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FINANCE

FIN2004

Lecture 3: Time Value of Money

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Time Value Analysis As we discussed, cash is crucial in financial valuation. One of the

main problems in finance is determining what value we should place

on prospective cash profits that will accrue to a project/ investment.

This has two components: time and uncertainty. For now we will

assume away uncertainty and concentrate onhow to devise

techniques for evaluating the worth of certain cash flows that will

come in at various dates in the future.

This is crucial, because it will allow us to compare the value ofprofits that may be coming in at very different times.

The main insight that we will start with is thata dollar paid today is

worth more than a dollar paid tomorrow.

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Learning Objectives Be able to compute thefuture value of an investmentmade

today Be able to compute thepresent value of cash to be received

at somefuture date

Be able to draw and explain the use of a time line

Be able to solve time value of money problems

Be able to explain what is an amortized loan and calculate

theperiodic payments of such a loan

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Lecture Outline

1. Basic Definitions and Concepts

2. Future Value and Compounding

3. Present Value and Discounting

4. Multiple Cash Flows

5. Annuities

6. Effective Annual Rate vs. Nominal Rate

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$1 received today ispreferredto $1 received

some time in the future. Why?

lost earnings: can invest the money to earn

interest

loss of purchasing power: because of the presence

of inflation

trade-off depends on therate of return

What is the Time Value of Money?

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Year 0 Year 1 Year 2

Today

End of

year 1

End of

year 2

Beginning

of year 1

Beginning

of year 2

Beginning

of year 3

0 1 2 3

Timelines

Drawing timelines can be helpful in understanding time value of

money problems

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Basic Definitions: PV and FV Present Value (PV) the value of something today. On

a timeline t = 0. Present Value is also referred to as the

market value of a cash flow to be received in the future.

Translating a value that comes at some point in the

future to its value in the present is referred to as

discounting.

Future Value (FV) the value of a cash flow sometimein the future. On a timeline t > 0.

Translating a value to the future is referred to as

compounding.

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Meaning Of Future And Present Values

Keep in mind that all that present values and

future values do is to put cash flows which

come in at different times on a

COMPARABLE BASIS!

Once they are then in the same units we will

be able to compare and make decisions on

which pattern of cash flows are preferable.

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Annuities & PerpetuitiesAn annuity is the word used to describe a series of cash flows in

which thesame cash flow CF (or payment) takes place each

period for a set number of periods.

An ordinaryannuity is one in which the first cash flow occurs

one period from now (end of the period 1).

Anannuity due is an annuity in which the first cash flow occurs

immediately (at the beginning of period 1).

Aperpetuity is simply a set of equal payments that are paidforever.

Agrowing perpetuity, is a set of payments which grow at a

constant rate each period and continue forever.

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Theprincipalin a loan context is the original amount

borrowed.

Interest is the compensation for the opportunity cost

of funds and the uncertainty of repayment of the

amount borrowed. Sometimes, referred to as:

Discount rate

Cost of capital

Opportunity cost of capital

Required return

More Definitions

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More Timelines

CF0 CF1 CF3CF2

0 1 2 3i%

Time line for a $100 lump sum due at the end of Year 2

100

0 1 2 Year i%

General Timeline

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Time line for an ordinary annuity of $100 for 3 years

100 100100

0 1 2 3i%

Time line foruneven CFs: -$50 at t = 0 and $100, $75, and

$50 at the end of Years 1 through 3.

100 5075

0 1 2 3i%

- 50

A negative sign means it is a cash outflow

More Timelines

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Back to Future ValuesWe can think of Future Value as an amount to

which an investment will grow after earninginterest.

Simple Interest: Interest earned only on the original

investment.

Compound Interest: In addition to interest earned on

the original investment, interest is also earned on

interest previously received (on the original

investment).

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Savings Example: Simple Interest Today you deposit $100 into a fixed deposit

account paying 5% simple interest. How muchshould you have in 5 years?

