Finance - Time value of money

5/19/2018 Finance - Time value of money

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Copyright 2012 Pearson Prentice Hall. All rights reserved.

Chapter 3

Time Value ofMoney: An

Introduction


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Copyright 2012 Pearson Prentice Hall. All rights reserved. 3-2

Chapter Outline

3.3 The Time Value of Money and InterestRates

3.4 Valuing Cash Flows at Different Points in

Time


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Learning Objectives

Understand the Valuation Principle, andhow it can be used to identify decisionsthat increase the value of the firm

Assess the effect of interest rates ontodays value of future cash flows

Calculate the value of distant cash flows inthe present and of current cash flows in

the future


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3.3 The Time Value of Money andInterest Rates

The Time Value of Money

In general, a dollar today is worth more than adollar in one year

If you have $1 today and you can deposit it in a bankat 10%, you will have $1.10 at the end of one year

The difference in value between money todayand money in the future the time value ofmoney.


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The Time Value of Money

Consider a firm's investment opportunity with acost of $100,000 today and a benefit of

$105,000 at the end of one year.


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Figure 3.1 Converting Between DollarsToday and Gold or Dollars in the Future


7/192Copyright 2012 Pearson Prentice Hall. All rights reserved. 3-7


The Interest Rate: Converting Cash AcrossTime

By depositing money, we convert money today

into money in the future By borrowing money, we exchange money

today for money in the future


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Interest Rate (r)

The rate at which money can be borrowed or lentover a given period

Interest Rate Factor (1 + r)

It is the rate of exchange between dollars today anddollars in the future

It has units of $ in one year/$ today


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Value of $100,000 Investment in One Year

If the interest is 10%, the cost of theinvestment is:

Cost = ($100,000 today) (1.10 $ in one year/$today)

= $110,000 in one year


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$110,000 is the opportunity cost of spending

$100,000 today The firm gives up the $110,000 it would have had in

one year if it had left the money in the bank

Alternatively, by borrowing the $100,000 from thesame bank, the firm would owe $110,000 in one year.


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The investments net value is difference

between the cost of the investment and thebenefit in one year:

$105,000 - $110,000 = -$5,000 in one year


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Value of $100,000 Investment Today

How much would we need to have in the bank todayso that we would end up with $105,000 in the bank inone year?

This is calculated by dividing by the interest rate factor:


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Value of $100,000 Investment Today

The investments net value is difference between thecost of the investment and the benefit in one year:$95,454.55 - $100,000 = -$4,545.55 today


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Present Versus Future Value

Present Value The value of a cost or benefit computed in terms of

cash today

(-$4,545.45)

Future Value

The value of a cash flow that is moved forward in time ($5,000)

(-$4,545.45 today) (1.10 $ in one year/$ today) = -$5,000in one year


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Discount Factors and Rates

Money in the future is worth less today so its pricereflects a discount

Discount Rate

The appropriate rate to discount a cash flow todetermine its value at an earlier time

Discount Factor The value today of a dollar received in the future,

expressed as:

1

1 r


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Example 3.4 Comparing Revenues atDifferent Points in Time

Problem: The launch of Sonys PlayStation 3 was delayed

until November 2006, giving Microsofts Xbox 360a full year on the market without competition.

Imagine that it is November 2005 and you are themarketing manager for the PlayStation. Youestimate that if the PlayStation 3 were ready tobe launched immediately, you could sell $2 billionworth of the console in its first year. However, ifyour launch is delayed a year, you believe thatMicrosofts head start will reduce your first-yearsales by 20%. If the interest rate is 8%, what isthe cost of a delay in terms of dollars in 2005?


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Solution:

Plan: Revenues if released today: $2 billion

Revenue decrease if delayed: 20% Interest rate: 8%

We need to compute the revenues if the launch isdelayed and compare them to the revenues from

launching today. However, in order to make a faircomparison, we need to convert the futurerevenues of the PlayStation if delayed into anequivalent present value of those revenues today.


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Execute: If the launch is delayed to 2006, revenues will

drop by 20% of $2 billion, or $400 million, to$1.6 billion.

To compare this amount to revenues of $2 billionif launched in 2005, we must convert it using theinterest rate of 8%:$1.6 billion in 2006 ($1.08 in 2006/$1 in 2005) = $1.481billion in 2005

Therefore, the cost of a delay of one year is

$2 billion - $1.481 billion = $0.519 billion ($519million).


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Evaluate: Delaying the project for one year was equivalent to giving

up $519 million in cash.

In this example, we focused only on the effect on the first

years revenues. However, delaying the launch delays theentire revenue stream by one year, so the total cost wouldbe calculated in the same way by summing the cost of delayfor each year of revenues.


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Example 3.4a Comparing Revenuesat Different Points in Time

Problem: The launch of Sonys PlayStation 3 was delayed

until November 2006, giving Microsofts Xbox 360a full year on the market without competition.

Imagine that it is November 2005 and you are themarketing manager for the PlayStation. Youestimate that if the PlayStation 3 were ready tobe launched immediately, you could sell $3 billionworth of the console in its first year. However, ifyour launch is delayed a year, you believe thatMicrosofts head start will reduce your first-yearsales by 35%. If the interest rate is 6%, what isthe cost of a delay in terms of dollars in 2005?


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Solution:

Plan: Revenues if released today: $3 billion, Revenue

decrease if delayed: 35% , Interest rate: 6% We need to compute the revenues if the launch is

delayed and compare them to the revenues fromlaunching today.

