Question

Calculate the accumulated value at time 5 years of payments that are received continuously over each year with payment of $100 during 1st year, $110 in 2nd year, $130 in 3rd year, $140 in 4th year and $200 in 5th year. Assume EAR = 5%.

Answer 1

If interest is compounded monthly,

Accumulated value = $100 * ( 1 + 0.05/12) ^ 60 + $100 * ( 1 + 0.05/12) ^ 59 + ..... + $100 * ( 1 + 0.05/12) ^ 49 + $ 110 * ( 1 + 0.05/12) ^ 48 + ..... + $110 * (1 + 0.05/12) ^ 37 + $130 * (1 + 0.05/12) ^ 36 + .... + $130 * ( 1 + 0.05/12) ^ 25 + $140 * (1 +0.05/12) ^ 24 + ... + $140 * ( 1 + 0.05/12) ^ 13 + $200 * ( 1 + 0.05/12) ^12 + ... + $200 * ( 1 + 0.05/12) ^ 1

= $100 * (1 + 0.05/12) ^49 * [ (1 + 0.05/12) ^ 12 - 1] / [ 1 + 0.05/12 - 1] + $110 * (1 + 0.05/12) ^37 * [ (1 + 0.05/12) ^ 12 - 1] / [ 1 + 0.05/12 - 1] + $130 * (1 + 0.05/12) ^25 * [ (1 + 0.05/12) ^ 12 - 1] / [ 1 + 0.05/12 - 1] + $140 * (1 + 0.05/12) ^13 * [ (1 + 0.05/12) ^ 12 - 1] / [ 1 + 0.05/12 - 1] + $200 * (1 + 0.05/12) * [ (1 + 0.05/12) ^ 12 - 1] / [ 1 + 0.05/12 - 1]

= $100 * 1.226 * *12.229 + $110 * 1.166 * 12.229 + $130 * 1.11 * 12.229 + $140 * 1.056 * 12.229 + $200 * 1.0042 * 12.229

= $1,499.2754 + $1,568.4915 + $ 1,764.6447 + $1,807.9354 + $ 2,546.0734

= $9,168.42

If interest is compounded yearly,

Accumulated value = $100*12* ( 1 + 0.05) ^ 4 + $110 * 12 * ( 1 + 0.05) ^ 3 + $130 * 12 * ( 1 + 0.05) ^ 2 + $140 * 12 * ( 1+ 0.05) ^ 1 + $200 * 12 * ( 1 + 0.05) ^ 0

= $1,458.6075 + $1,528.065 + $1,719.90 + $1,764 + $2,400

= $8,870.57