Fundamentals of Transportation/Evaluation

Evaluation

A benefit-cost analysis (BCA) is often required in determining whether a project should be approved, and is useful for comparing similar projects. It determines the stream of quantifiable economic benefits and costs that are associated with a project or policy. If the benefits exceed the costs, the project is worth doing, if the benefits fall short of the costs, the project is not. Benefit-cost analysis is appropriate where the technology is known and well understood, or a minor change from existing technologies. BCA is not appropriate when the technology is new and untried because the effects of the technology cannot be easily measured or predicted. However, just because something is new in one place, does not necessarily make it new, so benefit-cost analysis would be appropriate, e.g., for a light-rail or commuter rail line in a city without rail, or for any road project, but would not be appropriate (at the time of this writing) for something truly radical like teleportation.

The identification of the costs, and more particularly the benefits, is the chief component of the “art” of Benefit-Cost Analysis. This component of the analysis is different for every project. Furthermore, care should be taken to avoid double counting; especially counting cost savings in both the cost and the benefit columns. However, a number of benefits and costs should be included at a minimum. In transportation these costs should be separated for users, transportation agencies, and the public at large. Consumer benefits are measured by consumer’s surplus. It is important to recognize that the demand curve is downward sloping, so there a project may produce benefits both to existing users in terms of a reduction in cost and to new users by making travel worthwhile where previously it was too expensive.

Agency benefits come from profits. But since most agencies are non-profit, they receive no direct profits. Agency construction, operating, maintenance, or demolition costs may be reduced (or increased) by a new project; these cost savings (or increases) can either be considered in the cost column, or the benefit column, but not both.

Society is impacted by transportation project by an increase or reduction of negative and positive externalities. Negative externalities, or social costs, include air and noise pollution and accidents. Accidents can be considered either a social cost or a private cost, or divided into two parts, but cannot be considered in total in both columns.

If there are network externalities (i.e. the benefits to consumers are themselves a function of the level of demand), then consumers’ surplus for each different demand level should be computed. Of course this is easier said than done. In practice, positive network externalities are ignored in Benefit Cost Analysis.

Discount rate

Both the costs and benefits flowing from an investment are spread over time. While some costs are one-time and borne up front, other benefits or operating costs may be paid at some point in the future, and still others received as a stream of payments collected over a long period of time. Because of inflation, risk, and uncertainty, a dollar received now is worth more than a dollar received at some time in the future. Similarly, a dollar spent today is more onerous than a dollar spent tomorrow. This reflects the concept of time preference that we observe when people pay bills later rather than sooner. The existence of real interest rates reflects this time preference. The appropriate discount rate depends on what other opportunities are available for the capital. If simply putting the money in a government insured bank account earned 10% per year, then at a minimum, no investment earning less than 10% would be worthwhile. In general, projects are undertaken with those with the highest rate of return first, and then so on until the cost of raising capital exceeds the benefit from using that capital. Applying this efficiency argument, no project should be undertaken on cost-benefit grounds if another feasible project is sitting there with a higher rate of return.

Three alternative bases for the setting the government’s test discount rate have been proposed:

1. The social rate of time preference recognizes that a dollar's consumption today will be more valued than a dollar's consumption at some future time for, in the latter case, the dollar will be subtracted from a higher income level. The amount of this difference per dollar over a year gives the annual rate. By this method, a project should not be undertaken unless its rate of return exceeds the social rate of time preference.
2. The opportunity cost of capital basis uses the rate of return of private sector investment, a government project should not be undertaken if it earns less than a private sector investment. This is generally higher than social time preference.
3. The cost of funds basis uses the cost of government borrowing, which for various reasons related to government insurance and its ability to print money to back bonds, may not equal exactly the opportunity cost of capital.

Typical estimates of social time preference rates are around 2 to 4 percent while estimates of the social opportunity costs are around 7 to 10 percent.

Generally, for Benefit-Cost studies an acceptable rate of return (the government’s test rate) will already have been established. An alternative is to compute the analysis over a range of interest rates, to see to what extent the analysis is sensitive to variations in this factor. In the absence of knowing what this rate is, we can compute the rate of return (internal rate of return) for which the project breaks even, where the net present value is zero. Projects with high internal rates of return are preferred to those with low rates.

Determine a present value

The basic math underlying the idea of determining a present value is explained using a simple compound interest rate problem as the starting point. Suppose the sum of $100 is invested at 7 percent for 2 years. At the end of the first year the initial $100 will have earned $7 interest and the augmented sum ($107) will earn a further 7 percent (or $7.49) in the second year. Thus at the end of 2 years the $100 invested now will be worth $114.49.

The discounting problem is simply the converse of this compound interest problem. Thus, $114.49 receivable in 2 years time, and discounted by 7 per cent, has a present value of $100.