Solution: $100 + 5 years * 100(5%) = $125

Interest earned per year: $5

Principal

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Example: Simple Interest

Year 1: 5% of $100 = $5 + $100 = $105

Year 2: 5% of $100 = $5 + $105 = $110

Year 3: 5% of $100 = $5 + $110 = $115

Year 4: 5% of $100 = $5 + $115 = $120Year 5: 5% of $100 = $5 + $120 = $125

Interest earned per year: $5

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Simple Interest

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Example: Compound Interest

Suppose now you deposit your $100 into a Savings Deposit where interest is

earned at 5% on the previous years balance:

Interest Earned Per Year =Prior Year Balance x 5%

After 1 year:

FV1 = PV + INT1 = PV + PV (i)

= PV(1 + i) = $100(1.05)= $105.00

After 2 years (Displaying the Meaning of Compounding):

In General, FVn = PV(1 + i)n

Original invested amount

FV2 = PV(1 + i)2

= PV(1 + 2i + i * i)= PV(Principal + 2 years of Simple Interest + Interest on Interest)

= $100(1.05)2 = $110.25

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Compound Interest

Year 1: 5% of $100.00 = $5.00 + $100.00 = $105.00

Year 2: 5% of $105.00 = $5.25 + $105.00 = $110.25

Year 3: 5% of $110.25 = $5.51 + $110.25 = $115.76

Year 4: 5% of $115.76 = $5.79 + $115.76 = $121.55

Year 5: 5% of $121.55 = $6.08 + $121.55 = $127.63

Interest earned per year increases from $5

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Compound Interest

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Effects of CompoundingSimple interest vs. Compound interest

Consider the previous example of a 2 year investment:

FV with simple interest = 100 + 5 + 5 = 110

FV with compound interest = 110.25

The extra 0.25 comes from the interest of .05(5) = 0.25

earned on the first interest payment

The effect of compounding is small for a small number of

periods, but increases as the number of periods increases.

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Another Compounding Example Suppose one of your ancestors deposited $10 at

5.5% interest 200 years ago. How much would theinvestment be worth today?

FV = 10(1.055)

200

= 447,189.84 What is the effect of compounding?

Simple interest = 10 + 200(10)(.055) = 120.00

Compounding added $447,069.84 to the value of the

investment!

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Future Values: General Formula

FV = PV(1 + r)t = PV (1+i)n

FV = future value

PV = present value

r = i = period interest rate, expressed as a decimal (e.g. 5% =.05)

t = n = number of periods

Future value interest factor = (1 + r)t(FVIF)

FV = ?

0 1 2 35%

CF

Finding FVs (moving to the right on a time line) is calledcompounding.

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Example: Future Value as a General Growth

Suppose your company expects to increase unit

sales by 15% per year for the next 5 years. If you

currently sell 3 million cars in one year, how

many cars do you expect to sell in 5 years?

FV = 3,000,000(1.15)5 = 6,034,072

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1

23

456

789

1 01 11 21 3

1 41 51 6

1 71 8

1 92 02 12 2

2 32 42 5

2 62 72 8

2 93 03 1

3 23 33 43 5

3 63 7

3 83 94 0

4 14 2

4 34 4

A B C D E

In it ia l d e p o s it 1 0 0In te re s t ra te 0 % 6 % 1 2 %

Y e a r F V a t 0 % F V a t 6 % F V a t 1 2 %0 1 0 0 .0 0 1 0 0 .0 0 1 0 0 .0 0 < - - = $ B $ 2 * (1 + D $ 3 )^ $ A 6

1 1 0 0 .0 0 1 0 6 .0 0 1 1 2 .0 0 < - - = $ B $ 2 * (1 + D $ 3 )^ $ A 72 1 0 0 .0 0 1 1 2 .3 6 1 2 5 .4 43 1 0 0 .0 0 1 1 9 .1 0 1 4 0 .4 9

4 1 0 0 .0 0 1 2 6 .2 5 1 5 7 .3 55 1 0 0 .0 0 1 3 3 .8 2 1 7 6 .2 36 1 0 0 .0 0 1 4 1 .8 5 1 9 7 .3 87 1 0 0 .0 0 1 5 0 .3 6 2 2 1 .0 7

