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The Time Value Of

Money

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2

Introduction

This module introduces the

concepts and skills necessary tounderstand the time value ofmoney and its applications.

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Learning Objectives

1. Construct cash flow timelines toorganize your analysis of time value ofmoney problems and learn three

techniques for solving time value ofmoney problems.

2. Understand compounding and calculate

the future value of cash flow usingmathematical formulas, financial tablesand an Excel worksheet.

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Learning Objectives (cont.)

3. Understand discounting and calculate thepresent value of cash flows usingmathematical formulas, financial tables

and an Excel spreadsheet.

4. Understand how interest rates are quoted

and how to make them comparable.

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Learning Objectives

5. Distinguish between an ordinary annuityand an annuity due, and calculatepresent and future values of each.

6. Calculate the present value of a levelperpetuity and a growing perpetuity.

7. Calculate the present and future valuesof complex cash flow streams.

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6

We know that receiving $1 today is worthmorethan $1 in the future. This is due toopportunity costs.

The opportunity cost of receiving $1 inthe future is the interest we could haveearned if we had received the $1 sooner.

Today Future

The Time Value Concept

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If we can measure this opportunity

cost, we can:

Translate $1 today into its equivalent in thefuture (compounding).

Translate $1 in the future into its equivalenttoday (discounting).

?

Today Future

Today

?

Future

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Using Timelines to Visualize

Cashflows A timeline identifies the timing andamount of a stream of cash flows alongwith the interest rate.

A timeline is typically expressed in years,but it could also be expressed as months,

days or any other unit of time.

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Time Line Example

i=10%

Years

Cash flow -$100 $30 $20 -$10 $50

The 4-year timeline illustrates the following:

The interest rate is 10%.

A cash outflow of $100 occurs at the beginning of the first year (attime 0), followed by cash inflows of $30 and $20 in years 1 and 2, acash outflow of $10 in year 3 and cash inflow of $50 in year 4.

0 1 2 3 4

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Checkpoint 5.1

Creating a Timeline

Suppose you lend a friend $10,000 today to help himfinance a new Jimmy Johns Sub Shop franchise and inreturn he promises to give you $12,155 at the end ofthe fourth year. How can one represent this as atimeline? Note that the interest rate is 5%.

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Checkpoint 5.1

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Checkpoint 5.1

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13

The Time Value of Money

Compounding and

Discounting Single Sums

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Simple Interest and Compound

Interest What is the difference between simple

interest and compound interest?

Simple interest: Interest is earned only on theprincipal amount.

Compound interest: Interest is earned on boththe principal and accumulated interest of priorperiods.

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Interest (cont.)

Example 5.1: Suppose that youdeposited $500 in your savings

account that earns 5% annualinterest. How much will you havein your account after two years

using (a) simple interest and (b)compound interest?

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Interest (cont.)

Simple Interest

Interest earned

= 5% of $500 = .05500 = $25 per year Total interest earned = $252 = $50

Balance in your savings account:

= Principal + accumulated interest = $500 + $50 = $550

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Interest (cont.) Compound interest

Interest earned in Year 1

= 5% of $500 = $25

Interest earned in Year 2

= 5% of ($500 + accumulated interest)

= 5% of ($500 + 25) = .05525 = $26.25

Balance in your savings account:

= Principal + interest earned

= $500 + $25 + $26.25 = $551.25

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Present Value and Future Value

Time value of money calculations involvePresent value(what a cash flow would beworth to you today) and Future value

(what a cash flow will be worth in thefuture).

In example 5.1, Present value is $500 andFuture value is $551.25.

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Copyright 2011 Pearson Prentice Hall. All rights reserved.

COMPOUNDING AND

FUTURE VALUE

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Future Value Equation

Value at the end of year n for a sumcompounded at interest rate i is:

FVn = PV0 (1 + i)n

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Future Value Equation (cont.)

We can apply equation 5-1a toexample 5.1

FV2 = PV(1+i)n

= 500(1.05)2

= $551.25

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Continue example 5.1 where youdeposited $500 in savings account

earning 5% annual interest. Showthe amount of interest earned forthe first five years and the value of

your savings at the end of fiveyears.

