# INFLATION RATE

Inflation occurs when the cost of goods and services increases from one period to the next. The interest rate i defines the cost of money, and the inflation rate X measures the increase in the cost of goods and services. Therefore, the future cost of a commodity FC is higher than the present cost PC of the same commodity:

Over a lifetime N, the future cost of a commodity increases exponentially:

Similar to interest rates, inflation rates are typically assumed to be constant over the lifetime of the energy retrofit project. Table 3.2 presents historical data for inflation rates for selected countries. The escalation of energy cost is an important factor to consider in evaluating energy retrofit projects. The energy escalation rate can be considered as one form of inflation rate.

If the interest charges are compounded at the same periods during which inflation occurs, the future worth can be determined from the present value P as follows:

TABLE 3.2

Average Inflation Rates for Selected Countries

 Period/Year France Germany Japan United States Period 1971-1980 9.8 5.0 8.8 7.1 1981-1990 6.2 2.5 2.1 4.7 1990-1995 1.9 2.8 1.0 2.6 Year 1995 1.8 1.2 -0.1 2.8 2000 1.8 1.4 -0.5 3.4 2005 1.9 1.9 -0.6 3.4 2010“ 0.7 0.4 -1.4 1.0

Source: OECD, Economic Statistics, http://www.ocde.org, 2009. a Based on predictions.

The expression above for F can be rearranged as follows:

A composite interest rate 0 can be defined to account for the fact that inflation decreases the buying power of money due to increases in the cost of commodities:

It should be mentioned that theoretically the composite interest rate can be negative. In this case, the money loses its value with time. Example 3.2 estimates the impact of the inflation rate on the future value of a principle deposited in a bank.

Example 3.2

Determine the actual value of the \$10,000 investment for the building owner in Example 3.1 if the economy experiences an annual inflation rate of 4 percent.

SOLUTION

The composite interest rate can be determined using Eq. (3.11):

Using Eq. (3.10) with P = \$10,000 and N = 10, the investment will accumulate to the total amount F:

## Tax Rate

In most economies, the interest that is received from an investment is subject to taxation. If this taxation has a rate t over a period that coincides with the interest period, then the amount of taxes T to be collected from an investment P with an interest rate /' is determined as follows:

Therefore, the net return from the investment P to the investor after tax deductions is:

An effective interest rate io' can then be defined to account for the loss of income due to taxation:

Therefore, the composite interest rate defined by Eq. (3.11) can be generalized to account for both inflation and tax rates related to present and future values:

Example 3.3 illustrates the future value of a principle deposited in a bank when the combined impacts of interest, inflation, and tax rates are considered.

Example 3.3

If the building owner is in the 28 percent tax bracket, determine the actual value of his \$10,000 investment considered in Example 3.1 if the economy experiences an annual inflation rate of 4 percent.

SOLUTION

The composite interest rate can be determined using Eq. (3.15):

Using Eq. (3.10) with P = \$10,000 and N = 10, the investment will accumulate to the total amount F:

## Cash Flows

In evaluating energy-efficiency projects, it is important to account for the total cash receipts and disbursements due to the implementation of an energy conservation measure (such as the installation of a new boiler) for each period during the entire lifetime of the project. The difference between the total cash receipts (inflows) and total cash disbursements (outflows) for a given period of time is called a cash flow.

Over the lifetime of a project, an accurate accounting of all the cash flows should be performed. For energy-efficiency improvement projects, the cash flow accounting can be in a tabular format as illustrated in Table 3.3, which lists the cash flows attributed to the installation of a new steam boiler in a hospital. The table accounts for the cost related to the initial cost of a new boiler installation (counted as a disbursement for Year 0) and the cost savings due to the higher energy efficiency of the new boiler (counted as receipts in Year 1 through Year 10). The reduction in the yearly receipts is attributed to the aging of the equipment.

Note that the cash flows are positive when they are inflows (i.e., receipts) and negative when they are outflows (i.e., disbursements). To better visualize the evolution over time of the cash flows, a cash flow diagram as depicted in Figure 3.1 is used. Note that in this figure, the initial cash flow, C0 = -\$400,000 (disbursement), is represented by a downward-pointing arrow. Meanwhile, the cash flows occurring later, Cj through Сл, with N= 10, are receipts and are represented by upward-pointing arrow's.

TABLE 3.3

Cash Flows for an Installation of a New Boiler over a Lifetime of Ten Years

 End of Year Total Cash Receipts Total Cash Disbursements Total Cash Flows Comments 0 \$0 \$400,000 -\$400,000 Installation cost of a new boiler 1 \$40,000 \$0 +\$40,000 Net cost savings through Year 10 2 \$38,000 \$0 +\$38,000 3 \$36,000 \$0 +\$36,000 4 \$34,000 \$0 +\$34,000 5 \$33,000 \$0 +\$33,000 6 \$32,000 \$0 +\$32,000 7 \$31,000 \$0 +\$31,000 8 \$30,500 \$0 +\$30,500 9 \$30,000 \$0 +\$30,000 10 \$29,500 \$0 +\$29,500

FIGURE 3.1 Typical cash flow diagram.

It should be noted again that the cash flows cannot simply be added because the value of money changes from one period to the next. In the next section, various factors are defined to correlate cash flows occurring at different periods.