Interest

9-1Compound

Interest

Compound

Interest 99 99

McGraw-Hill Ryerson©

CompoundCompound

Chapter 9

9-2Compound

Interest

Compound

Interest 99 99


Calculate the…

Learning ObjectivesLearning Objectives

After completing this chapter, you will be able to:

… effective interest rate when given a nominal interest rate

… interest rate and term in compound interest

applications

… equivalent interest rate at another compounding frequency when given a nominal interest rate

LO-1LO-1

LO-2LO-2

LO-3LO-3

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Some key Keys!Some key Keys!

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The nominal interest rate (Interest/Year)

1. Number of compoundings (for lump payments)

2. Number of payments (for annuities)

Represents the Periodic Annuity Payment

(used in chapter 10)

Tells the calculator to compute (CPT)

Present Value or initial(first) lump sum

Find the following KEYS:Find the following KEYS:

Future Value or terminal(last) lump sum

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However, we can now input the number of compoundings per year into the financial calculator. This can be performed by using

the symbol


…it is rare for interest to be compounded only once per year!

…it is rare for interest to be compounded only once per year!

Previously, it was noted that

To access this symbol use:

…and you will see

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The 12 is a

default setting

The 12 is a

default setting

This display is referred to as “the worksheet”.

… represents the number of Payments per year

… represents the number of Compoundings per year

To access use:

Note: You can override these values by entering new ones!

…more…more

Appearsautomatically

Appearsautomatically

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The Power function Key. Used to calculate the

value of exponents.

Used to access symbols located “above”

another key, i.e. its acts like the

SHIFT key on a computer keyboard.

Use a calculator and algebraic sequencingUse a calculator and algebraic sequencing

Changes the sign of the data value of the number

being displayed.

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Some calculators have the yx symbol above the calculator key.

(1.04)8 is…

The key stroke sequence to evaluate an EXPONENT that is…

1.04 8

1.368569

0.73069

PositivePositive



NegativeNegative (1.04)-8 is…1.04 8

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This calculator can store up to 10 values.



Used to Store or save displayed values.

Used to Recall the saved values.

Let’s PractiseLet’s Practise

Therefore, the calculator must be informed as to

where the values are to be stored.

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Using the key

Using the key e.g. you want to store the value ’45’.

The key stroke sequence ‘to store’ is:45

..choose from 0 - 9

…’clear’ display

The key stroke sequence ‘to recall’ is:

…where you stored the value!

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What amount must you invest now at 6% compounded daily to accumulate to $5000 after 1 year?

n = 1 x 365 = 365 i = .06/365 FV = $5000

Solve…

PV = FV(1 + i)-n

Formula Formula

Extract necessary data...

LO-1LO-1

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.06

1

365

5000

= $4708.85= $4708.85

Solve…

PV = FV(1 + i)-n

Formula Formula

4,708.85

n = 1 x 365 = 365 i = .06/365 FV = $5000

365

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1

5000

0365

4,708.85

= $4708.85= $4708.85

6

n = 1 x 365 = 365 i = .06/365 FV = $5000

365

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Suppose you have $5000 and you want it to grow to $9000 in 7 years so you could buy a

car. What rate of interest do you need to accomplish your goal? Assume interest is

compounded annually.Solving for i this becomes… i = (FV/PV)1/n - 1

PV = $5000 n = 7 x 1 = 7FV = $9000

FV = PV(1 + i)n

Formula Formula


See next slide for steps in developing this formulaSee next slide for steps in developing this formula

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Formula Formula i = (FV/PV)1/n - 1

Developing the Formula

FV = PV(1 + i)n

2. Divide both sides by PV

1. Base formula

FV = PV(1 + i)n

PV PV

FVPV

3. To remove n as exponent raise both sides to the power 1/n = (1 + i)n1/n 1/n

4. Complete reducing FVPV

1/n = 1 + i

i =FVPV

1/n - 1

i = (FV/PV)1/n - 1

FVPV = (1 + i)n

Back to Solving the problem

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1.0876

7

9000 1

8.76% 8.76%

Solve…

5000

.0876Suppose you have $5000 and you want it to grow to $9000

in 7 years so you could buy a car.

What rate of interest do you

need to accomplish your goal? Assume

interest is compounded

annually.


PV = $5000 n = 1 x 7 = 7 FV = $9000

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5000

7

9000

0

1

8.76%8.76%

I/Y = 8.7596Suppose you have $5000 and you want it to grow to $9000





annually.

PV = $5000 n = 1 x 7 = 7 FV = $9000

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1.3% per month

1.9% per quarter

0.05041% daily

1.3 * 12 =

1.9 * 4 =

0.05041 * 365 =

Find the Nominal Rate if the Periodic Rate is:

The formula gives the Periodic Rate (i) but in practice we usually refer to

the Nominal(annual)Rate

The formula gives the Periodic Rate (i) but in practice we usually refer to

the Nominal(annual)Rate

15.6% per year

7.6% per year

18.39965= 18.4% per year

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Suppose you have $5000 and you want it to grow to $9000 in 7 years so you could buy a

car. What rate of interest do you need to

accomplish your goal?

