Perpetual Cash flow evaluation

If some cash flow paid \$100, 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?

• your list of numbers does not match your stated pattern? Oct 26 '18 at 20:26

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=\$4000. 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 \$300, the exact answer would be somewhere between \$4000 and \$4300. 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.

• Nah, just 2 perpetuities. \$100 * (1/1.1 + 1/1.1^2 + ...) + \$300 * (1/1.1^5 + 1/1.1^10 + ...)
– RJM
Oct 26 '18 at 17:19
• was it every 3rd? I guess i thought every 5th. Question was a bit unclear.
– RJM
Oct 27 '18 at 0:21
• My understanding was every 3rd but the first "bonus" is in year 5. The text seems a little inconsistent. I forgot about that first 4 years business when I did my calculation, but whatever. I think we've addressed the principle. (No pun intended.)
– Jay
Oct 27 '18 at 21:18

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.

Sum of converging series: for this sequence of cash flows (\$100, 100, 100, 100, 400, 100, 100,... ) discounted @ 10%, pv = 100/0.1 + 300/((1.1^5)-1) = \$1491.39.

EDIT:

If the question was for a \$400 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) = \$1906.34

If the question was for a perpetual sequence of cash flows of a repeating pattern \$400,\$100,\$100,\$400,\$100,\$100,..., then the PV is the sum of the following convergent geometric series.

\$100*(1/1.1 + 1/1.1^2 + ...) and (\$300/1.1)*(1+1/1.1^3+1/1.1^6+...)

Therefore PV = \$100/0.1 + (\$300*1.1^2)/(1.1^3-1) = \$2096.68.

• But it’s not a convergent series. The sum is unbounded. Oct 26 '18 at 4:45
• When you discount it every period in perpetuity it definitely is.
– RJM
Oct 26 '18 at 4:47
• @Lawrence Larry, if i were to promise give you \$100 every 6 months forever, how much would you pay me for that deal, infinite dollars. Let's discuss that deal with your calculation.
– RJM
Oct 26 '18 at 5:07
• The use of the term “convergent series” in your answer confused me. Oct 26 '18 at 5:11
• You're welcome. The cash flows are all discounted, so it becomes a (really two) geometric series with "r" < 1. I figured i would give more than just the answer, but was too busy to write it out.
– RJM
Oct 26 '18 at 5:15