The document discusses cost analysis for engineering projects. It covers:
- Investment costs, which are incurred during construction.
- Annual costs, which are ongoing like labor, materials, etc.
- The importance of analyzing multiple variants to select the lowest cost option.
- Converting costs over time to a common year using discounting, to compare total lifetime costs.
- Capitalizing annual costs to determine their present value equivalent to investment costs.
This document discusses methods for combining cash flow factors to solve more complex engineering economy problems involving shifted uniform series and gradients. It explains how to combine factors to find the present, future, or annual worth for cash flows that do not begin immediately or that change over time. Examples are provided to illustrate combining factors for shifted uniform series, non-standard shifted series, shifted gradients, and decreasing gradients. The use of spreadsheet functions to simplify the calculations is also discussed.
The document provides information on capital investment problems and cash flows, focusing on factors related to how time and interest affect money. It discusses single payment factors to find the future or present value of a single amount. It also covers uniform series factors to relate the present worth, future worth, or annual equivalent of a uniform series of cash flows. Finally, it examines gradient formulas, including arithmetic gradients where cash flows change by a constant amount each period, and geometric gradients where they change by a constant percentage each period. Formulas and examples are provided for each type of cash flow problem.
This chapter discusses various factors used in engineering economy to analyze cash flows over time under conditions of interest. It introduces the single-payment compound amount factor (SPCAF) and single-payment present worth factor (SPPWF) for analyzing single payments compounded over time. It also discusses the uniform series present worth factor (USPWF) and capital recovery factor (CRF) for analyzing uniform series of cash flows. Examples are provided to illustrate the calculation of future and present values using these factors under simple and compound interest conditions. Arithmetic and geometric gradient factors are also introduced for cash flows that regularly increase or decrease by a constant amount or percentage.
This document discusses concepts related to corporate income taxes and cash flow analysis. It defines key terms like gross income, taxable income, depreciation, taxes owed, and net profit after tax. It also discusses how to calculate cash flow before taxes and after taxes for each year of a project. The document provides an example cash flow analysis for a project using straight-line, double declining balance, and MACRS depreciation methods to demonstrate how different depreciation approaches can affect taxes paid over the life of a project. It emphasizes that accelerated depreciation methods and shorter recovery periods can lower the present value of total taxes paid.
This document provides information on engineering economics concepts related to cash flow, discount factors, equivalence, nonannual compounding, comparison of alternatives, depreciation, tax considerations, bonds, break-even analysis, inflation, and additional examples. Some key points include:
- Cash flow diagrams present cash flows as arrows on a timeline scaled to the magnitude of the cash flow. Expenses are down arrows and receipts are up arrows.
- Present worth, future worth, annual worth, and uniform gradient factors are used to convert between cash flows occurring at different times.
- Nonannual interest rates are converted to effective annual rates for analysis.
- Alternatives are compared using present worth, capitalized costs,
The document discusses the effects of inflation on engineering economy calculations. It defines inflation and presents two methods for accounting for inflation: 1) converting cash flows to constant value dollars using the real interest rate, and 2) using future dollars and the inflation-adjusted interest rate. The key effects of inflation are a reduction in purchasing power over time as money buys fewer goods/services. The document also discusses how inflation impacts present worth, future worth, and capital recovery calculations.
Lecture # 4 gradients factors and nominal and effective interest ratesBich Lien Pham
This document discusses gradients, which are cash flows that change by a regular pattern. It covers arithmetic gradients, where the cash flow increases or decreases by a constant amount each period, and geometric gradients, where the cash flow changes by a constant percentage each period. It also discusses shifted gradients, where the present value point is not at time 0, and how to calculate present values for these types of cash flows using factors or the NPV function in Excel.
The document discusses various methods for evaluating the economic profitability of capital investment projects, including present worth, future worth, annual worth, internal rate of return, and external rate of return. It provides examples of how to use these methods to calculate metrics like net present value, internal rate of return, payback period, and compare them to required rates of return to determine if projects are economically justified. The goal is to evaluate if a project's revenues over time can recover its costs and provide an adequate return given the risks.
This document discusses methods for combining cash flow factors to solve more complex engineering economy problems involving shifted uniform series and gradients. It explains how to combine factors to find the present, future, or annual worth for cash flows that do not begin immediately or that change over time. Examples are provided to illustrate combining factors for shifted uniform series, non-standard shifted series, shifted gradients, and decreasing gradients. The use of spreadsheet functions to simplify the calculations is also discussed.
The document provides information on capital investment problems and cash flows, focusing on factors related to how time and interest affect money. It discusses single payment factors to find the future or present value of a single amount. It also covers uniform series factors to relate the present worth, future worth, or annual equivalent of a uniform series of cash flows. Finally, it examines gradient formulas, including arithmetic gradients where cash flows change by a constant amount each period, and geometric gradients where they change by a constant percentage each period. Formulas and examples are provided for each type of cash flow problem.
This chapter discusses various factors used in engineering economy to analyze cash flows over time under conditions of interest. It introduces the single-payment compound amount factor (SPCAF) and single-payment present worth factor (SPPWF) for analyzing single payments compounded over time. It also discusses the uniform series present worth factor (USPWF) and capital recovery factor (CRF) for analyzing uniform series of cash flows. Examples are provided to illustrate the calculation of future and present values using these factors under simple and compound interest conditions. Arithmetic and geometric gradient factors are also introduced for cash flows that regularly increase or decrease by a constant amount or percentage.
This document discusses concepts related to corporate income taxes and cash flow analysis. It defines key terms like gross income, taxable income, depreciation, taxes owed, and net profit after tax. It also discusses how to calculate cash flow before taxes and after taxes for each year of a project. The document provides an example cash flow analysis for a project using straight-line, double declining balance, and MACRS depreciation methods to demonstrate how different depreciation approaches can affect taxes paid over the life of a project. It emphasizes that accelerated depreciation methods and shorter recovery periods can lower the present value of total taxes paid.
This document provides information on engineering economics concepts related to cash flow, discount factors, equivalence, nonannual compounding, comparison of alternatives, depreciation, tax considerations, bonds, break-even analysis, inflation, and additional examples. Some key points include:
- Cash flow diagrams present cash flows as arrows on a timeline scaled to the magnitude of the cash flow. Expenses are down arrows and receipts are up arrows.
- Present worth, future worth, annual worth, and uniform gradient factors are used to convert between cash flows occurring at different times.
- Nonannual interest rates are converted to effective annual rates for analysis.
- Alternatives are compared using present worth, capitalized costs,
The document discusses the effects of inflation on engineering economy calculations. It defines inflation and presents two methods for accounting for inflation: 1) converting cash flows to constant value dollars using the real interest rate, and 2) using future dollars and the inflation-adjusted interest rate. The key effects of inflation are a reduction in purchasing power over time as money buys fewer goods/services. The document also discusses how inflation impacts present worth, future worth, and capital recovery calculations.
Lecture # 4 gradients factors and nominal and effective interest ratesBich Lien Pham
This document discusses gradients, which are cash flows that change by a regular pattern. It covers arithmetic gradients, where the cash flow increases or decreases by a constant amount each period, and geometric gradients, where the cash flow changes by a constant percentage each period. It also discusses shifted gradients, where the present value point is not at time 0, and how to calculate present values for these types of cash flows using factors or the NPV function in Excel.
The document discusses various methods for evaluating the economic profitability of capital investment projects, including present worth, future worth, annual worth, internal rate of return, and external rate of return. It provides examples of how to use these methods to calculate metrics like net present value, internal rate of return, payback period, and compare them to required rates of return to determine if projects are economically justified. The goal is to evaluate if a project's revenues over time can recover its costs and provide an adequate return given the risks.
The document provides solutions to an engineering economics assignment involving time value of money calculations. It includes calculations for problems involving compound interest, effective interest rates, future values, and annuity payments. For each problem, it clearly shows the relevant cash flow diagram, formulas, calculations, and solutions. It also discusses ensuring the appropriate time periods are used for weekly and monthly cash flows.
This document discusses nominal and effective interest rates. It begins by defining key terms like nominal rate, effective rate, compounding period, and payment period. It then explains how to convert between nominal and effective rates for different compounding frequencies. The document provides examples of calculating future values for single payments and series of payments when the payment period is greater than or less than the compounding period. It also covers calculations for continuous compounding and situations when interest rates vary over time.
Chapter 7 ror analysis for a single alternativeBich Lien Pham
1. The document discusses methods for calculating rate of return (ROR) for projects and investments, including dealing with multiple ROR values and calculating external ROR.
