Compound Interest

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Compound Compound

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Interest Interest

Compound Chapter 8 McGraw-Hill Ryerson©

Compound Compound

8

Interest Interest

8-2 To better understand how Compound Interest is calculated, let’s review how we calculate Simple Interest! The formula on which we base our calculation is…

Formula Formula

I = Prt

Here we have an amount, the Principal, which is multiplied by the Interest Rate and the Time over which the Interest is earned! As we will now see, Compound Interest uses the Sum of P & I as a base on which to calculate

new Interest! McGraw-Hill Ryerson©

Compound Compound

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Compound Interest

8-3

- Future Value

Interest Interest

…the interest on the principal

plus the

interest of prior periods e.g. Principal + prior period interest = $1100.00 $1000.00

$100.00

Interest for the next period is calculated on $1100.00. This method will continue over the life of the loan or investment. (See later example) McGraw-Hill Ryerson©

Compound Compound

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Interest Interest

Compound Interest

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- Future Value

…is the compounded amount and is the FINAL amount of the loan or investment at the end of the last period! Contrast this with… ...is the value of a loan or investment TODAY!

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Compound Compound

8

Interest Interest

Compound Interest

8-5

- Future Value

…the calculation of interest over the life of the loan or investment Let’s assume that the interest rate is 10% pa.

Example: Principal + prior period interest = $1100.00 Interest is now calculated on $1100.00 Principal(Compounded) * 0.10 = $110.00 New P $1210.00 to start next period

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Graphically…

Compound Compound

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Compound Interest

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- Future Value

Interest Interest

Interest Interest

Interest Interest

Interest Interest

Interest Interest

Amount$1000 $1000 Amount 133.1

1331 1210 1100 1000

100

110 100

121

121

11 0 100

11 0 100

Compoundin Compoundin Compoundin g Period g Period g Period

0 McGraw-Hill Ryerson©

1

2 Time(Years)

3

Compoundin g Period

4

Compound Compound

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Interest Interest

Compound Interest

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- Future Value

What happens if the interest rate changes during the life of an investment?

Example… Example… McGraw-Hill Ryerson©

Compound Compound

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Interest Interest

Compound Interest

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- Future Value

You hold an investment for a period of 4 years. Rates of return for each year are 4%, 8%, -10% and 9% respectively. If you invested $1000 at the beginning of the term, how much will you have at the end of the last year?

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Compound Compound

8

Interest Interest

Compound Interest

8-9

- Future Value

You hold an investment for a period of 4 years. Rates of return for each year are 4%, 8%, -10% and 9% respectively. If you invested $1000 at the beginning of the term, how much will you have at the end of the last year?

Year 1

Year 2

Year 3

Year 4

$1000

$1040

$1123.20

$1010.88

$1000 * (1 + .04) = $1040 McGraw-Hill Ryerson©

$1040 * $1123.20 * (1 + .08) (1 - .10) = $1123.20 = $1010.88

$1010.88 * (1 +.09) = $1101.86 …Alternative …Alternative

Compound Compound

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Interest Interest

Compound Interest

8 - 10

- Future Value You hold an investment for a period of 4 years. Rates of return for each year are 4%, 8%, -10% and 9% respectively. If you invested $1000 at the beginning of the term, how much will you have at the end of the last year?

1000(1.04)(1.08)(.90)(1.09) = $1101.86 Solving Solving Alternative Alternative 1 -10% Solve for all Solve for all years at at 44 years It isis rare rare for forinterest interest to to be be It once! once! compounded only once per year! compounded only once per year!

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8 - 11

Compound Compound

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Interest Interest

Compounding Frequencies and Periods Frequency Frequency

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No. per per Year Year No.

Period Period

Annually

1

1 year

Semiannually

2

6 months

Quarterly

4

3 months

Monthly

12

1 month

Daily

365

1 day

Compound Compound

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Interest Interest

8 - 12

Development of a Formula Formula

Nominal or Annual Rate Number of compoundings per year Periodic Rate per period

Total Number of

Periods Periods

(j)

m (i)

n

Determining values for n and i McGraw-Hill Ryerson©

Compound Compound

8 - 13

Formulae Formulae

8

Interest Interest

To Determine Determine To

Time(Years)

*

n n

# of Compounding Frequencies p.a.(m)

To Determine Determine To

ii

Annual Interest Rate(j) # of Compounding Frequencies p.a. (m) McGraw-Hill Ryerson©

Compound Compound

Determining values for n

8

Interest Interest

If you compounded $100 for 3 years at 6% annually, semiannually, or quarterly, what are the values for n and i ?

Formula

Time(Years) *

# of Compounding Frequencies per year (m)

Annually 3* Semiannually 3 * Quarterly 3 * McGraw-Hill Ryerson©

No.

n

1

= 3 = 6 = 12

2 4

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Compound Compound

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Interest Interest

Determining values for

8 - 15

i

If you compounded $100 for 3 years at 6% annually, semiannually, or quarterly, what are the values for n and i ?

