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Dr. Fernando Gonzalez, University of Central Florida, Orlando Florida 1 Economics Notes The following notes are used for the economics portion of Senior Design 1. The material and examples are extracted from “Engineering Economic Analysis” 6th Edition by Donald G. Newnan, Engineering Press. Notation i Interest rate per period. n Number of periods P Present sum of money F Future sum of money A End-of-period cash receipt or disbursement, (ex. like monthly payment) G Uniform period-by-period increases or decreases of “A” (ex. increase your monthly payment by $10 each month) g Uniform rate increases or decreases of “A” (ex. increase your monthly payment by 10% each month) r Nominal interest rate per period Interest: Monetary fee for renting or loaning money. Compounding: Compounding is when the interest is charged or credited to the original balance at the end of each period. Since the new balance is now different the next interest will also be different.Dr. Fernando Gonzalez, University of Central Florida, Orlando Florida 2 Simple Interest Simple interest is the interest computed on the original sum without compounding. If you borrowed P Dollars for n years at an interest rate of i per year with no compounding then you will owe Pininerest= In interest and adding the original sum will be PPinF+= Or )1( inPF+= Example: You borrow $3000 for 5 year from a friend who charges you 12% annual simple interest. How much will you owe at the end of the 5 years. 4800$))5)(12.0(1(3000$ =+=FDr. Fernando Gonzalez, University of Central Florida, Orlando Florida 3 Compounded Interest Compounded interest is the interest charged on the current sum. The current sum changes as the interest fee is added periodically. Suppose you will borrow P Dollars for 4 years compounded yearly then: After the 1st year you balance is )1(1iPF += After the 2nd year your balance is 212)1()1)(1()1( iPiiPiFF +=++=+= After the 3rd year your balance is 323)1()1)(1)(1()1( iPiiiPiFF +=+++=+= And finally after the 4th year your balance will be 43)1()1)(1)(1)(1()1( iPiiiiPiFF +=++++=+= PF So in general niPF )1( += We refer to this formula as the single payment compound amount factor and is written as: ),,/( niPF And the single payment compound amount formula is then as: ),,/( niPFPF=Dr. Fernando Gonzalez, University of Central Florida, Orlando Florida 4 Example: You borrow $3000 for 5 years from a Bank who charges you 12% interest per year compounded annually. How much will you owe at the end of the 5 years. F3000 03.5287$))12.0(1(3000$5=+=F This sum is more than the $4800 because your friend does not charge you interest on the interest you owe him while the bank does. Example: The same problem as before but now the bank compounds monthly instead of yearly. How much will you owe at the end of the year. 09.5450$)01.1(3000$)1212.01(3000$60)12(5=+=+=F This is more because compounded yearly means that within the year it is simple interest where as now you only get simple interest within each month instead of each year. We could also have used a table lookup. First find the value of ),,/( niPF for i = 0.01 and n = 60. Then multiply this amount by $3000.Dr. Fernando Gonzalez, University of Central Florida, Orlando Florida 5 Uniform Series Compound Amount Uniform payment series is where you have periodical payments. Like you pay back the loan using monthly payments. Suppose you will deposit A Dollars each year for 4 years into a savings account that pays i interest per year compounded yearly. How much will you have at the end of the 4 years? AAAAF If we only consider the 1st year’s deposit then after the 4 years we will have: AAF1F2AF3AF4+++AAAAF= 31)1( iAF += We rise to the power of 3 instead of 4 because we will make our 1st deposit at the end of the 1st year and by then we will only have 3 years to left. The convention with uniform payments, “A,” is to make the 1st payment or deposit after the 1st period. Now if we only consider the 2nd year’s deposit then after the 4 years we will have: 22)1( iAF += Considering only the 3rd year’s deposit then after the 4 years we will have: 13)1( iAF += And finally considering only the 4th year’s deposit then after the 4 years we will have: AiAF =+=04)1(Dr. Fernando Gonzalez, University of Central Florida, Orlando Florida 6 Now if we consider all 4 deposits we simply add the 4 F’s. We get AiAiAiAF ++++++= )1()1()1(23 Factor out the A we get []1)1()1()1(23++++++= iiiAF Now we create a new equation multiplying by )1( i+ [])1()1()1()1()1(234iiiiAFi +++++++=+ Now we subtract the two equations so that we can eliminate all the terms but the first and last. [][]1)1()1()1()1()1()1()1()1(23234++++++=−+++++++=+iiiAFiiiiAFi To get []1)1(4−+= iAiF And simplify: −+=iiAF1)1(4 In the general case we have n periods instead of 4 so we have: −+=iiAFn1)1( We refer to this factor as the uniform series compound amount factor and is written as: ),,/( niAF So the uniform series compound amount formula is: ),,/( niAFAF=Dr. Fernando Gonzalez, University of Central Florida, Orlando Florida 7 Example: You deposit $500 annually for 7 year into a savings account that pays 6% interest per year compounded yearly. How much will you have in this account at the end of the 7 years? Note the first payment is in 1 year. AAAAAAAF 92.196,4$06.01)06.01(500$7=−+=F Or you could use the lookup table for ),,/( niAF using 0.06 for i and 7 for n. Then multiply this factor by $500 to get F.Dr. Fernando Gonzalez, University of Central Florida, Orlando Florida 8 Uniform Series Sinking Fund If we take the inverse of the uniform series compound amount factor to get A given F then we have: −+=1)1(niiFA We refer to this factor as the uniform series sinking fund factor and is written as: ),,/( niFA So the uniform series sinking fund formula is: ),,/( niFAFA= Example: You want to save $4000 for a trip you will take in 3 years by making 3 yearly deposits into a savings account that pays 6% annually compounded annually. How much will I have to deposit each year? Note the first payment is in 1 year. AAAF 44.1256$1)06.01(06.04000$3=−+=A Or you could use the lookup table for ),,/( niFA using 0.06 for i and 3 for n. Then multiply this factor by $4000 to get A.Dr. Fernando Gonzalez, University of Central Florida, Orlando


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UCF EEL 4914 - Lecture Economics

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