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YEAR PV orBeginning

Value

Interest Earned (5%) FV orEnding Value

1 $500.00 $500*.05 = $25 $525

2 $525.00 $525*.05 = $26.25 $551.25

3 $551.25 $551.25*.05=$27.56

$578.81

4 $578.81 $578.81*.05=$28.94 $607.75

5 $607.75 $607.75*.05=$30.39 $638.14

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We will obtain the same answer usingequation 5-1 i.e. FV = PV(1+i)n

= 500(1.05)5

= $638.14

So the balance in savings account at the endof 5 years will equal $638.14. The total

interest earned on the original principalamount of $500 will equal $138.14 (i.e.$638.14 minus $500.00).

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Tables have Three Variables

Interest factors (IF)

Time periods (n)

Interest rates per period (i)

If you know any two, you can solvealgebraically for the third variable.

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Solving By Table:

FV3

= PV0

(FVIF5%,5

)

= $500(1.2763)

= $638.15

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Power of Time

Figure 5.1 Future Value and Compound Interest Illustrated

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Power of Interest Rate

Figure 5.1 Future Value and Compound Interest Illustrated

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What Do We Learn from Panel B

and C of Figure 5-1?1. Power of Time: Future value of original

investment increases with time.However, there will be no change in

future value if the interest rate is equal tozero.

2. Power of Rate of Interest: An increase inrate of interest leads to an increase infuture value.

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Checkpoint 5.2 - Calculating the

Future Value of a Cash Flow

You are put in charge of managing yourfirms working capital. Your firm has

$100,000 in extra cash on hand anddecides to put it in a savings accountpaying 7% interest compounded annually.How much will you have in your accountin 10 years?

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Checkpoint 5.2

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Checkpoint 5.2

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Checkpoint 5.2

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FV Applications in Other Areas

Example 5.2 A DVD rental firm is currentlyrenting 8,000 DVDs per year. How manyDVDs will the firm be renting in 10 years if

the demand for DVD rentals is expected toincrease by 7% per year?

Using Equation 5-1a, FV = 8000(1.07)10 = $15,737.21

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Compound Interest with Shorter

Compounding Periods (cont.) Example 5.4 You invest $500 for seven years

to earn an annual interest rate of 8%, and theinvestment is compounded semi-annually.

What will be the future value of thisinvestment?

We will use equation 5-1b to solve theproblem.

This equation adjusts the number ofcompounding periods and interest rate toreflect the semi-annual compounding.

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Compound Interest with Shorter

Compounding Periods (cont.)

FV = PV(1+i/2)m*2

= 500(1+.08/2)7*2

= 500(1.7317)= $865.85

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Checkpoint 5.3

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Checkpoint 5.3: Check yourself

If you deposit $50,000 in an accountthat pays an annual interest rate of

10% compounded monthly, what willyour account balance be in 10years?

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DISCOUNTING AND

PRESENT VALUE

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The Key Question

What is value today of cash flow to bereceived in the future?

The answer to this question requirescomputing the present value i.e. the valuetoday of a future cash flow, and the process ofdiscounting, determining the present value of

an expected future cash flow.

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PV Equation (cont.)

Example 5.5 How much will $5,000 to bereceived in 10 years be worth today if theinterest rate is 7%?

PV = FV (1/(1+i)n )

= 5000 (1/(1.07)10)

= 5000 (.5083)= $2,541.50

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Impact of Interest Rates on PV

If the interest rate (or discount rate) ishigher (say 9%), the PV will be lower.

PV = 5000*(1/(1.09)10) = 5000*(0.4224)

=$2,112.00 If the interest rate (or discount rate) is

lower (say 2%), the PV will be higher.

PV = 5000*(1/(1.02)

10

) = 5000*(0.8203)= $4,101.50

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Checkpoint 5.4 - Solving for the

Present Value of a Future Cash Flow

Your firm has just sold a piece ofproperty for $500,000, but under the salesagreement, it wont receive the $500,000until ten years from today. What is thepresent value of $500,000 to be receivedten years from today if the discount rate

is 6% annually?

