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Arithmetic Gradient Series

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Arithmetic Gradient Series

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Suppose that there is a series of "n" payments uniformly spaced but differing from one period to the next by a constant. The change or "gradient" from one period to the next is denoted "G." Let A1 be the payment at EOY 1.

EOY = End of year

NCF = Net Cash Flow

Example 1:

EOY NCF ($)
1 100
2 150
3 200
4 250

To find the Present Worth, at EOY 0, of a gradient series that begins EOY 1, use

A1 = $100; G = + $50; i = 7%

P = A1 (P/A,i%,n) + G (P/G,i%,n)

Note that you must subtract the annual amount, A1, from all annual amounts before applying the gradient factor.

P = 100 (P/A,7%, 4) + 50 (P/G, 7%, 4)

P = 100 (3.387) + 50 (4.795)

P = 578.45

Example 2:

EOY NCF ($)
1 100
2 90
3 80
4 70

To find the annual equivalent (A series) of a gradient series that begins EOY 1, use

A1 = 100; G = - $10; i = 10%

A = A1 + G (A/G,i%,n)

A = 100 - 10 (A/G, 10%, 4)

A = 100 - 10 (1.388 )

A = $86.12

Now, find the present value at 10% interest of the series of payments given in Example 2 above.

P = $100 (P/A,10%,4) + ( - $10) (P/G,10%,4)

= $100 (3.170) - $10 (4.378) = $273

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Arithmetic Gradient Series

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Question 1.

Using an interest table gradient series factor, write an equation to find the Present Worth (at EOY 0) of the following series of payments. Interest is 5%.

EOY NCF ($)
0 -1,000
1 500
2 600
3 700
4 800
5 900

Choose an answer by clicking on one of the letters below, or click on "Review topic" if needed.

A P = - 1,000 + 500 (P/G,5%,5)

B P = - 1,000 + 500 (P/A,5%,5) + 100 (P/G,5%,5)

C P = - 1,000 + 500 (P/A,5%,5) + 100 (P/G,5%,4)

D P = - 1,000 + 500 (P/F,5%,1) + 100 (P/G,5%,5)

Review topic

Question 2.

Using an interest table gradient series factor, write an equation to find the Present Worth (at EOY 0) of the following series of payments. Interest is 7%.

EOY NCF ($)
0 -1,000
1 0
2 0
3 400
4 800
5 1200

Choose an answer by clicking on one of the letters below, or click on "Review topic" if needed.

A P = - 1,000 + 400 (P/G,7%,4) (P/F,7%,1)

B P = - 1,000 + 400 (P/G,7%,4)

C P = - 1,000 + 400 (P/A,7%,3) + 400 (P/G,7%,3)

D P = - 1,000 + 400 (P/G,7%,3) (P/F,7%,2)

Review topic



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