2. ROR is the interest rate that makes the net present value of a project's cash flows equal to zero. Multiple ROR values can exist if the cash flows change sign more than once.
3. External ROR removes the assumption that positive cash flows are reinvested at the project's ROR by considering external borrowing and investment rates. It can be calculated using the modified internal rate of return or return on invested capital approaches.
This document is a slide set on after-tax economic analysis that was developed by Dr. Don Smith of Texas A&M University. It covers key topics related to after-tax cash flows and economic evaluation including terminology, taxes, depreciation, replacement analysis, and international tax considerations. The learning objectives are to understand after-tax concepts and how to incorporate taxes into economic analysis using spreadsheets. Examples are provided to illustrate after-tax calculations and how the results can differ from before-tax analysis.
This document discusses depreciation and provides details on various depreciation methods. It defines depreciation as how businesses can recover the lost value of capital assets over time by deducting the asset's value from taxes. The document outlines straight-line, declining balance, and MACRS depreciation methods. It explains key terms like book value, salvage value, and recovery period and provides examples of how to calculate depreciation using different methods.
1. The document discusses the time value of money concept in engineering economy and introduces related terminology and formulas.
2. Key points covered include equivalence of money over time with interest, definitions of interest rate and rate of return, and explanations of simple and compound interest.
3. Standard notation for interest factors is presented, including the general form of (X/Y,i%,n) to represent various interest factors. Cash flow diagrams and the symbols P, F, and A are also explained.
Chapter 2 factors, effect of time & interest on moneyBich Lien Pham
This document discusses factors related to time and interest rates that affect money. It covers single payment factors, uniform series factors, arithmetic and geometric gradients, and methods for finding unknown interest rates or time periods. Key learning outcomes include single payment, uniform series, and gradient factors as well as techniques for determining factor values for untabulated rates or periods. Examples are provided to illustrate concepts such as single payments, uniform series, arithmetic gradients, and finding unknown rates or time periods.
Lecture # 3 compounding factors effects of inflationBich Lien Pham
This document summarizes key concepts for determining unknown interest rates, inflation rates, and numbers of periods in engineering economy problems. It discusses using the IRR, RATE, and NPER functions in Excel to calculate unknown values. It also covers handling varying interest rates over time through period-by-period analysis or approximation using an average rate. The effects of inflation are explained, including how future costs are estimated using an inflation rate. Common inflation measures like the Consumer Price Index are also introduced.
This document discusses techniques for analyzing cash flows that involve shifting, combining factors, and gradients. It provides examples of how to calculate present and future values for shifted uniform series, series with single cash flows, and both positive and negative arithmetic and geometric gradients. The key steps involve renumbering cash flows to determine the applicable time periods, then using the appropriate present worth, future worth, or gradient factors and equations.
This document contains lecture notes on interest formulas including geometric series, uniform series, arithmetic gradient, geometric gradient, nominal and effective interest rates, and continuous compounding. It provides examples and explanations of formulas for present worth, future worth, and compound interest calculations for situations involving constant and increasing cash flows over time with single-rate and multiple compounding periods.
This document discusses various equity valuation approaches and methods. It begins by outlining the valuation process of understanding the business, forecasting performance, and selecting an appropriate valuation model. It then describes absolute valuation methods like discounted cash flow analysis using free cash flow to the firm (FCFF) or free cash flow to equity (FCFE). Relative valuation methods discussed include comparing price/earnings (P/E), price/book value (P/B), and enterprise value/EBITDA multiples to industry peers. The document also briefly discusses the residual income model and notes factors to consider in choosing the most appropriate valuation model.
This document discusses interest rates and time value of money concepts. It begins by defining simple and compound interest rates. Examples are provided to illustrate calculating interest and total amounts due using simple and compound interest formulas. The concept of economic equivalence is introduced, showing that different cash flows can be equivalent based on a common interest rate. The single payment compound interest formula is derived and used to solve examples of determining future or present values. Overall, the document provides an introduction to fundamental time value of money and interest rate concepts in engineering economics.
The document discusses various interest formulas and their applications. It introduces compound interest, where interest is calculated on both the principal amount and accumulated interest from previous periods. Formulas are provided to calculate the compound amount, present worth amount, equal payment series compound amount, equal payment series sinking fund, equal payment series present worth amount, and equal payment series capital recovery amount. An example calculation is shown for each formula to illustrate how to use it to solve investment problems.
1031191EE 200 Engineering Economics 2Time Value o.docxaulasnilda
10/31/19
1
EE 200: Engineering Economics 2
Time Value of Money
2
2
136 ENGINEERING ECONOMICS
ENGINEERING ECONOMICS
Factor Name Converts Symbol Formula
Single Payment
Compound Amount to F given P (F/P, i%, n) (1 + i)
n
Single Payment
Present Worth to P given F (P/F, i%, n) (1 + i)
–n
Uniform Series
Sinking Fund to A given F (A/F, i%, n) ( ) 11 + ni
i
Capital Recovery to A given P (A/P, i%, n)
( )
( ) 11
1
+
+
n
n
i
ii
Uniform Series
Compound Amount to F given A (F/A, i%, n)
( )
i
i n 11 +
Uniform Series
Present Worth to P given A (P/A, i%, n)
( )
( )n
n
ii
i
+
+
1
11
Uniform Gradient
Present Worth to P given G (P/G, i%, n)
( )
( ) ( )nn
n
ii
n
ii
i
++
+
11
11
2
Uniform Gradient †
Future Worth to F given G (F/G, i%, n)
( )
i
n
i
i n+
2
11
Uniform Gradient
Uniform Series to A given G (A/G, i%, n) ( ) 11
1
+ ni
n
i
NOMENCLATURE AND DEFINITIONS
A �������������Uniform amount per interest period
B �������������Benefit
BV �����������Book value
C �������������Cost
d��������������In ation ad usted interest rate per interest period
Dj ������������Depreciation in year j
EV �����������Expected value
F �������������Future worth, value, or amount
f �������������� eneral in ation rate per interest period
G �������������Uniform gradient amount per interest period
i ��������������Interest rate per interest period
ie �������������Annual effective interest rate
MARR ����Minimum acceptable/attractive rate of return
m �������������Number of compounding periods per year
n��������������Number of compounding periods; or the expected
life of an asset
P �������������Present worth, value, or amount
r ��������������Nominal annual interest rate
Sn ������������Expected salvage value in year n
Subscripts
j ����������� at time j
n����������� at time n
†����������� F/G = (F/A – n)/i = (F/A) × (A/G)
Risk
Risk is the chance of an outcome other than what is planned to
occur or expected in the analysis�
NON-ANNUAL COMPOUNDING
i m
r1 1e
m
= + -b l
BREAK-EVEN ANALYSIS
By altering the value of any one of the variables in a situation,
holding all of the other values constant, it is possible to find a
value for that variable that makes the two alternatives equally
economical� This value is the break-even point�
Break-even analysis is used to describe the percentage of
capacity of operation for a manufacturing plant at which
income will ust cover expenses.
The payback period is the period of time required for the profit
or other benefits of an investment to equal the cost of the
investment�
Time Value of Money Relating Future and
Present Worth Shows Unreasonable Costs
3
𝐹 is Future Worth 𝑃 is Present Worth 𝑛 is the number of compounding periods
𝐴 is Uniform Amount per interest period (Payment) 𝑖 is the interest rate per period
3
10/31/19
2
College Costs Have Historically Increased Faster
than Inflation: Compound Interest Model
• Umich: Total Annual Cost
o 1980: $3,500
o 2020: $30,0
The document discusses various interest and investment concepts including simple interest, compound interest, nominal and effective interest rates, present worth, discount, annuities, perpetuities, and capitalized costs. It defines key formulas used to calculate future and present values under different interest compounding assumptions such as discrete, continuous, and various time periods. Factors like Fa, Fb, Fc, Fd, Ce, and Cf are introduced to represent discount and compounding rates for different cash flow patterns.
This document discusses various money-time relationships and concepts of equivalence. It begins by defining money as a medium of exchange, store of value, and unit of account. It then discusses capital, the different types of capital (equity and debt), and interest. The remainder of the document discusses how interest rates are determined and various interest rate formulas for calculating present and future values of single and uniform cash flows under simple, compound, continuous and discrete compounding. It also covers economic equivalence and cash flow diagrams/notations.