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Checkpoint 5.4

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Checkpoint 5.4

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Checkpoint 5.4:Check Yourself

What is the present value of $100,000 to bereceived at the end of 25 years given a 5%discount rate?

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Solving for the Number of Periods

Key Question: How long will it take toaccumulate a specific amount in thefuture?

It is easier to solve for n using the financialcalculator or Excel rather than mathematicalformula.

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S l i f h b f i d

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(cont.)

Example 5.6 How many years will it takefor an investment of $7,500 to grow to

$23,000 if it is invested at 8% annually?

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(cont.)

Using a Financial Table

FVn = PV0 (FVIF8%,n)

23,000 = 7,500 (FVIF8%,n)

FVIF8%,n = 23,500 / 7,500 = 3.1333

n = between 14 and 15 years

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R l f 72

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Rule of 72

Rule of 72 is an approximate formula todetermine the number of years it will taketo double the value of your investment.

Rule of 72

N = 72/interest rate

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Rule of 72 (cont.)

Example 5.7 Using Rule of 72, determinehow long it will take to double yourinvestment of $10,000 if you are able to

generate an annual return of 9%.

N = 72/interest rate

N = 72/9 = 8years

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S l i f R f I

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Solving for Rate of Interest

Key Question: What rate of interest willallow your investment to grow to adesired future value?

We can determine the rate of interestusing mathematical equation, the financial

calculator or the Excel spread sheet.

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Solving for Rate of Interest (cont.)

Example 5.8 At what rate of interest mustyour savings of $10,000 be compoundedannually for it to grow to $22,000 in 8

years?

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FVn = PV0 (FVIFi%,8)

22,000 = 10,000 (FVIFi%,8)

FVIFi%,8 = 22,000 / 10,000 = 2.2

i = between 10% and 11%

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S l in f r R t f Int r t ( nt )

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Using an Excel Spreadsheet

=Rate(nper,pmt,pv,fv)

=Rate(8,0,-10000,22000)

=10.36%

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Ch kp i t 5 6 S l i f th

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Checkpoint 5.6 - Solving for the

Interest Rate, i

Lets go back to that Prius example inCheckpoint 5.5. Recall that the Prius

always costs $20,000. In 10 years, youdreally like to have $20,000 to buy a newPrius, but you only have $11,167 now. Atwhat rate must your $11,167 becompounded annually for it to grow to$20,000 in 10 years?

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Checkpoint 5 6

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Checkpoint 5.6

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MAKING INTEREST

RATES COMPARABLE

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Comparing Loans using EAR

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Comparing Loans using EAR

We cannot compare two loans based onAPR if they do not have the samecompounding period.

To make them comparable, we calculatetheir equivalent rate using an annualcompounding period. We do this bycalculating the effective annual rate (EAR)

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Ch kp int 5 7

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Checkpoint 5.7

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Checkpoint 5 7

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Checkpoint 5.7

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The Future Value of an Ordinary

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The Future Value of an Ordinary

Annuity (cont.)


FV = $3000 (FVIFA5%,10)

= $3,000 (12.5779)

= $37,734

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Solving for Interest Rate in an

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g

Ordinary Annuity (cont.)

We will have to substitute differentnumbers for i until we find the value of ithat makes the right hand side of theexpression equal to 40.

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g

in an Ordinary Annuity (cont.)

Example 6.4: Suppose you are investing$6,000 at the end of each year in anaccount that pays 5%. How long will it

take before the account is worth $50,000?

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g

in an Ordinary Annuity (cont.)

Using an Excel Spreadsheet

n = NPER(rate, pmt, pv, fv)

n = NPER(5%,-6000,0,50000)

n = 7.14 years

Thus it will take 7.13 years for annual deposits of$6,000 to grow to $50,000 at an interest rate of 5%

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The Present Value of an Ordinary

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Annuity (cont.)

