The document discusses time value of money concepts including compounding, discounting, future value, and present value as they relate to single sums and annuities. It provides examples of calculating future value and present value for single sums deposited or received at a future date at a given interest rate. It also gives an example of calculating the future value of an annuity where the same amount is deposited each period for a number of periods at a given interest rate.
The document discusses the concepts of compounding and discounting in the time value of money. It provides examples of calculating the future value and present value of single sums using formulas and financial calculators. Compounding translates a present value into its future equivalent value, while discounting translates a future value into its present equivalent value. The examples demonstrate computing future and present values for various interest rates, time periods, and compounding frequencies.
Here are the steps to solve this problem:
FV = PV(1 + r/n)nt
FV = $4,000(1 + 0.09)11
FV = $4,000(2.1435)
FV = $8,574
The future value of $4,000 invested for 11 years at 9% compounded annually is $8,574.
General 2 HSC Credit and Borrowing - Future ValueSimon Borgert
This document provides three examples of calculating future and present values of investments using compound interest formulas. The first two examples show calculating the future value after 7 years of an initial $4000 investment at 5.5% annual interest, with the second example compounding interest monthly rather than annually, resulting in a higher future value. The third example calculates the present value of an annuity worth $11,375 in 5 years at 6% interest per year.
Time lines
Future value / Present value of lump sum
FV / PV of annuity
Perpetuities
Uneven CF stream
Compounding periods
Nominal / Effective / Periodic rates
Amortization
This document provides an overview of time value of money concepts including simple and compound interest, future and present value, and annuities. Key points covered include:
- Compound interest earns interest on previous interest amounts as well as the principal, resulting in higher total returns over time compared to simple interest.
- Future value and present value formulas allow calculating the value of a single deposit or withdrawal at a future or present point in time using a given interest rate.
- Annuities represent a series of equal periodic cash flows, and formulas are provided to calculate the future and present value of ordinary annuities and annuities due.
1. The document discusses the concept of time value of money and how an amount of money received today is worth more than the same amount received in the future due to factors like opportunity cost, inflation, and ability to invest and earn interest.
2. It provides examples of calculating future and present value using formulas, interest tables, calculators, and spreadsheets. The future value of a single cash flow is calculated as PV x (1+i)n, where PV is the present value, i is the interest rate, and n is the number of periods. The present value is calculated as FV x (1/(1+i))n, where FV is the future value.
1. The gold mine is expected to be exhausted within 20 years. With annual costs of $750,000 and $120/ton, it earns $450/ton and has annual output of 25,000 tons. The property valuation is $53,745,535.
2. The timber land was bought for $8,000,000 and sold for $200,000 after 14 years of $1,400,000 average annual profits. The investment rate is 13.2%.
3. For a company with 4 machines, the total annual straight-line depreciation charge is $49,600 by group depreciation and $47,666.67 by composite depreciation.
The document discusses the concepts of compounding and discounting in the time value of money. It provides examples of calculating the future value and present value of single sums using formulas and financial calculators. Compounding translates a present value into its future equivalent value, while discounting translates a future value into its present equivalent value. The examples demonstrate computing future and present values for various interest rates, time periods, and compounding frequencies.
Here are the steps to solve this problem:
FV = PV(1 + r/n)nt
FV = $4,000(1 + 0.09)11
FV = $4,000(2.1435)
FV = $8,574
The future value of $4,000 invested for 11 years at 9% compounded annually is $8,574.
General 2 HSC Credit and Borrowing - Future ValueSimon Borgert
This document provides three examples of calculating future and present values of investments using compound interest formulas. The first two examples show calculating the future value after 7 years of an initial $4000 investment at 5.5% annual interest, with the second example compounding interest monthly rather than annually, resulting in a higher future value. The third example calculates the present value of an annuity worth $11,375 in 5 years at 6% interest per year.
Time lines
Future value / Present value of lump sum
FV / PV of annuity
Perpetuities
Uneven CF stream
Compounding periods
Nominal / Effective / Periodic rates
Amortization
This document provides an overview of time value of money concepts including simple and compound interest, future and present value, and annuities. Key points covered include:
- Compound interest earns interest on previous interest amounts as well as the principal, resulting in higher total returns over time compared to simple interest.
- Future value and present value formulas allow calculating the value of a single deposit or withdrawal at a future or present point in time using a given interest rate.
- Annuities represent a series of equal periodic cash flows, and formulas are provided to calculate the future and present value of ordinary annuities and annuities due.
1. The document discusses the concept of time value of money and how an amount of money received today is worth more than the same amount received in the future due to factors like opportunity cost, inflation, and ability to invest and earn interest.
2. It provides examples of calculating future and present value using formulas, interest tables, calculators, and spreadsheets. The future value of a single cash flow is calculated as PV x (1+i)n, where PV is the present value, i is the interest rate, and n is the number of periods. The present value is calculated as FV x (1/(1+i))n, where FV is the future value.
1. The gold mine is expected to be exhausted within 20 years. With annual costs of $750,000 and $120/ton, it earns $450/ton and has annual output of 25,000 tons. The property valuation is $53,745,535.
2. The timber land was bought for $8,000,000 and sold for $200,000 after 14 years of $1,400,000 average annual profits. The investment rate is 13.2%.
3. For a company with 4 machines, the total annual straight-line depreciation charge is $49,600 by group depreciation and $47,666.67 by composite depreciation.
Let k = annual rate of return
FV = PV(1+k)n
230 = 200(1+k)2
1.15 = (1+k)2
1.075 = 1+k
.075 = k
k = 7.5% annual return
Therefore, the annual return on this investment is 7.5%
Solving for k
Example: You invest $1,000 today and want it to grow to
$1,500 in 5 years. What rate of return is needed?
0 1 2
$1,000 $1,500
This document provides an overview of the time value of money concepts. It defines key terms like present value, future value, and annuities. It explains the differences between simple and compound interest and how to calculate future and present value for single deposits and streams of cash flows. The document also demonstrates how to use interest tables and financial calculators to solve time value of money problems. Overall, the document serves as an introduction to fundamental time value of money and interest rate concepts.
The document discusses key concepts related to time value of money including:
- Future value and present value calculations using compound interest formulas
- Applications of compound interest such as calculating future values over multiple periods and finding internal rates of return
- Types of cash flows including annuities, perpetuities, and their present and future value calculations
- Adjusting interest rates to make them comparable across different compounding periods by calculating effective annual rates
This document summarizes key concepts related to inflation and its impact on project cash flows. It provides examples of calculating equivalent values under inflation for items such as annuity payments, lump sums, interest payments, and deposits/withdrawals. It demonstrates how to determine inflation-adjusted interest rates and convert between actual and constant dollars. Calculations are shown for topics like equivalent present worth, annuity due, future worth, and establishing equivalence between actual and constant cash flows.
