understanding money management - engineering economics
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Engineering Economics
Understanding Money Management
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Road Map
Market Interest Rates
Calculating Effective Interest Rates Based onPayment Periods
Equivalence Calculations with Effective Interest
Rates Debt Management
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Market Interest Rates
The market interest: the interest rate quoted by thefinancial market, i.e. banks and financial institutions.
Interest rate: to consider any anticipated changes in
earning power as well as
purchasing power in the economy
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Market Interest Rates
Pay the minimum, pay for years Making minimum payments on your credit cards can cost
you a bundle over a lot of years. Here's what wouldhappen if you paid the minimum-or more-every month on
a $2.705 card balance, with a 18.38% interest rate
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Payment Rate How Long To Pay off Debt Interest Paid
2% 27 years, 2 months 11,047
4% 08 years, 5 months 2,707
8% 02 years, 1 months 594
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Nominal Interest Rates
Financial institution uses a unit of time other than a year-for example, a month or a quarter-when calculatinginterest payments.
Institution usually quotes the interest rate on an annual
basis. For example, "18% compounded monthly." This statement means simply that each month the bank
will charge 1.5% interest (12 months per year X 1.5%per month = 18% per year) on the unpaid balance.
18% is the nominal interest rate orannual percentagerate (APR) and that the compounding frequency ismonthly (12 times per year).
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Annual Effective Yields
APR commonly used by financial institutions and isfamiliar to many customers, not explain precisely theamount of interest that will accumulate in a year.
To explain the true effect of more frequent compounding
on annual interest amounts, the term effective interestrate, commonlyknown as annual effective yield, orannual percentage yield(APY)
Annual effective yield (oreffective annual interestrate): rate truly represents the interest earned in a year
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Annual Effective Yields
For example, to calculate a total interest paid for a credit carddebt of $1,000 paying an interest 1.5% monthly.
P = $1,000, i = 1.5%, and N = 12
F = P(1 + i)N = $1,000(1 + 0.015)12 = $1,195.62
Total interest paid 195.62 hay 19.56% > 18% The Annual Effective Yield
M = 1, ia = i
7
1)Mr(1ai
M
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Annual Effective Yields
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Nominal
RateAnnual Semi Annual Quarterly Monthly Daily
4% 4.00% 4.04% 4.06% Nhm 3 4.08%
5% 5.00%
6% 6.00%
7% 7.00%
8% 8.00% 8.16% 8.24%
9% 9.00%
10% 10.00%11% 11.00%
12% 12.00% 12.36% 12.55% 12.68% 12.74%
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Annual Effective Yields
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Nominal
RateAnnual Semi Annual Quarterly Monthly Daily
4% 4.00% 4.04% 4.06% 4.07% 4.08%
5% 5.00%
6% 6.00%
7% 7.00%
8% 8.00%
9% 9.00%
10% 10.00%11% 11.00%
12% 12.00% 12.36% 12.55% 12.68% 12.74%
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Determine a Compound Period
Given: r = 2.23%, ia = 2.5%, find M
0.025 = (1 + 0.0223/M)M
1
1.025 = (1 + 0.0223/M)M
By trial and error, we find that M = 365, which indicatesdaily compounding
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1)Mr(1ai
M
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Road Map
Market Interest Rates
Calculating Effective Interest Rates Based onPayment Periods
Equivalence Calculations with Effective Interest
Rates Debt Management
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Discrete Compounding
If cash flow transactions occurquarterly, but interest iscompounded monthly, to calculate the effective interestrate on a quarterly basis:
i = (1 + r/M)C - 1 or
i = (1 + r/CK)C
- 1
Where:
M = the number of interest periods per year,
C = the number of interest periods per payment period, and
K = the number of payment periods per year.Note that M = CK
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Effective Rate per Payment Period
Suppose that you make quarterly deposits into a savingsaccount that earns 8% interest compounded monthly.Compute the effective interest rate per quarter.
Given: r = 8%.
C = 3 interest periods per quarter,K = 4 quarterly paymentsper year, and
M = 12 interest periods per year.
Find: i
i= (1+8%/12)3 -1 = 2.013% per quarter
The annual effective interest rate
ia = (1 + 0.02013)4 = 8.24%
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Effective Rate per Payment Period
Suppose that you make quarterly deposits into a savingsaccount that earns 8% interest compoundedsemiannually. Compute the effective interest rate perquarter.
Given: r = 8%.C = 1/2 interest periods per quarter,
K = 4 quarterly paymentsper year, and
M = 2 interest periods per year.
Find: i
i= (1+8%/2)1/2 -1 = 1.98% per quarter
The annual effective interest rate
ia = (1 + 0.0198)4
= % 14
A li d S f P j M
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Continuous Compounding
To be competitive on the financial market, or to enticepotential depositors, some financial institutions offermore frequent compounding.