8 1 0 0 .0 0 1 5 9 .3 8 2 4 7 .6 09 1 0 0 .0 0 1 6 8 .9 5 2 7 7 .3 1

1 0 1 0 0 .0 0 1 7 9 .0 8 3 1 0 .5 8

1 1 1 0 0 .0 0 1 8 9 .8 3 3 4 7 .8 51 2 1 0 0 .0 0 2 0 1 .2 2 3 8 9 .6 0

1 3 1 0 0 .0 0 2 1 3 .2 9 4 3 6 .3 51 4 1 0 0 .0 0 2 2 6 .0 9 4 8 8 .7 11 5 1 0 0 .0 0 2 3 9 .6 6 5 4 7 .3 61 6 1 0 0 .0 0 2 5 4 .0 4 6 1 3 .0 4

1 7 1 0 0 .0 0 2 6 9 .2 8 6 8 6 .6 01 8 1 0 0 .0 0 2 8 5 .4 3 7 6 9 .0 01 9 1 0 0 .0 0 3 0 2 .5 6 8 6 1 .2 8

2 0 1 0 0 .0 0 3 2 0 .7 1 9 6 4 .6 3

F U T U R E V A L U E O F A S IN G L E P A Y M E N T A T D IF F E R E N T IN T E R E S T R A T E S

H o w $ 1 0 0 a t t im e 0 g r o w s a t 0 % , 6 % , 1 2 %

0

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

6 0 0

7 0 0

8 0 0

9 0 0

1 0 0 0

0 5 1 0 1 5 2 0

F V a t 0 %F V a t 6 %

F V a t 1 2 %

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Four Ways to Find FVs

1. Solve the equation with a regular calculator.

2. Use Future Value tables.

3. Use a financial/programmable calculator.

4. Use EXCEL spreadsheet.

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Future Value: Using a FV Table A loaf of bread today costs $1.28. If grocery

prices are going up at the rate of 6% per year, how

much will a loaf of bread cost in 5 years?

Solution: $1.28 x 1.3382 = $1.71

This future value factor was found from the FV

Table: look under the column for interest of6%

and the row for period of5 years

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Future Value Table

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Financial Calculator Solution

Financial calculators solve this equation:

There are 4 variables. If 3 are known, the calculator will

solve for the 4th.

ni1PVFV

Excel: For an Excel Solution Example, please see slide 23.

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Quick Review Part 1 What is the difference between simple interest

and compound interest?

Suppose you have $500 to invest and you believe

that you can earn 8% per year over the next 15

years.

How much would you have using simple interest?

$1100 How much would you have at the end of 15 years

using compound interest? $1,586.08

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Present Values

How much do I have to invest today to have some amount in

the future?

We re-arrange FV = PV(1 + r)n

to solve forPV = FV / (1 + r)n

When we talk aboutdiscounting, we mean finding the

present value of some future amount. It is the reverse of

compounding.

When we simply talk about the value of something, we are

talking about thepresent value. If we want future value, we

specifically indicate that we want the future value.

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n

n i+1

1FV=

i+1

FV=PV

PV = $1001

1.10

= $100 0.7513 = $75.13.

3

PV Factor

PV Factor for n = 3 and i = 10%

Present Value Generally:

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Present Value One Period Example Suppose you need $10,000 in one year for the down

payment on a new car. If you can earn 7% annually,how much do you need to invest today?

PV = 10,000 / (1.07)1 = 9345.79

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Present Value Example 2

You want to begin saving for your daughters

university education and you estimate that she will

need $150,000 in 17 years. If you feel confident

that you can earn 8% per year, how much do youneed to invest today?

FV = $150,000, r = 8%, n = 17

PV = 150,000 / (1.08)17 = 40,540.34

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Present Values Example 3 Your parents set up a trust fund for you 10 years ago

that is now worth $19,671.51. If the fund earned 7%per year, how much did your parents invest?

We know the value today is $19,671.51 and we wantto know what it was worth 10 years ago if the annual

compound interest it has been receiving is 7%, so

PV = 19,671.51 / (1.07)10 = 10,000

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Present Value Important Relationship 1 For a given amount to be received in the future and

for a given interest rate available the longer into thefuture that amount is to be received, the lower the

present value of that amount

E.g., What is the present value of $500 to be

received in 5 years? 10 years? The discount rate is

10%

5 years: PV = 500 / (1.1)5 = 310.46

10 years: PV = 500 / (1.1)10 = 192.77

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Quick Review Part 2

What is the relationship between present value

and future value? Suppose you need $15,000 in 3 years. If you can

earn 6% annually, how much do you need toinvest today? $12,594.29

If you could invest the money at 8%, would youhave to invest more or less than if you invested it

at 6%? How much? Less $686.81

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Multiple Cash Flows Example

Suppose you plan to deposit $100 into an account

in one years time and $300 into the account in

three years from now. How much will be in the

account in five years if the interest rate is 8%?

