Chapter 5 - PowerPoint Presentation

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Chapter 5 - PPT Presentation

The Time Value of Money Laurence Booth Sean Cleary and Pamela Peterson Drake Outline of the chapter 51 Time value of money Simple interest Simple interest is interest that is paid only on the principal amount ID: 437019

interest 000 annuity year 000 interest year annuity years rate today problem account cash ear compounding 6000 suppose compounded

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Slide1

Chapter 5The Time Value of Money

Laurence Booth, Sean Cleary and Pamela Peterson DrakeSlide2

Outline of the chapterSlide3

5.1 Time value of moneySlide4

Simple interest

is interest that is paid only on the principal amount.

Interest = rate × principal amount of loanSlide5

Simple interest: example

2-year loan of $1,000 at 6% simple interest

At the end of the first year,

interest = 6% × $1,000 = $60

At the end of the second year,

interest = 6% × $1,000 = $60

and loan

repayment of $1,000Slide6

Compound interest

is interest paid on both the principal and any accumulated interest.

Interest =

Slide7

Compounding

is translating a present value into a future value, using compound interest.

Future value interest factor

is also referred to as the

compound factor

Slide8

Terminology and notation

Term

Notation

Meaning

Future value

Value at some specified future point in time

Present value

Value today

Interest

Compensation for the use of funds

Number of periods

nNumber of periods between the present value and the future valueCompound factor(1 + i)

Translates a present value into a future valueSlide9

Compare: simple versus compound

Suppose you deposit $5,000 in an account that pays 5% interest per year. What is the balance in the account at the end of four years if interest is:

Simple interest?

Compound interest?Slide10

Simple interest

Year

Beginning

Add interest

Ending

$5,000.00

+ (5% × $5,000) =

$5,250.00

+ (5% × $5,000) =

$5,500.00

+ (5% × $5,000) =

$5,750.00

+ (5% × $5,000) =

$6,000.00Slide11

Compound interest

Year

Beginning

Compounding

Ending

$5,000.00

× 1.05 =

$5,250.00

× 1.05 =

$5,512.50

× 1.05 =

$5,788.13

× 1.05 =

$6,077.53Slide12

Interest on interest

How much interest on interest?

Interest on interest =

compound

–

simple

Interest on interest = $6,077.53 – 6,000.00 =

$77.53Slide13

Comparison

Year

End

of year balance

Simple interest

Compound interest

$5,000.00

$5,250.00

$5,500.00

$5,512.50

$5,750.00

$5,788.13

$6,000.00

$6,077.53Slide14

Try it: Simple v. compound

Suppose you are comparing two accounts:

The Bank A account pays 5.5% simple interest.

The Bank B account pays 5.4% compound interest.

If you were to deposit $10,000 in each, what balance would you have in each bank at the end of five years?Slide15

Try it: Answer

Bank A: $12,750.00

Bank B: $13,007.78Slide16

A note about interest

Because compound interest is so common, assume that interest is compounded unless otherwise indicated.Slide17

Short-cuts

Example:

Consider

1,000

deposited for

three years

per year

.Slide18

The long way

= $1,000.00 × (1.06) = $1,060.00

1,060.00 ×

(

1.06)

= $1,123.60FV3 = $1,123.60 × (1.06) = $1,191.02orFV3

= $1,000 × (1.06)

= $1,191.02 orFV3 = $1,000 × 1.191016 = $1,191.02

Future value factorSlide19

Short-cut: Calculator

Known values:

PV = 1,000

n = 3

i = 6%

Solve for:

FVSlide20

Input three known values, solve for the one unknown

HP10B

BAIIPlus

HP12C

TI83/84

1000 +/- PV

3 N

6 I/

1000 +/- PV

3 N

6 I/

1000 CHS PV

3 n

[APPS] [Finance]

[

TVM

Solver]

N =3

I%=6

PV = -1000

FV [Alpha] [Solve}Known: PV, i , nUnknown: FVSlide21

Short-cut: spreadsheetMicrosoft Excel or Google Docs

(

RATE,NPER,PMT,PV,TYPE

)

TYPE default is 0, end of period

(.06,3,0,-1000)

-1000

(A1,A2,0,A3)Slide22

Problems Set 1Slide23

Suppose you deposit $2,000 in an account that pays 3.5% interest annually.

How much will be in the account at the end of three years?

How much of the account balance is interest on interest?

Problem 1.1

23Slide24

If you invest $100 today in an account that pays 7% each year for four years and 3% each year for five years, how much will you have in the account at the end of the nine years?

Problem 1.2

24Slide25

DiscountingSlide26

Discounting

is translating a future value into a present value.

