Engineering Economics Module 8

7/30/2019 Engineering Economics Module 8

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Module 8: Minimum Attractive Rate of Return (MARR)

Often we are less interested in determining an exact IRR than we are inknowing whether a proposed investment will earn a sufficiently high rate ofreturn.

For example, if I am considering investing in a 10-year Oregon generalobligation bond, I may want to consider only those bonds that have an IRR of4% or above. If I evaluate a bond and find its IRR to be < 3%, I dont have todo any further calculations because it clearly doesnt meet my criterion of IRR 4%. If I find a bond with an 4% < IRR < 5% then I know it would beacceptable since IRR > 4%.

The minimum rate I am willing to accept for an investment is called theMinimum Attractive Rate of Return (MARR).

Decision rule: When the MARR is known, you should accept the projector investment when IRR MARR.

How is the MARR determined? This varies with the type of organization orindividual decision-maker. MARR is usually set sufficiently high to allow forsome risk of estimation errors in future costs and benefits.

For example, if you would really like to guarantee a 10% return on aninvestment, you might set your MARR to 15%, 20% or even higher to allow for

these risks. We have certainly seen a lot of variability in the price of gasoline,electricity, and natural gas in recent years. If we are estimating these costsinto the future we run a risk of overestimating or underestimating these costs.Similarly, repair costs, insurance costs, and labor costs can be difficult toestimate. If we plug in our best estimates and calculate an IRR of only 10%we may find out after several years that our true rate of return was much lowerif costs were higher than we had estimated.

It is not unusual to find economic analyses using 30% and 40% MARRs toevaluate potential projects.

Module 8: Internal Rate of Return

Concept:The internal rate of return (IRR) is the interest rate paid on theunpaid balance of a loan that causes the loan principal to be fully repaidwhen the final payment is made.


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Until this point in our course, all problems you have been asked to solve havesupplied an interest rate. In this module you will be solving for the interest rategiven other information about costs, benefits, and analysis period.

Mathematically, this can be more difficult than the calculations you mastered

in earlier modules. At times you may have to revert to a trial and error processif your calculator does not have an IRR function built in. Alternatively, you mayfind a spreadsheet to be helpful. (Your textbook discusses the IRR and RATEfunctions in Excel that can be used.)

The IRR is the interest rate that balances the Present Worth of Costs(PWc) of an investment with the Present Worth of Benefits (PWb). Inother words, IRR is the interest rate that, when used to calculate PWband PWc, results in PWb = PWc (and, as a result, NPW = 0).

If PWb = PWc when calculated using the IRR as the interest rate, what mustalso be true about the relationship of EUAB and EUAC?

Reality check:If you cant answer this last question, you should re-visit theearlier discussions about the relationship of EUAB and PWb and therelationship of EUAC and PWc.

IRR Calculations: IRR problems solve for ias the unknown variable.

Example:

You borrow $8200 and pay pack $2000 per year for each of 5 years. Whatinterest rate have you paid? (Or, stated from the lenders perspective, youlend $8200 and receive 5 equal annual payments of $2000.) What is the rateof return on your investment?

Solution: This is a uniform series problem in which we know A, P, and n. Wecan set up our solution in one of two forms:

Solve for P given A:

P = A(P/A,i,n)

8200 = 2000 (P/A,i,5)

Isolate the (P/A,i,5) factor as your unknown by dividing both sides of theequation by 2000:


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8200/2000 = (P/A,i,5)

4.1 = (P/A,i,5)

Now its a matter of searching through the interest rate tables to find a rate for

which the (P/A,i,5) factor has a value of 4.1. This is found on the 7% table.This investment has earned a 7% rate of return.

The alternative form of the solution, given P,A, and n is to solve for A given P:

A = P (A/P,i,5)

2000 = 8200(A/P,i,5)

2000/8200 = (A/P,i,5)

0.2439 = (A/P,i,5)

The 7% table shows (A/P,7%,5) = 0.2439 so this investment is again shown tohave a 7% IRR.

Notice that the entire factor, whether we used (P/A,i,5) or (A/P,i,5) wasthe unknown in our equation. Once we determined its value throughalgebra, it was a matter of trial and error to find the appropriate interest ratetable that contained that value for a 5-year solution.

