Rate of Return and the Cost of Capital A big rate of return means you have to come up with a lot of extra money to get the investors to put-off their Dairy Queen Blizzards A small ROR means you only need a little extra

So Why All This Rate of Return Business? Remember all engineering econ problems involve –Write down the money that goes in and out of the project for each year in order (get your cash flow) –Multiply each number in the cash flow by a magic number –Add up the total and see whether the money pile is big enough Rate of Return Tells you how big the money pile has to be

Example If I put $1.00 in the bank at 5% interest, how much money will I have next year when I take the money out –5% of $1.00 is 5 cents –I will have $1.05 If I leave the money in the bank another year I will get 5% interest on $1.05 not just $1.00 –At the end of the year I will have (1.05)(1.05)=

Example continued If I leave the money in another year I will get 5% interest on $ –(1.1025)(1.05) = –and again the next year (1.1576)(1.05) = My interest is “Compounding” –Note that if I only got 5% each year on my dollar I would only have $1.20 The sneaky trick with interest is to multiply, not add (multiplication takes care of compounding)

Compounding Period In the previous example I got my interest every year and then I started compounding the interest on the interest. Why does the interest have to compound once a year - it doesn’t –Ever noticed CD rates at Banks 4.6% interest with a 4.75% yield? –They pay interest and compound it over shorter times so that by the end of the year the ROR is higher than the interest rate –Interest Rates are Usually Reported on an Annual Basis

The Credit Card Rip-Off Sammy Sucker gets a credit card offer from Spin on My Finger Bank and Trust –The interest rate is 18% (but they’ll give him a 5% purchase credit toward a new Turbo charged Volkswagon Beetle that will make all the girls think he is sexy) –Sammy goes out and maxes out his credit card at $10,000 –We’ll ignore his monthly minimum payments for a while

Sammy gets Spin on My Finger Bank and Trust divides the interest rate over 12 months –18%/12 months = 1.5% per month Month #1 Sammy doesn’t pay off his card –1.5% of $10,000 –(10000)*(1.015) = $10,150 or $10,150- $10,000 is $150 of interest

Sammy’s Adventure Month #2 Sammy doesn’t pay off his credit card –Spin on My Finger Bank and Trust compounds the interest –$10,150*(1.015) = $10, Month #3 Sammy doesn’t pay off his credit card –$10, * (1.015) = $10,456.78

This is Sammy’s Adventure - Not Ours I really love these calculations but if I have to do them 12 times I’m going to puke Enter Super formula –Note that all I’m doing is multiplying the original debt $10,000 by –Note that 1.015*1.015 is just (1.015) 2 –Note that 1.015*1.015*1.015 is just (1.015) 3 –Note that is just 1 plus the interest rate Magic formula (1 + i) n –where i is the interest rate –and n is the number of compounding periods

Now Lets Return to Sammy’s Saga After 1 year how much does Sammy owe? –He’s had 12 compounding periods at 1.5% interest each time –The magic formula is (1.015) 12 = Apply the formula to Sammy’s Debt –$10,000 * = $11,956 Note that Sammy paid – = or 19.56% interest because of compounding - not 18%

What Else is New Note that Sammy’s spending $10,000 is a cash flow number Note that we multiplied a cash flow number by a magic number Oh Cool! We just did our first engineering cash flow problem!

Magic Numbers There are many kinds of magic numbers –This one came from the formula (1 + i)^n –This one told us what the future debt would be from a present amount of money that Sammy Sucker spent This magic number is called a Future Value of a Present Amount factor –Common notation is F/P

Lets Pick on Sammy Some More Say Sammy Sucker goes all the way through College (he’s a little dense so it takes him 7 years) and never pays off that credit card –Sammy has gone 7 * 12 compounding periods (84) –Our formula says (1 + i {0.015}) 84 = –Sammy owes $34,925.90

F/P Factors You can see that the exact value of the magic F/P number depends on the interest rate and the number of compounding periods. We sometimes write F/P i,n Thus the F/P magic number for the end of 12 months would have been F/P 1.5, 12 The factor for after 7 years F/P 1.5,84