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ENGR 390 Lecture 7: Review Winter 2007 S.V. Atre 1 Effective Annual Interest Rate r = nominal interest rate per year i a = effective annual interest rate M = number of interest periods per year 1 ) / 1 ( + = M a M r i

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Engineering Economics 390. Courtesy of Atre, Sundar V Associate Professor, Oregon State University.

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ENGR 390 Lecture 7: Review Winter 2007

S.V. Atre 1

Effective Annual Interest Rate

r = nominal interest rate per yearia = effective annual interest rateM = number of interest periods per year

1)/1( −+= Ma Mri

ENGR 390 Lecture 7: Review Winter 2007

S.V. Atre 2

Interest Rates …EFFECTIVE INTEREST RATE

C = number of interest periods per payment periodK = number of payment periods per yearr/K = nominal interest rate per

payment period

1]/1[ −+= CCKri

where CK = number of compounding periods per year

continuous compounding =>

1]/1[ −+= CCKri

1)(]1)/1lim[(

/1 −=

−+=Kr

C

eCKri

C→∞

Continuous Compounding

ENGR 390 Lecture 7: Review Winter 2007

S.V. Atre 3

TimingTimingEqual Payment Series (Annuity)• Equal cash flows• Equal time between cash flows• First cash flow at end of first period• Last cash flow at end of last period

0 N

A

F ?

1 2 3

F = A(F/A,i,N)

0 N

A

P ?

1 2 3

P = A(P/A,i,N)

TimingTimingUniform (Linear) gradient of amount G• 0 at end of Period 1• G at end of Period 2• 2G at end of Period 3• (N-1)G at end of Period N

0 N

G

P ?

1 2 3

P = G(P/G,i,N)

ENGR 390 Lecture 7: Review Winter 2007

S.V. Atre 4

TimingTimingGeometric gradient of amounts A1 and g• A1 at end of Period 1• A1 * (1 + g) at end of Period 2• A1 * (1+ g) ^ 2 at end of Period 3

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.

ENGR 390 Lecture 7: Review Winter 2007

S.V. Atre 5

LoansLoans

• When we calculate the annual payment of a loan (A), the payment is actually composed of interest and payment on principal.

• The mechanics are best shown through an example.

Problem 1Problem 1

You borrow $1,000 to help pay for rent, food, and books. It is to be repaid in 3 equal, annual payments starting one year from now. Interest on the loan is 12% per year, compounded annually.

Determine the amount of the loan payments (A), the corresponding principal (P) and interest (I) amounts in each payment and the remaining balance (B) after each payment

ENGR 390 Lecture 7: Review Winter 2007

S.V. Atre 6

SolutionGiven: P = $1,000; i = 12%; N= 3Find: A, I1, I2, I3 ; P1, P2, P3 ; B1, B2, B3 A = $1,000(A/P, 12%, 3) = $416.35B0 = P = $1,000I1 = B0i = $1,000*0.12 = $120P1 = A – I1 = $416.35 - $120 = $296.35B1 = B0 – P1 = $1000 - $296.35 = $703.65I2 = B1i =$703.65*0.12 = $84.44P2 = = A – I2 = $416.35 - $84.44 = $331.91B2 = B1 – P2 = $703.65 - $331.91 = $371.74

Method 1: Generalizing

Bn = P – (A – Pi) (F/A, i, n)

In = (Bn-1)i

Pn = A - In

For the nth payment

ENGR 390 Lecture 7: Review Winter 2007

S.V. Atre 7

ApproachesApproaches……

Loan problems can be worked Loan problems can be worked two ways:two ways:

1.1. Create a TableCreate a Table……

2.2. Work problems in the same Work problems in the same manner wemanner we’’ve been using ve been using formulaeformulae

Create a TableCreate a Table……

With the following column headings:

• Year or payment number• Payment size• Interest payment• Cumulative interest payment• Principal payment• Remaining principal

ENGR 390 Lecture 7: Review Winter 2007

S.V. Atre 8

Solution

0.00371.74249.0544.61416.353

371.74331.91204.4484.44416.352

703.65296.35120.00120.00416.351

RemainingBalance

PrincipalPayment

Cumul.Interest

InterestPayment

PmtSize #

Problem 2Problem 2

You wish to payoff the loan at the end of 2 years after making your second loan payment.

How much do you owe?

371.74

ENGR 390 Lecture 7: Review Winter 2007

S.V. Atre 9

Method 2: Remaining Balance

Bn = A(P/A, i, N-n)

In = (Bn-1)i = A(P/A, i, N-n+1)i

Pn = A(P/F, i, N-n+1)

Problem 3Problem 3

A student borrowed $5,000, which she will repay in 30 equal monthly installments. After making her 25th

payment, she desires to pay the remainder of the loan in a single payment.

At 12% per year, compounded monthly, what is the amount of the payment?

ENGR 390 Lecture 7: Review Winter 2007

S.V. Atre 10

Solution Given: P = $5,000; i = 1%; N = 30 months

Find B25 = A(P/A,i, N-n)

A = $5,000(A/P,1%,30) = $193.74

After 25 payments:N-n = 30-25 = 5B25 = $193.74(P/A,1%,5)

= $939.64

01 2 3 30

$A

DIAGRAM:

$ 5 000

B25 = ?

