Calculations for Shifted Gradient

1

• Calculations for Shifted Gradient

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Shifted Gradient

• The present worth of an arithmetic gradient will
  always be located two periods before the gradient
  starts.
• To find the equivalent A series of a shifted
  gradient through all the periods, first find the
  present worth of the gradient at actual time 0,
  then apply the (A/P, i, n) factor.

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Example

• The by-product department of Peyton Packing has a
  cooker
• that has the cost stream below. If the interest
  rate is 18 per
• year, determine the present worth of the costs.
• Year Cost, Year Cost,
• 0 4000 6 8000
• 1 4000 7 9000
• 2 4000 8 9100
• 3 5000 9 9200
• 4 6000 10 9300
• 5 7000 11 9400

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Solution

• P40004000(P/F,18,1) 4000(P/A,18,6)100
  0(P/G,18,6(P/F,18,1)
• 9100(P/A,18,4)100(P/G,18,4)(P/F,18,7)
• P40004000(.8475)
• 4000(3.4976)1000(7.0834)(.8475)
• 9100(2.6901)100(3.4828)(.3139)
• P33,044

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Example

• Calculate the present worth for a machine that
  has an initial cost of 29,000, a life of 10
  years, and an annual operating cost of 13,000
  for the first 4 years, increasing by 10 per year
  thereafter. Use an interest rate of 20 per year.

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Solution

• PgA1 1-(1g) / (1i)n / (i-g)
• Pg -13000 1-(10.1) / (10.2)7 / (0.2-0.1)
• Pg -59,299
• P-29,000 -13,000(P/A,20,3) -
• 59,299(P/F,20,3)
• P-29,000 - 13,000(2.1065) -
• 59,299(0.5787)
• P -90,701

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Example (3.32, Page 119)

• Calculate the annual worth (years 1 through 7) of
  the
• following series of disbursement. Assume i12
  per
• year.
• Year Disbursement,
• 0 5000
• 1 3500
• 2 3500
• 3 3500
• 4 5000
• 5 5000
• 6 5000
• 7 5000

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Solution

• A 5000(A/P,12,7) 3500 1500(F/A,12,4)(A/F,1
  2,7)
• 5000(0.21912) 3500 1500 (4.7793)
• (0.09912)
• 5306.19

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Example (3.41, Page 120)

• For the cash flows below, find the value of x
  that makes the present worth
• in year 0 equal to 11,000 at an interest rate of
  12 per year.
• Year Cash Flow,
• 0 200
• 1 300
• 2 400
• 3 x
• 4 600
• 5 700
• 6 800
• 7 900
• 8 1000
• 9 1100

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Solution

• 11,000 200 300(P/A,12,9) 100(P/G,12,9)
• 500(P/F,12,3) x(P/F,12,3)
• 11,000 200 300(5.3282) 100(17.3563)
  500(0.7118)
• x(0.7118)
• x 10,989