Unit Cost Problems to Simplify Analysis

1. Unit Cost Problems to Simplify Analysis • Some problems can be awkward to solve • Example - • IDOT plans to make 127 from Murphysboro to Interstate 64 into a 4 lane • IDOT considers building a concrete roadway • The road will cost 85 million spread in three equal payments over a 3 year construction period (first payment now) • After construction the road will have to be restriped every 5 years for $200,000

2. IDOTs Concrete Baby • After 20 years of use the surface would become pitted and the road would have to be resurfaced at a cost of $30,000,000. • After another 20 years the road would have to be resurfaced again for $30,000,000 • After 50 total years of use the highway base will begin to break-up and the entire roadway will have to be taken up and rebuilt from scratch for $95,000,000

3. Another Bid • Ryan Buddies Inc. has brought IDOT another proposal to build an black-top highway instead. • The highway would cost only $50,000,000 and could be built over two years (two payments of $25,000,000 the first being now) • The road would have to be striped every 5 years of use for $200,000. • After 6 years of use the highway will need $1,000,000 in chuck hole repairs • The next year (7) the chuck holes will cost $2,000,000 to repair.

4. Cuddling Black Top • In year 8 the chuck holes would cost $3,000,000 to repair • In year 9 the chuck holes would cost $4,000,000 to repair • At the end of 10 years service the road would have to be completely resurfaced for $25,000,000 • After 5 years of post resurfacing use the chuck hole saga would repeat again. • After 20 total years of surface the roadbase would fail and the entire road will have to be ripped up and rebuilt at a cost of $55,000,000.

5. IDOTS Dilemma • Which type of highway should they build? • What Kind of Problem does this look like? All Cost Alternatives

6. Cash Flow for Two Alternatives

7. Lets Pick Black Top

8. How Do You Like My Cash Flow? • Like many comparisons of very different alternatives this cash flow swings negative to positive (wrecking IRR) • One of the features is that my Black Top choice saves me a lot of money after the first 20 years • But that’s because the Black Top highway is dead and gone - the concrete highway is still serving • Does that sound like valid data for decision making?

9. The Unequal Lives Problem • Very often when comparing alternatives for doing something there is a cheap version that has a short life and lot of maintenance and an expensive version with low maintenance and long life • My first attempt to saying I’ll get something run it till it dies and then leave the business is usually not the way people do things.

10. Standard Solutions • Try to do enough replacements of the cheap choice to get an equal life to the long lived item • Getting a number with an integer number of long life and short life choices can be a joke - try to get a common life on my roadway. • The Salvage Value Approach • Run the analysis to the life of the short lived alternative • Turn the long lived item in for salvage (a positive value)

11. The Salvage Value Approach • Issue #1 - Unlikely in real life that people will trade in well built equipment every time cheap equipment would have worn out • When you do things in cash flows that are vastly different from real life you risk getting wrong answers • Issue #2 - Most people buy equipment because it produces value for them. • Salvage market seldom pays the value of what a long lived item could still produce - (When was the last time you know of someone who had their car totaled and thought the insurance company gave them a fair settlement?)

12. More Salvage Value Issues • Salvage Markets are highly variable and regional for used heavy equipment • May mean that where a plant is located makes a big difference because of a salvage market that no one will really use • Some things don’t salvage well • Especially things that don’t move well (a lot of civil engineering structures are not very mobile) • There may not be a salvage market (such as salvage for a concrete highway with 30 years life left)

13. Truncating the Problem • Remember for the endless life problems I said usually the first 20 to 30 years makes the NPV - just cut it off • Issues with truncation for unequal life • often the cheap choice cuts off before end of decisive time for cash flow • In the case of highway projects were low cost government financing may be involved the critical life can be longer than 30 years

14. Problems with Truncation • Can be prone to manipulation • May try to cut-off just before a big expense or earning from one alternative or the other • Even if not manipulating you may not know how close to a major cost you can get without distorting result • Perhaps best truncation choice is to do multiple cycles of the cheap product till someplace close to long life if that is beyond the NPV forming years.

15. The Most Clear Conclusion • Problems comparing long and short lived investments (especially all cost alternatives problems) force people to make shaky approximations and assumptions to handle unequal lives • If I have a big list the chances that some of the choices will create iffy cash flows increases • Civil Projects for Highways can be especially prone to trigger problem patterns

16. $200,000 $200,000 $1,000,000 $1,000,000 $2,000,000 $2,000,000 $3,000,000 $3,000,000 $4,000,000 $4,000,000 The Unit Cost Solution Pick an interest rate and discount a full life cycle of each roads costs back to the start of the road life. 0 1 5 6 7 8 9 10 15 16 17 18 19 2 Interest Rate 4.5% for tax free bonds $25,000,000 $25,000,000 -$82,615,801

17. Stretch this money into equal annual Payments Over the Life of the Road Convert to Annual Cost * A/P4.5,20 = -$6,351,184/year -$82,615,801 0.07688

18. Do the Same to the Concrete Road • Annual Cost of Concrete Road • -$5,661,386/year • Compare this to the Black Top Road • -$6,351,184/year • Which Road is Most Cost Effective? • We converted an All Cost Alternatives Problem with a different lives problem into a unit cost problem

19. Unit Cost Problems • Problems of this type are sometimes called “Total Life Cycle Cost” in highway engineering • Most professions have some type of arrangement for unit cost • Get all the money into an annual cost • Divide the money by the number of units of interest you get (in highways it’s the service year) • Compare the cost/unit

20. Why Do We Do Unit Cost Problems? • Its not really even its own type of problem (most are just all cost alternatives problems) • It’s a method of solution • Done in most fields because it presents the answer in one number easy for someone in the business to understand • If I tell you it costs 25 cents/mile to own and operate an automobile you understand fast • Done because it covers up nasty practical problems with simple All Cost Alternatives • Especially the infamous unequal lives problem

21. Summary of 5 Types of Problems • Invest and Earn Problem • All Cost Alternatives Problem • Incremental Investment Problem • Competing Investments Problem • Unit Cost Problem

22. Basics Needed • Identify the investor and build a cash flow showing money in and out of his pocket • Identify the point of decision and put the pot at that location • identify needed locations for any temporary pots

23. The Six Magic Numbers • P/F • F/P • P/A • A/P • F/A • A/F • There are a few other minor numbers

24. Interest Rates • Interest Rates are almost always reported annually • can be adjusted to other compounding periods so they can be used as i in magic numbers • Example - Convert to Monthly Interest • Annual Rate% / 12 (convert to months) / 100 (convert from percent to fraction)

25. Components of Interest • Safe Rate (about 2%) • Inflation Rate (now around 4%) • Risk Premium (depends on investment) • Motivation Premium (usually small) • Dealt with by Multiplication • (1.02)(1.04)(1.09)(1.001) = 1.1574 • 15.74% • If inflation is not included = Real Rate • If inflation is included = Nominal Rate