Compound Interest ©Dr. B. C. Paul 2001 revisions 2008 Note – The subject covered in these slides is considered to be “common knowledge” to those familiar with the subject and books or articles covering the concepts are widespread.

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

Lets Meet one of Our Silly Six Mistakes  Interest rates are reported at an annual rate  Interest is often compounded several times during a year  You need to get a period interest rate  This is done by dividing the annual interest rate by the number of compounding periods in a year.  The MISTAKE  This last 7 year problem had 84 compounding periods  People take the annual interest rate and divide by the number of compounding periods in the problem  It should be the number of compounding periods in ONE YEAR

Lets Try Doing the Problem With Class Assistant First We’ll Look at The period interest Rate (Lets do the problem Assuming daily Compounding)

Lets Zoom in Close Enough to Read There is the 18% interest rate the credit card reported Since they compound daily there are 365 Compounding periods in a year (most banks Don’t deal with leap year – except to charge you extra interest) Out comes our period interest rate - just under 0.05% daily

Lets See What Sammy’s Actual Interest Rate is We put in the period interest rate we just got We compound 365 times each year Zipes !!! The actual effective interest rate is nearly 20%!!

Lets Get That F/P and Find Out What Sammy will Owe in 7 years Interest Rate Compounding Periods per year Number of days in 7 years (ie 365*7) Outcomes our F/P Using the F/P - $10,000 * = $35, Note that Sammy got scr_ _ _ _ worse when they compounded more times Each year (I wonder why credit cards use an average daily balance to do Their interest calculations)

Now Its Your Turn Do Homework Assignment #2 You will take a credit card reported interest rate and figure the actual percentage yield on that card