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L5: Interest Formulas – Equal Payment Series

ECON 320 Engineering EconomicsMahmut Ali GOKCEIndustrial Systems EngineeringComputer Sciences


Equal Payment Series

0 1 2 N

0 1 2 N

A A A

F

P

0 N


Equal Payment Series – Compound Amount Factor

0 1 2 N

0 1 2 NA A A

F

0 1 2N

A A A

F


Compound Amount Factor

0 1 2 N 0 1 2 N

A A A

F

A(1+i)N-1

A(1+i)N-2

1 2(1 ) (1 )N NF A i A i A


Equal Payment Series Compound Amount Factor (Future Value of an annuity)

F Ai

iA F A i N

N

( )

( / , , )

1 1

Example 2.9: Given: A = $5,000, N = 5 years, and i = 6% Find: F Solution: F = $5,000(F/A,6%,5) = $28,185.46

0 1 2 3N

F

A


Validation

F =?

0 1 2 3 4 5

$5,000 $5,000 $5,000 $5,000 $5,000

i = 6%

4

3

2

1

0

$5,000(1 0.06) $6,312.38

$5,000(1 0.06) $5,955.08

$5,000(1 0.06) $5,618.00

$5,000(1 0.06) $5,300.00

$5,000(1 0.06) $5,000.00

$28.185.46


Finding an Annuity Value

Example: Given: F = $5,000, N = 5 years, and i = 7% Find: A Solution: A = $5,000(A/F,7%,5) = $869.50

0 1 2 3N

F

A = ?

A Fi

i

F A F i N

N

( )

( / , , )

1 1


Example 2.10 Handling Time Shifts in a Uniform Series

F = ?

0 1 2 3 4 5

$5,000 $5,000 $5,000 $5,000 $5,000

i = 6%

First deposit occurs at n = 0


5 $5,000( / ,6%,5)(1.06)

$29,876.59

F F A

Annuity Due

Excel Solution

=FV(6%,5,5000,0,1)Beginning period


Sinking Fund Factor

Example 2.11 – College Savings Plan: Given: F = $100,000, N = 8 years, and i = 7% Find: A Solution:

A = $100,000(A/F,7%,8) = $9,746.78

0 1 2 3N

F

A

A Fi

i

F A F i N

N

LNM

OQP

( )

( / , , )

1 1


Excel Solution

Given: F = $100,000 i = 7% N = 8 years

0

1 2 3 4 5 6 7 8

$100,000

i = 8%

A = ?

Current age: 10 years old

• Find:

=PMT(i,N,pv,fv,type)=PMT(7%,8,0,100000,0)=$9,746.78


Capital Recovery Factor

Example 2.12: Paying Off Education Loan Given: P = $21,061.82, N = 5 years, and i = 6% Find: A Solution: A = $21,061.82(A/P,6%,5) = $5,000

1 2 3N

P

A = ?

0

A Pi i

i

P A P i N

N

N

( )

( )

( / , , )

1

1 1


P =$21,061.82

0 1 2 3 4 5 6

A A A A A

i = 6%

0 1 2 3 4 5 6

A’ A’ A’ A’ A’

i = 6%

P’ = $21,061.82(F/P, 6%, 1)

Grace period

Example 2.14 Deferred Loan Repayment Plan


Two-Step Procedure

' $21,061.82( / ,6%,1)

$22,325.53

$22,325.53( / ,6%,5)

$5,300

P F P

A A P


Present Worth of Annuity Series

Example 2.14:Powerball Lottery Given: A = $7.92M, N = 25 years, and i = 8% Find: P Solution: P = $7.92M(P/A,8%,25) = $84.54M

1 2 3N

P = ?

A

0

P Ai

i i

A P A i N

N

N

( )

( )

( / , , )

1 1

1


Excel Solution

Given: A = $7.92M i = 8% N = 25

Find: P

=PV(8%,25,7.92,0)

= $84.54M

0

A = $7.92 million

i = 8%

251 2

P = ?


0 1 2 3 4 5 6 7 8 9 10 11 12

44

Option 2: Deferred Savings Plan

$2,000

Example 2.15 Early Savings Plan – 8% interest

0 1 2 3 4 5 6 7 8 9 10

44

Option 1: Early Savings Plan

$2,000

?

?


Option 1 – Early Savings Plan

10

44

$2,000( / ,8%,10)

$28,973

$28,973( / ,8%,34)

$396,645

F F A

F F P

0 1 2 3 4 5 6 7 8 9 10

44


$2,000

?

6531Age



44 $2,000( / ,8%,34)

$317,233

F F A

0 11 12

44


$2,000

?


At What Interest Rate These Two Options Would be Equivalent?

44

44

Option 1:

$2,000( / , ,10)( / , ,34)

Option 2:

$2,000( / . ,34)

Option 1 = Option 2

$2,000( / , ,10)( / , ,34) $2,000( / . ,34)

Solve for

F F A i F P i

F F Ai

F A i F P i F Ai

i


123456789101112131415161718192021224041424344454647

A B C D E F

Year Option 1 Option 201 (2,000)$ 2 (2,000)$ Interest rate 0.083 (2,000)$ 4 (2,000)$ FV of Option 1 396,645.95$ 5 (2,000)$ 6 (2,000)$ FV of Option 2 317,253.34$ 7 (2,000)$ 8 (2,000)$ Target cell 79,392.61$ 9 (2,000)$

10 (2,000)$ 11 (2,000)$ 12 (2,000)$ 13 (2,000)$ 14 (2,000)$ 15 (2,000)$ 16 (2,000)$ 17 (2,000)$ 18 (2,000)$ 19 (2,000)$ 37 (2,000)$ 38 (2,000)$ 39 (2,000)$ 40 (2,000)$ 41 (2,000)$ 42 (2,000)$ 43 (2,000)$ 44 (2,000)$


Using Excel’s Goal Seek Function


Result


0 1 2 3 4 5 6 7 8 9 10

44

0 1 2 3 4 5 6 7 8 9 10 11 12

44



$2,000

$2,000

$396,644

$317,253

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