The document discusses cash flow diagrams and economic equivalence. It defines key terms like cash inflows, outflows, present value, and future value. Cash flow diagrams visually represent the timing of money coming into and going out of a project. Economic equivalence establishes that different amounts of money at different points in time can be considered equal based on interest rates and the time value of money. The document provides examples of how to calculate future value given present value, and vice versa, as well as examples for uniform series and gradients of cash flows. It demonstrates calculations for present and future equivalent values.
This document provides an introduction to value engineering interest formulas and their applications. It defines key terms like value, cost value, exchange value, use value, esteem value and different types of functions. It describes the value analysis/value engineering procedure and discusses when it should be used. Various interest formulas are explained including compound amount, present worth, uniform series and capital recovery. Examples are provided to demonstrate how to use the formulas to calculate future values, single payments, and equal installment amounts.
This document provides an introduction to value engineering interest formulas and their applications. It defines key terms like value, cost value, exchange value, use value, esteem value, primary function, secondary function, and tertiary function. It describes the value analysis/value engineering procedure and discusses types of values, functions, and differences between value analysis and value engineering. The document also presents various interest formulas for single payment compound amount, single payment present worth, uniform gradient series method, effective interest rate, equal payment series present worth amount, and equal payment series capital recovery amount. It includes examples of problems solved using these formulas.
This document discusses time value of money calculations using multiple factors. It provides examples of calculating present worth for non-uniform cash flows using factors like present worth (P/F), compound amount (F/P), uniform series present worth (P/A), and uniform series compound amount (F/A). Methods include calculating individual amounts and summing, or finding an equivalent uniform annual series. The document also discusses remembering that P/A places present worth one year before payments, while F/A places future worth in the year of the last payment.
This document provides information about a project assignment for a course on the time value of money. It includes details such as the student's name, registration number, course, and semester. The objectives are to discuss techniques for investment evaluation like present value, future value, and annuities. Formulas are provided for present value, future value, ordinary annuities, growing annuities, perpetuities, and their applications in areas like loans, savings, and project evaluation. The conclusion emphasizes the importance of understanding these time value of money concepts for efficient financial management in construction projects.
The document provides solutions to an engineering economics assignment involving time value of money calculations. It includes calculations for problems involving compound interest, effective interest rates, future values, and annuity payments. For each problem, it clearly shows the relevant cash flow diagram, formulas, calculations, and solutions. It also discusses ensuring the appropriate time periods are used for weekly and monthly cash flows.
This document discusses nominal and effective interest rates. It begins by defining key terms like nominal rate, effective rate, compounding period, and payment period. It then explains how to convert between nominal and effective rates for different compounding frequencies. The document provides examples of calculating future values for single payments and series of payments when the payment period is greater than or less than the compounding period. It also covers calculations for continuous compounding and situations when interest rates vary over time.
Chapter 7 ror analysis for a single alternativeBich Lien Pham
1. The document discusses methods for calculating rate of return (ROR) for projects and investments, including dealing with multiple ROR values and calculating external ROR.
2. ROR is the interest rate that makes the net present value of a project's cash flows equal to zero. Multiple ROR values can exist if the cash flows change sign more than once.
3. External ROR removes the assumption that positive cash flows are reinvested at the project's ROR by considering external borrowing and investment rates. It can be calculated using the modified internal rate of return or return on invested capital approaches.
This document is a slide set on after-tax economic analysis that was developed by Dr. Don Smith of Texas A&M University. It covers key topics related to after-tax cash flows and economic evaluation including terminology, taxes, depreciation, replacement analysis, and international tax considerations. The learning objectives are to understand after-tax concepts and how to incorporate taxes into economic analysis using spreadsheets. Examples are provided to illustrate after-tax calculations and how the results can differ from before-tax analysis.
This document discusses depreciation and provides details on various depreciation methods. It defines depreciation as how businesses can recover the lost value of capital assets over time by deducting the asset's value from taxes. The document outlines straight-line, declining balance, and MACRS depreciation methods. It explains key terms like book value, salvage value, and recovery period and provides examples of how to calculate depreciation using different methods.
1. The document discusses the time value of money concept in engineering economy and introduces related terminology and formulas.
2. Key points covered include equivalence of money over time with interest, definitions of interest rate and rate of return, and explanations of simple and compound interest.
3. Standard notation for interest factors is presented, including the general form of (X/Y,i%,n) to represent various interest factors. Cash flow diagrams and the symbols P, F, and A are also explained.
Chapter 2 factors, effect of time & interest on moneyBich Lien Pham
This document discusses factors related to time and interest rates that affect money. It covers single payment factors, uniform series factors, arithmetic and geometric gradients, and methods for finding unknown interest rates or time periods. Key learning outcomes include single payment, uniform series, and gradient factors as well as techniques for determining factor values for untabulated rates or periods. Examples are provided to illustrate concepts such as single payments, uniform series, arithmetic gradients, and finding unknown rates or time periods.
Lecture # 3 compounding factors effects of inflationBich Lien Pham
This document summarizes key concepts for determining unknown interest rates, inflation rates, and numbers of periods in engineering economy problems. It discusses using the IRR, RATE, and NPER functions in Excel to calculate unknown values. It also covers handling varying interest rates over time through period-by-period analysis or approximation using an average rate. The effects of inflation are explained, including how future costs are estimated using an inflation rate. Common inflation measures like the Consumer Price Index are also introduced.
This document discusses techniques for analyzing cash flows that involve shifting, combining factors, and gradients. It provides examples of how to calculate present and future values for shifted uniform series, series with single cash flows, and both positive and negative arithmetic and geometric gradients. The key steps involve renumbering cash flows to determine the applicable time periods, then using the appropriate present worth, future worth, or gradient factors and equations.
This document contains lecture notes on interest formulas including geometric series, uniform series, arithmetic gradient, geometric gradient, nominal and effective interest rates, and continuous compounding. It provides examples and explanations of formulas for present worth, future worth, and compound interest calculations for situations involving constant and increasing cash flows over time with single-rate and multiple compounding periods.
This document discusses various equity valuation approaches and methods. It begins by outlining the valuation process of understanding the business, forecasting performance, and selecting an appropriate valuation model. It then describes absolute valuation methods like discounted cash flow analysis using free cash flow to the firm (FCFF) or free cash flow to equity (FCFE). Relative valuation methods discussed include comparing price/earnings (P/E), price/book value (P/B), and enterprise value/EBITDA multiples to industry peers. The document also briefly discusses the residual income model and notes factors to consider in choosing the most appropriate valuation model.
This document discusses interest rates and time value of money concepts. It begins by defining simple and compound interest rates. Examples are provided to illustrate calculating interest and total amounts due using simple and compound interest formulas. The concept of economic equivalence is introduced, showing that different cash flows can be equivalent based on a common interest rate. The single payment compound interest formula is derived and used to solve examples of determining future or present values. Overall, the document provides an introduction to fundamental time value of money and interest rate concepts in engineering economics.
The document discusses various interest formulas and their applications. It introduces compound interest, where interest is calculated on both the principal amount and accumulated interest from previous periods. Formulas are provided to calculate the compound amount, present worth amount, equal payment series compound amount, equal payment series sinking fund, equal payment series present worth amount, and equal payment series capital recovery amount. An example calculation is shown for each formula to illustrate how to use it to solve investment problems.
1031191EE 200 Engineering Economics 2Time Value o.docxaulasnilda
10/31/19
1
EE 200: Engineering Economics 2
Time Value of Money
2
2
136 ENGINEERING ECONOMICS
ENGINEERING ECONOMICS
Factor Name Converts Symbol Formula
Single Payment
Compound Amount to F given P (F/P, i%, n) (1 + i)
n
Single Payment
Present Worth to P given F (P/F, i%, n) (1 + i)
–n
Uniform Series
Sinking Fund to A given F (A/F, i%, n) ( ) 11 + ni
i
Capital Recovery to A given P (A/P, i%, n)
( )
( ) 11
1
+
+
n
n
i
ii
Uniform Series
Compound Amount to F given A (F/A, i%, n)
( )
i
i n 11 +
Uniform Series
Present Worth to P given A (P/A, i%, n)
( )
( )n
n
ii
i
+
+
1
11
Uniform Gradient
Present Worth to P given G (P/G, i%, n)
( )
( ) ( )nn
n
ii
n
ii
i
++
+
11
11
2
Uniform Gradient †
Future Worth to F given G (F/G, i%, n)
( )
i
n
i
i n+
2
11
Uniform Gradient
Uniform Series to A given G (A/G, i%, n) ( ) 11
1
+ ni
n
i
NOMENCLATURE AND DEFINITIONS
A �������������Uniform amount per interest period
B �������������Benefit
BV �����������Book value
C �������������Cost
d��������������In ation ad usted interest rate per interest period
Dj ������������Depreciation in year j
EV �����������Expected value
F �������������Future worth, value, or amount
f �������������� eneral in ation rate per interest period
G �������������Uniform gradient amount per interest period
i ��������������Interest rate per interest period
ie �������������Annual effective interest rate
MARR ����Minimum acceptable/attractive rate of return
m �������������Number of compounding periods per year
n��������������Number of compounding periods; or the expected
life of an asset
P �������������Present worth, value, or amount
r ��������������Nominal annual interest rate
Sn ������������Expected salvage value in year n
Subscripts
j ����������� at time j
n����������� at time n
†����������� F/G = (F/A – n)/i = (F/A) × (A/G)
Risk
Risk is the chance of an outcome other than what is planned to
occur or expected in the analysis�
NON-ANNUAL COMPOUNDING
i m
r1 1e
m
= + -b l
BREAK-EVEN ANALYSIS
By altering the value of any one of the variables in a situation,
holding all of the other values constant, it is possible to find a
value for that variable that makes the two alternatives equally
economical� This value is the break-even point�
Break-even analysis is used to describe the percentage of
capacity of operation for a manufacturing plant at which
income will ust cover expenses.
