Compound InterestCompound Interest

©Dr. B. C. Paul 2001 revisions 2008©Dr. B. C. Paul 2001 revisions 2008

Note – The subject covered in these slides is considered to be “common 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 knowledge” to those familiar with the subject and books or articles covering the concepts are widespread.concepts are widespread.

Rate of Return and the Rate of Return and the Cost of CapitalCost of Capital

A big rate of return means you have to A big rate of return means you have to come up with a lot of extra money to get come up with a lot of extra money to get the investors to put-off their Dairy Queen the investors to put-off their Dairy Queen BlizzardsBlizzards

A small ROR means you only need a A small ROR means you only need a little extralittle extra

So Why All This Rate of So Why All This Rate of Return Business?Return Business?

Remember all engineering econ problems Remember all engineering econ problems involveinvolve Write down the money that goes in and out of the Write down the money that goes in and out of the

project for each year in order (get your cash flow)project for each year in order (get your cash flow) Multiply each number in the cash flow by a magic Multiply each number in the cash flow by a magic

numbernumber Add up the total and see whether the money pile is big Add up the total and see whether the money pile is big

enoughenough

Rate of Return Tells you how big the money Rate of Return Tells you how big the money pile has to bepile has to be

ExampleExample

If I put $1.00 in the bank at 5% interest, how If I put $1.00 in the bank at 5% interest, how much money will I have next year when I much money will I have next year when I take the money outtake the money out 5% of $1.00 is 5 cents5% of $1.00 is 5 cents I will have $1.05I will have $1.05

If I leave the money in the bank another year If I leave the money in the bank another year I will get 5% interest on $1.05 not just $1.00I will get 5% interest on $1.05 not just $1.00 At the end of the year I will have (1.05)(1.05)= At the end of the year I will have (1.05)(1.05)=

1.10251.1025

Example continuedExample continued If I leave the money in another year I will get If I leave the money in another year I will get

5% interest on $1.10255% interest on $1.1025 (1.1025)(1.05) = 1.1576(1.1025)(1.05) = 1.1576 and again the next year (1.1576)(1.05) = 1.2155and again the next year (1.1576)(1.05) = 1.2155

My interest is “Compounding”My interest is “Compounding” Note that if I only got 5% each year on my dollar Note that if I only got 5% each year on my dollar

I would only have $1.20I would only have $1.20

The sneaky trick with interest is to multiply, The sneaky trick with interest is to multiply, not add (multiplication takes care of not add (multiplication takes care of compounding)compounding)

Compounding PeriodCompounding Period In the previous example I got my interest In the previous example I got my interest

every year and then I started every year and then I started compounding the interest on the interest.compounding the interest on the interest.

Why does the interest have to compound Why does the interest have to compound once a year - it doesn’tonce a year - it doesn’t Ever noticed CD rates at Banks 4.6% interest with a Ever noticed CD rates at Banks 4.6% interest with a

4.75% yield?4.75% yield? They pay interest and compound it over shorter They pay interest and compound it over shorter

times so that by the end of the year the ROR is times so that by the end of the year the ROR is higher than the interest ratehigher than the interest rate

Interest Rates are Usually Reported on an Annual Interest Rates are Usually Reported on an Annual BasisBasis

The Credit Card Rip-OffThe Credit Card Rip-Off

Sammy Sucker gets a credit card offer from Sammy Sucker gets a credit card offer from Spin on My Finger Bank and TrustSpin on My Finger Bank and Trust The interest rate is 18% (but they’ll give him a 5% The interest rate is 18% (but they’ll give him a 5%

purchase credit toward a new Turbo charged purchase credit toward a new Turbo charged Volkswagon Beetle that will make all the girls Volkswagon Beetle that will make all the girls think he is sexy)think he is sexy)

Sammy goes out and maxes out his credit card at Sammy goes out and maxes out his credit card at $10,000$10,000