The document discusses periodic compound interest, where interest is compounded over regular time intervals called periods. It defines the key terms like principal (P), interest rate (i), number of periods (N), and accumulation (A). The periodic compound interest formula is A = P(1 + i)N. An example calculates how much money would accumulate over 60 years with monthly interest of 1% on a $1000 principal. The document also discusses how more frequent compounding results in higher returns approaching continuous compounding.
The document provides examples and explanations for solving problems involving patterns and functions. It presents a five-step plan for problem solving: 1) read the problem, 2) plan how to set up and solve, 3) do the work, 4) answer the question, and 5) check the answer. It then works through three examples step-by-step to find patterns in tables and write equations to describe the relationships. The homework assignment is to complete problems 1-20 on page 274.
1) The document discusses rate-of-return analysis and concepts including internal rate of return (IRR), breakeven interest rate (i*), simple vs. non-simple investments, and cash flow sign rules.
2) It provides examples of calculating i* for various projects using cash flows, present worth analysis, and plotting the net present worth.
3) Mixed investments are discussed along with calculating the rate of return considering both positive and negative cash flows.
The document discusses continuous compound interest and how the accumulated value increases as the compounding period decreases. It provides an example where $1000 is deposited at 8% annual interest. When compounded 4 times per year, the accumulated value after 20 years is $4875.44. When compounded 100, 1000, and 10000 times per year, the accumulated values gradually increase to $4949.87, $4952.72, and $4953 respectively, demonstrating that more frequent compounding results in higher returns. In the limit of compounding continuously, the accumulated value approaches $4953.03.
The document defines and provides examples of different types of annuities. It discusses ordinary annuities, deferred annuities, annuities due, perpetuities, and uniform gradients. Examples are provided to illustrate calculations for present value, future value, payment amounts, and capitalized costs for various annuity scenarios involving lump sums, installments, and perpetual payments over different time periods.
The document discusses different types of commission-based pay structures: straight commission based solely on sales, salary plus commission with a base salary and commission on sales above a threshold, and graduated commission with increasing rates for different tiers of sales. It provides examples of calculating commissions and gross pay for employees under these different structures.
This document discusses interest rate and economic equivalence concepts. It covers types of interest including simple and compound interest. It also discusses using present/future value factors to solve for single and uneven payment series. Examples are provided to illustrate calculating future or present value of lump sums, annuities, and other cash flows using interest rate conversion factors.
The document discusses methods for calculating annual equivalence (AE) for projects with different cash flows, interest rates, lengths, and other factors. Several examples are provided of calculating AE for various investment scenarios. Key factors considered include capital recovery costs, operating and maintenance costs, revenues, break-even points, and calculating costs on a per unit basis.
1) The document discusses nominal and effective interest rates, compounding frequencies, and cash flows with interest.
2) Examples are provided to calculate interest rates, future/present values, and payment amounts for various scenarios such as loans, leases, investments, and transfers between accounts.
3) Key concepts covered include the differences between nominal and effective rates, continuous compounding, solving for unknown amounts like deposits, installments, and balances over time.
Chapter 2 introduction to valuation - the time value of moneyKEOVEASNA5
This document provides an introduction to the time value of money concepts of future value, present value, interest rates, and compounding. It defines key terms and formulas. Several examples are provided to illustrate how to use the future value and present value formulas to calculate future or present values when given other relevant information such as principal, interest rate, and time period. The effects of compounding versus simple interest are demonstrated. The relationships between present/future values and interest rates/time periods are discussed. Methods for calculating implied interest rates and time periods are also presented.
1. The document provides examples of time value of money problems involving present value, future value, interest rates, and number of periods calculations. It gives the calculations for various cash flows received or paid in different periods with different interest rates.
2. It includes problems calculating NPV for projects such as factories, machines, and ships where the cash flows include costs, revenues, operating costs, refit costs, and salvage value over multiple periods.
3. The last examples calculate the amount of annual savings needed to buy a boat in 5 years and the annual annuity payment an individual can expect based on investing a present value over their life expectancy.
time value of money, future value with exercises, present value exercises. annuity, annuity due exercises, mixed flows, rule of 72 with exercise, unknown interest rate and time period with exercises. present value and future value with discounting monthly, quarterly, semi-annually, annually etc
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.
Comm370 lecture 3 - financial planning and growthnelsonpoon
This document discusses financial planning and growth. It defines key variables like assets, debt, equity, net income, and retention ratio. It then shows how to calculate a company's external funding needs, internal growth rate, and sustainable growth rate using these variables and common accounting metrics. The internal growth rate is the maximum rate supported without external financing, while the sustainable growth rate is the maximum that maintains a constant debt-to-equity ratio with external borrowing. Calculating and comparing a planned growth rate to these rates provides insight into future financing needs.
The document discusses the time value of money concepts of future value and present value for single sums and annuities. It provides examples of calculating future value and present value for single cash flows using formulas and a financial calculator. It also discusses the differences between ordinary annuities, where cash flows occur at the end of periods, and annuity dues, where cash flows occur at the beginning of periods.
Understanding the Time Value of Money; Single PaymentDIANN MOORMAN
- Calculators must be set to 4 decimal places and 1 payment per year for calculations.
- The document discusses the time value of money and how interest, compound interest, and inflation affect the value of money over time. It provides examples of calculating future and present value using the future and present value equations. It also discusses how to adjust for inflation when making time value of money calculations.
The document discusses various time value of money concepts including future value, present value, perpetuities, and annuities. It provides examples of calculations for future and present value of a single sum, as well as present value calculations for perpetuities and annuities. Uneven cash flows are discussed as being the sum of present values of regular cash flows. Key steps in solving time value problems are identified as drawing the timeline, identifying the cash flows, determining what value is being calculated, and using the appropriate formula.
Let k = annual rate of return
FV = PV(1+k)n
230 = 200(1+k)2
1.15 = (1+k)2
1.075 = 1+k
.075 = k
k = 7.5% annual return
Therefore, the annual return on this investment is 7.5%
Solving for k
Example: You invest $1,000 today and want it to grow to
$1,500 in 5 years. What rate of return is needed?
0 1 2
$1,000 $1,500
This document provides an overview of the time value of money concepts. It defines key terms like present value, future value, and annuities. It explains the differences between simple and compound interest and how to calculate future and present value for single deposits and streams of cash flows. The document also demonstrates how to use interest tables and financial calculators to solve time value of money problems. Overall, the document serves as an introduction to fundamental time value of money and interest rate concepts.
The document discusses key concepts related to time value of money including:
- Future value and present value calculations using compound interest formulas
- Applications of compound interest such as calculating future values over multiple periods and finding internal rates of return
- Types of cash flows including annuities, perpetuities, and their present and future value calculations
- Adjusting interest rates to make them comparable across different compounding periods by calculating effective annual rates
This document summarizes key concepts related to inflation and its impact on project cash flows. It provides examples of calculating equivalent values under inflation for items such as annuity payments, lump sums, interest payments, and deposits/withdrawals. It demonstrates how to determine inflation-adjusted interest rates and convert between actual and constant dollars. Calculations are shown for topics like equivalent present worth, annuity due, future worth, and establishing equivalence between actual and constant cash flows.