FV = 100(1.08)

4

+ 300(1.08)

2

= 136.05 + 349.92= 485.97

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Example: Multiple Cash Flows0 1 2 3 4

200 400 600 800

i = 12%

Find the PV of each cash flow and then sum the cash flows Year 1 CF: 200 / (1.12)1 = 178.57

Year 2 CF: 400 / (1.12)2 = 318.88

Year 3 CF: 600 / (1.12)3 = 427.07

Year 4 CF: 800 / (1.12)4 = 508.41

Total PV = 178.57 + 318.88 + 427.07 + 508.41 = 1432.93

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Example: Multiple Cash Flows Timeline

0 1 2 3 4

200 400 600 800178.57

318.88

427.07

508.41

1432.93 = PV

i = 12%

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Ordinary Annuity

PMT PMTPMT

0 1 2 3i%

PMT PMT

0 1 2 3i%

PMT

Annuity Due

Difference Between Annuity &Annuity Due:

PV FV

The difference between the two is (1+i) .

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Example: PV of an Ordinary Annuity

100 100100

0 1 2 310%

90.91

82.64

75.13

248.69 = PV

Whats the PV of a 3-year ordinary annuity of

$100 at 10%?

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Have payments but no lump sum FV, so enternothing or 0 for future value

3 10 100 0

N I/YR PV PMT FV

-248.69

INPUTS

OUTPUT

Example: PV of an Ordinary Annuity

Using a Financial Calculator:

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PV of an Ordinary Annuity

PV Annuity Due = PV Annuity * (1+r)

Annuity PV =

r

FactorPV-1*PMT

RegularLump Sum

PV Factor=

tr1

1

nn rr

PMTrrrr )1(

11*

1*

)1(

1...

)1(

1

1

1*PMT

FactorPV-1*PMT

2

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DEPOSITS AT END OF YEAR

Beginning

of year 1

Beginning

of year 2

Beginning

of year 3

Beginning

of year 4

Beginning

of year 5

Beginning

of year 6

Beginning

of year 7

Beginning

of year 8

Beginning

of year 9

Beginning

of year 10

End of

year 10

0 1 2 3 4 5 6 7 8 9 10

$100 $100 $100 $100 $100 $100 $100 $100 $100 $100

168.95

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Recall that aperpetuity is a set of equal payments that are paid forever.

If the periodic payment is $C, then the present value of the perpetuity

is:

Agrowing perpetuity, is a set of payments which grow at a constantrate forever. For example:

If the first payment is $C1, then the present value of the perpetuity is:

r

CPV

gr

CPV 1

Perpetuities Revisited

C C g2 1 1 ( ) C C g C g3 2 121 1 ( ) ( )and

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Implied Discount Rate Sometimes we will want to know what the implied

interest rate is for an investment We can re-arrange the basic PV and FV equations and

then solve for the implied interest rate i

FV = PV(1 + i)n

i = (FV / PV)1/n 1

More on discount rates later!

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Effective Annual Rate (EAR) The effective annual rate of interest refers to the actual rate

paid (or received) after taking into consideration any

compounding that may occurduring the year. If interest is

compoundedmore than once a year, then the stated rate will

be different than the effective rate. If interest is compounded

(or applied) exactly once a year, then the effective annual rate

will be equal to the stated rate

If you want to compare two alternative investments with

different compounding periods you need to compute the EAR

and use that for comparison.

Annual Percentage Rate (APR)

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Annual Percentage Rate (APR)

(Nominal Annual Rate or Quoted Rate or Stated Rate)

This is the annual rate that is quoted by law

By definition APR = period rate * the number of periodsper year

Consequently, to get the period rate we rearrange the APR

equation:

Period rate = APR / number of periods per year

Note that you should NEVER divide the effective rate by

the number of periods per year it will NOT give you the

period rate

C ti APR

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Computing APRs

What is the APR if the monthly rate is 0.5%?

