: Perpetual Cash flow evaluation If some cash flow paid 0, 100, 100, 100, 400, 100, 100,... perpetually (the pattern is 400,100,100) the first year is skipped. The discount rate is 10%. How do you find the PV of this?

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Sum of converging series: for this sequence of cash flows (0, 100, 100, 100, 400, 100, 100,... ) discounted @ 10%, pv = 100/0.1 + 300/((1.1^5)-1) = 91.39.

EDIT:

If the question was for a 0 cash flow every 3rd period as determined by Jay, then the PV @ 10% opportunity cost of capital is

100/0.1 + 300/((1.1^3)-1) = 06.34

If the question was for a perpetual sequence of cash flows of a repeating pattern 0,0,0,0,0,0,..., then the PV is the sum of the following convergent geometric series.

0*(1/1.1 + 1/1.1^2 + ...) and (0/1.1)*(1+1/1.1^3+1/1.1^6+...)

Therefore PV = 0/0.1 + (0*1.1^2)/(1.1^3-1) = 96.68.

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You could get a close approximation by just taking the average. (400+100+100)/3=200. What discount rate are we assuming? Say 5%. So 200/.05=00. It would actually be a little higher than that because of the bigger payout the first year. After that it would average out. As the "excess" the first year is 0, the exact answer would be somewhere between 00 and 00. With varying numbers like that, it would be easier to do it with a spreadsheet than with a formula, I'd think.

Update *

Oh, thanks RJM, good point. Just treat it as two series, 100 every period and 300 every 3rd period.

So again assuming 5% discount rate:

So NPV(100,.05, 1)=100/.05=2000

NPV(400,.05, 3)=300*1.05^2/(1.05^3-1)=300*6.99~=2100

So total=2000+2100=4100.

The first function is pretty routine. I got the second like this:

S=1/r+1/r^4+1/r^7+ ...
1/r^3*S=1/r^4+1/r^7+1/r^10+...
S-1/r^3*S=1/r
S(1-1/r^3)=1/r
S=(1/r)/(1-1/r^3)
=(1/r)/((r^3-1)/r^3)
=1/r*r^3/(r^3-1)
=r^2/(r^3-1)

Maybe there's an easier way to do it. That's the first way that occurred to me.

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If a payment of one dollar one year from now has a NPV of d, then after k triplets (that is, after 3k years), you have an NPV of 400*d^(3k+1)+100*d^(3k+2)+100*d^(3k+3), assuming payments come at the end of the year. You can factor out d^(3k+1) and get d^(3k+1)[400+100d+100d^2] or (d^3k)*d[400+100d+100d^2]. You now have a geometric sequence with r = d^3. So it would be d[400+100d+100d^2]/(1-d^3). In other words, you can treat it as getting a payment of d[400+100d+100d^2] every three years, and the discount rate over those three year is the cube of the yearly discount rate. Taking d=.9, this gives 1896.31.