Problem 4Problem 4

A company has obtained a $10,000 loan at an interest rate of 15% per year, compounded annually. The loan requires $500 payments at the end of each of the next 3 years (starting one year from now).

Determine how much must be paid 4 years from now in order to payoff the loan.

ENGR 390 Lecture 7: Review Winter 2007

S.V. Atre 11

SolutionGiven: P = $10,000; i = 15%; A = $500Find: B4B0 = P = $10,000I1 = B0i = $10,000*0.15 = $1500B1 = $10,000 + $1,500 - $500 = $11,000I2 = B1i =$11,000*0.15 = $1,650B2 = $11,000 + $1,650 - $500 = $12,150I3 = B2i = $12,150*0.15 = $1,822.5B3 = B2 + I3 - $500 = $13,472.5I4 = B3i = $13,472.5*0.15 = 2020.85B4 = B3 + I4 = $15,493.375

Alternative Solution

Bn = P – (A – Pi) (F/A, i, n) + A

B4 = $10,000 – (500-10,000*0.15)(F/A,15%,4) + A

B4 = $10,000 + $5000 + $500= $15,500

ENGR 390 Lecture 7: Review Winter 2007

S.V. Atre 12

Perspective Perspective ……

Lender is indifferent between loan Lender is indifferent between loan payments and investing loan payments and investing loan amount at interest rate.amount at interest rate.

ΣΣ Payments = Interest Paid Payments = Interest Paid ++(Beginning (Beginning –– Ending Principal)Ending Principal)

PW of Loans = PW of PaymentsPW of Loans = PW of Payments

HW 1 Problem 1

4.7 a What is the amount accumulated by $7000 in 8 years at 9% compounded annually?

ENGR 390 Lecture 7: Review Winter 2007

S.V. Atre 13

HW 1 Problem 1

HW 1 Problem 2

4.18 b. What is the future worth of $2,000 at the end of each year for 10 years at 8.25% compounded annually?

ENGR 390 Lecture 7: Review Winter 2007

S.V. Atre 14

HW 1 Problem 2

HW 1 Problem 3

4.26. An individual deposits an annual bonus into a savings account that pays 6% interest, compounded annually. The size of the bonus increases by $1,000 each year, and the initial bonus amount was $3,000. Determine how much will be in the account immediately after the 5th

deposit.

ENGR 390 Lecture 7: Review Winter 2007

S.V. Atre 15

HW 1 Problem 3

HW 1 Problem 4

4.30a An oil well is expected to produce 100,000 barrels of oil during its first year. Its subsequent production is expected to decrease by 10% over the previous year’s production. The oil well has a reserve of 1,000,000 barrels. If the price per barrel remains steady at $30, what is the present worth of the anticipated stream of revenue at an annual interest of 12% compounded annually over the next 7 years?

ENGR 390 Lecture 7: Review Winter 2007

S.V. Atre 16

HW 1 Problem 4

HW 1 Problem 5

4.53. You have $10,000 available for investment in stock. You are looking for a growth stock whose value is $35,000 over 5 years. What kind of growth rate are you looking for?

ENGR 390 Lecture 7: Review Winter 2007

S.V. Atre 17

HW 1 Problem 5

HW 2 Problem 1

A store offers a credit card that charges 0.95% per month, compounded monthly. What is the nominal interest rate for this credit card? What is the effective annual interest rate?

ENGR 390 Lecture 7: Review Winter 2007

S.V. Atre 18

HW 2 Problem 1

HW 2 Problem 2

5.6. A company is advertising a 24-month lease of a car for $520 payable at the beginning of each month. There is a $2,500 down payment, plus a $500 refundable deposit. Alternatively, it offers a 24-month lease with a single up-front payment of $12,780 plus a $500 refundable security deposit. If the interest rate is 6% compounded monthly, which lease is preferred?

ENGR 390 Lecture 7: Review Winter 2007

S.V. Atre 19

HW 2 Problem 2

HW 2 Problem 3

5.20a. What is the future worth of $3,000 at the end of each 6 month period for 10 years at 6% compounded semi-annually?

ENGR 390 Lecture 7: Review Winter 2007

S.V. Atre 20

HW 2 Problem 3

HW 2 Problem 4

5.27. What is the amount A, such that you will be able to withdraw the amounts shown in the cash flow diagram if the interest rate is 8% compounded quarterly?

ENGR 390 Lecture 7: Review Winter 2007

S.V. Atre 21

HW 2 Problem 4

HW 2 Problem 5

5.34. Sam plans to retire in 15 years. He deposits equal amounts quarterly in a bank that gives him an interest of 8% compounded quarterly. If he want to withdraw $25,000 semi-annually over the 5 years of his retirement how much should he deposit quarterly? The first withdrawal occurs at the end of 6 months after his retirement.

ENGR 390 Lecture 7: Review Winter 2007

S.V. Atre 22

HW 2 Problem 5

HW 2 Problem 5 (continued)