Earlier we said that the IRR was the interest rate that set PWb = PWc. Doesthat apply in this problem? Yes. From the borrowers perspective, PWb =$8200. This is the money you borrowed in year 0 and is a positive cash flowfor you. (Presumably you used the money to buy something that has benefit toyou.) Your annual costs are the $2000 repayments (cash outflows). PWc wascalculated to be $8200 when i= 7%. (Thats what we found in the first solutionwhen we found P given A.) So, the PWc of the 5 years payments of $2000was $8200. PWb indeed equals PWc at 7%.

If youre still not convinced, try solving this problem using 6% or 8% interest:

At i= 8%:

PWc = 2000(P/A,8%,5)

PWc = 2000(3.993)


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PWc = $7986.

At i= 8%, PWb = $8200 and PWc = $7986 so PWb > PWc.

At i= 6%:

PWc = 2000(P/A,6%,5)

PWc = 2000(4.212)

PWc = $8424.

At i= 6%, PWb = $8200 and PWc = $8424 so PWb < PWc.

We know from our solution above that at i= 7%, PWc = 2000(4.100) = $8200

so PWb = PWc when i= 7%.

This does not tell us whether 7% is a good rate of return on our investmentor a reasonable interest rate for our borrowing. It simply tells us that 7% is therate of return.

Example:

You spend $700 to upgrade your heating system. You save the following inheating costs each year as a result:

Year Savings

1 $100

2 175

3 250

4 325

What is the IRR of this project?

Solution:

Find the interest rate that sets EUAB = EUAC or sets PWb = PWc. It doesntmatter whether you solve for present worth or the uniform amount becausethe correct interest rate will work for both. Sometimes the nature of the cashflows can make one approach easier than the other. Well solve it both ways.

Solution method 1: Set PWb = PWc.


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PWc = $700 (spent on the system)

PWb = 100(P/A,i,4) + 75(P/G,i,4) (notice how we constructed a uniform series+ a gradient)

100(P/A,i,4) + 75(P/G,i,4) = 700

Since this is a single equation with two unknowns ((P/A,i,4) and (P/G,i,4) areboth unknown), this can only be approached by trial and error:

guess an interest rate

find the values of the factors from the interest rate table

evaluate the resulting expression

Lets guess 5%:

(P/A,5%,4) = 3.546

(P/G,5%,4) = 5.013

100(3.546) +75(5.013) = $730.58 so PWb > PWc at i= 5%

In our earlier example, we found that when PWb > PWc, our IRR was greater

than the interest rate we had tried. We can conclude here that the IRR > 5%.

Go ahead and try a higher rate and re-evaluate PWb. Well give you theanswer after showing you the second way to solve this problem, namelysetting EUAB = EUAC.

Solution method 2: Set EUAB = EUAC.

EUAC = 700(A/P,i,4) (Remember, we need to annualize the initial purchaseprice)

EUAB = 100 + 75(A/G,i,4) (This is annualizing the benefits of 100 plus thegradient of 75)

100 + 75(A/G,i,4) = 700(A/P,i,4)

Again, we have 2 unknowns in a single equation so we must resort to trial anderror as we did with the PWb = PWc equation in Solution method 1.


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Guess 8% (since we know 5% is too low from our first trial and error attempt):

EUAB = 100 + 75(1.404) = $205.30

EUAC = 700(.3087) = $216.09

At i= 8%, EUAB < EUAC. This tells us that IRR < 8%.

So far we have determined that 5% < IRR < 8% so were narrowing in ourchase.

Lets try i= 7%:

EUAB = 100 + 75(1.416) = $206.20

EUAC = 700(.2952) = $206.64

Within rounding, EUAB = EUAC when i= 7% so IRR = 7%.

Further comment on the estimation procedure:

The two problems above had solutions found on the interest rate tables.However, most real world problems end up with IRRs that are not found onthe tables, such as 2.6% or 17.4%. What do you do if youve evaluated PWband PWc (or EUAB and EUAC) and discover that one tables rate is too low

and the next tables rate is too high? Depending on the problem you wouldconsider one of the following options:

State that the IRR is between x% and y% and do not try to estimate anymore precisely. For example, you may find that 9% < IRR < 10%. Thatmay be close enough for the nature of the problem, especially ifalternative investments have IRRs significantly below or significantlyabove this range.