0.5% * (12) = 6%

What is the APR if the semiannual rate is 4%?

4% * (2) = 8%

What is the monthly rate if the APR (based on the monthlyrate) is 12%?

12% / 12 = 1%

Can you divide the above APR by 2 to get the semiannualrate? NO!!! You need an APR based on semiannual

compounding to find the semiannual rate.

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Things to Remember You ALWAYS need to make sure that the interest rate

and the time periodmatch.

If you are looking at annual periods, you need an annual

rate.

If you are looking at monthly periods, you need a monthlyrate.

E.g., If you have an APR based on monthly

compounding, you have to use monthly periods for lumpsums, or adjust the interest rate appropriately if you have

payments other than monthly

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Example: Computing EARs

Suppose you can earn 1% per month on $1 invested today.

What is the APR? 1%*(12) = 12%

How much are you effectively earning?

FV = 1(1.01)12 = 1.1268

Rate = (1.1268 1) / 1 = .1268 = 12.68% Suppose if you put it in another account, you earn 3% per quarter.

What is the APR? 3%*(4) = 12%

How much are you effectively earning?

FV = 1(1.03)4 = 1.1255

Rate = (1.1255 1) / 1 = .1255 = 12.55%

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EAR General Formula

1

m

APR1EAR

m

m = compounding frequency per year APR is the quoted or stated rate

EAR = - 1(1 + )APR

m

m

= ( 1 + )2 - 10.102

= (1.05)2 - 1.0

= 0.1025 = 10.25%.

Example: How do we find EAR for a nominal rate of 10%, compounded

semiannually?

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Example: Computing Effective Annual Rates

Provided below is the amount earned on a dollar invested

at APR = 8% at different compounding intervals:

Compounding

Intervals

Final Sum EAR

1 $1.0800 8.00%

2 $1.0816 8.16%

12 $1.0830 8.30%

365 $1.0833 8.33%

$1.0833 8.33%

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Comparing Savings Accounts You are looking at two savings accounts. One pays 5.25%,

with daily compounding. The other pays 5.3% withsemiannual compounding. Which account should you use?

First account:

EAR = (1 + .0525/365)365 1 = 5.39%

Second account:

EAR = (1 + .053/2)2 1 = 5.37%

Which account should you choose and why?

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Finding the Number of Periods

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Finding the Number of Periods

Start with basic equation and solve for t

(remember your logs)

FV = PV(1 + r)t

ln FV = ln PV(1+r)t = ln PV + ln (1+r)t

ln FV ln PV = t * ln (1+r)

t = ln(FV / PV) / ln(1 + r)

You can use the financial keys on the calculator as

well.

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Number of Periods Example Suppose you want to buy a new house. You currently have

$15,000. To buy the house youneedto have a 10% down

paymentplus an additional 5% in closing costs.* If the

type of house you want costs about $150,000 and you can

earn 7.5% per year, how long will it be before you haveenough money for the down payment and closing costs?

*Note: closing costs are only paid on the loan amount, not onthe total amount paid for the house.

Number of Periods Example Continued

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Number of Periods Example Continued

How much do you need to have in the future?

Down payment = .1(150,000) = 15,000

Closing costs = .05(150,000 15,000) = 6,750

Total needed = 15,000 + 6,750 = 21,750

Compute the number of periods

PV = -15,000

FV = 21,750

I/Y = 7.5

Compute N = 5.14 years

Using the formula

t = ln(21,750 / 15,000) / ln(1.075) = 5.14 years

Summary

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Appendix: Additional Slides

Please review these slides on your own

Different Types of Loans

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Different Types of Loans

a. Pure Discount Loans

No interest; principal paid at maturity; issued at discount

b. Interest Only Loans

Interest paid throughout the loan period; principal paid at

maturity

c. Loans with Fixed Principal Payments

Interest and fixed amount of principal paid throughout the

loan period.

d. Amortized Loans

Interest and a portion of the principal paid throughout the

loan period

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a. Pure Discount Loans

PrincipalPayment

Yr 1 Yr 2 Yr 3

$

InterestLoan Inception

Pure Discount Loans - Example

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Pure Discount Loans - Example

Treasury bills are excellent examples of pure

discount loans. The principal amount is repaid atsome future date, without any periodic interest

payments.

