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CHAPTER 9 Time Value of Money

Future value

Present value

Annuities

Rates of return

Amortization

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Time lines

Show the timing of cash flows.

Tick marks occur at the end of periods, so Time 0 is today; Time 1 is the end of the first period (year, month, etc.) or the beginning of the second period.

CF0

CF1

CF3

CF2

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Drawing time lines

100

3 year $100 ordinary annuity

100

$100 lump sum due in 2 years

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Future Value of Money

If you deposit $1,000 today at 10%, how much will you have after 15 years?

Interest($) = Principal ∙ Interest Rate(%)

Simple Interest

The original principal stays the same.

There is no interest on interest. The interest is only on the original principal.

Compound Interest

The principal changes through time.

There is “interest on interest”. The interest is on the new principal.

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Simple Interest

Interest($) = Principal($) ∙ Interest Rate(%) = V0 ∙ I

V1 = V0 + Interest = V0 + V0 ∙ I = V0(1 + I)

V2 = V1 + Interest = V1 + V0 ∙ I = V0(1 + I) + V0 ∙ I

= V0(1 + I + I) = V0(1 + 2I)

V3 = V2 + Interest = V2 + V0 ∙ I = V0(1 + 2I) + V0 ∙ I

= V0(1 + 2I + I) = V0(1 + 3I)

Vn = V0(1 + nI)

FVn = PV(1 + nI)

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Compound Interest

Interest ($) = Principal ($) ∙ Interest Rate (%) = V ∙ I

V1 = V0 + Interest = V0 + V0 ∙ I = V0(1 + I)

V2 = V1 + Interest = V1 + V1 ∙ I = V1(1 + I)

V3 = V2 + Interest = V2 + V2 ∙ I = V2(1 + I)

V2 = V1(1 + I) = V0(1 + I)(1 + I) = V0(1 + I)2

V3 = V2(1 + I) = V0(1 + I)2(1 + I) = V0(1 + I)3

Vn = V0 (1 + I)n

FVn = PV(1 + I)n = PV∙FVIF

V2 = V1 + Interest = V1 + (V0 + Interest) ∙ I

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Example

What is the future value of $20 invested for 2 years at 10%?

Simple: FV = PV(1+nI)

= 20(1+2I) = 20(1+0.2) = $24

Compound: FV = PV(1+I)n

= 20(1+I)2 = 20(1+0.1)2 = $24.2

What is the future value of $20 invested for 100 years at 10%?

Simple: FV = 20(1+ ) =

Compound : FV = 20(1.1)100 = 275,612.25

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The Power of Compounding

The Value of Manhattan

In 1626, the land was bought from American Indians at $24.

In 2018, value = $24(1+I)392

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Solving for FV: The formula method

Solve the general FV equation:

FVN = PV∙(1 + I)N = PV ∙ FVIF

FV15 = PV∙(1 + I)15 = $1,000∙(1.10)15 = $4,177.25

= $1,000∙4.177 = $4,177

(Table A)

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Present Value of Money

If you want to have $4,177.25 after 15 years, how much do you have to deposit today at 10%?

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PV = ?

4,177.25

Present Value of Money

Finding the PV of a cash flow or series of cash flows is called discounting (the reverse of compounding).

2 …

10%

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Solving for PV: The formula method

Solve the general FV equation for PV:

PV = = ∙

= FVN ∙ PVIF

PV = = = $4,177.25 ∙

= $4,177.25∙0.239

(Table C)

= $998.36, but $4,177.25∙0.2394 = $1,000

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Examples

If you want $10,000 after 10 years, how much do you have to deposit today at 5%?

If you want $100,000 someday for a world tour, how long will it take at 7% if you deposit $10,000 today?

If you want $50,000 in 10 years, at what rate do you have to invest your money if you have $10,000 today?

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Calculation

You deposit $1000 today at 6%. After one year (t=1), you withdraw $300, after two years (t=2), you deposit $500 more, and no deposit or withdrawal after that, then how much will you have in year 5 (t=5) ?

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Answer

Step by step solution

0 1 2 5

$1,000 -$300 $500 ?

1000 PV, 6 I/Y, 1 N, CPT FV =>1060

1060

( )PV, 6 I/Y, ( )N, FV => ( )

( ) + ( ) = ( )

( ) PV, 6 I/Y, ( ) N, FV => ( )

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Annuity

A series of cash flows of the same amount with fixed intervals for a specified number of periods.

