### More Interest Formulas

#### 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 A_{1} 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

A_{1} = $100; G = +
$50; i = 7%

P = A_{1} (P/A,i%,n)
+ G (P/G,i%,n)

Note that you must subtract
the annual amount, A_{1}, 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

A_{1} = 100; G = -
$10; i = 10%

A = A_{1} + 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

### More Interest Formulas

#### Arithmetic Gradient Series

Question 1

Question 2

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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