This page gives the core formulas used in the Reserve Sense Benchmark Method.

It is a reference page, not a step-by-step explanation. For that, see the Walkthrough.

Timing assumptions

For the timing rules behind these formulas, see Modeling Assumptions. These formulas assume that:

• expenditures are applied before interest is earned
• interest is earned during the year
• contributions are added at the end of the year

Variables

CRC = Current Replacement Cost
FRC = Future Replacement Cost
LS = Lifespan
EA = Effective Age
RL = Remaining Life
i = Long-term construction cost inflation rate
g = Long-term CPI inflation rate
j = Long-term interest rate
IC = Ideal Contribution for the fiscal year being modeled
IOB = Ideal Opening Balance at the start of the fiscal year
ICB = Ideal Closing Balance at the end of the fiscal year
E = Expenditure for the component during the fiscal year

Remaining Life

Remaining Life is the portion of the lifespan that has not yet been used up.

$$RL=LS-EA$$

For a newly installed component, EA = 0. For that component, RL = LS.

Future Replacement Cost

The Future Replacement Cost is the estimated cost of the component in the year of its next expected replacement.

$$FRC=CRC\cdot (1+i{)}^{RL}$$

This formula applies construction cost inflation over the component's remaining life.

Ideal Contribution

The Ideal Contribution is the mathematically fair contribution for the relevant fiscal year.

When j ≠ g

$$IC=FRC\cdot \frac{(j-g)(1+g{)}^{EA}}{(1+j{)}^{LS}-(1+g{)}^{LS}}$$

A separate form is needed when j = g because the standard expressions otherwise collapse into a zero denominator.

When j = g

$$IC=\frac{FRC}{LS}\cdot (1+j{)}^{1-RL}$$

This is the annual contribution for a given year, not necessarily the first year's contribution for a brand-new component.

Ideal Opening Balance

The Ideal Opening Balance is the fair amount that should be in the reserve fund for a component at the start of the relevant fiscal year.

When j ≠ g

$$IOB=FRC\cdot \frac{(1+j{)}^{EA}-(1+g{)}^{EA}}{(1+j{)}^{LS}-(1+g{)}^{LS}}$$

When j = g

$$IOB=\frac{FRC}{LS}\cdot EA\cdot (1+j{)}^{-RL}$$

Ideal Closing Balance

The Ideal Closing Balance is the fair amount that should be in the reserve fund for this component at the end of the relevant fiscal year.

When j ≠ g

$$ICB=FRC\cdot \frac{(1+j{)}^{EA+1}-(1+g{)}^{EA+1}}{(1+j{)}^{LS}-(1+g{)}^{LS}}$$

When j = g

$$ICB=\frac{FRC}{LS}\cdot (EA+1)\cdot (1+j{)}^{1-RL}$$

Interest relationship

The formulas above already produce the correct opening and closing Benchmark balances. But it can still be helpful to show the yearly relationship directly.

If a component has an expenditure during the fiscal year, and that expenditure is written as E, then:

$$Interest=j\cdot (IOB-E)$$

and:

$$ICB=IOB-E+Interest+IC$$

If there is no expenditure during the year, then E = 0.

Fund-level totals

The Benchmark is calculated separately for each component. The total reserve fund Benchmark for a given year is then found by summing the component results for that same year.

Here, k indexes the components in the study, n is the total number of components, and t is the year being modeled.

For year $t$:

$$TotalI{C}_t=\sum _{k=1}^{n}I{C}_{k,t}$$ $$TotalIO{B}_t=\sum _{k=1}^{n}IO{B}_{k,t}$$ $$TotalIC{B}_t=\sum _{k=1}^{n}IC{B}_{k,t}$$

Each component has its own lifespan, effective age, inflation path, and replacement timing. The method therefore does not calculate one blended formula for the reserve fund as a whole.

Instead, Benchmark values are calculated separately for each component in each year of the study, and fund-level totals are found by summing component values from that same year.

Where to go next

This page shows only the formulas.

• Go to Benchmark Overview to understand what a Benchmark Analysis is.
• Go to Walkthrough to see the method step by step.
• Go to Modeling Assumptions for the timing rules behind the method.
• Go to Legacy Method: A Simple Check to follow one component and see where the commonly used Legacy Method breaks down.