For example, we will compute thePV of ordinary annuity if we wish

to answer the question: what is thevalue today or lump sumequivalent of receiving $3,000

every year for the next 30 years ifthe interest rate is 5%?

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The Present Value of an Ordinary

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Annuity (cont.)

PMT = annuity payment deposited orreceived at the end of each period.

i = discount rate (or interest rate) on a

per period basis. n = number of periods for which the

annuity will last.

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Checkpoint 6.2

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119

Checkpoint 6.2

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120


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What is the present value of anannuity of $10,000 to be received at

the end of each year for 10 yearsgiven a 10 percent discount rate?

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Amortized Loans

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An amortized loan is a loan paid off inequal payments consequently, the

loan payments are an annuity.

Examples: Home mortgage loans, Auto

loans

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Amortized Loans (cont.)

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In an amortized loan, the present valuecan be thought of as the amountborrowed, nis the number of periods the

loan lasts for, iis the interest rate perperiod, future valuetakes on zerobecause the loan will be paid of after nperiods, and paymentis the loan paymentthat is made.

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Example 6.5 Suppose you plan to get a$9,000 loan from a furniture dealer at 18%annual interest with annual payments that

you will pay off in over five years. Whatwill your annual payments be on thisloan?

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PV = PMT (PVIFA18%,5)

9,000 = PMT (3.1272)PMT = $2,878

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The Loan Amortization Schedule

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Year Amount Owed

on Principal atthe Beginningof the Year(1)

Annuity

Payment(2)

Interest

Portionof theAnnuity(3) = (1)

18%

Repayment

of thePrincipalPortion oftheAnnuity(4) =

(2)

(3)

Outstanding

LoanBalance atYear end,After theAnnuityPayment (5)

=(1)

(4)

1 $9,000 $2,878 $1,620.00 $1,258.00 $7,742.00

2 $7,742 $2,878 $1,393.56 $1,484.44 $6,257.56

3 $6257.56 $2,878 $1,126.36 $1,751.64 $4,505.92

4 $4,505.92 $2,878 $811.07 $2,066.93 $2,438.98

5 $2,438.98 $2,878 $439.02 $2,438.98 $0.00

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The Loan Amortization Schedule

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(cont.)

We can observe the following from thetable:

Size of each payment remains the same. However, Interest payment declines each year

as the amount owed declines and more of theprincipal is repaid.

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Amortized Loans with Monthly

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Payments

Many loans such as auto and home loansrequire monthly payments. This requires

converting nto number of months andcomputing the monthly interest rate.

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Payments (cont.)

Mathematical Formula

Here annual interest rate = .06, number ofyears = 30, m=12, PV = $300,000

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( )

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Payments (cont.)

$300,000= PMT x

$300,000 = PMT [166.79]

$300,000 166.79 = $1798.67

1- 1/(1+.06/12)360

.06/12

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Checkpoint 6.3 - Determining the

O di B l f L

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Outstanding Balance of a Loan

Lets say that exactly ten years ago you took out a$200,000, 30-year mortgage with an annual interest rateof 9 percent and monthly payments of $1,609.25. Butsince you took out that loan, interest rates havedropped. You now have the opportunity to refinanceyour loan at an annual rate of 7 percent over 20 years.You need to know what the outstanding balance onyour current loan is so you can take out a lower-

interest-rate loan and pay it off. If you just made the120th payment and have 240 payments remaining,whats your current loan balance?

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Checkpoint 6.3

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133

Checkpoint 6.3

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134

Checkpoint 6.3

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Annuities Due

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Annuity due is an annuity in whichall the cash flows occur at the

beginning of the period. Forexample, rent payments onapartments are typically annuity

due as rent is paid at the beginningof the month.

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Annuities Due: Future Value

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Computation of future value of anannuity due requires compoundingthe cash flows for one additionalperiod, beyond an ordinaryannuity.

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( )

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(cont.)

Recall Example 6.1 where we calculatedthe future value of 10-year ordinaryannuity of $3,000 earning 5 per cent to be

$37,734.