The payback period is the period of time required for the profit
or other benefits of an investment to equal the cost of the
investment�
Time Value of Money Relating Future and
Present Worth Shows Unreasonable Costs
3
𝐹 is Future Worth 𝑃 is Present Worth 𝑛 is the number of compounding periods
𝐴 is Uniform Amount per interest period (Payment) 𝑖 is the interest rate per period
3
10/31/19
2
College Costs Have Historically Increased Faster
than Inflation: Compound Interest Model
• Umich: Total Annual Cost
o 1980: $3,500
o 2020: $30,0
The document discusses various interest and investment concepts including simple interest, compound interest, nominal and effective interest rates, present worth, discount, annuities, perpetuities, and capitalized costs. It defines key formulas used to calculate future and present values under different interest compounding assumptions such as discrete, continuous, and various time periods. Factors like Fa, Fb, Fc, Fd, Ce, and Cf are introduced to represent discount and compounding rates for different cash flow patterns.
This document discusses various money-time relationships and concepts of equivalence. It begins by defining money as a medium of exchange, store of value, and unit of account. It then discusses capital, the different types of capital (equity and debt), and interest. The remainder of the document discusses how interest rates are determined and various interest rate formulas for calculating present and future values of single and uniform cash flows under simple, compound, continuous and discrete compounding. It also covers economic equivalence and cash flow diagrams/notations.
The document discusses cash flow diagrams and economic equivalence. It defines key terms like cash inflows, outflows, present value, and future value. Cash flow diagrams visually represent the timing of money coming into and going out of a project. Economic equivalence establishes that different amounts of money at different points in time can be considered equal based on interest rates and the time value of money. The document provides examples of how to calculate future value given present value, and vice versa, as well as examples for uniform series and gradients of cash flows. It demonstrates calculations for present and future equivalent values.
This document provides an introduction to value engineering interest formulas and their applications. It defines key terms like value, cost value, exchange value, use value, esteem value and different types of functions. It describes the value analysis/value engineering procedure and discusses when it should be used. Various interest formulas are explained including compound amount, present worth, uniform series and capital recovery. Examples are provided to demonstrate how to use the formulas to calculate future values, single payments, and equal installment amounts.
This document provides an introduction to value engineering interest formulas and their applications. It defines key terms like value, cost value, exchange value, use value, esteem value, primary function, secondary function, and tertiary function. It describes the value analysis/value engineering procedure and discusses types of values, functions, and differences between value analysis and value engineering. The document also presents various interest formulas for single payment compound amount, single payment present worth, uniform gradient series method, effective interest rate, equal payment series present worth amount, and equal payment series capital recovery amount. It includes examples of problems solved using these formulas.
This document discusses time value of money calculations using multiple factors. It provides examples of calculating present worth for non-uniform cash flows using factors like present worth (P/F), compound amount (F/P), uniform series present worth (P/A), and uniform series compound amount (F/A). Methods include calculating individual amounts and summing, or finding an equivalent uniform annual series. The document also discusses remembering that P/A places present worth one year before payments, while F/A places future worth in the year of the last payment.
This document provides information about a project assignment for a course on the time value of money. It includes details such as the student's name, registration number, course, and semester. The objectives are to discuss techniques for investment evaluation like present value, future value, and annuities. Formulas are provided for present value, future value, ordinary annuities, growing annuities, perpetuities, and their applications in areas like loans, savings, and project evaluation. The conclusion emphasizes the importance of understanding these time value of money concepts for efficient financial management in construction projects.
The document discusses cost-benefit analysis and various methods used to evaluate costs and benefits of projects. It defines key terms like tangible/intangible and direct/indirect costs and benefits. Several evaluation methods are described - net benefit analysis, present value analysis, net present value, payback period analysis, break-even analysis and cash flow analysis. Their formulas, examples and advantages/disadvantages are provided. The document concludes that cost-benefit analysis involves identifying, categorizing and evaluating costs and benefits to interpret results and take action regarding alternative systems.
Presentación con Temas Relacionados a la Ingeniería Económica.
Valor Presente, Valor Neto, Tasa Interna de Retorno, Análisis Costo Beneficio, Inversión Social , Inversión Privada
This document discusses economics and interest factors that are important for engineering decisions. It covers topics such as interest, present and future worth factors, bonds, taxes, and depreciation. Equations are provided for calculating interest on lump sums, uniform series of amounts, and non-uniform amounts. Examples show how to apply these concepts to problems involving loans, investments, and evaluating potential projects.
This document discusses techniques for analyzing cash flows that begin or change at times other than the standard uniform series. It introduces methods for shifted uniform series using P/A and F/A factors, combinations of series and single cash flows, and shifted arithmetic and geometric gradients, including negative gradients. Examples are provided to demonstrate the application of these methods to calculate present, future, and equivalent uniform worth for nonstandard cash flows.
This document provides formulas for calculating key project management metrics such as net present value, PERT estimates, expected monetary value, communication channels, float, earned value, variance, performance indexes, estimates, and burn rate. Formulas allow calculating the time, cost, and schedule performance of a project for monitoring and control purposes.
Here are the key steps to solve this problem:
* Future sum (F) = Rs. 1,00,000
* Interest rate (i) = 15%
* Number of periods (n) = 10 years
* Interest is compounded annually
* To find the present worth (P):
* P = F / (1 + i)^n
* P = Rs. 1,00,000 / (1 + 0.15)^10
* P = Rs. 1,00,000 / 2.472
* P = Rs. Rs. 40,320
Therefore, the single payment that should be made now to receive Rs. 1,00,000 after 10 years at
This document discusses key concepts related to the time value of money, including:
- Calculating the future and present value of a single amount and an annuity using compound interest formulas.
- Examples of using financial calculators and spreadsheets to solve time value of money problems.
- Additional topics covered include annuities due, perpetuities, non-annual periods, and effective annual rates. Students are encouraged to use financial calculators to simplify solving discounted cash flow problems.
This document discusses key concepts related to the time value of money, including:
- Calculating the future and present value of a single amount and an annuity using compound interest formulas.
- Examples of using financial calculators and spreadsheets to solve time value of money problems.
- Additional topics covered include annuities due, perpetuities, non-annual periods, and effective annual rates. Students are encouraged to use financial calculators to simplify solving discounted cash flow problems.
The document discusses key concepts related to the time value of money, including formulas for calculating the future value and present value of single amounts and annuities. It provides examples of using these formulas to solve for unknown values like interest rates, time periods, or cash flow amounts. The document also covers topics like perpetuities, non-annual interest compounding, and effective annual rates.
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1. COSTOS DE OBRAS (PROYECTOS)
COSTOS DE INVERSIÓN
COSTOS ANUALES
SELECCIÓN DE PROYECTOS
2. COSTOS DE INVERSIÓN Y COSTOS ANUALES
Los costos de inversión son aquellos en que se incurre durante el proceso inversionista.
Son costos, por lo general, de envergadura y están compuestos en su mayor parte por
costos de fabricación y costos de adquisición de equipos e instalaciones.
Los costos anuales son aquellos que ocurren de una forma continua y reiterada,
compuestos por costos de mano de obra, materia prima, combustible, trabajo intelectual,
administración, etc., y cuyo orden de magnitud es relativamente menor que el de los
costos de inversión.
