chapter 5 - time value of money

Prepared byPrepared byKen Hartviksen and Robert IronsideKen Hartviksen and Robert Ironside

INTRODUCTION TOINTRODUCTION TO

CORPORATE FINANCECORPORATE FINANCELaurence Booth Laurence Booth •• W. Sean Cleary W. Sean Cleary

Chapter 5 – Time Value of MoneyChapter 5 – Time Value of Money

CHAPTER 5CHAPTER 5 Time Value of MoneyTime Value of Money

CHAPTER 5 – Time Value of Money 5 - 3

Lecture AgendaLecture Agenda

• Learning ObjectivesLearning Objectives• Important TermsImportant Terms• Types of CalculationsTypes of Calculations• CompoundingCompounding• DiscountingDiscounting• Annuities and LoansAnnuities and Loans• PerpetuitiesPerpetuities• Effective Rates of ReturnEffective Rates of Return• Summary and ConclusionsSummary and Conclusions

– Concept Review QuestionsConcept Review Questions


Learning ObjectivesLearning Objectives

• Understand the importance of the time value of moneyUnderstand the importance of the time value of money• Understand the difference between simple interest and Understand the difference between simple interest and

compound interest compound interest • Know how to solve for present value, future value, time or Know how to solve for present value, future value, time or

raterate• Understand annuities and perpetuitiesUnderstand annuities and perpetuities• Know how to construct an amortization tableKnow how to construct an amortization table


Important Chapter TermsImportant Chapter Terms

• AmortizeAmortize• AnnuityAnnuity• Annuity dueAnnuity due• Basis pointBasis point• Cash flowsCash flows• Compound interestCompound interest• Compound interest factor Compound interest factor

(CVIF)(CVIF)• Discount rateDiscount rate• DiscountingDiscounting• Effective rate Effective rate

• LesseeLessee• Medium of exchangeMedium of exchange• MortgageMortgage• Ordinary annuitiesOrdinary annuities• PerpetuitiesPerpetuities• Present value interest Present value interest

factor (PVIF)factor (PVIF)• ReinvestedReinvested• Required rate of returnRequired rate of return• Simple interestSimple interest• Time value of moneyTime value of money

Types of CalculationsTypes of Calculations

Time Value of MoneyTime Value of Money


Before We Get StartedBefore We Get StartedTypes of CalculationsTypes of Calculations

Ex Ante:Ex Ante:– Calculations done ‘before-the-fact’Calculations done ‘before-the-fact’– It is a forecast of what might happenIt is a forecast of what might happen– All forecasts require assumptionsAll forecasts require assumptions

• It is important to understand the assumptions underlying any It is important to understand the assumptions underlying any formula used to ensure that those assumptions are consistent with formula used to ensure that those assumptions are consistent with the problem being solved.the problem being solved.

– As a forecast, while you may be able to calculate the answer to As a forecast, while you may be able to calculate the answer to a high degree of accuracy…it is probably best to round off the a high degree of accuracy…it is probably best to round off the answer so that users of your calculations are not misled.answer so that users of your calculations are not misled.

Ex Post:Ex Post:– Calculation done ‘after-the-fact’Calculation done ‘after-the-fact’– It is an analysis of what has happenedIt is an analysis of what has happened– It is usually possible, and perhaps wise to express the result as It is usually possible, and perhaps wise to express the result as

accurately as possible.accurately as possible.

The Basic ConceptThe Basic Concept



The Time Value of Money ConceptThe Time Value of Money Concept

• Cannot directly compare $1 today with $1 to Cannot directly compare $1 today with $1 to be received at some future datebe received at some future date– Money received today can be invested to earn a rate of returnMoney received today can be invested to earn a rate of return

– Thus $1 today is worth more than $1 to be received at some future dateThus $1 today is worth more than $1 to be received at some future date

• The interest rate or discount rate is the The interest rate or discount rate is the variable that equates a present value today variable that equates a present value today with a future value at some later date with a future value at some later date


Opportunity CostOpportunity Cost

Opportunity cost = Alternative useOpportunity cost = Alternative use

– The opportunity cost of money is the interest rate that The opportunity cost of money is the interest rate that would be earned by investing it.would be earned by investing it.

– It is the underlying reason for the time value of moneyIt is the underlying reason for the time value of money– Any person with money today knows they can invest Any person with money today knows they can invest

those funds to be some greater amount in the future.those funds to be some greater amount in the future.– Conversely, if you are promised a cash flow in the Conversely, if you are promised a cash flow in the

future, it’s present value today is less than what is future, it’s present value today is less than what is promised!promised!


Choosing from Investment AlternativesChoosing from Investment Alternatives Required Rate of Return or Discount RateRequired Rate of Return or Discount Rate

• You have three choices:You have three choices:1.1. $20,000 received today$20,000 received today

2.2. $31,000 received in 5 years$31,000 received in 5 years

3.3. $3,000 per year indefinitely$3,000 per year indefinitely

• To make a decision, you need to know what To make a decision, you need to know what interest rate to use.interest rate to use.– This interest rate is known as your This interest rate is known as your required rate of required rate of

returnreturn or or discount rate.discount rate.

Simple InterestSimple Interest




Simple interest is interest paid or received on Simple interest is interest paid or received on only the initial investment (or principal).only the initial investment (or principal).

At the end of the investment period, the At the end of the investment period, the principal plus interest is received.principal plus interest is received.

0 1 2 3 … n

I1 I2 I3 In+P

0 1 2 3 … n

I1 I2 I3 In+P


Simple InterestSimple Interest ExampleExample

PROBLEM:PROBLEM:

Invest $1,000 today for a Invest $1,000 today for a five-year term and receive five-year term and receive 8 percent annual simple 8 percent annual simple interest.interest.

How much will you How much will you accumulate by the end of accumulate by the end of five years?five years?

Year Beginning Amount Ending Amount1 $1,000 $1,0802 1,080 1,1603 1,160 1,2404 1,240 1,3205 1,320 $1,400

400,1$

400$000,1$

)80$5(000,1$

)08.000,1$5(000,1$5

Value

k)P(nPe n)Value (tim


Simple InterestSimple InterestGeneral FormulaGeneral Formula

k)P(nPe n)Value (tim [ 5-1]

Where:P = principal investedn = number of yearsk = interest rate



Simple interest problems are rare.Simple interest problems are rare.

In finance we are most interested in In finance we are most interested in COMPOUND INTEREST.COMPOUND INTEREST.

Compound InterestCompound Interest



Compound InterestCompound Interest Compounding (Computing Future Values)Compounding (Computing Future Values)

Compound interest is interest that is earned Compound interest is interest that is earned on the principal amount invested and on any on the principal amount invested and on any accrued interest.accrued interest.


Compound InterestCompound Interest ExampleExample

PROBLEM:PROBLEM:

Invest $1,000 today for a five-year term and receive 8 Invest $1,000 today for a five-year term and receive 8 percent annual percent annual compound interestcompound interest. How much will the . How much will the accumulated value be at time 5.accumulated value be at time 5.

