<MATH>
What I don't understand is how you derived the formula
for the residual value. R = FreeCashFlow(next year)/(k -
g). R is the residual value. k is our discount rate. In
percent it's k%, in decimals it's 0.01 times k. g is the
growth rate.
Without looking back at Pirah's post, I don't know exactly
which years he's talking about for the residual value, but
here's how you'd calculate the present value of a cash
stream that will give you FCF in one year, FCF * (1+g) the
next year, and so on, with a discount rate of (1+k). For
consistency, I'll use the same variables as in your quote.
R = Sum(from n = 0 to infinity) of FCF * (1+g)^n / (1+k)^(n+1)
R = FCF/(1+k) + Sum(from n = 1 to infinity) of FCF * (1+g)^n / (1+k)^(n+1)
R = FCF/(1+k) + Sum(from n = 0 to infinity) of FCF * (1+g)^(n+1) / (1+k)^(n+2)
R = FCF/(1+k) + ((1+g)/(1+k))* Sum(from n = 0 to infinity) of FCF * (1+g)^n / (1+k)^(n+1)
R = FCF/(1+k) + ((1+g)/(1+k)) * R
R * (1 - (1+g)/(1+k)) = FCF/(1+k)
R * ((1+k)/(1+k) - (1+g)/(1+k)) = FCF/(1+k)
R * ((k-g)/(1+k)) = FCF/(1+k)
R = FCF/(k-g)
</MATH>
The general idea is that a dollar in one year is worth $1/(1+k) today. So if you can guarantee that the cash stream will grow faster than its value depreciates, that is indeed worth an infinite amount of money (supposedly). In my opinion, this is one problem with DCF theory, but I can't really wrap my brain around it. In the very long run, few companies seem able to grow cash flow faster than the discount rate (KO may be an exception, which may be why Buffett says "hold forever").
Incidentally, I think that DCF is a really interesting and important way to value certain types of stocks. IMO, though, it's just not that useful to us since long-run growth rates of this class of tech stocks are so hard to predict. I doubt that many DCFs of EMC in 1990 said that its DCF was 1,000 times greater than its price, although that turned out to be roughly the case (OK, off by (1+k)^10).
Also, real options theory may be more useful for tech stocks, but it's really hard to do (I think--I don't actually know how to do it), and maybe not worth the effort. I think The Fool used to apply it to Amazon, which hasn't really helped the theory's credibility as a valuation tool...
ardethan@caveatemptor.net |