PV = $5000 FV = $9000 n = 7 x 12 = 84

Solve…

Assume interest is compounded monthly.



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1 .0070

84

90001

.0070 per month .0070 per month

Solve…

5000

.0070

12

8.43% nominal rate

8.43% nominal rate

.0843Suppose you have $5000 and you want it to grow to $9000





monthly.


PV = $5000 n = 7 x 12 = 84 FV = $9000

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5000

9000

0

12

8.43%8.43%

I/Y = 8.4264

7 Suppose you have $5000 and you want it to grow to $9000





monthly.

12

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Suppose you have $19,500 and you want it to grow to $23,000 so you could buy a

car. You know that you can get 3% interest compounded daily. How long will it

take to reach your goal?Solving for n this becomes…n = ln(FV/PV) / ln(1+ i)

PV = $19500 i = .03/ 365 = .0001 FV = $23000

FV = PV(1 + i)n

Formula Formula


See next slide for steps in developing this formulaSee next slide for steps in developing this formula

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Back to Solving the problem

n = ln(FV/PV) / ln(1+ i)Formula Formula

FV = PV(1 + i)n1. Base formula

Developing the Formula

2. To isolate n, divide both sides by PV

FV = PV(1 + i)n

PV PVFVPV = (1 + i)n

3. Take the natural logarithm (LN or ln) of both sides

FVPV

= n * ln(1 + i)ln

4. Complete reducingFVPVln

ln (1 + i)n =

n = ln(FV/PV) / ln(1+ i)

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.03 23000

or 2009daysor 2009days

Solve…19500

Formula Formula n = ln(FV/PV) / ln(1+ i)

365

1

0.0001 0.1651 2008.55

Suppose you have $19,500 and you

want it to grow to

$23,000 so you could buy a car.

You know that you can get 3% interest compounded daily. How long will it take to reach your goal?

PV = $19500 i = .03/ 365 = .0001 FV = $23000

…

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19500

3

23000

0

365

Suppose you have $19,500 and you

want it to grow to

$23,000 so you could buy a car.

You know that you can get 3%

compounded daily. How long will it

take to reach your goal?

N = 2008.55

2009days 2009days

Conversion of Days

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Conversion of DaysConversion of Days

365 2008.55

2008.55

5.50288 5.50288(years)

5 (years)

0.50288

126.0346 Months

5 Years 6 months5 Years 6 months

6.03465 Years

6 Months

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How long will it take at 4.2% compounded monthly for $800 to grow to $1100?


PV = $800 FV = $1100 i = .042/ 12 = .0033

Solve…


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.042 1100

91.1months 91.1months

Solve…800


12

1

0.0035 .3185 91.1459

7 years 7.1 months

PV = $800 FV = $1100 i = .042/ 12 = .0033

How long will it take at

4.2% compounded monthly for

$800 to grow to $1100?

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800

4.2

1100

0

12

N = 91.1459

How long will it take at

4.2% compounded monthly for

$800 to grow to $1100? 91.1months 91.1months

7 years 7.1 months

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How long will it take for money to triple when invested at 7.2%

compounded semi-annually?


PV = $1 FV = $3 i = .072/ 2 = .036

Solve…

Note:When no FV or PV given use the easiest number!

Note:When no FV or PV given use the easiest number!

$1$1

Extract necessary data...FV =

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.072 3

31.1sa periods 31.1sa periods

Solve…

12

1

0.0354 1.0986 31.0631

15 years 3.2 months

How long will it take

for money to triple when invested at

7.2%

compounded semi-

annually?


PV = $1 FV = $3 i = .072/ 2 = .036

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1

7.2

3

0

2

N = 31.0631How long will it take

for money to triple when invested at

7.2%

compounded semi-

annually?

31.1sa periods 31.1sa periods

15 years 3.2 months

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How long will it take for money to double when invested at 10.0%

compounded annually?

Note:This results in a “rule of thumb” that you can use for

comparisons or ballparking your other solutions.

Note:This results in a “rule of thumb” that you can use for

comparisons or ballparking your other solutions.


PV


i = .010/ 2 = .005FV = $2

Solve…

Use $1Use $1

= $1

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1

10

2

0

1

N = 7.2725How long will it take for money to

double when invested at

10.0% compounded annually?

7.3 Years 7.3 Years

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Rule of 72Rule of 72…Rule of thumb to quickly estimate the time needed to double your money!