Present values can be calculated by the following equation:

(1) $P=\frac{F}{{\left(1+i\right)}^n}$

where:

• F = future money sum
• P = present value
• i = discount rate per time period (i.e. years) in decimal form (e.g. 0.07)
• n = number of time periods before the sum is received (or cost paid, e.g. 2 years)

Illustrating our example with equations we have:

$P=\frac{F}{{\left(1+i\right)}^n}=\frac{114.49}{{\left(1+0.07\right)}^2}=100.00$

The present value, in year 0, of a stream of equal annual payments of $A starting year 1, is given by the reciprocal of the equivalent annual cost. That is, by:

(2) $P=A\left[\frac{{\left(1+i\right)}^n-1}{i{\left(1+i\right)}^n}\right]$

where:

• A = Annual Payment

For example: 12 annual payments of $500, starting in year 1, have a present value at the middle of year 0 when discounted at 7% of: $3971

$P=A\left[\frac{{\left(1+i\right)}^n-1}{i{\left(1+i\right)}^n}\right]=500\left[\frac{{\left(1+0.07\right)}^{12}-1}{0.07{\left(1+0.07\right)}^{12}}\right]=3971$

The present value, in year 0, of m annual payments of $A, starting in year n + 1, can be calculated by combining discount factors for a payment in year n and the factor for the present value of m annual payments. For example: 12 annual mid-year payments of $250 in years 5 to 16 have a present value in year 4 of $1986 when discounted at 7%. Therefore in year 0, 4 years earlier, they have a present value of $1515.

$P_{Y=4}=A\left[\frac{{\left(1+i\right)}^n-1}{i{\left(1+i\right)}^n}\right]=250\left[\frac{{\left(1+0.07\right)}^{12}-1}{0.07{\left(1+0.07\right)}^{12}}\right]=1986$

$P_{Y=0}=\frac{F}{{\left(1+i\right)}^n}=\frac{P_{Y=4}}{{\left(1+i\right)}^n}=\frac{1986}{{\left(1+0.07\right)}^4}=1515$

Evaluation criterion

Three equivalent conditions can tell us if a project is worthwhile

1. The discounted present value of the benefits exceeds the discounted present value of the costs
2. The present value of the net benefit must be positive.
3. The ratio of the present value of the benefits to the present value of the costs must be greater than one.

However, that is not the entire story. More than one project may have a positive net benefit. From the set of mutually exclusive projects, the one selected should have the highest net present value. We might note that if there are insufficient funds to carry out all mutually exclusive projects with a positive net present value, then the discount used in computing present values does not reflect the true cost of capital, rather it is too low.

There are problems with using the internal rate of return or the benefit/cost ratio methods for project selection, though they provide useful information. The ratio of benefits to costs depends on how particular items (for instance, cost savings) are ascribed to either the benefit or cost column. While this does not affect net present value, it will change the ratio of benefits to costs (though it cannot move a project from a ratio of greater than one to less than one).

Benefit Cost Application: A Sample Problem

This problem, adapted from Watkins (1996), illustrates how a Benefit Cost Analysis might be applied to a project such as a highway widening. The improvement of the highway saves travel time and increases safety (by bringing the road to modern standards). But there will almost certainly be more total traffic than was carried by the old highway. This example excludes external costs and benefits, though their addition is a straightforward extension. The data for the “No Expansion” can be collected from off-the-shelf sources, however the “Expansion” column’s data requires the use of forecasting and modeling.

Table 1: Data
		No Expansion	Expansion
Peak
Passenger Trips	(per hour)	18,000	24,000
Trip Time	(minutes)	50	30
Off-peak
Passenger Trips	(per hour)	9,000	10,000
Trip Time	(minutes)	35	25
Traffic Fatalities	(per year)	2	1

note: the operating cost for a vehicle is unaffected by the project, and is $4.

Table 2: Model Parameters
Peak Value of Time	($/minute)	$0.15
Off-Peak Value of Time	($/minute)	$0.10
Value of Life	($/life)	$3,000,000

Benefits

Figure 1: Change in Consumers' Surplus

A 50 minute trip at $0.15/minute is $7.50, while a 30 minute trip is only $4.50. So for existing users, the expansion saves $3.00/trip. Similarly in the off-peak, the cost of the trip drops from $3.50 to $2.50, saving $1.00/trip.

Consumers’ surplus increases both for the trips which would have been taken without the project and for the trips which are stimulated by the project (so-called “induced demand”), as illustrated above in Figure 1. Our analysis is divided into Old and New Trips, the benefits are given in Table 3.

Table 3: Hourly Benefits
TYPE	Old trips	New Trips	Total
Peak	$54,000	$9000	$63,000
Off-peak	$9,000	$500	$9,500

Note: Old Trips: For trips which would have been taken anyway the benefit of the project equals the value of the time saved multiplied by the number of trips. New Trips: The project lowers the cost of a trip and public responds by increasing the number of trips taken. The benefit to new trips is equal to one half of the value of the time saved multiplied by the increase in the number of trips. There are 250 weekdays (excluding holidays) each year and four rush hours per weekday there are 1000 peak hours per year. With 8760 hours per year, we get 7760 offpeak hours per year. These numbers permit the calculation of annual benefits (shown in Table 3).