The document discusses periodic compound interest, where interest is compounded over regular time intervals called periods. It defines the key terms like principal (P), interest rate (i), number of periods (N), and accumulation (A). The periodic compound interest formula is A = P(1 + i)N. An example calculates how much money would accumulate over 60 years with monthly interest of 1% on a $1000 principal. The document also discusses how more frequent compounding results in higher returns approaching continuous compounding.
The document provides examples and explanations for solving problems involving patterns and functions. It presents a five-step plan for problem solving: 1) read the problem, 2) plan how to set up and solve, 3) do the work, 4) answer the question, and 5) check the answer. It then works through three examples step-by-step to find patterns in tables and write equations to describe the relationships. The homework assignment is to complete problems 1-20 on page 274.
1) The document discusses rate-of-return analysis and concepts including internal rate of return (IRR), breakeven interest rate (i*), simple vs. non-simple investments, and cash flow sign rules.
2) It provides examples of calculating i* for various projects using cash flows, present worth analysis, and plotting the net present worth.
3) Mixed investments are discussed along with calculating the rate of return considering both positive and negative cash flows.
The document discusses continuous compound interest and how the accumulated value increases as the compounding period decreases. It provides an example where $1000 is deposited at 8% annual interest. When compounded 4 times per year, the accumulated value after 20 years is $4875.44. When compounded 100, 1000, and 10000 times per year, the accumulated values gradually increase to $4949.87, $4952.72, and $4953 respectively, demonstrating that more frequent compounding results in higher returns. In the limit of compounding continuously, the accumulated value approaches $4953.03.
The document defines and provides examples of different types of annuities. It discusses ordinary annuities, deferred annuities, annuities due, perpetuities, and uniform gradients. Examples are provided to illustrate calculations for present value, future value, payment amounts, and capitalized costs for various annuity scenarios involving lump sums, installments, and perpetual payments over different time periods.
The document discusses different types of commission-based pay structures: straight commission based solely on sales, salary plus commission with a base salary and commission on sales above a threshold, and graduated commission with increasing rates for different tiers of sales. It provides examples of calculating commissions and gross pay for employees under these different structures.
This document discusses interest rate and economic equivalence concepts. It covers types of interest including simple and compound interest. It also discusses using present/future value factors to solve for single and uneven payment series. Examples are provided to illustrate calculating future or present value of lump sums, annuities, and other cash flows using interest rate conversion factors.
The document discusses methods for calculating annual equivalence (AE) for projects with different cash flows, interest rates, lengths, and other factors. Several examples are provided of calculating AE for various investment scenarios. Key factors considered include capital recovery costs, operating and maintenance costs, revenues, break-even points, and calculating costs on a per unit basis.
1) The document discusses nominal and effective interest rates, compounding frequencies, and cash flows with interest.
2) Examples are provided to calculate interest rates, future/present values, and payment amounts for various scenarios such as loans, leases, investments, and transfers between accounts.
3) Key concepts covered include the differences between nominal and effective rates, continuous compounding, solving for unknown amounts like deposits, installments, and balances over time.
Chapter 2 introduction to valuation - the time value of moneyKEOVEASNA5
This document provides an introduction to the time value of money concepts of future value, present value, interest rates, and compounding. It defines key terms and formulas. Several examples are provided to illustrate how to use the future value and present value formulas to calculate future or present values when given other relevant information such as principal, interest rate, and time period. The effects of compounding versus simple interest are demonstrated. The relationships between present/future values and interest rates/time periods are discussed. Methods for calculating implied interest rates and time periods are also presented.
1. The document provides examples of time value of money problems involving present value, future value, interest rates, and number of periods calculations. It gives the calculations for various cash flows received or paid in different periods with different interest rates.
2. It includes problems calculating NPV for projects such as factories, machines, and ships where the cash flows include costs, revenues, operating costs, refit costs, and salvage value over multiple periods.
3. The last examples calculate the amount of annual savings needed to buy a boat in 5 years and the annual annuity payment an individual can expect based on investing a present value over their life expectancy.
time value of money, future value with exercises, present value exercises. annuity, annuity due exercises, mixed flows, rule of 72 with exercise, unknown interest rate and time period with exercises. present value and future value with discounting monthly, quarterly, semi-annually, annually etc
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.
Comm370 lecture 3 - financial planning and growthnelsonpoon
This document discusses financial planning and growth. It defines key variables like assets, debt, equity, net income, and retention ratio. It then shows how to calculate a company's external funding needs, internal growth rate, and sustainable growth rate using these variables and common accounting metrics. The internal growth rate is the maximum rate supported without external financing, while the sustainable growth rate is the maximum that maintains a constant debt-to-equity ratio with external borrowing. Calculating and comparing a planned growth rate to these rates provides insight into future financing needs.
The document discusses the time value of money concepts of future value and present value for single sums and annuities. It provides examples of calculating future value and present value for single cash flows using formulas and a financial calculator. It also discusses the differences between ordinary annuities, where cash flows occur at the end of periods, and annuity dues, where cash flows occur at the beginning of periods.
Understanding the Time Value of Money; Single PaymentDIANN MOORMAN
- Calculators must be set to 4 decimal places and 1 payment per year for calculations.
- The document discusses the time value of money and how interest, compound interest, and inflation affect the value of money over time. It provides examples of calculating future and present value using the future and present value equations. It also discusses how to adjust for inflation when making time value of money calculations.
The document discusses various time value of money concepts including future value, present value, perpetuities, and annuities. It provides examples of calculations for future and present value of a single sum, as well as present value calculations for perpetuities and annuities. Uneven cash flows are discussed as being the sum of present values of regular cash flows. Key steps in solving time value problems are identified as drawing the timeline, identifying the cash flows, determining what value is being calculated, and using the appropriate formula.
The document discusses the concept of time value of money and how to calculate future and present values of cash flows. It provides examples of calculating the future and present value of a single cash flow using formulas, tables, and a financial calculator. It also discusses calculating future and present values for annuities, as well as solving for unknown variables like interest rate or number of periods. Formulas, tables, and a calculator can all be used to solve these types of time value of money problems.
The document discusses various topics related to bond valuation and interest rates, including:
- Characteristics of bonds such as fixed coupon payments and paying par value at maturity.
- How the intrinsic value of a bond is calculated by discounting its expected cash flows at the required rate of return.
- Examples of how the intrinsic value changes when interest rates change.
- The concept of yield to maturity and examples of calculating YTM for different types of bonds.