We’ll ignore his monthly minimum payments for a We’ll ignore his monthly minimum payments for a whilewhile

Sammy gets -------Sammy gets -------

Spin on My Finger Bank and Trust Spin on My Finger Bank and Trust divides the interest rate over 12 monthsdivides the interest rate over 12 months 18%/12 months = 1.5% per month18%/12 months = 1.5% per month

Month #1 Sammy doesn’t pay off his cardMonth #1 Sammy doesn’t pay off his card 1.5% of $10,0001.5% of $10,000 (10000)*(1.015) = $10,150 or $10,150- (10000)*(1.015) = $10,150 or $10,150-

$10,000 is $150 of interest$10,000 is $150 of interest

Sammy’s AdventureSammy’s Adventure

Month #2 Sammy doesn’t pay off his Month #2 Sammy doesn’t pay off his credit cardcredit card Spin on My Finger Bank and Trust Spin on My Finger Bank and Trust

compounds the interestcompounds the interest $10,150*(1.015) = $10,302.25$10,150*(1.015) = $10,302.25

Month #3 Sammy doesn’t pay off his Month #3 Sammy doesn’t pay off his credit cardcredit card $10,302.25 * (1.015) = $10,456.78$10,302.25 * (1.015) = $10,456.78

This is Sammy’s This is Sammy’s Adventure - Not OursAdventure - Not Ours I really love these calculations but if I have to do I really love these calculations but if I have to do

them 12 times I’m going to pukethem 12 times I’m going to puke

Enter Super formulaEnter Super formula Note that all I’m doing is multiplying the original debt Note that all I’m doing is multiplying the original debt

$10,000 by 1.015$10,000 by 1.015 Note that 1.015*1.015 is just (1.015)Note that 1.015*1.015 is just (1.015)22

Note that 1.015*1.015*1.015 is just (1.015)Note that 1.015*1.015*1.015 is just (1.015)33

Note that 1.015 is just 1 plus the interest rateNote that 1.015 is just 1 plus the interest rate

Magic formula (1 + i)Magic formula (1 + i)nn

where i is the interest ratewhere i is the interest rate and n is the number of compounding periodsand n is the number of compounding periods

Now Lets Return to Now Lets Return to Sammy’s SagaSammy’s Saga

After 1 year how much does Sammy owe?After 1 year how much does Sammy owe? He’s had 12 compounding periods at 1.5% He’s had 12 compounding periods at 1.5%

interest each timeinterest each time The magic formula is (1.015)The magic formula is (1.015)1212 = 1.1956 = 1.1956

Apply the formula to Sammy’s DebtApply the formula to Sammy’s Debt $10,000 * 1.1956 = $11,956$10,000 * 1.1956 = $11,956

Note that Sammy paidNote that Sammy paid 1.1956 - 1 = 0.1956 or 19.56% interest 1.1956 - 1 = 0.1956 or 19.56% interest

because of compounding - not 18%because of compounding - not 18%

What Else is NewWhat Else is New

Note that Sammy’s spending $10,000 is Note that Sammy’s spending $10,000 is a cash flow numbera cash flow number

Note that we multiplied a cash flow Note that we multiplied a cash flow number by a magic numbernumber by a magic number

Oh Cool! We just did our first engineering Oh Cool! We just did our first engineering cash flow problem!cash flow problem!

Magic NumbersMagic Numbers

There are many kinds of magic numbersThere are many kinds of magic numbers This one came from the formula (1 + i)^nThis one came from the formula (1 + i)^n This one told us what the future debt would This one told us what the future debt would

be from a present amount of money that be from a present amount of money that Sammy Sucker spentSammy Sucker spent

This magic number is called a Future This magic number is called a Future Value of a Present Amount factorValue of a Present Amount factor Common notation is F/PCommon notation is F/P