Doubling time (in years) = 72

% annual rate of return

… investment earning 9% compounded

annually will double in approx… 72/9 = 8 years


annually will double in

approx…

72/12 = 6 years


annually will

double in approx… 72/4 = 18 years

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LO-2LO-2

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… or Stated Rate is the rate on which the bank calculates interest

… is the annually compounded rate that produces the same Future Value as the given

nominal rate (j)Effective Rate = Interest for 1 year

Principal

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Ending Balance

Ending Balance

Compounding Period

Compounding Period

$1,000$1,000

Nominal Rates of Interest Compared

$1,060.00

$1,060.90

$1,061.36

$1,061.83

Beginning Balance

Beginning Balance

Nominal Rate

Nominal Rate

+ 6%+ 6%

Annual

Semiannual

Quarterly

Daily

Effective RateAPY of Interest

Effective RateAPY of Interest

6.00%

6.09%

6.14%

6.18%

Invested for One Year

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What is the effective rate of interest corresponding to 9% pa compounded daily?

Formula Formula f = (1 + i)m - 1

Extract necessary data... j = .09 m = 365 i = .09/ 365 = .0002

f = (1 + .09/365)365 – 1 = .0942 = 9.42%

Solve…

Solve…

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FV = 109.4162

365

0

= 9.42%= 9.42%

100

9365

Solve…

What is the effective rate

of interest corresponding

to 9% pa compounded

daily?

Use $100Use $100

109.4162 - 100

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ComparisonsComparisons

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Which is the most attractive of the following interest rates offered on 3 year GICs? A. 5.8% compounded semi-annually B. 5.68% compounded monthly C. 5.4% compounded daily

TipTip Faster if you use

Comparing…Comparing…


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ComparisonsComparisonsApproach Approach

Invest $100 at each of the rates for 1 year

and compare

Invest $100 at each of the rates for 1 year

and compare

A. 5.8% compounded semi-annually

B. 5.68% compounded monthly

C. 5.4% compounded daily

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Invest $100 for 1 year


A. 5.8% compounded

semi-annually

A. 5.8% compounded

semi-annually

FV = 105.882

0

100

5.82

= 5.88%= 5.88% 105.88 - 100

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Invest $100 for 1 year FV = 105.83

12

5.68

12

= 5.83%= 5.83% 105.83 - 100

B. 5.68% compounded

monthly

B. 5.68% compounded

monthly

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Invest $100 for 1 year FV = 105.55

365

5.4

365

= 5.55%= 5.55%105.55 - 100

C. 5.4% compounded

daily

C. 5.4% compounded

daily

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A. 5.8% compounded semi-annually 5.88%5.88%

B. 5.68% compounded monthly 5.83%5.83%

C. 5.4% compounded daily 5.55%5.55%


the 5.8% compounded semi-annually provides for the best

rate of return on investment!

the 5.8% compounded semi-annually provides for the best

rate of return on investment!

ResultsResults

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Interest rates that produce the

same future value after one year

Equivalent Interest RatesEquivalent Interest Rates

LO-3LO-3

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Compounded quarterlyCompounded quarterlyRefers to compounding

frequency …NOT ‘per quarter’!

per quarterper quarter

Refers to the compounding

period!

ExampleExample

4% compounded

quarterly equals

1% for each

quarter

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For a given interest rate of 8% compounded quarterly, what is the equivalent nominal rate of interest

with monthly compounding?

i2 = (1+i1)m1/m

2 - 1

i1 and m1 represent the given values of i and m

Formula Formula

i2 and m2 represent the equivalent values of i and m

Explanation

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Explanation

For a given interest rate of 8% i

compounded quarterly m

Represented by

ValueGiven Equivalent

i1

m1

what is the equivalent nominal rate of interest

with monthly compounding?

i2

m2

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For a given interest rate of 8% pa, compounded quarterly, what is the equivalent nominal rate of interest with monthly compounding?

i2 = (1+i1)m1/m

2 - 1Formula Formula

i2 = (1+ .08 )4/ 12 - 1/4

i2 = .0066nominal rate = j = .0066 * 12

= 7.95%Note…

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is a fraction change it to a decimal number and STORE it before continuing.


Since the exponent

4 /12 = 0.3333

i2 = (1+.08 )4/ 12 - 1/4Formula Formula

Solve…

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i2 = (1+.08 )4/ 12 - 1/4


1

0.3333

12

For a given interest rate of 8% pa, compounded

quarterly, what is the equivalent

nominal rate of interest with

monthly compounding?

1.021.00660.0066 0.0795

4

12

.08

4

1

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FV = 108.24

4

100

84


quarterly, what is the

equivalent



Calculate the FV of $100 invested for 1 year as given

= 8.24%= 8.24%FV = 108.24 - 100

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I/Y = 7.947312

12


quarterly, what is the

equivalent



Compute the value of i2 that also makes $100 grow to the same as

FV in m2 compoundings

7.95% compounded monthly = 8% compounded quarterly

7.95% compounded monthly = 8% compounded quarterly

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This completes Chapter 9This completes Chapter 9

Interest

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