SOLUTION:SOLUTION:

YearBeginning

AmountEnding Amount

1 $1,000.00 $1,080.00

2 1,080.00 1,166.40

3 1,166.40 1,259.71

4 1,259.71 1,360.49

5 1,360.49 1,469.33 33.469,1$)08.1(

49.360,1$)08.1()08.1)(08.1)(08.1)(08.1(

71.259,1$)08.1()08.1)(08.1)(08.1(

40.166,1$)08.1()08.1)(08.1(

080,1$)08.1(

1

55

44

33

22

11

PFV

PPFV

PPFV

PPFV

PFV

k)(PValueFuture n

$1,469.338)$1,000(1.0FV

1

:step simple onein solution The

55

0

nn k)(PVFV


Compound InterestCompound Interest Example of Interest Earned on InterestExample of Interest Earned on Interest

PROBLEM:PROBLEM:Invest $1,000 today for a five-year term and receive 8 percent annual Invest $1,000 today for a five-year term and receive 8 percent annual compound interest.compound interest.

The Interest-earned-on-Interest Effect:The Interest-earned-on-Interest Effect:Interest (year 1) = $1,000 Interest (year 1) = $1,000 × .08 = $80× .08 = $80

Interest (year 2 ) =($1,000 + $80)×.08 = $86.40Interest (year 2 ) =($1,000 + $80)×.08 = $86.40

Interest (year 3) = ($1,000+$80+$86.40) × .08 = $93.31Interest (year 3) = ($1,000+$80+$86.40) × .08 = $93.31

Year Beginning Amount Ending AmountInterest earned

in the year1 $1,000.00 $1,080.00 $80.002 1,080.00 1,166.40 $86.403 1,166.40 1,259.71 $93.314 1,259.71 1,360.49 $100.785 1,360.49 1,469.33 $108.84


Compound InterestCompound InterestGeneral FormulaGeneral Formula

Where:FV= future valueP = principal investedn = number of yearsk = interest rate

[ 5-2] 10n

n k)(PVFV


Compound InterestCompound InterestGeneral FormulaGeneral Formula

CVIFfactorinterest compound theasknown is 1 nk)(

[ 5-2] 10n

n k)(PVFV


Compound InterestCompound Interest Simple versus Compound InterestSimple versus Compound Interest

Compounding of interest magnifies the returns Compounding of interest magnifies the returns on an investment.on an investment.

Returns are magnified:Returns are magnified:• The The longer they are compoundedlonger they are compounded• The higher the rate they are compoundedThe higher the rate they are compounded

(See Figure 5-1 to compare simple and compound interest effects over (See Figure 5-1 to compare simple and compound interest effects over time)time)


Compound InterestCompound Interest Simple versus Compound InterestSimple versus Compound Interest

5-1 FIGURE

DO

LL

AR

S

Simple Compound

8,000

7,000

6,000

5,000

4,000

3,000

2,000

1,000

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

YEARS


Compound InterestCompound Interest Compound Interest at Varying RatesCompound Interest at Varying Rates

Compounding of interest magnifies the returns Compounding of interest magnifies the returns on an investment.on an investment.

Returns are magnified:Returns are magnified:• The longer they are compoundedThe longer they are compounded• The The higher the ratehigher the rate they are compounded they are compounded

(See Table 5-1 that demonstrates the cumulative effect of higher rates (See Table 5-1 that demonstrates the cumulative effect of higher rates of return earned over time.)of return earned over time.)


Compound InterestCompound Interest Compounded Returns over Time for Various Asset ClassesCompounded Returns over Time for Various Asset Classes

Annual Arithmetic

Average (%)

Annual Geometric Mean (%)

Yeark-End Value, 2005 ($)

Government of Canada treasury bills 5.20 5.11 $29,711Government of Canada bonds 6.62 6.24 61,404Canadian stocks 11.79 10.60 946,009U.S. stocks 13.15 11.76 1,923,692

Source: Data are from the Canadian Institute of Actuaries

Table 5-1 Ending Wealth of $1,000 Invested From 1938 to 2005 in Various Asset Classes


Compound InterestCompound InterestSolution Using a Financial Calculator (TI BA II Plus)Solution Using a Financial Calculator (TI BA II Plus)

PMT PV I/Y N

Input the following variables:

0 → ; -1,000 → ; 10 → ; and 5 →

CPT FVPress (Compute) and then

PMT refers to regular paymentsFV is the future valueI/Y is the period interest rateN is the number of periods

PV is entered with a negative sign to reflect investors must pay money now to get money in the future.

Answer = $1,610.51

$1,610.510)$1,000(1.1

1

.%10000,1$

5

0

n

nn

FV

k)(PVFV

yearsfiveforatinvestedofvalueFuture


Compound InterestCompound InterestSolution Using a Excel SpreadsheetSolution Using a Excel Spreadsheet

• Electronic spreadsheets have built-in Electronic spreadsheets have built-in formulae that can assist in the solution of formulae that can assist in the solution of problemsproblems

• Electronic spreadsheets can also be created Electronic spreadsheets can also be created to solve complex problems using both built-in to solve complex problems using both built-in functions, defined mathematical algorithms functions, defined mathematical algorithms and relationships. and relationships.


Compound InterestCompound InterestSolution Using a Excel Spreadsheet Built-in FormulaSolution Using a Excel Spreadsheet Built-in Formula

Determining the Future Value of $1,000 Determining the Future Value of $1,000 invested for forty years at 10%:invested for forty years at 10%:

1.1. Place cursor in cell on spreadsheetPlace cursor in cell on spreadsheet

2.2. Using the pull-down menu, choose, INSERT, FUNCTIONUsing the pull-down menu, choose, INSERT, FUNCTION

3.3. Choose financial functionsChoose financial functions

4.4. Choose FVChoose FV

5.5. Fill in the appropriate function arguments as follows:Fill in the appropriate function arguments as follows:

=FV (rate, nper, pmt, pv, type)=FV (rate, nper, pmt, pv, type)

=FV (0.10, 40, 0, 1000,0) which yields =FV (0.10, 40, 0, 1000,0) which yields → -45,259.26→ -45,259.26

(The answer is expressed as a negative because we entered the (The answer is expressed as a negative because we entered the investment as a positive number. )investment as a positive number. )


Using Excel to Solve for FVUsing Excel to Solve for FV Built-in Formula Function Arguments and SolutionBuilt-in Formula Function Arguments and Solution


Compound InterestCompound Interest Underlying AssumptionsUnderlying Assumptions

Notice the compound interest assumptions that are Notice the compound interest assumptions that are embodied in the basic formula:embodied in the basic formula:

FVFV22 = = $1,000 × (1+k$1,000 × (1+k11) × (1+k) × (1+k22) ) FVFVnn= PV= PV00 × (1+k) × (1+k)nn

Assumptions:Assumptions:• The rate of interest does not change over the periods of The rate of interest does not change over the periods of

compound interestcompound interest• Interest is earned and reinvested at the end of each periodInterest is earned and reinvested at the end of each period• The principal remains invested over the life of the The principal remains invested over the life of the

investmentinvestment• The investment is started at time 0 (now) and we are The investment is started at time 0 (now) and we are

determining the compound value of the whole investment at determining the compound value of the whole investment at the end of some time period (t= 1, 2, 3, 4,…)the end of some time period (t= 1, 2, 3, 4,…)