Formula Formula

Annual Interest

Rate (j)

# of Compounding Frequencies per year(m)

6% / Annually Semiannually 6% / Quarterly 6% /

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No. 1 2 4

Rate -- ii Rate = 6% = 3% = 1.5%

8 - 16

Compound Compound

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Interest Interest

Development of a Formula Formula for Future Value FV = PV(1 + i)n Where…

PV= Present Value(Principal) i = rate per period n = number of periods McGraw-Hill Ryerson©

Compound Compound

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Interest Interest

Compound Interest

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- Future Value n FV = PV(1 + i) Formula Formula

Steve Smith deposited $1,000 in a savings account for 4 years at a rate of 8% compounded semiannually. What is Steve’s interest and compounded amount?

Extract necessary data...

PV = $1000 n = 4X2=8 i = .08/2 = .04 McGraw-Hill Ryerson©

Solve…

Compound Compound

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Interest Interest

Compound Interest - Future Value n FV = PV(1 + i) Formula Formula

Solve… Using PV = $1000 n = 8 i= .04

FV = $1000(1 + .04)8 = $1000(1.368569) = $1,368.57 Principal $1,000.00 + Interest 368.57 Compounded $1,368.57 McGraw-Hill Ryerson©

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Compound Compound

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Interest Interest

What is the effect on the Future Value of different Compounding Periods of Interest? McGraw-Hill Ryerson©

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Compound Compound

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Interest Interest

Compound Interest

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- Future Value

If you compounded $100 for 3 years at 6% annually, semiannually, or quarterly, what are the final amounts that you would have at the end of the three (3) years ?

Annual Annual

FVA = 100(1.06)3

SemiSemi-

FVS = 100(1.03)6

$119.10 $119.10 $119.41 $119.41

Semi = 6%/2 Quarterly Quarterly

FVQ = 100(1.015)12 Quarterly = 6%/4

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$119.56 $119.56

Compound Compound

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Interest Interest

Simple Vs Compound Interest

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AlJones Jonesdeposited deposited$1,000 $1,000in inaasavings savingsaccount account Al for55years yearsat at10% 10%p.a.. p.a.. for

AnnualSSimple impleIInterest nterest Annual Rateof of10% 10% Rate

AnnualC Compound ompound Annual Rateof of10% 10% Rate

WhatisisAl’s Al’s What impleIInterest nterestand and SSimple

WhatisisAl’s Al’s What nterestand and IInterest

Maturity aturityV Value? alue? M

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Compounded ompoundedV Value? alue? C

Compound Compound

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Interest Interest

Simple Vs Compound Interest

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AlJones Jonesdeposited deposited$1,000 $1,000in inaasavings savingsaccount accountfor for55years yearsat at10% 10% Al Simple Simple

n = 5 * 1 = 5 i = .10

I = Prt I = $1,000 * .10 * 5 = $500 FV = $1,000 + $500

= $1,500 McGraw-Hill Ryerson©

Formulae Formulae

Compound Compound

FV = PV(1 + i)n I = FV – PV

Compare

= $1610.51 - $1000

Compare

FV = $1000(1.1)5 = $1,000 *1.6105

= $610.51

Compound Compound

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Interest Interest

Beginning Beginning Balance Balance

$1,000 $1,000

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Nominal Rates of Interest Compared Nominal Nominal Rate Rate

6% ++6%

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Compounding Compounding Period Period

Ending Ending Balance Balance

Annual Semiannual

$1,060.00 $1,060.90

Quarterly

$1,061.36

Daily

$1,061.83

Compound Compound

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Interest Interest

Compounding Daily Interest

8 - 24

Calculate the Future Value of $2,000 compounded daily for 4 years at 4.5%.

n FV = PV(1 + i) Formula Formula n = 4 * 365 = 1460 i = .045 /365 = 0.0001232

FV = $2000(1+ .045/365)1460 $2,000 ** 1.1972 1.1972 == $2,394.41 $2,394.41 == $2,000 McGraw-Hill Ryerson©

Compound Compound

8 - 25

8

Interest Interest

You invested $6000 at 4.5% compounded quarterly. After 2 years, the rate changed to 5.2% compounded monthly. What amount will you have 41/2 years after the initial investment?

Prepare a ‘time-line’ as part of the solution

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8 - 26

Compound Compound

8

Interest Interest

You invested $6000 at 4.5% compounded quarterly. After 2 years, the rate changed to 5.2% compounded monthly. What amount will you have 41/2 years after the initial investment?

