Lecture 5

ENGR 390 Lecture 5: Understanding Money & Its Management - 2

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 and Effective Interest Rates with Different Compounding Periods Effective Rates

S.V. Atre

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 5: Understanding Money & Its Management - 2

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

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

S.V. Atre

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ENGR 390 Lecture 5: Understanding Money & Its Management - 2

Winter 2007

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.

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

S.V. Atre

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ENGR 390 Lecture 5: Understanding Money & Its Management - 2

Winter 2007

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?

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 5: Understanding Money & Its Management - 2

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 5: Understanding Money & Its Management - 2

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

S.V. Atre

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ENGR 390 Lecture 5: Understanding Money & Its Management - 2

Winter 2007

Problem 4 You will deposit $1000 every 3 months for 4 years into an account that pays interest of 8% per year, compounded monthly. The first deposit will be in 3 months. How much will be in the account in 4 years? GIVEN: A = $1,000 r = 8%/yr C = 3 periods/qr K = 4 payments/yr M = 12 mo/yr

i = [1 + 0 .08 /( 3)( 4 )] 3 − 1 = 2 .0130 %

FIND F: DIAGRAM: 0

1

F? 2

3

4 yrs

N = 4(4) = 16 F = $1,000 (F/A, 2.013%, 16) = $18,658.12

$1,000

Continuous Compounding

i = [1 + r / CK ]C − 1 where CK = number of compounding periods per year continuous compounding => C → ∞

i = lim[( 1 + r / CK ) C − 1] = er /K −1

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ENGR 390 Lecture 5: Understanding Money & Its Management - 2

Winter 2007

Case 4: 8% compounded continuously Payment Period = Quarter Interest Period = Continuously 1st Q

2nd Q

3rd Q

4th Q

∞ interest periods

Given r = 8%, K = 4 payments per year

i = er / K −1 = e 0 .02 − 1 = 2 .0201 % per quarter

Summary: Effective interest rate per quarter

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Case 1

Case 2

Case 3

Case 4

8% compounded quarterly

8% compounded monthly

8% compounded weekly

8% compounded continuously

Payments occur quarterly

Payments occur quarterly

Payments occur quarterly

Payments occur quarterly

2.000% per quarter

2.013% per quarter

2.0186% per quarter

2.0201% per quarter

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ENGR 390 Lecture 5: Understanding Money & Its Management - 2

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 C

i = [1 + r / CK ] − 1

•For continuous compounding

i = er/K − 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

Continuous Case: Quarterly Deposits with Continuous 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: K = 4 payment periods/year

C = ∞ interest periods per quarter

Step 2: i

i = e0.12/ 4 − 1 = 3.045% per quarter

Step 3: Step 4:

N = 4(3) = 12 F = $1,000 (F/A, 3.045%, 12)

= $14,228.37

S.V. Atre

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ENGR 390 Lecture 5: Understanding Money & Its Management - 2

Winter 2007

Problem 5 Determine the total amount accumulated in an account paying interest at the rate of 10% per year, compounded continuously if deposits of $1,000 are made at the end of each of the next 5 years. DIAGRAM: 0

1

2

GIVEN:

F? 3

A = $1,000 r = 10%/yr C = ∞ K = 1 payment/yr N = 5 yrs

5 yrs FIND F: $1,000

Problem 5 GIVEN: A = $1,000 r = 10%/yr C = ∞ K = 1 payment/yr N = 5 yrs FIND F:

i = e0.10/1 −1 DIAGRAM: 0

1

2

F? 3

5 yrs $1,000

S.V. Atre

= 10.517 % per year N=5 F = $1,000 (F/A, 10.517%, 5) = $6,168.25

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ENGR 390 Lecture 5: Understanding Money & Its Management - 2

Winter 2007

Problem 6 A firm pays back a $10,000 loan with quarterly payments over the next 5 years. The $10,000 returns 4% APR compounded monthly. What is the quarterly payment amount ? DIAGRAM:

GIVEN:

$10,000 1

2

3

P = $10,000 r = 4%/yr C = 3 interest periods/quarter K = 4 payments/yr M = 12 compounding periods/yr

5 yrs = 20 qtr

0

FIND A: $A = ?

Problem 6 GIVEN: P = $10,000 r = 4%/yr C = 3 interest periods/quarter K = 4 payment periods/yr M = 12 compounding periods/yr DIAGRAM:

FIND A:

i = [1 + 0 .04 /( 3)( 4 )] 3 − 1

$10,000 1

2

3

5 yrs = 20 qtr

= 1 .0033 %

0 $A = ?