Compound InterestCompound Interest Underlying Assumptions – Timing of Cash FlowsUnderlying Assumptions – Timing of Cash Flows

Time = 0 Time = 1 Time = 2

Time of Investment


Compound Interest FormulaCompound Interest Formula(For a single cash flow)(For a single cash flow)

FVFVnn=PV=PV00 (1+k) (1+k)nn

Where:Where:FVFVnn= the future value (sum of both interest and principal) of the = the future value (sum of both interest and principal) of the

investment at some time in the future investment at some time in the future PVPV00= the original principal invested= the original principal investedk= the rate of return earned on the investmentk= the rate of return earned on the investmentn = the time or number of periods the investment is allowed to grow n = the time or number of periods the investment is allowed to grow


CVIFCVIFk,nk,n(For a single cash flow)(For a single cash flow)

Tables of Compound Tables of Compound Value Interest Factors Value Interest Factors can be created:can be created:

Period 1% 2% 3% 4% 5% 6% 7%1 1.0100 1.0200 1.0300 1.0400 1.0500 1.0600 1.07002 1.0201 1.0404 1.0609 1.0816 1.1025 1.1236 1.14493 1.0303 1.0612 1.0927 1.1249 1.1576 1.1910 1.22504 1.0406 1.0824 1.1255 1.1699 1.2155 1.2625 1.31085 1.0510 1.1041 1.1593 1.2167 1.2763 1.3382 1.40266 1.0615 1.1262 1.1941 1.2653 1.3401 1.4185 1.50077 1.0721 1.1487 1.2299 1.3159 1.4071 1.5036 1.60588 1.0829 1.1717 1.2668 1.3686 1.4775 1.5938 1.71829 1.0937 1.1951 1.3048 1.4233 1.5513 1.6895 1.838510 1.1046 1.2190 1.3439 1.4802 1.6289 1.7908 1.9672

6289.1

)05.1( 1010%,5

yearsnkCVIF



The table shows that the longer you invest…the greater the The table shows that the longer you invest…the greater the amount of money you will accumulate.amount of money you will accumulate.

It also shows that you are better off investing at higher rates It also shows that you are better off investing at higher rates of return. of return.

Period 1% 2% 3% 4% 5% 6% 7%1 1.0100 1.0200 1.0300 1.0400 1.0500 1.0600 1.07002 1.0201 1.0404 1.0609 1.0816 1.1025 1.1236 1.14493 1.0303 1.0612 1.0927 1.1249 1.1576 1.1910 1.22504 1.0406 1.0824 1.1255 1.1699 1.2155 1.2625 1.31085 1.0510 1.1041 1.1593 1.2167 1.2763 1.3382 1.40266 1.0615 1.1262 1.1941 1.2653 1.3401 1.4185 1.50077 1.0721 1.1487 1.2299 1.3159 1.4071 1.5036 1.60588 1.0829 1.1717 1.2668 1.3686 1.4775 1.5938 1.71829 1.0937 1.1951 1.3048 1.4233 1.5513 1.6895 1.838510 1.1046 1.2190 1.3439 1.4802 1.6289 1.7908 1.9672



How long does it take to double or triple your investment? At How long does it take to double or triple your investment? At

5%...at 10%?5%...at 10%?

Period 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%1 1.0100 1.0200 1.0300 1.0400 1.0500 1.0600 1.0700 1.0800 1.0900 1.10002 1.0201 1.0404 1.0609 1.0816 1.1025 1.1236 1.1449 1.1664 1.1881 1.21003 1.0303 1.0612 1.0927 1.1249 1.1576 1.1910 1.2250 1.2597 1.2950 1.33104 1.0406 1.0824 1.1255 1.1699 1.2155 1.2625 1.3108 1.3605 1.4116 1.46415 1.0510 1.1041 1.1593 1.2167 1.2763 1.3382 1.4026 1.4693 1.5386 1.61056 1.0615 1.1262 1.1941 1.2653 1.3401 1.4185 1.5007 1.5869 1.6771 1.77167 1.0721 1.1487 1.2299 1.3159 1.4071 1.5036 1.6058 1.7138 1.8280 1.94878 1.0829 1.1717 1.2668 1.3686 1.4775 1.5938 1.7182 1.8509 1.9926 2.14369 1.0937 1.1951 1.3048 1.4233 1.5513 1.6895 1.8385 1.9990 2.1719 2.357910 1.1046 1.2190 1.3439 1.4802 1.6289 1.7908 1.9672 2.1589 2.3674 2.593711 1.1157 1.2434 1.3842 1.5395 1.7103 1.8983 2.1049 2.3316 2.5804 2.853112 1.1268 1.2682 1.4258 1.6010 1.7959 2.0122 2.2522 2.5182 2.8127 3.138413 1.1381 1.2936 1.4685 1.6651 1.8856 2.1329 2.4098 2.7196 3.0658 3.452314 1.1495 1.3195 1.5126 1.7317 1.9799 2.2609 2.5785 2.9372 3.3417 3.797515 1.1610 1.3459 1.5580 1.8009 2.0789 2.3966 2.7590 3.1722 3.6425 4.177216 1.1726 1.3728 1.6047 1.8730 2.1829 2.5404 2.9522 3.4259 3.9703 4.5950


The Rule of 72The Rule of 72

• If you don’t have access to time value of money tables or a If you don’t have access to time value of money tables or a financial calculator but want to know how long it takes for financial calculator but want to know how long it takes for your money to double…use the rule of 72!your money to double…use the rule of 72!

years 164.5

72

:in double it willmoney your on rate 4.5% aearn expect toyou If

rateinterest compound Annual

72 double toyears ofNumber



Let us predict what happens with an investment if it is invested Let us predict what happens with an investment if it is invested at 5% …show the accumulated value after t=1, t=2, t=3, at 5% …show the accumulated value after t=1, t=2, t=3, etc. etc. Period 1% 2% 3% 4% 5%

1 1.0100 1.0200 1.0300 1.0400 1.05002 1.0201 1.0404 1.0609 1.0816 1.10253 1.0303 1.0612 1.0927 1.1249 1.15764 1.0406 1.0824 1.1255 1.1699 1.21555 1.0510 1.1041 1.1593 1.2167 1.27636 1.0615 1.1262 1.1941 1.2653 1.34017 1.0721 1.1487 1.2299 1.3159 1.40718 1.0829 1.1717 1.2668 1.3686 1.47759 1.0937 1.1951 1.3048 1.4233 1.551310 1.1046 1.2190 1.3439 1.4802 1.6289

FV

0.0000

0.2000

0.4000

0.6000

0.8000

1.0000

1.2000

1.4000

1.6000

1.8000

1 2 3 4 5 6 7 8 9 10

Year

FV

of

$1.0

0



Let us predict what happens with an investment if it is invested at Let us predict what happens with an investment if it is invested at 5% and 10% …show the accumulated value after t=1, t=2, t=3, 5% and 10% …show the accumulated value after t=1, t=2, t=3, etc. etc.