The document discusses time value of money concepts like present value, future value, present value of annuity, and future value of annuity. It provides formulas to calculate these values and includes examples showing how to apply the formulas to calculate PV, FV, PVA and FVA given cash flows, interest rates, and time periods. Sample questions are included along with step-by-step solutions demonstrating the calculations.
This document provides an overview of time value of money concepts including simple and compound interest, present and future value, and annuities. Some key points include:
- Compound interest earns interest on interest, resulting in higher total interest compared to simple interest over time.
- The future value of a single deposit or investment can be calculated using a formula that compounds the principal by the interest rate over multiple periods.
- The present value of a future amount can be calculated by discounting the future value back using the interest rate.
- Annuities represent a series of equal periodic payments or receipts, and their future or present value can be calculated using annuity formulas that take into account all
The document discusses the concepts of time value of money, interest, and annuities. It defines key terms like present value, future value, simple interest, compound interest, and ordinary annuity. It provides examples of calculating simple interest, compound interest, future value, present value, and future value of annuities using standard formulas. Various questions and solutions are given to illustrate time value of money calculations.
What is the 'Time Value of Money - TVM'
The time value of money (TVM) is the idea that money available at the present time is worth more than the same amount in the future due to its potential earning capacity. This core principle of finance holds that, provided money can earn interest, any amount of money is worth more the sooner it is received. TVM is also referred to as present discounted value.
BREAKING DOWN 'Time Value of Money - TVM'
Money deposited in a savings account earns a certain interest rate. Rational investors prefer to receive money today rather than the same amount of money in the future because of money's potential to grow in value over a given period of time. Money earning an interest rate is said to be compounding in value.
BREAKING DOWN 'Compound Interest'
Compound Interest Formula
Compound interest is calculated by multiplying the principal amount by one plus the annual interest rate raised to the number of compound periods minus one.The total initial amount of the loan is then subtracted from the resulting value.
This document discusses the time value of money concept through examples of simple and compound interest, present and future value calculations for single amounts, annuities, and mixed cash flows. It provides formulas, examples, and guidelines for solving time value of money problems involving deposits, loans, and returns over time discounted or compounded at given interest rates.
This document discusses the time value of money concepts of simple and compound interest, present and future value, and annuities. It provides formulas and examples for calculating future and present value of single deposits using tables or calculators. It also covers calculating the future value of annuities and using annuity tables. Key concepts covered include compound interest earning interest on interest, and the higher growth it provides over time compared to simple interest.
This document discusses the time value of money and various time value of money concepts. It begins by explaining that money has time value because it can earn interest over time and because purchasing power changes with inflation over time. It then discusses the role of time value in finance decisions and provides examples comparing cash flows received at different points in time. The document reviews concepts of future value, present value, interest, compounding, discounting, and provides examples of calculations for these topics. It also covers annuities, the difference between ordinary and due annuities, and calculations for future and present value of annuities.
This document discusses the time value of money concept in finance. It defines key terms like present value, future value, simple interest, and compound interest. It provides formulas for calculating future value and present value of single deposits. Examples are given to demonstrate calculating interest using simple interest formulas versus compound interest formulas. Tables are presented to allow looking up interest factors instead of using formulas. The document also introduces the concepts of amortization schedules and using a financial calculator for time value of money problems.
The document provides an overview of key concepts related to the time value of money, including simple and compound interest, present and future value calculations, and annuities. It explains time value of money principles, interest rates, and how to use tables or calculators to solve for unknown values like future or present worth when given other known amounts. The chapter aims to help readers understand how to adjust the value of cash flows to a single point in time using interest rates.
The time value of money is the concept that money available at the present time is worth more than the identical sum in the future due to its potential earning capacity. This core principle of finance holds that, provided money can earn interest, any amount of money is worth more the sooner it is received. Time Value of Money is also sometimes referred to as present discounted value.
this is a lecture on time value of money which explains the topic time value of money in a very easy and simple way... it also explains some examples on the topic... plus definition of rate of return, real rate of return, inflation premium, nominal interest rate,market risk, maturity risk,liquidity risk,and default risk,
The document discusses time value of money concepts including future value, present value, compound interest, and annuities. It provides examples of using the compound interest formula and financial calculators to solve for future value, present value, interest rates, number of periods, and payment amounts of cash flows. It also discusses amortization tables and constructing an amortization schedule for a loan.
CHAPTER 9Time Value of MoneyFuture valuePresent valueAnn.docxtiffanyd4
CHAPTER 9
Time Value of Money
Future value
Present value
Annuities
Rates of return
Amortization
9-‹#›
1
Time lines
Show the timing of cash flows.
Tick marks occur at the end of periods, so Time 0 is today; Time 1 is the end of the first period (year, month, etc.) or the beginning of the second period.
CF0
CF1
CF3
CF2
0
1
2
3
I%
9-‹#›
2
Drawing time lines
100
100
100
0
1
2
3
I%
3 year $100 ordinary annuity
100
0
1
2
I%
$100 lump sum due in 2 years
9-‹#›
3
Future Value of Money
If you deposit $1,000 today at 10%, how much will you have after 15 years?
Interest($) = Principal ∙ Interest Rate(%)
Simple Interest
The original principal stays the same.
There is no interest on interest. The interest is only on the original principal.
Compound Interest
The principal changes through time.
There is “interest on interest”. The interest is on the new principal.
9-‹#›
9-‹#›
9-‹#›
Simple Interest
Interest($) = Principal($) ∙ Interest Rate(%) = V0 ∙ I
V1 = V0 + Interest = V0 + V0 ∙ I = V0(1 + I)
V2 = V1 + Interest = V1 + V0 ∙ I = V0(1 + I) + V0 ∙ I
= V0(1 + I + I) = V0(1 + 2I)
V3 = V2 + Interest = V2 + V0 ∙ I = V0(1 + 2I) + V0 ∙ I
= V0(1 + 2I + I) = V0(1 + 3I)
.
.
Vn = V0(1 + nI)
FVn = PV(1 + nI)
9-‹#›
Compound Interest
Interest ($) = Principal ($) ∙ Interest Rate (%) = V ∙ I
V1 = V0 + Interest = V0 + V0 ∙ I = V0(1 + I)
V2 = V1 + Interest = V1 + V1 ∙ I = V1(1 + I)
V3 = V2 + Interest = V2 + V2 ∙ I = V2(1 + I)
V2 = V1(1 + I) = V0(1 + I)(1 + I) = V0(1 + I)2
V3 = V2(1 + I) = V0(1 + I)2(1 + I) = V0(1 + I)3
Vn = V0 (1 + I)n
FVn = PV(1 + I)n = PV∙FVIF
V2 = V1 + Interest = V1 + (V0 + Interest) ∙ I
9-‹#›
Example
What is the future value of $20 invested for 2 years at 10%?
Simple: FV = PV(1+nI)
= 20(1+2I) = 20(1+0.2) = $24
Compound: FV = PV(1+I)n
= 20(1+I)2 = 20(1+0.1)2 = $24.2
What is the future value of $20 invested for 100 years at 10%?
