bpbiv -- I was hoping to close this with the commonsense observation that earnings growth shouldn't make the stock *less* valuable. I was hoping you might think about that odd prediction from your earlier post and come back having found your own error.
Instead I guess you wanna go into the algebra. So here's your algebraic mistake: instead of the correct formula (which does appear in your Abrams reference):
for net present value of future returns that grow at rate g discounted at rate r, you made a sign error:
where r+g gives your mistaken 26.25%. Unfortunately, for reasons that should be obvious (if not, run some spreadsheets), the Gordon formula for present value of a stream of earnings in perpetuity doesn't work when g > r.
In addition, by even counting earnings during the next five years, you're ignoring my statement that earnings during the growth period are reinvested in equity, not collected as cash flows. In fact we'd be double-counting if we included them in cash flows.
In the second stage for the happy case, I assumed a CONSTANT (g=0) $2.50 in earnings per year, beginning at y/e 2008 (exactly 5 years from now). Since g=0, Gordon's formula reduces to
1/(r-0) = 1/r.
I happened to set r = 6.25%, so 1/(1-r) = 16 in the case I calculated. Happy to consider other values of r. $2.50 x 16 = $40, but we have to discount back 5 years so I divided by (1+r)^5. to get about $29.50.
Again, for my "Happy" case I assumed (a) NO cash flows during the next five years as all cash to be reinvested in the business during those years. (b) CONSTANT cash flows of $2.50 beginning in year 6 and continuing indefinitely.
I think I can guess why you've added g+r, because if you (correctly) subtracted g from r you'd get a negative number. But (as Gordon clearly indicates) the present value would be unbounded if g > r indefinitely. That's fine, and it's why I avoided the trap you got stuck in by using a two-stage model. Nevertheless, It's not ok to just change the sign of the growth to that of an earnings decay!
I hope this helps. I've taken some time here; please do me the courtesy of giving this some careful thought on your end before responding.
<<I was hoping to close this with the commonsense observation that earnings growth shouldn't make the stock *less* valuable.>>
I never said earnings growth makes the stock less valuable. I merely pointed out that the cap rate and the discount rate are not the same thing. You called it a discount rate, but then seemed to use it like a cap rate.
<<I was hoping you might think about that odd prediction from your earlier post and come back having found your own error.>>
I made no prediction.
<<instead of the correct formula...: 1/(r-g) for net present value of future returns that grow at rate g discounted at rate r, you made a sign error: 1/(r+g).>>
I made no sign error. You calculated yr 5 value by taking $2.50/6.25%. That does not "discount" the future earnings from that point on, it capitalizes the earnings as of then, making it a cap rate, not a discount rate. You also assumed 20% growth per year. If cap rate = discount rate minus growth rate, the converse is that discount rate = cap rate PLUS growth rate.
<<Unfortunately, for reasons that should be obvious...the Gordon formula...doesn't work when g > r.>>
No argument from me there.
<<In addition, by even counting earnings during the next five years, you're ignoring my statement that earnings during the growth period are reinvested in equity, not collected as cash flows. In fact we'd be double-counting if we included them in cash flows.>>
Your post didn't address cash flow; only EPS. You said EPS was reinvested (not thrown away). Reinvestment typically adds value. I assume that the company reinvests the earnings in equipment, R&D, and working capital. Equipment and R&D hit EPS through depreciation & amort in years after the cash is spent, and EPS reinvested in working capital increases the cash on hand. If the company makes $2.50/sh net of increased depn and amort, then future years' cash flow/share should be higher than EPS. Thus EPS in the first 5 years MUST be considered, even if reinvested. You apparently are considering CFPS for the first 5 years and EPS. If you use CFPS for the first 5 years, you should be consistent in subsequent years.
As for "commonsense observations", is it reasonable to assume that a stock that reinvests 100% of the next 5 yr EPS, hits $2.50 EPS, then has 0 growth after that warrants a 6.25% discount rate?
Typical discount rate build up would be: Risk free rate-- say 4% Add risk premium for equity-- usually about 7% (represents large cap premium) Add risk premium for size-- say 3% (smaller cap companies statistically have higher expected returns than large-- Add risk premium for company-specific risks (like patent expiration, possible negative litigation, trendy products, etc.) - say 6%. Those add up to 20%, (to me a minimum reasonable discount rate for a company like NLS.) I believe 6.25% for a 0-growth company (your assumption, not mine) is unsupportable.
<<I've taken some time here; please do me the courtesy of giving this some careful thought on your end before responding.>>
And I have also taken some time here both on this post and the previous one. Please do me the favor of not assuming that I don't think about what I post.