Period 5% 10%1 1.0500 1.10002 1.1025 1.21003 1.1576 1.33104 1.2155 1.46415 1.2763 1.61056 1.3401 1.77167 1.4071 1.94878 1.4775 2.14369 1.5513 2.357910 1.6289 2.5937

Future Value

0.0000

1.0000

2.0000

3.0000

4.0000

5.0000

6.0000

7.0000

8.0000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Time

FV

of

$1

.00

Notice: compound interest creates an exponential curve and there will be a substantial difference over the long term when you can earn higher rates of return.

Notice: compound interest creates an exponential curve and there will be a substantial difference over the long term when you can earn higher rates of return.

Types of Problems in Compounding Types of Problems in Compounding

Time Value of Money SkillsTime Value of Money Skills


Types of Compounding ProblemsTypes of Compounding Problems

• There are really only four different things you can be There are really only four different things you can be asked to find using this basic equation:asked to find using this basic equation:


– Find the initial amount of money to invest (PVFind the initial amount of money to invest (PV00))– Find the Future value (FVFind the Future value (FVnn))– Find the rate (k)Find the rate (k)– Find the time (n)Find the time (n)


Types of Compounding ProblemsTypes of Compounding ProblemsSolving for the Rate (k)Solving for the Rate (k)

• Your have asked your father for a loan of $10,000 to get Your have asked your father for a loan of $10,000 to get you started in a business. You promise to repay him you started in a business. You promise to repay him $20,000 in five years time.$20,000 in five years time.

• What compound rate of return are you offering to pay?What compound rate of return are you offering to pay?• This is an ex ante calculation.This is an ex ante calculation.

FVFVtt=PV=PV00 (1+k) (1+k)nn

$20,000= $10,000 (1+r)$20,000= $10,000 (1+r)55

2=(1+r)2=(1+r)55

221/51/5=1+r=1+r1.14869=1+r1.14869=1+rr = 14.869%r = 14.869%


Types of Compounding ProblemsTypes of Compounding ProblemsSolving for Time (n) or Holding PeriodsSolving for Time (n) or Holding Periods

• You have $150,000 in your RRSP (Registered Retirement You have $150,000 in your RRSP (Registered Retirement Savings Plan). Assuming a rate of 8%, how long will it take Savings Plan). Assuming a rate of 8%, how long will it take to have the plan grow to a value of $300,000?to have the plan grow to a value of $300,000?– This is an ex ante calculationThis is an ex ante calculation

FVFVtt=PV=PV00(1+k)(1+k)nn

$300,000= $150,000 (1+.08)$300,000= $150,000 (1+.08)nn

2=(1.08)2=(1.08)nn

ln 2 =ln 1.08 ln 2 =ln 1.08 × n× n0.69314 = .07696 0.69314 = .07696 × × nn

t = 0.69314 / .076961041 = 9.00 yearst = 0.69314 / .076961041 = 9.00 years


Types of Compounding ProblemsTypes of Compounding ProblemsSolving for Time (n) – using logarithmsSolving for Time (n) – using logarithms

• You have $150,000 in your RRSP (Registered Retirement You have $150,000 in your RRSP (Registered Retirement Savings Plan). Assuming a rate of 8%, how long will it take Savings Plan). Assuming a rate of 8%, how long will it take to have the plan grow to a value of $300,000?to have the plan grow to a value of $300,000?– This is an ex ante calculation.This is an ex ante calculation.


$300,000= $150,000 (1+.08)$300,000= $150,000 (1+.08)nn

2=(1.08)2=(1.08)nn

log 2 =log 1.08 log 2 =log 1.08 × n× n0.301029995 = 0.033423755 0.301029995 = 0.033423755 × × nn

t = 9.00 yearst = 9.00 years


Types of Compounding ProblemsTypes of Compounding ProblemsSolving for the Future Value (FVSolving for the Future Value (FVnn))

• You have $650,000 in your pension plan today. Because You have $650,000 in your pension plan today. Because you have retired, you and your employer will not make any you have retired, you and your employer will not make any further contributions to the plan. However, you don’t plan further contributions to the plan. However, you don’t plan to take any pension payments for five more years so the to take any pension payments for five more years so the principal will continue to grow.principal will continue to grow.

• Assuming a rate of 8%, forecast the value of your pension Assuming a rate of 8%, forecast the value of your pension plan in 5 years.plan in 5 years.– This is an ex ante calculation.This is an ex ante calculation.


FVFV55= $650,000 (1+.08)= $650,000 (1+.08)55

FVFV55 = $650,000 = $650,000 × 1.469328077× 1.469328077

FVFV55 = $955,063.25 = $955,063.25


Types of Compounding ProblemsTypes of Compounding ProblemsFinding the amount of money to invest (PVFinding the amount of money to invest (PV00))

• You hope to save for a down payment on a home. You hope to save for a down payment on a home. You hope to have $40,000 in four years time; You hope to have $40,000 in four years time; determine the amount you need to invest now at 6%determine the amount you need to invest now at 6%– This is a process known as discountingThis is a process known as discounting– This is an ex ante calculationThis is an ex ante calculation


$40,000= PV$40,000= PV00 (1.1) (1.1)44

PVPV00 = $40,000/1.4641=$27,320.53 = $40,000/1.4641=$27,320.53


Compound InterestCompound Interest Discounting (Computing Present Values)Discounting (Computing Present Values)

1

1

)1(0 nnnn

k)(FV

k

FVPV

[ 5-3]

AnnuitiesAnnuities

Time Value of Money ConceptsTime Value of Money Concepts


AnnuityAnnuity

• An annuity is a finite series of equal and An annuity is a finite series of equal and periodic cash flows.periodic cash flows.


Annuities and PerpetuitiesAnnuities and Perpetuities Ordinary Annuity FormulaOrdinary Annuity Formula

)1(

11

0

kk

PMTPVn

[ 5-5]


Ordinary AnnuityOrdinary AnnuityInvolve end-of-period payments – First cash flow occurs at n=1Involve end-of-period payments – First cash flow occurs at n=1

An annuity is a finite series of equal and periodic cash flows

where PMT1=PMT2=PMT3=…=PMTn

An annuity is a finite series of equal and periodic cash flows

where PMT1=PMT2=PMT3=…=PMTn

Time = n

PMTn

Time = 0

Time of Investment n=0

Time = 1

PMT1

Time = 2

PMT2

Time = 3

PMT3


Future Value of An Ordinary AnnuityFuture Value of An Ordinary Annuity

• An example of a compound annuity would be An example of a compound annuity would be where you save an equal sum of money in where you save an equal sum of money in each period over a period of time to each period over a period of time to accumulate a future sum.accumulate a future sum.


Annuities and PerpetuitiesAnnuities and Perpetuities Ordinary AnnuitiesOrdinary Annuities

Compound Value Annuity Formula (CVAF)Compound Value Annuity Formula (CVAF)

11

PMT(CVAF) k

k)(PMTFV

n

n

[ 5-4]


Future Value of An AnnuityFuture Value of An Annuity

Example:

How much will you have at the end of three years if you save $1,000 each year for three years at a rate of 10%?

FV3 = $1,000 × {[(1.1)3 - 1].1} =$1,000 × 3.31 = $3,310

11

k

k)(PMTFV

n

n


Future Value of An AnnuityFuture Value of An Annuity

Example:

How much will you have at the end of three years if you save $1,000 each year for three years at a rate of 10%?