S.V. Atre

N = 4(5) = 20 A = $10,000 (A/P, 1.0033%, 20) = $554.30

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ENGR 390 Lecture 5: Understanding Money & Its Management - 2

Winter 2007

Complex Cash Flows Complex Cash Flows – Separate complex cash flows into component cash flows in order to use the standard formulas. Remember: You can only combine cash flows if they occur at the same point in time.

Problem 1 A construction firm is considering the purchase of an air compressor. The compressor has the following expected end of year maintenance costs: Year 1 $800 Year 2 $800 Year 3 $900 Year 4 $1000 Year 5 $1100 Year 6 $1200 Year 7 $1300 Year 8 $1400 What is the present equivalent maintenance cost if the interest rate is 12% per year compounded annually?

S.V. Atre

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ENGR 390 Lecture 5: Understanding Money & Its Management - 2

Winter 2007

Problem 1 – Alt Soln 1 GIVEN: MAINT COST1-8 PER YEAR TABLE i = 12%/YR, CPD ANNUALLY FIND P: P = PA + PG + PF = A(P|A,i,N) + G(P|G,i,N) + F(P|F,i,N) = $700(P|A,12%,8) + $100(P|G,12%,8) + $100(P|F,12%,1) DIAGRAM: P? 1

2

= $700(4.9676) + $100(14.4715) + $100(0.8929) = $5,014 3

4

PA ? 1

N=8 $700

0

2

3

4

N=8 PF ? N=1

0 $700

$100 $100 $200

$300

PG ? 1

$700

0 2

4

3

$100

N=8

0

NOTE: CAN SEPARATE INTO 3 CASH FLOWS: $100 $200 ANNUAL, LINEAR GRADIENT, AND FUTURE

$700

$300

Problem 1 – Alt Soln 2

GIVEN: MAINT COST1-8 PER DIAGRAM i = 12%/YR, CPD ANNUALLY FIND P: P = PA + PG(PPG) = A(P|A,i,N) + G(P|G,i,N-1)(P|F,i,1)

= $800(P|A,12%,8) + $100(P|G,12%,7)(P|F,12%,1) DIAGRAM: P? 1 2

= $800(4.9676) + $100(11.6443)(0.8929) = $5,014 3

4

N=8

0 $800 $100

PA ? 1

4

N=8 $800

PG ? 0

$600 0 NOTE: PG MUST BE OFFSET ONE YEAR – SO BRING THE OFFSET YEAR BACK TO TIME ZERO

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3

0 PPG ?

$200

2

1

2

1 PG ? $100

3

N=7 $200 $600

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ENGR 390 Lecture 5: Understanding Money & Its Management - 2

Winter 2007

Problem 2 A young couple has decided to make advance plans for financing their 3 year old daughter’s college education. Money can be deposited at 8% per year, compounded annually. What annual deposit on each birthday, from the 4th to the 17th (inclusive), must be made to provide $7,000 on each birthday from the 18th to the 21st (inclusive)? DIAGRAM: 4 0

$7,000 17 18 21 yrs A?

GIVEN: WITHDRAWALS18-21 = $7 000 i = r = 8%/YR, CPD YEARLY FIND A4-17: P17 = A(P|A,i,N) = A(F|A,i,N) = 7 000(P|A,8%,4) = A(F|A,8%,14)

STRATEGY: CAN BREAK INTO 2 CASH FLOWS, SO PICK A CONVENIENT POINT IN TIME AND SET = 7 000(3.3121) = A(24.2149) DEPOSITS EQUAL TO WITHDRAWALS…

A = $957

Problem 3 Anita Plass-Tuwurk, who owns an engineering consulting firm, bought an old house to use as her business office. She found that the ceiling was poorly insulated and that the heat loss could be cut significantly if 6 inches of foam insulation were installed. She estimated that with the insulation she could cut the heating bill by $40 per month and the air conditioning cost by $25 per month. Assuming that the summer season is 3 months (June, July, August) of the year and the winter season is another 3 months (December, January, and February) of the year, how much can she spend on insulation if she expects to keep the property for 5 years? Assume that neither heating nor air conditioning would be required during the fall and spring seasons. She is making this decision in April about whether to install the insulation in May. If the insulation is installed, it will be paid for at the end of May. Anita’s interest rate is 9%, compounded monthly.