El orden de magnitud de los costos anuales puede resultar comparable, y hasta exceder
con creces al de los costos de inversión cuando se considera el monto resultante de
aquellos, a lo largo de toda la vida útil de la obra
3. EL CÁLCULO DE VARIANTES
La idea del cálculo de los costos de una obra, para el ingeniero, está vinculada indisolublemente a
la idea de la optimización. El ingeniero no puede limitarse a formular una sola variante de una
obra y calcular sus costos, ya que su verdadera función como tal es la de obtener la variante de
menor costo posible.
Esto quiere decir que hace falta estudiar distintas variantes de una obra calcular los costos de
todas ellas y establecer una comparación para tomar una decisión sobre cuál es la mejor. Es
bueno ilustrar esto con un ejemplo numérico. Sean las siguientes variantes de una obra cuyos costos
de inversión y anuales se han calculado; si el beneficio reportado a la sociedad es el mismo para
todas las variantes, ¿cuál es la mejor?
Variante Inversión Costo Anual Vida Útil
1 1.56E6 30000 20 años
2 1.42E6 54000 20 años
3 1.23E6 70000 20 años
4. OPCIONES DE DECISIÓN INICIALES (NO PRECISAMENTE ADECUADAS)
(1) Opción 1, aplicar la siguiente definición:
𝐶𝑜𝑠𝑡𝑜 𝑇𝑜𝑡𝑎𝑙 = 𝐼𝑛𝑣𝑒𝑟𝑠𝑖ó𝑛 + 𝐶𝑜𝑠𝑡𝑜 𝐴𝑛𝑢𝑎𝑙 𝑉𝑖𝑑𝑎 Ú𝑡𝑖𝑙
(1) Opción 2, Otra definición:
𝐶𝑜𝑠𝑡𝑜 𝑇𝑜𝑡𝑎𝑙 =
𝐼𝑛𝑣𝑒𝑟𝑠𝑖ó𝑛
𝑉𝑖𝑑𝑎 Ú𝑡𝑖𝑙
+ 𝐶𝑜𝑠𝑡𝑜 𝐴𝑛𝑢𝑎𝑙
Variante Inversión Costo
Anual
Vida
Útil
Costo
Total 1
Costo
Total 2
1 1.56E6 30000 20 años 2.16E6 108 000
2 1.42E6 54000 20 años 2.50E6 125 000
3 1.23E6 70000 20 años 2.63E6 131 500
SE SUMAN CANTIDADES QUE SON DESEMBOLSADAS EN DISTINTOS TIEMPOS
5. REPRESENTACIÓN GRÁFICA DE COSTOS EN EL TIEMPO
I
A A A A A A A A
0 1 2 3 n
I: Inversión A: Costo Anual, se considera constante
Se tiene como origen (cero) el momento de entrada en producción de la obra (fin del proceso
inversionista), y se acepta convencionalmente que en ese punto (señalado como origen) ocurre la
inversión I que dio lugar a la obra.
En los años que siguen, la obra está en producción y al final de cada uno se contabilizan los costos
anuales a, los cuales se ubican -de forma convencional también- al final del año en que han ocurrido.
Como se desprende del gráfico explicado, la inversión I y cada uno de los n costos anuales A, ocurren
en años diferentes, lo cual, como se verá más adelante, implica que antes de sumarlos es necesario
ajustarlos todos a un instante de tiempo común
6. VALOR PRESENTE Y FUTURO DE UN CAPITAL
Es práctica común en economía referir los beneficios anuales al capital invertido inicial mediante
el cálculo de un factor, que será denominado en lo sucesivo como, r. Así, se dice que un capital I
produce beneficios anuales por valor de r I.
Si se suman estos beneficios durante todos los n años, se obtiene:
𝐹 = 𝐼 + 𝑟𝐼 + 𝑟𝐼 + ⋯ + 𝑟𝐼 (𝑛 𝑣𝑒𝑐𝑒𝑠) (1.1)
Donde F, es el Valor Futuro del Capital, I.
n, el tiempo en años
r, Tasa ce crecimiento del capital o el Interés anual
Sumando y agrupando:
𝐹 = 1 + 𝑛𝑟 𝐼 (1.2)
Se observa que el capital I, se ha incrementado en la cantidad nrI , y esto se interpreta como un
fenómeno de crecimiento de capital con el tiempo, siempre que ese capital se invierta en el
proceso de producción porque, lógicamente, un capital ocioso no puede crecer. La fórmula (1.2)
se conoce con el nombre de Fórmula del Interés Simple y, a r, se le llama tasa de interés. El valor
F se dice que es el valor futuro del capital I, después de n años creciendo a una tasa r de interés
simple.
7. Si los beneficios anuales rI, como sucede en la realidad, son utilizados cada año como capital
en otras inversiones que también crecen con el mismo ritmo o tasa r, entonces el valor de F,
para una sucesión de años, se escribe:
𝐹 = 𝐼 + 𝑟𝐼 = 1 + 𝑟 𝐼 Para el año 1
𝐹 = 1 + 𝑟 𝐼 + 𝑟 1 + 𝑟 𝐼 = 1 + 𝑟 𝐼 Para el año 2
𝐹 = 1 + 𝑟 𝐼 + 𝑟 1 + 𝑟 𝐼 = 1 + 𝑟 𝐼 Para el año 3
𝐹 = 1 + 𝑟 𝐼 Para el año n (1.3)
Esta fórmula se conoce con el nombre de fórmula del interés compuesto. El valor F se dice
ahora que es el valor futuro de un capital I después de n años creciendo a una tasa r de
interés compuesto. En lo sucesivo se preferirá el uso de esta fórmula (1.3) por reproducir
mejor el proceso real
Por otra parte, de la Ecuación (1.3), puede obtenerse la siguiente expresión para calcular el
Valor Presente de la Inversión o Capital:
𝐼 = 𝐹 (1.4)
8. Generalizando las dos situaciones expresadas por las ecuaciones (1.3) y (1.4), puede
llegarse a la conclusión de que, conociendo la tasa r, es posible transformar el valor de un
capital calculado en una fecha dada a su equivalente en otra fecha, bien sea esta posterior
o anterior a la primera, como se observa en la siguiente figura:
F
I
P
4 2 0 2 4 6 8 10 12
n=4 n=12
Si se conoce el valor de un capital I a fines de 2021, por ejemplo, es posible calcular, para
una tasa r dada, su valor equivalente F a fines de 2032 aplicando (1.3) con n = 12 años.
Si lo que se quiere es calcular su valor equivalente F a fines de 2017, se aplica entonces la
(1.4) con n = 4 años
9. VALOR FUTURO DE UNA ANUALIDAD
Si cada una de las cantidades anuales A del gráfico de costos de la figura se utiliza como un capital
que crece al ritmo r, se plantea entonces calcular a cuánto equivale el valor de la serie completa de
las n cantidades, al final de los n años. Para hacer este cálculo se considera cada uno de los valores
A por separado (como capitales independientes) y se aplica la fórmula (4.3) para determinar su
crecimiento desde el año en que ocurre hasta el año n.
Empezando con la del primer año y terminando con la del año n (que no crece), el valor F deseado
será:
𝐹 = 𝐴 1 + 𝑟 + 𝐴 1 + 𝑟 + ⋯ + 𝐴 1 + 𝑟 + 𝐴 1 + 𝑟 + 𝐴 (1.5)
Multiplicando, por (1+r) ambos miembros de la ecuación (1.5) y luego restando miembro a miembro
ese producto con la misma ecuación (1.5), se obtiene:
𝐹𝑟 = 𝐴 1 + 𝑟 − 𝐴
𝐹 = 𝐴 (1.6)
P
A A A A A A
0 1 2 3 4 n
F
A A A A A A
0 1 2 3 4 n
10. VALOR PRESENTE (CAPITALIZACIÓN) DE UNA ANUALIDAD
P
A A A A A A
0 1 2 3 4 n
F
A A A A A A
0 1 2 3 4 n
Si en la Ec. (1.4), sustituimos la expresión (1.6), se obtiene:
𝐼 = 𝐹 ∴ 𝐹 = 𝐴, nombrando por otra parte, a la Inversión I, por el Valor Presente de la
misma, P, se tiene:
𝑃 = 𝐴 (1.7)
La relación entre una anualidad, A, y el Valor Presente equivalente P, que ha crecido durante n años, a
una tasa r, de interés compuesto.
