4.4 Advanced Financial Evaluation: Annual Worth, SIR, Depreciation, and Interest Formulas
Key Takeaways
- The six core interest factors—(F/P), (P/F), (F/A), (A/F), (P/A), and (A/P)—are the algebraic foundation of every energy-finance calculation; the capital-recovery factor (A/P,i,n) converts a present cost into an equal annual series.
- Annual Worth (AW), also called Equivalent Uniform Annual Cost (EUAC), restates a project's life-cycle cost as a level yearly amount, enabling comparison of alternatives with different service lives.
- The Savings-to-Investment Ratio (SIR) = PV(savings) ÷ PV(incremental investment); an SIR > 1.0 is cost-effective, and federal agencies rank projects by SIR to allocate limited capital.
- Depreciation (straight-line, declining-balance, sum-of-years-digits, and MACRS) is a non-cash tax deduction that lowers taxable income; MACRS accelerates deductions, improving after-tax project economics through a larger early-year tax shield.
- Interest factor tables (and their formulas) let the CEM solve payback, NPV, EUAC, and SIR by hand in the open-book exam without relying on a spreadsheet.
Advanced Financial Evaluation: Annual Worth, SIR, Depreciation, and Interest Formulas
Simple payback (Section 4.2) and NPV/IRR (Section 4.3) answer "is this project worth it?" But when a CEM must rank competing projects with different lives, different scales, or different tax treatments, those two tools are not enough. The CEM Body of Knowledge adds Interest Formulas and Tables, Annual Worth, the Savings-to-Investment Ratio, and Depreciation Methods to complete the financial toolkit.
Interest Formulas and Factors
All time-value-of-money math reduces to six factors relating a present sum (P), a future sum (F), and a uniform annual series (A) at interest rate i over n periods. With the factor notation (X/Y, i, n) = "find X given Y":
| Factor | Symbol | Formula | Use |
|---|---|---|---|
| Future given Present | (F/P,i,n) | (1+i)^n | Project a cost forward |
| Present given Future | (P/F,i,n) | 1/(1+i)^n | Discount a future sum |
| Future given Annual | (F/A,i,n) | [(1+i)^n − 1]/i | Accumulate annual savings |
| Annual given Future | (A/F,i,n) | i/[(1+i)^n − 1] | Sinking fund |
| Present given Annual | (P/A,i,n) | [(1+i)^n − 1]/[i(1+i)^n] | Capitalize an annual stream |
| Annual given Present | (A/P,i,n) | [i(1+i)^n]/[(1+i)^n − 1] | Capital recovery |
The capital recovery factor (A/P,i,n) is the workhorse: it converts a first cost into the equal annual payment that recovers it with interest. The CEM exam provides interest tables; the candidate must pick the correct factor and apply it.
Annual Worth (Equivalent Uniform Annual Cost)
Annual Worth (AW), or Equivalent Uniform Annual Cost (EUAC) when costs-only, restates a project's entire life-cycle cash flow as a single level yearly amount. Because it is expressed per year, it lets the CEM compare alternatives with different service lives on equal footing—something NPV cannot do directly without repeated cycles.
For an asset with first cost P, salvage value S, life n, and discount rate i:
EUAC = (P − S)(A/P,i,n) + S·i
The capital-recovery portion (P−S)(A/P) recovers the depreciable base; the S·i term accounts for the opportunity cost of the salvage tied up over the life. Savings-side Annual Worth (AW) is computed similarly for the annual savings stream and compared against EUAC: a project is economic when AW(savings) ≥ EUAC.
Worked Example: EUAC of a Chiller Replacement
A chiller costs $80,000, has a salvage value of $8,000 after 15 years, and the discount rate is 8%.
- (A/P, 8%, 15) = [0.08(1.08)^15]/[(1.08)^15 − 1] = 0.1168
- EUAC = (80,000 − 8,000)(0.1168) + 8,000(0.08) = 72,000 × 0.1168 + 640 = $8,410 + $640 = $9,050/yr
The chiller's owning cost is ~$9,050 per year before any energy is consumed. Comparing this EUAC against the annual energy savings of a competing high-efficiency unit tells the CEM which wins on a level-year basis.
Savings-to-Investment Ratio (SIR)
SIR = PV(operating savings) ÷ PV(incremental investment cost). It is a dimensionless ratio, so it scales across project sizes. Rules:
- SIR > 1.0 → cost-effective (savings exceed the investment).
- Higher SIR = better return per dollar invested → federal agencies and ESCOs rank projects by SIR to ration limited capital.
SIR complements NPV: NPV picks the project with the largest absolute net gain; SIR picks the one that squeezes the most savings from each dollar. For a constrained budget, ranking by SIR maximizes total savings achieved.
Depreciation Methods
Depreciation is a non-cash deduction that spreads an asset's first cost over its useful life for tax purposes. It does not change cash flow directly, but the tax shield (depreciation × tax rate) lowers taxable income and improves after-tax project economics. The CEM should know four methods:
| Method | Pattern | Notes |
|---|---|---|
| Straight-line | Equal deduction each year | (P − S)/n; simplest, lowest early shield |
| Declining balance | Fixed % of remaining book | Accelerated; larger early deductions |
| Sum-of-years-digits | Fraction n, n−1, … of depreciable base | Accelerated, moderate |
| MACRS | IRS statutory percentages | Most accelerated; U.S. tax standard for equipment |
Modified Accelerated Cost Recovery System (MACRS) is the U.S. federal depreciation system: it assigns each asset class a recovery period and a statutory percentage table that front-loads deductions. Bonus depreciation (e.g., 100%/80% first-year under the IRA trajectory) and Section 179 expensing further accelerate the shield. Because a dollar of deduction today is worth more than one in year 10, accelerated depreciation raises a project's after-tax IRR and is a real lever in the investment decision (the CEM exam treats an investment tax credit as effectively reducing first cost, not as a tax-rate cut).
How the Four Tools Fit Together
A complete financial case stacks the tools: interest factors fund the math, NPV/IRR flag viability, Annual Worth equalizes lives for comparison, SIR ranks the portfolio, and depreciation adjusts for taxes. A CEM who stops at simple payback leaves both money and rank-order accuracy on the table.
An asset has a first cost of $80,000, salvage of $8,000, a 15-year life, and i = 8%. Using (A/P, 8%, 15) = 0.1168, what is the EUAC of owning it?
What does a Savings-to-Investment Ratio (SIR) of 1.4 indicate, and how is it used?
Which depreciation method is the U.S. federal tax standard for equipment and front-loads deductions most aggressively?