Arithmetic gradient.ppt

1. EGR 312 – Spring ‘05 1
Arithmetic Gradient Factors (P/G, A/G)
Cash flows that increase or decrease by a constant
amount are considered arithmetic gradient cash
flows. The amount of increase (or decrease) is
called the gradient.
$100
$125
$150
$175
G = $25
Base = $100
0 1 2 3 4
$2000
$1500
$1000
$500
G = -$500
Base = $2000
0 1 2 3 4

2. EGR 312 – Spring ‘05 2
Arithmetic Gradient Factors (P/G, A/G)
Equivalent cash flows:
 +
Note: the gradient series by
convention starts in year 2.
$100
$125
$150
$175
G = $25
Base = $100
0 1 2 3 4 0 1 2 3 4
$100
0 1 2 3 4
$25
$50
$75

3. EGR 312 – Spring ‘05 3
Arithmetic Gradient Factors (P/G, A/G)
To find P for a gradient cash flow that starts at the end
of year 2 and end at year n:
or P = G(P/G,i,n)
where (P/G,i,n) =
0 1 2 3 …
n
G
2G
nG
P
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 n
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G
P
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4. Problems
1. The maintenance on a machine is
expected to be $155 at the end of the
first year, and increasing $35 each year
for the following seven years. What
present of money would need to be set
aside now to pay the maintenance for the
eight year period ? Assume 6% of
interest.
EGR 312 – Spring ‘05 4

5. 2. The tuition fee in a school are expect to
inflate at the rate of 8%per year and the
tuition at year 1 is P60,000. The parents are
then given a financial planning by the bank
that says they must deposit an amount of
money today that will cover for their child’s
whole 4-year education. If the amount
deposited has an interest rate of 5% per year
compounded annually, how much money
must be deposited to the bank today.
EGR 312 – Spring ‘05 5

6. EGR 312 – Spring ‘05 6
Arithmetic Gradient Factors (P/G, A/G)
To find P for the arithmetic gradient cash flow:
 +
P = $100(P/A,i,4) + $25(P/G,i,4)
$155
$125
$150
$175
0 1 2 3 4 0 1 2 3 4
$155
0 1 2 3 4
$35
$70
$105

7. EGR 312 – Spring ‘05 7
Arithmetic Gradient Factors (P/G, A/G)
To find P for the declining arithmetic
gradient cash flow:
 --
P = $2000(P/A,i,4) - $500(P/G,i,4)
$2000
$1500
$1000
$500
0 1 2 3 4 0 1 2 3 4
$2000
0 1 2 3 4
$500
$1000
$1500

8. EGR 312 – Spring ‘05 8
Arithmetic Gradient Factors (P/G, A/G)
To find the uniform annual series, A, for an arithmetic
gradient cash flow G:

A = G(P/G,i,n) (A/P,i,4)
= G(A/G,i,n)
Where (A/G,i,n) =
0 1 2 3 …
n
G
2G
nG
0 1 2 3 … n
A
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1
)
1
(
1
n
i
n
i

9. EGR 312 – Spring ‘05 9
Geometric Gradient Factors (Pg /A)
A Geometric gradient is when the periodic payment is
increasing (decreasing) by a constant percentage:
A1 = $100, g = 0.1
A2 = $100(1+g)
A3 = $100(1+g)2
An = $100(1+g)n-1
$100
$110
$121
$133
0 1 2 3 4

10. EGR 312 – Spring ‘05 10
Geometric Gradient Factors (Pg /A)
To find the Present Worth, Pg , for a geometric gradient
cash flow G:
Pg
$100
$110
$121
$133
0 1 2 3 4
i
g
i
n
A
P
i
g
g
i
i
g
A
P
g
n
g
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1
1
1
1
1
1

11. EGR 312 – Spring ‘05 11
Determining Unknown Interest Rate
To find an unknown interest rate from a single-payment
cash flow or uniform-series cash flow, the following
methods can be used:
1) Use of Engineering Econ. Formulas.
2) Use of factor tables (and interpolation)
3) Spreadsheet (Excel)
a) =IRR(first cell: last cell)
b) =RATE(number_years,A,P,F)

12. EGR 312 – Spring ‘05 12
Determining Unknown Interest Rate
Example: The list price for a vehicle is stated as $25,000.
You are quoted a monthly payment of $658.25 per month
for 4 years. What is the monthly interest rate? What
interest rate would be quoted (yearly interest rate)?
Using factor table:
$25000 = $658.25(P/A,i,48)
37.974 = (P/A,i,48)
i = 1% from table 4, pg 705
0r 12% annually

13. EGR 312 – Spring ‘05 13
Determining Unknown Interest Rate
Example: The list price for a vehicle is stated as $25,000.
You are quoted a monthly payment of $658.25 per month
for 4 years. What is the monthly interest rate? What
interest rate would be quoted (yearly interest rate)?
Using formula:
Use Excel trial and error method to find i.
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 48
48
)
1
(
1
i)
(1
$658.25
$25000
i
i
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


 48
48
)
1
(
1
i)
(1
37.9795
i
i

14. EGR 312 – Spring ‘05 14
Determining Unknown Number of Periods (n)
To find an unknown number of periods for a single-
payment cash flow or uniform-series cash flow, the
following methods can be used:
1) Use of Engineering Econ. Formulas.
2) Use of factor tables
3) Spreadsheet (Excel)
a) =NPER(i%,A,P,F)

15. EGR 312 – Spring ‘05 15
Determining Unknown Number of Periods (n)
Example: Find the number of periods required such that
an invest of $1000 at 5% has a future worth of $5000.
P = F(P/F,5%,n)
$1000 = $5000(P/F,5%,n)
0.2 = (P/F,5%,n)
n ~ 33 periods