Chapter 4 - 2_Time Value of Money Part II(1)

7/29/2019 Chapter 4 - 2_Time Value of Money Part II(1)

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Chapter 4 Part II

The Time Value of Money - Part II

Instructor: Joonyup Eun

1 Purdue School of Industrial Engineering -

Fall 2013, IE343 Engineering Economics


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Contents

2

A Uniform Series (Annuity) to Its Present and FutureEquivalent Values

Deferred Annuities (Uniform Series)

Equivalence Calculations Involving Multiple Interest

Formulas

Purdue School of Industrial Engineering -



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3

A Uniform Series

(Annuity) to Its Presentand Future Equivalent

Values




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Definition of Annuity

4

An annuity is a series of equal (uniform) cash flowsoccurring at fixed time intervals.

The first cash flow occurs at the end of the first period.

The last cash flow occurs at the end of the last period

0 1 2 3 N-1 N




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PreliminaryGeometric Series

5

Lets define , , , , , =

Calculate

1 1 1

1




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Finding F when Given A

6

0 1 2 3 N-1 N

1

1

1 1

Find the future equivalent at the end ofth period of anannuity, a series of.

1 1 1 1 1 1

1 1 1 1

=

1

1




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Finding F when Given A

7

Let ,%, +

. Then

1 1

, %, Example:

Interest at 8% per year

Receiving $502 each year for the next 3 years.

0 1 2 3 8%

$502 $502 $502 ?

+ 502+.

. 1629.693 , %, 502 ,8%,3 502 3.2464 1629.693




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Finding P when Given A

8

Find the present equivalent of an annuity, a series of.

0 1 2 3 N-1 N

1 1

1

1 ()

1

1 1 1 1 1 1 1 1

1 1

1 1

1

=

1

1




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Finding P when Given A

9

Let ,%, +

+ . Then , %,

Example:

Interest at 8% per year Receiving $502 each year for the next 3 years.

0 1 2 3 8%

$502 $502 $502 ?

+ + 502+. . . 1293.703

, %, 502 , 8%, 3 502 2.5771 1293.704Purdue School of Industrial Engineering -



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Relationship between and

10

, %, 1293.703 1629.693 , 8%, 3 1629.693 0.7938 1293.703 , %, , %, , %, , %, , %, , %, , %, , %,

+ +

+

+

+ +

0 1 2 3

8%

$502 $502 $502 1293.703

1629.693




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Finding A when Given F

11

+

+ Let , %, + , Then , %, Example: You need $1,000,000 ten years later. You will make 10

yearly deposits in a saving account at the end of each year. The

interest rate is 5% per year. How much money you need to pay

every year?




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Finding A when Given F

12

Example: You need $1,000,000 ten years later. You will make 10yearly deposits in a saving account at the end of each year. The

interest rate is 5% per year. How much money you need to

pay every year?

0 1 2 3 9 10 5%

1,000,000

, %, 1,000,000 ,5%,10 1,000,000 0.0795 79,500




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Finding A when Given P

13

+

+ +

+

Let , %, + + , then , %, Example: You plan to borrow a loan of $100,000 which you will

repay with equal annual payments for the next 5 years.

Suppose the interest rate you are charged is 8% per year and

you will make the first payment one year after receiving the

loan. How much is your annual payment?




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Finding A when Given P

14

Example: You plan to borrow a loan of $100,000 which you will

repay with equal annual payments for the next 5 years.

Suppose the interest rate you are charged is 8% per year and

you will make the first payment one year after receiving the

loan. How much is your annual payment?

0 1 2 3 4 5

$100,000

8%

, %, 100,000 , 8%, 5 100,0000.2505 25,050




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Other consideration

15

For the problems that Finding the number of cash flows in an Annuity Given A,P and I

Example



0 1 2 3 N-1 N? 5%

1,000

1,000,000

1,000,000 1,000 , 5%, 1,000 +. . +. N?