11. EJEMPLO
Sea una Anualidad, A, de 40,000 $USD/año, como se muestra en el gráfico. Encontrar el Valor Presente, P,
de dicha anualidad, para el año 2018 y el Valor Futuro para el año 2032. Siendo la tasa de interés del
12.5% anual
F
P I
A A A A A
2018 2019 2020 2021 2022 2023 2024 2025 2026 2027 2028 2029 2030 2031 2032
Solución:
Las anualidades, primero deben convertirse en Valores Puntuales, equivalentes. Como por ejemplo una
Inversión, I.
Luego, el Valor Puntual, se debe trasladar al tiempo que se requiere, sea éste, Futuro o Pasado / Presente.
12. Aplicando la Ec. (1.7), para calcular el Valor Presente de la Anualidad, es decir, la Inversión, I:
𝑃 = 𝐼 =
1 + 𝑟 − 1
𝑟 1 + 𝑟
𝐴 = 142,422.73 $𝑈𝑆𝐷
Con este valor puntual de la Inversión (semejante al Valor Presente), calculamos el Valor Presente P, para el año
2018, con la Ec. (1.4)
𝐼 =
1
1 + 𝑟
𝐹
En esta ecuación, en valor I, representa el valor más anterior en el tiempo, mientras que F, el valor más posterior
o futuro. Por lo que en este caso, F, será 142,422.73 $USD mientras que I, será el “Valor Presente”, para el año
2018:
𝐼 = 𝑃 2018 =
1
1 + 0.125
142422.73 = 79,034.50 $𝑈𝑆𝐷
Para calcular el Valor Futuro (para el 2032), utilizamos la misma ecuación (1.4), despejando F, o la Ec. (1.3):
𝐹 = 1 + 𝑟 𝐼 = 1 + 0.125 79,034.50 = 411,104.29$𝑈𝑆𝐷
13. CAPITALIZACIÓN Y ANUALIZACIÓN DE LOS COSTOS
CAPITALIZACIÓN Y ANUALIZACIÓN DE LOS COSTOS
Una vez estudiada la influencia del tiempo en la
valoración de los costos, es conveniente continuar el
desarrollo del ejemplo tratado anteriormente d
comparación de alternativas. Resultan evidentes ahora
los pasos necesarios para obtener el costo total de las
variantes:
a) Escoger el instante de tiempo común para
transformar el capital, I y la anualidad, A.
b) Transformar el capital I, a su equivalente en el
instante de tiempo escogido.
c) Transformar la anualidad A, a su equivalente en el
mismo instante de tiempo.
d) Sumar ambos valores transformados.
Después de realizados estos cálculos para todas las
variantes, se obtendrán los costos totales equivalentes,
y entonces sí será posible la selección a base de
escoger la de costo total equivalente (CTE), mínimo
como la mejor.
P
A A A A A A
0 1 2 3 4 n
Variante Inversión Costo
Anual
Vida Útil
1 1.56E6 30000 20 años
2 1.42E6 54000 20 años
3 1.23E6 70000 20 años
Sea r=12.5% anual
14. Para la Variante 1, capitalizamos en el tiempo / año 0, el valor anualizado, con la Ec. (1.7)
𝑃 =
1 + 𝑟 − 1
𝑟 1 + 𝑟
𝐴 =
1 + 0.125 − 1
0.125 1 + 0.125
30000 = 217,240.60$𝑈𝑆𝐷
Siendo que la anualidad, ha sido capitalizada al año 0, que es el mismo tiempo de la Inversión; ambos
valores, pueden ser sumados, obteniendo el Costo Total Equivalente, CTE (También se denomina Costo
Capitalizado total CCT):
𝐶𝑇𝐸 = 𝐼 + 𝑃
Donde I0, es la Inversión en el año 0
𝑃 , es el valor capitalizado de la anualidad A, del proyecto, en el año 0
Obteniendo CTE, de la primera variante:
𝐶𝑇𝐸 = 1560000 + 217240.60 = 1,777,240.60 $𝑈𝑆𝐷
Realizamos las mismas operaciones para las variantes 2 y 3 y las ordenamos en el siguiente cuadro:
Variante Inversión Costo Anual 𝑷𝟎
𝑨 CTE
1 1.56E6 30,000 217,240.60 1,777,240.60
2 1.42E6 54,000 391,033.08 1,811,033.08
3 1.23E6 70,000 506,894.74 1,736,894.74
La Variante seleccionada, es la 3.
15. Otra forma de realizar la comparación de variantes, consiste en convertir el capital I, en una anualidad
equivalente, bajo la misma tasa r y sumarla a la anualidad real de los costos anuales.
Para transformar capitales en anualidades equivalentes, se obtiene la siguiente expresión, de la Ec. (1.6):
𝐴 = 𝐹 (1.8)
Donde A(F), es la Anualidad equivalente del valor futuro F, [$/año]
n, el tiempo [años]
r, la tasa de interés compuesto
Asimismo, para anualizar el valor presente o el valor de la inversión:
𝐴 = 𝑃 (1.9)
Donde: A(P), es la anualidad de la Inversión o del valor presente P, [$/año]
n, tiempo [años]
r, tasa de interés compuesto
Ya obteniendo el valor anualizado de la Inversión, A(P), se puede sumar a la anualidad, A. A este resultado,
se denomina, Costo Anualizado Total (CAT):
𝐶𝐴𝑇 = 𝐴 + 𝐴 (1.10)
16. EJEMPLO
Resolveremos el mismo ejemplo ya visto, con este método
Variante Inversión Costo
Anual
Vida Útil
1 1.56E6 30000 20 años
2 1.42E6 54000 20 años
3 1.23E6 70000 20 años
Sea r=12.5% anual
Solución:
Aplicando la Ec. (1.9), para Anualizar el
valor de la Inversión, para cada variante y
sumando los valores anualizados, con la Ec.
(1.10), obtendremos el CAT, para cada una
de ellas
Para la variante 1, se tiene:
𝐴 =
𝑟 1 + 𝑟
1 + 𝑟 − 1
𝑃 =
0.125 1 + 0.125
1 + 0.125 − 1
1.56𝐸6 = 215,429.34 $𝑈𝑆𝐷/𝑎ñ𝑜
Luego
𝐶𝐴𝑇 = 𝐴 + 𝐴 = 215,429.34 + 30,000.00 = 245,429.34 $𝑈𝑆𝐷/𝑎ñ𝑜
Realizando los mismos cálculos para cada una de las variantes se obtiene el cuadro que se muestra a
continuación:
17. Variante Inversión Costo Anual 𝑨
𝑷 CAT
1 1.56E6 30,000 215,429.34 245,429.34
2 1.42E6 54,000 196,095.94 250,095.94
3 1.23E6 70,000 169,857.75 239,857.75
Con este método, al igual que con el anterior, la variante 3, es la seleccionada
18. EJERCICIO 1
El economista de una empresa constructora calculó que los fondos productivos de ésta, tendrían un valor de
1E6 $USD a fines de 2036. Si el propio economista determinó que el ritmo promedio de crecimiento de
dichos fondos en los últimos 10 años fue del 5 % ¿a cuánto ascendía el valor de los fondos a finales de
2021?
F
P
2021 2022 2023 2024 2025 2026 2027 2028 2029 2030 2031 2032 2033 2034 2035 2036
𝐼 = 𝐹 (1.4)
𝐼 =
1
1 + 0.05
1𝐸6 = 481,017.10 $𝑈𝑆𝐷
19. EJERCICIO 2
Sobre la base de un crédito, un organismo recibía del banco $USD 50 000 anuales desde finales de
2018 y durante 4 años. Si el banco le cobra 6.5 % de interés, ¿a cuánto asciende la deuda del organismo
a fines de 2021?
F
A A A A
0 2018 2019 2020 2021
SOLUCIÓN
La ecuación que relaciona el Valor Futuro y una anualidad, es la ecuación (1.6):
𝐹 =
1 + 𝑟 − 1
𝑟
𝐴 =
1 + 0.065 − 1
𝑟
50000 = 220,358.73 $𝑈𝑆𝐷
20. EJERCICIO 3
A finales de 2021, una persona, depositará una suma en el banco, que después irá retirando en cantidades
iguales de 120,000 Bs, durante los próximos 5 años. Si el banco paga el 4% de interés anual, ¿qué suma
deberá depositarse ?
P
A A A A A
2021 2022 2023 2024 2025 2026
(Año cero)
La ecuación que relaciona el valor presente (a depositarse) de una anualidad, es la Ec. (1.7)
𝑃 =
1 + 𝑟 − 1
𝑟 1 + 𝑟
𝐴 =
1 + 0.04 − 1
0.04 1 + 0.04
120000 = 534,218.68 𝐵𝑠
21. EJERCICIO 4
Una planta de potabilización entró en producción a fines de 2018 con un costo de inversión de 3.5E6
$USD a cuenta del presupuesto (POA), de la empresa prestadora del servicio. Si la tasa de interés
aplicable es del 8 % y el Estado establece que el pago por fondos fijos sea tal que equivalga al valor de
la planta en un plazo de 25 años. ¿qué tanto por ciento de la inversión representa este pago?