The

discount factor

is the inverse of the compound factor:

To translate a future value into a present value, PV=

Slide27

Example

Suppose you have a goal of saving $100,000 three years from today. If your funds earn 4% per year, what lump-sum would you have to deposit today to meet your goal?Slide28

Example, continued

Known values:

= $100,000

n = 3

i = 4%

Unknown: PVSlide29

Example, continued

PV =

PV = $100,000 × 0.8889964

PV = $88,899.64

Check:

= $88,899.64 × (1 + 0.04)

= $100,000

Slide30

Short-cut: Calculator

HP10B

BAIIPlus

HP12C

TI83/84

100000 +/- PV

3 N

4 I/

100000

+/- PV

3 N

4 I/

100000 CHS PV

[APPS] [Finance] [

TVM

Solver]

N =3

I%=4

= 100000

PV [Alpha] [Solve]Slide31

Short-cut: spreadsheetMicrosoft Excel or Google Docs

=PV(

RATE,NPER,PMT,PV,TYPE

)

TYPE default: end of period

=PV(.06,3,0,-1000)

100000

=PV(A1,A2,0,A3)Slide32

Try it: Present value

What is the today’s value of $10,000 promised ten years from now if the discount rate is 3.5%?Slide33

Try it: Answer

Given:

= $10,000

N = 10

I = 3.5%

Solve for PV

PV =

= $7,089.19

Slide34

Frequency of compounding

If interest is compounded more than once per year, we need to make an adjustment in our calculation.

The

stated rate

or nominal rate of interest is the

annual percentage rate

(

APR

The rate per period depends on the frequency of compounding.Slide35

Discrete compounding:Adjustments

Adjust the number of periods and the rate per period.

Suppose the nominal rate is 10% and compounding is quarterly:

The rate per period is 10%

 4 = 2.5%

The number of periods is

number of years × 4Slide36

Continuous compounding:Adjustments

The compound factor is

APR

x n

The discount factor is

Suppose the nominal rate is 10%.

For five years, the continuous compounding factor is e

0.10 x 5

= 1.6487

The continuous compounding discount factor for five years is 1 ÷

0.10 x 5 = 0.60653 Slide37

Try it: Frequency of compounding

If you invest $1,000 in an investment that pays a nominal 5% per year, with interest compounded semi-annually, how much will you have at the end of 5 years?Slide38

Try it: Answer

Given:

PV = $1,000

n = 5 × 2 = 10

i = 0.05

 2 =

0.25

Solve for

= $1,000

× (1 + 0.025)10 = $1,280.08Slide39

Problem Set 2Slide40

Suppose you set aside an amount today in an account that pays 5% interest per year, for five years. If your goal is to have $1,000 at the end of five years, what would you need to set aside today?

Problem 2.1

40Slide41

Suppose you set aside an amount today in an account that pays 5% interest per year, compounded quarterly, for five years. If your goal is to have $1,000 at the end of five years, what would you need to set aside today?

Problem 2.2

41Slide42

Suppose you set aside an amount today in an account that pays 5% interest per year, compounded continuously, for five years. If your goal is to have $1,000 at the end of five years, what would you need to set aside today?

Problem 2.3

42Slide43

5.2 Annuities and Perpetuities

PV?

?Slide44

What is an annuity?

annuity

is a periodic cash flow.

Same amount each period

Regular intervals of time

The different types depend on the timing of the first cash flow.Slide45

Type of annuities

Type

First cash flow

Examples

Ordinary

One period

from today

Mortgage

Annuity due

Immediately

Lottery payments

Rent

Deferred annuity

Beyond one period from today

Retirement savingsSlide46

Time lines: 4-payment annuity

Ordinary

Annuity due

Deferred annuity

FVSlide47

Key to valuing annuities

The key to valuing annuities is to get the timing of the cash flows correct.

When in doubt, draw a time line.Slide48

Example: PV of an annuity

What is the present value of a series of three cash flows of $4,000 each if the discount rate is 6%, with the first cash flow one year from today?

$4,000

$4,000Slide49

Example: PV of an annuityThe long way

$4,000

$3,773.58



3,559.99



3,358.48



$10,692.05Slide50

Example: PV of an annuityIn table form

Year

Cash flow

Discount factor

Present value

$4,000.00

0.94340

$3,773.58

$4,000.00

0.89000

3,559.99

$4,000.00

0.83962

3,358.48

2.67301

$10,692.05

PV = $4,000.00 × 2.67301 = $10,692.05 Slide51

Example: PV of an annuityFormula short-cuts

PV =

PV = $4,000 × 2.67301

PV = $10,692.05

Slide52

Example: PV of an annuityCalculator short cuts

Given:

PMT

= $4,000

i = 6%

N = 3

Solve for PVSlide53

Example: PV of an annuitySpreadsheet short-cuts

PV(

RATE,NPER,PMT,FV,TYPE

)

=PV(.06,3,4000,0)

Note: Type is important for annuities

If Type is left out, it is assumed a 0

0 is for an ordinary annuity

1 is for an annuity dueSlide54

Example: FV of an annuity

What is the future value of a series of three cash flows of $4,000 each if the discount rate is 6%, with the first cash flow one year from today?