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For the problems that Finding the interest rate I, Given A, F and N

Example

Other consideration

0 1 2 3 49 50 ?

1,000

1,000,000

1,000,000 1,000 ,%,50 1,000 + ?


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Other consideration

For the problems that Finding the interest rate I, Given A, F and N

Finding the number of cash flows in an Annuity Given A,P and I

No closed form (analytical) solution is known.

We can solve these problems numerically.

Please refer to Excel help session.


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18

Example

Annual interest rate is

10%. You have 3 plans to pay the balance back.

Plan 1: Pay interest, $1,000, due at end of each year and principal, $10,000,at end of third year. PV= FV=

Plan 2: Pay principal and interest, $13,310, in one payment at the end ofthird year PV= FV=

Plan 3: Pay off the debt in 3 equal end-of-year payments of$4,021PV= FV=

0 1 2 3 10%

$10,000

$1,000 $1,000 $1,000

Plan 1 1000 , 10%, 3 10,000 , 10%, 3 10,000

?Purdue School of Industrial Engineering -



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19

Example



Plan 1: Pay interest, $1,000, due at end of each year and principal, $10,000,at end of third year. PV=10,000 FV=


Plan 3: Pay off the debt in 3 equal end-of-year payments of$4,021PV= FV=

0 1 2 3 10%

$10,000

$1,000 $1,000 $1,000

Plan 1 1000 ,10%,3 10,000 13,310




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20

Example



Plan 1: Pay interest, $1,000, due at end of each year and principal, $10,000,at end of third year. PV=10,000 FV=13,310


Plan 3: Pay off the debt in 3 equal end-of-year payments of$4,021PV= FV=Plan 2 13,310 ,10%,3 10,000

0 1 2 3 10%

$13,310 ?Purdue School of Industrial Engineering -



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21

Example




Plan 2: Pay principal and interest, $13,310, in one payment at the end ofthird year PV=10,000 FV=

Plan 3: Pay off the debt in 3 equal end-of-year payments of$4,021PV= FV=Plan 2 13,310

0 1 2 3 10%

$13,310 ?Purdue School of Industrial Engineering -



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22

Example




Plan 2: Pay principal and interest, $13,310, in one payment at the end ofthird year PV=10,000 FV=13,310

Plan 3: Pay off the debt in 3 equal end-of-year payments of$4,021PV= FV=Plan 3

0 1 2 3 10%

$4,021 $4,021 $4,021

4,021 ,10%,3 10,000




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23

Example





Plan 3: Pay off the debt in 3 equal end-of-year payments of$4,021PV=10,000 FV=Plan 3

0 1 2 3 10%

$4,021 $4,021 $4,021

4,021 ,10%,3 13,310




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24

Example





Plan 3: Pay off the debt in 3 equal end-of-year payments of$4,021PV=10,000 FV=13,310

All three plans are economically equivalent.

Purdue School of Industrial Engineering -Fall 2013, IE343 Engineering Economics


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25

Example

You are analyzing a project with 5-year life. The project requires a

capital investment of $50,000 now, and it will generate uniform

annual revenue of $6,000 at the end of each year. Further, the

project will have a salvage value of $4,500 at the end of the fifth year

and it will require $3,000 each year for the operation. Appropriate

interest rate is 10%. What is the PV?

0 1 2 3 4 5

50,0006,000 , 10%, 5 3,000 ,10%,5 4,500 ,10%,5 35,833.49



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26

Example






interest rate is 10%. What is the FV?

0 1 2 3 4 5

$50,000

$6,000 $6,000 $6,000 $6,000 $6,000

$3,000 $3,000 $3,000 $3,000 $3,000

$4500

50,000 , 10%, 5 6,000 ,10%,5 3,000 ,10%,5 4,500 57,710.20

10% ?Purdue School of Industrial Engineering -



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27

Example






interest rate is 10%. What is the PV?

