How can I compute the minimum possible bank angle for a turn with a given radius?
Given a radius $R$ and a velocity $V$, the bank angle can be computed by: 

$\phi = \textrm{atan}\left(\frac{V^2}{R\cdot g }\right)$,

where $g$ is the acceleration by gravity. This shows that the lower the speed becomes, the lower the bank angle becomes. The lowest attainable speed is the stall speed, however we have to take into account the load factor during the turn. 

The load factor $n$ during the turn is:

$n=\frac{1}{\cos \phi}$

Due to the increased load factor, the stall speed increases in the turn:

$V_{stall,turn} = V_{stall} \cdot \sqrt n $

The minimum bank angle for a turn with radius R will occur when the speed is equal to the stall speed for that angle. 

Solving this mathematically for $\phi$ results in:

$\phi_{min} = \textrm{atan}\left(\frac{V_{stall}^2}{\sqrt{R^2g^2-V_{stall}^4}}\right)$

Note that is the theoretical approach; in practice you would not fly at the stall speed, especially not during turns. 

Units are all in SI, you have to apply conversion factors if you want to use aviation units.



Example:


$V_{stall}$ = 50 m/s (approx. 97 knots) 
$R$ = 400 m 
$\phi_{min}$ = ~39.6 degrees 
$V$ = ~57 m/s (approx. 111 knots)