If a T-bill promises to repay $10,000 in 12

months and the market interest rate is 7 percent,

how much will the bill sell for in the market?PV = 10,000 / 1.07 = 9345.79

b Interest Only Loan

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b. Interest Only Loan

Yr 1

$

Interest

PrincipalPayment

Yr 2 Yr 3

Interest InterestInterest

b. Interest Only Loan - Example

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b. Interest Only Loan Example

Consider a 3-year, interest only loan with a 7%

interest rate. The principal amount is $10,000.

Interest is paid annually.

What would the stream of cash flows be?

Years 1 2: Interest payments of .07(10,000)= 700

Year 3: Interest + principal = 10,700

This cash flow stream is similar to the cash flows

on corporate bonds.

c. Loan with Fixed Principal Payment

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Each payment covers the interest expense plus a

fixedprincipal portion

$

Interest

Principal Payments

Yr 1 Yr 2 Yr 3

Interest

Interest

c. Loan with Fixed Principal Payment

Loan with Fixed Principal Payment -

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p y

Example

Consider a $50,000, 10 year loan at 8% interest. The

loan agreement requires the firm to pay $5,000 in

principal each year plus interest for that year.

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d. Amortized Loan

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Each equalpayment covers the interest expense

plus reduces principal

$Interest

Principal Payments

Yr 1 Yr 2 Yr 3

Interest

Interest

Amortized Loan - Example

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p

Each payment covers the interest expense plus

reduces principal

Consider a 4 year loan with annual payments. The

interest rate is 8% and the principal amount is $5000.

What is the annual payment (via financial calculator)?

4

8 5000

-1509.60

Fixed Payment Amortization Schedule

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Year BeginningBalance

TotalPayment

InterestPaid

PrincipalPaid

EndingBalance

1 5,000.00 1,509.60 400.00 1,109.60 3,890.40

2 3,890.40 1,509.60 311.23 1,198.37 2,692.03

3 2,692.03 1,509.60 215.36 1,294.24 1,397.79

4 1,397.79 1,509.60 111.82 1,397.78 0.01

Totals 6,038.40 1,038.41 4,999.99

Quick Review Part 3

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Q

What is a pure discount loan? What is a good

example of a pure discount loan?

What is an interest only loan? What is a good

example of an interest only loan?

What is an amortized loan? What is a good

example of an amortized loan?

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Implied Discount Rate Example 2

Suppose you are offered an investment that will

allow you to double your money in 6 years. What

is the implied rate of interest?

FV = PV (1+r)t

FV = 2, PV = 1, t = 6

r = (2/1)1/6 1 = .122462 = 12.25%

Implied Discount Rate Example 3

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Suppose you have a 1-year old daughter and you

want to provide $75,000 in 17 years towards hercollege education. You currently have $5000 to

invest. What interest rate must you earn to have

the $75,000 when you need it?

FV = PV (1+r)

t

FV = 75, PV = 5, t = 17

r = (75,000 / 5,000)

1/17

1 = .172688 = 17.27%

Final Example

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Whats the value at the end of Year 3 of the following CF stream

if the quoted interest rate is 10%, compounded semiannually?

0 1

100

2 3

5%

4 5 6 6 monthperiods

100 100

APR

Payments occur annually, but compounding occurs each 6

months.

So we cant use normal annuity valuation techniques.

1st Method: Can Compound Each Cash Flow

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p

0 1

100

2 3

5%

4 5 6

100 100.00110.25121.55331.80

FV3 = $100(1.05)4 + $100(1.05)2 + $100= $331.80

Well get back to this again in the next slide.

2nd Method: Can Treat as an Annuity

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Find the EAR for the quoted rate:

y

EAR = (1 + ) - 1 = 10.25%.0.1022

Whats the PV of this stream of cash flows?

0

100

15%

2 3

100 100

90.70

82.2774.62

247.59

and PV = $331.80 * PVIF 3,10.25% = $247.59

PV =