Summary of comparison methods we know so far:

Present worth

Future worth

Annual worth

Still to come:

Rate of return

Cost/benefit

Payback period

Comparing Alternatives

In comparing alternatives, any cost common to all the alternativescan be ignored.

This will sometimes result in all the alternatives having negativepresent worth. Don’t worry – just choose the least negative.

In many situations – for example, Question 2.5 – you will need tobe careful to set up the comparison fairly.

Study Period

The Chopper lawnmower costs $100, requires $10 in fuel every year,and will last 4 years. The LawnBoy lawnmower costs $120, uses $15in fuel every year, and lasts 6 years. If I expect to be cutting lawns for the rest of my life, which is the better buy?

Either determine a salvage value:

PW(Chopper) = -100 - 10(P/A, i, 4) PW(LawnBoy) = -120 - 15(P/A, i, 4) + S(P/F, i, 4)

Or consider the lowest common multiple of lives:

PW(Chopper) = -100 - 10(P/A, i, 12) – 100(P/F,i,4) – 100(P/F,i,8) PW(LawnBoy) = -120 - 15(P/A, i, 12) - 120(P/F, i, 6)

The Chopper lawnmower costs $100, requires $10 in fuel every year,and will last 4 years. The LawnBoy lawnmower costs $120, uses $15in fuel every year, and lasts 6 years. If I expect to be cutting lawns for the rest of my life, which is the better buy?

…Or look at Equivalent Uniform Annual Worth:

EUAC(Chopper) = -100(A/P, i, 4) – 10

EUAC(LawnBoy) = -120(A/P, i, 6) - 15

Sets of Alternatives

Alternatives may be independent, contingent, or exclusive

Independent: Buying socks and buying a tie

Contingent: Buying shoes and buying socks

Exclusive: Buying a Lamborghini and buying a Ferrari

In a complex situation, arrange the possibilities into mutually exclusivesubsets and pick the best subset.

Example: You can do A, B, C , D or E. If you do A, you can’t do B. To doD, you must do either B or C. If you do E, you must do A. You can’t do both B and C, and if you do D, you can’t do E.

What are the possible subsets?

Rates of Return

What is your interest rate?

The MARR

This is your Minimum Acceptable Rate of Return

If you’re in business at all, it’s because you think youcan make more money from your investment than thebank can.

So, your MARR must be at least as great as the bankrate.

The MARR

If all the initial investment in the business is yours,your MARR is whatever you think it should be.

If you’ve borrowed some of the money from the bank,your MARR must be enough to pay the interest on theloan.

If your business is owned by its shareholders…

your rate of return must be high enough to keep them happy.

The Internal Rate of Return

One way of comparing projects is to calculate theirrates of return.

If the project’s rate of return is less than your MARR,don’t do it.

If you have several projects, the one with the highestrate of return might be the best…

The Internal Rate of Return

If your project requires a single present investment, P, and yields a single future payout, F, in N years time, then its rate of return is the solution to:

P = F(P/F, i, N)

The Internal Rate of Return

This can be re-stated as:

PW = P - F(P/F, i, N) = 0

More generally, the internal rate of return, or IRR,is the interest rate which makes the total present worth of the project cashflows equal to zero.

Example:

We build a bicycle factory at an initial costof $800 000. We hire workers at $100 000 ayear, payable at year’s end. In the first fiveyears of the project, our year-end sales are:

End of Year Sales ($ 000)

1 80

2 200

3 500

4 1000

5 1200

What is the rate of return for the project?

A good way of solving these problems is tobuild a spreadsheet.

We work out the value of:

PW = - 800 - 20(P/F,i,1) + 100(P/F,i,2)

+ 400(P/F,i,3) + 900(P/F,i,4) + 1100(P/F,i,5)

For different values of i and plot a graph.

i -800 -20(P/F,i,1) 100(P/F,i,2) 400(P/F,i,3) 900(P/F,i,4) 1100(P/F,i,5) PW0.00 -800 -20 100 400 900 1100 16800.05 -800 -19 91 346 740 862 1220

0.10 -800 -18 83 301 615 683 8630.15 -800 -17 76 263 515 547 5830.20 -800 -17 69 231 434 442 3600.25 -800 -16 64 205 369 360 1820.30 -800 -15 59 182 315 296 370.35 -800 -15 55 163 271 245 -810.40 -800 -14 51 146 234 205 -1790.45 -800 -14 48 131 204 172 -260