Simple: FV = 20(1+ ) =
Compound : FV = 20(1.1)100 = 275,612.25
9-‹#›
The Power of Compounding
The Value of Manhattan
In 1626, the land was bought from American Indians at $24.
In 2018, value = $24(1+I)392
9-‹#›
Solving for FV:
The formula method
Solve the general FV equation:
FVN = PV∙(1 + I)N = PV ∙ FVIF
FV15 = PV∙(1 + I)15 = $1,000∙(1.10)15 = $4,177.25
= $1,000∙4.177 = $4,177
(Table A)
9-‹#›
Present Value of Money
If you want to have $4,177.25 after 15 years, how much do you have to deposit today at 10%?
9-‹#›
PV = ?
4,177.25
Present Value of Money
Finding the PV of a cash flow or series of cash flows is called discounting (the reverse of compounding).
0
1
2 …
15
10%
9-‹#›
13
Solving for PV:
The formula method
Solve the general FV equat.
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1. The Time Value of MoneyThe Time Value of Money
Compounding andCompounding and
Discounting Single SumsDiscounting Single Sums
2. We know that receiving $1 today is worthWe know that receiving $1 today is worth
moremore than $1 in the future. This is duethan $1 in the future. This is due
toto opportunity costsopportunity costs..
The opportunity cost of receiving $1 inThe opportunity cost of receiving $1 in
the future is thethe future is the interestinterest we could havewe could have
earned if we had received the $1earned if we had received the $1
sooner.sooner.
Today Future
3. If we can measure this opportunityIf we can measure this opportunity
cost, we can:cost, we can:
Translate $1 today into its equivalent in the futureTranslate $1 today into its equivalent in the future
(compounding)(compounding)..
Translate $1 in the future into its equivalent todayTranslate $1 in the future into its equivalent today
(discounting)(discounting)..
?
Today Future
Today
?
Future
5. Future Value - single sumsFuture Value - single sums
If you deposit $100 in an account earning 6%, howIf you deposit $100 in an account earning 6%, how
much would you have in the account after 1 year?much would you have in the account after 1 year?
Mathematical Solution:Mathematical Solution:
FV = PV (FVIFFV = PV (FVIF i,ni,n ))
FV = 100 (FVIFFV = 100 (FVIF .06,1.06,1 ) (use FVIF table, or)) (use FVIF table, or)
FV = PV (1 + i)FV = PV (1 + i)nn
FV = 100 (1.06)FV = 100 (1.06)11
== $106$106
00 11
PV = -100PV = -100 FV =FV = 106106
6. Future Value - single sumsFuture Value - single sums
If you deposit $100 in an account earning 6%, howIf you deposit $100 in an account earning 6%, how
much would you have in the account after 5 years?much would you have in the account after 5 years?
Mathematical Solution:Mathematical Solution:
FV = PV (FVIFFV = PV (FVIF i,ni,n ))
FV = 100 (FVIFFV = 100 (FVIF .06,5.06,5 ) (use FVIF table, or)) (use FVIF table, or)
FV = PV (1 + i)FV = PV (1 + i)nn
FV = 100 (1.06)FV = 100 (1.06)55
== $$133.82133.82
00 55
PV = -100PV = -100 FV =FV = 133.133.8282
7. Mathematical Solution:Mathematical Solution:
FV = PV (FVIFFV = PV (FVIF i,ni,n ))
FV = 100 (FVIFFV = 100 (FVIF .015,20.015,20 )) (can’t use FVIF table)(can’t use FVIF table)
FV = PV (1 + i/m)FV = PV (1 + i/m) mxnmxn
FV = 100 (1.015)FV = 100 (1.015)2020
== $134.68$134.68
00 2020
PV = -100PV = -100 FV =FV = 134.134.6868
Future Value - single sumsFuture Value - single sums
If you deposit $100 in an account earning 6% withIf you deposit $100 in an account earning 6% with
quarterly compoundingquarterly compounding, how much would you have, how much would you have
in the account after 5 years?in the account after 5 years?
8. Mathematical Solution:Mathematical Solution:
FV = PV (FVIFFV = PV (FVIF i,ni,n ))
FV = 100 (FVIFFV = 100 (FVIF .005,60.005,60 )) (can’t use FVIF table)(can’t use FVIF table)
FV = PV (1 + i/m)FV = PV (1 + i/m) mxnmxn
FV = 100 (1.005)FV = 100 (1.005)6060
== $134.89$134.89
00 6060
PV = -100PV = -100 FV =FV = 134.134.8989
Future Value - single sumsFuture Value - single sums
If you deposit $100 in an account earning 6% withIf you deposit $100 in an account earning 6% with
monthly compoundingmonthly compounding, how much would you have, how much would you have
in the account after 5 years?in the account after 5 years?
9. 00 100100
PV = -1000PV = -1000 FV =FV = $2.98m$2.98m
Future Value - continuous compoundingFuture Value - continuous compounding
What is the FV of $1,000 earning 8% withWhat is the FV of $1,000 earning 8% with
continuous compoundingcontinuous compounding, after 100 years?, after 100 years?
Mathematical Solution:Mathematical Solution:
FV = PV (eFV = PV (e inin
))
FV = 1000 (eFV = 1000 (e .08x100.08x100
) = 1000 (e) = 1000 (e 88
))
FV =FV = $2,980,957.$2,980,957.9999
11. Mathematical Solution:Mathematical Solution:
PV = FV (PVIFPV = FV (PVIF i,ni,n ))
PV = 100 (PVIFPV = 100 (PVIF .06,1.06,1 ) (use PVIF table, or)) (use PVIF table, or)
PV = FV / (1 + i)PV = FV / (1 + i)nn
PV = 100 / (1.06)PV = 100 / (1.06)11
== $94.34$94.34
PV =PV = -94.-94.3434
FV = 100FV = 100
00 11
Present Value - single sumsPresent Value - single sums
If you receive $100 one year from now, what is theIf you receive $100 one year from now, what is the
PV of that $100 if your opportunity cost is 6%?PV of that $100 if your opportunity cost is 6%?
12. Mathematical Solution:Mathematical Solution:
PV = FV (PVIFPV = FV (PVIF i,ni,n ))
PV = 100 (PVIFPV = 100 (PVIF .06,5.06,5 ) (use PVIF table, or)) (use PVIF table, or)
PV = FV / (1 + i)PV = FV / (1 + i)nn
PV = 100 / (1.06)PV = 100 / (1.06)55
== $74.73$74.73
Present Value - single sumsPresent Value - single sums
If you receive $100 five years from now, what is theIf you receive $100 five years from now, what is the
PV of that $100 if your opportunity cost is 6%?PV of that $100 if your opportunity cost is 6%?
