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7148979 Engineering Economics Fundamentals

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Engineering Economics – Fundamentals John P. Greaney, PE Copyright 2004 All rights reserved (Revised 02/19/2004) 1.0. Time Value of Money Understanding the time value of money is the central theme of Engineering Economics. Interest is the cost of borrowing money from a lender (e.g., a bank or other financial institution, or an individual) Even if an owner doesn’t take on debt to finance a project, the opportunity cost of not investing the funds elsewhere is considered in the
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  Engineering Economics – Fundamentals John P. Greaney, PE Copyright 2004 All rights reserved (Revised 02/19/2004) 1.0. Time Value of Money Understanding the time value of money is the central theme of Engineering Economics. Interest is the cost of borrowing money from a lender (e.g., a bank or other financial institution, or an individual) Even if an owner doesn’t take on debt to finance a project, the opportunity cost of not investing the funds elsewhere is considered in the economic analysis of the project. This interest rate equivalent is often called the discount rate . 1.1. Future Value With Compound Annual Interest Calculation Virtually every commercial lending transaction makes use of compound interest. The general form of the Future Value equation takes this into consideration FV = P (1+i/t) nt Where FV = Future Value P = initial Principal i = effective interest rate n = term of the investment in years t = number of times interest is compounded during year Example 1:  What is the Future Value of $1,000 invested at 8% interest compounded annually for 20 years? FV = P (1+i/t) nt  = $1,000 (1 + (0.08/1)) (20 x 1)  = $4,660.96 Interest may be compounded monthly, daily, hourly, or even continuously Compounded monthly: FV = $1,000 (1 + (0.08/12)) (20 x 12)  = $4,926.80 Compounded daily: FV = $1,000 (1 + (0.08/365)) (20 x 365)  = $4,952.16 Compounded hourly: FV = $1,000 (1 + (0.08/8760)) (20 x 8760)  = $4,953.00  As the period for compounding becomes infinitely small (i.e. continuous compounding), the Future Value formula is expressed as: FV = Pe in Where FV = Future Value   P = initial Principal e = constant (2.71828183) i = effective interest rate n = term of the investment in years (Note: e is a constant and the base of the natural logarithm. Mathematicians often call it Euler’s (sounds like “oiler”) Number in honor of the Swiss mathematician Leonhard Euler. Financial types usually just call it the “compound interest constant”.) Example 2:  What is the Future Value of $1,000 invested at 8% interest compounded continuously for 20 years? FV = Pe in  = $1,000 (2.71828) (0.08 x 20)  = $4,953.03 The effect of increasing the frequency of compounding is depicted graphically in the following chart. Over a 20-year term, there’s little increase in the end value of the investment if interest is compounded more frequently than daily. Compound Interest $1,000 invested @ 8.00% for 20 years $4,500$4,600$4,700$4,800$4,900$5,000    Y  e  a  r   l  y   M  o  n   t   h   l  y   D  a   i   l  y   H  o  u  r   l  y   C  o  n   t . Frequency of Compounding    V  a   l  u  e   A   f   t  e  r   2   0   Y  e  a  r  s   1.2. Nominal and Effective Rates of Interest  The phenomenon of compound interest may also be considered by calculating the effective rate of interest. In Examples 1 and 2 above, the nominal rate is  8%. The effective rate of interest grows as the period of compounding shortens. The formula for the effective rate of interest is as follows. i = (1+ ((i (t) )/t)) t )-1 Where i = effective interest rate i (t)  = nominal interest rate t = number of times interest is compounded during year Note that in the term i (t)  , (t)   is a superscript, not “i” to the “t” power. The variable for the nominal rate of interest with monthly compounding would be denoted i (12) . For daily compounding, the variable would be i (365)   Example 3:  What is the effective rate of interest for an 8% nominal annual rate of interest compounded annually? i = (1+ ((i (t) )/t)) t )-1 = (1+ ((0.08)/1)) 1 )-1 = 0.0800000 = 8.00%  In the case of annual compounding, the nominal and effective rates of interest are the same. Example 4:  What is the effective rate of interest for an 8% nominal annual rate of interest compounded monthly? i = (1+ ((i (t) )/t)) t )-1 = (1+ ((0.08)/12)) 12 )-1 = 0.0829995 = 8.30%    As we increase the frequency of compounding from daily to hourly to an infinite number of periods (i.e. continuous compounding), the effective interest rate equation becomes Where i = effective interest rate i (t)  = nominal interest rate (where t approaches infinity) e = constant (2.71828183) t = number of times interest is compounded during year Example 5:  What is the effective rate of interest for an 8% nominal annual rate of interest compounded continuously? i   = (2.71828183) 0.08 ) -1 = 0.083287 = 8.33%    1.3. Average Annual Return vs. Average Annualized Return Many people don’t realize that there’s a big difference between average annual returns and average annualized returns. Average annual return usually provides a poor description of investment performance.  Average annual return is simply the sum of the annual change in the value of an investment divided by the number of years. R ave  = ( ! R n ) / n Where R ave  = Average Annual Return R n  = Annual Return for Year (n) n = term of the investment in years Let’s look at two examples. Example 6:  What is the average annual return for a $1,000 investment that gains 30% the first year, loses 30% the second year, and then gains 30% the third year? How much money do you have at the end of three years? R ave  = ( ! R n ) / n = (30%+(-30%)+30%)/3 = 10%   Starting Annual Annual Year End   Balance Return (%)Return ($)Balance Year 1 $ 1,000 30% $ 300 $ 1,300  Year 2 $ 1,300 -30% $ (390) $ 910  Year 3 $ 910 30% $ 273 $ 1,183   Example 7:  What is the average annual return for a $1,000 investment that gains 10% per year for three years? How much money do you have at the end of three years? R ave  = ( ! R n ) / n = (10%+10%+10%)/3 = 10%   Starting Annual Annual Year End   Balance Return (%)Return ($)Balance Year 1 $ 1,000 10% $ 100 $ 1,100  Year 2 $ 1,100 10% $ 110 $ 1,210  Year 3 $ 1,210 10% $ 121 $ 1,331 1.4. Compound Annual Growth Rate (CAGR)
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