FV3 = $1,000 × {[(1.1)3 - 1] / .1} =$1,000 × 3.31 = $3,310

What does the formula assume?

$1,0001 × (1.1) × (1.1) = $1,210

+ $1,0002 × (1.1) = $1,100

+ $1,0003 = $1,000

Sum = = $3,310


Future Value of An AnnuityFuture Value of An AnnuityAssumptionsAssumptions

FVA3 = $1,000 × {[(1.1)3 - 1].1} =$1,000 × 3.31 = $3,310

What does the formula assume?

$1,0001 × (1.1) × (1.1) = $1,210

+ $1,0002 × (1.1) = $1,100

+ $1,0003 = $1,000

Sum = = $3,310

The CVAF assumes that time zero (t=0) (today) you decide to invest, but you don’t make the first investment until one year from today. The Future Value you forecast is the value of the entire fund (a series of investments together with the accumulated interest) at the end of some year n = 1 or n = 2 …in this case n = 3. NOTE: the rate of interest is assumed to remain unchanged throughout the forecast period.

If these assumptions

don’t hold…you can’t use the

formula.


Adjusting your solution to the Adjusting your solution to the circumstances of the problemcircumstances of the problem

• The time value of money formula can be applied to The time value of money formula can be applied to any situation…what you need to do is to understand any situation…what you need to do is to understand the assumptions underlying the formula…then adjust the assumptions underlying the formula…then adjust your approach to match the problem you are trying to your approach to match the problem you are trying to solve.solve.

• In the foregoing problem…ít isn’t too logical to start a In the foregoing problem…ít isn’t too logical to start a savings program…and then not make the first savings program…and then not make the first investment until one year later!!!investment until one year later!!!


Example of AdjustmentExample of Adjustment(An Annuity Due)(An Annuity Due)

You plan to invest $1,000 today, $1,000 one You plan to invest $1,000 today, $1,000 one year from today and $1,000 two years from year from today and $1,000 two years from today.today.

What sum of money will you accumulate at What sum of money will you accumulate at time 3 if your money is assumed to earn 10%.time 3 if your money is assumed to earn 10%.

This is known as an annuity due rather than a regular annuity.This is known as an annuity due rather than a regular annuity.


Annuity DueAnnuity DueFirst cash flow occurs at n=0First cash flow occurs at n=0

An annuity due is a finite series of equal and periodic cash

flows where PMT1=PMT2=PMT3=…=PMTn but the first payment occurs at time=0.

An annuity due is a finite series of equal and periodic cash

flows where PMT1=PMT2=PMT3=…=PMTn but the first payment occurs at time=0.

Time = n

PMTn

Time = 0 Time = 1

PMT1

Time = 2

PMT2

Time = 3

PMT3 No PMT


Example of AdjustmentExample of AdjustmentAn Annuity DueAn Annuity Due

You plan to invest $1,000 today, $1,000 one year from today and $1,000 two You plan to invest $1,000 today, $1,000 one year from today and $1,000 two years from today.years from today.What sum of money will you accumulate in three years if your money is What sum of money will you accumulate in three years if your money is assumed to earn 10%.assumed to earn 10%.

You should know that there is a simple way of adjusting a normal You should know that there is a simple way of adjusting a normal annuity to become an annuity due…just multiply the normal annuity to become an annuity due…just multiply the normal annuity result by (1+k) and you will convert to an annuity due!annuity result by (1+k) and you will convert to an annuity due!

FVFV3 3 (Annuity due)= $1,000 × {[(1.1)3 - 1].1}(Annuity due)= $1,000 × {[(1.1)3 - 1].1}× (1+ k)× (1+ k)

=$1,000 × 3.31 =$1,000 × 3.31 × 1.1× 1.1

= $3,310 = $3,310 × 1.1 = $3,641× 1.1 = $3,641

$1,0001 × (1.1) × (1.1) × (1.1) = $1,331+ $1,0002 × (1.1) × (1.1) = $1,210+ $1,0003 × (1.1) = $1,100Sum = = $3,641


Annuities and PerpetuitiesAnnuities and Perpetuities Future Value of an Annuity Due FormulaFuture Value of an Annuity Due Formula

)111

k (k

k)(PMTFV

n

n

[ 5-6]


Annuities and PerpetuitiesAnnuities and Perpetuities Present Value of an Annuity DuePresent Value of an Annuity Due

k)(1 )1(

11

0

kk

PMTPVn

[ 5-7]

Discounting Cash FlowsDiscounting Cash Flows

Time Value of Money …Time Value of Money …


What is Discounting?What is Discounting?

• Discounting is the inverse of compounding.Discounting is the inverse of compounding.

nnk

nk kCVIFPVIF

)1(

11

,,


Example of DiscountingExample of Discounting

You will receive $10,000 one year from today. If you had You will receive $10,000 one year from today. If you had the money today, you could earn 8% on it.the money today, you could earn 8% on it.

What is the present value of $10,000 received one year What is the present value of $10,000 received one year from now at 8%?from now at 8%?

PVPV00=FV=FV11 × PVIF× PVIFk,n k,n = $10,000 × (1/ 1.08= $10,000 × (1/ 1.0811))PVPV00 = $10,000 × 0.9259 = $9,259.26 = $10,000 × 0.9259 = $9,259.26

NOTICE: A present value is always less than the absolute value NOTICE: A present value is always less than the absolute value of the cash flow unless there is no time value of money. If there of the cash flow unless there is no time value of money. If there is no rate of interest then PV = FVis no rate of interest then PV = FV


PVIFPVIFk,nk,n(For a single cash flow)(For a single cash flow)

Tables of present value interest factors can be created:Tables of present value interest factors can be created:

Period 1% 2% 3% 4% 5% 6% 7%1 0.9901 0.9804 0.9709 0.9615 0.9524 0.9434 0.93462 0.9803 0.9612 0.9426 0.9246 0.9070 0.8900 0.87343 0.9706 0.9423 0.9151 0.8890 0.8638 0.8396 0.81634 0.9610 0.9238 0.8885 0.8548 0.8227 0.7921 0.76295 0.9515 0.9057 0.8626 0.8219 0.7835 0.7473 0.71306 0.9420 0.8880 0.8375 0.7903 0.7462 0.7050 0.66637 0.9327 0.8706 0.8131 0.7599 0.7107 0.6651 0.62278 0.9235 0.8535 0.7894 0.7307 0.6768 0.6274 0.58209 0.9143 0.8368 0.7664 0.7026 0.6446 0.5919 0.543910 0.9053 0.8203 0.7441 0.6756 0.6139 0.5584 0.5083

nnk kPVIF

)1(

1,



Notice – the farther away the receipt of the cash flow from today…Notice – the farther away the receipt of the cash flow from today…the lower the present value…the lower the present value…

Notice – the higher the rate of interest…the lower the present value.Notice – the higher the rate of interest…the lower the present value.