0 1 2 3 4

$20 $20 $30 $20

$20 $30 $40 $50

$20 $20 $0 $20

$0 $0 $20 $20

$20 $30 $20 $30

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FV of Annuity

100

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FV of Annuity

FVA = 100(1+I)2 + 100(1+I) + 100

= 100[(1+I)2 + (1+I) + 1]

= 100∙

For n periods,

FVA = PMT∙

= PMT∙FVAIF

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PV of Annuity

100

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PV of Annuity

PVA = + +

= 100 [ + + ]

= 100

For n periods,

PVA = PMT

= PMT∙PVAIF

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Examples

If you deposit $3,000 a year for 10 years at 7%, how much will you have after 10 years?

If you want to receive $5o,000 per year for 20 years, how much do you have to deposit today at 5%?

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More examples

You need $100,000 in year 15 to start your own business. If your bank’s interest rate is 6%, how much do you have to deposit each year to get $100,000?

You need $100,000 for a world tour. If you deposit $10,000 each year, how long will it take for you to accumulate $100,000 at 7%?

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Investment Choice

You have $10,000 to invest. There are two choices for your investment.

Choice A: Buying an annuity at $10,000 and receiving $1,000 for 20 years.

Choice B: Depositing $10,000 in a bank that pays 8% interest rate.

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Harder Problem

You need to accumulate $11,000. To do so, you plan to deposit $1,350 per year in a bank that pays 6% interest. Your last deposit will be less than $1,350 if less is needed to round out to $11,000.

A. How many years will it take to reach your goal?

B. How large will the last deposit be for you to have exactly $11,000 in your account?

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Answer

Two ways

A: The FV at year 6 will be $9,416.68, and the money will grow in the account for a year to $9,981.68.

B: The FV at year 7 will be $11,331.68.

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What is the difference between an ordinary annuity and an annuity due?

Ordinary Annuity

PMT

Annuity Due

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Ordinary Annuity and Annuity Due

FVADUE = FVA(1+I)

PVADUE = PVA(1+I)

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PV of Annuity Due

What is the PV of 3-year annuity due of $100 payments at 10%?

Now, $100 payments occur at the beginning of each period.

PVAdue= PVA (1+I) = $248.69(1.1) =$273.55.

Alternatively, set calculator to “BEGIN” mode and solve for the FV of the annuity:

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FV of Annuity Due

What is the FV of 3-year annuity due of $100 payments at 10%?

Now, $100 payments occur at the beginning of each period.

FVAdue= FVA (1+I) = $331(1.1) = $364.10.

Alternatively, set calculator to “BEGIN” mode and solve for the FV of the annuity:

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What is the (future) value of this annuity at t = 1? At t = 2?

100(1+I) + 100 +

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Question

If you can receive $50,000 per year forever, how much are you willing to pay for that?

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Perpetuity

PV =

If I = 10%, PV = = $0.5M

If I= 5%, PV = = $1M

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Growing Perpetuity

If the payments grow at a constant rate, g, it is a growing perpetuity.

PV =

Example: If I = 10%, g = 5%,

PMT0 = $50,000,

PV = = $1.05M

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The Power of Compound Interest

A 20-year-old student wants to save $3 a day for her retirement. Every day she places $3 in a drawer. At the end of the year, she invests the accumulated savings ($1,095) in a brokerage account with an expected annual return of 12%.

How much money will she have when she is 65 years old?

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Solving for FV:

If she begins saving today, how much will she have when she is 65?

If she sticks to her plan, she will have $( ) when she is 65.

( ) N, 12 I/YR, -1095 PMT, FV =>

( )

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Solving for FV:

If you don’t start saving until you are 40 years old, how much will you have at 65?

If a 40-year-old investor begins saving today, and sticks to the plan, he or she will have $146,000.59 at age 65. This is $1.3 million less than if starting at age 20.

Lesson: It pays to start saving early.

25 N, 12 I/Y, 1095 PMT,

FV =› 146,000.59

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Solving for PMT:

How much must the 40-year old deposit annually to catch the 20-year old?

To find the required annual contribution, enter the number of years until retirement and the final goal of $1,487,261.89, and solve for PMT.

25 N, 12 I/Y, 1,487,261.89 FV,

PMT = › 11,154.42

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What is the PV of this uneven cash flow stream?

100

300

10%

-50

90.91

247.93

225.39

-34.15

530.08 = PV

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Solving for PV: Uneven cash flow stream

Input cash flows in the calculator’s “CF” register:

CF0 = 0

C01 = 100

F01 = 1

C02 = 300

F02 = 2

C03 = -50

Press NPV button, then enter I = 10, and hit CPT=> $530.087. (Here NPV = PV.)

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NPV (Net Present Value)

NPV is calculated net of costs.

If your project’s PV of cash inflows is greater than the PV of cash outflows, the project will enhance your company’s profitability.

In chapter 9, generally there is no cost at time 0, so the NPVs are positive, but in chapter 12, when we evaluate projects, the NPVs can be negative.

Example: 0 1 2 3 (I = 10%)

-1000 300 300 500

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