3.5E6 $USD
A A A A A A
2018
(Año cero) n=25 años
Se busca obtener la anualidad de una Inversión, en un tiempo de 25 años.
La ecuación que relaciona una anualidad con una Inversión, es la Ec. (1.9)
𝐴 =
𝑟 1 + 𝑟
1 + 𝑟 − 1
𝑃 =
0.08 1 + 0.08
1 + 0.08 − 1
3.5𝐸6 = 327,875.73 $𝑈𝑆𝐷/𝑎ñ𝑜
Finalmente, para obtener el porcentaje de la inversión, obtenemos el cociente
327,875.73
3.5𝐸6
100 = 9.4%
Respuesta: Representa el 9.4% de la Inversión
22. EJERCICIO 5
Se construye una planta de producción y se demora tres años en entrar en producción. Si la Inversión fue de
3 M$USD y los costos anuales alcanzan a 120,000 $USD/año, a partir de que entró en producción durante
12 años, calcule el Costo Anualizado Total, considerando una tasa de interés de 10% anual.
SOLUCIÓN
Se realiza el gráfico de costos según indica el problema:
Una solución, es transportar la Inversión, al año 2. Luego anualizarla y sumar el valor anualizado de la
Inversión, con el costo anualizado que tenemos como dato. Este resultado es el CAT.
Procediendo con la solución, para transportar la Inversión, al año 2, se hace uso de la Ec. (1.3):
𝐹 = 1 + 𝑟 𝐼 = 1 + 0.10 3𝐸6 = 3,630,000.00 $𝑈𝑆𝐷
Para anualizar este valor, aplicamos la Ec. (1.9)
𝐴 =
𝑟 1 + 𝑟
1 + 𝑟 − 1
𝑃 =
0.10 1 + 0.10
1 + 0.10 − 1
3,630,000 = 532,750.83$𝑈𝑆𝐷/𝑎ñ𝑜
CAT = A(P) + A = 652,750.83 $USD/año
3E6 $USD
A A A A A A A A A A A A
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
23. COSTOS DE INVERSIÓN
Los costos de inversión son aquellos en que se incurre durante el proceso inversionista. Si se analizan las
etapas de este proceso es posible reconocer qué tipos de costos se producen.
En la primera y segunda fases (fundamentación y proyecto) del proceso, los resultados que se obtienen
revisten la forma de documentos técnico-económicos especializados. Son el producto del trabajo
intelectual y del trabajo de administración. Además, como en estas fases pueden haberse requerido
investigaciones de campo, de laboratorio y otras, es posible que incidan también en cierta medida todos
los demás costos (materia prima, combustible, equipos e instalaciones, transporte, mano de obra),
excluyendo quizás el costo de fabricación, que aparecería sólo en casos muy excepcionales.
En la tercera fase es donde ocurre el grueso del costo de inversión y es donde sus componentes y su
importancia relativa dependen, en mayor medida, del tipo de obra que se construye, sin perjuicio de que
casi siempre aparecen todos los tipos de costos estudiados.
El presupuesto general de la obra (que no es más que el estimado anticipado del costo de inversión),
puede organizarse de dos maneras fundamentales: en una de ellas, se divide la obra en sus partes
principales y para cada una se hace el desglose según los tipos de costos que inciden, obteniéndose como
resultado el presupuesto individual de cada parte; la otra manera de hacerlo es considerando la
distribución de la inversión en el tiempo, es decir, una vez que se ha elaborado el proyecto de
organización de la obra, se tiene un cronograma detallado que establece la secuencia de construcción, y
sobre esta base es posible determinar qué tipos de costos ocurren en cada año, trimestre, mes, etc., con lo
cual se obtiene el presupuesto individual de cada período de tiempo
24. Supóngase que en una obra determinada la distribución del fondo de inversión ocurre como en la figura
siguiente:
El período de construcción de la obra es de 5 años y los fondos I1, I2, …I5 son cantidades anuales que se
van invirtiendo en la construcción y montaje de los distintos elementos de la obra según el cronograma de
construcción.
En la composición de cada uno de estos fondos entran los distintos tipos de costos que se han estudiado
(materia prima, combustible, equipos e instalaciones, transporte, mano de obra, fabricación) y aunque, en
realidad, estos costos se producen en el transcurso del año, se suman todos y se consideran
convencionalmente situados al final del año en que ocurrieron. Esto es así porque el interés se paga, según
lo definido hasta aquí, en períodos discretizados de un año
25. Se quiere calcular ahora el valor de la inversión total, el cual, evidentemente no es igual a la suma de los
valores anuales, debido a que interviene el fenómeno del crecimiento del capital con el tiempo, que en este
caso se debe al efecto de inmovilización de recursos, ya que estos fondos anuales durante el período de
construcción no son productivos.
De acuerdo con lo anterior es necesario aplicar la Ec. (1.3), que convierte valor presente en valor futuro, a
cada uno de los fondos anuales por separado para calcular su valor futuro en un instante de tiempo común y
luego obtener la suma. No es posible aplicar la fórmula para el cálculo del valor futuro de una anualidad
porque los desembolsos anuales no son necesariamente iguales, es decir, el ritmo de inversión anual en una
obra no tiene que ser constante. El instante de tiempo común para hallar los valores futuros es el momento de
entrada en producción de la obra, el cual, generalmente coincide con el final del último año de inversiones.
El fondo total de inversión de la obra de la figura, se calcularía entonces, de la siguiente manera:
𝐼 = 𝐼 1 + 𝑟 + 𝐼 1 + 𝑟 + 𝐼 1 + 𝑟 + 𝐼 1 + 𝑟 + 𝐼
la cual puede generalizarse para un período de construcción de M años en la siguiente fórmula con notación
compacta:
𝐼 = ∑ 𝐼 1 + 𝑟 (1.11)
26. COMPARACIÓN DE COSTOS
Uno de los aspectos más importantes a tener en cuenta cuando se realiza una comparación de variantes
es que se lleve a cabo sobre una base común de cálculo. Eso quiere decir que no existan diferencias entre
los distintos parámetros y fechas, de las variantes, que favorezcan ficticiamente a una u otra. Se
entenderá que las variantes se comparan sobre una base común de cálculo cuando se cumplan los
siguientes requisitos:
1. Que la tasa de interés r, para el cálculo de los factores de conversión sea la misma en todas las
variantes.
2. Que la fecha de entrada en producción, o sea, el instante de tiempo en que se calcula el valor de la
inversión, sea el mismo para todas las variantes. Si esto no ocurre en la realidad, es necesario efectuar
las conversiones correspondientes según lo estudiado.
3. Que la fecha de inicio de las anualidades (costos anuales y beneficios anuales) sea la misma para
todas las variantes y esté ubicada un año después de la fecha en que se situó el costo total de
inversión.
27. Si existen diferencias entre las fechas de inicio de las anualidades en las variantes, será necesario proceder
a un "traslado" de las que estén desplazadas de la fecha correcta. Este "traslado" puede realizarse
mediante las fórmulas estudiadas anteriormente. Supóngase la situación de la figura siguiente:
Una solución, para poder realizar la comparación entre estos dos proyectos, es la de Capitalizar [en el año
cero], la anualidad A1, empleando la Ec. (1.7):
𝑃
( )
=
1 + 𝑟 − 1
𝑟 1 + 𝑟
𝐴
Luego, obtenemos el Valor o Costo Capitalizado Total, de la primera variante, [en el año cero], con la Ec.
𝐶𝐶𝑇
( )
= 𝑃
( )
+ 𝑃
P1
A1 A1 A1 A1 A1
0 1 2 3 4 5
P2
A2 A2 A2 A2 A2
0 1 2 3 4 5 6 7
28. Por otra parte, podemos también obtener el Costo Capitalizado Total, de la Variante 2, aplicando las mismas
ecuaciones anteriores, obteniendo:
𝐶𝐶𝑇
( )
= 𝑃
( )
+ 𝑃
Aunque se han obtenido los Costos Capitalizados de ambas variantes, éstas no pueden ser comparadas, debido
a que están en diferentes puntos d tiempo.