S.V. Atre

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ENGR 390 Lecture 5: Understanding Money & Its Management - 2

Winter 2007

Problem 3

GIVEN: SAVINGS = $40/MO(DEC,JAN, FEB); $25/MO (JUN, JUL, AUG) r = 9%/YR, CPD MONTHLY FIND P(SAVINGS OVER 5 YEARS): 5 YR DIAGRAM: PA

1ST YR DIAGRAM: 0

$40

$25

0 1 2 3 4 5 6 7 8 9 10 11 12 MO PA ? α β

1 2 3 4 5 YRS P?

i = r = 0.09 = 0.75% / MO M 12 PA = Pα + Pβ(PPβ) = Aα(P|A,i,Nα) + Aβ(P|A,i,Nβ)(P|F,i,6) = $25(P|A,0.75%,3) + $40(P|A,0.75%,3)(P|F,0.75%,6) = $25(2.9556) + $40(2.9556)(0.9562) = $186.94

Problem 3

GIVEN: SAVINGS = $40/MO(DEC,JAN, FEB); $25/MO (JUN, JUL, AUG) r = 9%/YR, CPD MONTHLY FIND P(SAVINGS OVER 5 YEARS):

$186.94

5 YR DIAGRAM: $186.94 0

=

1 2 3 4 5 YRS P? C = 12 K =1 C

r ⎞ ⎛ i = ⎜1 + ⎟ −1 ⎝ CK ⎠ 12

⎛ 0.09 ⎞ = ⎜1 + ⎟ ⎝ 12(1) ⎠ = 9.38%

S.V. Atre

−1

0

$186.94 +

1 P0 ?

2 3 4 5 YRS

0

1 2 3 4 5 YRS PA ?

P = P0 + PA = A + A(P | A, i,N) ⎡ (1 + i)N − 1⎤ = A + A⎢ N ⎥ ⎣ i(1 + i) ⎦ = $186.94 + $186.94(P | A,9.38%,4) ⎡ (1 + 0.0938 )4 − 1 ⎤ = $186.94 + $186.94 ⎢ 4⎥ ⎣ 0.0938(1 + 0.0938 ) ⎦ = $186.94 + $186.94[3.21288 ] = $787.56

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ENGR 390 Lecture 5: Understanding Money & Its Management - 2

Winter 2007

Credit Card Debt † Annual fees † Annual

percentage rate † Grace period † Minimum payment † Finance charge

Methods of Calculating Interest on Credit Card Method

S.V. Atre

Description

Interest You Owe

Adjusted Balance

The bank subtracts the amount of your payment from the beginning balance and charges you interest on the remainder. This method costs you the least.

Your beginning balance is $3,000. With the $1,000 payment, your new balance will be $2,000. You pay 1.5% on this new balance, which will be $30.

Average Daily Balance

The bank charges you interest on the average of the amount you owe each day during the period. So the larger the payment you make, the lower the interest you pay.

Your beginning balance is $3,000. With your $1,000 payment at the 15th day, your balance will be reduced to $2,000. Therefore, your average balance will be (1.5%)($3,000+$2,000)/2=$37.50.

Previous Balance

The bank does not subtract any payments you make from your previous balance. You pay interest on the total amount you owe at the beginning of the period. This method costs you the most.

Regardless of your payment size, the bank will charge 1.5% on your beginning balance $3,000: (1.5%)($3,000)=$45.

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ENGR 390 Lecture 5: Understanding Money & Its Management - 2

Winter 2007

Buying versus Lease Decision Option 1 Debt Financing

Option 2 Lease Financing

Price

$14,695

$14,695

Down payment

$2,000

0

APR (%) Monthly payment Length

3.6% $372.55

$236.45

36 months

36 months

Fees

$495

Cash due at lease end

$300

Purchase option at lease end Cash due at signing

S.V. Atre

$8.673.10 $2,000

$731.45

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ENGR 390 Lecture 5: Understanding Money & Its Management - 2

Winter 2007

Which Option is Better? † Debt Financing:

Pdebt = $2,000 + $372.55(P/A, 0.5%, 36) - $8,673.10(P/F, 0.5%, 36) = $6,998.47 † Lease Financing:

Please = $495 + $236.45 + $236.45(P/A, 0.5%, 35) + $300(P/F, 0.5%, 36) = $8,556.90

Summary † Financial institutions often quote interest

rate based on an APR. † In all financial analysis, we need to convert the APR into an appropriate effective interest rate based on a payment period. † When payment period and interest period differ, calculate an effective interest rate that covers the payment period.

S.V. Atre

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