00 55
PV =PV = -74.-74.7373
FV = 100FV = 100
13. Mathematical Solution:Mathematical Solution:
PV = FV (PVIFPV = FV (PVIF i,ni,n ))
PV = 100 (PVIFPV = 100 (PVIF .07,15.07,15 ) (use PVIF table, or)) (use PVIF table, or)
PV = FV / (1 + i)PV = FV / (1 + i)nn
PV = 100 / (1.07)PV = 100 / (1.07)1515
== $362.45$362.45
Present Value - single sumsPresent Value - single sums
What is the PV of $1,000 to be received 15 yearsWhat is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?from now if your opportunity cost is 7%?
00 1515
PV =PV = -362.-362.4545
FV = 1000FV = 1000
14. 00 55
PV =PV = FV =FV =
Present Value - single sumsPresent Value - single sums
If you sold land for $11,933 that you bought 5If you sold land for $11,933 that you bought 5
years ago for $5,000, what is your annual rate ofyears ago for $5,000, what is your annual rate of
return?return?
15. Mathematical Solution:Mathematical Solution:
PV = FV (PVIFPV = FV (PVIF i,ni,n ))
5,000 = 11,933 (PVIF5,000 = 11,933 (PVIF ?,5?,5 ))
PV = FV / (1 + i)PV = FV / (1 + i)nn
5,000 = 11,933 / (1+ i)5,000 = 11,933 / (1+ i)55
.419 = ((1/ (1+i).419 = ((1/ (1+i)55
))
2.3866 = (1+i)2.3866 = (1+i)55
(2.3866)(2.3866)1/51/5
= (1+i)= (1+i) i =i = .19.19
Present Value - single sumsPresent Value - single sums
If you sold land for $11,933 that you bought 5If you sold land for $11,933 that you bought 5
years ago for $5,000, what is your annual rate ofyears ago for $5,000, what is your annual rate of
return?return?
16. Present Value - single sumsPresent Value - single sums
Suppose you placed $100 in an account that paysSuppose you placed $100 in an account that pays
9.6% interest, compounded monthly. How long9.6% interest, compounded monthly. How long
will it take for your account to grow to $500?will it take for your account to grow to $500?
00
PV =PV = FV =FV =
17. Present Value - single sumsPresent Value - single sums
Suppose you placed $100 in an account that paysSuppose you placed $100 in an account that pays
9.6% interest, compounded monthly. How long9.6% interest, compounded monthly. How long
will it take for your account to grow to $500?will it take for your account to grow to $500?
Mathematical Solution:Mathematical Solution:
PV = FV / (1 + i)PV = FV / (1 + i)nn
100 = 500 / (1+ .008)100 = 500 / (1+ .008)NN
5 = (1.008)5 = (1.008)NN
ln 5 = ln (1.008)ln 5 = ln (1.008)NN
ln 5 = N ln (1.008)ln 5 = N ln (1.008)
1.60944 = .007968 N1.60944 = .007968 N N = 202 monthsN = 202 months
18. Hint for single sum problems:Hint for single sum problems:
In every single sum present value andIn every single sum present value and
future value problem, there are fourfuture value problem, there are four
variables:variables:
FVFV,, PVPV,, ii andand nn..
When doing problems, you will be givenWhen doing problems, you will be given
three variables and you will solve for thethree variables and you will solve for the
fourth variable.fourth variable.
Keeping this in mind makes solving timeKeeping this in mind makes solving time
value problems much easier!value problems much easier!
19. The Time Value of MoneyThe Time Value of Money
Compounding and DiscountingCompounding and Discounting
Cash Flow StreamsCash Flow Streams
0 1 2 3 4
20. Annuity:Annuity: a sequence ofa sequence of equalequal cashcash
flows, occurring at the end of eachflows, occurring at the end of each
period.period.
0 1 2 3 4
AnnuitiesAnnuities
21. If you buy a bond, you willIf you buy a bond, you will
receive equal semi-annual couponreceive equal semi-annual coupon
interest payments over the life ofinterest payments over the life of
the bond.the bond.
If you borrow money to buy aIf you borrow money to buy a
house or a car, you will pay ahouse or a car, you will pay a
stream of equal payments.stream of equal payments.
Examples of Annuities:Examples of Annuities:
22. 0 1 2 3
Future Value - annuityFuture Value - annuity
If you invest $1,000 each year at 8%, how muchIf you invest $1,000 each year at 8%, how much
would you have after 3 years?would you have after 3 years?
23. Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 8I = 8 N = 3N = 3
PMT = -1,000PMT = -1,000
FV =FV = $3,246.40$3,246.40
Future Value - annuityFuture Value - annuity
If you invest $1,000 each year at 8%, how muchIf you invest $1,000 each year at 8%, how much
would you have after 3 years?would you have after 3 years?
0 1 2 3
10001000 10001000 10001000
24. Mathematical Solution:Mathematical Solution:
FV = PMT (FVIFAFV = PMT (FVIFA i,ni,n ))
FV = 1,000 (FVIFAFV = 1,000 (FVIFA .08,3.08,3 )) (use FVIFA table, or)(use FVIFA table, or)
FV = PMT (1 + i)FV = PMT (1 + i)nn
- 1- 1
ii
FV = 1,000 (1.08)FV = 1,000 (1.08)33
- 1 =- 1 = $3246.40$3246.40
.08.08
Future Value - annuityFuture Value - annuity
If you invest $1,000 each year at 8%, how muchIf you invest $1,000 each year at 8%, how much
would you have after 3 years?would you have after 3 years?
25. 0 1 2 3
Present Value - annuityPresent Value - annuity
What is the PV of $1,000 at the end of each of theWhat is the PV of $1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%?next 3 years, if the opportunity cost is 8%?
26. Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 8I = 8 N = 3N = 3
PMT = -1,000PMT = -1,000
PV =PV = $2,577.10$2,577.10
0 1 2 3
10001000 10001000 10001000
Present Value - annuityPresent Value - annuity
What is the PV of $1,000 at the end of each of theWhat is the PV of $1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%?next 3 years, if the opportunity cost is 8%?
27. Mathematical Solution:Mathematical Solution:
PV = PMT (PVIFAPV = PMT (PVIFA i, ni, n ))
PV = 1,000 (PVIFAPV = 1,000 (PVIFA .08,3.08, 3 ) (use PVIFA table, or)) (use PVIFA table, or)
11
PV = PMT 1 - (1 + i)PV = PMT 1 - (1 + i)nn
ii
11
PV = 1000 1 - (1.08 )PV = 1000 1 - (1.08 )33
== $2,577.10$2,577.10
.08.08
Present Value - annuityPresent Value - annuity
What is the PV of $1,000 at the end of each of theWhat is the PV of $1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%?next 3 years, if the opportunity cost is 8%?
28. Other Cash Flow PatternsOther Cash Flow Patterns
0 1 2 3
The Time Value of Money
29. PerpetuitiesPerpetuities
Suppose you will receive a fixedSuppose you will receive a fixed
payment every period (month, year,payment every period (month, year,
etc.) forever. This is an example ofetc.) forever. This is an example of
a perpetuity.a perpetuity.
