MIE 754 - Class #3 MIE 754 - Class #3 Manufacturing & Engineering Manufacturing & Engineering

EconomicsEconomics

•Concerns and QuestionsConcerns and Questions•Quick Recap of Previous ClassQuick Recap of Previous Class•Today’s Focus:Today’s Focus:

–Chap 2 - Computations Involving Chap 2 - Computations Involving Interest Interest (continued) (continued)

•Hmwk #2 Due next class:Hmwk #2 Due next class:–Chap 1 - Probs: 1, 6, 11, 12, 13Chap 1 - Probs: 1, 6, 11, 12, 13

Concerns and Questions?Concerns and Questions?

Quick Recap of Previous ClassQuick Recap of Previous Class

TerminologyTerminology EquivalenceEquivalence Time Value of MoneyTime Value of Money Simple and Compound InterestSimple and Compound Interest Present and Future ValuesPresent and Future Values

Cash Flow DiagramsCash Flow Diagrams

F occurs n periods after PF occurs n periods after P

What if you know F and want to find P?What if you know F and want to find P?

F=P(1+i)F=P(1+i)nn P=?P=?

Example of moving money through Example of moving money through time:time:

Suppose I=6%, F=1191.00, and n=3Suppose I=6%, F=1191.00, and n=3

How much is this equivalent to now?How much is this equivalent to now?

Additional examplesAdditional examples

Cashflow Series: Cashflow Series: UniformUniform

P occurs one time period before the P occurs one time period before the 1st uniform cashflow (A)1st uniform cashflow (A)

F occurs at the same time as the last F occurs at the same time as the last uniform cashflow (A)uniform cashflow (A)

(P/A i,n), (A/P i,n), (F/A i,n), (A/F i,n)(P/A i,n), (A/P i,n), (F/A i,n), (A/F i,n) Be consistent in time units for i and nBe consistent in time units for i and n n is ALWAYS = # of A’s of uniform n is ALWAYS = # of A’s of uniform

cashflow seriescashflow series

Cashflow Series: Cashflow Series: UniformUniform

P

F

A A A A A A A A

Lotto ExampleLotto Example

If you win $5,000,000 in the Virginia If you win $5,000,000 in the Virginia lottery, how much will you be paid lottery, how much will you be paid each year? How much money must each year? How much money must the lottery commission have on hand the lottery commission have on hand at the time of the award? Assume at the time of the award? Assume interest = 3% per year and that there interest = 3% per year and that there are 20 payments.are 20 payments.

Several Additional In-Class Several Additional In-Class ExamplesExamples

Cashflow Series: Cashflow Series: GradientGradient

Cash flows change by a constant Cash flows change by a constant amount each time periodamount each time period

Easiest if align with a uniform seriesEasiest if align with a uniform series Then follow guidelines for uniform Then follow guidelines for uniform

seriesseries We’ll only be concerned with two We’ll only be concerned with two

factors (P/G i,n) and (A/G i,n)factors (P/G i,n) and (A/G i,n)

Cashflow Series: Cashflow Series: GradientGradient

4 Types of Gradient4 Types of Gradient• positive and increasingpositive and increasing• positive and decreasingpositive and decreasing• negative and increasingnegative and increasing• negative and decreasingnegative and decreasing

Several ExamplesSeveral Examples

Cashflow Series: Cashflow Series: Special ConsiderationsSpecial Considerations

Deferred CashflowsDeferred Cashflows• In-class examplesIn-class examples

Changing Interest Over Study PeriodChanging Interest Over Study Period• In-class examplesIn-class examples

Nominal and Effective Interest RatesNominal and Effective Interest Rates

r% compounded ‘x’-lyr% compounded ‘x’-ly i = r/m interest per compounding i = r/m interest per compounding

period, where r= nominal rate and period, where r= nominal rate and m= # of compounding periods m= # of compounding periods

12% compounded quarterly is12% compounded quarterly is 3% every quarter (every 3 months)3% every quarter (every 3 months)

12% compounded semi-annually 12% compounded semi-annually means?means?

ExampleExample

You deposit $500 into a savings You deposit $500 into a savings account earning 6% compounded account earning 6% compounded semiannually. How much money is semiannually. How much money is in the account after 4 years?in the account after 4 years?

Effective Interest RateEffective Interest Rate

i i effeff = (1 + r/M) = (1 + r/M)mm - 1 - 1

where where M = number of interest periods per M = number of interest periods per

yearyear r = nominal interest rater = nominal interest rate m = number of interest periods of m = number of interest periods of

concernconcern

ExamplesExamples

Given Nominal Effective/year

Effective/period

18%compoundedannually18%compoundedsemiannually18%compoundedquarterly18%compoundedmonthly

More ExamplesMore Examples

What is the effective 1/2 year (semi-What is the effective 1/2 year (semi-annual) interest rate for 12% annual) interest rate for 12% compounded quarterly?compounded quarterly?

More ExamplesMore Examples

A variety of cashflow diagram types and A variety of cashflow diagram types and effective interesteffective interest

In-class examples for:In-class examples for:• UniformUniform• GradientGradient• Single Single

Where does the iWhere does the ieffeff formula come formula come

from?from?

Economic EffectivenessEconomic Effectiveness

Present Worth - equivalent worth of Present Worth - equivalent worth of entire cash flow at t= 0entire cash flow at t= 0

Annual Worth - equivalent uniform Annual Worth - equivalent uniform annual seriesannual series

Future Worth - equivalent worth at Future Worth - equivalent worth at t=nt=n

Rate of Return (Chap 4)Rate of Return (Chap 4) Benefit-Cost Ratio (Chap 7)Benefit-Cost Ratio (Chap 7)