BJT

Choose 20Khz input frequency, as output attenuates at low and high frequencies.

Peak to peak AC voltage at base of transistor = 0.0193v.

Peak to peak AC voltage at collector of transistor = 4.5409v.
Gain = 4.5409/0.0193 = 235.279. (from simulation).
Gain = -gm*(r0||Rc) ~ -gm*Rc. (formula) (r0 resistor within hybrid pi model of transistor, usually large).
gm = Ic/VT = 662uA/26mv = 0.025461.
Gain = 254.6 (from formula) close to simulation.
Note: R3/R5 = R4/R6 to keep transistor in operation mode.

change R5 to 20K, transistor in saturation mode. (collector voltage = VCC).

Collector output binded.

Change R5 to 5K, transistor close to cutoff. (base current in small).

AC gain is reduced dramatically.