Cost Benefit Analysis 笔记

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本文是成本效益分析的笔记，教材是 Suzanne Bonner 的「Social Cost Benefit Analysis and Economic Evaluation」

L1 - Intro to Social CBA

Cost-Benefit Analysis (CBA) is: A process of identifying, measuring and comparing the social benefits and social costs of an investment project, program or policy intervention, from a public interest perspective.

Economic evaluations focus on societal perspective

A private or public sector project has implications for:

  • Employment
  • Government expenditure – provision of services
  • The Economy
  • The Environment
  • Government revenue – taxes, charges

Types of Market Failure

  • Externalities
  • Public goods
  • Incomplete information
  • Uncompetitive market structures

Net Social Benefit (NSB) = Total Social Benefit - Total Social Cost

  • NSB > 0 improves welfare
  • Larger NSB = more efficient resource allocation
  • a potential Pareto improvement

Types of CBA Analyses

  • CBA is used for both prospective (appraisal) and retrospective (evaluation)
  • Ex ante (or prospective) CBA
    • conducted prior to the decision to implement a new policy
    • Ex ante analysis is most useful
  • Ex post (or retrospective) CBA
    • conducted at the end of the intervention and all of the
    • effects have been realized
  • In medias res CBA
    • conducted during the intervention - after the decision to proceed, but before all impacts have occurred
    • is continuation of this policy a good idea?
  • Comparative CBA
    • compares the ex ante analysis to an in medias res or ex post analysis of the same project (very few of these comparisons have been conducted)

Not all costs and benefits can be easily quantified in monetary terms

L2 - Discounting and Decision Rules

shadow price: when market prices are adjusted to reflect true values

The discount rate is NOT the inflation rate

  • Discounting is based on the concept of time preference
  • Inflation refers to changes in prices over time

Discounting Single Period

Future Value Analysis

  • 𝐹𝑉 = 𝑋(1 + 𝑖)

Present Value Analysis

  • 𝑃𝑉 = 𝑌 / (1 + 𝑖)
  • Perpetuity 𝑃𝑉 = 𝑌/𝑖

Net Present Value Analysis

  • all the benefits (B) minus all costs (C) of a project (including the initial investment)
  • 𝑁𝑃𝑉 = 𝑃𝑉(𝐵) – 𝑃𝑉(𝐶)

Discounting Multiple Periods

Projects Over Multiple Years

  • FV=PV(1+i)tFV=PV(1+i)^{t}
  • PV=FV(1+i)tPV=\frac{FV}{(1+i)^{t}}
  • Discount Factor (or PV factor) = 1(1+i)t\frac{1}{(1+i)^{t}}

NPV=∑t=0nNB(1+i)tNPV=\sum_{t=0}^{n}{\frac{NB}{(1+i)^{t}}}

  • where NB (Net Benefit) = B - C

Decision Rule

  • NPV > 0: Undertake (Accept)
  • NPV < 0: Reject

Internal Rate of Return (IRR): The discount rate at which the NPV = 0

  • Can have multiple solutions for IRR

The IRR Decision Rule I

  • IRR > r: Accept
  • IRR < r: Reject

Hurdle Rate

  • the lowest rate of return a project or investment must achieve before a manager or investor deems it acceptable
  • often used in the private sector / private perspective CBA

Crossover Rate: the rate of return (or WACC) at which the NPV of two projects are equal

Benefit-Cost Ratio (BCR): 𝐵𝐶𝑅 = 𝑃𝑉(𝐵)/𝑃𝑉(𝐶)

  • Accept if BCR >= 1

L3 - Discounting Continued

Beginning of Year Discounting: PV=FV(1+i)t−1PV=\frac{FV}{(1+i)^{t-1}}

Mid-Year Discounting: PV=FV(1+i)t−0.5PV=\frac{FV}{(1+i)^{t-0.5}}

The End of Year Discounting result is more conservative.

