There is more to MV=PQ than meets the eye.
Central banks can expand M but this does not mean they can expand MV. V can always fall to compensate for the increase in M, as has happened for most of the period 2008 – 2014. The maintenance or expansion of V is dependent on a number of things. Necessary conditions include a functioning fractional reserve banking system with sufficient capital and appropriate reserve ratios to transmit the increases in V. Sufficient conditions include a health demand for credit which is dependent on business sentiment.
Assuming that it is possible to increase MV, the impact on PQ and its constituents remain complicated. While PQ is a scalar, P and Q are in fact vectors. Q is a list of all the possible stuff you can spend money on, and P is the corresponding vector of prices. A couple of things to note about Q are that it includes goods, services, and assets, indeed, anything you can allocate money to, and that while the scalar PQ must rise if MV does, its not clear a) which good, service or asset market is experiencing rising nominal output or b) for a given good, service or asset market experiencing rising nominal output whether price, real output or both are rising. In other words, even if MV and therefore PQ is rising, some markets may experience falling nominal output while others may experience rising nominal output and in markets with rising nominal output it could all be due to inflation or real growth or both, but you couldn’t control which.
All things being equal, if nominal output in a particular market is not falling, and real output is falling or decelerating, then price must increase or accelerate. An example where this might be happening is the stock market, where companies are buying back stock. All things being equal, this will put upward pressure on stock prices. M&A is also another mechanism which may reduce the float.
The bond market is a bit more interesting. Prices are rising while output (issuance) is rising. That means that the bond (and loan) markets are responsible for diverting a large amount of liquidity away from other markets such as those for goods and services.
Asset inflation could divert liquidity away from current consumption resulting in lower inflation. Conversely, a recovery in inflation could signal a diversion of liquidity away from asset markets. There is a scenario under which inflation expectations driven by QE cause investors to divert capital away from savings in fixed income assets such as bank deposits and government bonds towards risky assets such as equities and high yield bonds in an attempt to hedge future inflation. Conventional wisdom seems to imply inflation encourages current consumption but the recovery in capital goods ahead of consumer goods seems to support some extend of substitution away from current consumption to future consumption albeit in assets that provide a positive linkage to inflation. If risky corporate assets are seen as claims on future production, there is an argument that they are inflation hedges. QE therefore causes inflation of these future claims at the expense of current inflation.
Fundamentals should not be ignored. However, fundamentals can be framed within the above concept as providing the relative attractiveness of each good or asset market in the allocation problem. The global allocation is still driven by the almost mechanical and physical allocation of liquidity.
At a more mundane level:
Monetary expansion can lead to asset price inflation.
For a given level of monetary expansion, assets and or goods and services can experience inflation. Excessive inflation in assets can explain lack of inflation in goods and services markets.
A withdrawal from expansionary monetary policy could cause deflation or slower inflation in assets or goods and services.
Under neutral monetary policy, goods and services inflation implies declining asset prices.