Estimate a new interest rate within the range youve identified, say 9.5%,

and use the calculator-based formulas or a spreadsheet to continue toevaluate PWb and PWc (or EUAB and EUAC) using this more preciserate. Continue to narrow down the rate in this manner until you find, to thenumber of decimal places you or your boss finds appropriate, the IRR.

Use linear interpolation to estimate the IRR within the range youveidentified. This is a cruder, less precise estimation method than the


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calculator-based method but it will be closer than simply stating the rangeof x% < IRR < y%. We will present an example of using linearinterpolation when we discuss bond yields later in this module.

Module 8: Bond Analysis

This section discusses the relationship between the rate of return on a bondand the price of the bond.

What is a bond?

A bond is a legal debt instrument. It obligates the bond issuer to repaythe bond holder a particular amount called the Face Value at a knowntime in the future (the maturity date). It may also obligate the bondissuer to repay the bond holder a specific amount of money, either

annually or semi-annually (twice a year) until the bond reaches itsmaturity date.

Bonds are issued by corporations or different branches of government. Bondsissued by state or local governments are known as municipal bonds and theinterest payments received by the bond holder over the life of the bond areexempt from federal income tax (and state and local income tax if the bondholder resides in the city or state that issued the bond). Interest paymentsreceived on corporate bonds are fully taxable on federal and state tax returns.

You are probably familiar with US Savings Bonds, issued by the federalgovernment. US Savings Bonds do not pay any periodic interest but hold allpayment until maturity. Bonds that pay no periodic interest are known as zerocoupon bonds.

Here are the key components of a bond:

Purchase price: The price the bond holder pays for the bond when buying itfrom the original issuer or in the bond market from another seller.

Maturity: The date when the bond will be paid in full by the bond issuer to thecurrent bond holder. At the time of maturity, the bond holder will receive thefinal interest payment plus the face value amount of the bond and the bondwill be retired (fully paid off). Maturities may be as short as several months oras long as 30 years.


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Face value (or denomination): The amount the bond holder will receive atmaturity.

Interest rate (or coupon rate): The percent of the face value that will bepaid as interest to the bond holder each year until maturity. For example, if the

bond has a $5000 face value and pays 5% interest, the bond holder willreceive .05*$5000 = $250 interest every year until maturity. If the bond paysinterest semi-annually, the bond holder will receive half this amount ($125)every six months.

Current yield (or yield to maturity): The rate of return the bond holderreceives on this bond. If the purchase price paid by the bond holder equalsthe face value of the bond, the current yield will be the same as the couponrate. However, it is possible to pay more or less than the face value whenbuying the bond so the current yield is usually different from the coupon rate.

(As you will see in this section, if the bond holder pays less than the facevalue, the current yield will be greater than the coupon rate. If the bond holderpays more than the face value, the current yield will be less than the couponrate. Bond price and current yield are, therefore, inversely related.)

Example:

A bond is issued that will mature in 10 years. It has a $1000 face value and a 4%interest rate, interest to be paid semi-annually. If the bond sells for $1000, what is itsyield (rate of return)?

Here is the time-line for this problem:

Period

Payment madeorreceived by

bondholder

0 -$1000 bought the bond

1 20 interest received at end of 6 months = .5 * .04 * 1000

2 20 interest received at end of 1 year

3 20 interest received at end of 1.5 years

etc.19 20 interest received at end of 9.5 years

20 1020 interest for final 6 months plus return of face value

To find the rate of return on this bond, set PWb = PWc

PWc = $1000 (the purchase price of the bond)


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PWb = 20(P/A,i,20) + 1000(P/F,i,20)

Notice that PWb consists of a uniform series of the $20 semi-annual interestpayments for 10 years (20 periods) plus the one-time final payment of $1000at the end of period 20. The 10-year horizon is a 20-period problem because

interest payments are made semi-annually.

Caution:

When you solve this problem as stated, you are solving for the nominalsemi-annual (6-month) interest rate. To find the nominal annual rate ofreturn, you must multiply this value by 2 to convert from semi-annual toannual.

Try guessing 2% for the semi-annual rate of return (half of the 4% annual

interest rate):

at i= 2%:

PWb = 20(P/A,2%,20) + 1000(P/F,2%,20)

PWb = 20(16.351) + 1000(.6730)

PWb = 327.02 + 673 = $1000.02

Within rounding, PWb = PWc at i= 2% so the semi-annual rate of return is 2%and the annual rate of return (yield) is 4%.