Para poder comparar las variantes, procedemos, por ejemplo a “transportar” el CTT1, al año 2, con la Ec. (1.3):
𝐹 = 1 + 𝑟 𝐼, la cual, la vamos a escribir de manera descriptiva:
𝐹
( )
= 1 + 𝑟 𝐶𝐶𝑇
( )
= 𝐶𝐶𝑇
( )
De esta manera ambas variantes de proyecto, pueden compararse para decidir cuál de las variantes elegir
para la ejecución.
29. En caso de que las variantes no se presenten de esta forma, una solución que se ha aplicado en la práctica es
escoger, como período de cálculo, aquella que tiene la vida útil más corta y aplicar una corrección a las demás,
para hacerlas equivalentes a la primera. Esta corrección se aplica solamente al fondo de inversión, razonando
que a medida que la vida útil disminuye le debe corresponder una inversión menor. Distintos autores, proponen,
a modo de corrección, calcular el valor residual de la obra al final del período de cálculo escogido, llevarlo a
valor presente y descontarlo del fondo de inversión. Para calcular el valor residual, estos autores utilizan el
criterio de depreciación lineal, como se ilustra en la figura siguiente:
En la figura, n1, es el período de cálculo (igual a la Vida Útil de la alternativa 1). N2 es la Vida Útil de la
variante 2, que se está analizando. I2, el costo de la variante que se analiza. R, el valor residual de la variante
2.
I2
Variación del valor de la obra en el tiempo
R
n1
n2
30. Calculamos el valor de R, aplicando semejanza de triángulos:
𝑅 = 𝐼 (1.12)
Luego, el valor “corregido” de I2, será el siguiente:
𝐼
( )
= 𝐼 − (1.13)
Una vez hecha la corrección, en los cálculos del costo total capitalizado (o anualizado) para la
comparación se utiliza el valor I(2)n1, como fondo de inversión y la anualidad existente durante los
primeros n1 años.
ANUALIDADES NO UNIFORMES
En caso de que las anualidades no sean uniformes se deben convertir en anualidades uniformes antes de
proceder a la comparación. El procedimiento de conversión en este caso depende de la forma que tenga la
anualidad considerada. Para el caso de variación lineal, más adelante se ofrecen métodos de cálculo.
Algunos autores, incluyen casos más complejos. En definitiva el método general sería, por ejemplo, llevar
cada valor anual a valor futuro (o valor presente) y sumarlos todos, y luego convertir este valor futuro total
en una anualidad uniforme
31. En la práctica puede suceder que las anualidades se presenten con un valor que no es constante. Esto sucede
porque muchas veces existe un período de puesta en explotación de la obra que dura varios años, y por
tanto, los costos y beneficios anuales pueden ir creciendo paulatinamente hasta alcanzar el máximo previsto.
Estas anualidades, se muestran en la siguiente figura:
G
A0
0 1 2 n
a) Anualiad Creciente
G
A0
0 1 2 n
b) Anualidad Decreciente
El valor más pequeño de la anualidad se representa
por Ao, y la variación anual constante por G
(gradiente). Aparentemente, la forma más cómoda de
trabajar con este tipo de anualidades es
convirtiéndolas en anualidades constantes
equivalentes. Para ello puede tomarse cada valor
anual, llevarlo a valor presente, sumarlos y luego
convertir el valor presente total en la anualidad
constante deseada; todo esto mediante las fórmulas
ya conocidas
El procedimiento para convertir estas anualidades en anualidades constantes equivalentes, se muestra a
continuación.
32. G
0 1 2 n
a) Anualidad Creciente
G
0 1 2 n
b) Anualidad Decreciente
G
A0-G
A0-G
G
Se separa la anualidad, en dos partes,
como se muestra en la figura.
La parte inferior, en ambos casos, es una
anualidad constante (A0 – G).
Por lo que únicamente es necesario
convertir, la parte superior, triangular.
Denominemos BC, a la Anualidad
equivalente, a la anualidad creciente y
BD, a la Anualidad equivalente, a la
anualidad decreciente.
Luego, las Anualidades Equivalentes
totales, para anualidades crecientes y
decrecientes, se calcularán, con las
siguientes expresiones:
𝐴 = 𝐴 − 𝐺 + 𝐵 (1.14)
𝐴 = 𝐴 − 𝐺 + 𝐵 (1.15)
Los valores (A0 – G), puede ser positivo, negativo o nulo
33. Para calcular BC y BD, se emplean las siguientes ecuaciones:
𝐵 = 1 + 𝑟 − (1.16)
𝐵 = − 1 (1.17)
34. PARÁMETROS ECONÓMICOS PARA ANÁLISIS DE ALTERNATIVAS
La existencia de varias formas diferentes de combinar el costo y el beneficio no ofrecería complicaciones si no
fuera por las contradicciones que se pueden presentar entre ellos cuando se usan según las definiciones clásicas.
En lo que sigue se discutirá sobre los tres parámetros principales, que son: la ganancia neta, la relación beneficio-
costo y la tasa de reproducción interna o Tasa Interna de Retorno. En su cálculo se empleará explícitamente la
fórmula del interés compuesto, y la determinación de cualquiera de ellos presupone el conocimiento de los valores
de todos los costos, beneficios y la tasa de interés aplicable, así como que antes de aplicarlos a la selección de
variantes se ha aplicado a todas ellas la base común de cálculo. Las definiciones que se ofrecen a continuación son
las empleadas usualmente.
LA GANANCIA NETA
La ganancia neta, para una obra dada, se define como la diferencia entre el beneficio y el costo de la misma
tomados desde el punto de vista de la empresa. En el costo entran todos los componentes como el costo anual real,
pago de amortización, pago por fondos fijos, pago de intereses, impuestos y seguros. Se calcula con la siguiente
expresión:
𝐺 = 𝐵 − 𝐶 (1.18)
Donde: GN, Ganancia Neta [$/año]
BA, Beneficios Anuales [$/año]
CA, Costos Anuales [$/año]
35. También pueden utilizarse los valores del beneficio y el costo capitalizado, con tal que se utilicen en la misma
forma para todas las variantes, ya que el valor numérico es mayor. La expresión sería:
𝐺 = 𝐵 − 𝐶 (1.19)
Donde: GK, Ganancia Capitalizada [$]
BK, Beneficios Capitalizados [$]
CK, Costos Capitalizados [$]
Según la definición de este parámetro (dado un conjunto de variantes excluyentes a comparar entre sí), se
debe calcular la ganancia neta para cada una y seleccionar la que da el valor máximo.
Para el caso de variantes compatibles, se calculan todos los B−C, se colocan en orden decreciente del valor
de B−C y se aceptan, en este orden, todas aquellas que quepan dentro de los limites de la capacidad de
inversión disponible. En el caso de que una variante diera B < C, se dice que es una variante antieconómica.
36. RELACIÓN BENEFICIO – COSTO
Se obtiene a través de los mismos valores básicos usados para el parámetro anterior, pero en lugar de
restarse se dividen. También se pueden usar valores anuales o capitalizados, pero ahora se cumple que:
𝑅 / = = (1.20)
El criterio de selección de variantes es el de seleccionar la de máximo B/C para el caso de excluyentes, y
aceptar en orden decreciente de B/C para el caso de compatibles. Si una variante da un valor B/C < 1 (que
es lo mismo que B < C ) se dice que es antieconómica.
37. TASA DE REPRODUCCIÓN INTERNA O TASA INTERNA DE RETORNO
La tasa r, pudiera interpretarse como una especie de intensidad de producción de beneficios que se le exige
a la obra (recordar que el pago de interés es un beneficio que recibe el que concede el préstamo). De esta
forma, a medida que se le exige una mayor intensidad, la obra responde con una disminución de la ganancia
neta remanente (y del B/C también).
El valor numérico de la tasa r, que produce B−C = 0 (B/C = 1) puede concebirse como la máxima intensidad
que se le puede exigir a una obra sin que produzca pérdidas. A este valor de r es al que se le denomina
tasa de reproducción interna (TRI).
La TRI de una obra se define entonces como aquel valor de r que hace B−C = 0 (B/C = l), o sea, iguala los
costos a los beneficios. Como es evidente, para calcular la TRI no hace falta conocer previamente el valor de
la tasa aplicable ro y sólo se necesitan los gráficos de la inversión, costos y beneficios de la obra en cuestión.
Para una obra en la que se conocen n, I, B y CA y estos dos últimos son constantes, la ecuación para el cálculo
de la TRI es:
𝐵 − 𝐶 − 𝐼 = 0 (1.21)
En la Ec. (1.21), la incógnita, es r, que se denomina TRI o TIR.