interest formulas – equal payment series lecture no.5 professor c. s. park fundamentals of...
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Interest Formulas – Equal Payment Series
Lecture No.5Professor C. S. ParkFundamentals of Engineering EconomicsCopyright © 2005
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
Option 1: Early Savings Plan
$2,000
?
6531Age
Option 2: Deferred Savings Plan
44 $2,000( / ,8%,34)
$317,233
F F A
0 11 12
44
Option 2: Deferred Savings Plan
$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
Option 1: Early Savings Plan
Option 2: Deferred Savings Plan
$2,000
$2,000
$396,644
$317,253