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Find P using gradient formula

Short answer

Break the series into uniform and gradient parts, move all to year 2 at 10% interest using P/F, P/A, and P/G factors. The equivalent present worth is P = 90.02.

Find the value of P if i = 10%. Use gradient formula also. [Figure: cash flow diagram, years 0 to 6; receipts 10 at year 0, 12 at year 1, 14 at year 2, then 16 at years 3, 4, 5 and 6; P = ? as a downward arrow at year 2]

Assumption

P is the single equivalent amount at year 2 (where the arrow is drawn) of all the receipts: 10 at year 0, 12 at year 1, 14 at year 2, and 16 at years 3, 4, 5 and 6. i = 10%.

                     16   16   16   16
           12   14    ^    ^    ^    ^
      10    ^    ^    |    |    |    |
       ^    |    |    |    |    |    |
  -----+----+----+----+----+----+----+---> year
       0    1    2    3    4    5    6
                 |
                 v
               P = ?

Splitting the series

Receipts = uniform 10 (years 0-6) + gradient 0, 2, 4, 6 (years 0-3, G = 2) + uniform 6 (years 4-6).

Value at year 2 of each part

  1. Uniform 10 at years 0, 1, 2: 10 (F/A, 10%, 3) = 10 × 3.310 = 33.100
  2. Uniform 10 at years 3 to 6 (4 payments): 10 (P/A, 10%, 4) = 10 × 3.1699 = 31.699
  3. Gradient 0, 2, 4, 6 at years 0 to 3. Its (P/G, 10%, 4) value (4.3781) falls one period before the first term, i.e. at year −1, so 2 (P/G, 10%, 4)(F/P, 10%, 3) = 2 × 4.3781 × 1.3310 = 11.655
  4. Uniform 6 at years 4, 5, 6: 6 (P/A, 10%, 3) = 6 × 2.4869 = 14.921 at year 3, so at year 2: 14.921 (P/F, 10%, 1) = 14.921 × 0.9091 = 13.565

P = 33.100 + 31.699 + 11.655 + 13.565 = 90.02

Check by direct compounding to year 2: 10(1.21) + 12(1.1) + 14 + 16(0.9091 + 0.8264 + 0.7513 + 0.6830) = 12.1 + 13.2 + 14 + 50.72 = 90.02.

Answer: P = 90.02 (in the units of the figure), equivalent to the whole series at year 2.

✓ Answer checked by Claude Opus 5.5 (high effort)

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