As the number of compounding periods (M) becomes
very large, the interest rate per compounding period(r/M) becomes very small. As M approaches infinity andr/M approaches zero. we approximate the situation ofcontinuous compounding
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1ei
1)CKr(1limi
1])CKr
[(1limi
r/k
C
CK
C
CK
A li d S ft P j t M t
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CaIcuIating an Effective Interest Rate
Find the effective interest rate per quarter at a nominal rateof 8% compounded (a) weekly. (b) daily, and (c)continuously.
Given: r = 8%, K = 4 payments per year. Find: i per quarter.
a) WeeklyC = M/K = 52/4 = 13
ia = (1 + r/CK)C1 = (1 + 8%/52)131 =
b) Daily
C = M/K = 365/4 = 91.25ia = (1 + r/CK)C 1 = (1 + 8%/365)91.25 1 =
c) Continuously
i = er/k-1 = e8%/4 - 1 =
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A li d S ft P j t M t
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Road Map
Market Interest Rates Calculating Effective Interest Rates Based on
Payment Periods
Equivalence Calculations with Effective Interest
Rates Debt Management
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A li d S ft P j t M t
E i l C l l i i h
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Applied Software Project ManagementEquivalence Calculations withEffective Interest Rates
When calculating equivalent values, we need to identifyboth the interest period and the payment period.
If the time interval for compounding is different from thetime interval for cash transaction (or payment), we needto find the effective interest ratethat covers the paymentperiod
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Calculating Auto Loan Payments
Dealers' newspaper advertisements:8.5% Annual Percentage Rate!
48-month financing on all Mustangs in stock.
60 cars to choose from
ALL READY FOR DELIVERY!Prices starting as low as $21,599
You just add sales tax and 1 % for dealer's freight.
We will pay the tag, title, and license
Add 4% sales tax = $863.96
Add 1% dealer's freight = $215.99
Total purchase price = $22,678.95
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Calculating Auto Loan Payments
You can afford to make a down payment of $2,678.95,so the net amount to be financed is $20,000.
a) What would the monthly payment be?
b) After the 25th payment, you want to pay off the
remaining loan in a lump sum amount. What is therequired amount of this lump sum?
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Calculating Auto Loan Payments
a) What would the monthly payment be?The advertisement does not specify a compounding period,
but in automobile financing, the interest and the paymentperiods are almost always monthly. Thus, the 8.5% APRmeans 8.5% compounded monthly
Given: P = $20,000, r = 8.5% per year, K = 12 paymentsper year, N = 48 months, M = 12 interest periods/year.
Find: A
i = 8.5%/12 = 0.7083%
A = P(A/P,i,N) = 20,000(A/P, 0.7083%,48)
A = 492.97
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]1Ni)(1
Ni)i(1P[A
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Calculating Auto Loan Payments
b) After the 25th payment, you want to pay off theremaining loan in a lump sum amount. What is therequired amount of this lump sum?
Calculating the equivalent worth of the remaining 23payments at the end of the 25th month, with the timescale shifted by 25 months:
Given:A = $492.97, i = 0.7083% per month, and N = 23months,
Find: Remaining balance after 25 months (B25
)
B25 = A(P/A,i,N) = 492.97(P/A,0.7083%,23)
B25 = 10,428.96
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Compounding Occurs at a Different Rate
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Applied Software Project ManagementCompounding Occurs at a Different Ratethan That at Which Payments Are Made
Procedure for dealing with compounding periods and
payment periods that cannot be compared is as follows:1. Identify the number of compounding periods per year
(M), the number of payment periods per year (K). andthe number of interest periods per payment period (C)
2. Compute the effective interest rate per payment period: For discrete compounding, compute
i = ( 1 + r /M)C - 1
For continuous compounding, compute
i = er/K
- 13. Find the total number of payment periods:
N = K X (number of years).
4. Use i and N in the appropriate formulas
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Compounding Occurs More
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Applied Software Project ManagementCompounding Occurs MoreFrequently than Payments Are Made
1. Identify the parameter values for M, K, and C. where M = 12 compounding periods per year,
K = four payment periods per year, and
C = three interest periods per payment period.
2. To compute effective interest: i = ( 1 + r /M)C -1 = (1 + 0.12/12)3 1
i = 3.030% per quarter
3. Find the total number of payment periods, N, where N = K (number of years) = 4 x 3 = 12 quarters.
4. Use i and N in the appropriate equivalence formulas:F = A(F/A, i, N) = 1000(F/A,3.030%,12)
F = 14,216.24
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N)A(F/A,i,]i
1Ni)(1A[F
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Road Map
Market Interest Rates Calculating Effective Interest Rates Based on
Payment Periods
Equivalence Calculations with Effective Interest
Rates Debt Management
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Borrowing with Credit Cards
Consumer's perspective: does not have to wait for apaycheck to reach the bank before making a purchase.