Table 4: Annual Travel Time Benefits
TYPE	Old trips	New Trips	Total
Peak	$54,000,000	$9,000,000	$63,000,000
Off-peak	$69,840,000	$3,880,000	$73,720,000
Total	$123,840,000	$12,880,000	$136,720,000

The safety benefits of the project are the product of the number of lives saved multiplied by the value of life. Typical values of life are on the order of $3,000,000 in US transportation analyses. We need to value life to determine how to trade off between safety investments and other investments. While your life is invaluable to you (that is, I could not pay you enough to allow me to kill you), you don’t act that way when considering chance of death rather than certainty. You take risks that have small probabilities of very bad consequences. You do not invest all of your resources in reducing risk, and neither does society. If the project is expected to save one life per year, it has a safety benefit of $3,000,000. In a more complete analysis, we would need to include safety benefits from non-fatal accidents.

The annual benefits of the project are given in Table 5. We assume that this level of benefits continues at a constant rate over the life of the project.

Table 5: Total Annual Benefits
Type of Benefit	Value of Benefits Per Year
Time Saving	$136,720,000
Reduced Risk	$3,000,000
Total	$139,720,000

Costs

Highway costs consist of right-of-way, construction, and maintenance. Right-of-way includes the cost of the land and buildings that must be acquired prior to construction. It does not consider the opportunity cost of the right-of-way serving a different purpose. Let the cost of right-of-way be $100 million, which must be paid before construction starts. In principle, part of the right-of-way cost can be recouped if the highway is not rebuilt in place (for instance, a new parallel route is constructed and the old highway can be sold for development). Assume that all of the right-of-way cost is recoverable at the end of the thirty-year lifetime of the project. The $1 billion construction cost is spread uniformly over the first four-years. Maintenance costs $2 million per year once the highway is completed.

The schedule of benefits and costs for the project is given in Table 6.

Table 6: Schedule Of Benefits And Costs ($ millions)
Time (year)	Benefits	Right-of-way costs	Construction costs	Maintenance costs
0	0	100	0	0
1-4	0	0	250	0
5-29	139.72	0	0	2
30	139.72	-100	0	2

Conversion to Present Value

The benefits and costs are in constant value dollars. Assume the real interest rate (excluding inflation) is 2%. The following equations provide the present value of the streams of benefits and costs.

To compute the Present Value of Benefits in Year 5, we apply equation (2) from above.

$P=A\left[\frac{{\left(1+i\right)}^n-1}{i{\left(1+i\right)}^n}\right]=139.72\left[\frac{{\left(1+0.02\right)}^{26}-1}{0.02{\left(1+0.02\right)}^{26}}\right]=2811.31$

To convert that Year 5 value to a year 1 value, we apply equation (1)

$P=\frac{F}{{\left(1+i\right)}^n}=\frac{2811.31}{{\left(1+0.02\right)}^4}=2597.21$

The present value of right-of-way costs is computed as today’s right of way cost ($100 M) minus the present value of the recovery of those costs in year 30, computed with equation (1):

$P=\frac{F}{{\left(1+i\right)}^n}=\frac{100}{{\left(1+0.02\right)}^{30}}=55.21$

$100-55.21=44.79$

The present value of the construction costs is computed as the stream of $250M outlays over four years is computed with equation (2):

$P=A\left[\frac{{\left(1+i\right)}^n-1}{i{\left(1+i\right)}^n}\right]=250\left[\frac{{\left(1+0.02\right)}^4-1}{0.02{\left(1+0.02\right)}^4}\right]=951.93$

Maintenance Costs are similar to benefits, in that they fall in the same time periods. They are computed the same way, as follows: To compute the Present Value of $2M in Maintenance Costs in Year 5, we apply equation (2) from above.

$P=A\left[\frac{{\left(1+i\right)}^n-1}{i{\left(1+i\right)}^n}\right]=2\left[\frac{{\left(1+0.02\right)}^{26}-1}{0.02{\left(1+0.02\right)}^{26}}\right]=40.24$

To convert that Year 5 value to a year 1 value, we apply equation (1)

$P=\frac{F}{{\left(1+i\right)}^n}=\frac{40.24}{{\left(1+0.02\right)}^4}=37.18$

As table 7 shows, the benefit/cost ratio of 2.5 and the positive net present value of $1563.31 million indicate that the project is worthwhile under these assumptions (value of time, value of life, discount rate, life of the road). Under a different set of assumptions, (e.g. a higher discount rate), the outcome may differ.

Table 7: Present Value of Benefits and Costs ($ millions)
	Present Value
Benefits	2,597.21
Costs
Right-of-Way	44.79
Construction	951.93
Maintenance	37.18
Costs SubTotal	1,033.90
Net Benefit (B-C)	1,563.31
Benefit/Cost Ratio	2.5