Apologies for going on and on and on when it comes to this topic, but I think it's important. Central banks set macroeconomic policy, partly through setting a policy rate, and this affects the environment in which the yields on securities, i.e. interest rates, are set. But they do not directly control much except for their own policy rates (in the UK, a discount rate at which it lends to banks and pays on reserves they hold with it; in the US and Eurozone other rates are used in preference to or as well as a discount rate).
I have made various logical and empirical arguments for this claim in the posts I link in the first line; this post was spurred by my discovery of a 2013 paper from Eugene Fama (joint winner of2013's economics Nobel) making the case yet more rigorously. Entitled "Does the Fed Control Interest Rates?" it answers 'probably not':
To what extent does TF, the target Federal funds rate set by the Fed, influence other rates? There is lots of variation in rates unrelated to TF, and any effects of TF on rates dissipate quickly for longer maturities. For short rates, all the tests have interpretations in terms of: (i) a Fed that has the power to control rates and uses it, and (ii) a Fed that has little power over rates or chooses not to exercise its power. In the end, there is no conclusive evidence (here or elsewhere) on the role of the Fed versus market forces in the long-term path of interest rates.
A key piece of evidence is the fact that large spreads often emerge between private securities like commercial paper and the Fed-driven rate (i.e. Banks do not lend at the rate the Fed is targeting, despite its best efforts):
Fama argues that the evidence should be a cautionary tale for those who believe the Fed to have great powers of controlling the economy, making the case instead the market forces are usually the key driver, and I agree with him as far as it goes. But I think that monetary policy doesn't work necessarily by affecting real yields but by devaluing money to clear markets.
I think the central bank's control of the monetary base is what gives it power: it can vary the supply of money elastically (like free banks would) to stop money demand shocks from interacting with sticky wages and causing recessions. What it cannot do is produce any extra prosperity above the amount you get from monetary stability and neutrality.