You can think of a perpetuity as anYou can think of a perpetuity as an
annuityannuity that goes onthat goes on foreverforever..
30. Present Value of aPresent Value of a
PerpetuityPerpetuity
When we find the PV of anWhen we find the PV of an annuityannuity,,
we think of the followingwe think of the following
relationship:relationship:
PV = PMT (PVIFAPV = PMT (PVIFA i, ni, n ))
31. Mathematically,Mathematically,
(PVIFA i, n ) =(PVIFA i, n ) =
We said that a perpetuity is anWe said that a perpetuity is an
annuity where n = infinity. Whatannuity where n = infinity. What
happens to this formula whenhappens to this formula when nn
gets very, very large?gets very, very large?
1 -1 -
11
(1 + i)(1 + i)nn
ii
32. When n gets very large,When n gets very large,
this becomes zero.this becomes zero.
So we’re left with PVIFA =So we’re left with PVIFA =
1
i
1 -
1
(1 + i)n
i
33. PMT
i
PV =
So, the PV of a perpetuity is verySo, the PV of a perpetuity is very
simple to find:simple to find:
Present Value of a Perpetuity
34. What should you be willing to pay inWhat should you be willing to pay in
order to receiveorder to receive $10,000$10,000 annuallyannually
forever, if you requireforever, if you require 8%8% per yearper year
on the investment?on the investment?
PMT $10,000PMT $10,000
i .08i .08
= $125,000= $125,000
PV = =PV = =
36. Begin Mode vs. End ModeBegin Mode vs. End Mode
1000 1000 10001000 1000 1000
4 5 6 7 84 5 6 7 8
year year year
5 6 7
37. Begin Mode vs. End ModeBegin Mode vs. End Mode
1000 1000 10001000 1000 1000
4 5 6 7 84 5 6 7 8
year year year
5 6 7
PVPV
inin
ENDEND
ModeMode
FVFV
inin
ENDEND
ModeMode
38. Begin Mode vs. End ModeBegin Mode vs. End Mode
1000 1000 10001000 1000 1000
4 5 6 7 84 5 6 7 8
year year year
6 7 8
39. Begin Mode vs. End ModeBegin Mode vs. End Mode
1000 1000 10001000 1000 1000
4 5 6 7 84 5 6 7 8
year year year
6 7 8
PVPV
inin
BEGINBEGIN
ModeMode
FVFV
inin
BEGINBEGIN
ModeMode
40. Earlier, we examined thisEarlier, we examined this
“ordinary” annuity:“ordinary” annuity:
Using an interest rate of 8%, weUsing an interest rate of 8%, we
find that:find that:
TheThe Future ValueFuture Value (at 3) is(at 3) is
$3,246.40$3,246.40..
TheThe Present ValuePresent Value (at 0) is(at 0) is
$2,577.10$2,577.10..
0 1 2 3
10001000 10001000 10001000
41. What about this annuity?What about this annuity?
Same 3-year time line,Same 3-year time line,
Same 3 $1000 cash flows, butSame 3 $1000 cash flows, but
The cash flows occur at theThe cash flows occur at the
beginningbeginning of each year, ratherof each year, rather
than at thethan at the endend of each year.of each year.
This is anThis is an “annuity due.”“annuity due.”
0 1 2 3
10001000 10001000 10001000
42. 0 1 2 3
-1000-1000 -1000-1000 -1000-1000
Future Value - annuity dueFuture Value - annuity due
If you invest $1,000 at the beginning of each of theIf you invest $1,000 at the beginning of each of the
next 3 years at 8%, how much would you have atnext 3 years at 8%, how much would you have at
the end of year 3?the end of year 3?
Calculator Solution:Calculator Solution:
Mode = BEGIN P/Y = 1Mode = BEGIN P/Y = 1 I = 8I = 8
N = 3N = 3 PMT = -1,000PMT = -1,000
FV =FV = $3,506.11$3,506.11
43. Future Value - annuity dueFuture Value - annuity due
If you invest $1,000 at the beginning of each of theIf you invest $1,000 at the beginning of each of the
next 3 years at 8%, how much would you have atnext 3 years at 8%, how much would you have at
the end of year 3?the end of year 3?
Mathematical Solution:Mathematical Solution: Simply compound the FV of theSimply compound the FV of the
ordinary annuity one more period:ordinary annuity one more period:
FV = PMT (FVIFAFV = PMT (FVIFA i,ni,n ) (1 + i)) (1 + i)
FV = 1,000 (FVIFAFV = 1,000 (FVIFA .08,3.08,3 ) (1.08)) (1.08) (use FVIFA table, or)(use FVIFA table, or)
FV = PMT (1 + i)FV = PMT (1 + i)nn
- 1- 1
ii
FV = 1,000 (1.08)FV = 1,000 (1.08)33
- 1 =- 1 = $3,506.11$3,506.11
(1 + i)(1 + i)
(1.08)(1.08)
44. Calculator Solution:Calculator Solution:
Mode = BEGIN P/Y = 1Mode = BEGIN P/Y = 1 I = 8I = 8
N = 3N = 3 PMT = 1,000PMT = 1,000
PV =PV = $2,783.26$2,783.26
0 1 2 3
10001000 10001000 10001000
Present Value - annuity duePresent Value - annuity due
What is the PV of $1,000 at the beginning of eachWhat is the PV of $1,000 at the beginning of each
of the next 3 years, if your opportunity cost is 8%?of the next 3 years, if your opportunity cost is 8%?
45. Present Value - annuity duePresent Value - annuity due
Mathematical Solution:Mathematical Solution: Simply compound the FV of theSimply compound the FV of the
ordinary annuity one more period:ordinary annuity one more period:
PV = PMT (PVIFAPV = PMT (PVIFA i,ni,n ) (1 + i)) (1 + i)
PV = 1,000 (PVIFAPV = 1,000 (PVIFA .08,3.08,3 ) (1.08)) (1.08) (use PVIFA table, or)(use PVIFA table, or)
11
PV = PMT 1 - (1 + i)PV = PMT 1 - (1 + i)nn
ii
11
PV = 1000 1 - (1.08 )PV = 1000 1 - (1.08 )33
== $2,783.26$2,783.26
.08.08
(1 + i)(1 + i)
(1.08)(1.08)
46. Is this anIs this an annuityannuity??
How do we find the PV of a cash flowHow do we find the PV of a cash flow
stream when all of the cash flows arestream when all of the cash flows are
different? (Use a 10% discount rate.)different? (Use a 10% discount rate.)
Uneven Cash FlowsUneven Cash Flows
00 11 22 33 44
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
47. Sorry! There’s no quickie for this one.Sorry! There’s no quickie for this one.
We have to discount each cash flowWe have to discount each cash flow
back separately.back separately.