$4,000

$4,000Slide55

Example: FV of an annuity

The long way

$4,000.00



4,240.00



4,494.40

$12,734.40Slide56

Example: FV of an annuity

In table form

Year

Cash flow

Compound factor

Future value

$4,000.00

1.1236

$4,494.40

$4,000.00

1.0600

4,240.00

$4,000.00

1.0000

4,000.00

3.1836

$12,734.40

PV = $4,000.00 × 3.1836 = $12,734.40 Slide57

Example: FV of an annuity

Calculator short cuts

CALCULATOR

Given:

PMT

= $4,000

i = 6%

N = 3

Solve for

FVSlide58

Example: FV of an annuity

Spreadsheet short-cuts

(

RATE,NPER,PMT,PV,type

)

(.06,3,4000,0)Slide59

Annuity due

Consider a

series of three cash flows of $4,000 each if the discount rate is 6%, with the first cash flow

today.

What is the present value of this annuity?

What is the future value of this annuity?Slide60

The time line

$4,000.00

PV?

This is an annuity dueSlide61

Valuing an annuity due

Present value

Year

End of year cash flow

Compound factor

Present value of cash flow

$4,000.00

1.00000

$4,000.00

0.94340

3,773.58

$4,000.00

0.89000

3,559.99

2.83339

$11,333.57

Future value

Year

End of year cash flow

Factor

Future value

$4,000.00

1.19102

$4,764.06

$4,000.00

1.12360

4,494.40

$4,000.00

1.06000

4,240.00

3.37462

$13,498.46Slide62

Valuing an annuity due: Using calculators

Present value

PMT

= 4000

N = 3

I = 6%

BEG mode

Solve for PV

Future value

PMT

= 4000

N = 3

I = 6%

BEG mode

Solve for

FVSlide63

Valuing an annuity due: Using spreadsheets

Present value

=PV(

RATE,NPER,PMT,FV,TYPE

)

=PV(0.06,3,4000,0,1)

Future value

=PV(

RATE,NPER,PMT,FV,TYPE

)

=PV(0.06,3,4000,0,1

)Slide64

Any other way?

There is one period difference between an ordinary annuity and an annuity due. Therefore:

annuity

due

ordinary

annuity

× (1 + i)

and

FVannuity due = FVordinary annuity × (1 + i)Slide65

Valuing a deferred annuity

deferred annuity

is an annuity that begins beyond one year from today.

That means that it could begin 2, 3, 4, … years from today, so each problem is unique.Slide66

Valuing a deferred annuity

4-payment ordinary annuity, then discount value one period

←PV

4-payment annuity due, then discount value two periods

←PV

2Slide67

Example: Deferred annuity

What is the value today of a series of five cash flows of $6,000 each, with the first cash flow received four years from today, if the discount rate is 8%?

PV?

CFSlide68

Example, cont.

Using an ordinary annuity:

= $23,956.26

= $19,017.25

Using an annuity due:

= $25,872.76PV0 = $19,017.25Discount 3 periods at 8%

Discount 4 periods at 8%Slide69

Example: Deferred annuityCalculator solutions

HP10B

BAIIPlus

TI83/84

0 CF

6000 CF

6000 CF6000 CF8 i

NPV

0 CF ↑ 1

0 CF ↑ 1 F10 CF ↑ 1 F20 CF ↑ 1 F36000 CF ↑ 1 F4

6000 CF ↑ 1 F5

6000 CF ↑ 1 F6

6000 CF ↑ 1 F7

6000 CF ↑ 1 F8

8 i

NPV

] {

0 0 0 6000 6000 6000 6000 6000}

STO

nd] L1[APPS] [Finance][ENTER] 7NPV(.08,0,L1)[ENTER]Slide70

Example: Deferred annuitySpreadsheet solutions

Year

Cash flow

$6000

=PV(0.08,3,0,PV(0.08,5,6000,0))

=PV(0.08,4,0,PV(0.08,5,6000,0,1))

NPV

(0.08,A1:A9)Slide71

Perpetuities

perpetuity

is an even cash flows that occurs at regular intervals of time, forever.