Comparing Non Mutually Exclusive Projects

Project A B C D E
PV Capital Cost 400 500 300 150 100
PV Ongoing Cost 50 25 30 10 10
PV Benefits 1000 1200 200 300 125
Net Benefits (NPV) 550 675 -130 140 15
NB/PV(CapCost) 1.38 1.35 -0.43 0.93 0.15
Rank 1 2 5 3 4

Rank using ratio of net benefit to capital cost

Different budgets:

  • $900 - A + B
  • $1000 - A + B + E
  • $500 - B (> A + E)

Comparing Projects with Different Time Frames

Equivalent annual net benefit method (EANB)

Equivalent annual net benefit (EANB) identifies a yearly annuity received for the life of a project.

  • It is possible to convert any given amount or, any cash flow into an annuity

EANB = NPV / Annuity Factor (AF)

Annuity Factor: the sum of the discount factors over the life of the project where the stream of net benefits follows a fixed annuity format

DF=1(1+dr)tDF = \frac{1}{(1+dr)^t}

Year Net Benefit DF
0 -800
1 400 0.9434
2 400 0.89
3 400 0.8396

Therefore, AF = 2.6730

Since this is a geometric series, we can also calculate with the following formula

AF=1−(1+dr)−tdrAF = \frac{1-(1+dr)^-t}{dr}

NPV = $269.20

EANB = 296.20 / 2.6730 = 100.71 per year

If annuity in perpetuity: AF = 1 / dr

Pros of EANB

  • it annualizes the NPV allowing for a per year net social benefit comparison between projects on a per year common metric
  • results in a consistent decision when comparing projects with unequal lives

Limitation

  • project comparison is the selection of the discount rate

Example EANB using Annuity Factors

Project A is a solar farm. It has NPV of $20 and a life of 25 years

Project B is a hydroelectric plant has a NPV $30 million and continues in perpetuity

The annuity factor, at 4 percent is:

  • 15.62 for 25 year project
  • 25 for perpetuity

EANB A = 20/15.62 = 1.28 million

EANB B =30/25 = 1.20 million

If the projects are mutually exclusive, project A should be selected as it has a higher EANB

Roll-Over Method

Project X – has a life of six years, an initial cost of $60,000 and a yearly net benefit stream of $15,000

Project Y – has a life of three years, an initial cost of $40,000 and a yearly net benefit stream of $20,000

Assuming dr = 6%

Year Project X Project Y
0 ($60,000) ($40,000)
1 $15,000 $20,000
2 $15,000 $20,000
3 $15,000 $20,000
4 $15,000 ($20,000)
5 $15,000 $20,000
6 $15,000 $20,000

Note: Project Y Year 4 = -$40000 + $20000

NPVrollover=NPV1+∑NPV1(1+dr)(t−n)NPV_{roll over}=NPV_1+\sum\frac{NPV_1}{(1+dr)^{(t-n)}}

  • NPV1NPV_1 = initial NPV of the project for the normal life span
  • dr = discount rate
  • t = total number of periods required for the roll-over
  • n = initial project duration

NPV_X = $13,759.86

NPV_Y = $13,460.24 + $13,460.24/(1.06)^3 = $24,761.72

Terminal (Horizon) Values

The costs or benefits associated with a project may continue years into the
future beyond the project scope

Horizon Values

  • considers cost / benefit far future seperately
    • in some cases the scrap value is used as the horizon value
  • long "near future" discounting period
    • = assuming a zero horizon value

If horizon value is perpetuity: PV = FV / dr

Dealing with Inflation

nominal dollars - nominal discount rate

real dollars - real discount rate

Real interest rate: r = (i - m) / (1 + m)

  • r = the real interest rate
  • i = nominal interest rate
  • m = inflation rate

Example: If the nominal interest rate is 5% and the inflation rate is 2% then the real interest rate is?

r = (0.05 - 0.02) / (1 + 0.02) = 2.94%