0 $6000

2 years

FV1 = PV2 i = .045/4 n = (2*4) = 8

FV1 = 6000(1+.045/4)8 = 6000(1.0936) = 6561.75 McGraw-Hill Ryerson©

4.5 years

FV2 i = .052/12 n = 2.5*12 = 30 FV2 = 6561.75(1+.052/12)30 = 6561.75(1.1385) = $7470.61

8 - 27

Compound Compound

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Interest Interest

You borrowed $5000 at 7% compounded monthly. On the first and second anniversaries of the loan, you made payments of $2500. What is the balance outstanding immediately following the second payment?

Prepare a ‘time-line’ as part of the solution

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Compound Compound

8 - 28

8

Interest Interest

You borrowed $5000 at 7% compounded monthly. On the first and second anniversaries of the loan, you made payments of $2500. What is the balance outstanding immediately following the second payment? 1 year 2 years 0 $5000 FV1 - $2500 = PV2 FV2 i = .07/12 n = 12 i = .07/12 n = 12 12 FV1 = 5000(1+.07/12) FV2 = 2861.45 (1+.07/12)12 = 5000(1.072290) = 2861.45(1.072290) = 5361.45 = $3068.30 PV2 = 5361.45 – 2500.00 = $3068.30 – 2500.00 = 2861.45 NewBalance Balance = $568.30 New McGraw-Hill Ryerson©

Compound Compound

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Interest Interest

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8 - 29

Compound Compound

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Interest Interest

8 - 30

Calculating Present Present Value Value Calculating

You expect to need $1,500 in 3 years. Your bank offers 4% interest compounded semiannually. How much money must you put in the bank today (PV) to reach your goal in 3 years?

Prepare the solution…(a) algebraically, and (b) by financial calculator

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8 - 31

Compound Compound

Calculating Present Present Value Value Calculating

8

Interest Interest

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

You expect to need $1,500 in 3 years. Your bank offers 4% interest compounded semiannually. How much money must you put in the bank today (PV) to reach your goal in 3 years? n=3*2=6

i = .04/2 = .02

(a) PV = $1500(1+.02)-6 = $1500 * .8880

1,331.96 0.88797 1.02 6

= $1,331.96 1500 McGraw-Hill Ryerson©

8 - 32

Compound Compound

Calculating Present Present Value Value Calculating

8

Interest Interest

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

What amount must you invest now at 5% compounded daily to accumulate to $6000 after 1 year? j = 5% m = 365

PV = $6000(1+.05/365)-365 = $6000 * .9512

i = .05/365 n = 1*365 = 365 FV = $6000

= $5,707.40

5,707.40 0.0001 0.9512 1.001 .05 365 1 365

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6000

Compound Compound

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Interest Interest

8 - 33

Equivalent Payments Payments Equivalent

Two payments of $2200 each must be made 1 and 4 years from now. If money can earn 5% compounded monthly, what single payment 3 years from now would be equivalent to the two scheduled payments? Step 1 Draw a Time-line Draw a Time-line Step 1 Step 22 Find Findthe theFV FVof ofthe thepayment paymentthat that Step movedfrom fromYear Year11to toYear Year33 isismoved Step 33 Find Findthe thePV PVof ofthe thepayment paymentthat that Step movedfrom fromYear Year44to toYear Year33 isismoved Prepare the solution…(a) algebraically, and (b) by financial calculator McGraw-Hill Ryerson©

Compound Compound

8 - 34

Equivalent Payments Payments Equivalent

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Interest Interest

Step 11 DrawaaTime-line Time-line Draw Step Two payments of $2200 each must be made 1 and 4 years from now. If money can earn 5% compounded monthly, what single payment 3 years from now would be equivalent to the two3scheduled payments? 0 1 year 2 years years 4 years PV1 $2200 $2200 FV2 FV i = .05/12 Step 22 Step

Findthe theFV FVof of Find thepayment payment the thatisismoved moved that fromYear Year11to to from Year33 Year McGraw-Hill Ryerson©

1

n = 2*12 = 24 PV 2

(a) FV1 = 2200(1+.05/12)24 = 2200(1.1049) = 2430.87

2430.87 Now

0

Compound Compound

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Interest Interest

Making aa choice!… choice!… Making

8 - 35

Suppose a bank quotes nominal annual interest rates of 6.6% compounded annually, 6.5% compounded semi-annually, and 6.4% compounded monthly on five-year GICs. Which rate should an investor choose for an investment of $1,000?

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Compound Compound

Comparisons

8

Interest Interest

8 - 36

Results Results j = 6.6% compounded annually j = 6.5%

1376.53 1376.53

compounded semi-annually j = 6.4%

1376.89 1376.89

compounded monthly

1375.96 1375.96

the 6.5% 6.5% compounded compounded semi-annually semi-annually the

provides for forthe the best best provides rate of of return return on on investment! investment! rate McGraw-Hill Ryerson©