However, in order to make a fair comparison, weneed to convert the future revenues of thePlayStation if delayed into an equivalent presentvalue of those revenues today.


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Execute: If the launch is delayed to 2006, revenues will

drop by 35% of $3 billion, or $1.05 billion, to

$1.95 billion. To compare this amount to revenues of $3 billion

if launched in 2005, we must convert it using theinterest rate of 6%:

$1.95 billion in 2006 ($1.06 in 2006/$1 in 2005) =$1.840 billion in 2005

Therefore, the cost of a delay of one year is

$3 billion - $1.840 billion = $1.160 billion


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Evaluate: Delaying the project for one year was equivalent to giving

up $1.16 billion in cash.

In this example, we focused only on the effect on the first

years revenues. However, delaying the launch delays the entire revenue

stream by one year, so the total cost would be calculated inthe same way by summing the cost of delay for each year ofrevenues.


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Timelines

Constructing a Timeline

Identifying Dates on a Timeline

Date 0 is today, the beginning of the first year Date 1 is the end of the first year


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Timelines Distinguishing Cash Inflows from Outflows

Representing Various Time Periods Just change the label from Year to Month if

monthly


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3.4 Valuing Cash Flows at DifferentPoints in Time

Rule 1: Comparing and Combining Values

It is only possible to compare or combinevalues at the same point in time


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Rule 2: Compounding

To calculate a cash flows future value, youmust compound it


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Rule 2: Compounding

Compound Interest

The effect of earning interest on interest


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Figure 3.2 The Composition of Interestover Time


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Rule 3: Discounting

To calculate the value of a future cash flow atan earlier point in time, we must discount it.


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If $826.45 is invested today for two yearsat 10% interest, the future value will be$1000


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If $1000 were three years away, thepresent value, if the interest rate is 10%,will be $751.31


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Example 3.5 Present Value of aSingle Future Cash Flow

Problem: You are considering investing in a savings bond that will pay

$15,000 in ten years. If the competitive market interestrate is fixed at 6% per year, what is the bond worth today?


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Solution:

Plan: First set up your timeline. The cash flows for this

bond are represented by the following timeline:

Thus, the bond is worth $15,000 in ten years. Todetermine the value today, we compute thepresent value using Eq. 3.2 and our interest rateof 6%.


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Execute:

10

15,000$8,375.92 today

1.06

PV

Given: 10 6 0 15,000

Solve for: -8,375.92

Excel Formula: =PV(RATE,NPER, PMT, FV) = PV(0.06,10,0,15000)


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Evaluate: The bond is worth much less today than its final payoff

because of the time value of money.


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Example 3.5a Present Value of aSingle Future Cash Flow

Problem: XYZ Company expects to receive a cash flow of $2 million in

five years. If the competitive market interest rate is fixed at4% per year, how much can they borrow today in order to

be able to repay the loan in its entirety with that cash flow?


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Solution:

Plan: First set up your timeline. The cash flows for the loan are

represented by the following timeline:

Thus, XYZ Company will be able to repay the loan with itsexpected $2 million cash flow in five years. To determine thevalue today, we compute the present value using Eq. 3.2 andour interest rate of 4%.


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Execute:

Given: 5 4 0 2,000,000

Solve for: -1,643,854.21



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Evaluate: The loan is much less than the $2 million the company will

pay back because of the time value of money.


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Example 3.5b Present Value of aSingle Future Cash Flow

Problem: You are considering investing in a savings bond that will pay

$20,000 in twenty years. If the competitive market interestrate is fixed at 5% per year, what is the bond worth today?


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Solution

Plan: First set up your timeline. The cash flows for this

bond are represented by the following timeline:

Thus, the bond is worth $20,000 in twenty years.To determine the value today, we compute thepresent value using Eq. 3.2 and our interest rateof 5%.


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Execute:

20

20,000$7,537.79 today

1.05PV

Given: 20 5 0 20,000

Solve for: -7,357.79



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Evaluate: The bond is worth much less today than its final payoff

because of the time value of money.


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Chapter Quiz

1. If crude oil trades in a competitive market, wouldan oil refiner that has a use for the oil value itdifferently than another investor would?

2. How do we determine whether a decisionincreases the value of the firm?

3. Is the value today of money to be received in oneyear higher when interest rates are high or when

interest rates are low?4. What do you need to know to compute a cash

flows present or future value?


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Chapter 4

Time Valueof Money:

Valuing CashFlowStreams


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Chapter Outline

4.1 Valuing a Stream of Cash Flows

4.2 Perpetuities

4.3 Annuities

4.4 Growing Cash Flows4.5 Solving for Variables Other Than Present Value

or Future Value


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Learning Objectives

Value a series of many cash flows

Value a perpetual series of regular cash flows called aperpetuity

Value a common set of regular cash flows called an annuity

Value both perpetuities and annuities when the cash flowsgrow at a constant rate

Compute the number of periods, cash flow, or rate of returnin a loan or investment


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Rules developed in Chapter 3:

Rule 1: Only values at the same point in timecan be compared or combined.

Rule 2: To calculate a cash flows future value,we must compound it.

Rule 3: To calculate the present value of afuture cash flow, we must discount it.


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Applying the Rules of Valuing Cash Flows

Suppose we plan to save $1,000 today, and$1,000 at the end of each of the next two years.

If we earn a fixed 10% interest rate on oursavings, how much will we have three years fromtoday?


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We can do this in several ways.

