7.2 The Basics: Time Value of Money and Rates

Reading Assignment

J.R. Brownson, Solar Energy Conversion Systems (SECS), Chapter 10 (Focus on the Introduction and the Time Value of Money.)
W. Short et al. (1995) Manual for the Economic Evaluation of Energy Efficiency and Renewable Energy Technologies [1]. NREL Technical Report TP-462-5173. (Read pp. 1-22: from Introduction through Taxes, skim pp. 10-11.)

Just as in other lessons, we will devote some time to getting everyone up to speed on the basics of financial analysis. Now remember, as a team, you will be working with peers who may have greater expertise than you with financal topics. I would like you to understand the general concepts of cash flows, inflation rates, fixed costs versus variable costs of a system, unit cost for a system, and taxes. This is the pragmatic side of seeking out a high solar utility for our clients, from a financial perspective.

Fundamentals

We will be considering cash flows (i.e., revenues - expenses; or savings - costs) in a process called Life Cycle Cost Analysis. As we have seen in the reading, cash flows can be developed for systems operations, for investment decisions, and for financing. We will be representing cash flows in a simple, discrete pattern called end-of-period cash flow, where the periodicity is 1 year and the compounding or discounting that occurs uses an annual rate.

Given: our SECSs will tend to have life-spans that are quite long, often well beyond the 25 years of the PV module warrantees.
Also given: a lot can happen financially in 30 years. The USA has had three recessions since 1988 (list of U.S. recessions) [2], and fluctuations in the rate of inflation between 2-6%.
And finally: SECSs are still "fresh" to many consumers; they're going to be foreign systems to most clients in the beginning of a project development.
The result is energy systems that have a long horizon of life, a long financial period of evaluation to assess, and yet the installed systems exist within a dynamic financial setting: increased uncertainty and risk (without better information available).

Did you see that last bit? Clients will perceive increased uncertainty and risk without better information available. That's your job! To provide better information and transparent project evaluation, which demonstrates an understanding of both the solar resource and the financials associated with a proposed SECS. In Chapter 10 of the textbook, we demonstrate how conveying the financial metrics of the project within a proposal is one way to provide useful information in a transparent manner.

Time Value of Money

We call the process of evaluating a project the Life Cycle Cost Analysis (LCCA), and one of the important criteria is the period of analysis, or period of evaluation. The "period" conveys a time horizon for your LCCA. If we recall our microeconomic drivers affecting the elasticity of demand, we know that the time horizon is an important factor. In our case, SECS will tend to have long life-spans.

As such, we distinguish between the concept of "value" at various points in time.

Present Value (PV, not photovoltaics this time!): specifies worth for assets like SECSs, for money, or for periodic cash flows, where the worth is in today’s dollars, provided the rate of return is specified (as "d"). The value is processed from year "n" back to "year zero" (meaning the present).

$P V = \frac{F V}{{(1 + d)}^{n}}$

Future Value (FV): specifies the worth for things as a dollar value in the future. We use FV for Fuel Costs (FC) and Fuel Savings (FS) in our LCCA. Costs are represented as "C" and Savings as "S." The rate of inflation is specified here as "i."

$F V = C \cdot {(1 + i)}^{n - 1}$

Present Worth in year n (PW_n): This is the ratio of the future costs with respect to the discount rate over time.

$P W_{n} = \frac{C \cdot {(1 + i)}^{n - 1}}{{(1 + d)}^{n}}$

You will notice that the same topics are discussed in detail in the assigned reading of the Manual for Economic Evaluation by Short et al. (1995).

Discount Rates

There are two ways to represent discount rates, and you will observe both in the SAM simulation software or similar financial analysis tools. Using these rates, we can produce a discounted cash flow model (DFM) to compare projects.

Nominal Discount Rate ($d_n$): discount rates for time value of money that are not adjusted for the effects of inflation. (Nominal = not inflation-adjusted).
Real Discount Rate ($d_r$): the discount rate where the rate of inflation has been adjusted, by excluding the effect of inflation. As such a real discount rate will be a lower value than the nominal discount rate for inflation. (Real = inflation-adjusted).
Caveat: If the inflation rate is negative (deflation) then the real discount rate would actually be higher than the nominal rate.

$(1 + d_{n}) = (1 + d_{r}) \cdot (1 + i)$ $d_{n} = [(1 + d_{r}) \cdot (1 + i)] - 1$ $d_{r} = [\frac{(1 + d_{n})}{(1 + i)}] - 1$

You will note in the Short et al. document that the nominal discount rate has a loose approximation of $d_{r} \approx d_{n} - i$ . But I want you to think, will fuel inflation rates be the same as labor inflation rates, and insurance inflation rates? We will have an example in the discussion where we pull apart different inflation rates and use real discount rates in our analysis of a solar hot water system.

Taxes and Depreciation

We have already seen that the DSIRE website [3] for the states and federal government of the USA is a useful resource for incentives. Part of those incentives is tied in to tax credits, and there is a significant portion of your reading devoted to the concept of depreciation.

Depreciation: the use of income tax deductions to recover the costs of property used in trade/business or for the production of income. Depreciation does not include land.
MACRS: Modified Accelerated Cost Recovery System [4]. You should observe that the Wikipedia site and your reading from Short et al. will be quite similar. MACRS is used in the SAM simulation software.

Net Salvage Value

One of the things that occurs in an LCCA at the end of the Period of Analysis is the question of how to finish the summation. This is like the Monty Python movie, The Holy Grail [5], where the old fellow says: "I'm not dead!" At the end of your 15-25 year evaluation for LCCA, you will no doubt have a fully functional SECS still! They don't just break down and fall apart, and in fact they will likely last for decades beyond your evaluation period. So how do we assess the value of the system at the end of the period?

We assume that the system has a net salvage value (a resale value) that is a fraction of its initial value, translated into present dollars. In our discussion, we will assume a 20-year-old solar hot water system still has 30% of its initial value, framed in present dollars for year 20.