00 11 22 33 44
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
Uneven Cash FlowsUneven Cash Flows
48. Sorry! There’s no quickie for this one.Sorry! There’s no quickie for this one.
We have to discount each cash flowWe have to discount each cash flow
back separately.back separately.
00 11 22 33 44
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
Uneven Cash FlowsUneven Cash Flows
50. Annual Percentage Yield (APY)Annual Percentage Yield (APY)
Which is the better loan:Which is the better loan:
8%8% compoundedcompounded annuallyannually, or, or
7.85%7.85% compoundedcompounded quarterlyquarterly??
We can’t compare these nominal (quoted)We can’t compare these nominal (quoted)
interest rates, because they don’t include theinterest rates, because they don’t include the
same number of compounding periods persame number of compounding periods per
year!year!
We need to calculate the APY.We need to calculate the APY.
51. Annual Percentage Yield (APY)Annual Percentage Yield (APY)
Find the APY for the quarterly loan:Find the APY for the quarterly loan:
The quarterly loan is more expensive thanThe quarterly loan is more expensive than
the 8% loan with annual compounding!the 8% loan with annual compounding!
APY =APY = (( 1 +1 + )) mm
- 1- 1quoted ratequoted rate
mm
APY =APY = (( 1 +1 + )) 44
- 1- 1
APY = .0808, or 8.08%APY = .0808, or 8.08%
.0785.0785
44
53. ExampleExample
00 11 22 33 44 55 66 77 88
$0$0 00 00 00 4040 4040 4040 4040 4040
Cash flows from an investment areCash flows from an investment are
expected to beexpected to be $40,000$40,000 per year at theper year at the
end of years 4, 5, 6, 7, and 8. If youend of years 4, 5, 6, 7, and 8. If you
require arequire a 20%20% rate of return, what israte of return, what is
the PV of these cash flows?the PV of these cash flows?
54. This type of cash flow sequence isThis type of cash flow sequence is
often called aoften called a ““deferred annuitydeferred annuity.”.”
00 11 22 33 44 55 66 77 88
$0$0 00 00 00 4040 4040 4040 4040 4040
55. How to solve:How to solve:
1)1) Discount each cash flow back toDiscount each cash flow back to
time 0 separately.time 0 separately.
Or,Or,
00 11 22 33 44 55 66 77 88
$0$0 00 00 00 4040 4040 4040 4040 4040
56. 2)2) Find the PV of the annuity:Find the PV of the annuity:
PVPV3:3: End mode; P/YR = 1; I = 20;End mode; P/YR = 1; I = 20;
PMT = 40,000; N = 5PMT = 40,000; N = 5
PVPV33== $119,624$119,624
00 11 22 33 44 55 66 77 88
$0$0 00 00 00 4040 4040 4040 4040 4040
57. Then discount this single sum back toThen discount this single sum back to
time 0.time 0.
PV: End mode; P/YR = 1; I = 20;PV: End mode; P/YR = 1; I = 20;
N = 3; FV = 119,624;N = 3; FV = 119,624;
Solve: PV =Solve: PV = $69,226$69,226
119,624119,624
00 11 22 33 44 55 66 77 88
$0$0 00 00 00 4040 4040 4040 4040 4040
58. The PV of the cash flowThe PV of the cash flow
stream isstream is $69,226$69,226..
69,22669,226
00 11 22 33 44 55 66 77 88
$0$0 00 00 00 4040 4040 4040 4040 4040
119,624119,624
59. Retirement ExampleRetirement Example
After graduation, you plan to investAfter graduation, you plan to invest
$400$400 per month in the stock market.per month in the stock market.
If you earnIf you earn 12%12% per year on yourper year on your
stocks, how much will you havestocks, how much will you have
accumulated when you retire in 30accumulated when you retire in 30
years?years?
00 11 22 33 . . . 360. . . 360
400 400 400 400400 400 400 400
61. Retirement ExampleRetirement Example
If you invest $400 at the end of each month for theIf you invest $400 at the end of each month for the
next 30 years at 12%, how much would you have atnext 30 years at 12%, how much would you have at
the end of year 30?the end of year 30?
62. Retirement ExampleRetirement Example
If you invest $400 at the end of each month for theIf you invest $400 at the end of each month for the
next 30 years at 12%, how much would you have atnext 30 years at 12%, how much would you have at
the end of year 30?the end of year 30?
Mathematical Solution:Mathematical Solution:
FV = PMT (FVIFAFV = PMT (FVIFA i,ni,n ))
FV = 400 (FVIFAFV = 400 (FVIFA .01,360.01,360 )) (can’t use FVIFA table)(can’t use FVIFA table)
FV = PMT (1 + i)FV = PMT (1 + i)nn
- 1- 1
ii
FV = 400 (1.01)FV = 400 (1.01)360360
- 1 =- 1 = $1,397,985.65$1,397,985.65
63. House Payment ExampleHouse Payment Example
If you borrowIf you borrow $100,000$100,000 atat 7%7% fixedfixed
interest forinterest for 3030 years in order toyears in order to
buy a house, what will be yourbuy a house, what will be your
monthly house payment?monthly house payment?
66. Team AssignmentTeam Assignment
Upon retirement, your goal is to spendUpon retirement, your goal is to spend 55
years traveling around the world. Toyears traveling around the world. To
travel in style will requiretravel in style will require $250,000$250,000 perper
year at theyear at the beginningbeginning of each year.of each year.
If you plan to retire inIf you plan to retire in 3030 yearsyears, what are, what are
the equalthe equal monthlymonthly payments necessarypayments necessary
to achieve this goal? The funds in yourto achieve this goal? The funds in your
retirement account will compound atretirement account will compound at
10%10% annually.annually.
67. How much do we need to have byHow much do we need to have by
the end of year 30 to finance thethe end of year 30 to finance the
trip?trip?
PVPV3030 = PMT (PVIFA= PMT (PVIFA .10, 5.10, 5) (1.10) =) (1.10) =
= 250,000 (3.7908) (1.10) == 250,000 (3.7908) (1.10) =
== $1,042,470$1,042,470
2727 2828 2929 3030 3131 3232 3333 3434 3535
250 250 250 250 250250 250 250 250 250
69. Now, assuming 10% annualNow, assuming 10% annual
compounding, what monthlycompounding, what monthly
payments will be required for youpayments will be required for you
to haveto have $1,042,466$1,042,466 at the end ofat the end of
year 30?year 30?
2727 2828 2929 3030 3131 3232 3333 3434 3535
250 250 250 250 250250 250 250 250 250
1,042,4661,042,466
71. So, you would have to placeSo, you would have to place $461.17$461.17 inin
your retirement account, which earnsyour retirement account, which earns
10% annually, at the end of each of the10% annually, at the end of each of the
next 360 months to finance the 5-yearnext 360 months to finance the 5-year
world tour.world tour.