Lets Pick on Sammy Lets Pick on Sammy Some MoreSome More

Say Sammy Sucker goes all the way Say Sammy Sucker goes all the way through College (he’s a little dense so it through College (he’s a little dense so it takes him 7 years) and never pays off takes him 7 years) and never pays off that credit cardthat credit card Sammy has gone 7 * 12 compounding Sammy has gone 7 * 12 compounding

periods (84)periods (84) Our formula says (1 + i {0.015})Our formula says (1 + i {0.015})8484 = 3.49259 = 3.49259 Sammy owes $34,925.90Sammy owes $34,925.90

F/P FactorsF/P Factors

You can see that the exact value of the You can see that the exact value of the magic F/P number depends on the interest magic F/P number depends on the interest rate and the number of compounding rate and the number of compounding periods.periods.

We sometimes write F/PWe sometimes write F/Pi,ni,n

Thus the F/P magic number for the end of Thus the F/P magic number for the end of 12 months would have been F/P12 months would have been F/P1.5, 121.5, 12

The factor for after 7 years F/PThe factor for after 7 years F/P1.5,841.5,84

Lets Meet one of Our Silly Lets Meet one of Our Silly Six MistakesSix Mistakes

Interest rates are reported at an annual rateInterest rates are reported at an annual rate Interest is often compounded several times during a Interest is often compounded several times during a

yearyear You need to get a period interest rateYou need to get a period interest rate

This is done by dividing the annual interest rate by the number This is done by dividing the annual interest rate by the number of compounding periods in a year.of compounding periods in a year.

The MISTAKEThe MISTAKE This last 7 year problem had 84 compounding periodsThis last 7 year problem had 84 compounding periods People take the annual interest rate and divide by the number People take the annual interest rate and divide by the number

of compounding periods in the problemof compounding periods in the problem It should be the number of compounding periods in ONE It should be the number of compounding periods in ONE

YEARYEAR

Lets Try Doing the Problem Lets Try Doing the Problem With Class AssistantWith Class Assistant

First We’ll Look atThe period interestRate

(Lets do the problemAssuming dailyCompounding)

Lets Zoom in Close Lets Zoom in Close Enough to ReadEnough to Read

Period Interest Rate Adjustment

Enter Annual Interest Rate as a PercentageDo not use the % key during entry

Annual Int Rate 18365 Enter number of compounding periods/year

(Would be 12 for months 365 for daily)in % in decimal

Period Int Rate 0.049315068 0.000493

There is the 18% interest rate the credit card reported

Since they compound daily there are 365Compounding periods in a year (most banksDon’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 Lets See What Sammy’s Actual Interest Rate isActual Interest Rate is

Period Interest Rate to an Annual Interest Rate

Enter Your Period Interest Rate as a PercentageDo not use the % key during entryPeriod Interest Rate 0.049315

365 Enter number of compounding periods/year

Annual Interest Rate 18Effective Interest Rate 19.71642

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 Lets Get That F/P and Find Out What Sammy will Owe Out What Sammy will Owe in 7 yearsin 7 years

Magic # Calculator

Enter Annual Interest Rate in %Do not use the % key during data entry

Annual Int Rate 18365 Enter the number of compounding periods/yearin % in decimal

Period Int Rate 0.049315068 0.000493

Enter # Compouning Periods to Move Cash (value of n) 2555The value should be an interger

F/P 3.524326722 (used to move one cash flow element n compounging period into the future)

Interest RateCompounding Periods per year

Number of days in 7 years(ie 365*7)

Outcomes our F/PUsing the F/P - $10,000 * 3.524326722 = $35,243.27

Note that Sammy got scr_ _ _ _ worse when they compounded more timesEach year (I wonder why credit cards use an average daily balance to doTheir interest calculations)

Now Its Your TurnNow Its Your Turn

Do Homework Assignment #2Do Homework Assignment #2

You will take a credit card reported interest rate and figure You will take a credit card reported interest rate and figure the actual percentage yield on that cardthe actual percentage yield on that card