Example:

Now suppose the same bond is bought for $880 instead of $1000. What is therate of return (yield) on the bond?

PWc = $880

PWb = 20(P/A,i,20) + 1000(P/F,i,20)

Note that PWb doesnt change in format. The bond holder will still receive $20interest every 6 months plus the $1000 face value at maturity. The only thingthat changes is the price of the bond which is PWc.


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If you pay less than the face value for the bond, your rate of return will behigher than the stated interest rate. So, we know that annual i> 4% and semi-annual i> 2%. Lets try 2.5% to evaluate PWb.

at i= 2.5%:

PWb = 20(P/A,2.5%, 20) + 1000(P/F,2.5%,20)

PWb = 20(15.589) + 1000(.6103) = $922.08

This is closer to $880 than PWb when i= 2% but it may not be close enough.Try at i= 3%:

at i= 3%:

PWb = 20(P/A,3%,20) + 1000(P/F,3%,20) = $851.24

The semi-annual interest rate that sets PWb = PWc is between 2.5% and 3%,meaning the annual rate of return is between 5% and 6%. This is a prettylarge spread. To be more precise, we can use linear interpolation to refineour estimate.

Linear interpolation assumes our answer lies on a straight line between i=2.5% and i = 3%. This is not exactly correct since interest rates arecompounding (hence growing geometrically, not linearly) but it will give us a

close enough approximation when the boundaries of the known rates are fairlyclose as in this example (where the difference between 2.5% and 3% is only1/2%).

Here is a formula for linear interpolation:

Notice that the denominator of the above equation calculates the difference in

value between PWb when i= 2.5% (which was $922.08) and PWb when i=(2.5% + 0.5%) which was $851.24. The numerator determines the differencebetween PWb when i= 2.5% and the actual PWc that was paid for the bond.Since the denominator spans the effect of adding a full 0.5% interest rate toour base of 2.5%, the numerator effectively tells us what fraction of thatdistance (or what fraction of 0.5%) we actually covered.


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Solving this equation:

i= 2.797%The semi-annual rate of return is 2.8% the annual rate of return (currentyield) = 5.6%.

Why does a bond sell at a price below its face value?

A bond that sells below its face value is said to sell at a discount. As the market rate ofinterest rises (often for very complex reasons that go beyond the scope of this course),a potential seller of a bond must price the bond to be competitive with other similarbonds in the market. This means that corporate bonds with similar risk characteristicsshould be priced similarly to each other. Municipal bonds should be priced similar toother municipal bonds with similar risk characteristics. (Bond risk is also a complicatedpart of financial analysis and beyond the scope of this course.)

Since the face value and semi-annual interest amounts cannot change over the life of

the bond (F and A are constant), the only way to make the bond competitive is to lowerits price. As long as an investor is able to earn 5.6% on a bond, he or she would notcare if the bond is priced at $880 as in this example or if it were priced at $1000 but paid$28 interest every six months (instead of the $20 in our example) for 10 years.

The second bond would also yield 5.6% with the following PWb and PWc:

PWc = $1000

PWb = 28(P/A,i,20) + 1000(P/F,i,20)

If you solve the PWb equation using the calculator-based formulas where youcan use interest rates not on the tables, the semi-annual interest rate will be2.8%. Try it and see for yourself.

Why does a bond sell at a price above its face value?


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If market interest rates fall, a bonds price could rise above its face value. Thatbond would be said to be selling at a premium. Again, this would happen tokeep the yield on this bond competitive with the yield on similar bonds in themarket. The higher price increases PWc. Since F and A (the face value andperiodic interest amounts) are constant, imust fall to let PWc = PWb.

Example:

The same 10 year $1000 bond paying 4% interest semi-annually is sold for$1042. What is the bonds current yield?

PWc = $1042

PWb = 20(P/A,i,20)+ 1000(P/F,i,20)

Try i = 1.75% semi-annually (you know i< 2% because PWc > face value)and 20/1000 = 2%)

PWb = 20(P/A,1.75%,20) + 1000(P/F,1.75%,20)

PWb = 20(16.753) + 1000(.7068)

PWb = 335.06 + 706.80 = $1041.86

The bonds semi-annual yield = 1.75% the annual rate of return (current

yield) = 3.5%.

Engineering Economics Module 8

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