First, take the deposit at date 0 and move itforward to date 1.

Combine those two amounts and move thecombined total forward to date 2.


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Continuing in the same fashion, we can solve theproblem as follows:


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Another approach is to compute the future valuein year 3 of each cash flow separately.

Once all amounts are in year 3 dollars, combinethem.


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Consider a stream of cash flows: C0atdate 0, C1at date 1, and so on, up to CNatdate N.

We compute the present value of this cash

flow stream in two steps.


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First, compute the present value of each cashflow.

Then combine the present values.

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Example 4.1Present Value of a Stream of Cash Flows

Problem: You have just graduated and need money to buy a new car.

Your rich Uncle Henry will lend you the money so long asyou agree to pay him back within four years.

You offer to pay him the rate of interest that he wouldotherwise get by putting his money in a savings account.

Example 4 1


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Example 4.1Present Value of a Stream of Cash Flows(contd)

Problem: Based on your earnings and living expenses, you think you

will be able to pay him $5000 in one year, and then $8000each year for the next three years.

If Uncle Henry would otherwise earn 6% per year on hissavings, how much can you borrow from him?

Example 4 1


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Solution:

Plan: The cash flows you can promise Uncle Henry are as follows:

Uncle Henry should be willing to give you an amount equal

to these payments in present value terms.

Example 4 1


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Plan: We will:

Solve the problem using equation 4.1

Verify our answer by calculating the future value of thisamount.

Example 4 1


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Execute: We can calculate the PV as follows:

2 3 45000 8000 8000 80001.06 1.06 1.06 1.06

4716.98 7119.97 6716.95 6336.75

$24,890.65

PV

Example 4 1


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Execute: Now, suppose that Uncle Henry gives you the money, and

then deposits your payments in the bank each year.

How much will he have four years from now?

Example 4 1


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Execute: We need to compute the future value of the annual

deposits.

One way is to compute the bank balance each year.

Example 4.1


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Execute: To verify our answer, suppose your uncle kept his

$24,890.65 in the bank today earning 6% interest.

In four years he would have:

FV= $24,890.65(1.06)4=$31,423.87 in 4 years

Example 4.1


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Evaluate: Thus, Uncle Henry should be willing to lend you $24,890.65

in exchange for your promised payments.

This amount is less than the total you will pay him

($5000+$8000+$8000+$8000=$29,000) due to the timevalue of money.

Example 4 1a


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Example 4.1aPresent Value of a Stream of Cash Flows

Problem: You have just graduated and need money to pay

the deposit on an apartment.

Your rich aunt will lend you the money so long asyou agree to pay her back within six months.

You offer to pay her the rate of interest that shewould otherwise get by putting her money in asavings account.

Example 4.1a


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Example 4.1aPresent Value of a Stream of Cash Flows(contd)

Problem: Based on your earnings and living expenses, you think you

will be able to pay her $70 next month, $85 in each of thenext two months, and then $900 each month for months 4through 6.

If your aunt would otherwise earn 6% per year on hersavings, how much can you borrow from her?

Example 4.1a


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Solution:

Plan: The cash flows you can promise your aunt are as follows:

She should be willing to give you an amount equal to these

payments in present value terms.

Example 4.1a


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Plan: We will:

Solve the problem using equation 4.1

Verify our answer by calculating the future value of thisamount.

Example 4 1a Present Value of a Stream


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Example 4.1a Present Value of a Streamof Cash Flows (contd)

Execute: We can calculate the PV as follows:

2 3 4 5 670 85 85 90 90 901.005 1.005 1.005 1.005 1.005 1.005

$69.65 $84.16 $83.74 $88.22 $87.78 $87.35

$500.90

PV

Example 4.1a


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a p e 4 aPresent Value of a Stream of Cash Flows(contd)

Execute: Now, suppose that your aunt gives you the money, and

then deposits your payments in the bank each month.

How much will she have six months from now?

Example 4.1a


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pPresent Value of a Stream of Cash Flows(contd)

Execute: We need to compute the future value of the monthly

deposits.

One way is to compute the bank balance each month.

Example 4.1a


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Execute: To verify our answer, suppose your aunt kept her $500.90

in the bank today earning 6% interest.

In six months she would have:

FV= $500.90(1.005)6=$516.11 in 6 months

Example 4.1a


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Evaluate: Thus, your aunt should be willing to lend you $500.90 in

exchange for your promised payments.

This amount is less than the total you will pay her($70+$85+$85+$90+$90+$90=$510) due to the time

value of money.


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Using a Financial Calculator: Solving for Presentand Future Values

Financial calculators and spreadsheets have theformulas pre-programmed to quicken the process.

There are five variables used most often: N

PV

PMT

FV I/Y


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Example 1: Suppose you plan to invest $20,000 in an account

paying 8% interest.

How much will you have in the account in 15

years? To compute the solution, we enter the four

variables we know and solve for the one we wantto determine, FV.


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Example 1: For the HP-10BII or the TI-BAII Plus calculators:

Enter 15 and press the N key.

Enter 8 and press the I/Y key (I/YR for the HP)

Enter -20,000 and press the PV key. Enter 0 and press the PMT key.

Press the FV key (for the TI, press CPT and then FV).


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Given: 15 8 -20,000 0

Solve for: 63,443

Excel Formula: = FV(0.08,15,0,-20000)

Notice that we entered PV (the amount were putting in to thebank) as a negative number and FV is shown as a positive number(the amount we take out of the bank).

It is important to enter the signs correctly to indicate the directionthe funds are flowing.

