Lecture 4

ENGR 390 Lecture 4: Understanding Money & Its Management - 1

Winter 2007

Chapter 5: Understanding Money & Its Management 1. If interest period is other than annual, how do we calculate economic equivalence? 2. If payments occur more frequently than annual, how do we calculate economic equivalence?

Nominal Versus Effective Interest Rates Nominal Interest Rate: Interest rate quoted based on an annual period

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Effective Interest Rate: Actual interest earned or paid in a year or some other time period

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ENGR 390 Lecture 4: Understanding Money & Its Management - 1

Winter 2007

18% Compounded Monthly

Interest period

Nominal interest rate

Annual percentage rate (APR)

18% compounded monthly † Question: Suppose that you invest $1 for 1 year at an

18% compounded monthly. How much interest would you earn? † Solution:

F = $ 1(1 + i )12 = $ 1(1 + 0 .015 )12

= $1.1956 ia = 0.1956 or 19.56% 18%

: 1.5%

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ENGR 390 Lecture 4: Understanding Money & Its Management - 1

Winter 2007

Single Cash Flow Formula F = P(1 + i) N F = P( F / P, i, N)

† Single payment

compound amount factor (growth factor) † Given: i = 18 / 12 = 1 . 5 % N = 12 months P = $1 † Find:

F

0 N

F = 1(1.1956 )

P

F = $1.1956

From Interest Table

Nominal and Effective Interest Rates with Different Compounding Periods Effective Rates

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Nominal Rate

Compounding Annually

Compounding Semi-annually

Compounding Quarterly

Compounding Monthly

Compounding Daily

4%

4.00%

4.04%

4.06%

4.07%

4.08%

5

5.00

5.06

5.09

5.12

5.13

6

6.00

6.09

6.14

6.17

6.18

7

7.00

7.12

7.19

7.23

7.25

8

8.00

8.16

8.24

8.30

8.33

9

9.00

9.20

9.31

9.38

9.42

10

10.00

10.25

10.38

10.47

10.52

11

11.00

11.30

11.46

11.57

11.62

12

12.00

12.36

12.55

12.68

12.74

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ENGR 390 Lecture 4: Understanding Money & Its Management - 1

Winter 2007

Effective Annual Interest Rates (9% compounded quarterly) First quarter

Base amount + Interest (2.25%)

$10,000 + $225

Second quarter

= New base amount + Interest (2.25%)

= $10,225 +$230.06

Third quarter

= New base amount + Interest (2.25%)

= $10,455.06 +$235.24

Fourth quarter

= New base amount + Interest (2.25 %) = Value after one year

= $10,690.30 + $240.53 = $10,930.83

Equivalence Calculations with More Frequent Compounding r = nominal, annual interest rate and compounding period Examples: 12% per year compounded monthly (1) 10% per year compounded quarterly (2) i

= interest rate per compounding period = annual interest rate ________________________________________________________________________

( # of compounding periods per year) Examples: 12% / 12 months = 1% compounded monthly 10% / 4 quarters = 2.5% compounded quarterly

(1) (2)

Which would I rather have: 12% compounded annually or 12% compounded monthly?

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ENGR 390 Lecture 4: Understanding Money & Its Management - 1

Winter 2007

Let’s Show It! EFFECTIVE INTEREST RATE ia = effective interest rate per year compounded annually = ( 1 + interest rate per period) # of periods - 1 Example: r = 12% per year compounded monthly i = 12% / 12 months = 1 % compounded monthly ia = (1 + .01)12 – 1 = 12.68% compounded annually

Another example… r = 12% per year compounded semi-annually isemi-annual

= 12% / 2 = 6% per 6 months

ia = (1 + .06)2 – 1 = .1236 = 12.36% per year compounded annually As the compounding period gets smaller, does the effective interest rate increase or decrease?

