Since the market perfectly captures all the information, is it true that any anticipated monetary policy has no impact on equity prices? Or the “information” has an immediate impact but the “event” itself has no impact.
We begin our story at time $t-1$, where we have a Goldilocks economy (not too hot nor too cold). At time $t$, the economy gets worse. At time $t+1$, it becomes obvious that the central bank will cut the policy rate by 25 basis points at the next meeting and that this is sufficient to move the economy back to where it was at $t-1$. At time $t+2$, the central bank cuts the policy rate by 25 basis points. If markets are semi-strong efficient and the worsening of the economy is public information (along with the monetary policy action), then we would expect stocks to fall on $t$, rise on $t+1$ to the same level as $t-1$, and be unchanged on $t+2$ (assuming nothing else is happening in the economy).
However, this does not make monetary policy useless in this situation. If the monetary policy authority, meeting on $t+2$, left rates unchanged (instead of cutting them), then we would expect equities to fall on $t+2$. While there are many monetary policy channels, in most of them, the changing of the policy rate actually improves the economy. This can raise equity prices by boosting firm cash flows.
When the central bank cuts rates, this isn't useless, even if expected, because it caused the economy to be better, which also boosted the value of financial assets. Yes, if the rate change is fully anticipated, there is no effect seen on the day of the rate change, but that is not because it is useless. Rather, it is because it is already priced into the value of financial assets before the rate change actually occurs.
It is important to understand that the Efficient Market Hypothesis is not a law. It is a hypothesis in the truest sense - it relies on a number of important assumptions to hold before it can be declared valid. It is generally acknowledged that those assumptions do not hold at all times.
In real markets, the EMH has trouble explaining the fact that markets often shift on sentiment rather than fundamentals. So-called rational expectations are routinely violated, and this violation has been shown to present market-timing opportunities. For example, and very pertinent to your question, in Monetary Momentum, by Neuhierl and Weber), evidence is presented for a market-timing opportunity that results from monetary policy surprises. They argue that "A simple trading strategy exploiting the drift around FOMC meetings increases Sharpe ratios relative to a buy-and-hold investment by a factor of 4." This is just one way in which monetary policy can directly (though not immediately) impact equities markets.
Another is a matter of current concern - that the persistent low-interest rate environment supported by Fed overnight lending activities (e.g., through repo markets) is flooding the banking system with cheap currency. The theory behind why this is troublesome runs something like the following:
Banks find themselves in the awkward position of not being able to leave this money on the table due to the low lending rates ("it's too good to pass up!"). In a healthy economy with good fundamentals that support long-term growth, banks would take this money and immediately loan it out at slightly higher interest, making money on the spread. But if - as is argued - the economy is not doing so well, banks are reluctant to add loans to their balance sheet. The question then becomes what can they do with this nearly-free money that will create an interest rate spread?
The suggested answer is that they invest it in the equities market. Since the repeal of the Glass-Steagal Act in the US in the late 1990s, this was possible because there was no longer a legal requirement to separate a bank's commercial lending activities from its investment banking activities.
This leads (if true) to artificial demand inflation for securities, decoupling prices from fundamentals and rendering the EMH invalid.
Long story short: The EMH, like other highly idealized models in economics, is unlikely to occur in the real world, but is a useful benchmarking tool for understanding what is really going on and identifying potential issues needing policy attention.
I’m going to stay away from the Efficient Market Hypothesis (since I believe that there’s a number of formulations), but just comment on monetary policy.
The theory is that a central bank is supposed to have a reaction function (based on economic data) as to how it sets interest rates. (For example, following a variant of the Taylor Rule.) The risk-free curve (e.g. government bond yield curve) is supposed to be priced so that forward rates follow the expected path of the policy rate (based on the reaction function) - plus a term premium.
This risk-free rate is an input to computing the fair value of equities.
If monetary policy evolves as anticipated, actual short rates end up where forwards predict. The implication is that bond prices end up where forwards predict. This means that this input to equity prices is effectively unchanged, and thus should not affect equity prices.