Example 4 2


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Example 4.2Computing the Future Value

Problem: Lets revisit the savings plan we considered earlier. We plan

to save $1000 today and at the end of each of the next twoyears.

At a fixed 10% interest rate, how much will we have in the

bank three years from today?

Example 4 2 Computing the Future


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Example 4.2 Computing the FutureValue (contd)

Solution:Plan: Well start with the timeline for this savings plan:

Lets solve this in a different way than we did inthe text, while still following the rules.

Example 4.2 Computing the Future


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Plan: First well compute the present value of the cash flows.

Then well compute its value three years later (its futurevalue).



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Execute: There are several ways to calculate the present value of the

cash flows.

Here, we treat each cash flow separately an then combinethe present values.



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Execute: Saving $2735.54 today is equivalent to saving $1000 per

year for three years.

Now lets compute future value in year 3 of that $2735.54:



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Copyright 2012 Pearson Prentice Hall. All rights reserved.3-83


Evaluate: The answer of $3641 is precisely the same result we found

earlier.

As long as we apply the three rules of valuing cash flows,we will always get the correct answer.

4 2 P t iti


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4.2 Perpetuities

The formulas we have developed so far allow usto compute the present or future value of anycash flow stream.

Now we will consider two types of cash flow

streams: Perpetuities

Annuities

4 2 P t iti


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4.2 Perpetuities

Perpetuities A perpetuity is a stream of equal cash flows

that occur at regular intervals and last forever.

Here is the timeline for a perpetuity:

the first cash flow does not occur immediately;it arrives at the end of the first period

4 2 P t iti


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4.2 Perpetuities

Using the formula for present value, thepresent value of a perpetuity withpayment C and interest rate r is given by:

Notice that all the cash flows are thesame.

Also, the first cash flow starts at time 1.

2 3C C C ......(1 r) (1 r) (1 r)PV=

4 2 P t iti


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4.2 Perpetuities

Lets derive a shortcut by creating our ownperpetuity.

Suppose you can invest $100 in a bankaccount paying 5% interest per year

forever. At the end of the year youll have $105 in

the bank your original $100 plus $5 ininterest.

4 2 Perpet ities


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4.2 Perpetuities

Suppose you withdraw the $5 and reinvestthe $100 for another year.

By doing this year after year, you can

withdraw $5 every year in perpetuity:

4 2 Perpetuities


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3-89

4.2 Perpetuities

To generalize, suppose we invest anamount P at an interest rate r.

Every year we can withdraw the interest

we earned, C=r P, leaving P in the bank. Because the cost to create the perpetuity

is the investment of principal, P, the valueof receiving C in perpetuity is the upfront

cost, P.

4 2 Perpetuities


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4.2 Perpetuities

PV(Cin perpetuity)

C

r(Eq. 4.4)

Present Value of a Perpetuity

Example 4.3


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3-91

pEndowing a Perpetuity

Problem: You want to endow an annual graduation party at your alma

mater. You want the event to be a memorable one, so youbudget $30,000 per year forever for the party.

If the university earns 8% per year on its investments, and

if the first party is in one years time, how much will youneed to donate to endow the party?

Example 4.3


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Solution:Plan: The timeline of the cash flows you want to

provide is:

This is a standard perpetuity of $30,000 peryear. The funding you would need to give theuniversity in perpetuity is the present value ofthis cash flow stream

pEndowing a Perpetuity (contd)

Example 4.3


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Execute: From the formula for a perpetuity,

PV C/r $30,000/0.08 $375,000 today

Example 4.3


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Evaluate: If you donate $375,000 today, and if the university invests

it at 8% per year forever, then the graduates will have$30,000 every year for their graduation party.

Example 4.3a


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3-95

Endowing a Perpetuity

Problem: You just won the lottery, and you want to endow a

professorship at your alma mater.

You are willing to donate $4 million of your winnings for thispurpose.

If the university earns 5% per year on its investments, andthe professor will be receiving her first payment in one year,how much will the endowment pay her each year?

Example 4.3a


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3-96

Solution:Plan: The timeline of the cash flows you want to

provide is:

This is a standard perpetuity. The amount she canwithdraw each year and keep the principal intact

is the cash flow when solving equation 4.4.

Endowing a Perpetuity (contd)

Example 4.3a


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Execute: From the formula for a perpetuity,

Example 4.3a


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Evaluate: If you donate $4,000,000 today, and if the university

invests it at 5% per year forever, then the chosen professorwill receive $200,000 every year.

4 3 Annuities


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3-99

4.3 Annuities

Annuities An annuity is a stream of N equal cash flows

paid at regular intervals.

The difference between an annuity and

a perpetuity is that an annuity ends aftersome fixed number of payments

4 3 Annuities


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3-100

4.3 Annuities

Present Value of An Annuity Note that, just as with the perpetuity, we

assume the first payment takes place oneperiod from today.

To find a simpler formula, use the sameapproach as we did with a perpetuity:create your own annuity.

2 3 N

C C C C......

(1 r) (1 r) (1 r) (1 r)PV=

4.3 Annuities


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3-101

4.3 Annuities

With an initial $100 investment at 5%interest, you can create a 20-year annuityof $5per year, plus you will receive anextra $100 when you close the account atthe end of 20years:

4 3 Annuities


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3-102Copyright 2009 Pearson Prentice Hall. All rights reserved.

