Benefit-Cost Ratio and Payback Period

Benefit-Cost Ratio and Payback Period

Learning Objectives

Understand and calculate the conventional and modified Benefit-Cost Ratios for public sector projects.
Perform incremental Benefit-Cost Analysis to select between mutually exclusive alternatives.
Compute Simple and Discounted Payback Periods and understand their limitations.

Benefit-Cost Ratio Analysis

Benefit-Cost Ratio (B/C)

A ratio used to summarize the overall value for money of a project or proposal. It is the major method legally mandated for the evaluation of public sector projects (e.g., dams, highways, airports, flood control) where the objective is to maximize social welfare rather than corporate profit.

Conventional B/C Ratio Formula

Calculates the ratio of the present worth of net benefits to the present worth of total costs.

B/C = \frac{PW(Benefits) - PW(Disbenefits)}{PW(Initial Cost) + PW(O\&M Costs) - PW(Salvage Value)}

Variables

Symbol	Description	Unit
$B/C$	Conventional Benefit-Cost Ratio	unitless
$Benefits$	Advantages, savings, or revenues experienced by the public	$
$Disbenefits$	Disadvantages, losses, or costs experienced by the public as a consequence of the project	$
$Initial Cost$	Initial capital investment incurred by the government sponsor	$
$O\&M Costs$	Ongoing Operations & Maintenance costs	$
$Salvage Value$	Expected value at the end of the project's life	$

Decision Criterion:

If $B/C \ge 1.0$ , the project is justified (benefits outweigh costs).
If $B/C < 1.0$ , the project is not economically justified.

Modified B/C Ratio

The Modified B/C ratio subtracts annual Operations & Maintenance (O&M) costs from the numerator (benefits) rather than adding them to the denominator (costs). This formulation isolates the net annual value generated against the sheer initial capital investment required from the government budget. It will always yield the same accept/reject decision as the conventional ratio (i.e., if one is $\ge 1$ , the other is also $\ge 1$ ).

Modified B/C Ratio

Isolates the net annual value generated against the sheer initial capital investment.

B/C_{mod} = \frac{PW(Benefits) - PW(Disbenefits) - PW(O\&M)}{PW(Initial Investment) - PW(Salvage Value)}

Variables

Symbol	Description	Unit
$B/C_{mod}$	Modified Benefit-Cost Ratio	unitless
$Benefits$	Present worth of public advantages or savings	$
$Disbenefits$	Present worth of public disadvantages or losses	$
$O\&M$	Present worth of ongoing Operations & Maintenance costs	$
$Initial Investment$	Present worth of the initial capital investment	$
$Salvage Value$	Present worth of the salvage value	$

\Delta

Exactly like Rate of Return analysis, when choosing among mutually exclusive public projects, you cannot simply select the project with the highest individual B/C ratio. You must perform an incremental analysis.

Procedure

STEP-BY-STEP

Order the acceptable alternatives from lowest initial cost to highest initial cost.
Set the lowest-cost acceptable alternative as the Defender and the next higher as the Challenger.
Calculate the incremental costs ( $\Delta C$ ) and incremental benefits ( $\Delta B$ ) between them: $\Delta = \text{Challenger} - \text{Defender}$ .
Calculate the incremental ratio: $\Delta B/C = \frac{\Delta B}{\Delta C}$ .
If $\Delta B/C \ge 1.0$ , the extra investment is justified; the Challenger becomes the new Defender. If $\Delta B/C < 1.0$ , the Defender remains.

Payback Period Method

Payback Period ( $N_p$ )

The amount of time required for the cumulative net cash inflows generated by an investment to exactly equal the initial capital investment. It measures liquidity and how quickly capital is recovered, acting as a proxy for risk.

Simple Payback Period

The Conventional Payback Period ignores the time value of money (interest rate = 0%). It simply asks how long it takes to recoup the nominal dollars spent.

If cash flows are uneven, you simply sum the cash flows year by year until the cumulative total reaches zero. If the net annual cash flow is uniform (constant every year), you can use the formula below.

Conventional Payback Period Formula

Calculates the time required to recover the initial investment ignoring the time value of money, assuming uniform cash flows.

N_p = \frac{\text{Initial Investment } (P)}{\text{Net Annual Cash Flow } (A)}

Variables

Symbol	Description	Unit
$N_p$	Payback Period	years
$P$	Initial Investment	$
$A$	Net Annual Cash Flow	$/year

Discounted Payback Period

The Discounted Payback Period considers the time value of money ( $i > 0$ ). It is the time required for the present worth of the future cash inflows to equal the initial investment. Because future dollars are discounted (worth less today), the discounted payback period will always be longer than the simple payback period.

If $\frac{i \cdot P}{A} \ge 1$ , the project will never pay back its initial investment at that interest rate (the argument of the logarithm becomes negative or zero).

Discounted Payback Period Formula

Calculates the time required to recover the initial investment considering the time value of money, for a uniform series.

N_p = -\frac{\ln(1 - \frac{i \cdot P}{A})}{\ln(1 + i)}

Variables

Symbol	Description	Unit
$N_p$	Discounted Payback Period	years
$P$	Initial Investment	$
$A$	Net Annual Cash Flow	$/year
$i$	Interest rate per compounding period	decimal

Visualizing Payback Period

Use the interactive chart below to visualize how cumulative cash flow changes over time. The precise point where the cumulative cash flow curve crosses the horizontal axis ( $y=0$ ) is the exact payback period. Notice how increasing the interest rate stretches the curve further to the right, lengthening the discounted payback time.

Interactive Simulation

Experiment with the interest rate and cash flows below to see how they impact both Simple and Discounted payback periods.

Payback Period Analyzer

Cash Flows

Yr 0

Yr 1

Yr 2

Yr 3

Yr 4

Yr 5

Payback Period:3.20 Years

Loading chart...

Key Takeaways

Public Sector Focus: B/C ratio is the standard for evaluating government-funded projects where benefits accrue to the public rather than generating direct corporate revenue.
The Core Metric: A project is economically justified if the ratio of the Present Worth (or Annual Worth) of its net benefits to its costs is $\ge 1.0$ .
Incremental B/C: Never pick a mutually exclusive project simply because it has the highest individual B/C ratio; you must evaluate the $\Delta B/C$ of the difference in costs.
Risk and Liquidity Metric: Payback period measures how fast you get your money back, which is a proxy for risk.
Simple vs. Discounted: Simple payback ignores the time value of money (interest rate = 0%). Discounted payback is more accurate.
The Major Flaw: Payback period entirely ignores any cash flows (good or bad) that occur after the payback point is reached. It should never be the sole criterion for selecting projects.