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## Suppose that an asset’s cash flow decreases at a constant rate of 4% per year. Use the geometric progression.

a. If the asset’s initial cash flow is \$1,000 (a1 = 1000), what will the asset’s cash flow be in 23 years? (a23 = ?)
b. What is the total sum of the asset’s cash flows for 50 years (starting from \$1,000 initial cash flow)? (S50 = ?)

Hi, I'm pretty sure I did this right, I am in my 3rd year as an actuarial science student and studying for the Actuarial FM exam.

If your cash flows decrease every year by 4%, that means your cash flows are 96% of what they were each year. (100% - 4% = 96%)

So initially your cash flow is 1000, then after year one is 1000(.96), after year 2 is 1000(.96)(.96) and after year 3 is 1000(.96)(.96)(.96) so on and so forth.  Rewritten the sum of the cash flows is:

1000 + 1000(.96) +1000(.96)2 +1000(.96)3 ...

The formula for a sum of geometric progression is a(1 - rm) / (1-r)

so then sum of the initial and 23 following cash flows = 1000(1 - .9623) / (1- .96)
the sum of the initial and 22 following cash flows = 1000(1 - .9622) / (1 - .96)

If we subtract the 2 we should get the amount of the 23rd year cash flow.

Part 2. use the same formula but use 50 for m. 1000(1 - .9650) / (1 - .96)

My answers: 15,223.61 - 14,816.26 = 407.35 cash flow after 23 years
21,752.86 sum of the cash flows after 50 years