue (due to energy operating cost savings), *CF _{k} = A,* the net present worth

*(NPW)*is reduced to:

For the project to be economically viable, the *NPW* has to be positive or at worst zero *(NPW* > 0). Obviously, the higher the NPW, the more economically sound the project.

The NPW value method is often called the net savings method because the revenues are often due to the cost savings from implementing the project.