Compound InterestCompound Interest

©Dr. B. C. Paul 2001 revisions 2008, 2011©Dr. B. C. Paul 2001 revisions 2008, 2011

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

A small ROR means you only need a little extraA small ROR means you only need a little extra When we last left our story we had figured out When we last left our story we had figured out

how to determine our rate of returnhow to determine our rate of return

So What Do We Do With So What Do We Do With it Now?it Now?

Remember that money at different points in Remember that money at different points in time has a different value.time has a different value. We know if we don’t get our money for many We know if we don’t get our money for many

years then part of the money pays us for years then part of the money pays us for inflation, risk, waiting, and beating the other guyinflation, risk, waiting, and beating the other guy

If we knew how much of the money covered If we knew how much of the money covered this we would know how to count up money this we would know how to count up money from different points in timefrom different points in time

The rate of return can be used to calculate The rate of return can be used to calculate this ratio.this ratio.

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 This one came from the formula This one told us what the future debt would be This one told us what the future debt would be

from a present amount of money that Sammy from a present amount of money that Sammy Sucker spentSucker spent

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

)1( in

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

This is the Screw UpThis is the Screw Up

I have 18% interest compounded monthly I have 18% interest compounded monthly with interest accumulating for 84 monthswith interest accumulating for 84 months To screw up the period interest rateTo screw up the period interest rate 18%/84 = 0.214%18%/84 = 0.214% Of course we are too smart to do thisOf course we are too smart to do this

The Right Way to Do ItThe Right Way to Do It

18% interest compounded monthly and 18% interest compounded monthly and accumulated for 84 months.accumulated for 84 months.

To get the period interest rate we divide To get the period interest rate we divide by the number of compounding periods in by the number of compounding periods in 1 year.1 year. Lets see – There are 12 months in a yearLets see – There are 12 months in a year 18%/12 = 1.5%18%/12 = 1.5% This is the correct way to get a period This is the correct way to get a period

interest rate.interest rate.

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)

What Might the FE Exam do What Might the FE Exam do to test your knowledge?to test your knowledge?

Many of the problems in the first part of Many of the problems in the first part of the FE exam test whether.the FE exam test whether. You know how to convert money to different You know how to convert money to different

points in time.points in time. You need to recognize that the solution is You need to recognize that the solution is

simply a given dollar amount * a magic simply a given dollar amount * a magic numbernumber

You need to know which magic number to You need to know which magic number to use to move the moneyuse to move the money

FE Like ExampleFE Like Example

$100 is deposited in the bank at 5% $100 is deposited in the bank at 5% interest. How much money will be in the interest. How much money will be in the account in 6 years?account in 6 years? (a)- $75(a)- $75 (b)- $134(b)- $134 (c)- $137(c)- $137 (d)- $121(d)- $121

Problem is Easy IfProblem is Easy If

You recognize the solution isYou recognize the solution is

There is a trick thoughThere is a trick though The F/E exam does not let you use a computer, The F/E exam does not let you use a computer,

class assistant or a programmable calculator to get class assistant or a programmable calculator to get F/PF/P

PF

years

*100$6%,5

So How Do We Get F/PSo How Do We Get F/P

Choice #1 is the formula methodChoice #1 is the formula method The FE exam gives you formula for all the The FE exam gives you formula for all the

magic numbers you need (and a few others magic numbers you need (and a few others to confuse you if you are just guessing)to confuse you if you are just guessing)

F/P formula is (1+i)F/P formula is (1+i)nn

Plug in i= 0.05 (note I had to know to convert Plug in i= 0.05 (note I had to know to convert a % to a decimal form)a % to a decimal form)

N = 6N = 6 (1+.05)(1+.05)66 = 1.3401 = 1.3401

Choice #2 for F/P is a Choice #2 for F/P is a table look uptable look up

Tables are provided in the FE Exam Booklet that is provided to you to use whileYou take the test

Trick #1 to Use An Trick #1 to Use An Interest TableInterest Table

Pick the table based on the Correct Interest Rate

If that number were 7% - You’d get the wrong answer!

Trick #2 – Pick the Trick #2 – Pick the Correct Magic Number.Correct Magic Number.

We want F/P

Now Read Down the Now Read Down the Number of Compounding Number of Compounding PeriodsPeriods

I want 6 compoundingperiods

I grab 1.34 as the F/P value

Just as an IncidentalJust as an Incidental

Magic # CalculatorNote - It is recommended that you go to the Magic # Calculator Tab for advice on how to use this feature

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

Annual Int Rate 51 Enter the number of compounding periods/year

in % in decimalPeriod Int Rate 5 0.05

Enter # Compouning Periods to Move Cash (value of n) 6The value should be an interger

F/P 1.340095641 (used to move one cash flow element n compounging period into the future)

Class Assistant would have given theSame F/P (no surprise since theSpreadsheet contains the F/P formula)

Put in the interest rate

Put in the number ofCompounding periods

Pull out 1.34 as the F/P value

Now Lets Get the AnswerNow Lets Get the Answer

$100 * 1.34 = $134$100 * 1.34 = $134

Wow – that math was toughWow – that math was tough Check our choicesCheck our choices

(a)- $75(a)- $75 (b)- $134(b)- $134 (c)- $137(c)- $137 (d)- $121(d)- $121

It looks like the winner is (b)- $134It looks like the winner is (b)- $134

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