4-

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4.3 Annuities

The Law of One Price tells us that becauseit only took an initial investment of $100to create the cash flows on the timeline,the present value of these cash flows is$100:

$100 20 - year annuity of $5 per year $100 in 20 yearsPV( ) PV( )

4 3 Annuities


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3-103

4.3 Annuities

Rearranging:

20

20 - year annuity of $5 per year $100 $100 in 20 years

$100$100 - $100 $37.69 $62.31

(1.05)

PV( ) PV( )

=

4 3 Annuities


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3-104

4.3 Annuities

We usually want to know the PV as afunction of C, r, and N.

Since C can be written as $100(0.05)=$5,

we can further re-arrange:

20 20

20

$5$5 $5 10.0520 - year annuity of $5 per year - 1

0.05 (1.05) 0.05 1.05

1 1$5 1(0.05) 1.05

PV( )=

4.3 Annuities


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4.3 Annuities

In general:

1 1(annuity of C for N periods with interest rate r) 1

(1 N

PV C

r r)

Example 4.4P esent Val e of a Lotte P i e


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Present Value of a Lottery PrizeAnnuity

Problem: You are the lucky winner of the $30 million state lottery.

You can take your prize money either as (a) 30 payments of$1 million per year (starting today), or (b) $15 million paid

today. If the interest rate is 8%, which option should you take?

Example 4.4Present Value of a Lottery Prize


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Present Value of a Lottery PrizeAnnuity (contd)

Solution:

Plan: Option (a) provides $30 million in prize money but paid over

time. To evaluate it correctly, we must convert it to apresent value. Here is the timeline:



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Plan (contd): Because the first payment starts today, the last payment

will occur in 29 years (for a total of 30 payments).

The $1 million at date 0 is already stated in present value

terms, but we need to compute the present value of theremaining payments.

Fortunately, this case looks like a 29-year annuity of $1million per year, so we can use the annuity formula.



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Execute: From the formula for an annuity,

PV(29-year annuity of $1million) $1 million1

0.081

1

1.0829

$1 million 11.16

$11.16 million today



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Execute (contd): Thus, the total present value of the cash flows is $1 million

+ $11.16 million = $12.16 million. In timeline form:



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Present Value of a Lottery PrizeAnnuity (contd)Execute (contd): Financial calculators or Excel can handle annuities easily

just enter the cash flow in the annuity as the PMT:

Given: 29 8.0 1,000,000 0

Solve for: -11,158,406


Example 4.4Present Value of a Lottery Prize Annuity


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Present Value of a Lottery Prize Annuity(contd)

Evaluate: The reason for the difference is the time value of money.

If you have the $15 million today, you can use $1 millionimmediately and invest the remaining $14 million at an 8%

interest rate. This strategy will give you $14 million 8% = $1.12 millionper year in perpetuity!

Alternatively, you can spend $15 million $11.16 million =$3.84 million today, and invest the remaining $11.16million, which will still allow you to withdraw $1 million eachyear for the next 29 years before your account is depleted.

Example 4.4aP t V l f A it


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Present Value of an Annuity

Problem: Your parents have made you an offer you cant refuse.

Theyre planning to give you part of your inheritance

early.

Theyve given you a choice. Theyll pay you $10,000 per year for each of the next

seven years (beginning today) or theyll give you their

2007 BMW M6 Convertible, which you can sell for

$61,000 (guaranteed) today. If you can earn 7% annually on your investments,

which should you choose?

Example 4.4aP t V l f A it ( td)


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Present Value of an Annuity (contd)

Solution:Plan: Option (a) provides $10,000 paid over time. To evaluate it

correctly, we must convert it to a present value. Here is the

timeline:

Example 4.4aP t V l f A it ( td)


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Plan (contd): The $10,000 at date 0 is already stated in present value

terms, but we need to compute the present value of theremaining payments.

Fortunately, this case looks like a 6-year annuity of $10,000per year, so we can use the annuity formula.

Example 4.4aP esent Val e of an Ann it (contd)


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Execute: From the formula for a annuity,

Example 4.4aPresent Value of an Annuity (contd)


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Execute (contd): Thus, the total present value of the cash flows is $10,000 +

$47,665. In timeline form:



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Execute (contd): Financial calculators or Excel can handle annuities easily

just enter the cash flow in the annuity as the PMT:

Given: 6 7 10000 0

Solve for: -47,665




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Present Value of an Annuity (cont d)

Evaluate: Lucky you!

Even if you dont want to keep it, the fact that you can sellit for more than the annuity is worth means youre better offtaking the BMW.

4.3 Annuities


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(annuity) (1

11 (1 )

(1 )

1

((1 ) 1)

N

N

N

N

FV PV r)

Cr

r r

C rr

(Eq. 4.6)

Future Value of an Annuity

Example 4.5Retirement Savings Plan Annuity


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Retirement Savings Plan Annuity

Problem: Ellen is 35 years old, and she has decided it is time to plan

seriously for her retirement.

At the end of each year until she is 65, she will save$10,000 in a retirement account.

If the account earns 10% per year, how much will Ellenhave saved at age 65?



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Retirement Savings Plan Annuity(contd)

Solution

Plan: As always, we begin with a timeline. In this case, it is helpful

to keep track of both the dates and Ellens age:



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Retirement Savings Plan Annuity(contd)

Plan (contd):

Ellens savings plan looks like an annuity of$10,000 per year for 30 years.

(Hint: It is easy to become confused when youjust look at age, rather than at both dates andage. A common error is to think there are only65-36= 29 payments. Writing down both datesand age avoids this problem.)