What will be the future value if the

deposits of $3,000 were made at thebeginning of the year i.e. the cash flowswere annuity due?

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( t )

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(cont.)

FV = $3000 {[ (1+.05)10 - 1] (.05)} (1.05)

= $3,000 { [0.63] (.05) } (1.05)

= $3,000 {12.58}(1.05)

= $39,620

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Annuities Due: Present Value

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Since with annuity due, each cash flow isreceived one year earlier, its present valuewill be discounted back for one lessperiod.

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( t )

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(cont.)

Recall checkpoint 6.2 Check yourselfproblem where we computed the PV of 10-year ordinary annuity of $10,000 at a 10percent discount rate to be equal to$61,446.

What will be the present value if $10,000 isreceived at the beginning of each year i.e.the cash flows were annuity due?

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( t )

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(cont.)

PV = $10,000 {

[1-(1/(1.10)10

] (.10)} (1.1)

= $10,000 {[ 0.6144] (.10)}(1.1)

= $10,000 {6.144) (1.1)

= $ 67,590

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Annuities Due

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The examples illustrate that boththe future value and present valueof an annuity due are larger thanthat of an ordinary annuitybecause, in each case, allpayments are received or paidearlier.

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Perpetuities

Perpetuities

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A perpetuity is an annuity that continuesforever or has no maturity. For example, adividend stream on a share of preferredstock. There are two basic types ofperpetuities:

Growing perpetuity in which cash flows growat a constant rate, g, from period to period.

Level perpetuity in which the payments areconstant rate from period to period.

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Present Value of a Level Perpetuity

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PV = the present value of a level

perpetuity PMT = the constant dollar amount

provided by the perpetuity

i = the interest (or discount) rate perperiod

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Present Value of a Level Perpetuity

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Example 6.6 What is the present value of$600 perpetuity at 7% discount rate?

PV = $600 .07 = $8,571.43

148

Checkpoint 6.4

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The Present Value of a Level PerpetuityWhat is the present value of a perpetuity of$500 paid annually discounted back to the

present at 8 percent?

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Checkpoint 6.4

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What is the present value of streamof payments equal to $90,000 paid

annually and discounted back tothe present at 9 percent?

152

Present Value of a Growing

Perpetuity

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Perpetuity

In growing perpetuities, theperiodic cash flows grow at a

constant rate each period.

The present value of a growing

perpetuity can be calculated usinga simple mathematical equation.

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Present Value of a Growing

Perpetuity (cont )

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Perpetuity (cont.)

PV = Present value of a growing perpetuity

PMTperiod1 = Payment made at the end of firstperiod

i = rate of interest used to discount thegrowing perpetuitys cash flows

g = the rate of growth in the payment of cashflows from period to period

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Checkpoint 6.5

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156

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What is the present value of astream of payments where the year 1

payment is $90,000 and the futurepayments grow at a rate of 5% peryear? The interest rate used to

discount the payments is 9%.

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Complex Cash Flow

Streams

Complex Cash Flow Streams

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The cash flows streams in the businessworld may not always involve one type ofcash flows. The cash flows may have amixed pattern. For example, different cashflow amounts mixed in with annuities.

For example, figure 6-4 summarizes the

cash flows for Marriott.

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Complex Cash Flow Streams

(cont )

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(cont.)

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Checkpoint 6.6 - The Present Value

of a Complex Cash Flow Stream

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of a Complex Cash Flow Stream

What is the present value of cashflows of $500 at the end of years

through 3, a cash flow of a negative$800 at the end of year 4, and cashflows of $800 at the end of years 5

through 10 if the appropriatediscount rate is 5%?

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Checkpoint 6.6

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Checkpoint 6.6

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Step 3 cont.

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Checkpoint 6.6

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What is the present value of cashflows of $300 at the end of years 1through 5, a cash flow of negative$600 at the end of year 6, andcash flows of $800 at the end ofyears 7-10 if the appropriatediscount rate is 10%?

05 06 Time Value

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