A Student's Guide to Cost-Benefit Analysis for Natural Resources

**Lesson 5 - Project Selection Criteria**

Introduction

We have introduced discounted cash flow analysis. We will examine investment criteria for selecting a project (i.e., formulae): Net Present Value (NPV), Benefit-Cost Ratio (B/C ratio), Internal Rate of Return (IRR) and for projects of unequal length (i.e., Equivalent Annual Net Benefits and Common Multiples of Duration).

**Net Present Value** **Criterion**

The Net Present Value (NPV) criterion is the principal government investment project evaluation criterion. The cash flows consist of a mixture of costs and benefits occurring over time. Net present value is merely the algebraic difference between discounted benefits and discounted costs as they occur over time. (You must think of the terms Anet present value@ and Anet present benefits@ as being interchangeable.) The formula for NPV is:

Where: NPV, t = year, B = benefits, C = cost, i=discount rate.

Two sample problem:

Problem #1) NPV; road repair project; 5 yrs.; i = 4% (real discount rates, constant dollars)

year = |
1 |
2 |
3 |
4 |
5 |

Benefits |
$0 |
1200 |
1200 |
1200 |
1200 |

- Cost |
$3000 |
0 |
0 |
500 |
0 |

B-C |
$-3000 |
1200 |
1200 |
700 |
1200 |

Disc. Factor |
1.04^1=1.04 |
1.04^2= 1.082 |
1.04^3= 1.125 |
1.04^4= 1.169 |
1.04^5= 1.217 |

Disc. Annual Cash Flows |
-$2884.61 |
1109.06 |
1066.67 |
601.89 |
986.03 |

Sum NPV = $879.04. Q: Go or no go? A: a single project with a positive NPV is a Ago.@

Problem #2) NPV, 5 yrs; i = 7.8%. Begins in time = 0.

Year= |
0 |
1 |
2 |
3 |
4 |
5 |

Benefit |
$0 |
2500 |
2500 |
2500 |
3000 |
3000 |

Cost |
$10,000 |
500 |
500 |
500 |
500 |
500 |

Net |
-$10,000 |
2000 |
2000 |
2000 |
2500 |
2500 |

Disc. Factor |
1.078^0=1 |
1.078^1= 1.078 |
1.078^2= 1.162 |
1.078^3= 1.253 |
1.078^4= 1.35 |
1.078^5= 1.45 |

Disc Cash Flow |
-$10,000 |
1855.28 |
1721.17 |
1596.17 |
1851.85 |
1724.14 |

Sum NPV = ($1125.39). Decision: Result is negative, hence no go.

Benefit/Cost Ratio

Most have heard of B/C ratio. Although not the preferred evaluation criterion, the B/C ratio does serve a useful purpose which we will discuss later. B/C formula:

Problem #3) Plant grass to reclaim a strip mine site and use for livestock grazing. 5 year project, i = 10% , begin time 0.

Year = |
0 |
1 |
2 |
3 |
4 |
5 |

Benefits |
0 |
0 |
0 |
0 |
$5000 |
20000 |

Disc Factor |
1.1^0=1 |
1.1^1=1.1 |
1.1^2=1.21 |
1.1^3= 1.331 |
1.1^4 = 1.464 |
1.1^5 = 1.61 |

PVBen |
0 |
0 |
0 |
0 |
$3415 |
$12,422 |

Cost |
$6,000 |
4000 |
1000 |
1000 |
1000 |
1000 |

PVC |
$6,000 |
3636 |
826 |
763 |
683 |
621 |

Sum benefits = $15,837

Sum costs = $12,529

B/C ratio = $15,837/$12,529 = 1.26. Q: Go or no go? A: for a single project go. But we=ll say more on B/C ratio and multiple project comparisons later.

Internal Rate of Return

The IRR is used more for private sector projects, but it is important to know.

IRR is different than our other project evaluation criteria. In our previous formula, i was a known and we solved for the discounted cash flows. With IRR,** **i is the unknown. IRR is the annual earnings rate of the project.