To determine the amount Ellen will have in thebank at age 65, well need to compute the futurevalue of this annuity.



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Execute:

Using Financial calculators or Excel:

FV $10,0001

0.10(1.10

30 1)

$10,000 164.49

$1.645 million at age 65

Given: 30 10.0 0 -10,000Solve for: -1,644,940

Excel Formula: =FV(RATE,NPER, PMT, PV) = FV(0.10,30,10000,0)



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Evaluate: By investing $10,000 per year for 30 years (a total of

$300,000) and earning interest on those investments, the

compounding will allow her to retire with $1.645 million.

Example 4.5aRetirement Savings Plan Annuity


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Problem: Adam is 25 years old, and he has decided it is time to plan

seriously for his retirement.

He will save $10,000 in a retirement account at the end ofeach year until he is 45.

At that time, he will stop paying into the account, though hedoes not plan to retire until he is 65.

If the account earns 10% per year, how much will Adamhave saved at age 65?



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SolutionPlan: As always, we begin with a timeline. In this case, it is

helpful to keep track of both the dates and Adams age:



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Adams savings plan looks like an annuity of $10,000 peryear for 20 years.

The money will then remain in the account until Adam is 6520 more years.

To determine the amount Adam will have in the bank at age

45, well need to compute the future value of this annuity. Then well compound the future value into the future 20

more years to see how much hell have at 65.



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Execute:


Given: 20 10.0 0 -10,000Solve for: $572,750

Excel Formula: =FV(RATE,NPER, PMT, PV) = FV(0.10,20,10000,0)



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Execute:


Given: 20 10.0 -$572,750 0

Solve for: $3,853,175

Excel Formula: =FV(RATE,NPER, PMT, PV) = FV(0.10,20,0,-572750)



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Evaluate: By investing $10,000 per year for 20 years (a

total of $200,000) and earning interest on those

investments, the compounding will allow him to

retire with $3.85 million.

Even though he invested for 10 fewer years than

Ellen did, Adam will end up with more than twice

as much money because hes starting hisretirement plan ten years earlier than she will.

4.4 Growing Cash Flows


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A growing perpetuity is a stream of cashflows that occur at regular intervals andgrow at a constant rate forever.

For example, a growing perpetuity with afirst payment of $100 that grows at a rateof 3% has the following timeline:



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PV(growing perpetuity) Cr g(Eq. 4.7)

Present Value of a GrowingPerpetuity

Example 4.6Endowing a Growing Perpetuity


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Endowing a Growing Perpetuity

Problem: In Example 4.3, you planned to donate money to your alma

mater to fund an annual $30,000 graduation party.

Given an interest rate of 8% per year, the required donationwas the present value of PV=$30,000/0.08=$375,000.

Before accepting the money, however, the studentassociation has asked that you increase the donation toaccount for the effect of inflation on the cost of the party infuture years.

Although $30,000 is adequate for next years party, thestudents estimate that the partys cost will rise by 4% per

year thereafter. To satisfy their request, how much do you need to donate

now?



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g g p y(contd)

Solution:

Plan: The cost of the party next year is $30,000, and the cost

then increases 4% per year forever. From the timeline, we

recognize the form of a growing perpetuity and can value itthat way.



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g g p y(contd)

Execute:

To finance the growing cost, you need toprovide the present value today of:

PV $30,000 /(0.08 0.04) $750,000 today



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g g p y(contd)

Evaluate:

You need to double the size of your gift!

Example 4.6aEndowing a Growing Perpetuity


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Endowing a Growing Perpetuity

Problem: In Example 4.3a, you planned to donate $4 million to your

alma mater to fund an endowed professorship.

Given an interest rate of 7% per year, the professor wouldbe able to collect $200,000 per year from your generosity.

The inflation rate is expected to be 2% per year. How much can the professor be paid in the first year in

order to allow her annual salary to increase by 2% eachyear and keep the principal intact?



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g g(contd)

Solution:

Plan: The salary needs to increase 2% per year forever. From the

timeline, we recognize the form of a growing perpetuity and

can value it that way.



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g g p y(contd)

Evaluate: She can only withdraw $120,000 in her first year.

In the second year, her payment will be $120,000X 1.02 = $122,400 and the payments will

continue to increase each year.



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Present Value of a Growing Annuity A growing annuity is a stream of N

growing cash flows, paid at regularintervals

It is a growing perpetuity that eventuallycomes to an end.



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The following timeline shows a growingannuity with initial cash flow C, growing ata rate of gevery period until period N:



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Present Value of a Growing Annuity:

N

1 1

1 1

g

PV= C r - g r

Example 4.7Retirement Savings with a Growing


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Annuity

Problem: In Example 4.5, Ellen considered saving $10,000 per year

for her retirement. Although $10,000 is the most she cansave in the first year, she expects her salary to increaseeach year so that she will be able to increase her savings by

5% per year. With this plan, if she earns 10% per year onher savings, how much will Ellen have saved at age 65?



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Annuity (contd)

Solution:Plan:Her new savings plan is represented by the following

timeline:

This example involves a 30-year growing annuity with a growthrate of 5% and an initial cash flow of $10,000. We can use Eq.4.8 to solve for the present value of a growing annuity.