Most credit cards operate as revolving credit, i.e.
a line of borrowing that you can tap into at will
pay back as quickly or slowly as you want as long asyou pay the minimum required each month.
Four things affect card-based credit costs: annual fees,finance charges, the grace period, and the method of
calculating interest Three different ways to compute interest charges:
Adjusted Balance, Average Daily Balance, PreviousBalance
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Borrowing with Credit Cards
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Method Description Example of the Interest You Owe, Given aBeginning Balance of $3,000 at 18%
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.
With a $1,000 payment, your new balance
will be $2,000. You pay 1.5% interest for
the month on this new balance. which
comes out to (1.5%) ($2.000) = $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.
With a $1.000 payment on the 15th day,
your balance reduced to $2,000. Thereforethe interest on your average daily balance
for the month will be(l.5%)($3,000 +$2,000)/2 = $37.50.
Previous Balance
The bank does not subtract anypayments 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.
The annual interest rate is 18%compounded monthly. Regardless of your
payment size, the bank will charge 1.5%
on your beginning balance of $3.000.
Therefore, your interest for the month is
(1.5%)($3,000) = $45
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Paying Off Cards Saves a Bundle
Suppose that you owe $2,000 on a credit card thatcharges 18% APR, and you make either the minimum10% payment or $20, whichever is larger, every month.
How long will it take to pay off debt? Assume that thebank uses the adjusted-balance method to calculateyour interest.
Given:APR = 18% (or 1.5% per month). beginningbalance = $2.000, and monthly payment = 10% ofoutstanding balance or $20, whichever is larger.
Find: Number of months to pay off the loan, assumingthat no new purchases are made during this paymentperiod. (open excel file)
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Commercial Loans Calculating
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Applied Software Project ManagementCommercial Loans CalculatingPrincipal and Interest Payments
One of the most important applications of compoundinterest involves loans: paid off in installments over time.
Repaid in equal periodic amounts (e.g., weekly, monthly,quarterly, or annually): amortized loan.
Examples: automobile loans, loans for appliances. homemortgage loans, and most business debts
Most commercial loans: interest compounded monthly.
Two factors determine what borrowing will cost you: the
finance charge and the length of the loan. The cheapest loan is not necessarily the loan with the
lowest payments, or even the loan with the lowest interestrate.
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Commercial Loans Calculating
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Applied Software Project ManagementCommercial Loans CalculatingPrincipal and Interest Payments
Instead, you have to look at the total cost of borrowing.which depends on the interest rate, fees, and the term(i.e., the length of time it takes you to repay the loan).
While you probably cannot influence the rate and fees.you may be able to arrange for a shorter term
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Using Excel to Determine a Loan's
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Applied Software Project ManagementUsing Excel to Determine a Loan sBalance, Principal, and lnterest
Suppose you secure a home improvement loan in theamount of $5,000 from a local bank. The loan officergives you the following loan terms:
Contract amount = $5,000;
Contract period = 24 months; Annual percentage rate = 12%
Monthly installment = $235.37
Given: P = $5,000, A = $235.37 per month, r = 12% per
pear, M = 12 compounding periods per year, and N = 24months.
Find: Bn, and In, for n = 1 to 24.
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Using Excel to Determine a Loan's
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pp ed So a e ojec a age eUsing Excel to Determine a Loan sBalance, Principal, and lnterest
We can easily see how the bank calculated the monthlypayment of $235.37235.37(P/A, 1, 24) = 235.57 * 21.2431 = 5,000
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Comparing Different Financing Options
Decision to pay cash, take out a loan, or sign a lease (acar) depends on a number of personal as well aseconomic factors.
1. Have enough money to buy: purchase in cash
lose the opportunity to earn interest on the amount you spend
2. Debt financing: own the vehicle at the end of your financing term
Paying the interest will drive up the real cost
3. Leasing: length of lease agreement, monthly payments,the yearly mileage allowance tailored to driving needs
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Buying a Car: Paying in Cash versus
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pp j gBuying a Car: Paying in Cash versusTaking a Loan
Option A: Purchase the vehicle at the normal price of$26,200, and pay for the vehicle over 36 months withequal monthly payments at 1.9% APR financing.
Option B: Purchase the vehicle at a discounted price of$24,048 to be paid immediately. The funds that would beused to purchase the vehicle are presently earning 5%annual interest compounded monthly
Which option is more economically sound?
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Buying a Car: Paying in Cash versus
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pp j gBuying a Car: Paying in Cash versusTaking a Loan
Option A:
With the 1.9% interest, your monthly payments will be
A = 26,200(A/P,1.9%/12,36) = 749.29
PA = 749.29(P/A, 5%/12, 36) = 25,000
Option B:PB = 24,048
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Tutorial
Do end chapter problems: 3.3, 3.10, 3.14, 3.15, 3.18, 3.20, 3.34, 3.37, 3.40, 3.42, 3.43,
3.50, 3.61
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