Sensitivity is sometimes denoted by a number, for example, 10 dB. The notation describes the sound pressure level (related to perceived loudness) a speaker will produce at a distance of 1 meter when 1 watt of power is applied.
For example, a 10 dB horn speaker produces 10 decibels when 1 watt is applied, measured 1 meter away from the speaker. The specification can also be given as 10dB (1W/1m).
But here's the thing: If you increase the sensitivity of your speaker horn by double (+3 dB), the loudness doesn't necessarily double as well. Increasing the sensitivity by double only increases the speaker output by 3dB. In other words, if the output of a speaker is 81dB and the power rating is 10W, if you double the power to 20W, the loudness will reach only 84dB. If you again double the power to 40 watts, it will produce 87dB. And 80 watts will give you 90dB.
It takes an SPL increase of 10dB to double the perceived loudness of the sound coming out of a speaker.
That's the math on power (W) and sensitivity (dB). That is how loud a loudspeaker gets if you increase its power rating.
Bigger magnets, in theory, increase speaker sensitivity. The bigger the magnet is, the more reactive and powerful the driver will be. A powerful permanent magnet also translates to a more sensitive loudspeaker.
Neodymium is a powerful natural magnet. A small piece of neodymium has stronger electromagnetic currents than an inferior magnet of the same size. This is the reason why powerful speakers have become more lightweight. So it's less about the size and more about the magnetic strength.
Larger/stronger magnets produce stronger internal magnetic fields, which will interact with the voice coil to cause stronger diaphragm movement, increasing the speaker's excursion and loudness.