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Annuity (contd)

Execute:The present value of Ellens growing annuity isgiven by:

30

1 1.05$10,000 10.10 - 0.05 1.10

$10,000 15.0463

$150,463today

PV=



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Annuity (contd)

30$150,463 1.10

$2.625

FV=

million in 30 years

Execute:Ellens proposed savings plan is equivalent tohaving $150,463 in the bank today. To determine

the amount she will have at age 65, we need tomove this amount forward 30 years:



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Annuity (contd)

Evaluate:Ellen will have saved $2.625 million at age 65 usingthe new savings plan. This sum is almost $1 million

more than she had without the additional annualincreases in savings.

Because she is increasing her savings amount each

year and the interest on the cumulative increasescontinues to compound, her final savings is muchgreater.

Example 4.7aRetirement Savings with a Growing


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Annuity

Problem: In Example 4.5a, Adam considered saving $10,000 per year

for his retirement. Although $10,000 is the most he cansave in the first year, he expects his salary to increase eachyear so that he will be able to increase his savings by 4%

per year. With this plan, if he earns 10% per year on hissavings, how much will Adam have saved at age 65?



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Annuity (contd)

Solution:Plan:His new savings plan is represented by the following

timeline:

This example involves a 20-year growing annuity with a growthrate of 4% and an initial cash flow of $10,000. We can use Eq.4.8 to solve for the present value of a growing annuity.


i ( d)


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Annuity (contd)

Execute:The present value of Adams growing annuity isgiven by:

20

1 1.04$10,000 1

0.10 - 0.04 1.10

$10,000 11.2384

$112,384today

PV=

Example 4.7aRetirement Savings with a GrowingA i ( d)


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Annuity (contd)

40$112,384 1.10

$5,086,416

FV=

in 40 years

Execute:Adams proposed savings plan is equivalent tohaving $112,384 in the bank today. To determine

the amount he will have at age 65, we need to movethis amount forward 40 years:


i ( d)


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Annuity (contd)

Evaluate:Adam will have saved $5.086 million at age 65using the new savings plan. This sum is over $1

million more than he had without the additionalannual increases in savings.

Because he is increasing his savings amount each

year and the interest on the cumulative increasescontinues to compound, his final savings is muchgreater.

4.5 Solving for Variables Other ThanPresent Value or Future Value


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In some situations, we use the presentand/or future values as inputs, and solvefor the variable we are interested in.

We examine several special cases in thissection.



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CP

1r

1 1(1 r)N

(Eq. 4.8)

Solving for Cash Flows

Example 4.8Computing a Loan Payment


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Problem: Your firm plans to buy a warehouse for $100,000.

The bank offers you a 30-year loan with equal annualpayments and an interest rate of 8% per year.

The bank requires that your firm pay 20% of the purchaseprice as a down payment, so you can borrow only $80,000.

What is the annual loan payment?

Example 4.8Computing a Loan Payment (contd)


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Solution:Plan: We start with the timeline (from the banks

perspective):

Using Eq. 4.8, we can solve for the loan payment,C, given N=30, r = 8% (0.08) and P=$80,000



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Execute: Eq. 4.8 gives the payment (cash flow) as follows:

C

P

1

r1

1

(1 r)N

80, 000

1

0.081

1

(1.08)30

$7106.19



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Execute (contd): Using a financial calculator or Excel:

Given: 30 8.0 -80,000 0

Solve for: 7106.19

Excel Formula: =PMT(RATE,NPER, PV, FV) =PMT(0.08,30,-80000,0)



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Evaluate: Your firm will need to pay $7,106.19 each year to repay the

loan.

The bank is willing to accept these payments because thePV of 30 annual payments of $7,106.19 at 8% interest rate

per year is exactly equal to the $80,000 it is giving youtoday.

Example 4.8aComputing a Loan Payment


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Problem: Suppose you accept your parents offer of a 2007 BMW M6convertible, but thats not the kind of car you want.

Instead, you sell the car for $61,000, spend $11,000 on aused Corolla, and use the remaining $50,000 as a downpayment for a house.

The bank offers you a 30-year loan with equal monthlypayments and an interest rate of 6% per year, and requiresa 20% down payment.

How much can you borrow, and what will be the paymenton the loan?

Example 4.8aComputing a Loan Payment (contd)


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Solution:Plan: To calculate the amount we can borrow, we need to find out

what amount $50,000 is 20% of:

$50,000 = .2 X Value

Value = $50,000/.2 = $250,000

Because youll be putting $50,000 down, your loan amountwill be $250,000 - $50,000 = $200,000.



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Solution:Plan: We start with the timeline:

Note, we need to use the monthly interest rate. Since thequoted rate is an APR, we can just divide the annual rate by

12:r = .06/12 = .005



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Execute: Eq. 4.8 gives the payment (cash flow) as follows:

= $1,199.10



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Execute (contd): Using a financial calculator or Excel:

Given: 360 0.5 200,000 0

Solve for: -1199.10

Excel Formula: =PMT(RATE,NPER, PV, FV) =PMT(0.005,360,200000,0)



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Rate of Return The rate of return is the rate at which the

present value of the benefits exactly offsets thecost.



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Suppose you have an investmentopportunity that requires a $1000investment today and will pay $2000 in sixyears.

What interest rate, r, would you need sothat the present value of what you get isexactly equal to the present value of whatyou give up?

6

20001000(1 r)



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Rearranging:

6

1

6

1000 (1 r) 2000

20001 r

1000

1.1225,or

r 12.25%



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Suppose your firm needs to purchase anew forklift.

The dealer gives you two options:

A price for the forklift if you pay cash

($40,000)

The annual payments if you take out a loanfrom the dealer (no money down and fourannual payments of $15,000).