Period 1% 2% 3% 4% 5% 6% 7%1 0.9901 0.9804 0.9709 0.9615 0.9524 0.9434 0.93462 0.9803 0.9612 0.9426 0.9246 0.9070 0.8900 0.87343 0.9706 0.9423 0.9151 0.8890 0.8638 0.8396 0.81634 0.9610 0.9238 0.8885 0.8548 0.8227 0.7921 0.76295 0.9515 0.9057 0.8626 0.8219 0.7835 0.7473 0.71306 0.9420 0.8880 0.8375 0.7903 0.7462 0.7050 0.66637 0.9327 0.8706 0.8131 0.7599 0.7107 0.6651 0.62278 0.9235 0.8535 0.7894 0.7307 0.6768 0.6274 0.58209 0.9143 0.8368 0.7664 0.7026 0.6446 0.5919 0.543910 0.9053 0.8203 0.7441 0.6756 0.6139 0.5584 0.5083

5083.0)07.1(

11010%,7

nkPVIF



If someone offers to pay you a sum 50 or 60 years hence…that If someone offers to pay you a sum 50 or 60 years hence…that promise is ‘pretty-much’ worthless!!!promise is ‘pretty-much’ worthless!!!

nnk kPVIF

)1(

1,

Period 5% 10% 15% 20% 25% 30% 35%60 0.0535 0.0033 0.0002 0.0000 0.0000 0.0000 0.000070 0.0329 0.0013 0.0001 0.0000 0.0000 0.0000 0.000080 0.0202 0.0005 0.0000 0.0000 0.0000 0.0000 0.000090 0.0124 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000

100 0.0076 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000110 0.0047 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

The present value of $10 million promised 100 years fromtoday at a 10% discount rate is = $10,000,000 * 0.0001 = $1,000!!!!

The Reinvestment Rate The Reinvestment Rate



The Nature of Compound InterestThe Nature of Compound Interest

• When we assume compound interest, we are implicitly When we assume compound interest, we are implicitly assuming that any credited interest is reinvested in the assuming that any credited interest is reinvested in the next period, hence, the growth of the fund is a function of next period, hence, the growth of the fund is a function of interest on the principal, and a growing interest upon interest on the principal, and a growing interest upon interest stream….interest stream….

• This principal is demonstrated when we invest $10,000 at This principal is demonstrated when we invest $10,000 at 8% per annum over a period of say 4 years…the future 8% per annum over a period of say 4 years…the future value of this investment can be decomposed as follows...value of this investment can be decomposed as follows...


FVFV44 of $10,000 @ 8% of $10,000 @ 8%

Rate of Interest = 8.00%

Time

Principal at Beginning of the Year Interest

End of Period Value of the

Fund (Principal plus Interest)

1 $10,000.00 $800.00 $10,800.002 $10,800.00 $864.00 $11,664.003 $11,664.00 $933.12 $12,597.124 $12,597.12 $1,007.77 $13,604.89

Of course we can find the answer using the formula:

FV4 =$10,000(1+.08)4

=$10,000(1.36048896)

=$13,604.89


Annuity AssumptionsAnnuity Assumptions

• When using the unadjusted formula or table When using the unadjusted formula or table values for annuities (whether future value or values for annuities (whether future value or present value) we always assume:present value) we always assume:– the focal point is time 0the focal point is time 0– the first cash flow occurs at time 1the first cash flow occurs at time 1– intermediate cash flows are reinvested at the rate of interest for intermediate cash flows are reinvested at the rate of interest for

the remaining time periodthe remaining time period– the interest rate is unchanging over the period of the analysis.the interest rate is unchanging over the period of the analysis.


FV of an Annuity DemonstratedFV of an Annuity Demonstrated

When determining the Future Value of an Annuity…we assume we are standing at time zero, the first cash flow will occur at the end of the year and we are trying to determine the accumulated future value of a series of five equal and periodic payments as demonstrated in the following time line...

0 1 2 3 4 5

$2,000 $2,000 $2,000 $2,000 $2,000



We could be trying find out how much we would accumulate in a savings fund…if we saved $2,000 per year for five years at 8%…but we won’t make the first deposit in the fund for one year...

0 1 2 3 4 5

$2,000 $2,000 $2,000 $2,000 $2,000



The time value of money formula assumes that each payment will be invested at the going rate of interest for the remaining time to maturity….

This final $2,000 is contributed to the fund, but is assumed not to earn any interest.

$2,000 invested at 8% for 4 years



$2,000 invested at 8% for 1 year

0 1 2 3 4 5

$2,000 $2,000 $2,000 $2,000 $2,000



Annuity Assumptions: A demonstration

- focal point is time zero- the first cash flow occurs at time one

Future value of a $2,000 annuity at the end of five years at 8%:

Time Cashflow CVIF Future Value

01 $2,000 1.3605 $2,720.982 $2,000 1.2597 $2,519.423 $2,000 1.1664 $2,332.804 $2,000 1.0800 $2,160.005 $2,000 1.0000 $2,000.00

Future Value of Annuity = FV(5) $11,733.20

CVIF for 4 years at 8% (4 years is the remaining time to maturity.)

Notice that the final cashflow is just received, it doesn't receive any interest.





You can always discount or compound the individual cash flows…however it may be quicker to use an annuity formula.

Future value of a $2,000 annuity at the end of five years at 8%:

Time Cashflow CVIF Future Value

01 $2,000 1.3605 $2,720.982 $2,000 1.2597 $2,519.423 $2,000 1.1664 $2,332.804 $2,000 1.0800 $2,160.005 $2,000 1.0000 $2,000.00

Future Value of Annuity = FV(5) $11,733.20

Using the formula: FV(5) = PMT(CVAF t=5, r=8%) = $2,000 [(((1 + r)t)-1) / r] = $2,000(5.8666) = $11,733.20

CVIF for 4 years at 8% (4 years is the remaining time to maturity.)

Notice that the final cashflow is just received, it doesn't receive any interest.



In summary the assumptions are:In summary the assumptions are:

– focal point is time zerofocal point is time zero– we assume the cash flows occur at the end of every we assume the cash flows occur at the end of every

yearyear– we assume the interest rate does not change during we assume the interest rate does not change during

the forecast periodthe forecast period– the interest received is reinvested at that same rate of the interest received is reinvested at that same rate of

interest for the remaining time until maturity.interest for the remaining time until maturity.


PV of an Annuity DemonstratedPV of an Annuity Demonstrated



You can always discount or compound the individual cash flows…however it may be quicker to use an annuity formula.

Present value of a five year $2,000 annual annuity at 8%:

Time Cashflow PVIF Present Value

01 $2,000 0.9259 $1,851.852 $2,000 0.8573 $1,714.683 $2,000 0.7938 $1,587.664 $2,000 0.7350 $1,470.065 $2,000 0.6806 $1,361.17

Present Value of Annuity = $7,985.42

Using the formula: PV = PMT(PVIFA n=5, k=85) = $2,000 [1- 1/(1 + k)n] / k = $2,000(3.9927) = $7,985.40

PVIF for 1 year at 8%


The Reinvestment Rate AssumptionThe Reinvestment Rate Assumption

• It is crucial to understand the reinvestment rate assumption It is crucial to understand the reinvestment rate assumption that is built-in to the time value of money.that is built-in to the time value of money.

• Obviously, when we forecast, we must make assumptions…Obviously, when we forecast, we must make assumptions…however, if that assumption not realistic…it is important however, if that assumption not realistic…it is important that we take it into account.that we take it into account.