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ENGR 390 Lecture 4: Understanding Money & Its Management - 1

Winter 2007

Let’s Illustrate the Answer… r = 12% per year compounded daily idaily

= 12%/365 = .000329

ia = (1 + .000329)365 – 1 = .12747 = 12.747% per year compounded annually What happens if we let the compounding period get infinitely small?

Effective Annual Interest Rate

ia = (1 + r / M ) − 1 M

r = nominal interest rate per year ia = effective annual interest rate M = number of interest periods per year

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ENGR 390 Lecture 4: Understanding Money & Its Management - 1

Winter 2007

Problem 1 The local bank branch pays interest on savings accounts at the rate of 6% per year, compounded monthly. What is the effective annual rate of interest paid on accounts? GIVEN: r = 6%/yr M = 12mo/yr

DIAGRAM: NONE NEEDED!

FIND ia:

M

r⎞ ⎛ ia = ⎜ 1 + ⎟ − 1 ⎝ M⎠ 12

⎛ 0.06 ⎞ = ⎜1 + ⎟ 12 ⎠ ⎝

− 1 = 6.17%

Equivalence Analysis using Effective Interest Rate † Identify the compounding period (e.g.,

annually, quarterly, monthly † Identify the payment period (e.g., annual, quarter, month, week, etc), etc) † Find the effective interest rate that covers the payment period.

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ENGR 390 Lecture 4: Understanding Money & Its Management - 1

Winter 2007

When Payment Periods and Compounding Periods Coincide Step 1: Identify the number of compounding periods (M) per year Step 2: Compute the effective interest rate per payment period (i) i = r/M Step 3: Determine the total number of payment periods (N) N = M (number of years) Step 4: Use the appropriate interest formula using i and N above

Dollars Up in Smoke What three levels of smokers who bought cigarettes every day for 50 years at $1.75 a pack would have if they had instead banked that money each week: Level of smoker

Would have had

1 pack a day

$169,325

2 packs a day

$339,650

3 packs a day

$507,976

Note: Assumes constant price per pack, the money banked weekly and an annual interest rate of 5.5% compounded weekly Source: USA Today, Feb. 20, 1997

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ENGR 390 Lecture 4: Understanding Money & Its Management - 1

Winter 2007

Sample Calculation: One Pack per Day Step 1: Identify compounding period “5.5% interest compounded weekly”

Step 2: Compute the effective interest rate per payment period. Payment period = weekly i = 5.5%/52 = 0.10577% per week

Step 3: Find the total number of deposit periods N = (52 weeks/yr.)(50 years) = 2600 weeks

Step 4: Use the appropriate formula F = A(F|A,I,N) Weekly deposit amount, A = $1.75 x 7 = $12.25 per week F = $12.25 (F/A, 0.10577%, 2600) = $169,325

Problem 2 What amount must be deposited in an account paying 6% per year, compounded monthly in order to have $2,000 in the account at the end of 5 years? GIVEN: F5 = $2,000 r = 6%/yr M = 12 mo/yr

DIAGRAM:

FIND P: $2,000

0

1

2 5 yrs

P?

⎛ 12 mos ⎞ N = (M )(# yrs ) = ⎜ ⎟( 5 yrs ) = 60 mos ⎝ yr ⎠ r ⎛ 0 .06 ⎞⎛ 1yr ⎞ =⎜ ⎟⎜ ⎟ = 0 .5 % / mo M ⎝ yr ⎠⎝ 12 mo ⎠ P = $ 2000 (P | F,0 .5 %, 60 ) = $ 2000 ( 0 . 7414 ) i=

= $ 1482 . 80

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ENGR 390 Lecture 4: Understanding Money & Its Management - 1

Winter 2007

Problem 3 $5,000 is to be repaid in equal monthly payments over the next 2 years. The first payment is due in 1 month. Determine the payment amount at a nominal interest rate of 12% per year, compounded monthly. GIVEN:

DIAGRAM: $5,000

⎛ 12 mos ⎞ P = $5,000 ⎟⎟ ( 2 yrs ) = 24 mos N = ( M )(# yrs ) = ⎜⎜ r = 12%/yr ⎝ yr ⎠ M = 12 mo/yr ⎛ 0 . 12 ⎞ ⎛ 1 yr ⎞ r ⎟⎜ i= = ⎜⎜ ⎟ = 1 % / mo FIND A: M ⎝ yr ⎟⎠ ⎝ 12 mo ⎠ 1

2 yrs

A = $ 5000 ( A | P ,1 %, 24 ) = $ 5000 ( 0 .0471 ) = $ 235 .50

0 A?

Effective Interest Rate per Payment Period (i)

i = [1 + r / CK ]C − 1 C = number of interest periods per payment period K = number of payment periods per year r/K = nominal interest rate per payment period What happens when payment period is not annual?

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ENGR 390 Lecture 4: Understanding Money & Its Management - 1

Winter 2007

12% compounded monthly Payment Period = Quarter Compounding Period = Month

1st Qtr 1% 1%

3rd Qtr

2nd Qtr

4th Qtr

1%

3.030 %

One-year • Effective interest rate per quarter i = (1 + 0 . 01 ) 3 − 1 = 3 . 030 % • Effective annual interest rate 12 i a = ( 1 + 0 .0 1 )

− 1 = 1 2 .6 8 %

i a = ( 1 + 0 .0 3 0 3 0 ) 4 − 1 = 1 2 .6 8 %

Case 1: 8% compounded quarterly Payment Period = Quarter Interest Period = Quarterly 1st Q

2nd Q 1 interest period

3rd Q

4th Q

Given r = 8%, K = 4 payments per year C = 1 interest periods per quarter M = 4 interest periods per year

i = [1 + r / C K ] C − 1 = [1 + 0 . 0 8 / (1 ) ( 4 ) ] 1 − 1 = 2 . 0 0 0 % p e r q u a r te r

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ENGR 390 Lecture 4: Understanding Money & Its Management - 1

Winter 2007

Case 2: 8% compounded monthly Payment Period = Quarter Interest Period = Monthly 1st Q

2nd Q 3 interest periods

3rd Q

4th Q

Given r = 8%, K = 4 payments per year C = 3 interest periods per quarter M = 12 interest periods per year

i = [1 + r / C K ] C − 1 = [1 + 0 . 0 8 / ( 3 ) ( 4 ) ] 3 − 1 = 2 . 0 1 3 % p e r q u a r te r

Case 3: 8% compounded weekly Payment Period = Quarter Interest Period = Weekly 1st Q

2nd Q 13 interest periods

3rd Q

4th Q

Given r = 8%, K = 4 payments per year C = 13 interest periods per quarter M = 52 interest periods per year

i = [1 + r / C K ] C − 1 = [1 + 0 . 0 8 / (1 3 )( 4 )]1 3 − 1 = 2 . 0 1 8 6 % p e r q u a rte r

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ENGR 390 Lecture 4: Understanding Money & Its Management - 1

Winter 2007

Payment Periods Differ from Compounding Periods Step 1: Identify the following parameters M = No. of compounding periods per year K = No. of payment periods per year C = No. of interest periods per payment period Step 2: Compute the effective interest rate per payment period •For discrete compounding

i = [1 + r / CK ] C − 1 Step 3: Find the total no. of payment periods N = K (no. of years) Step 4: Use i and N in the appropriate equivalence formula

Discrete Case: Quarterly Deposits with Monthly Compounding Year 1 r = 12%,

0

1

2 3 4

Year 2 5

6

7 8

F3 = ?

Year 3 9 10 11

12

Quarters

A = $1,000 Step 1: M = 12 compounding periods/year

K = 4 payment periods/year C = 3 interest periods per quarter Step 2: i

i = [1 + 0 .12 /( 3)( 4 )] 3 − 1 = 3 .030 %

Step 3: Step 4:

N = 4(3) = 12 F = $1,000 (F/A, 3.030%, 12)

= $14,216.24

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