The valuation of a perpetuity is simple:

+…

Slide72

Problem Set 3Slide73

Which do you prefer if the appropriate discount rate is 6% per year:

An annuity of $4,000 for four annual payments starting today.

An annuity of $4,100 for four annual payments, starting one year from today.

An annuity of $4,200 for four annual payments, starting two years from today.

Problem 3.1

73Slide74

5.3 Nominal and effective rates

iSlide75

APR & EAR

The

annual percentage rate

(

APR

) is the nominal or stated annual rate.

The APR ignores compounding within a year.

The APR understates the true, effective rate.

The

effective annual rate

(

EAR) incorporates the effect of compounding within a year.Slide76

APR EAR

EAR =

Suppose interest is stated as 10% per years, compounded quarterly.

EAR

EAR = 10.3813%

Slide77

EAR with continuous compounding

EAR =

Suppose interest is stated as 10% per years, compounded continuously.

EAR

EAR = 10.5171%

Slide78

Frequency of compounding

If interest is compounded more frequently than annually, then this is considered in compounding and discounting.

There are two approaches

Adjust the i and n; or

Calculate the EAR and use thisSlide79

Example: EAR & compounding

Suppose you invest $2,000 in an investment that pays 5% per year, compounded quarterly. How much will you have at the end of 4 years?Slide80

Example: EAR & compounding

Method 1:

= $2,000 (1 + 0.0125)

= $2,439.78

Method 2:

EAR = (1 +

)

– 1 = 5.0945%

FV = $2,000 (1 + 0.050945)4

= $2,439.78

Slide81

Try it: APR & EAR

Suppose a loan has a stated rate of 9%, with interest compounded monthly. What is the effective annual rate of interest on this loan?Slide82

Try it: Answer

EAR

EAR =

9.3807%

Slide83

Problem Set 4Slide84

What is the effective interest rate that corresponds to a 6% APR when interest is compounded monthly?

Problem 4.1

84Slide85

What is the effective interest rate that corresponds to a 6% APR when interest is compounded continuously?

Problem 4.2

85Slide86

5.4 ApplicationsSlide87

Saving for retirement

Suppose you estimate that you will need $60,000 per year in retirement. You plan to make your first retirement withdrawal in 40 years, and figure that you will need 30 years of cash flow in retirement. You plan to deposit funds for your retirement starting next year, depositing until the year before retirement. You estimate that you will earn 3% on your funds.

How much do you need to deposit each year to satisfy your plans?Slide88

Deferred annuity time line

…

||||

D = Deposit (39 in total)

W = Withdrawal (30 in total)Slide89

Deferred annuity time line

…

||||

← Ordinary annuity

↓

Ordinary

annuity

→

…

DSlide90

Two steps

Step 1: Present value of ordinary annuity

N = 30; i = 3%;

PMT

= $60,000

= $1,176,026.48

Step 2: Solve for payment in an ordinary annuity

N = 39; i = 3%;

= $1,176,026.48PMT = $16,280.74Slide91

What does this mean?

If there are 39 annual deposits of $16,280.74 each and the account earns 3%, there will be enough to allow for 30 withdrawals of $60,000 each, starting 40 years from today.Slide92

Balance in retirement accountSlide93

Practice problems



€





£Slide94

Problem 1

What is the future value of $2,000 invested for five years at 7% per year, with interest compounded annually?Slide95

Problem 2

What is the value today of €10,000 promised in four years if the discount rate is 4%?Slide96

Problem 3

What is the present value of a series of five end-of-year cash flows of $1,000 each if the discount rate is 4%?Slide97

Problem 4

Suppose you plan to save $3,000 each year for ten years. If you earn 5% annual interest on your savings, how much more will you have at the end of ten years if you make your payments at the beginning of the year instead of the end of the year?Slide98

Problem 5

Sue plans to deposit $5,000 in a savings account each year for thirty years, starting ten years from today. Yan plans to deposit $3,500 in a savings account each year for forty years, starting at the end of this year. If both Sue and Yan earn 3% on their savings, who will have the most saved at the end of forty years?Slide99

Problem 6

Suppose you have two investment opportunities:

Opportunity 1: APR of 12%, compounded monthly

Opportunity 2: APR of 11.9%, compounded continuously

Which opportunity provides the better return?Slide100

Problem 7

If you can earn 5% per year, what would you have to deposit in an account today so that you have enough saved to allow withdrawals of $40,000 each year for twenty years, beginning thirty years from today?Slide101

Problem 8

Suppose you deposit ¥50000 in an account that pays 4% interest, compounded continuously. How much will you have in the account at the end of ten years if you make no withdrawals?Slide102

The end

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