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Setting the present value of the cash flowsequal to zero requires that the presentvalue of the payments equals the purchaseprice:

The solution for r is the interest ratecharged by the dealer, which you can

compare to the rate charged by your bank.

4

1 140,000 15,000 1r (1 r)



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There is no simple way to solve for theinterest rate.

The only way to solve this equation is toguess at values for r until you find the

right one.

An easier solution is to use a financialcalculator or a spreadsheet.



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Given: 4 40,000 -15,000 0

Solve for: 18.45

Excel Formula: =RATE(NPER,PMT,PV,FV)=Rate(4,-25000,40000,0)

Example 4.9Computing the Rate of Return with aFinancial Calculator


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Problem: Lets return to the lottery example (Example 4.4).

How high of a rate of return do you need to earn investingon your own in order to prefer the $15 million payout?



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Solution:Plan: We need to solve for the rate of return that

makes the two offers equivalent.

Anything above that rate of return would makethe present value of the annuity lower than the$15 million lump sum payment and

anything below that rate of return would make itgreater than the $15 million.



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Execute:

The rate equating the two options is 5.72%.

Given: 29-

14,000,0001,000,000 0

Solve for: 5.72

Excel Formula: =RATE(NPER, PMT, PV,FV) =RATE(29,1000000,-14000000,0)



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Evaluate: 5.72%is the rate of return that makes giving up the $15

million payment and taking the 30installments of $1million exactly a zero NPV action.

If you could earn more than 5.72%investing on your own,

then you could take the $15 million, invest it and generatethirty installments that are each more than $1 million.

If you could not earn at least 5.72%on your investments,you would be unable to replicate the $1 millioninstallments on your own and would be better off taking the

installment plan.

Example 4.9aComputing the Internal Rate of Returnwith a Financial Calculator


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Problem: Lets return to the BMW example (Example 4.4a).

What rate of return would make you indifferentbetween the car and the $10,000 per year payout

(even if the car is your favorite color and has HDradio)?



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Solution:Plan: We need to solve for the rate of return that

makes the two offers equivalent.

Anything above that rate of return would makethe present value of the annuity lower than the$61,000 car and

anything below that rate of return would make itgreater than the $61,000.



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Execute:

The rate equating the two options is 4.85%.

Given: 6 -51,000 10,000 0

Solve for: 4.85%

Excel Formula: =RATE(NPER, PMT, PV,FV) = RATE(6,10000,-61000,0)



180/192


Evaluate: 4.85% is the rate of return that makes giving up the

$61,000 car and taking the 7 installments of $10,000exactly a zero NPV action.

If you can earn more than 4.85% investing on your own,

then you can take the $61,000, invest it and generateseven installments that are each more than $10,000.

If you can not earn at least 4.85% on your investments,you would be unable to replicate the $10,000 installmentson your own and would be better off taking the generous

payments your parents have offered.



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Solving for the Number of Periods In addition to solving for cash flows or the

interest rate, we can solve for the amount oftime it will take a sum of money to grow to a

known value. In this case, the interest rate, present value,

and future value are all known.

We need to compute how long it will take for

the present value to grow to the future value.

Example 4.10Solving for the Number of Periods in aSavings Plan


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Problem: Lets return to saving for a down payment on a

house.

Imagine that some time has passed and you have

$10,050 saved already, and you can now affordto save $5,000 per year at the end of each year.

Also, interest rates have increased so that younow earn 7.25% per year on your savings.

How long will it take you to get to your goal of$60,000?



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Solution:Plan: The timeline for this problem is



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Plan (contd): We need to find N so that the future value of our current

savings plus the future value of our planned additional

savings (which is an annuity) equals our desired amount.

There are two contributors to the future value: the initial

lump sum $10,050 that will continue to earn interest, and

the annuity contributions of $5,000 per year that will earn

interest as they are contributed.

Thus, we need to find the future value of the lump sum plus

the future value of the annuity



185/192


Execute:

Given: 7.25 -10,050 -5,000 60,000

Solve for: 7

Excel Formula: =NPER(RATE,PMT, PV, FV) =NPER(0.0725,-5000,-10050,60000)



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Evaluate: It will take seven years to save the down payment.

Example 4.10aSolving for the Number of Periods in aSavings Plan


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Problem: Lets return to Ellen and Adam.

Suppose Ellen decides she will continue workinguntil she has as much at retirement as her

brother, Adam, will have when he retires. She will continue to contribute $10,000 each year

to her retirement account.

How much longer will she need to work to tie the

competition with her brother?



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Solution:Plan:

The timeline for this problem is



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Plan (contd): We need to find N so that the FV of the $1,645,000 shellhave at age 65 plus the $10,000 shell contribute each year

is equal to $3,850,000.

Remember, shes earning 10% on her investments.



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Execute:

Given: 10 -1645000 -10,000 3850000

Solve for: 8.57

Excel Formula: =NPER(RATE,PMT, PV, FV) =NPER(0.10,-10000,-1645000,3850000)



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Evaluate: Ellen will have to work until shes 73 years old. (Heres hoping she really loves her job!)

Chapter Quiz


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1. How do you calculate the present value of a cash flowstream?

2. What is the intuition behind the fact that an infinitestream of cash flows has a finite present value?

3. What are some examples of annuities?

4. What is the difference between an annuity and aperpetuity?

5. What is an example of a growing perpetuity?

6. How do you calculate the cash flow of an annuity?

Finance - Time value of money

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