• This reinvestment rate assumption in particular, is This reinvestment rate assumption in particular, is important in the important in the yield-to-maturityyield-to-maturity calculations in bonds… calculations in bonds…and in the and in the Internal Rate of ReturnInternal Rate of Return (IRR) calculation in (IRR) calculation in capital budgeting.capital budgeting.

PerpetuitiesPerpetuities



PerpetuitiesPerpetuities PerpetuitiesPerpetuities

• A perpetuity is an infinite annuityA perpetuity is an infinite annuity• An infinite series of payments where each An infinite series of payments where each

payment is equal and periodic.payment is equal and periodic.• Examples of perpetuities in financial markets Examples of perpetuities in financial markets

includes:includes:– Common stock (with a no growth in dividend Common stock (with a no growth in dividend

assumption)assumption)– Preferred stockPreferred stock– Consol bonds (bonds with no maturity date)Consol bonds (bonds with no maturity date)


PerpetuityPerpetuityInvolve end-of-period payments – First cash flow occurs at n=1Involve end-of-period payments – First cash flow occurs at n=1

A perpetuity is an infinite series of equal and periodic cash

flows where PMT1=PMT2=PMT3=…=PMTα

A perpetuity is an infinite series of equal and periodic cash

flows where PMT1=PMT2=PMT3=…=PMTα

Time = α

PMTα

Time = 0

Time of Investment n=0

Time = 1

PMT1

Time = 2

PMT2

Time = 3

PMT3


PerpetuitiesPerpetuities Perpetuity FormulaPerpetuity Formula

0 k

PMTPV [ 5-8]

Where:

PV0 = Present value of the perpetuity

PMT = the periodic cash flow

k = the discount rate


Perpetuity: An ExamplePerpetuity: An Example

While acting as executor for a distant relative, While acting as executor for a distant relative, you discover a $1,000 Consol Bond issued by you discover a $1,000 Consol Bond issued by Great Britain in 1814, issued to help fund the Great Britain in 1814, issued to help fund the Napoleonic War. If the bond pays annual interest Napoleonic War. If the bond pays annual interest of 3.0% and other long U.K. Government bonds of 3.0% and other long U.K. Government bonds are currently paying 5%, what would each $1,000 are currently paying 5%, what would each $1,000 Consol Bond sell for in the market?Consol Bond sell for in the market?


Perpetuity: SolutionPerpetuity: Solution

0

$1,000 0.03

0.05$30

0.05$600

PMTPV

k

Nominal Versus Effective RatesNominal Versus Effective Rates



Nominal Versus Effective Interest RatesNominal Versus Effective Interest Rates

• So far, we have assumed annual So far, we have assumed annual compoundingcompounding

• When rates are compounded annually, the When rates are compounded annually, the quoted rate and the effective rate are equalquoted rate and the effective rate are equal

• As the number of compounding periods per As the number of compounding periods per year increases, the effective rate will become year increases, the effective rate will become larger than the quoted ratelarger than the quoted rate


Nominal versus Effective RatesNominal versus Effective Rates General Formula for Effective Annual Rate General Formula for Effective Annual Rate

1)1( m

m

QRk[ 5-9]


Calculating the Effective RateCalculating the Effective Rate

1 1m

Effective

QRk

m

Where:

kEffective = Effective annual interest rate

QR = the quoted interest rate

M = the number of compounding periods per year


Example: Effective Rate CalculationExample: Effective Rate Calculation

• A bank is offering loans at 6%, compounded monthly. A bank is offering loans at 6%, compounded monthly. What is the effective annual interest rate on their What is the effective annual interest rate on their loans?loans?

12

1 1

.061 1

12

6.17%

m

Effective

QRk

m


Nominal versus Effective RatesNominal versus Effective Rates Continuous Compounding FormulaContinuous Compounding Formula

1 QRek[ 5-10]


Continuous CompoundingContinuous Compounding

• When compounding occurs continuously, we When compounding occurs continuously, we calculate the effective annual rate using e, calculate the effective annual rate using e, the base of the natural logarithms the base of the natural logarithms (approximately 2.7183)(approximately 2.7183)

1QREffectivek e


10% Compounded At Various Frequencies10% Compounded At Various Frequencies

Compounding Compounding FrequencyFrequency

Effective Annual Effective Annual Interest RateInterest Rate

22 10.25%10.25%

44 10.3813%10.3813%

1212 10.4713%10.4713%

5252 10.5065%10.5065%

365365 10.5156%10.5156%

ContinuousContinuous 10.5171%10.5171%


Calculating the Quoted RateCalculating the Quoted Rate

• If we know the effective annual interest rate, (kIf we know the effective annual interest rate, (kEffEff) ) and we know the number of compounding periods, and we know the number of compounding periods, (m) we can solve for the Quoted Rate, as follows:(m) we can solve for the Quoted Rate, as follows:

1

1 1mEffQR k m


When Payment & Compounding Periods DifferWhen Payment & Compounding Periods Differ

• When the number of payments per year is When the number of payments per year is different from the number of compounding different from the number of compounding periods per year, you must calculate the periods per year, you must calculate the interest rate per payment period, using the interest rate per payment period, using the following formula following formula

1 1

m

f

PerPeriod

QRk

m

Where:f = the payment frequency per year


Nominal versus Effective RatesNominal versus Effective Rates Formula for Effective Rates for “Any” PeriodFormula for Effective Rates for “Any” Period

11 -)m

QR(k f

m

[ 5-11]

Loans and Loan Amortization TablesLoans and Loan Amortization Tables



Loan AmortizationLoan Amortization

– A blended payment loan is repaid in equal periodic A blended payment loan is repaid in equal periodic paymentspayments

– However, the amount of principal and interest varies However, the amount of principal and interest varies each periodeach period

– Assume that we want to calculate an amortization Assume that we want to calculate an amortization table showing the amount of principal and interest table showing the amount of principal and interest paid each period for a $5,000 loan at 10% repaid in paid each period for a $5,000 loan at 10% repaid in three equal annual instalments. three equal annual instalments.


Blended Interest and Principal Loan Blended Interest and Principal Loan Payments - formulaPayments - formula

kk)(1

11

PMT Principal

)PMT(PVAFPrincipal

n

nk,

Where:

Pmt = the fixed periodic payment

t= the amortization period of the loan

r = the rate of interest on the loan


Blended Interest and Principal Loan Blended Interest and Principal Loan Payments - examplePayments - example

52.018,1$818147.9

000,10$Pmt

.08)08.1(

11

Pmt000,10$

r)1(

11

PMT Principal

20

nk

Where:

Pmt = unknown

t= 20 years

r = 8%

Calculator Approach:10,000 PV

0 FV20 N8 I/YCPT PMT $1,018.52


How are Loan Amortization Tables Used?How are Loan Amortization Tables Used?

• To separate the loan repayments into their constituent To separate the loan repayments into their constituent components.components.– Each level payment is made of interest plus a repayment of Each level payment is made of interest plus a repayment of

some portion of the principal outstanding on the loan.some portion of the principal outstanding on the loan.– This is important to do when the loan has been taken out for the This is important to do when the loan has been taken out for the

purposes of earning taxable income…as a result, the interest is purposes of earning taxable income…as a result, the interest is a tax-deductible expense.a tax-deductible expense.


