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I guess I didn't make the question clear enough:
What CAPE level (or similar long-term value metric) for US equities
should value-oriented investors look for when seeking generally
favorable market conditions for investing new money?
The historical median? The median minus 10%? One standard deviation below the ten-year high?

I don't think any of those things will tell you when you're seeing a market bottom.
That's because fairly long-lasting market bottoms happen at a wide variety of valuation levels.

Valuation will tell you an excellent central expectation for the average
stock in the next while, where "next while" is defined as a longish interval
starting now, ending some time past the current market cycle (so normal
timing methods are useless) and ending at any month NOT at unusually
high or unusually low in the next cycle or the one after that.
So, fantastic for retirement planning, wonderful for asset allocation
for those who eschew individual security selection and market timing,
but not very good for telling you when to buy or when to sell.

Bottoms are best identified in behavioural terms—times when absolutely everybody has given up.
They are screaming into their phones at their brokers saying "I don't
care what it is, I don't care what the price is, just sell it all, now"!
You get insane spikes in the number of stocks hitting new lows, and
importantly you get extraordinarily few stocks hitting new highs at the exact same time.
If there are stocks like inverse funds or bond funds in the sample
you're using then you have to adjust accordingly.
So, NYSE figures are a bit tricky to use since under half of the
listings are actually plain vanilla US companies in recent years.

The more extreme this metric, the more "major" the bottom you're seeing.
My short term bottom indicator simply compares the current situation
to what has been experienced in the last 8-10-12 months.
As a result, the direness of its signals is somewhat in proportion to
how bad things have been in the last year. The last year has been very happy,
so a signal now would not mean all that much, since it's relative to recent experience.
Its last two signals were 2013-06-24 and 2014-02-05. Good days to buy a little.

My major bottom indicator compares it to the prior 20 years. Something like that—long time since I tuned it.
The last signal of that model, a medium strength one, was 2011-10-04.
That's the model that was screaming "buy" at the bottom in March 2009.
It had done so a few times on the way down, but I used a few other
methods to decide that that particular one was finally the bottom.
(as it happens I was off by one, the final bottom being a few days later than my call)
There seemed to be a limit to how many false alarms it can give in a row before one of them is right.
And presumably a limit to how low some things can go before buyers will
come out of the woodwork, but I don't know how to put a number on that.

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