0 1 2 3 4 5

50,0003,000 , 10%, 4 7,500 ,10%,5 35,833.49



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28

Example







0 1 2 3 4 5

$50,000

$3,000 $3,000$3,000 $3,000$7,500

10%

50,000 , 10%, 5 3,000 ,10%,4 7,500



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29

Example







0 1 2 3 4 5

$50,000

$3,000 $3,000$3,000 $3,000$7,500

10%

50,000 , 10%, 5 3,000 ,10%,4 7,500 ,10%,1 57,710.20



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30

Example







0 1 2 3 4 5

$50,000

$3,000 $3,000$3,000 $3,000

10%

$4500$3,000

50,000 , 10%, 5 3,000 ,10%,5 4,500 57,710.20



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31

Deferred Annuities(Uniform Series)



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Deferred Annuities

32

Annuity series where the first of the uniform cash flows

occurs at the end of period J+1 instead of at the end of

period 1 and there are N-J such cash flows.

Ordinary Annuities

Deferred Annuities

0 2 4 6

1 3 5

N-4 N-2 N

N-5 N-3 N-1

0 2 4 61 3 5 J+1 N

J J+2 N-1



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Present Value of Deferred Annuities

33

, %, , %,

, %, , %,

0 2 4 61 3 5 J+1 N

J J+2 N-1



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Example

34

A father, on the day his son is born, wishes to determine what

lump amount would have to be paid into an account bearing

interest of 12% per year to provide withdrawals of $2,000 on

each of the sons 18th, 19th, 20th, and 21st birthdays.

0 2 4 61 3 5 18 2117 19 20

2,000 2,000

12%

2000 , 12%, 4 6,074.60 ,12%,17 6,074.600.1456 884.46



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Future Value of Deferred Annuities

35

, %,

0 2 4 61 3 5 J+1 N

J J+2 N-1

?



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Deferred Future Value of Annuities

36

, %, , %,

, %, , %,

0 2 J-1 J+11 J N

N-1

?



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Example

37

When you take your first job, you decide to start saving right away for

your retirement. You put $5,000 per year into the companys 401(k)plan, which averages 8% interest per year. Five years later, you move to

another job and start a new 401(k) plan. You never get around to

merging the funds in the two plans. If the first plan continued to earn

interest at the rate of 8% per year for 35 years after you stopped

making contributions, how much is the account worth?

0 2 4 61 3 5 18 4017 19 39

5,000 5,000 8%

? ? 5,000 ,8%,5 5,0005.8666 $29,333.3 , 8%, 35 29,333.3 14.7853 $433,697


$433,697 in the account


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38

Equivalence Calculations

Involving Multiple InterestFormulas



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Example

39

The present equivalent expenditure The future equivalent expenditure

The annual equivalent expenditure

0 2 4 61 3 5 87 20%$100

$200$500

$400 $400 $400 $400 $400



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Example (Present equivalent expenditure )

40

0 2 4 61 3 5 87 20%$100

$200$500

$400 $400 $400 $400 $400

100 , 20%, 1 200 , 20%, 2 500 , 20%, 3 400 ,20%,5

83.33

138.88

289.35

692.26

1,203.82 ,20%,3



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Example (Future equivalent expenditure )0 2 4 61 3 5 87 20%

$100$200

$500$400 $400 $400 $400 $400

100 , 20%, 7 200 ,20%,6 500 , 20%, 5 400 , 20%, 5 , 20%, 8 5,176.19

1,203.82

1,203.82 4.2998



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Example (Annual equivalent expenditure )0 2 4 61 3 5 87 20%

$100 $200

$500$400 $400 $400 $400 $400

1,203.82

, 20%, 8 1,203.82 0.2606 313.73

5,176.19

, 20%, 8 5,176.19 0.0606 313.73

?

Chapter 4 - 2_Time Value of Money Part II(1)

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