Loan Amortization TablesLoan Amortization TablesUsing an Excel SpreadsheetUsing an Excel Spreadsheet

Principal = $100,000

Rate = 8.0%

Term = 5

PVAF = 3.99271

Payment = $25,045.65

Retired Ending

Year Principal Interest Payment Principal Balance

1 100,000.00 8,000.00 25,045.65 17,045.65 82,954.35

2 82,954.35 6,636.35 25,045.65 18,409.30 64,545.06

3 64,545.06 5,163.60 25,045.65 19,882.04 44,663.02

4 44,663.02 3,573.04 25,045.65 21,472.60 23,190.41

5 23,190.41 1,855.23 25,045.65 23,190.41 0.00


Loan or Mortgage ArrangementsLoan or Mortgage Arrangements Effective Rate for Any Period FormulaEffective Rate for Any Period Formula

11 -)m

QR(k f

m

Eff [ 5-11]


Loan AmortizationLoan Amortization Example with SolutionExample with Solution

• First calculate the annual paymentsFirst calculate the annual payments

3

1 1

1 1

5,000

1 1.10

0.10

$2,010.57

n

Annuity

Annuity

n

kPV PMT

k

PVPMT

k

k


0 FV3 N10 I/YCPT PMT $2,010.57


Amortization TableAmortization Table

Period Principal: Start of Period

Payment

Interest Principal Principal:End of Period

1 5,000.00 2,010.57

500.00 1,510.57 3,489.43

2 3,489.43 2,010.57

348.94 1,661.63 1,827.80

3 1,827.80 2,010.57

182.78 1,827.78 0


Calculating the Balance O/SCalculating the Balance O/S

• At any point in time, the balance outstanding At any point in time, the balance outstanding on the loan (the principal not yet repaid) is on the loan (the principal not yet repaid) is the PV of the loan payments not yet made.the PV of the loan payments not yet made.

• For example, using the previous example, we For example, using the previous example, we can calculate the balance outstanding at the can calculate the balance outstanding at the end of the first year, as shown on the next end of the first year, as shown on the next pagepage


Calculating the Balance O/S after the 1Calculating the Balance O/S after the 1stst Year Year

1

2

1 1

1 1.102,010.57

.10

$3,489.42

n

t

kPV PMT

k


Canadian Residential MortgagesCanadian Residential Mortgages

• A Canadian residential mortgage is a loan A Canadian residential mortgage is a loan with one special featurewith one special feature– By law, banks in Canada can only compound the By law, banks in Canada can only compound the

interest twice per year on a conventional mortgage, interest twice per year on a conventional mortgage, but payments are typically made at least monthlybut payments are typically made at least monthly

• To solve for the payment, you must first To solve for the payment, you must first calculate the correct periodic interest ratecalculate the correct periodic interest rate


Canadian Residential MortgagesCanadian Residential Mortgages

• For example, suppose we want to calculate the For example, suppose we want to calculate the monthly payment on a $100,000 mortgage amortized monthly payment on a $100,000 mortgage amortized over 25 years with a 6% annual interest rate.over 25 years with a 6% annual interest rate.

• First, calculate the monthly interest rate:First, calculate the monthly interest rate:

2

12

1 1

.061 1

2

.004938622 0.4938622%

m

f

PerPeriod

QRk

m

or


Calculating the Monthly PaymentCalculating the Monthly Payment

• Now, calculate the monthly payment on the mortgageNow, calculate the monthly payment on the mortgage

0

0

300

1 1

1 1

100,000

1 1.004938622

.004938622

$639.81

n

t

tn

kPV PMT

k

PVPMT

k

k


0FV300 N.4938622 I/YCPT PMT $639.81


Monthly Mortgage Loan Amortization Monthly Mortgage Loan Amortization TableTable

Principal = $100,000

Quoted rate = 6.0%

Effective annual Rate = 6.090% (Assuming semi-annual compounding)

Effective monthly Rate = 0.49386%

Term = 25 years

Term in months = 300

PVAF = 156.297225

Payment = $639.81

Retired Ending

Month Principal Interest Payment Principal Balance

1 100,000.00 493.86 639.81 145.94 99,854.06

2 99,854.06 493.14 639.81 146.67 99,707.39

3 99,707.39 492.42 639.81 147.39 99,560.00

4 99,560.00 491.69 639.81 148.12 99,411.88

5 99,411.88 490.96 639.81 148.85 99,263.03


Summary and ConclusionsSummary and Conclusions

In this chapter you have learned:In this chapter you have learned:

– To compare cash flows that occur at different points in timeTo compare cash flows that occur at different points in time– To determine economically equivalent future values from values To determine economically equivalent future values from values

that occur in previous periods through compounding.that occur in previous periods through compounding.– To determine economically equivalent present values from cash To determine economically equivalent present values from cash

flows that occur in the future through discountingflows that occur in the future through discounting– To find present value and future values of annuities, andTo find present value and future values of annuities, and– To determine effective annual rates of return from quoted To determine effective annual rates of return from quoted

interest rates.interest rates.

Concept Review QuestionsConcept Review Questions



Concept Review Question 1Concept Review Question 1Quoted versus Effective RatesQuoted versus Effective Rates

Why can effective rates often be very Why can effective rates often be very different from quoted rates?different from quoted rates?

The more frequently interest is compounded the higher the The more frequently interest is compounded the higher the effective rate of return.effective rate of return.

Because financial institutions are legally only required to Because financial institutions are legally only required to quote APR (Annual Percentage Rates) that are stated quote APR (Annual Percentage Rates) that are stated (nominal) the published rate is often much lower than the (nominal) the published rate is often much lower than the actual rate charged depending on the frequency of actual rate charged depending on the frequency of compounding. compounding.

This is why reading the fine print is so important!This is why reading the fine print is so important!


Internet LinksInternet Links

• Planning tools and online courses throughPlanning tools and online courses through TD Canada TrustTD Canada Trust• Online tools and calculators throughOnline tools and calculators through RBC Royal BankRBC Royal Bank


CopyrightCopyright

Copyright © 2007 John Wiley & Copyright © 2007 John Wiley & Sons Canada, Ltd. All rights Sons Canada, Ltd. All rights reserved. Reproduction or reserved. Reproduction or translation of this work beyond that translation of this work beyond that permitted by Access Copyright (the permitted by Access Copyright (the Canadian copyright licensing Canadian copyright licensing agency) is unlawful. Requests for agency) is unlawful. Requests for further information should be further information should be addressed to the Permissions addressed to the Permissions Department, John Wiley & Sons Department, John Wiley & Sons Canada, Ltd.Canada, Ltd. The purchaser may The purchaser may make back-up copies for his or her make back-up copies for his or her own use only and not for distribution own use only and not for distribution or resale.or resale. The author and the The author and the publisher assume no responsibility publisher assume no responsibility for errors, omissions, or damages for errors, omissions, or damages caused by the use of these files or caused by the use of these files or programs or from the use of the programs or from the use of the information contained herein.information contained herein.

chapter 5 - time value of money

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