To find IRR we want to know: Awhat is the discount rate (i) that will equate a time series of benefits and costs?@ Or, otherwise stated: PVB = PVC; or where PVB - PVC = 0

or

Once the unknown Ai@ has been determined, you can compare i to the best available alternative rate of return. If the calculated i (IRR) is greater than the minimum acceptable rate of return (MARR) (i.e., you won=t accept an earning rate less than the MARR) then you will Ago@ with your project. Note: Calculated Ai@ = internal rate of return; MARR = external rate of return.

A word on computational difficulties: One problem with IRR is that it cannot be solved for in a direct algebraic fashion. Why? Recall from algebra, you need one equation for each unknown in order to solve. With IRR you have more unknowns than equations. Thus, you cannot solve for i.

Hence, IRR must be solved for in iterative Atrial-and-error@ fashion.

Procedure for trial and error:

1) set-up your annual benefits and costs separately

2) put in an initial discount rate, discount all benefits and cost,

3) examine to see if B=C

4) if not, repeat calculations with a new discount rate,

5) repeat calculations with a new i until B–C (to first decimal place).

IRR Problem #4) We take a series of annual cash flows, begin with 7% discount rate:

Year |
1 |
2 |
3 |
4 |
5 |

Cost |
$85,000 |
5000 |
5000 |
5000 |
5000 |

Disc Fact. |
1.07 |
1.14 |
1.22 |
1.31 |
1.4 |

PVC |
$79,439 |
4385 |
4098 |
3816 |
3571 |

Benefits |
$0 |
20000 |
25000 |
35000 |
50000 |

PVB |
$0 |
17534 |
20491 |
26717 |
35714 |

At 7% discount rate: sum PVB = $100456 - sum PVC = $95309 = $5147.

Decision: increase or decrease i? A; if B>C, the increase i and try again at 8% B-C = $2710. at 9% B-C = $586; at 9.3% benefits = costs, thus IRR = 9.3%

IRR is the annual earning rate of the project. Rule: accept project if IRR>MARR.

Projects of Unequal Duration

Thus far, we discussed projects without much discussion regarding the project duration.

The duration of projects is important, however, when you are comparing alternative projects. The rule: you cannot compare the NPVs of projects with unequal durations. You must make some adjustment for duration to make the comparable.

For example:

1) project A is 10 yrs. w/ NPV of $45,000; i=6%

2) project B is 15 years w/ NPV of $50,000; i=6%

Project B would seem to be the choice, but we cannot say because they are of unequal duration. You cannot compare projects of unequal service length.

Two Methods for Comparing Projects of Unequal Length:

1. EANB - compute equivalent annual net benefits (EANB). EANB restates NPV as a series of equivalent annual payments. It computes the amount needed to payoff a specified sum (NPV) in a series of equal periodic (e.g. annual) payments. Thus, 2 AEA projects@ are made comparable because their returns are annualized. The formula (NPV x Acapital recovery factor@):

Problem: 2 projects A&B.

A: project NPV =$45,000, t = 10, i = 6%

B: project NPV = $50,000, t = 15, i =6%

Question: which one should you undertake?

B seems better with higher NPV, but the 2 projects are of unequal length so you cannot compare just yet. You must use the EANB method. Work through this example. You will see that project A has the highest EANB, thus is the favored project.

2. Common Multiples of Project Duration: A second method of comparing projects of unequal duration is to compute the NPV using common multiples of project duration. Same problem:

1) project A is 10 yrs. w/ NPV of $45,000; i=6%

2) project B is 15 years w/ NPV of $50,000; i=6%

Steps:

1) find the common multiple in years of the 2 project lengths (in this case 30 years).

2) common multiple = 30 years. Thus 3-project As = 2-project Bs

3) for project A, the NPV of $45000 will cover the first 10 years. Another $45000 will be received 10 years hence, another $45000 is received 20 years hence. These 3 projects cover the 30 years.

4) Discount as follows:

Project A: NPV = $45,000 + $45,000/(1.06)^{10} + $45,000/(1.06)^{20} = $84,158

5) Project B NPV is $50,000 for first 15 years. Another $50,000 is received 15 years hence and this covers the last 15 years.

6) Discount:

Project B: NPV = $50,000 + $50,000(1.06)^{15} = $70,863

Decision:

Project A: NPV = $84,158 > Project B: NPV = $70,863

Conclude: accept project A.

Q: are EANB and common multiples methods consistent? Yes, they are consistently in their ranking of projects.

Links: