Why Unit Cost Why Unit Cost ProblemsProblems©Dr. B. C. Paul 2001©Dr. B. C. Paul 2001

Unit Cost Problems to Unit Cost Problems to Simplify AnalysisSimplify Analysis

Some problems can be awkward to solveSome problems can be awkward to solve Example -Example -

IDOT plans to make 127 from Murphysboro to IDOT plans to make 127 from Murphysboro to Interstate 64 into a 4 laneInterstate 64 into a 4 lane

IDOT considers building a concrete roadwayIDOT considers building a concrete roadway The road will cost $561 million spread in three The road will cost $561 million spread in three

equal payments over a 3 year construction period equal payments over a 3 year construction period (first payment now)(first payment now)

After construction the road will have to be restriped After construction the road will have to be restriped every 5 years for $200,000every 5 years for $200,000

IDOTs Concrete BabyIDOTs Concrete Baby

After 20 years of use the surface would become pitted After 20 years of use the surface would become pitted and the road would have to be resurfaced at a cost of and the road would have to be resurfaced at a cost of $150,000,000.$150,000,000.

After another 20 years the road would have to be After another 20 years the road would have to be resurfaced again for $150,000,000resurfaced again for $150,000,000

After 50 total years of use the highway base will begin After 50 total years of use the highway base will begin to break-up and the entire roadway will have to be to break-up and the entire roadway will have to be taken up and rebuilt from scratch for $625,000,000taken up and rebuilt from scratch for $625,000,000

Another BidAnother Bid

Ryan Buddies Inc. has brought IDOT another proposal to Ryan Buddies Inc. has brought IDOT another proposal to build an black-top highway instead.build an black-top highway instead.

The highway would cost only $400,000,000 and could be The highway would cost only $400,000,000 and could be built over two years (two payments of $200,000,000 the built over two years (two payments of $200,000,000 the first being now)first being now)

The road would have to be striped every 5 years of use The road would have to be striped every 5 years of use for $200,000.for $200,000.

After 6 years of use the highway will need $10,000,000 After 6 years of use the highway will need $10,000,000 in chuck hole repairsin chuck hole repairs

The next year (7) the chuck holes will cost $20,000,000 The next year (7) the chuck holes will cost $20,000,000 to repair.to repair.

Cuddling Black TopCuddling Black Top

In year 8 the chuck holes would cost $30,000,000 to In year 8 the chuck holes would cost $30,000,000 to repairrepair

In year 9 the chuck holes would cost $40,000,000 to In year 9 the chuck holes would cost $40,000,000 to repairrepair

At the end of 10 years service the road would have to be At the end of 10 years service the road would have to be completely resurfaced for $120,000,000completely resurfaced for $120,000,000

After 5 years of post resurfacing use the chuck hole saga After 5 years of post resurfacing use the chuck hole saga would repeat again.would repeat again.

After 20 total years of surface the roadbase would fail After 20 total years of surface the roadbase would fail and the entire road will have to be ripped up and rebuilt and the entire road will have to be ripped up and rebuilt at a cost of $455,000,000.at a cost of $455,000,000.

IDOTS DilemmaIDOTS Dilemma

Which type of highway should they build?Which type of highway should they build? What Kind of Problem does this look What Kind of Problem does this look

like?like?

All Cost Alternatives

The Classic ProblemThe Classic Problem

We have an All Cost Alternatives We have an All Cost Alternatives Problem with an Unequal Lives Problem with an Unequal Lives LandmineLandmine 50 year concrete highway50 year concrete highway 20 year asphalt highway20 year asphalt highway

Standard Solution OneStandard Solution One Turn the long lived asset in for salvageTurn the long lived asset in for salvage

How does one role up and resell a highway with How does one role up and resell a highway with 30 years of life left?30 years of life left?

Standard Solution TwoStandard Solution Two

Replace the low cost alternative to match Replace the low cost alternative to match the life of the long lived alternativethe life of the long lived alternative To get a common life for 20 and 50 years we To get a common life for 20 and 50 years we

need a 100 year road planneed a 100 year road plan Is that realistic?Is that realistic?

The Unit Cost SolutionThe Unit Cost Solution

Pick an interest rate and discount a full life cycle of each roads costs back to thestart of the road life.

0 1 7 8 9 10 11 12 17 18 19 20 21

$200,000$10,000,000

$20,000,000$30,000,000

$40,000,000

$120,000,000

$200,000$10,000,000

$20,000,000$30,000,000

$40,000,000

2

Interest Rate4.5% for taxfree bonds

-$620,651,717$200,000,000

Convert to Annual CostConvert to Annual Cost

-$620,651,717

Stretch this money into equal annual Payments Over the Life of the Road

* A/P4.5,20 =

0.07688

-$47,713,311/year

Lets Do This With Class Assistant Lets Do This With Class Assistant on the Concrete Highway!on the Concrete Highway!

-$187,000,000 each

0 1 2 3 8 13 18 23 28 33 38 43 48

$200,000 $200,000 $200,000

$150,000,000$150,000,000

Go to the Cash Flow Go to the Cash Flow Analyzer SectionAnalyzer Section

Set in our 4.5%Interest rate

Set it for annualcompounding

Enter the Cash FlowEnter the Cash Flow

Set Up the Total Life Set Up the Total Life Cycle Cost BoxesCycle Cost Boxes

To Use the ERR and Modified IRR you must specify the rate of interest available in your external investmentsEnter the rate of interest on external investments as a % (don't put %) 4

Period interest rate 0.04 (in decimal form)ERR (version 1) 0 % annual NPV investment 539699704 NFV earningsERR (version 2) 0 % annual NPV investment 537188869Mod. IRR #NUM! % annual

To Use the Total Life Cycle Cost you must specify the compounding period where the project goes into service andthe number of compounding periods that the project remains in service after starting

Start of Project Life 3Useful Life 50 PV at Start of Project -701590484

Total Life Cycle Cost -35501984

Tell it what year the road goesInto service.

Tell it how long the road lasts.

Read Off the AnswerRead Off the Answer

To Use the ERR and Modified IRR you must specify the rate of interest available in your external investmentsEnter the rate of interest on external investments as a % (don't put %) 4

Period interest rate 0.04 (in decimal form)ERR (version 1) 0 % annual NPV investment 539699704 NFV earningsERR (version 2) 0 % annual NPV investment 537188869Mod. IRR #NUM! % annual

To Use the Total Life Cycle Cost you must specify the compounding period where the project goes into service andthe number of compounding periods that the project remains in service after starting

Start of Project Life 3Useful Life 50 PV at Start of Project -701590484

Total Life Cycle Cost -35501984

The number by the Total Life CycleCost Heading is the Total Life CycleCost. - in this case $35,501,984

For interest the PVOf the cash flow atThe start of the projectIs also given.

Do the Same to the Do the Same to the Concrete RoadConcrete Road

Annual Cost of Concrete RoadAnnual Cost of Concrete Road -$35,501,984/year-$35,501,984/year

Compare this to the Black Top RoadCompare this to the Black Top Road -$47,713,311/year-$47,713,311/year

Which Road is Most Cost Effective?Which Road is Most Cost Effective? We converted an All Cost Alternatives We converted an All Cost Alternatives

Problem with a different lives problem Problem with a different lives problem into a unit cost probleminto a unit cost problem

Unit Cost ProblemsUnit Cost Problems

Problems of this type are sometimes called Problems of this type are sometimes called “Total Life Cycle Cost” in highway “Total Life Cycle Cost” in highway engineeringengineering

Most professions have some type of Most professions have some type of arrangement for unit costarrangement for unit cost Get all the money into an annual costGet all the money into an annual cost Divide the money by the number of units of Divide the money by the number of units of

interest you get (in highways it’s the service interest you get (in highways it’s the service year)year)

Compare the cost/unitCompare the cost/unit

Why Do We Do Unit Cost Why Do We Do Unit Cost Problems?Problems?

Its not really even its own type of problem (most are Its not really even its own type of problem (most are just all cost alternatives problems)just all cost alternatives problems) It’s a method of solutionIt’s a method of solution

Done in most fields because it presents the answer Done in most fields because it presents the answer in one number easy for someone in the business to in one number easy for someone in the business to understandunderstand If I tell you it costs 25 cents/mile to own and operate an automobile If I tell you it costs 25 cents/mile to own and operate an automobile

you understand fastyou understand fast

Done because it covers up nasty practical problems Done because it covers up nasty practical problems with simple All Cost Alternativeswith simple All Cost Alternatives Especially the infamous unequal lives problemEspecially the infamous unequal lives problem

Summary of 5 Types of Summary of 5 Types of ProblemsProblems

Invest and Earn ProblemInvest and Earn Problem All Cost Alternatives ProblemAll Cost Alternatives Problem Incremental Investment ProblemIncremental Investment Problem Competing Investments ProblemCompeting Investments Problem Unit Cost ProblemUnit Cost Problem

Basics NeededBasics Needed

Identify the investor and build a cash flow Identify the investor and build a cash flow showing money in and out of his pocketshowing money in and out of his pocket

Identify the point of decision and put the Identify the point of decision and put the pot at that locationpot at that location identify needed locations for any temporary identify needed locations for any temporary

potspots

The Six Magic NumbersThe Six Magic Numbers

P/FP/F F/PF/P P/AP/A A/PA/P F/AF/A A/FA/F There are a few other minor numbersThere are a few other minor numbers

Interest RatesInterest Rates

Interest Rates are almost always Interest Rates are almost always reported annuallyreported annually can be adjusted to other compounding can be adjusted to other compounding

periods so they can be used as i in magic periods so they can be used as i in magic numbersnumbers

Example - Convert to Monthly InterestExample - Convert to Monthly Interest Annual Rate% / 12 (convert to months) / 100 Annual Rate% / 12 (convert to months) / 100

(convert from percent to fraction)(convert from percent to fraction)

Components of InterestComponents of Interest

Safe Rate (about 2%)Safe Rate (about 2%) Inflation Rate (now around 4%)Inflation Rate (now around 4%) Risk Premium (depends on investment)Risk Premium (depends on investment) Motivation Premium (usually small)Motivation Premium (usually small) Dealt with by MultiplicationDealt with by Multiplication

(1.02)(1.04)(1.09)(1.001) = 1.1574(1.02)(1.04)(1.09)(1.001) = 1.1574 15.74%15.74%

If inflation is not included = Real RateIf inflation is not included = Real Rate If inflation is included = Nominal RateIf inflation is included = Nominal Rate